Finite Group Databasebeta

← #29740059

Subgroups — #29740059

Order 15616.

918 conjugacy classes of subgroups.

The subgroup lattice has 696 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16839960C211yesmaximal, characteristic
2#21449578C211yesmaximal, characteristic
3Q8×#11758196C211yesmaximal, characteristic, complemented
4#21447349C211yesmaximal, characteristic, complemented
5#21447318C211yesmaximal, characteristic, complemented
6#21447225C211yesmaximal, characteristic, complemented
7#21447051C211yesmaximal, characteristic
8#21446502C211yesmaximal, characteristic, complemented
9#21446376C211yesmaximal, characteristic, complemented
10#21446176C211yesmaximal, characteristic, complemented
11#21446174C211yesmaximal, characteristic, complemented
12#21446172C211yesmaximal, characteristic, complemented
13#21446085C211yesmaximal, characteristic, complemented
14#21446080C211yesmaximal, characteristic, complemented
15C2×C4×SD16×C61C211yesmaximal, characteristic, complemented
16C22×#11758201C2211yescharacteristic
17C2×#14519773C2211yescharacteristic
18C2×#14519760—21no—
19C2×Q8×Dic61C2211yescharacteristic
20C2×Q8×Dic61C2211yescharacteristic
21#16839960—24no—
22C2×#14519708C2211yescharacteristic
23C2×#14519706C2211yescharacteristic
24C2×#14519701C2211yescharacteristic
25C2×#14519699C2211yescharacteristic
26C2×#14519699C2211yescharacteristic
27C2×#14519698C2211yescharacteristic
28C2×#14519698C2211yescharacteristic
29C2×#14519697C2211yescharacteristic
30#16839677—21no—
31#16839677—21no—
32#16839665C2211yescharacteristic
33#16839664C2211yescharacteristic
34#16839663C2211yescharacteristic
35#16839660—21no—
36#16839660C2211yescharacteristic
37#16839658—21no—
38#16839658C2211yescharacteristic
39#16839658C2211yescharacteristic
40#16839641—21no—
41#16839631—21no—
42#16839631C2211yescharacteristic
43C4×#11758196—21no—
44C4×#11758196C2211yescharacteristic
45#16839525C2211yescharacteristic
46#16839525C2211yescharacteristic
47#16839524C2211yescharacteristic
48#16839524C2211yescharacteristic
49#16839517C2211yescharacteristic
50#16839516C2211yescharacteristic
51#16839516C2211yescharacteristic
52#16839516C2211yescharacteristic
53C22×SD16×C61C2211yescharacteristic
54C2×C4×Q8×C61C2211yescharacteristic
55C2×C4×D4×C61C2211yescharacteristic
56C4×SD16×C61—24no—
57C2×#106×C61C2211yescharacteristic
58C2×#103×C61C2211yescharacteristic
59C2×#102×C61C2211yescharacteristic
60C2×C4×C8×C61C2211yescharacteristic
61C22×D244C2311yescharacteristic
62C22×C22⋊D61—21no—
63C22×C22⋊D61—21no—
64C2×#11758201—22no—
65C2×#11758201—22no—
66C2×#11758201—22no—
67C22×#92418C2311yescharacteristic
68C22×#92418C2311yescharacteristic
69#14519773—22no—
70#14519773—41no—
71#14519760—44no—
72Q8×Dic61—22no—
73Q8×Dic61—22no—
74Q8×Dic61—22no—
75Q8×Dic61—41no—
76C2×C4⋊Dic61—21no—
77C2×C4⋊Dic61—21no—
78C2×C4⋊Dic61C2311yescharacteristic
79C2×C4⋊Dic61C2311yescharacteristic
80C2×C4⋊Dic61C2311yescharacteristic
81C2×#11758196—21no—
82C2×#11758196—21no—
83C2×#11758196—21no—
84C2×#11758196—21no—
85C2×#11758196—21no—
86C2×#11758196D411yescharacteristic
87C2×#11758196D411yescharacteristic
88C2×C22⋊Dic61—21no—
89C2×C22⋊Dic61—21no—
90C2×#11758198—21no—
91C2×#11758198—21no—
92C2×#11758198—21no—
93C2×#11758198—21no—
94C2×C4×Dic61—21no—
95C2×C4×Dic61—21no—
96C2×C4×Dic61C2311yescharacteristic
97C2×C4×Dic61C2311yescharacteristic
98#14519708—41no—
99#14519706—22no—
100#14519701—22no—
101#14519699—22no—
102#14519699—22no—
103#14519698—22no—
104#14519698—22no—
105#14519697—41no—
106#14519684—21no—
107#14519684—21no—
108#14519684—21no—
109#14519684D411yescharacteristic
110#14519684D411yescharacteristic
111C22×Q8×C61C2311yescharacteristic
112C22×D4×C61C2311yescharacteristic
113C23×C4×C61—21no—
114C2×SD16×C61—22no—
115C2×SD16×C61—22no—
116C2×SD16×C61D414yes—
117C22×C8×C61C2311yescharacteristic
118C22×C8×C61C2311yescharacteristic
119C4×Q8×C61—22no—
120C4×Q8×C61—41no—
121C4×D4×C61—22no—
122C4×D4×C61—41no—
123C2×C4⋊C4×C61—21no—
124C2×C4⋊C4×C61C2311yescharacteristic
125C2×C4⋊C4×C61C2311yescharacteristic
126C2×C22⋊C4×C61—21no—
127C2×C42×C61—21no—
128C2×C42×C61C2311yescharacteristic
129#106×C61—22no—
130#103×C61—22no—
131#102×C61—22no—
132C4×C8×C61—22no—
133C24×C61—21no—
134C23×D61—21no—
135C2×Q8×C61—21no—
136C2×Q8×C61—21no—
137C2×Q8×C61—21no—
138C2×Q8×C61C2×D412yes—
139C2×Q8×C61#4112yes—
140C2×Q8×C61#4112yes—
141C2×D4×C61—21no—
142C2×D4×C61—21no—
143C2×D4×C61—21no—
144C2×D4×C61—21no—
145C2×D4×C61—21no—
146C2×D4×C61C2×D412yes—
147C22×C4×C61—21no—
148C22×C4×C61—21no—
149C22×C4×C61—21no—
150C22×C4×C61—42no—
151C22×C4×C61—42no—
152C22×C4×C61—81no—
153C22×C4×C61C2×D411yescharacteristic
154C22×C4×C61C2×D411yescharacteristic
155C22×C4×C61C2411yescharacteristic
156C2×C22⋊D61—42no—
157C2×C22⋊D61—42no—
158C2×C22⋊D61—42no—
159C2×C22⋊D61—42no—
160C2×C22⋊D61—81no—
161C2×C22⋊D61—81no—
162C22×Dic61—21no—
163C22×Dic61—21no—
164C22×Dic61—21no—
165C22×Dic61C2×D411yescharacteristic
166C22×Dic61C2×D411yescharacteristic
167C22×Dic61C2×D411yescharacteristic
168C22×Dic61C2×D411yescharacteristic
169C2×D244—21no—
170C2×D244—21no—
171C2×D244—21no—
172C2×D244—21no—
173C2×D244—21no—
174C2×D244—21no—
175SD16×C61—28no—
176C2×C8×C61—21no—
177C2×C8×C61—21no—
178C2×C8×C61—21no—
179C2×C8×C61—21no—
180C2×C8×C61—21no—
181C2×C8×C61C2×D412yes—
182C4⋊C4×C61—21no—
183C4⋊C4×C61—21no—
184C4⋊C4×C61—21no—
185C4⋊C4×C61—21no—
186C4⋊C4×C61—42no—
187C22⋊C4×C61—42no—
188C42×C61—21no—
189C42×C61—21no—
190C42×C61—42no—
191C22⋊Dic61—42no—
192C22⋊Dic61—81no—
193#11758201—44no—
194#11758198—42no—
195#11758198—42no—
196#11758198—81no—
197#11758198—81no—
198C4⋊Dic61—24no—
199C4⋊Dic61—21no—
200C4⋊Dic61—21no—
201C4⋊Dic61—21no—
202C4⋊Dic61—21no—
203C4⋊Dic61—21no—
204C4⋊Dic61—42no—
205C4⋊Dic61#4112yes—
206#11758196—41no—
207#11758196—42no—
208#11758196—42no—
209#11758196—81no—
210#11758196—81no—
211#11758196—81no—
212#11758196SD1614yescomplemented
213C4×Dic61—24no—
214C4×Dic61—21no—
215C4×Dic61—21no—
216C4×Dic61—21no—
217C4×Dic61—21no—
218C4×Dic61—42no—
219C2×#92418—21no—
220C2×#92418—21no—
221C2×#92418—21no—
222C2×#92418—21no—
223C2×#92418—21no—
224C2×#92418—21no—
225C23×C61—22no—
226C23×C61—41no—
227C23×C61—41no—
228C23×C61—41no—
229C23×C61—41no—
230C23×C61—41no—
231C23×C61—41no—
232C23×C61C22×D411yescharacteristic
233C22×D61—41no—
234C22×D61—41no—
235C22×D61—41no—
236C22×D61—41no—
237C22×D61—41no—
238C22×D61—41no—
239C22×D61—41no—
240Q8×C61—22no—
241Q8×C61—22no—
242Q8×C61—22no—
243Q8×C61#12314yescomplemented
244D4×C61—22no—
245D4×C61—22no—
246D4×C61—41no—
247D4×C61—41no—
248C2×C4×C61—22no—
249C2×C4×C61—22no—
250C2×C4×C61—21no—
251C2×C4×C61—22no—
252C2×C4×C61—21no—
253C2×C4×C61—21no—
254C2×C4×C61—21no—
255C2×C4×C61—21no—
256C2×C4×C61—41no—
257C2×C4×C61—41no—
258C2×C4×C61—41no—
259C2×C4×C61—41no—
260C2×C4×C61—41no—
261C2×C4×C61—41no—
262C2×C4×C61—84no—
263C2×C4×C61C2×SD1612yes—
264C2×C4×C61C22×D411yescharacteristic
265C2×C4×C61C2×#4111yescharacteristic
266C2×C4×C61C2×#4111yescharacteristic
267C2×C4×C61C2×#4111yescharacteristic
268C2×C4×C61H(5,𝔽2)11yescharacteristic
269C2×C4×C612−1+411yescharacteristic
270C22⋊D61—84no—
271C22⋊D61—84no—
272C2×Dic61—21no—
273C2×Dic61—21no—
274C2×Dic61—21no—
275C2×Dic61—21no—
276C2×Dic61—21no—
277C2×Dic61—22no—
278C2×Dic61—21no—
279C2×Dic61—21no—
280C2×Dic61—21no—
281C2×Dic61—21no—
282C2×Dic61—21no—
283C2×Dic61—41no—
284C2×Dic61—41no—
285C2×Dic61—41no—
286C2×Dic61—41no—
287C2×Dic61—41no—
288C2×Dic61—41no—
289C2×Dic61—41no—
290C2×Dic61—41no—
291C2×Dic61C2×SD1612yes—
292C2×Dic61C2×SD1612yes—
293D244—41no—
294D244—41no—
295D244—41no—
296D244—41no—
297C8×C61—22no—
298C8×C61—41no—
299#92418—41no—
300#92418—41no—
301#26163—611nomaximal, Sylow
302C22×C61—42no—
303C22×C61—42no—
304C22×C61—42no—
305C22×C61—81no—
306C22×C61—81no—
307C22×C61—81no—
308C22×C61—81no—
309C22×C61#52511yescharacteristic
310C22×C61D4211yescharacteristic
311C22×C61#54811yescharacteristic
312C22×C61C22×SD1611yescharacteristic
313C22×C61#57511yescharacteristic
314C22×C61#57611yescharacteristic
315C22×C61#57711yescharacteristic
316C2×D61—81no—
317C2×D61—81no—
318C2×D61—81no—
319C2×D61—81no—
320C2×D61—81no—
321C2×D61—81no—
322C2×D61—81no—
323C4×C61—24no—
324C4×C61—22no—
325C4×C61—41no—
326C4×C61—81no—
327C4×C61—81no—
328C4×C61C2×#12311yescharacteristic
329C4×C61#53911yescharacteristic
330C4×C61#54711yescharacteristic
331C4×C61#55311yescharacteristic
332Dic61—22no—
333Dic61—22no—
334Dic61—41no—
335Dic61—42no—
336Dic61—41no—
337Dic61—81no—
338Dic61—81no—
339C2×#460—611no—
340C2×C4×SD16D6111yescharacteristic, complemented
341#2445—611no—
342Q8×C4⋊C4—611no—
343#2094—611no—
344#2088—611no—
345#2071—611no—
346#2041—611no—
347#1934—611no—
348#1916—611no—
349#1880—611no—
350#1880—611no—
351#1880—611no—
352#1865—611no—
353#1864—611no—
354C2×C61—82no—
355C2×C61#244311yescharacteristic
356C2×C61#310311yescharacteristic
357C2×C61#311911yescharacteristic
358C2×C61#326811yescharacteristic
359C2×C61D4×SD1611yescharacteristic
360C2×C61#336211yescharacteristic
361C2×C61#337111yescharacteristic
362D61—161no—
363C22×SD16—611no—
364C22×SD16C2×D6111yescharacteristic
365C2×#127—611no—
366C2×#123—1221no—
367C2×C4×Q8—611no—
368C2×C4×Q8—611no—
369C2×C4×Q8C2×D6111yescharacteristic
370C2×C4×D4C2×D6111yescharacteristic
371#460—1224no—
372C4×SD16—24no—
373C2×#106—611no—
374C2×#106C2×D6111yescharacteristic
375C2×C4⋊C8—611no—
376C2×#103—611no—
377C2×#103C2×D6111yescharacteristic
378C2×#102—611no—
379C2×#102—611no—
380C2×#102—611no—
381C2×#102—611no—
382C2×#102—611no—
383C2×#102C2×D6111yescharacteristic
384C2×C4×C8C2×D6111yescharacteristic
385#394—1221no—
386#394—1221no—
387#390—611no—
388#390—611no—
389#390—611no—
390#389—611no—
391#389—611no—
392#389—611no—
393#389—1221no—
394#389—1221no—
395#386—1221no—
396#384—611no—
397#384—1221no—
398C4×C4⋊C4—611no—
399C4×C4⋊C4—1221no—
400#340—611no—
401#340—611no—
402#340—611no—
403#340—611no—
404#336—611no—
405#336—611no—
406#336—611no—
407#336—611no—
408C61#2616311yescharacteristic, complemented, Sylow
409C22×Q8C22×D6111yescharacteristic
410C22×D4—611no—
411C22×D4—1221no—
412C22×D4—1221no—
413C22×D4C22×D6111yescharacteristic
414C23×C4—21no—
415C2×SD16—22no—
416C2×SD16—22no—
417C2×SD16—1222no—
418C2×SD16—1222no—
419C2×SD16—1222no—
420C2×SD16C22⋊D6114yes—
421C22×C8—611no—
422C22×C8—611no—
423C22×C8C22×D6111yescharacteristic
424C22×C8C22×D6111yescharacteristic
425#127—1222no—
426#127—2441no—
427#123—2444no—
428C4×Q8—22no—
429C4×Q8—41no—
430C4×Q8—1222no—
431C4×Q8—1222no—
432C4×Q8—1222no—
433C4×Q8—2441no—
434C4×D4—22no—
435C4×D4—41no—
436C2×C4⋊C4—21no—
437C2×C4⋊C4—611no—
438C2×C4⋊C4—611no—
439C2×C4⋊C4—611no—
440C2×C4⋊C4—611no—
441C2×C4⋊C4—611no—
442C2×C4⋊C4—1221no—
443C2×C4⋊C4—1221no—
444C2×C4⋊C4—1221no—
445C2×C4⋊C4—1221no—
446C2×C4⋊C4—1221no—
447C2×C4⋊C4—1221no—
448C2×C4⋊C4—1221no—
449C2×C4⋊C4C22×D6111yescharacteristic
450C2×C4⋊C4C22×D6111yescharacteristic
451C2×C22⋊C4—21no—
452C2×C22⋊C4—1221no—
453C2×C22⋊C4—1221no—
454C2×C22⋊C4—1221no—
455C2×C22⋊C4—1221no—
456C2×C22⋊C4—1221no—
457C2×C22⋊C4—1221no—
458C2×C42—21no—
459C2×C42—611no—
460C2×C42—611no—
461C2×C42—1221no—
462C2×C42—1221no—
463C2×C42C22×D6111yescharacteristic
464#106—22no—
465#106—2441no—
466C4⋊C8—1222no—
467#103—22no—
468#103—1222no—
469#102—22no—
470#102—1222no—
471#102—1222no—
472#102—1222no—
473#102—1222no—
474#102—2441no—
475C4×C8—22no—
476#95—611no—
477#95—611no—
478#95—1221no—
479#95—1221no—
480#95—1221no—
481C24—21no—
482C24—1221no—
483C2×Q8—21no—
484C2×Q8—21no—
485C2×Q8—21no—
486C2×Q8#1175822212yes—
487C2×Q8#1175822212yes—
488C2×Q8C2×C22⋊D6112yes—
489C2×D4—21no—
490C2×D4—21no—
491C2×D4—21no—
492C2×D4—21no—
493C2×D4—21no—
494C2×D4—1221no—
495C2×D4—1221no—
496C2×D4—1221no—
497C2×D4—1221no—
498C2×D4—1221no—
499C2×D4—1221no—
500C2×D4—4881no—
501C2×D4—4881no—
502C2×D4—2442no—
503C2×D4—2442no—
504C2×D4—2442no—
505C2×D4—2442no—
506C2×D4C2×C22⋊D6112yes—
507C22×C4—21no—
508C22×C4—21no—
509C22×C4—21no—
510C22×C4—42no—
511C22×C4—42no—
512C22×C4—81no—
513C22×C4—611no—
514C22×C4—611no—
515C22×C4—611no—
516C22×C4—611no—
517C22×C4—1221no—
518C22×C4—1221no—
519C22×C4—1221no—
520C22×C4D4×D6111yescharacteristic
521C22×C4D4×D6111yescharacteristic
522C22×C4C23×D6111yescharacteristic
523SD16—28no—
524SD16—2444no—
525C2×C8—21no—
526C2×C8—21no—
527C2×C8—21no—
528C2×C8—21no—
529C2×C8—21no—
530C2×C8—1221no—
531C2×C8—1221no—
532C2×C8—1221no—
533C2×C8—1221no—
534C2×C8—1221no—
535C2×C8—1221no—
536C2×C8C2×C22⋊D6112yes—
537C4⋊C4—21no—
538C4⋊C4—21no—
539C4⋊C4—21no—
540C4⋊C4—21no—
541C4⋊C4—42no—
542C4⋊C4—614no—
543C4⋊C4—612no—
544C4⋊C4—1224no—
545C4⋊C4—1221no—
546C4⋊C4—1221no—
547C4⋊C4—1221no—
548C4⋊C4—1221no—
549C4⋊C4—1221no—
550C4⋊C4—4881no—
551C4⋊C4—4881no—
552C4⋊C4—4881no—
553C4⋊C4—2441no—
554C4⋊C4—2442no—
555C4⋊C4—2442no—
556C4⋊C4—2442no—
557C22⋊C4—42no—
558C22⋊C4—4881no—
559C22⋊C4—4881no—
560C22⋊C4—4881no—
561C22⋊C4—2442no—
562C22⋊C4—2442no—
563C22⋊C4—2442no—
564C42—21no—
565C42—21no—
566C42—42no—
567C42—1224no—
568C42—1221no—
569C42—1221no—
570C42—1221no—
571C42—1221no—
572C42—2442no—
573C23—22no—
574C23—41no—
575C23—41no—
576C23—41no—
577C23—41no—
578C23—41no—
579C23—41no—
580C23—2441no—
581C23—2441no—
582C23—2441no—
583C23—2441no—
584C23—2441no—
585C23—2441no—
586C23—2441no—
587C23C2×D4×D6111yescharacteristic
588Q8—22no—
589Q8—22no—
590Q8—22no—
591Q8#1451976014yescomplemented
592D4—22no—
593D4—22no—
594D4—41no—
595D4—41no—
596D4—4884no—
597D4—4884no—
598D4—2441no—
599D4—2441no—
600D4—2441no—
601D4—2441no—
602C2×C4—22no—
603C2×C4—21no—
604C2×C4—21no—
605C2×C4—22no—
606C2×C4—22no—
607C2×C4—21no—
608C2×C4—21no—
609C2×C4—21no—
610C2×C4—41no—
611C2×C4—41no—
612C2×C4—41no—
613C2×C4—41no—
614C2×C4—41no—
615C2×C4—41no—
616C2×C4—84no—
617C2×C4—612no—
618C2×C4—612no—
619C2×C4—1221no—
620C2×C4—1221no—
621C2×C4—1221no—
622C2×C4—1221no—
623C2×C4—1222no—
624C2×C4—1221no—
625C2×C4—1221no—
626C2×C4—1221no—
627C2×C4—1221no—
628C2×C4—1221no—
629C2×C4—1221no—
630C2×C4—2441no—
631C2×C4—2441no—
632C2×C4—2441no—
633C2×C4—2441no—
634C2×C4—2441no—
635C2×C4—2441no—
636C2×C4—2441no—
637C2×C4—2441no—
638C2×C4SD16×D6112yes—
639C2×C4C22×C22⋊D6111yescharacteristic
640C2×C4#41×D6111yescharacteristic
641C2×C4C2×#1175822211yescharacteristic
642C2×C4C2×#1175822211yescharacteristic
643C2×C4#1451982211yescharacteristic
644C2×C4#1451982411yescharacteristic
645C8—22no—
646C8—41no—
647C8—2441no—
648C8—2441no—
649C22—42no—
650C22—42no—
651C22—42no—
652C22—81no—
653C22—81no—
654C22—81no—
655C22—81no—
656C22—4881no—
657C22—4881no—
658C22—4881no—
659C22—4881no—
660C22—4881no—
661C22—4881no—
662C22—4881no—
663C22#1684027111yescharacteristic
664C22D4×C22⋊D6111yescharacteristic
665C22#1684042111yescharacteristic
666C22C2×SD16×D6111yescharacteristic
667C22#1684055911yescharacteristic
668C22#1684056411yescharacteristic
669C22#1684056811yescharacteristic
670C4—24no—
671C4—22no—
672C4—41no—
673C4—81no—
674C4—81no—
675C4—1222no—
676C4—1222no—
677C4—4881no—
678C4—4881no—
679C4—2441no—
680C4—2442no—
681C4—2441no—
682C4C2×#1451976011yescharacteristic
683C4#1684035411yescharacteristic
684C4#1684041111yescharacteristic
685C4#1684046111yescharacteristic
686C2—82no—
687C2—9761no—
688C2#2144954811yescharacteristic
689C2#2145584811yescharacteristic
690C2#2145602411yescharacteristic
691C2#2145735911yescharacteristic
692C2SD16×C22⋊D6111yescharacteristic
693C2#2145848811yescharacteristic
694C2#2145858311yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC2×C4×SD16×C61C2
socleC23×C61C22×D4
prime radicalC2×#14519773C22
hypercenterC2×C4×SD16D61
hypocenterC61#26163
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×C4×SD16

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
3. Q8×#11758196C2semidirect
4. #21447349C2semidirect
5. #21447318C2semidirect
6. #21447225C2semidirect
8. #21446502C2semidirect
9. #21446376C2semidirect
10. #21446176C2semidirect
11. #21446174C2semidirect
12. #21446172C2semidirect
13. #21446085C2semidirect
14. #21446080C2semidirect
15. C2×C4×SD16×C61C2semidirect
212. #11758196SD16semidirect
243. Q8×C61#123semidirect
340. C2×C4×SD16D61semidirect
408. C61#26163semidirect
591. Q8#14519760semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×#168399602
#214495782
Q8×#117581962
#214473492
#214473182
#214472252
#214470512
#214465022
#214463762
#214461762
#214461742
#214461722
#214460852
#214460802
C2×C4×SD16×C612
#2616361