Finite Group Databasebeta

← #29702313

Subgroups — #29702313

Order 15616.

958 conjugacy classes of subgroups.

The subgroup lattice has 662 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16839986C211yesmaximal, characteristic
2C2×#16839847C211yesmaximal, characteristic
3#21448886C211yesmaximal, characteristic
4#21448737C211yesmaximal, characteristic, complemented
5#21446627C211yesmaximal, characteristic
6#21446584C211yesmaximal, characteristic, complemented
7#21446502C211yesmaximal, characteristic
8#21446492C211yesmaximal, characteristic, complemented
9#21446174C211yesmaximal, characteristic
10#21446162C211yesmaximal, characteristic, complemented
11#21446087C211yesmaximal, characteristic, complemented
12#21446063C211yesmaximal, characteristic
13C8×#11758196C211yesmaximal, characteristic, complemented
14#103×Dic61C211yesmaximal, characteristic, complemented
15C2×C4×Q16×C61C211yesmaximal, characteristic, complemented
16C22×#11758201C2211yescharacteristic
17C2×#14519766C2211yescharacteristic
18C2×Q8×Dic61C2211yescharacteristic
19C2×Q8×Dic61C2211yescharacteristic
20C2×#14519739C2211yescharacteristic
21#16839986C418yescomplemented
22#16839847—24no—
23C2×#14519708C2211yescharacteristic
24C2×#14519706C2211yescharacteristic
25C2×#14519701C2211yescharacteristic
26C2×#14519701C2211yescharacteristic
27C2×#14519701C2211yescharacteristic
28C2×#14519699C2211yescharacteristic
29C2×#14519699C2211yescharacteristic
30C2×#14519697C2211yescharacteristic
31C2×C8×Dic61C2211yescharacteristic
32C2×C8×Dic61C2211yescharacteristic
33#16839669C2211yescharacteristic
34#16839664C2211yescharacteristic
35#16839660—21no—
36#16839660C2211yescharacteristic
37#16839658—21no—
38#16839658C2211yescharacteristic
39#16839653—21no—
40#16839652—21no—
41#16839633—21no—
42#16839627—21no—
43#16839627C2211yescharacteristic
44C4⋊C4×Dic61C2211yescharacteristic
45C4×#11758196—21no—
46C4×#11758196C2211yescharacteristic
47C4×#11758198—21no—
48#16839525C2211yescharacteristic
49#16839525C2211yescharacteristic
50#16839524C2211yescharacteristic
51#16839524C2211yescharacteristic
52#16839517C2211yescharacteristic
53#16839516C2211yescharacteristic
54C22×Q16×C61C2211yescharacteristic
55C2×C4×Q8×C61C2211yescharacteristic
56C2×C4×Q8×C61C2211yescharacteristic
57C4×Q16×C61—24no—
58C2×#107×C61C2211yescharacteristic
59C2×#103×C61C2211yescharacteristic
60C2×#103×C61C2211yescharacteristic
61C2×C4×C8×C61C2211yescharacteristic
62C22×D244C2311yescharacteristic
63C22×C4×D61—21no—
64C2×#11758201—22no—
65C2×#11758201—22no—
66C2×#11758201C2×C414yes—
67C22×#92418C2311yescharacteristic
68C22×#92418C2311yescharacteristic
69#14519766—22no—
70#14519766C2×C414yes—
71Q8×Dic61—22no—
72Q8×Dic61—22no—
73Q8×Dic61—41no—
74Q8×Dic61C2×C414yescomplemented
75#14519739—22no—
76#14519739—41no—
77C2×C4⋊Dic61—21no—
78C2×C4⋊Dic61—21no—
79C2×C4⋊Dic61C2311yescharacteristic
80C2×C4⋊Dic61C2311yescharacteristic
81C2×C4⋊Dic61C2311yescharacteristic
82C2×#11758196—21no—
83C2×#11758196—21no—
84C2×#11758196D411yescharacteristic
85C2×#11758196D411yescharacteristic
86C2×#11758198—21no—
87C2×#11758198—21no—
88C2×#11758198—21no—
89C2×#11758198—21no—
90C2×#11758198—21no—
91C2×C4×Dic61—21no—
92C2×C4×Dic61—21no—
93C2×C4×Dic61—21no—
94C2×C4×Dic61C2311yescharacteristic
95C2×C4×Dic61C2311yescharacteristic
96#14519708—22no—
97#14519706C2×C414yescomplemented
98#14519701—22no—
99#14519701—41no—
100#14519701C2×C414yescomplemented
101#14519699—22no—
102#14519699C2×C414yes—
103#14519697—41no—
104C8×Dic61—22no—
105C8×Dic61—22no—
106#14519684—21no—
107#14519684—21no—
108#14519684—21no—
109#14519684—21no—
110#14519684—21no—
111#14519684—21no—
112#14519684—21no—
113#14519684D411yescharacteristic
114#14519684D411yescharacteristic
115C22×Q8×C61C2311yescharacteristic
116C22×Q8×C61C2311yescharacteristic
117C2×Q16×C61—22no—
118C2×Q16×C61—22no—
119C2×Q16×C61C2×C414yescomplemented
120C22×C8×C61C2311yescharacteristic
121C22×C8×C61C2311yescharacteristic
122C4×Q8×C61—22no—
123C4×Q8×C61—22no—
124C4×Q8×C61—41no—
125C4×Q8×C61—41no—
126C2×C4⋊C4×C61—21no—
127C2×C4⋊C4×C61—21no—
128C2×C4⋊C4×C61C2311yescharacteristic
129C2×C4⋊C4×C61C2311yescharacteristic
130C2×C42×C61—21no—
131C2×C42×C61—21no—
132C2×C42×C61C2311yescharacteristic
133#107×C61—22no—
134#103×C61—22no—
135#103×C61D414yes—
136C4×C8×C61—22no—
137C23×D61—21no—
138C2×Q8×C61—21no—
139C2×Q8×C61—21no—
140C2×Q8×C61—21no—
141C2×Q8×C61—21no—
142C2×Q8×C61—21no—
143C2×Q8×C61—21no—
144C2×Q8×C61—21no—
145C2×Q8×C61—21no—
146C2×Q8×C61C22×C412yes—
147C2×Q8×C61C22×C412yes—
148C2×Q8×C61C2×D412yes—
149C2×Q8×C61#4112yes—
150C22×C4×C61—21no—
151C22×C4×C61—21no—
152C22×C4×C61—21no—
153C22×C4×C61—21no—
154C22×C4×C61C2×D411yescharacteristic
155C22×C4×C61C2×D411yescharacteristic
156C22×C4×C61C2411yescharacteristic
157C22×Dic61—21no—
158C22×Dic61—21no—
159C22×Dic61—21no—
160C22×Dic61C2×D411yescharacteristic
161C22×Dic61C2×D411yescharacteristic
162C22×Dic61C2×D411yescharacteristic
163C22×Dic61C2×D411yescharacteristic
164C2×D244—21no—
165C2×D244—21no—
166C2×D244—21no—
167C2×D244—21no—
168C2×D244—21no—
169C2×D244C22×C412yes—
170C2×C4×D61—42no—
171C2×C4×D61—42no—
172C2×C4×D61—81no—
173Q16×C61—28no—
174C2×C8×C61—21no—
175C2×C8×C61—21no—
176C2×C8×C61—21no—
177C2×C8×C61—21no—
178C2×C8×C61C22×C412yes—
179C2×C8×C61C2×D412yes—
180C4⋊C4×C61—21no—
181C4⋊C4×C61—21no—
182C4⋊C4×C61—21no—
183C4⋊C4×C61—42no—
184C4⋊C4×C61—42no—
185C4⋊C4×C61C2×D412yes—
186C42×C61—21no—
187C42×C61—21no—
188C42×C61—42no—
189C42×C61—42no—
190#11758201—28no—
191#11758198—24no—
192#11758198—24no—
193#11758198—42no—
194#11758198—42no—
195#11758198—81no—
196C4⋊Dic61—21no—
197C4⋊Dic61—24no—
198C4⋊Dic61—21no—
199C4⋊Dic61—21no—
200C4⋊Dic61—21no—
201C4⋊Dic61—21no—
202C4⋊Dic61—42no—
203C4⋊Dic61C22×C412yes—
204#11758196—22no—
205#11758196—22no—
206#11758196—42no—
207#11758196—42no—
208C4×Dic61—21no—
209C4×Dic61—21no—
210C4×Dic61—24no—
211C4×Dic61—42no—
212C4×Dic61—81no—
213C4×Dic61C22×C412yes—
214C4×Dic61#4112yes—
215C2×#92418—21no—
216C2×#92418—21no—
217C2×#92418—21no—
218C2×#92418—21no—
219C2×#92418—21no—
220C2×#92418C22×C412yes—
221C23×C61C22×D411yescharacteristic
222C22×D61—22no—
223C22×D61—41no—
224C22×D61—41no—
225C22×D61—41no—
226C22×D61—41no—
227C22×D61—41no—
228C22×D61—41no—
229Q8×C61—22no—
230Q8×C61—22no—
231Q8×C61—22no—
232Q8×C61—22no—
233Q8×C61—22no—
234Q8×C61—41no—
235Q8×C61—41no—
236Q8×C61C4×D414yes—
237C2×C4×C61—22no—
238C2×C4×C61—22no—
239C2×C4×C61—21no—
240C2×C4×C61—22no—
241C2×C4×C61—21no—
242C2×C4×C61—22no—
243C2×C4×C61—22no—
244C2×C4×C61—21no—
245C2×C4×C61—21no—
246C2×C4×C61—21no—
247C2×C4×C61—41no—
248C2×C4×C61—41no—
249C2×C4×C61—41no—
250C2×C4×C61—41no—
251C2×C4×C61—41no—
252C2×C4×C61—41no—
253C2×C4×C61—41no—
254C2×C4×C61#13512yes—
255C2×C4×C61C23×C411yescharacteristic
256C2×C4×C61C22×D411yescharacteristic
257C2×C4×C61C2×#4111yescharacteristic
258C2×C4×C61C2×#4111yescharacteristic
259C2×C4×C61H(5,𝔽2)11yescharacteristic
260C2×C4×C612−1+411yescharacteristic
261C2×Dic61—21no—
262C2×Dic61—21no—
263C2×Dic61—21no—
264C2×Dic61—21no—
265C2×Dic61—21no—
266C2×Dic61—21no—
267C2×Dic61—22no—
268C2×Dic61—21no—
269C2×Dic61—21no—
270C2×Dic61—21no—
271C2×Dic61—41no—
272C2×Dic61—41no—
273C2×Dic61—41no—
274C2×Dic61—41no—
275C2×Dic61—41no—
276C2×Dic61—41no—
277C2×Dic61—41no—
278C2×Dic61—41no—
279C2×Dic61C4×D412yes—
280C2×Dic61C4×D412yes—
281C2×Dic61#13512yes—
282D244—22no—
283D244—22no—
284D244—41no—
285D244—41no—
286C4×D61—84no—
287C8×C61—22no—
288C8×C61—22no—
289#92418—22no—
290#92418—41no—
291#23000—611nomaximal, Sylow
292C22×C61C2×C4×D411yescharacteristic
293C22×C61D4211yescharacteristic
294C22×C61#54811yescharacteristic
295C22×C61C2×#13511yescharacteristic
296C22×C61C2×#13511yescharacteristic
297C22×C61#57711yescharacteristic
298C22×C61#57811yescharacteristic
299C2×D61—42no—
300C2×D61—42no—
301C2×D61—42no—
302C2×D61—81no—
303C2×D61—81no—
304C2×D61—81no—
305C2×D61—81no—
306C4×C61—24no—
307C4×C61—22no—
308C4×C61—42no—
309C4×C61—42no—
310C4×C61—41no—
311C4×C61—81no—
312C4×C61C2×C4×D411yescharacteristic
313C4×C61#52011yescharacteristic
314C4×C61#54711yescharacteristic
315C4×C61#55311yescharacteristic
316Dic61—22no—
317Dic61—22no—
318Dic61—22no—
319Dic61—42no—
320Dic61—41no—
321Dic61—81no—
322Dic61—81no—
323C2×#464—611no—
324C2×C4×Q16D6111yescharacteristic, complemented
325C2×C4×SD16—611no—
326#2367—611no—
327#2351—611no—
328#1958—611no—
329#1949—611no—
330#1934—611no—
331#1933—611no—
332#1880—611no—
333#1878—611no—
334#1865—611no—
335#1862—611no—
336C8×C4⋊C4—611no—
337C4×#103—611no—
338C2×C61#236111yescharacteristic
339C2×C61C4×#13511yescharacteristic
340C2×C61#294711yescharacteristic
341C2×C61#328211yescharacteristic
342C2×C61#328511yescharacteristic
343C2×C61#336411yescharacteristic
344C2×C61#336511yescharacteristic
345D61—82no—
346C22×Q16C2×D6111yescharacteristic
347C22×SD16—611no—
348C2×#124—611no—
349C2×C4×Q8—611no—
350C2×C4×Q8—611no—
351C2×C4×Q8C2×D6111yescharacteristic
352C2×C4×Q8C2×D6111yescharacteristic
353C2×C4×D4—611no—
354#464—618no—
355C4×Q16—24no—
356C4×SD16—1224no—
357C2×#107C2×D6111yescharacteristic
358C2×#106—611no—
359C2×C4⋊C8—611no—
360C2×#103—611no—
361C2×#103—611no—
362C2×#103—611no—
363C2×#103C2×D6111yescharacteristic
364C2×#103C2×D6111yescharacteristic
365C2×#102—611no—
366C2×#102—611no—
367C2×#102—611no—
368C2×C4×C8—611no—
369C2×C4×C8—611no—
370C2×C4×C8C2×D6111yescharacteristic
371#391—611no—
372#390—611no—
373#389—611no—
374#389—611no—
375#389—1221no—
376#389—1221no—
377#388—1221no—
378#388—1221no—
379#385—1221no—
380#382—611no—
381#382—1221no—
382C4×C4⋊C4—611no—
383C4×C4⋊C4—611no—
384C4×C4⋊C4—1221no—
385C4×C22⋊C4—1221no—
386#340—611no—
387#340—611no—
388#340—611no—
389#340—611no—
390#336—611no—
391#336—611no—
392C61#2300011yescharacteristic, complemented, Sylow
393C22×Q8C22×D6111yescharacteristic
394C22×Q8C22×D6111yescharacteristic
395C22×D4—611no—
396C23×C4—1221no—
397C2×Q16—22no—
398C2×Q16—22no—
399C2×Q16C4×D6114yescomplemented
400C2×SD16—614no—
401C2×SD16—1222no—
402C2×SD16—1222no—
403C22×C8—611no—
404C22×C8—611no—
405C22×C8C22×D6111yescharacteristic
406C22×C8C22×D6111yescharacteristic
407#124—614no—
408#124—1222no—
409C4×Q8—22no—
410C4×Q8—22no—
411C4×Q8—41no—
412C4×Q8—41no—
413C4×Q8—614no—
414C4×Q8—1222no—
415C4×Q8—1222no—
416C4×Q8—2441no—
417C4×D4—1222no—
418C4×D4—2441no—
419C2×C4⋊C4—21no—
420C2×C4⋊C4—21no—
421C2×C4⋊C4—611no—
422C2×C4⋊C4—611no—
423C2×C4⋊C4—611no—
424C2×C4⋊C4—611no—
425C2×C4⋊C4—611no—
426C2×C4⋊C4—1221no—
427C2×C4⋊C4—1221no—
428C2×C4⋊C4—1221no—
429C2×C4⋊C4—1221no—
430C2×C4⋊C4C22×D6111yescharacteristic
431C2×C4⋊C4C22×D6111yescharacteristic
432C2×C22⋊C4—1221no—
433C2×C22⋊C4—1221no—
434C2×C22⋊C4—1221no—
435C2×C22⋊C4—1221no—
436C2×C22⋊C4—1221no—
437C2×C42—21no—
438C2×C42—21no—
439C2×C42—611no—
440C2×C42—611no—
441C2×C42—1221no—
442C2×C42—1221no—
443C2×C42—1221no—
444C2×C42C22×D6111yescharacteristic
445#107—22no—
446#106—1222no—
447C4⋊C8—614no—
448#103—22no—
449#103—614no—
450#103—1222no—
451#103—2441no—
452#103C22⋊D6114yes—
453#102—614no—
454#102—1222no—
455#102—2441no—
456C4×C8—22no—
457C4×C8—1222no—
458C4×C8—1222no—
459#95—611no—
460#95—611no—
461#95—1221no—
462#95—1221no—
463#95—1221no—
464#95—1221no—
465#95—1221no—
466#95—1221no—
467#95—1221no—
468C24—1221no—
469C2×Q8—21no—
470C2×Q8—21no—
471C2×Q8—21no—
472C2×Q8—21no—
473C2×Q8—21no—
474C2×Q8—21no—
475C2×Q8—21no—
476C2×Q8—21no—
477C2×Q8C2×C4×D6112yes—
478C2×Q8C2×C4×D6112yes—
479C2×Q8#1175822212yes—
480C2×Q8C2×C22⋊D6112yes—
481C2×D4—612no—
482C2×D4—1221no—
483C2×D4—1221no—
484C2×D4—1221no—
485C2×D4—1221no—
486C2×D4—1221no—
487C22×C4—21no—
488C22×C4—21no—
489C22×C4—21no—
490C22×C4—21no—
491C22×C4—611no—
492C22×C4—611no—
493C22×C4—611no—
494C22×C4—611no—
495C22×C4—1221no—
496C22×C4—1221no—
497C22×C4—1221no—
498C22×C4—4881no—
499C22×C4—2442no—
500C22×C4—2442no—
501C22×C4D4×D6111yescharacteristic
502C22×C4D4×D6111yescharacteristic
503C22×C4C23×D6111yescharacteristic
504Q16—28no—
505SD16—1228no—
506C2×C8—21no—
507C2×C8—21no—
508C2×C8—21no—
509C2×C8—21no—
510C2×C8—612no—
511C2×C8—1221no—
512C2×C8—1221no—
513C2×C8—1221no—
514C2×C8—1221no—
515C2×C8—1221no—
516C2×C8C2×C4×D6112yes—
517C2×C8C2×C22⋊D6112yes—
518C4⋊C4—21no—
519C4⋊C4—21no—
520C4⋊C4—21no—
521C4⋊C4—42no—
522C4⋊C4—42no—
523C4⋊C4—612no—
524C4⋊C4—1222no—
525C4⋊C4—1224no—
526C4⋊C4—1221no—
527C4⋊C4—1222no—
528C4⋊C4—1221no—
529C4⋊C4—1221no—
530C4⋊C4—1221no—
531C4⋊C4—1221no—
532C4⋊C4—2442no—
533C4⋊C4—2442no—
534C4⋊C4—2442no—
535C4⋊C4C2×C22⋊D6112yes—
536C22⋊C4—1224no—
537C22⋊C4—1224no—
538C22⋊C4—4881no—
539C22⋊C4—2442no—
540C22⋊C4—2442no—
541C42—21no—
542C42—21no—
543C42—42no—
544C42—42no—
545C42—612no—
546C42—612no—
547C42—1224no—
548C42—1221no—
549C42—1221no—
550C42—4881no—
551C42—2442no—
552C23—1222no—
553C23—2441no—
554C23—2441no—
555C23—2441no—
556C23—2441no—
557C23—2441no—
558C23—2441no—
559C23C2×D4×D6111yescharacteristic
560Q8—22no—
561Q8—22no—
562Q8—22no—
563Q8—22no—
564Q8—22no—
565Q8—41no—
566Q8—41no—
567Q8C4×C22⋊D6114yes—
568D4—1222no—
569D4—1222no—
570D4—2441no—
571D4—2441no—
572C2×C4—22no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—22no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—612no—
590C2×C4—612no—
591C2×C4—612no—
592C2×C4—1221no—
593C2×C4—1221no—
594C2×C4—1221no—
595C2×C4—1221no—
596C2×C4—1221no—
597C2×C4—1221no—
598C2×C4—1222no—
599C2×C4—1221no—
600C2×C4—1221no—
601C2×C4—1221no—
602C2×C4—4884no—
603C2×C4—2441no—
604C2×C4—2441no—
605C2×C4—2441no—
606C2×C4—2441no—
607C2×C4—2441no—
608C2×C4—2441no—
609C2×C4—2441no—
610C2×C4—2441no—
611C2×C4#1451980012yes—
612C2×C4C22×C4×D6111yescharacteristic
613C2×C4C22×C22⋊D6111yescharacteristic
614C2×C4#41×D6111yescharacteristic
615C2×C4C2×#1175822211yescharacteristic
616C2×C4#1451982211yescharacteristic
617C2×C4#1451982411yescharacteristic
618C8—22no—
619C8—22no—
620C8—1222no—
621C8—2441no—
622C22—4881no—
623C22—4881no—
624C22—4881no—
625C22—4881no—
626C22—2442no—
627C22—2442no—
628C22—2442no—
629C22C4×D4×D6111yescharacteristic
630C22D4×C22⋊D6111yescharacteristic
631C22#1684042111yescharacteristic
632C22#135×D6111yescharacteristic
633C22C2×#1451980011yescharacteristic
634C22#1684056911yescharacteristic
635C22#1684057511yescharacteristic
636C4—24no—
637C4—22no—
638C4—41no—
639C4—42no—
640C4—42no—
641C4—81no—
642C4—1222no—
643C4—1222no—
644C4—1222no—
645C4—4881no—
646C4—4881no—
647C4—2441no—
648C4—2442no—
649C4C2×C4×C22⋊D6111yescharacteristic
650C4#1684024111yescharacteristic
651C4#1684041111yescharacteristic
652C4#1684046111yescharacteristic
653C2—4882no—
654C2#2144883611yescharacteristic
655C2C4×#1451980011yescharacteristic
656C2#2145465711yescharacteristic
657C2#2145752011yescharacteristic
658C2#2145756011yescharacteristic
659C2#2145850511yescharacteristic
660C2#2145851311yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC2×C4×Q16×C61C2
socleC23×C61C22×D4
prime radicalC22×D244C23
cosocleC22×C4×C61C24
hypercenterC2×C4×Q16D61
hypocenterC61#23000
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×C4×Q16

Complements

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

NH ≅ G/Ntype
4. #21448737C2semidirect
6. #21446584C2semidirect
8. #21446492C2semidirect
10. #21446162C2semidirect
11. #21446087C2semidirect
13. C8×#11758196C2semidirect
14. #103×Dic61C2semidirect
15. C2×C4×Q16×C61C2semidirect
21. #16839986C4semidirect
74. Q8×Dic61C2×C4semidirect
97. #14519706C2×C4semidirect
100. #14519701C2×C4semidirect
119. C2×Q16×C61C2×C4semidirect
324. C2×C4×Q16D61semidirect
392. C61#23000semidirect
399. C2×Q16C4×D61semidirect

Maximal subgroups

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

subgroupindex
C2×#168399862
C2×#168398472
#214488862
#214487372
#214466272
#214465842
#214465022
#214464922
#214461742
#214461622
#214460872
#214460632
C8×#117581962
#103×Dic612
C2×C4×Q16×C612
#2300061