Finite Group Databasebeta

← #29740245

Subgroups — #29740245

Order 15616.

904 conjugacy classes of subgroups.

The subgroup lattice has 692 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16839962C211yesmaximal, characteristic
2C2×#16839860C211yesmaximal, characteristic, complemented
3#21449584C211yesmaximal, characteristic
4#21447343C211yesmaximal, characteristic, complemented
5#21447310C211yesmaximal, characteristic, complemented
6#21447233C211yesmaximal, characteristic, complemented
7#21447053C211yesmaximal, characteristic
8#21446515C211yesmaximal, characteristic, complemented
9#21446385C211yesmaximal, characteristic, complemented
10#21446174C211yesmaximal, characteristic, complemented
11#21446174C211yesmaximal, characteristic, complemented
12#21446174C211yesmaximal, characteristic, complemented
13#21446087C211yesmaximal, characteristic, complemented
14#21446079C211yesmaximal, characteristic, complemented
15C61×#2351C211yesmaximal, characteristic, complemented
16C22×#11758201C2211yescharacteristic
17C22×#11758201C2211yescharacteristic
18C2×#14519774C2211yescharacteristic
19C2×#14519762—21no—
20C2×C4×D244C2211yescharacteristic
21#16839962—24no—
22#16839860—24no—
23C2×#14519710C2211yescharacteristic
24C2×#14519708C2211yescharacteristic
25C2×C4⋊#92418C2211yescharacteristic
26C2×#14519701C2211yescharacteristic
27C2×#14519701C2211yescharacteristic
28C2×#14519697C2211yescharacteristic
29C2×#14519697C2211yescharacteristic
30C2×#14519697C2211yescharacteristic
31C2×#14519697C2211yescharacteristic
32C2×#14519697C2211yescharacteristic
33C2×#14519697C2211yescharacteristic
34C2×#14519688C2211yescharacteristic
35#16839680—21no—
36#16839679—21no—
37#16839664C2211yescharacteristic
38#16839664C2211yescharacteristic
39#16839664C2211yescharacteristic
40#16839660C2211yescharacteristic
41#16839640—21no—
42#16839524C2211yescharacteristic
43#16839524C2211yescharacteristic
44#16839524C2211yescharacteristic
45#16839524C2211yescharacteristic
46#16839517C2211yescharacteristic
47#16839517C2211yescharacteristic
48#16839517C2211yescharacteristic
49#16839517C2211yescharacteristic
50C2×C4×Q8×C61C2211yescharacteristic
51C2×C4×Q8×C61C2211yescharacteristic
52C2×C4×Q8×C61C2211yescharacteristic
53C61×#389—21no—
54C61×#389—21no—
55C61×#389C2211yescharacteristic
56C61×#389C2211yescharacteristic
57C61×#382—21no—
58C61×#382C2211yescharacteristic
59C4×C4⋊C4×C61—21no—
60C4×C4⋊C4×C61C2211yescharacteristic
61C22×D244—21no—
62C22×D244—21no—
63C22×D244C2311yescharacteristic
64C22×D244C2311yescharacteristic
65C22×C4×D61—21no—
66C2×#11758201—22no—
67C2×#11758201—22no—
68C2×#11758201—22no—
69C2×#11758201—22no—
70C2×#11758201—22no—
71C2×#11758201D414yes—
72C22×#92418C2311yescharacteristic
73C22×#92418C2311yescharacteristic
74C22×#92418C2311yescharacteristic
75C22×#92418C2311yescharacteristic
76#14519774—22no—
77#14519774—41no—
78#14519762—44no—
79C4×D244—22no—
80C4×D244—41no—
81C2×C4⋊Dic61C2311yescharacteristic
82C2×C4⋊Dic61C2311yescharacteristic
83C2×#11758196—21no—
84C2×#11758198—21no—
85C2×#11758198—21no—
86C2×#11758198—21no—
87C2×#11758198—21no—
88C2×#11758198—21no—
89C2×#11758198—21no—
90C2×#11758198—21no—
91#14519710—22no—
92#14519708—41no—
93C4⋊#92418—22no—
94#14519701—22no—
95#14519701—22no—
96#14519697—22no—
97#14519697—22no—
98#14519697—22no—
99#14519697—22no—
100#14519697—22no—
101#14519697—41no—
102#14519688—22no—
103#14519684—21no—
104C22×Q8×C61C2311yescharacteristic
105C4×Q8×C61—22no—
106C4×Q8×C61—22no—
107C4×Q8×C61—22no—
108C4×Q8×C61—22no—
109C4×Q8×C61—41no—
110C4×Q8×C61—41no—
111C2×C4⋊C4×C61—21no—
112C2×C4⋊C4×C61—21no—
113C2×C4⋊C4×C61—21no—
114C2×C4⋊C4×C61—21no—
115C2×C4⋊C4×C61—21no—
116C2×C4⋊C4×C61D411yescharacteristic
117C2×C4⋊C4×C61D411yescharacteristic
118C2×C4⋊C4×C61C2311yescharacteristic
119C2×C4⋊C4×C61C2311yescharacteristic
120C2×C4⋊C4×C61C2311yescharacteristic
121C2×C42×C61—21no—
122C2×C42×C61—21no—
123C2×C42×C61—21no—
124C2×C42×C61C2311yescharacteristic
125C2×C42×C61C2311yescharacteristic
126C2×C42×C61C2311yescharacteristic
127#95×C61—21no—
128#95×C61—21no—
129#95×C61—21no—
130#95×C61—21no—
131#95×C61D411yescharacteristic
132#95×C61D411yescharacteristic
133C23×D61—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—
141C22×C4×C61—21no—
142C22×C4×C61—21no—
143C22×C4×C61—21no—
144C22×C4×C61—21no—
145C22×C4×C61C2×D411yescharacteristic
146C22×C4×C61C2×D411yescharacteristic
147C22×C4×C61C2×D411yescharacteristic
148C22×C4×C61C2×D411yescharacteristic
149C22×C4×C61C2×D411yescharacteristic
150C22×C4×C61C2×D411yescharacteristic
151C22×C4×C61C2411yescharacteristic
152C22×Dic61—21no—
153C22×Dic61—21no—
154C2×D244—21no—
155C2×D244—21no—
156C2×D244—21no—
157C2×D244—21no—
158C2×D244—21no—
159C2×D244—21no—
160C2×D244—21no—
161C2×D244—21no—
162C2×D244—21no—
163C2×D244—21no—
164C2×D244—21no—
165C2×D244—42no—
166C2×D244—42no—
167C2×D244—42no—
168C2×D244—42no—
169C2×D244—81no—
170C2×D244—81no—
171C2×D244C2×D412yes—
172C2×C4×D61—42no—
173C2×C4×D61—42no—
174C2×C4×D61—81no—
175C4⋊C4×C61—24no—
176C4⋊C4×C61—22no—
177C4⋊C4×C61—21no—
178C4⋊C4×C61—21no—
179C4⋊C4×C61—21no—
180C4⋊C4×C61—21no—
181C4⋊C4×C61—21no—
182C4⋊C4×C61—41no—
183C4⋊C4×C61—42no—
184C4⋊C4×C61—42no—
185C4⋊C4×C61—42no—
186C4⋊C4×C61—42no—
187C4⋊C4×C61#4112yes—
188C42×C61—24no—
189C42×C61—21no—
190C42×C61—21no—
191C42×C61—21no—
192C42×C61—21no—
193C42×C61—21no—
194C42×C61—21no—
195C42×C61—42no—
196C42×C61—42no—
197#11758201—28no—
198#11758201—44no—
199#11758198—42no—
200#11758198—42no—
201#11758198—42no—
202#11758198—42no—
203#11758198—81no—
204#11758198—81no—
205#11758198—81no—
206C4⋊Dic61—21no—
207C4⋊Dic61—21no—
208C4⋊Dic61—21no—
209C4⋊Dic61—21no—
210#11758196—42no—
211C2×#92418—21no—
212C2×#92418—21no—
213C2×#92418—21no—
214C2×#92418—21no—
215C2×#92418—21no—
216C2×#92418—21no—
217C2×#92418—21no—
218C2×#92418—21no—
219C2×#92418—21no—
220C2×#92418—21no—
221C2×#92418—21no—
222C2×#92418C2×D412yes—
223C23×C61C22×D411yescharacteristic
224C22×D61—22no—
225C22×D61—41no—
226C22×D61—41no—
227C22×D61—41no—
228C22×D61—41no—
229C22×D61—41no—
230C22×D61—41no—
231C22×D61—41no—
232C22×D61—41no—
233C22×D61—41no—
234C22×D61—41no—
235C22×D61—41no—
236C22×D61—41no—
237C22×D61—41no—
238Q8×C61—22no—
239Q8×C61—22no—
240Q8×C61—22no—
241Q8×C61#12314yescomplemented
242C2×C4×C61—22no—
243C2×C4×C61—21no—
244C2×C4×C61—22no—
245C2×C4×C61—22no—
246C2×C4×C61—21no—
247C2×C4×C61—21no—
248C2×C4×C61—21no—
249C2×C4×C61—22no—
250C2×C4×C61—21no—
251C2×C4×C61—21no—
252C2×C4×C61—21no—
253C2×C4×C61—21no—
254C2×C4×C61—21no—
255C2×C4×C61—21no—
256C2×C4×C61—21no—
257C2×C4×C61—21no—
258C2×C4×C61—21no—
259C2×C4×C61—21no—
260C2×C4×C61—21no—
261C2×C4×C61—21no—
262C2×C4×C61—21no—
263C2×C4×C61—41no—
264C2×C4×C61—41no—
265C2×C4×C61—41no—
266C2×C4×C61—41no—
267C2×C4×C61—41no—
268C2×C4×C61—41no—
269C2×C4×C61—41no—
270C2×C4×C61—41no—
271C2×C4×C61AGL(1,ℤ/8)12yes—
272C2×C4×C61C22×D411yescharacteristic
273C2×C4×C61C2×#4111yescharacteristic
274C2×C4×C61C2×#4111yescharacteristic
275C2×C4×C61C2×#4111yescharacteristic
276C2×C4×C61H(5,𝔽2)11yescharacteristic
277C2×C4×C612−1+411yescharacteristic
278C2×Dic61—41no—
279C2×Dic61—41no—
280C2×Dic61—41no—
281C2×Dic61—41no—
282C2×Dic61—41no—
283C2×Dic61—41no—
284D244—22no—
285D244—22no—
286D244—41no—
287D244—41no—
288D244—41no—
289D244—41no—
290D244—41no—
291D244—41no—
292D244—84no—
293D244—84no—
294C4×D61—84no—
295#92418—22no—
296#92418—41no—
297#92418—41no—
298#92418—41no—
299#26175—611nomaximal, Sylow
300C22×C61#52511yescharacteristic
301C22×C61D4211yescharacteristic
302C22×C61#54811yescharacteristic
303C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
304C22×C61#57511yescharacteristic
305C22×C61#57711yescharacteristic
306C22×C61#57711yescharacteristic
307C2×D61—42no—
308C2×D61—42no—
309C2×D61—42no—
310C2×D61—81no—
311C2×D61—81no—
312C2×D61—81no—
313C2×D61—81no—
314C2×D61—81no—
315C2×D61—81no—
316C2×D61—81no—
317C2×D61—81no—
318C2×D61—81no—
319C2×D61—81no—
320C2×D61—81no—
321C4×C61—24no—
322C4×C61—22no—
323C4×C61—41no—
324C4×C61—42no—
325C4×C61—41no—
326C4×C61—41no—
327C4×C61—41no—
328C4×C61—41no—
329C4×C61—81no—
330C4×C61—81no—
331C4×C61C2×#12311yescharacteristic
332C4×C61#53911yescharacteristic
333C4×C61#54711yescharacteristic
334C4×C61#55311yescharacteristic
335Dic61—81no—
336Dic61—81no—
337C2×#460—611no—
338C2×#440—611no—
339#2445—611no—
340#2351D6111yescharacteristic, complemented
341#2092—611no—
342#2086—611no—
343#2073—611no—
344#2041—611no—
345#1936—611no—
346#1917—611no—
347#1880—611no—
348#1880—611no—
349#1880—611no—
350#1865—611no—
351#1864—611no—
352C2×C61#244311yescharacteristic
353C2×C61#311511yescharacteristic
354C2×C61#312611yescharacteristic
355C2×C61#326511yescharacteristic
356C2×C61#327511yescharacteristic
357C2×C61#336811yescharacteristic
358C2×C61#337111yescharacteristic
359D61—82no—
360D61—161no—
361C22×SD16—611no—
362C22×SD16—611no—
363C2×#127—611no—
364C2×#123—1221no—
365C2×C4×Q8C2×D6111yescharacteristic
366C2×C4×Q8C2×D6111yescharacteristic
367C2×C4×Q8C2×D6111yescharacteristic
368C2×C4×D4—611no—
369#460—1224no—
370#440—1224no—
371C2×#107—611no—
372C2×#106—611no—
373C2×C4⋊C8—611no—
374C2×#103—611no—
375C2×#103—611no—
376C2×#102—611no—
377C2×#102—611no—
378C2×#102—611no—
379C2×#102—611no—
380C2×#102—611no—
381C2×#102—611no—
382C2×#97—611no—
383#394—1221no—
384#394—1221no—
385#390—611no—
386#390—611no—
387#390—611no—
388#389—21no—
389#389—21no—
390#389—611no—
391#389C2×D6111yescharacteristic
392#389C2×D6111yescharacteristic
393#386—1221no—
394#382—21no—
395#382C2×D6111yescharacteristic
396C4×C4⋊C4—21no—
397C4×C4⋊C4C2×D6111yescharacteristic
398#340—611no—
399#340—611no—
400#340—611no—
401#340—611no—
402#336—611no—
403#336—611no—
404#336—611no—
405#336—611no—
406C61#2617511yescharacteristic, complemented, Sylow
407C22×Q8C22×D6111yescharacteristic
408C22×D4—611no—
409C22×D4—611no—
410C22×D4—1221no—
411C22×D4—1221no—
412C23×C4—1221no—
413C2×SD16—614no—
414C2×SD16—1222no—
415C2×SD16—1222no—
416C2×SD16—1222no—
417C2×SD16—1222no—
418C2×SD16—1222no—
419C22×C8—611no—
420C22×C8—611no—
421C22×C8—611no—
422C22×C8—611no—
423#127—1222no—
424#127—2441no—
425#123—2444no—
426C4×Q8—22no—
427C4×Q8—22no—
428C4×Q8—22no—
429C4×Q8—22no—
430C4×Q8—41no—
431C4×Q8—41no—
432C4×D4—1222no—
433C4×D4—2441no—
434C2×C4⋊C4—21no—
435C2×C4⋊C4—21no—
436C2×C4⋊C4—21no—
437C2×C4⋊C4—21no—
438C2×C4⋊C4—21no—
439C2×C4⋊C4—611no—
440C2×C4⋊C4—611no—
441C2×C4⋊C4—1221no—
442C2×C4⋊C4C22⋊D6111yescharacteristic
443C2×C4⋊C4C22⋊D6111yescharacteristic
444C2×C4⋊C4C22×D6111yescharacteristic
445C2×C4⋊C4C22×D6111yescharacteristic
446C2×C4⋊C4C22×D6111yescharacteristic
447C2×C22⋊C4—1221no—
448C2×C22⋊C4—1221no—
449C2×C22⋊C4—1221no—
450C2×C22⋊C4—1221no—
451C2×C22⋊C4—1221no—
452C2×C22⋊C4—1221no—
453C2×C22⋊C4—1221no—
454C2×C42—21no—
455C2×C42—21no—
456C2×C42—21no—
457C2×C42C22×D6111yescharacteristic
458C2×C42C22×D6111yescharacteristic
459C2×C42C22×D6111yescharacteristic
460#107—1222no—
461#106—2441no—
462C4⋊C8—1222no—
463#103—1222no—
464#103—1222no—
465#102—1222no—
466#102—1222no—
467#102—1222no—
468#102—1222no—
469#102—1222no—
470#102—2441no—
471#97—1222no—
472#95—21no—
473#95—21no—
474#95—21no—
475#95—21no—
476#95—1221no—
477#95C22⋊D6111yescharacteristic
478#95C22⋊D6111yescharacteristic
479C24—1221no—
480C24—1221no—
481C2×Q8—21no—
482C2×Q8—21no—
483C2×Q8—21no—
484C2×Q8#1175822212yes—
485C2×Q8D4×D6112yes—
486C2×Q8#1175822612yes—
487C2×D4—612no—
488C2×D4—1221no—
489C2×D4—1221no—
490C2×D4—1221no—
491C2×D4—1221no—
492C2×D4—1221no—
493C2×D4—1221no—
494C2×D4—1221no—
495C2×D4—1221no—
496C2×D4—1221no—
497C2×D4—1221no—
498C2×D4—1221no—
499C2×D4—4881no—
500C2×D4—4881no—
501C2×D4—2442no—
502C2×D4—2442no—
503C2×D4—2442no—
504C2×D4—2442no—
505C22×C4—21no—
506C22×C4—21no—
507C22×C4—21no—
508C22×C4—21no—
509C22×C4—1221no—
510C22×C4—1221no—
511C22×C4—4881no—
512C22×C4—2442no—
513C22×C4—2442no—
514C22×C4C2×C22⋊D6111yescharacteristic
515C22×C4C2×C22⋊D6111yescharacteristic
516C22×C4C2×C22⋊D6111yescharacteristic
517C22×C4C2×C22⋊D6111yescharacteristic
518C22×C4C2×C22⋊D6111yescharacteristic
519C22×C4C2×C22⋊D6111yescharacteristic
520C22×C4C23×D6111yescharacteristic
521SD16—1228no—
522SD16—2444no—
523C2×C8—612no—
524C2×C8—1221no—
525C2×C8—1221no—
526C2×C8—1221no—
527C2×C8—1221no—
528C2×C8—1221no—
529C2×C8—1221no—
530C2×C8—1221no—
531C2×C8—1221no—
532C2×C8—1221no—
533C2×C8—1221no—
534C2×C8—1221no—
535C4⋊C4—24no—
536C4⋊C4—21no—
537C4⋊C4—22no—
538C4⋊C4—21no—
539C4⋊C4—21no—
540C4⋊C4—21no—
541C4⋊C4—21no—
542C4⋊C4—41no—
543C4⋊C4—42no—
544C4⋊C4—42no—
545C4⋊C4—42no—
546C4⋊C4—42no—
547C4⋊C4—1221no—
548C4⋊C4—1221no—
549C4⋊C4—1221no—
550C4⋊C4—1221no—
551C4⋊C4—2442no—
552C4⋊C4#1175822212yes—
553C22⋊C4—4881no—
554C22⋊C4—4881no—
555C22⋊C4—4881no—
556C22⋊C4—2442no—
557C22⋊C4—2442no—
558C22⋊C4—2442no—
559C22⋊C4—2442no—
560C42—24no—
561C42—21no—
562C42—21no—
563C42—21no—
564C42—21no—
565C42—21no—
566C42—21no—
567C42—42no—
568C42—42no—
569C23—1222no—
570C23—2441no—
571C23—2441no—
572C23—2441no—
573C23—2441no—
574C23—2441no—
575C23—2441no—
576C23—2441no—
577C23—2441no—
578C23—2441no—
579C23—2441no—
580C23—2441no—
581C23—2441no—
582C23—2441no—
583C23C22×C22⋊D6111yescharacteristic
584Q8—22no—
585Q8—22no—
586Q8—22no—
587Q8#1451976214yescomplemented
588D4—1222no—
589D4—1222no—
590D4—4884no—
591D4—4884no—
592D4—2441no—
593D4—2441no—
594D4—2441no—
595D4—2441no—
596D4—2441no—
597D4—2441no—
598C2×C4—22no—
599C2×C4—21no—
600C2×C4—21no—
601C2×C4—22no—
602C2×C4—22no—
603C2×C4—21no—
604C2×C4—21no—
605C2×C4—22no—
606C2×C4—21no—
607C2×C4—21no—
608C2×C4—21no—
609C2×C4—21no—
610C2×C4—21no—
611C2×C4—21no—
612C2×C4—21no—
613C2×C4—21no—
614C2×C4—21no—
615C2×C4—21no—
616C2×C4—21no—
617C2×C4—21no—
618C2×C4—21no—
619C2×C4—41no—
620C2×C4—41no—
621C2×C4—41no—
622C2×C4—41no—
623C2×C4—41no—
624C2×C4—41no—
625C2×C4—41no—
626C2×C4—41no—
627C2×C4—4884no—
628C2×C4—2441no—
629C2×C4—2441no—
630C2×C4—2441no—
631C2×C4—2441no—
632C2×C4—2441no—
633C2×C4—2441no—
634C2×C4#1451980212yes—
635C2×C4C2×D4×D6111yescharacteristic
636C2×C4C2×#1175822211yescharacteristic
637C2×C4C2×#1175822211yescharacteristic
638C2×C4C2×#1175822611yescharacteristic
639C2×C4#1451982211yescharacteristic
640C2×C4#1451982511yescharacteristic
641C8—1222no—
642C8—2441no—
643C8—2441no—
644C8—2441no—
645C22—4881no—
646C22—4881no—
647C22—4881no—
648C22—4881no—
649C22—4881no—
650C22—4881no—
651C22—4881no—
652C22—4881no—
653C22—4881no—
654C22—4881no—
655C22—4881no—
656C22—2442no—
657C22—2442no—
658C22—2442no—
659C22#1684027311yescharacteristic
660C22D4×C22⋊D6111yescharacteristic
661C22#1684042311yescharacteristic
662C22C2×#1451980211yescharacteristic
663C22#1684056011yescharacteristic
664C22#1684057011yescharacteristic
665C22#1684057011yescharacteristic
666C4—24no—
667C4—22no—
668C4—41no—
669C4—41no—
670C4—41no—
671C4—42no—
672C4—41no—
673C4—41no—
674C4—81no—
675C4—81no—
676C4—4881no—
677C4—4881no—
678C4C2×#1451976211yescharacteristic
679C4#1684035911yescharacteristic
680C4#1684041311yescharacteristic
681C4#1684046011yescharacteristic
682C2—9761no—
683C2—4882no—
684C2#2144955411yescharacteristic
685C2#2145597011yescharacteristic
686C2#2145611011yescharacteristic
687C2#2145731511yescharacteristic
688C2#2145745111yescharacteristic
689C2#2145854811yescharacteristic
690C2#2145858711yescharacteristic
subgroupG/H
centerC23C22×C22⋊D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2351C2
socleC23×C61C22×D4
prime radicalC2×#14519774C22
hypercenter#2351D61
hypocenterC61#26175
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ #2351

Complements

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

NH ≅ G/Ntype
2. C2×#16839860C2semidirect
4. #21447343C2semidirect
5. #21447310C2semidirect
6. #21447233C2semidirect
8. #21446515C2semidirect
9. #21446385C2semidirect
10. #21446174C2semidirect
11. #21446174C2semidirect
12. #21446174C2semidirect
13. #21446087C2semidirect
14. #21446079C2semidirect
15. C61×#2351C2semidirect
241. Q8×C61#123semidirect
340. #2351D61semidirect
406. C61#26175semidirect
587. Q8#14519762semidirect

Maximal subgroups

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

subgroupindex
C2×#168399622
C2×#168398602
#214495842
#214473432
#214473102
#214472332
#214470532
#214465152
#214463852
#214461742
#214460872
#214460792
C61×#23512
#2617561