Finite Group Databasebeta

← #29749973

Subgroups — #29749973

Order 15616.

710 conjugacy classes of subgroups.

The subgroup lattice has 644 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456188C211yesmaximal, characteristic
2#21455788C211yesmaximal, characteristic
3#21455661C211yesmaximal, characteristic, complemented
4#21447183C211yesmaximal, characteristic, complemented
5#21447162C211yesmaximal, characteristic, complemented
6#21447115C211yesmaximal, characteristic
7#21446852C211yesmaximal, characteristic
8#21446619C211yesmaximal, characteristic, complemented
9#21446525C211yesmaximal, characteristic
10#21446474C211yesmaximal, characteristic, complemented
11#21446471C211yesmaximal, characteristic, complemented
12#21446212C211yesmaximal, characteristic, complemented
13#21446045C211yesmaximal, characteristic
14#21445699C211yesmaximal, characteristic, complemented
15C61×#2947C211yesmaximal, characteristic
16#16840390C2211yescharacteristic
17#16840333C2211yescharacteristic
18C2×#14519775C2211yescharacteristic
19#16840129—21no—
20#16840120—21no—
21#16840115—21no—
22#16840111—21no—
23#16840103—21no—
24#16840103—21no—
25#16840097—21no—
26#16840092—21no—
27#16840069—21no—
28#16840069—21no—
29#16840057—21no—
30#16840047—21no—
31C2×#14519712C2211yescharacteristic
32#16839805C2211yescharacteristic
33#16839802C2211yescharacteristic
34#16839798C2211yescharacteristic
35#16839789C2211yescharacteristic
36C2×#14519704C2211yescharacteristic
37#16839771C2211yescharacteristic
38#16839770C2211yescharacteristic
39#16839770C2211yescharacteristic
40#16839763C2211yescharacteristic
41C2×#14519702C2211yescharacteristic
42C2×#14519701C2211yescharacteristic
43C2×#14519700C2211yescharacteristic
44C2×#14519696C2211yescharacteristic
45C2×#14519695C2211yescharacteristic
46#16839729C2211yescharacteristic
47#16839717C2211yescharacteristic
48C2×C8×Dic61C2211yescharacteristic
49#16839631C2211yescharacteristic
50#16839531C2211yescharacteristic
51#16839530C2211yescharacteristic
52#16839527C2211yescharacteristic
53#16839523C2211yescharacteristic
54#16839521C2211yescharacteristic
55#16839521C2211yescharacteristic
56C2×#137×C61C2211yescharacteristic
57C61×#520C2211yescharacteristic
58C61×#519C2211yescharacteristic
59C61×#441—21no—
60C61×#440—21no—
61C4×Q16×C61—21no—
62C4×SD16×C61—21no—
63C61×#428C2211yescharacteristic
64C61×#416C2211yescharacteristic
65C2×#103×C61C2211yescharacteristic
66C61×#405C2211yescharacteristic
67C22×Dic122C2311yescharacteristic
68C2×#11758194C2311yescharacteristic
69C2×#11758194C2311yescharacteristic
70C2×#11758194C2311yescharacteristic
71C22×#92418C2311yescharacteristic
72#14519777—21no—
73#14519776—21no—
74#14519775—21no—
75#14519775—22no—
76#14519775—22no—
77#14519769—21no—
78#14519768—21no—
79#14519764—21no—
80#14519758—41no—
81#14519756—21no—
82#14519756—21no—
83#14519754—41no—
84#14519736C2311yescharacteristic
85C2×C4⋊Dic61C2311yescharacteristic
86C2×#11758196—41no—
87C2×#11758196—41no—
88C2×C4×Dic61C2311yescharacteristic
89#14519712—22no—
90#14519710—21no—
91#14519710—21no—
92#14519710D411yescharacteristic
93#14519710D411yescharacteristic
94#14519708—21no—
95#14519708—21no—
96#14519708D411yescharacteristic
97#14519708D411yescharacteristic
98C4⋊#92418—21no—
99C4⋊#92418—21no—
100#14519704—22no—
101#14519702—21no—
102#14519702—21no—
103#14519701—21no—
104#14519701—21no—
105#14519701—21no—
106#14519701—21no—
107#14519701—21no—
108#14519700—21no—
109#14519700—21no—
110#14519700—21no—
111#14519700—21no—
112#14519700—21no—
113#14519700—21no—
114#14519698—21no—
115#14519696—22no—
116#14519695—22no—
117C22⋊#92418—21no—
118C22⋊#92418—21no—
119#14519687—21no—
120C8×Dic61—21no—
121C8×Dic61—22no—
122C2×#41×C61C2311yescharacteristic
123C22×Q8×C61C2311yescharacteristic
124#137×C61—22no—
125#137×C61—22no—
126C2×Q16×C61—21no—
127C2×SD16×C61—21no—
128C2×M16×C61C2311yescharacteristic
129C22×C8×C61C2311yescharacteristic
130C4×Q8×C61—21no—
131C4×Q8×C61—21no—
132C4×Q8×C61—21no—
133C4×Q8×C61—21no—
134C4×Q8×C61—21no—
135C4×D4×C61—21no—
136C4×D4×C61—41no—
137#117×C61—21no—
138#117×C61—41no—
139#117×C61C2311yescharacteristic
140#117×C61C2311yescharacteristic
141C2×C4⋊C4×C61—21no—
142C2×C4⋊C4×C61C2311yescharacteristic
143#107×C61—21no—
144#106×C61—21no—
145#103×C61—21no—
146#103×C61—21no—
147#103×C61D411yescharacteristic
148#103×C61D411yescharacteristic
149#102×C61D411yescharacteristic
150#102×C61D411yescharacteristic
151#97×C61—21no—
152C4×C8×C61—21no—
153#41×C61—21no—
154#41×C61—21no—
155#41×C61—21no—
156#41×C61—21no—
157C2×Q8×C61—21no—
158C2×Q8×C61—21no—
159C2×Q8×C61—21no—
160C2×Q8×C61—21no—
161C2×Q8×C61—21no—
162C2×Q8×C61C2×D411yescharacteristic
163C2×Q8×C61#4112yes—
164C2×D4×C61—21no—
165C2×D4×C61C2×D411yescharacteristic
166C22×C4×C61—21no—
167C22×C4×C61—21no—
168C22×C4×C61—21no—
169C22×C4×C61—41no—
170C22×C4×C61C2411yescharacteristic
171C22×Dic61—21no—
172C22×Dic61—21no—
173C22×Dic61—21no—
174C2×Dic122—21no—
175C2×Dic122—21no—
176C2×Dic122—21no—
177C2×Dic122—21no—
178C2×Dic122—21no—
179C2×Dic122—21no—
180C2×Dic122—41no—
181Q16×C61—24no—
182SD16×C61—24no—
183M16×C61—21no—
184M16×C61#4112yes—
185C2×C8×C61—21no—
186C2×C8×C61—21no—
187C2×C8×C61—21no—
188C2×C8×C61C2×D411yescharacteristic
189C2×C8×C61C2×D411yescharacteristic
190C4⋊C4×C61—21no—
191C4⋊C4×C61—21no—
192C4⋊C4×C61—41no—
193C4⋊C4×C61—41no—
194C4⋊C4×C61—41no—
195C4⋊C4×C61C2×D411yescharacteristic
196C4⋊C4×C61C2×D411yescharacteristic
197C4⋊C4×C61C2×D411yescharacteristic
198C4⋊C4×C61C2×D411yescharacteristic
199C22⋊C4×C61—21no—
200C22⋊C4×C61—21no—
201C22⋊C4×C61—41no—
202C42×C61—21no—
203C42×C61—21no—
204C42×C61—41no—
205C42×C61—41no—
206C42×C61—41no—
207C22⋊Dic61—21no—
208C22⋊Dic61—41no—
209C22⋊Dic61—41no—
210C4⋊Dic61—21no—
211C4⋊Dic61—21no—
212C4⋊Dic61—41no—
213C4⋊Dic61C2×D411yescharacteristic
214C4⋊Dic61C2×D411yescharacteristic
215#11758196—44no—
216#11758196—41no—
217#11758196—41no—
218#11758196—41no—
219#11758196—41no—
220#11758196—82no—
221C4×Dic61—21no—
222C4×Dic61—21no—
223C4×Dic61#4112yes—
224#11758194—21no—
225#11758194—21no—
226#11758194—21no—
227#11758194—21no—
228#11758194—21no—
229#11758194—21no—
230C2×#92418—21no—
231C2×#92418—21no—
232C2×#92418—21no—
233C2×#92418—21no—
234C2×#92418—21no—
235C2×#92418C2×D411yescharacteristic
236C2×#92418C2×D411yescharacteristic
237C23×C61—21no—
238C23×C61H(5,𝔽2)11yescharacteristic
239Q8×C61—22no—
240Q8×C61—21no—
241Q8×C61—21no—
242Q8×C61—22no—
243Q8×C61—41no—
244Q8×C61—41no—
245D4×C61—21no—
246D4×C61—21no—
247D4×C61—41no—
248D4×C61—41no—
249C2×C4×C61—21no—
250C2×C4×C61—21no—
251C2×C4×C61—21no—
252C2×C4×C61—21no—
253C2×C4×C61—21no—
254C2×C4×C61—22no—
255C2×C4×C61—41no—
256C2×C4×C61—41no—
257C2×C4×C61—41no—
258C2×C4×C61—41no—
259C2×C4×C61—41no—
260C2×C4×C61—81no—
261C2×C4×C61—161no—
262C2×C4×C61C22×D411yescharacteristic
263C2×C4×C61C22×D411yescharacteristic
264C2×C4×C61C2×#4111yescharacteristic
265C2×C4×C61C2×#4111yescharacteristic
266C2×C4×C61C2×#4111yescharacteristic
267C2×C4×C612−1+411yescharacteristic
268C2×Dic61—22no—
269C2×Dic61—22no—
270C2×Dic61—22no—
271C2×Dic61—21no—
272C2×Dic61—21no—
273C2×Dic61—41no—
274C2×Dic61—41no—
275C2×Dic61—41no—
276C2×Dic61—41no—
277C2×Dic61—81no—
278Dic122—41no—
279Dic122—41no—
280Dic122—41no—
281Dic122—41no—
282Dic122—161no—
283C8×C61—22no—
284C8×C61—21no—
285C8×C61—21no—
286#92418—21no—
287#92418—21no—
288#92418—41no—
289#92418—41no—
290#92418—41no—
291#26849—611nomaximal, Sylow
292C22×C61—21no—
293C22×C61—21no—
294C22×C61—41no—
295C22×C61—41no—
296C22×C61—41no—
297C22×C61#54211yescharacteristic
298C22×C61D4211yescharacteristic
299C22×C61#55111yescharacteristic
300C4×C61—41no—
301C4×C61—42no—
302C4×C61—41no—
303C4×C61—41no—
304C4×C61—41no—
305C4×C61—41no—
306C4×C61—81no—
307C4×C61#52511yescharacteristic
308C4×C61#53211yescharacteristic
309C4×C61#54711yescharacteristic
310C4×C61#54811yescharacteristic
311Dic61—24no—
312Dic61—41no—
313Dic61—41no—
314Dic61—81no—
315Dic61—81no—
316#3137—611no—
317#3097—611no—
318#3082—611no—
319#2947D6111yescharacteristic
320#2064—611no—
321#2060—611no—
322#2052—611no—
323#2005—611no—
324#1956—611no—
325#1937—611no—
326#1930—611no—
327#1930—611no—
328#1886—611no—
329#1859—611no—
330#1772—611no—
331C2×C61—21no—
332C2×C61—21no—
333C2×C61—81no—
334C2×C61#246111yescharacteristic
335C2×C61#329111yescharacteristic
336C2×C61#329311yescharacteristic
337C2×#137C2×D6111yescharacteristic
338#544—611no—
339#536—611no—
340C2×C4⋊Q8—611no—
341#520C2×D6111yescharacteristic
342#519C2×D6111yescharacteristic
343#491—1221no—
344#489—1221no—
345#488—1221no—
346#487—1221no—
347#485—1221no—
348#485—1221no—
349#484—1221no—
350#483—1221no—
351#479—1221no—
352#479—1221no—
353#477—1221no—
354#475—1221no—
355#441—21no—
356#440—21no—
357C4×Q16—21no—
358C4×SD16—21no—
359C2×#108—611no—
360#428—611no—
361#428—611no—
362#428C2×D6111yescharacteristic
363#427—611no—
364#424—611no—
365C2×#104—611no—
366#419—611no—
367#419—611no—
368#419—611no—
369#417—611no—
370#416C2×D6111yescharacteristic
371C2×#103—611no—
372C2×#103—611no—
373C2×#103—611no—
374C2×#103C2×D6111yescharacteristic
375C2×#101—611no—
376C2×#101—611no—
377#408—611no—
378#405—611no—
379#405C2×D6111yescharacteristic
380C2×C4×C8—611no—
381#384—611no—
382#343—611no—
383#343—611no—
384#341—611no—
385#339—611no—
386#339—611no—
387#339—611no—
388C61#2684911yescharacteristic, complemented, Sylow
389C2×#41C22×D6111yescharacteristic
390C22×Q8—611no—
391C22×Q8C22×D6111yescharacteristic
392#137—22no—
393#137—22no—
394C2×Q16—21no—
395C2×SD16—21no—
396C2×M16—611no—
397C2×M16—611no—
398C2×M16—611no—
399C2×M16C22×D6111yescharacteristic
400C22×C8—611no—
401C22×C8C22×D6111yescharacteristic
402C4⋊Q8—1221no—
403C4⋊Q8—1222no—
404C4⋊Q8—1221no—
405C4⋊Q8—1222no—
406C4⋊Q8—1221no—
407#125—1221no—
408#125—1221no—
409#124—1221no—
410#123—2441no—
411C22⋊Q8—1221no—
412C22⋊Q8—1221no—
413C22⋊Q8—2441no—
414C4×Q8—21no—
415C4×Q8—21no—
416C4×Q8—21no—
417C4×Q8—21no—
418C4×Q8—21no—
419C4×D4—21no—
420C4×D4—41no—
421#117—21no—
422#117—41no—
423#117—611no—
424#117C22×D6111yescharacteristic
425#117C22×D6111yescharacteristic
426C2×C4⋊C4—21no—
427C2×C4⋊C4—611no—
428C2×C4⋊C4—2441no—
429C2×C4⋊C4—2441no—
430C2×C4⋊C4C22×D6111yescharacteristic
431C2×C42—611no—
432#108—1222no—
433#107—21no—
434#107—611no—
435#107—611no—
436#107—1221no—
437#107—1221no—
438#106—21no—
439#106—611no—
440#106—611no—
441#106—1221no—
442#106—1221no—
443C4⋊C8—1221no—
444C4⋊C8—1221no—
445#104—1222no—
446#103—21no—
447#103—21no—
448#103—1221no—
449#103—1221no—
450#103—1221no—
451#103—1221no—
452#103—1221no—
453#103—1221no—
454#103—1221no—
455#103—1221no—
456#103—1221no—
457#103—1221no—
458#103—1221no—
459#103—1221no—
460#103—1221no—
461#103C22⋊D6111yescharacteristic
462#103C22⋊D6111yescharacteristic
463#102—1221no—
464#102C22⋊D6111yescharacteristic
465#102C22⋊D6111yescharacteristic
466#101—1222no—
467#101—1222no—
468C22⋊C8—1221no—
469C22⋊C8—1221no—
470#97—21no—
471#97—1221no—
472C4×C8—21no—
473C4×C8—1222no—
474C4×C8—1221no—
475#41—21no—
476#41—21no—
477#41—21no—
478#41—21no—
479C2×Q8—21no—
480C2×Q8—21no—
481C2×Q8—21no—
482C2×Q8—21no—
483C2×Q8—21no—
484C2×Q8—1221no—
485C2×Q8—1221no—
486C2×Q8—1221no—
487C2×Q8—1221no—
488C2×Q8—1221no—
489C2×Q8—1221no—
490C2×Q8—2441no—
491C2×Q8#1175822212yes—
492C2×Q8C2×C22⋊D6111yescharacteristic
493C2×D4—21no—
494C2×D4C2×C22⋊D6111yescharacteristic
495C22×C4—21no—
496C22×C4—21no—
497C22×C4—21no—
498C22×C4—41no—
499C22×C4—1221no—
500C22×C4—1221no—
501C22×C4—1221no—
502C22×C4C23×D6111yescharacteristic
503Q16—24no—
504SD16—24no—
505M16—21no—
506M16—1221no—
507M16—1221no—
508M16—1221no—
509M16—1221no—
510M16—1221no—
511M16—1221no—
512M16#1175822212yes—
513C2×C8—21no—
514C2×C8—21no—
515C2×C8—21no—
516C2×C8—611no—
517C2×C8—611no—
518C2×C8—1221no—
519C2×C8—1221no—
520C2×C8—1221no—
521C2×C8—1221no—
522C2×C8—1221no—
523C2×C8C2×C22⋊D6111yescharacteristic
524C2×C8C2×C22⋊D6111yescharacteristic
525C4⋊C4—21no—
526C4⋊C4—21no—
527C4⋊C4—41no—
528C4⋊C4—41no—
529C4⋊C4—41no—
530C4⋊C4—611no—
531C4⋊C4—611no—
532C4⋊C4—1221no—
533C4⋊C4—1221no—
534C4⋊C4—4882no—
535C4⋊C4—2444no—
536C4⋊C4—2441no—
537C4⋊C4—2441no—
538C4⋊C4—2441no—
539C4⋊C4—2441no—
540C4⋊C4—2441no—
541C4⋊C4D4×D6111yescharacteristic
542C4⋊C4D4×D6111yescharacteristic
543C4⋊C4C2×C22⋊D6111yescharacteristic
544C4⋊C4C2×C22⋊D6111yescharacteristic
545C22⋊C4—21no—
546C22⋊C4—21no—
547C22⋊C4—41no—
548C22⋊C4—1221no—
549C22⋊C4—2441no—
550C22⋊C4—2441no—
551C42—21no—
552C42—21no—
553C42—41no—
554C42—41no—
555C42—41no—
556C42—612no—
557C42—1221no—
558C42—1221no—
559C23—21no—
560C23#1451982211yescharacteristic
561Q8—22no—
562Q8—21no—
563Q8—21no—
564Q8—22no—
565Q8—41no—
566Q8—41no—
567Q8—9761no—
568Q8—2441no—
569Q8—2441no—
570Q8—2441no—
571Q8—2441no—
572D4—21no—
573D4—21no—
574D4—41no—
575D4—41no—
576C2×C4—21no—
577C2×C4—21no—
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—81no—
588C2×C4—161no—
589C2×C4—1222no—
590C2×C4—1222no—
591C2×C4—1222no—
592C2×C4—1221no—
593C2×C4—1221no—
594C2×C4—4881no—
595C2×C4—2441no—
596C2×C4—2441no—
597C2×C4—2441no—
598C2×C4—2441no—
599C2×C4C2×D4×D6111yescharacteristic
600C2×C4C22×C22⋊D6111yescharacteristic
601C2×C4#41×D6111yescharacteristic
602C2×C4C2×#1175822211yescharacteristic
603C2×C4C2×#1175822211yescharacteristic
604C2×C4#1451982411yescharacteristic
605C8—22no—
606C8—21no—
607C8—21no—
608C8—1221no—
609C8—1221no—
610C8—2441no—
611C8—2441no—
612C8—2441no—
613C22—21no—
614C22—21no—
615C22—41no—
616C22—41no—
617C22—41no—
618C22#1684037711yescharacteristic
619C22D4×C22⋊D6111yescharacteristic
620C22#1684044111yescharacteristic
621C4—41no—
622C4—42no—
623C4—41no—
624C4—41no—
625C4—41no—
626C4—41no—
627C4—81no—
628C4—1224no—
629C4—4881no—
630C4—4881no—
631C4—2441no—
632C4—2441no—
633C4#1684027311yescharacteristic
634C4#1684030911yescharacteristic
635C4#1684041111yescharacteristic
636C4#1684042111yescharacteristic
637C2—21no—
638C2—21no—
639C2—81no—
640C2#2144978011yescharacteristic
641C2#2145761011yescharacteristic
642C2#2145762111yescharacteristic
subgroupG/H
centerC22D4×C22⋊D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2947C2
socleC22×C61D42
prime radicalC2×#41×C61C23
hypercenter#2947D61
hypocenterC61#26849
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ #2947

Complements

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

NH ≅ G/Ntype
3. #21455661C2semidirect
4. #21447183C2semidirect
5. #21447162C2semidirect
8. #21446619C2semidirect
10. #21446474C2semidirect
11. #21446471C2semidirect
12. #21446212C2semidirect
14. #21445699C2semidirect
388. C61#26849semidirect

Maximal subgroups

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

subgroupindex
#214561882
#214557882
#214556612
#214471832
#214471622
#214471152
#214468522
#214466192
#214465252
#214464742
#214464712
#214462122
#214460452
#214456992
C61×#29472
#2684961