Finite Group Databasebeta

← #29661509

Subgroups — #29661509

Order 15616.

892 conjugacy classes of subgroups.

The subgroup lattice has 560 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456843C211yesmaximal, characteristic, complemented
2C2×#16840172C211yesmaximal, characteristic, complemented
3#21456212C211yesmaximal, characteristic, complemented
4#21445153C214yesmaximal, complemented
5#21445136C214yesmaximal, complemented
6#21444430C211yesmaximal, characteristic, complemented
7#21444421C211yesmaximal, characteristic, complemented
8#21444407C211yesmaximal, characteristic, complemented
9C61×#3559C211yesmaximal, characteristic
10C2×#14519777C2211yescharacteristic
11#16840172—22no—
12#16840172C2214yes—
13#16840157C2212yes—
14#16840132C2212yes—
15#16840121C2214yes—
16#16840112—21no—
17#16840108—21no—
18#16839972—21no—
19#16839967—21no—
20#16839917—21no—
21#16839900—21no—
22#16839819C2211yescharacteristic
23C2×#14519710—21no—
24C2×#14519708—21no—
25#16839763—22no—
26C4×#11758194C2211yescharacteristic
27C2×#14519688C2211yescharacteristic
28#16839504C2214yes—
29#16839504C2212yes—
30#16839502C2212yes—
31#16839502C2214yes—
32C61×#563C2212yes—
33C61×#562C2212yes—
34C61×#554—21no—
35C61×#553—21no—
36C61×#553—21no—
37C61×#547—22no—
38C61×#547—21no—
39C61×#544—21no—
40C61×#540—21no—
41C61×#539—22no—
42C61×#539—21no—
43C61×#537—21no—
44C61×#532C2211yescharacteristic
45C61×#529C2211yescharacteristic
46C2×#125×C61C2211yescharacteristic
47C22×Dic122—21no—
48C2×#11758202—22no—
49C2×#11758200—22no—
50C2×#11758194—21no—
51C22×#92418—21no—
52#14519777—21no—
53#14519777—21no—
54#14519777—21no—
55#14519777C2312yes—
56C2×C4⋊Dic61—21no—
57C2×C4⋊Dic61—21no—
58#14519710—24no—
59#14519708—24no—
60C4⋊#92418—21no—
61#14519701—24no—
62#14519700—22no—
63#14519700—22no—
64#14519698—24no—
65C22⋊#92418—21no—
66#14519688—21no—
67#14519688C2312yes—
68#14519688C2312yes—
69#14519688C2312yes—
70C4×#92418—21no—
71C2×#41×C61—22no—
72C2×#41×C61D412yes—
73C4⋊Q8×C61—21no—
74C4⋊Q8×C61—22no—
75C4⋊Q8×C61D411yescharacteristic
76#127×C61D411yescharacteristic
77#126×C61—22no—
78#126×C61—22no—
79#126×C61—21no—
80#125×C61—22no—
81#125×C61—21no—
82#125×C61—21no—
83#125×C61—21no—
84#125×C61D412yes—
85#125×C61C2312yes—
86#125×C61C2311yescharacteristic
87#124×C61C2312yes—
88#124×C61C2311yescharacteristic
89#123×C61—22no—
90#123×C61—22no—
91#123×C61—22no—
92#123×C61—22no—
93#123×C61—21no—
94#123×C61—22no—
95#123×C61—22no—
96C22⋊Q8×C61—22no—
97C22⋊Q8×C61—22no—
98C22⋊Q8×C61—22no—
99C22⋊Q8×C61—22no—
100C22⋊Q8×C61—22no—
101C22⋊Q8×C61—22no—
102C22⋊Q8×C61D411yescharacteristic
103#121×C61—22no—
104#121×C61—22no—
105#121×C61—21no—
106#121×C61—21no—
107#121×C61—22no—
108#121×C61—22no—
109#121×C61—21no—
110#121×C61—21no—
111#121×C61D411yescharacteristic
112C4×Q8×C61D411yescharacteristic
113C4×D4×C61—22no—
114C4×D4×C61—22no—
115C4×D4×C61—21no—
116C4×D4×C61—21no—
117C4×D4×C61—21no—
118C4×D4×C61—21no—
119C4×D4×C61—21no—
120C4×D4×C61D411yescharacteristic
121#117×C61—21no—
122#117×C61—21no—
123C2×C4⋊C4×C61—22no—
124C2×C4⋊C4×C61—21no—
125C2×C4⋊C4×C61—21no—
126C2×C4⋊C4×C61—21no—
127C2×C4⋊C4×C61D411yescharacteristic
128C2×C4⋊C4×C61D411yescharacteristic
129C2×C42×C61C2311yescharacteristic
130#41×C61—44no—
131#41×C61—84no—
132C2×Q8×C61—22no—
133C2×Q8×C61C2×D411yescharacteristic
134C2×Q8×C61C2×D412yes—
135C2×D4×C61—22no—
136C2×D4×C61—21no—
137C2×D4×C61—22no—
138C2×D4×C61—22no—
139C2×D4×C61—21no—
140C2×D4×C61—21no—
141C2×D4×C61C2×D411yescharacteristic
142C2×D4×C61C2×D412yes—
143C22×C4×C61—21no—
144C22×C4×C61—22no—
145C22×C4×C61—22no—
146C22×C4×C61—22no—
147C22×C4×C61—21no—
148C22×C4×C61—21no—
149C22×C4×C61—22no—
150C22×C4×C61C2×D411yescharacteristic
151C22×C4×C61C2×D411yescharacteristic
152C22×C4×C61C2×D411yescharacteristic
153C22×Dic61—41no—
154C2×Dic122—22no—
155C2×Dic122—41no—
156C2×Dic122—82no—
157C4⋊C4×C61—22no—
158C4⋊C4×C61—22no—
159C4⋊C4×C61—22no—
160C4⋊C4×C61—22no—
161C4⋊C4×C61—22no—
162C4⋊C4×C61—21no—
163C4⋊C4×C61—21no—
164C4⋊C4×C61—21no—
165C4⋊C4×C61—22no—
166C4⋊C4×C61—21no—
167C4⋊C4×C61—21no—
168C4⋊C4×C61—21no—
169C4⋊C4×C61—22no—
170C4⋊C4×C61—22no—
171C4⋊C4×C61—21no—
172C4⋊C4×C61—21no—
173C4⋊C4×C61C2×D412yes—
174C4⋊C4×C61C2×D412yes—
175C4⋊C4×C61C2×D411yescharacteristic
176C4⋊C4×C61C2×D411yescharacteristic
177C22⋊C4×C61—22no—
178C22⋊C4×C61—22no—
179C22⋊C4×C61—22no—
180C22⋊C4×C61—22no—
181C22⋊C4×C61—22no—
182C22⋊C4×C61—21no—
183C22⋊C4×C61—21no—
184C22⋊C4×C61—21no—
185C22⋊C4×C61—21no—
186C22⋊C4×C61—22no—
187C22⋊C4×C61—22no—
188C22⋊C4×C61—21no—
189C42×C61—22no—
190C42×C61—21no—
191C42×C61C2×D411yescharacteristic
192C42×C61C2×D411yescharacteristic
193C42×C61C2×D411yescharacteristic
194C42×C61C2411yescharacteristic
195#11758202—82no—
196#11758200—82no—
197C4⋊Dic61—22no—
198C4⋊Dic61—22no—
199C4⋊Dic61—41no—
200C4⋊Dic61—41no—
201#11758194—42no—
202C2×#92418—24no—
203C2×#92418—22no—
204C2×#92418—22no—
205C2×#92418—22no—
206C23×C61—22no—
207C23×C61—21no—
208C23×C61#12011yescharacteristic
209Q8×C61—41no—
210Q8×C61—42no—
211Q8×C61—82no—
212D4×C61—41no—
213D4×C61—42no—
214D4×C61—44no—
215D4×C61—42no—
216D4×C61—42no—
217D4×C61—81no—
218D4×C61—82no—
219D4×C61—82no—
220C2×C4×C61—22no—
221C2×C4×C61—22no—
222C2×C4×C61—21no—
223C2×C4×C61—21no—
224C2×C4×C61—22no—
225C2×C4×C61—22no—
226C2×C4×C61—21no—
227C2×C4×C61—22no—
228C2×C4×C61—41no—
229C2×C4×C61—42no—
230C2×C4×C61—42no—
231C2×C4×C61—82no—
232C2×C4×C61—81no—
233C2×C4×C61—82no—
234C2×C4×C61—82no—
235C2×C4×C61—82no—
236C2×C4×C61#12011yescharacteristic
237C2×C4×C61#12011yescharacteristic
238C2×C4×C61#12011yescharacteristic
239C2×C4×C61C22×D411yescharacteristic
240C2×C4×C61C22×D411yescharacteristic
241C2×C4×C61C22×D411yescharacteristic
242C2×Dic61—42no—
243C2×Dic61—82no—
244Dic122—42no—
245Dic122—42no—
246Dic122—82no—
247#92418—24no—
248#92418—42no—
249#92418—42no—
250#19022—611nomaximal, Sylow
251C22×C61—21no—
252C22×C61—21no—
253C22×C61—42no—
254C22×C61—41no—
255C22×C61—41no—
256C22×C61—42no—
257C22×C61—42no—
258C22×C61—41no—
259C22×C61#45512yes—
260C22×C61C2×#12011yescharacteristic
261C4×C61—21no—
262C4×C61—21no—
263C4×C61—21no—
264C4×C61—21no—
265C4×C61—21no—
266C4×C61—42no—
267C4×C61—42no—
268C4×C61—41no—
269C4×C61—41no—
270C4×C61—42no—
271C4×C61—41no—
272C4×C61#45612yes—
273Dic61—82no—
274#3559D6111yescharacteristic
275#3218—611no—
276C2×#501—611no—
277#3141—611no—
278#1664—614no—
279#1661—614no—
280#1530—611no—
281#1528—611no—
282#1524—611no—
283C2×C61—22no—
284C2×C61—82no—
285C2×C61—81no—
286C2×C61C2×#45511yescharacteristic
287C2×C61C2×#45611yescharacteristic
288C2×C61#302111yescharacteristic
289#563C2×D6112yes—
290#562C2×D6112yes—
291#554—21no—
292#553—21no—
293#553—21no—
294#547—22no—
295#547—21no—
296#544—21no—
297#540—21no—
298#539—22no—
299#539—21no—
300#537—21no—
301#532C2×D6111yescharacteristic
302C2×C4⋊Q8—611no—
303#529C2×D6111yescharacteristic
304C2×#125C2×D6111yescharacteristic
305#501—614no—
306#501—1222no—
307#497—612no—
308#491—612no—
309#489—614no—
310#487—1221no—
311#486—1221no—
312#462—1221no—
313#461—1221no—
314#451—1221no—
315#448—1221no—
316#432—611no—
317C2×#107—1221no—
318C2×#106—1221no—
319#417—1222no—
320C4×M16—611no—
321C2×#97—611no—
322#330—614no—
323#330—612no—
324#329—612no—
325#329—614no—
326C61#1902211yescharacteristic, complemented, Sylow
327C2×#41—22no—
328C2×#41C22⋊D6112yes—
329C22×Q8—1221no—
330C2×Q16—1222no—
331C2×SD16—1222no—
332C2×M16—1221no—
333C22×C8—1221no—
334C4⋊Q8—21no—
335C4⋊Q8—22no—
336C4⋊Q8—612no—
337C4⋊Q8—1221no—
338C4⋊Q8—1221no—
339C4⋊Q8—1221no—
340C4⋊Q8C22⋊D6111yescharacteristic
341#127C22⋊D6111yescharacteristic
342#126—22no—
343#126—22no—
344#126—21no—
345#125—22no—
346#125—21no—
347#125—21no—
348#125—21no—
349#125C22⋊D6112yes—
350#125C22×D6112yes—
351#125C22×D6111yescharacteristic
352#124C22×D6111yescharacteristic
353#124C22×D6112yes—
354#123—22no—
355#123—22no—
356#123—22no—
357#123—22no—
358#123—21no—
359#123—22no—
360#123—22no—
361C22⋊Q8—22no—
362C22⋊Q8—22no—
363C22⋊Q8—22no—
364C22⋊Q8—22no—
365C22⋊Q8—22no—
366C22⋊Q8—22no—
367C22⋊Q8C22⋊D6111yescharacteristic
368#121—22no—
369#121—22no—
370#121—21no—
371#121—21no—
372#121—22no—
373#121—22no—
374#121—21no—
375#121—21no—
376#121C22⋊D6111yescharacteristic
377C4×Q8C22⋊D6111yescharacteristic
378C4×D4—22no—
379C4×D4—22no—
380C4×D4—21no—
381C4×D4—21no—
382C4×D4—21no—
383C4×D4—21no—
384C4×D4—21no—
385C4×D4C22⋊D6111yescharacteristic
386#117—21no—
387#117—21no—
388C2×C4⋊C4—22no—
389C2×C4⋊C4—21no—
390C2×C4⋊C4—21no—
391C2×C4⋊C4—21no—
392C2×C4⋊C4—1221no—
393C2×C4⋊C4—1221no—
394C2×C4⋊C4C22⋊D6111yescharacteristic
395C2×C4⋊C4C22⋊D6111yescharacteristic
396C2×C42C22×D6111yescharacteristic
397#107—1224no—
398#106—1224no—
399C4⋊C8—1221no—
400#103—1224no—
401#103—1222no—
402#103—1222no—
403#102—1224no—
404C22⋊C8—1221no—
405#97—612no—
406#97—612no—
407#97—612no—
408#97—1221no—
409C4×C8—1221no—
410#41—44no—
411#41—84no—
412C2×Q8—22no—
413C2×Q8—1222no—
414C2×Q8—4882no—
415C2×Q8—2441no—
416C2×Q8C2×C22⋊D6112yes—
417C2×Q8C2×C22⋊D6111yescharacteristic
418C2×D4—22no—
419C2×D4—21no—
420C2×D4—22no—
421C2×D4—22no—
422C2×D4—21no—
423C2×D4—21no—
424C2×D4C2×C22⋊D6112yes—
425C2×D4C2×C22⋊D6111yescharacteristic
426C22×C4—21no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—21no—
431C22×C4—21no—
432C22×C4—22no—
433C22×C4—2441no—
434C22×C4C2×C22⋊D6111yescharacteristic
435C22×C4C2×C22⋊D6111yescharacteristic
436C22×C4C2×C22⋊D6111yescharacteristic
437Q16—4882no—
438SD16—4882no—
439M16—2442no—
440C2×C8—1224no—
441C2×C8—1222no—
442C2×C8—1222no—
443C2×C8—1222no—
444C4⋊C4—22no—
445C4⋊C4—22no—
446C4⋊C4—22no—
447C4⋊C4—22no—
448C4⋊C4—22no—
449C4⋊C4—22no—
450C4⋊C4—21no—
451C4⋊C4—21no—
452C4⋊C4—22no—
453C4⋊C4—21no—
454C4⋊C4—21no—
455C4⋊C4—21no—
456C4⋊C4—22no—
457C4⋊C4—21no—
458C4⋊C4—21no—
459C4⋊C4—21no—
460C4⋊C4—1222no—
461C4⋊C4—1222no—
462C4⋊C4—2441no—
463C4⋊C4—2441no—
464C4⋊C4C2×C22⋊D6111yescharacteristic
465C4⋊C4C2×C22⋊D6111yescharacteristic
466C4⋊C4C2×C22⋊D6112yes—
467C4⋊C4C2×C22⋊D6112yes—
468C22⋊C4—22no—
469C22⋊C4—22no—
470C22⋊C4—22no—
471C22⋊C4—22no—
472C22⋊C4—22no—
473C22⋊C4—21no—
474C22⋊C4—21no—
475C22⋊C4—22no—
476C22⋊C4—22no—
477C22⋊C4—21no—
478C22⋊C4—21no—
479C22⋊C4—21no—
480C42—22no—
481C42—21no—
482C42C2×C22⋊D6111yescharacteristic
483C42C2×C22⋊D6111yescharacteristic
484C42C2×C22⋊D6111yescharacteristic
485C42C23×D6111yescharacteristic
486C23—22no—
487C23—21no—
488C23#1451974611yescharacteristic
489Q8—41no—
490Q8—42no—
491Q8—82no—
492Q8—4882no—
493Q8—2442no—
494Q8—2442no—
495D4—41no—
496D4—42no—
497D4—44no—
498D4—42no—
499D4—42no—
500D4—81no—
501D4—82no—
502D4—82no—
503C2×C4—22no—
504C2×C4—22no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—22no—
508C2×C4—22no—
509C2×C4—21no—
510C2×C4—22no—
511C2×C4—41no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—82no—
515C2×C4—81no—
516C2×C4—82no—
517C2×C4—82no—
518C2×C4—82no—
519C2×C4—4882no—
520C2×C4—2442no—
521C2×C4#1451974611yescharacteristic
522C2×C4#1451974611yescharacteristic
523C2×C4#1451974611yescharacteristic
524C2×C4C22×C22⋊D6111yescharacteristic
525C2×C4C22×C22⋊D6111yescharacteristic
526C2×C4C22×C22⋊D6111yescharacteristic
527C8—1224no—
528C8—2442no—
529C8—2442no—
530C22—21no—
531C22—21no—
532C22—42no—
533C22—41no—
534C22—41no—
535C22—42no—
536C22—42no—
537C22—41no—
538C22#1683994012yes—
539C22C2×#1451974611yescharacteristic
540C4—21no—
541C4—21no—
542C4—21no—
543C4—21no—
544C4—21no—
545C4—42no—
546C4—42no—
547C4—41no—
548C4—41no—
549C4—42no—
550C4—41no—
551C4—4882no—
552C4#1683994512yes—
553C2—22no—
554C2—82no—
555C2—81no—
556C2C2×#1683994011yescharacteristic
557C2C2×#1683994511yescharacteristic
558C2#2145519111yescharacteristic
subgroupG/H
centerC22C2×#14519746
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
Fitting, prime radicalC61×#3559C2
socleC22×C61C2×#120
hypercenter#3559D61
hypocenterC61#19022
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3559

Complements

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

NH ≅ G/Ntype
1. #21456843C2semidirect
2. C2×#16840172C2semidirect
3. #21456212C2semidirect
4. #21445153C2semidirect
5. #21445136C2semidirect
6. #21444430C2semidirect
7. #21444421C2semidirect
8. #21444407C2semidirect
326. C61#19022semidirect

Maximal subgroups

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

subgroupindex
#214568432
C2×#168401722
#214562122
#214451532
#214451362
#214444302
#214444212
#214444072
C61×#35592
#1902261