Finite Group Databasebeta

← #29572440

Subgroups — #29572440

Order 15616.

672 conjugacy classes of subgroups.

The subgroup lattice has 530 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21444529C211yesmaximal, characteristic, complemented
2#21444450C211yesmaximal, characteristic, complemented
3#21444396C211yesmaximal, characteristic, complemented
4#21444370C211yesmaximal, characteristic, complemented
5#21444366C211yesmaximal, characteristic, complemented
6#21444288C211yesmaximal, characteristic, complemented
7C61×#3554C211yesmaximal, characteristic
8#16839819C2211yescharacteristic
9#16839819C2211yescharacteristic
10#16839815C2211yescharacteristic
11#16839630C2211yescharacteristic
12#16839511—21no—
13#16839508—21no—
14#16839504—21no—
15#16839502—21no—
16#16839499—21no—
17#16839499—21no—
18#16839499—21no—
19#16839496—22no—
20#16839496—21no—
21D4⋊#92418—22no—
22C61×#562—22no—
23C61×#559—21no—
24C61×#559C411yescharacteristic, complemented
25C61×#557C411yescharacteristic, complemented
26C61×#554—21no—
27C61×#550C411yescharacteristic, complemented
28D4×Q8×C61C411yescharacteristic, complemented
29C61×#548—21no—
30C61×#547—21no—
31C61×#542—21no—
32C61×#542—21no—
33C61×#540—21no—
34C61×#539—22no—
35C61×#536—21no—
36C61×#533C2211yescharacteristic
37C61×#532C2211yescharacteristic
38C61×#529C2211yescharacteristic
39C4⋊#92418—21no—
40C4⋊#92418—21no—
41C4⋊#92418—21no—
42C22⋊#92418—21no—
43C22⋊#92418—21no—
44C22⋊#92418—21no—
45#14519688—21no—
46#14519688—21no—
47C4×#92418—21no—
48#14519684—21no—
49#14519684—21no—
50#14519684—21no—
51C2×#41×C61—21no—
52C2×#41×C61—21no—
53C22×Q8×C61—21no—
54C4⋊Q8×C61—21no—
55C4⋊Q8×C61—21no—
56C4⋊Q8×C61C2×C411yescharacteristic
57C4⋊Q8×C61D411yescharacteristic
58C4⋊Q8×C61D411yescharacteristic
59#127×C61—21no—
60#127×C61C2×C411yescharacteristic
61#126×C61—22no—
62#126×C61—21no—
63#125×C61—22no—
64#125×C61—21no—
65#125×C61—21no—
66#125×C61D411yescharacteristic
67#124×C61—22no—
68#124×C61—22no—
69#124×C61—21no—
70#124×C61—21no—
71#124×C61—21no—
72#124×C61D411yescharacteristic
73#123×C61—22no—
74#123×C61—22no—
75#123×C61—22no—
76#123×C61—22no—
77#123×C61—22no—
78#123×C61—22no—
79#123×C61—21no—
80C22⋊Q8×C61—22no—
81C22⋊Q8×C61—22no—
82C22⋊Q8×C61—22no—
83C22⋊Q8×C61—22no—
84C22⋊Q8×C61—21no—
85C22⋊Q8×C61—22no—
86C22⋊Q8×C61—21no—
87C22⋊Q8×C61D411yescharacteristic
88C22⋊Q8×C61D411yescharacteristic
89C22⋊Q8×C61D411yescharacteristic
90#121×C61—22no—
91#121×C61—22no—
92#121×C61—22no—
93#121×C61—21no—
94#121×C61—21no—
95#121×C61D412yes—
96#121×C61D411yescharacteristic
97C4×Q8×C61—21no—
98C4×Q8×C61—21no—
99C4×Q8×C61—21no—
100C4×Q8×C61C2×C411yescharacteristic
101C4×Q8×C61C2×C411yescharacteristic
102C4×Q8×C61C2×C411yescharacteristic
103C4×D4×C61—22no—
104C4×D4×C61—21no—
105C4×D4×C61—21no—
106C4×D4×C61—21no—
107C4×D4×C61—21no—
108C4×D4×C61—21no—
109C4×D4×C61—21no—
110C4×D4×C61C2×C411yescharacteristic
111C4×D4×C61D412yes—
112#117×C61—21no—
113#117×C61—21no—
114C2×C4⋊C4×C61—22no—
115C2×C42×C61C2311yescharacteristic
116#41×C61—82no—
117#41×C61—82no—
118C2×Q8×C61—22no—
119C2×Q8×C61—21no—
120C2×Q8×C61—21no—
121C2×Q8×C61—82no—
122C2×Q8×C61C22⋊C411yescharacteristic
123C2×Q8×C61C22⋊C411yescharacteristic
124C2×Q8×C61C22⋊C411yescharacteristic
125C2×D4×C61—22no—
126C2×D4×C61—22no—
127C2×D4×C61—21no—
128C2×D4×C61—22no—
129C2×D4×C61—21no—
130C2×D4×C61—21no—
131C2×D4×C61C22⋊C411yescharacteristic
132C2×D4×C61C22⋊C412yes—
133C22×C4×C61—22no—
134C22×C4×C61—22no—
135C22×C4×C61—22no—
136C22×C4×C61—21no—
137C22×C4×C61—22no—
138C22×C4×C61C2×D411yescharacteristic
139C22×C4×C61C2×D411yescharacteristic
140C22×C4×C61C2×D411yescharacteristic
141C22×Dic61—41no—
142C4⋊C4×C61—22no—
143C4⋊C4×C61—22no—
144C4⋊C4×C61—22no—
145C4⋊C4×C61—22no—
146C4⋊C4×C61—21no—
147C4⋊C4×C61—21no—
148C4⋊C4×C61—21no—
149C4⋊C4×C61—21no—
150C4⋊C4×C61—21no—
151C4⋊C4×C61—21no—
152C4⋊C4×C61—21no—
153C4⋊C4×C61—21no—
154C4⋊C4×C61—21no—
155C4⋊C4×C61—21no—
156C4⋊C4×C61—21no—
157C4⋊C4×C61—21no—
158C4⋊C4×C61C22⋊C411yescharacteristic
159C4⋊C4×C61C22⋊C411yescharacteristic
160C4⋊C4×C61C22⋊C411yescharacteristic
161C4⋊C4×C61C22⋊C412yes—
162C4⋊C4×C61C22⋊C411yescharacteristic
163C22⋊C4×C61—22no—
164C22⋊C4×C61—22no—
165C22⋊C4×C61—22no—
166C22⋊C4×C61—22no—
167C22⋊C4×C61—22no—
168C22⋊C4×C61—21no—
169C22⋊C4×C61—21no—
170C22⋊C4×C61—21no—
171C22⋊C4×C61—21no—
172C22⋊C4×C61—21no—
173C22⋊C4×C61—21no—
174C22⋊C4×C61—21no—
175C22⋊C4×C61—22no—
176C22⋊C4×C61—22no—
177C42×C61—22no—
178C42×C61—21no—
179C42×C61—21no—
180C42×C61—21no—
181C42×C61C22×C411yescharacteristic
182C42×C61C2×D411yescharacteristic
183C42×C61C2×D411yescharacteristic
184C42×C61C2×D411yescharacteristic
185C2×#92418—41no—
186C2×#92418—41no—
187C2×#92418—41no—
188C23×C61—22no—
189C23×C61—21no—
190C23×C61#12011yescharacteristic
191Q8×C61—42no—
192Q8×C61—41no—
193Q8×C61—41no—
194Q8×C61—41no—
195Q8×C61—81no—
196Q8×C61—82no—
197D4×C61—42no—
198D4×C61—41no—
199D4×C61—42no—
200D4×C61—42no—
201D4×C61—81no—
202D4×C61—82no—
203D4×C61—82no—
204D4×C61—82no—
205C2×C4×C61—21no—
206C2×C4×C61—21no—
207C2×C4×C61—21no—
208C2×C4×C61—21no—
209C2×C4×C61—21no—
210C2×C4×C61—22no—
211C2×C4×C61—21no—
212C2×C4×C61—21no—
213C2×C4×C61—41no—
214C2×C4×C61—41no—
215C2×C4×C61—41no—
216C2×C4×C61—42no—
217C2×C4×C61—82no—
218C2×C4×C61—82no—
219C2×C4×C61—82no—
220C2×C4×C61—82no—
221C2×C4×C61C2×C22⋊C411yescharacteristic
222C2×C4×C61C2×C22⋊C411yescharacteristic
223C2×C4×C61C2×C22⋊C411yescharacteristic
224C2×C4×C61#12011yescharacteristic
225C2×C4×C61#12011yescharacteristic
226C2×C4×C61#12011yescharacteristic
227C2×Dic61—81no—
228C2×Dic61—81no—
229C2×Dic61—81no—
230#92418—81no—
231#92418—81no—
232#92418—81no—
233#8503—611nomaximal, Sylow
234C22×C61—21no—
235C22×C61—21no—
236C22×C61—21no—
237C22×C61—42no—
238C22×C61—41no—
239C22×C61—41no—
240C22×C61—42no—
241C22×C61—42no—
242C22×C61—41no—
243C22×C61#37911yescharacteristic
244C4×C61—21no—
245C4×C61—21no—
246C4×C61—21no—
247C4×C61—21no—
248C4×C61—21no—
249C4×C61—21no—
250C4×C61—41no—
251C4×C61—41no—
252C4×C61—41no—
253C4×C61—41no—
254C4×C61—41no—
255C4×C61—42no—
256C4×C61—41no—
257C4×C61—41no—
258Dic61—161no—
259#3554D6111yescharacteristic
260#1554—611no—
261#1535—611no—
262#1521—611no—
263#1515—611no—
264#1514—611no—
265#1496—611no—
266C2×C61—41no—
267C2×C61—82no—
268C2×C61—81no—
269C2×C61#179411yescharacteristic
270C2×C61#179511yescharacteristic
271C2×C61#179611yescharacteristic
272#562—22no—
273#559—21no—
274#559Dic6111yescharacteristic, complemented
275#557Dic6111yescharacteristic, complemented
276#554—21no—
277#550Dic6111yescharacteristic, complemented
278D4×Q8Dic6111yescharacteristic, complemented
279#548—21no—
280#547—21no—
281#542—21no—
282#542—21no—
283#540—21no—
284#539—22no—
285#536—21no—
286#533C2×D6111yescharacteristic
287#532C2×D6111yescharacteristic
288#529C2×D6111yescharacteristic
289#432—611no—
290#432—611no—
291#431—611no—
292#383—611no—
293#333—1221no—
294#332—1221no—
295#330—1221no—
296#329—1221no—
297#328—1221no—
298#328—1221no—
299#328—1221no—
300#327—1222no—
301#327—1221no—
302D4⋊C8—1222no—
303C61#850311yescharacteristic, complemented, Sylow
304C2×#41—21no—
305C2×#41—21no—
306C22×Q8—21no—
307C4⋊Q8—21no—
308C4⋊Q8—21no—
309C4⋊Q8C2×Dic6111yescharacteristic
310C4⋊Q8C22⋊D6111yescharacteristic
311C4⋊Q8C22⋊D6111yescharacteristic
312#127—21no—
313#127C2×Dic6111yescharacteristic
314#126—22no—
315#126—21no—
316#125—22no—
317#125—21no—
318#125—21no—
319#125C22⋊D6111yescharacteristic
320#124—22no—
321#124—22no—
322#124—21no—
323#124—21no—
324#124—21no—
325#124C22⋊D6111yescharacteristic
326#123—22no—
327#123—22no—
328#123—22no—
329#123—22no—
330#123—22no—
331#123—22no—
332#123—21no—
333C22⋊Q8—22no—
334C22⋊Q8—22no—
335C22⋊Q8—22no—
336C22⋊Q8—22no—
337C22⋊Q8—21no—
338C22⋊Q8—22no—
339C22⋊Q8—21no—
340C22⋊Q8C22⋊D6111yescharacteristic
341C22⋊Q8C22⋊D6111yescharacteristic
342C22⋊Q8C22⋊D6111yescharacteristic
343#121—22no—
344#121—22no—
345#121—22no—
346#121—21no—
347#121—21no—
348#121C22⋊D6111yescharacteristic
349#121C22⋊D6112yes—
350C4×Q8—21no—
351C4×Q8—21no—
352C4×Q8—21no—
353C4×Q8C2×Dic6111yescharacteristic
354C4×Q8C2×Dic6111yescharacteristic
355C4×Q8C2×Dic6111yescharacteristic
356C4×D4—22no—
357C4×D4—21no—
358C4×D4—21no—
359C4×D4—21no—
360C4×D4—21no—
361C4×D4—21no—
362C4×D4—21no—
363C4×D4C2×Dic6111yescharacteristic
364C4×D4C22⋊D6112yes—
365#117—21no—
366#117—21no—
367C2×C4⋊C4—22no—
368C2×C42C22×D6111yescharacteristic
369C4⋊C8—1221no—
370C4⋊C8—1221no—
371C4⋊C8—1221no—
372C22⋊C8—1221no—
373C22⋊C8—1221no—
374C22⋊C8—1221no—
375#97—1221no—
376#97—1221no—
377C4×C8—1221no—
378#95—1221no—
379#95—1221no—
380#95—1221no—
381#41—82no—
382#41—82no—
383C2×Q8—22no—
384C2×Q8—21no—
385C2×Q8—21no—
386C2×Q8—82no—
387C2×Q8C22⋊Dic6111yescharacteristic
388C2×Q8C22⋊Dic6111yescharacteristic
389C2×Q8C22⋊Dic6111yescharacteristic
390C2×D4—22no—
391C2×D4—22no—
392C2×D4—21no—
393C2×D4—22no—
394C2×D4—21no—
395C2×D4—21no—
396C2×D4C22⋊Dic6111yescharacteristic
397C2×D4C22⋊Dic6112yes—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—21no—
402C22×C4—22no—
403C22×C4—2441no—
404C22×C4C2×C22⋊D6111yescharacteristic
405C22×C4C2×C22⋊D6111yescharacteristic
406C22×C4C2×C22⋊D6111yescharacteristic
407C2×C8—2441no—
408C2×C8—2441no—
409C2×C8—2441no—
410C4⋊C4—22no—
411C4⋊C4—22no—
412C4⋊C4—22no—
413C4⋊C4—22no—
414C4⋊C4—21no—
415C4⋊C4—21no—
416C4⋊C4—21no—
417C4⋊C4—21no—
418C4⋊C4—21no—
419C4⋊C4—21no—
420C4⋊C4—21no—
421C4⋊C4—21no—
422C4⋊C4—21no—
423C4⋊C4—21no—
424C4⋊C4—21no—
425C4⋊C4—21no—
426C4⋊C4C22⋊Dic6111yescharacteristic
427C4⋊C4C22⋊Dic6111yescharacteristic
428C4⋊C4C22⋊Dic6111yescharacteristic
429C4⋊C4C22⋊Dic6112yes—
430C4⋊C4C22⋊Dic6111yescharacteristic
431C22⋊C4—22no—
432C22⋊C4—22no—
433C22⋊C4—22no—
434C22⋊C4—22no—
435C22⋊C4—22no—
436C22⋊C4—21no—
437C22⋊C4—21no—
438C22⋊C4—21no—
439C22⋊C4—21no—
440C22⋊C4—21no—
441C22⋊C4—21no—
442C22⋊C4—21no—
443C22⋊C4—22no—
444C22⋊C4—22no—
445C42—22no—
446C42—21no—
447C42—21no—
448C42—21no—
449C42C22×Dic6111yescharacteristic
450C42C2×C22⋊D6111yescharacteristic
451C42C2×C22⋊D6111yescharacteristic
452C42C2×C22⋊D6111yescharacteristic
453C23—22no—
454C23—21no—
455C23#1451974611yescharacteristic
456Q8—41no—
457Q8—41no—
458Q8—41no—
459Q8—42no—
460Q8—81no—
461Q8—82no—
462D4—42no—
463D4—41no—
464D4—42no—
465D4—42no—
466D4—81no—
467D4—82no—
468D4—82no—
469D4—82no—
470C2×C4—21no—
471C2×C4—21no—
472C2×C4—21no—
473C2×C4—21no—
474C2×C4—21no—
475C2×C4—22no—
476C2×C4—21no—
477C2×C4—21no—
478C2×C4—41no—
479C2×C4—41no—
480C2×C4—41no—
481C2×C4—42no—
482C2×C4—82no—
483C2×C4—82no—
484C2×C4—82no—
485C2×C4—82no—
486C2×C4—4881no—
487C2×C4—4881no—
488C2×C4—4881no—
489C2×C4C2×C22⋊Dic6111yescharacteristic
490C2×C4C2×C22⋊Dic6111yescharacteristic
491C2×C4C2×C22⋊Dic6111yescharacteristic
492C2×C4#1451974611yescharacteristic
493C2×C4#1451974611yescharacteristic
494C2×C4#1451974611yescharacteristic
495C8—4881no—
496C8—4881no—
497C8—4881no—
498C22—21no—
499C22—21no—
500C22—21no—
501C22—42no—
502C22—41no—
503C22—41no—
504C22—42no—
505C22—42no—
506C22—41no—
507C22#1683961811yescharacteristic
508C4—21no—
509C4—21no—
510C4—21no—
511C4—21no—
512C4—21no—
513C4—21no—
514C4—41no—
515C4—41no—
516C4—41no—
517C4—41no—
518C4—41no—
519C4—42no—
520C4—41no—
521C4—41no—
522C4—9761no—
523C2—41no—
524C2—82no—
525C2—81no—
526C2#2144578311yescharacteristic
527C2#2144578911yescharacteristic
528C2#2144579411yescharacteristic
subgroupG/H
centerC22#16839618
commutatorC42×C61C22×C4
FrattiniC2×C42C22×D61
Fitting, prime radicalC61×#3554C2
socleC22×C61#379
cosocleC2×C42×C61C23
hypercenter#3554D61
hypocenterC61#8503
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3554

Complements

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

NH ≅ G/Ntype
1. #21444529C2semidirect
2. #21444450C2semidirect
3. #21444396C2semidirect
4. #21444370C2semidirect
5. #21444366C2semidirect
6. #21444288C2semidirect
24. C61×#559C4semidirect
25. C61×#557C4semidirect
27. C61×#550C4semidirect
28. D4×Q8×C61C4semidirect
274. #559Dic61semidirect
275. #557Dic61semidirect
277. #550Dic61semidirect
278. D4×Q8Dic61semidirect
303. C61#8503semidirect

Maximal subgroups

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

subgroupindex
#214445292
#214444502
#214443962
#214443702
#214443662
#214442882
C61×#35542
#850361