Finite Group Databasebeta

← #29572333

Subgroups — #29572333

Order 15616.

686 conjugacy classes of subgroups.

The subgroup lattice has 498 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21446813C211yesmaximal, characteristic, complemented
2#21444541C211yesmaximal, characteristic, complemented
3#21444426C211yesmaximal, characteristic, complemented
4#21444284C211yesmaximal, characteristic, complemented
5#21444236C211yesmaximal, characteristic, complemented
6#21444175C211yesmaximal, characteristic, complemented
7C61×#3559C211yesmaximal, characteristic
8#16839819C2211yescharacteristic
9#16839815C2211yescharacteristic
10#16839786C2211yescharacteristic
11C4×#11758194C2211yescharacteristic
12#16839531—21no—
13#16839531—21no—
14#16839508—21no—
15#16839506—21no—
16#16839504—21no—
17#16839502—21no—
18Q8⋊#92418—21no—
19D4⋊#92418—22no—
20D4⋊#92418—21no—
21#16839488—22no—
22C61×#563—21no—
23C61×#562—21no—
24C61×#554—21no—
25C61×#553—21no—
26C61×#553—21no—
27C61×#547—22no—
28C61×#547—21no—
29C61×#544—21no—
30C61×#540—21no—
31C61×#539—22no—
32C61×#539C412yes—
33C61×#537C412yes—
34C61×#532C2211yescharacteristic
35C61×#529C2211yescharacteristic
36C2×#125×C61C2211yescharacteristic
37C2×#11758194—21no—
38C2×#11758194—21no—
39C4⋊#92418—21no—
40C4⋊#92418—21no—
41C4⋊#92418—21no—
42C4⋊#92418—21no—
43C22⋊#92418—21no—
44C22⋊#92418—21no—
45#14519688—21no—
46#14519688—21no—
47C4×#92418—21no—
48C4×#92418—21no—
49C2×#41×C61—22no—
50C2×#41×C61—21no—
51C4⋊Q8×C61—21no—
52C4⋊Q8×C61—22no—
53C4⋊Q8×C61D411yescharacteristic
54#127×C61D411yescharacteristic
55#126×C61—22no—
56#126×C61—22no—
57#126×C61—21no—
58#125×C61—21no—
59#125×C61—22no—
60#125×C61—21no—
61#125×C61—21no—
62#125×C61—21no—
63#125×C61—21no—
64#125×C61D411yescharacteristic
65#124×C61—21no—
66#124×C61D411yescharacteristic
67#123×C61—22no—
68#123×C61—22no—
69#123×C61—22no—
70#123×C61—22no—
71#123×C61—22no—
72#123×C61—22no—
73#123×C61—21no—
74C22⋊Q8×C61—22no—
75C22⋊Q8×C61—22no—
76C22⋊Q8×C61—22no—
77C22⋊Q8×C61—22no—
78C22⋊Q8×C61—22no—
79C22⋊Q8×C61—22no—
80C22⋊Q8×C61C2×C411yescharacteristic
81#121×C61—21no—
82#121×C61—22no—
83#121×C61—21no—
84#121×C61—22no—
85#121×C61—22no—
86#121×C61—22no—
87#121×C61—21no—
88#121×C61C2×C412yes—
89#121×C61C2×C411yescharacteristic
90C4×Q8×C61D411yescharacteristic
91C4×D4×C61—22no—
92C4×D4×C61—21no—
93C4×D4×C61—21no—
94C4×D4×C61—22no—
95C4×D4×C61—21no—
96C4×D4×C61—21no—
97C4×D4×C61D412yes—
98C4×D4×C61D411yescharacteristic
99#117×C61—21no—
100#117×C61—21no—
101C2×C4⋊C4×C61—22no—
102C2×C4⋊C4×C61—21no—
103C2×C4⋊C4×C61C2×C412yes—
104C2×C4⋊C4×C61D412yes—
105C2×C4⋊C4×C61D411yescharacteristic
106C2×C4⋊C4×C61D411yescharacteristic
107C2×C42×C61C2311yescharacteristic
108#41×C61—82no—
109#41×C61—84no—
110C2×Q8×C61—22no—
111C2×Q8×C61—21no—
112C2×Q8×C61C22⋊C411yescharacteristic
113C2×D4×C61—22no—
114C2×D4×C61—21no—
115C2×D4×C61—22no—
116C2×D4×C61—22no—
117C2×D4×C61—21no—
118C2×D4×C61—21no—
119C2×D4×C61C22⋊C411yescharacteristic
120C2×D4×C61C22⋊C412yes—
121C22×C4×C61—22no—
122C22×C4×C61—22no—
123C22×C4×C61—22no—
124C22×C4×C61—21no—
125C22×C4×C61—22no—
126C22×C4×C61C22⋊C412yes—
127C22×C4×C61C22⋊C412yes—
128C22×C4×C61C22×C411yescharacteristic
129C22×C4×C61C2×D411yescharacteristic
130C22×C4×C61C2×D411yescharacteristic
131C4⋊C4×C61—22no—
132C4⋊C4×C61—22no—
133C4⋊C4×C61—22no—
134C4⋊C4×C61—22no—
135C4⋊C4×C61—22no—
136C4⋊C4×C61—22no—
137C4⋊C4×C61—21no—
138C4⋊C4×C61—21no—
139C4⋊C4×C61—21no—
140C4⋊C4×C61—22no—
141C4⋊C4×C61—21no—
142C4⋊C4×C61—21no—
143C4⋊C4×C61—22no—
144C4⋊C4×C61—21no—
145C4⋊C4×C61—21no—
146C4⋊C4×C61—21no—
147C4⋊C4×C61—21no—
148C4⋊C4×C61C22⋊C412yes—
149C4⋊C4×C61C22⋊C411yescharacteristic
150C4⋊C4×C61C22⋊C411yescharacteristic
151C22⋊C4×C61—22no—
152C22⋊C4×C61—22no—
153C22⋊C4×C61—22no—
154C22⋊C4×C61—22no—
155C22⋊C4×C61—22no—
156C22⋊C4×C61—21no—
157C22⋊C4×C61—21no—
158C22⋊C4×C61—22no—
159C22⋊C4×C61—21no—
160C22⋊C4×C61—21no—
161C22⋊C4×C61—21no—
162C22⋊C4×C61—22no—
163C42×C61—22no—
164C42×C61—21no—
165C42×C61C2×D411yescharacteristic
166C42×C61C2×D411yescharacteristic
167C42×C61C2×D411yescharacteristic
168C42×C61C2×D411yescharacteristic
169#11758194—22no—
170#11758194—41no—
171#11758194—42no—
172C2×#92418—41no—
173C2×#92418—41no—
174C2×#92418—41no—
175C2×#92418—41no—
176C23×C61—21no—
177C23×C61—22no—
178C23×C61C2×C22⋊C411yescharacteristic
179Q8×C61—41no—
180Q8×C61—81no—
181Q8×C61—82no—
182D4×C61—42no—
183D4×C61—41no—
184D4×C61—44no—
185D4×C61—42no—
186D4×C61—41no—
187D4×C61—41no—
188D4×C61—81no—
189D4×C61—82no—
190D4×C61—82no—
191C2×C4×C61—21no—
192C2×C4×C61—21no—
193C2×C4×C61—21no—
194C2×C4×C61—21no—
195C2×C4×C61—22no—
196C2×C4×C61—22no—
197C2×C4×C61—21no—
198C2×C4×C61—22no—
199C2×C4×C61—41no—
200C2×C4×C61—41no—
201C2×C4×C61—41no—
202C2×C4×C61—41no—
203C2×C4×C61—42no—
204C2×C4×C61—42no—
205C2×C4×C61—82no—
206C2×C4×C61—82no—
207C2×C4×C61—82no—
208C2×C4×C61—82no—
209C2×C4×C61C2×C22⋊C411yescharacteristic
210C2×C4×C61C2×C22⋊C411yescharacteristic
211C2×C4×C61#12011yescharacteristic
212C2×C4×C61#12011yescharacteristic
213C2×C4×C61#12011yescharacteristic
214C2×C4×C61#12011yescharacteristic
215#92418—42no—
216#92418—81no—
217#92418—81no—
218#92418—81no—
219#8486—611nomaximal, Sylow
220C22×C61—21no—
221C22×C61—21no—
222C22×C61—21no—
223C22×C61—41no—
224C22×C61—42no—
225C22×C61—42no—
226C22×C61—41no—
227C22×C61—42no—
228C22×C61—41no—
229C22×C61#37911yescharacteristic
230C4×C61—21no—
231C4×C61—21no—
232C4×C61—21no—
233C4×C61—21no—
234C4×C61—21no—
235C4×C61—41no—
236C4×C61—42no—
237C4×C61—42no—
238C4×C61—41no—
239C4×C61—42no—
240C4×C61—41no—
241C4×C61#45411yescharacteristic
242C4×C61#45611yescharacteristic
243#3559D6111yescharacteristic
244#1998—611no—
245#1557—611no—
246#1529—611no—
247#1495—611no—
248#1484—611no—
249#1470—611no—
250C2×C61—41no—
251C2×C61—81no—
252C2×C61—82no—
253C2×C61#179111yescharacteristic
254C2×C61#179311yescharacteristic
255C2×C61#179511yescharacteristic
256#563—21no—
257#562—21no—
258#554—21no—
259#553—21no—
260#553—21no—
261#547—22no—
262#547—21no—
263#544—21no—
264#540—21no—
265#539—22no—
266#539Dic6112yes—
267#537Dic6112yes—
268#532C2×D6111yescharacteristic
269#529C2×D6111yescharacteristic
270C2×#125C2×D6111yescharacteristic
271#432—611no—
272#431—611no—
273#423—611no—
274C4×M16—611no—
275#343—1221no—
276#343—1221no—
277#332—1221no—
278#331—1221no—
279#330—1221no—
280#329—1221no—
281Q8⋊C8—1221no—
282D4⋊C8—1222no—
283D4⋊C8—1221no—
284#324—1222no—
285C61#848611yescharacteristic, complemented, Sylow
286C2×#41—22no—
287C2×#41—21no—
288C2×M16—1221no—
289C2×M16—1221no—
290C4⋊Q8—21no—
291C4⋊Q8—22no—
292C4⋊Q8C22⋊D6111yescharacteristic
293#127C22⋊D6111yescharacteristic
294#126—22no—
295#126—22no—
296#126—21no—
297#125—21no—
298#125—22no—
299#125—21no—
300#125—21no—
301#125—21no—
302#125—21no—
303#125C22⋊D6111yescharacteristic
304#124—21no—
305#124C22⋊D6111yescharacteristic
306#123—22no—
307#123—22no—
308#123—22no—
309#123—22no—
310#123—22no—
311#123—22no—
312#123—21no—
313C22⋊Q8—22no—
314C22⋊Q8—22no—
315C22⋊Q8—22no—
316C22⋊Q8—22no—
317C22⋊Q8—22no—
318C22⋊Q8—22no—
319C22⋊Q8C2×Dic6111yescharacteristic
320#121—21no—
321#121—22no—
322#121—21no—
323#121—22no—
324#121—22no—
325#121—22no—
326#121—21no—
327#121C2×Dic6111yescharacteristic
328#121C2×Dic6112yes—
329C4×Q8C22⋊D6111yescharacteristic
330C4×D4—22no—
331C4×D4—21no—
332C4×D4—21no—
333C4×D4—22no—
334C4×D4—21no—
335C4×D4—21no—
336C4×D4C22⋊D6111yescharacteristic
337C4×D4C22⋊D6112yes—
338#117—21no—
339#117—21no—
340C2×C4⋊C4—22no—
341C2×C4⋊C4—21no—
342C2×C4⋊C4C2×Dic6112yes—
343C2×C4⋊C4C22⋊D6112yes—
344C2×C4⋊C4C22⋊D6111yescharacteristic
345C2×C4⋊C4C22⋊D6111yescharacteristic
346C2×C42C22×D6111yescharacteristic
347C4⋊C8—1221no—
348C4⋊C8—1221no—
349C4⋊C8—1221no—
350C4⋊C8—1221no—
351C22⋊C8—1221no—
352C22⋊C8—1221no—
353#97—1221no—
354#97—1221no—
355C4×C8—1221no—
356C4×C8—1221no—
357#41—82no—
358#41—84no—
359C2×Q8—22no—
360C2×Q8—21no—
361C2×Q8C22⋊Dic6111yescharacteristic
362C2×D4—22no—
363C2×D4—21no—
364C2×D4—22no—
365C2×D4—22no—
366C2×D4—21no—
367C2×D4—21no—
368C2×D4C22⋊Dic6111yescharacteristic
369C2×D4C22⋊Dic6112yes—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—21no—
374C22×C4—22no—
375C22×C4C22⋊Dic6112yes—
376C22×C4C22⋊Dic6112yes—
377C22×C4C22×Dic6111yescharacteristic
378C22×C4C2×C22⋊D6111yescharacteristic
379C22×C4C2×C22⋊D6111yescharacteristic
380M16—1222no—
381M16—2442no—
382M16—2441no—
383C2×C8—2441no—
384C2×C8—2441no—
385C2×C8—2441no—
386C2×C8—2441no—
387C4⋊C4—22no—
388C4⋊C4—22no—
389C4⋊C4—22no—
390C4⋊C4—22no—
391C4⋊C4—22no—
392C4⋊C4—22no—
393C4⋊C4—21no—
394C4⋊C4—21no—
395C4⋊C4—21no—
396C4⋊C4—22no—
397C4⋊C4—21no—
398C4⋊C4—21no—
399C4⋊C4—22no—
400C4⋊C4—21no—
401C4⋊C4—21no—
402C4⋊C4—21no—
403C4⋊C4—21no—
404C4⋊C4C22⋊Dic6112yes—
405C4⋊C4C22⋊Dic6111yescharacteristic
406C4⋊C4C22⋊Dic6111yescharacteristic
407C22⋊C4—22no—
408C22⋊C4—22no—
409C22⋊C4—22no—
410C22⋊C4—22no—
411C22⋊C4—22no—
412C22⋊C4—21no—
413C22⋊C4—21no—
414C22⋊C4—22no—
415C22⋊C4—21no—
416C22⋊C4—21no—
417C22⋊C4—21no—
418C22⋊C4—22no—
419C42—22no—
420C42—21no—
421C42C2×C22⋊D6111yescharacteristic
422C42C2×C22⋊D6111yescharacteristic
423C42C2×C22⋊D6111yescharacteristic
424C42C2×C22⋊D6111yescharacteristic
425C23—22no—
426C23—21no—
427C23C2×C22⋊Dic6111yescharacteristic
428Q8—41no—
429Q8—81no—
430Q8—82no—
431D4—42no—
432D4—41no—
433D4—44no—
434D4—42no—
435D4—41no—
436D4—41no—
437D4—81no—
438D4—82no—
439D4—82no—
440C2×C4—21no—
441C2×C4—21no—
442C2×C4—21no—
443C2×C4—21no—
444C2×C4—22no—
445C2×C4—22no—
446C2×C4—21no—
447C2×C4—22no—
448C2×C4—41no—
449C2×C4—41no—
450C2×C4—41no—
451C2×C4—41no—
452C2×C4—42no—
453C2×C4—42no—
454C2×C4—82no—
455C2×C4—82no—
456C2×C4—82no—
457C2×C4—82no—
458C2×C4C2×C22⋊Dic6111yescharacteristic
459C2×C4C2×C22⋊Dic6111yescharacteristic
460C2×C4#1451974611yescharacteristic
461C2×C4#1451974611yescharacteristic
462C2×C4#1451974611yescharacteristic
463C2×C4#1451974611yescharacteristic
464C8—4881no—
465C8—4881no—
466C8—4881no—
467C8—2442no—
468C22—21no—
469C22—21no—
470C22—21no—
471C22—42no—
472C22—41no—
473C22—41no—
474C22—42no—
475C22—42no—
476C22—41no—
477C22#1683961811yescharacteristic
478C4—21no—
479C4—21no—
480C4—21no—
481C4—21no—
482C4—21no—
483C4—42no—
484C4—41no—
485C4—41no—
486C4—42no—
487C4—42no—
488C4—41no—
489C4#1683993511yescharacteristic
490C4#1683994511yescharacteristic
491C2—41no—
492C2—82no—
493C2—81no—
494C2#2144577011yescharacteristic
495C2#2144577811yescharacteristic
496C2#2144578911yescharacteristic
subgroupG/H
centerC22#16839618
commutatorC22×C4×C61C22×C4
FrattiniC2×C42C22×D61
Fitting, prime radicalC61×#3559C2
socleC22×C61#379
cosocleC2×C42×C61C23
hypercenter#3559D61
hypocenterC61#8486
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3559

Complements

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

NH ≅ G/Ntype
1. #21446813C2semidirect
2. #21444541C2semidirect
3. #21444426C2semidirect
4. #21444284C2semidirect
5. #21444236C2semidirect
6. #21444175C2semidirect
285. C61#8486semidirect

Maximal subgroups

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

subgroupindex
#214468132
#214445412
#214444262
#214442842
#214442362
#214441752
C61×#35592
#848661