Finite Group Databasebeta

← #29572494

Subgroups — #29572494

Order 15616.

724 conjugacy classes of subgroups.

The subgroup lattice has 528 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21446777C211yesmaximal, characteristic, complemented
2#21446739C211yesmaximal, characteristic, complemented
3#21444514C211yesmaximal, characteristic, complemented
4#21444421C211yesmaximal, characteristic, complemented
5#21444344C211yesmaximal, characteristic, complemented
6#21444232C211yesmaximal, characteristic, complemented
7C61×#3559C211yesmaximal, characteristic
8#16839819C2211yescharacteristic
9C2×C4⋊#92418C2211yescharacteristic
10C2×#14519704—22no—
11C4×#11758194C2211yescharacteristic
12C4×C4⋊Dic61C2211yescharacteristic
13#16839524—21no—
14#16839524—21no—
15#16839511—21no—
16#16839506—21no—
17#16839504—22no—
18#16839499—21no—
19#16839496—21no—
20Q8⋊#92418—21no—
21D4⋊#92418—21no—
22C61×#563C412yescomplemented
23C61×#562C412yescomplemented
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×#539—21no—
33C61×#537—21no—
34C61×#532C2211yescharacteristic
35C61×#529C2211yescharacteristic
36C2×#125×C61C2211yescharacteristic
37C2×#11758194—21no—
38C22×#92418—21no—
39C2×C4⋊Dic61—21no—
40C2×C4×Dic61—21no—
41C4⋊#92418—21no—
42C4⋊#92418—21no—
43C4⋊#92418—21no—
44#14519704—44no—
45C22⋊#92418—21no—
46#14519688—21no—
47#14519688—21no—
48C4×#92418—21no—
49#14519684—21no—
50C2×#41×C61—22no—
51C2×#41×C61D412yes—
52C4⋊Q8×C61—21no—
53C4⋊Q8×C61—22no—
54C4⋊Q8×C61D411yescharacteristic
55#127×C61D411yescharacteristic
56#126×C61—22no—
57#126×C61—22no—
58#126×C61—21no—
59#125×C61—22no—
60#125×C61—21no—
61#125×C61—21no—
62#125×C61—21no—
63#125×C61C2×C412yes—
64#125×C61C2×C411yescharacteristic
65#125×C61D412yes—
66#124×C61C2×C412yes—
67#124×C61C2×C411yescharacteristic
68#123×C61—22no—
69#123×C61—22no—
70#123×C61—22no—
71#123×C61—22no—
72#123×C61—21no—
73#123×C61—22no—
74#123×C61—22no—
75C22⋊Q8×C61—22no—
76C22⋊Q8×C61—22no—
77C22⋊Q8×C61—22no—
78C22⋊Q8×C61—22no—
79C22⋊Q8×C61—22no—
80C22⋊Q8×C61—22no—
81C22⋊Q8×C61D411yescharacteristic
82#121×C61—22no—
83#121×C61—22no—
84#121×C61—21no—
85#121×C61—21no—
86#121×C61—22no—
87#121×C61—22no—
88#121×C61—21no—
89#121×C61—21no—
90#121×C61D411yescharacteristic
91C4×Q8×C61D411yescharacteristic
92C4×D4×C61—22no—
93C4×D4×C61—22no—
94C4×D4×C61—21no—
95C4×D4×C61—21no—
96C4×D4×C61—21no—
97C4×D4×C61—21no—
98C4×D4×C61—21no—
99C4×D4×C61D411yescharacteristic
100#117×C61—21no—
101#117×C61—21no—
102C2×C4⋊C4×C61—22no—
103C2×C4⋊C4×C61—21no—
104C2×C4⋊C4×C61—21no—
105C2×C4⋊C4×C61—21no—
106C2×C4⋊C4×C61D411yescharacteristic
107C2×C4⋊C4×C61D411yescharacteristic
108C2×C42×C61C2311yescharacteristic
109#41×C61—42no—
110#41×C61—42no—
111#41×C61—82no—
112#41×C61—82no—
113C2×Q8×C61—22no—
114C2×Q8×C61C22⋊C411yescharacteristic
115C2×Q8×C61C22⋊C412yes—
116C2×D4×C61—22no—
117C2×D4×C61—21no—
118C2×D4×C61—22no—
119C2×D4×C61—22no—
120C2×D4×C61—21no—
121C2×D4×C61—21no—
122C2×D4×C61C22⋊C411yescharacteristic
123C2×D4×C61C22⋊C412yes—
124C22×C4×C61—21no—
125C22×C4×C61—22no—
126C22×C4×C61—22no—
127C22×C4×C61—22no—
128C22×C4×C61—21no—
129C22×C4×C61—21no—
130C22×C4×C61—22no—
131C22×C4×C61C2×D411yescharacteristic
132C22×C4×C61C2×D411yescharacteristic
133C22×C4×C61C2×D411yescharacteristic
134C22×Dic61—41no—
135C4⋊C4×C61—22no—
136C4⋊C4×C61—22no—
137C4⋊C4×C61—22no—
138C4⋊C4×C61—22no—
139C4⋊C4×C61—22no—
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—22no—
148C4⋊C4×C61—21no—
149C4⋊C4×C61—21no—
150C4⋊C4×C61—21no—
151C4⋊C4×C61C22⋊C412yes—
152C4⋊C4×C61C22⋊C412yes—
153C4⋊C4×C61C22⋊C411yescharacteristic
154C4⋊C4×C61C22⋊C411yescharacteristic
155C22⋊C4×C61—22no—
156C22⋊C4×C61—22no—
157C22⋊C4×C61—22no—
158C22⋊C4×C61—22no—
159C22⋊C4×C61—22no—
160C22⋊C4×C61—21no—
161C22⋊C4×C61—21no—
162C22⋊C4×C61—22no—
163C22⋊C4×C61—22no—
164C22⋊C4×C61—21no—
165C22⋊C4×C61—21no—
166C22⋊C4×C61—21no—
167C42×C61—22no—
168C42×C61—21no—
169C42×C61C22×C411yescharacteristic
170C42×C61C2×D411yescharacteristic
171C42×C61C2×D411yescharacteristic
172C42×C61C2×D411yescharacteristic
173C4⋊Dic61—22no—
174C4⋊Dic61—41no—
175C4×Dic61—42no—
176#11758194—22no—
177#11758194—41no—
178C2×#92418—41no—
179C2×#92418—42no—
180C2×#92418—41no—
181C2×#92418—41no—
182C23×C61—22no—
183C23×C61—21no—
184C23×C61#12011yescharacteristic
185Q8×C61—41no—
186Q8×C61—42no—
187Q8×C61—82no—
188D4×C61—41no—
189D4×C61—42no—
190D4×C61—42no—
191D4×C61—41no—
192D4×C61—42no—
193D4×C61—41no—
194D4×C61—41no—
195D4×C61—41no—
196D4×C61—81no—
197D4×C61—82no—
198D4×C61—82no—
199C2×C4×C61—21no—
200C2×C4×C61—21no—
201C2×C4×C61—21no—
202C2×C4×C61—21no—
203C2×C4×C61—21no—
204C2×C4×C61—21no—
205C2×C4×C61—22no—
206C2×C4×C61—22no—
207C2×C4×C61—21no—
208C2×C4×C61—22no—
209C2×C4×C61—41no—
210C2×C4×C61—41no—
211C2×C4×C61—41no—
212C2×C4×C61—41no—
213C2×C4×C61—41no—
214C2×C4×C61—82no—
215C2×C4×C61—81no—
216C2×C4×C61—82no—
217C2×C4×C61—82no—
218C2×C4×C61—82no—
219C2×C4×C61C2×C22⋊C411yescharacteristic
220C2×C4×C61C2×C22⋊C411yescharacteristic
221C2×C4×C61C2×C22⋊C411yescharacteristic
222C2×C4×C61#12011yescharacteristic
223C2×C4×C61#12011yescharacteristic
224C2×C4×C61#12011yescharacteristic
225C2×Dic61—42no—
226C2×Dic61—81no—
227C2×Dic61—81no—
228#92418—42no—
229#92418—81no—
230#92418—81no—
231#8513—611nomaximal, Sylow
232C22×C61—21no—
233C22×C61—21no—
234C22×C61—42no—
235C22×C61—41no—
236C22×C61—41no—
237C22×C61—42no—
238C22×C61—42no—
239C22×C61—41no—
240C22×C61#37911yescharacteristic
241C22×C61#45311yescharacteristic
242C22×C61#45511yescharacteristic
243C4×C61—21no—
244C4×C61—21no—
245C4×C61—21no—
246C4×C61—21no—
247C4×C61—21no—
248C4×C61—42no—
249C4×C61—42no—
250C4×C61—41no—
251C4×C61—41no—
252C4×C61—42no—
253C4×C61—41no—
254C4×C61#45411yescharacteristic
255C4×C61#45611yescharacteristic
256Dic61—82no—
257#3559D6111yescharacteristic
258#1989—611no—
259#1980—611no—
260#1551—611no—
261#1528—611no—
262#1509—611no—
263#1483—611no—
264C2×C61—21no—
265C2×C61—21no—
266C2×C61—82no—
267C2×C61—81no—
268C2×C61#179211yescharacteristic
269C2×C61#179311yescharacteristic
270C2×C61#179411yescharacteristic
271#563Dic6112yescomplemented
272#562Dic6112yescomplemented
273#554—21no—
274#553—21no—
275#553—21no—
276#547—22no—
277#547—21no—
278#544—21no—
279#540—21no—
280#539—22no—
281#539—21no—
282#537—21no—
283#532C2×D6111yescharacteristic
284#529C2×D6111yescharacteristic
285C2×#125C2×D6111yescharacteristic
286#432—611no—
287C2×C4⋊C8—611no—
288C2×#104—1222no—
289C4×M16—611no—
290C4×C4⋊C4—611no—
291#340—1221no—
292#340—1221no—
293#333—1221no—
294#331—1221no—
295#330—1222no—
296#328—1221no—
297#327—1221no—
298Q8⋊C8—1221no—
299D4⋊C8—1221no—
300C61#851311yescharacteristic, complemented, Sylow
301C2×#41—22no—
302C2×#41C22⋊D6112yes—
303C2×M16—1221no—
304C22×C8—1221no—
305C4⋊Q8—21no—
306C4⋊Q8—22no—
307C4⋊Q8C22⋊D6111yescharacteristic
308#127C22⋊D6111yescharacteristic
309#126—22no—
310#126—22no—
311#126—21no—
312#125—22no—
313#125—21no—
314#125—21no—
315#125—21no—
316#125C2×Dic6112yes—
317#125C2×Dic6111yescharacteristic
318#125C22⋊D6112yes—
319#124C2×Dic6111yescharacteristic
320#124C2×Dic6112yes—
321#123—22no—
322#123—22no—
323#123—22no—
324#123—22no—
325#123—21no—
326#123—22no—
327#123—22no—
328C22⋊Q8—22no—
329C22⋊Q8—22no—
330C22⋊Q8—22no—
331C22⋊Q8—22no—
332C22⋊Q8—22no—
333C22⋊Q8—22no—
334C22⋊Q8C22⋊D6111yescharacteristic
335#121—22no—
336#121—22no—
337#121—21no—
338#121—21no—
339#121—22no—
340#121—22no—
341#121—21no—
342#121—21no—
343#121C22⋊D6111yescharacteristic
344C4×Q8C22⋊D6111yescharacteristic
345C4×D4—22no—
346C4×D4—22no—
347C4×D4—21no—
348C4×D4—21no—
349C4×D4—21no—
350C4×D4—21no—
351C4×D4—21no—
352C4×D4C22⋊D6111yescharacteristic
353#117—21no—
354#117—21no—
355C2×C4⋊C4—22no—
356C2×C4⋊C4—21no—
357C2×C4⋊C4—21no—
358C2×C4⋊C4—21no—
359C2×C4⋊C4—1221no—
360C2×C4⋊C4C22⋊D6111yescharacteristic
361C2×C4⋊C4C22⋊D6111yescharacteristic
362C2×C42—1221no—
363C2×C42C22×D6111yescharacteristic
364C4⋊C8—1221no—
365C4⋊C8—1221no—
366C4⋊C8—1221no—
367#104—2444no—
368C22⋊C8—1221no—
369#97—1221no—
370#97—1221no—
371C4×C8—1221no—
372#95—1221no—
373#41—42no—
374#41—42no—
375#41—82no—
376#41—82no—
377C2×Q8—22no—
378C2×Q8C22⋊Dic6111yescharacteristic
379C2×Q8C22⋊Dic6112yes—
380C2×D4—22no—
381C2×D4—21no—
382C2×D4—22no—
383C2×D4—22no—
384C2×D4—21no—
385C2×D4—21no—
386C2×D4C22⋊Dic6111yescharacteristic
387C2×D4C22⋊Dic6112yes—
388C22×C4—21no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—21no—
393C22×C4—21no—
394C22×C4—22no—
395C22×C4—2441no—
396C22×C4C2×C22⋊D6111yescharacteristic
397C22×C4C2×C22⋊D6111yescharacteristic
398C22×C4C2×C22⋊D6111yescharacteristic
399M16—1222no—
400M16—2441no—
401C2×C8—2442no—
402C2×C8—2441no—
403C2×C8—2441no—
404C2×C8—2441no—
405C4⋊C4—22no—
406C4⋊C4—22no—
407C4⋊C4—22no—
408C4⋊C4—22no—
409C4⋊C4—22no—
410C4⋊C4—22no—
411C4⋊C4—21no—
412C4⋊C4—21no—
413C4⋊C4—22no—
414C4⋊C4—21no—
415C4⋊C4—21no—
416C4⋊C4—21no—
417C4⋊C4—22no—
418C4⋊C4—21no—
419C4⋊C4—21no—
420C4⋊C4—21no—
421C4⋊C4—1222no—
422C4⋊C4—2441no—
423C4⋊C4C22⋊Dic6112yes—
424C4⋊C4C22⋊Dic6112yes—
425C4⋊C4C22⋊Dic6111yescharacteristic
426C4⋊C4C22⋊Dic6111yescharacteristic
427C22⋊C4—22no—
428C22⋊C4—22no—
429C22⋊C4—22no—
430C22⋊C4—22no—
431C22⋊C4—22no—
432C22⋊C4—21no—
433C22⋊C4—21no—
434C22⋊C4—22no—
435C22⋊C4—22no—
436C22⋊C4—21no—
437C22⋊C4—21no—
438C22⋊C4—21no—
439C42—22no—
440C42—21no—
441C42—2442no—
442C42C22×Dic6111yescharacteristic
443C42C2×C22⋊D6111yescharacteristic
444C42C2×C22⋊D6111yescharacteristic
445C42C2×C22⋊D6111yescharacteristic
446C23—22no—
447C23—21no—
448C23#1451974611yescharacteristic
449Q8—41no—
450Q8—42no—
451Q8—82no—
452D4—41no—
453D4—42no—
454D4—42no—
455D4—41no—
456D4—42no—
457D4—41no—
458D4—41no—
459D4—41no—
460D4—81no—
461D4—82no—
462D4—82no—
463C2×C4—21no—
464C2×C4—21no—
465C2×C4—21no—
466C2×C4—21no—
467C2×C4—21no—
468C2×C4—21no—
469C2×C4—22no—
470C2×C4—22no—
471C2×C4—21no—
472C2×C4—22no—
473C2×C4—41no—
474C2×C4—41no—
475C2×C4—41no—
476C2×C4—41no—
477C2×C4—41no—
478C2×C4—82no—
479C2×C4—81no—
480C2×C4—82no—
481C2×C4—82no—
482C2×C4—82no—
483C2×C4—4881no—
484C2×C4—4881no—
485C2×C4—2442no—
486C2×C4C2×C22⋊Dic6111yescharacteristic
487C2×C4C2×C22⋊Dic6111yescharacteristic
488C2×C4C2×C22⋊Dic6111yescharacteristic
489C2×C4#1451974611yescharacteristic
490C2×C4#1451974611yescharacteristic
491C2×C4#1451974611yescharacteristic
492C8—4881no—
493C8—4881no—
494C8—2442no—
495C22—21no—
496C22—21no—
497C22—42no—
498C22—41no—
499C22—41no—
500C22—42no—
501C22—42no—
502C22—41no—
503C22#1683961811yescharacteristic
504C22#1683993011yescharacteristic
505C22#1683994011yescharacteristic
506C4—21no—
507C4—21no—
508C4—21no—
509C4—21no—
510C4—21no—
511C4—42no—
512C4—42no—
513C4—41no—
514C4—41no—
515C4—42no—
516C4—41no—
517C4—4882no—
518C4#1683993511yescharacteristic
519C4#1683994511yescharacteristic
520C2—21no—
521C2—21no—
522C2—82no—
523C2—81no—
524C2#2144577311yescharacteristic
525C2#2144577811yescharacteristic
526C2#2144578311yescharacteristic
subgroupG/H
centerC22#16839618
commutatorC42×C61C22×C4
FrattiniC2×C42C22×D61
Fitting, prime radicalC61×#3559C2
socleC22×C61#379
cosocleC2×C42×C61C23
hypercenter#3559D61
hypocenterC61#8513
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. #21446777C2semidirect
2. #21446739C2semidirect
3. #21444514C2semidirect
4. #21444421C2semidirect
5. #21444344C2semidirect
6. #21444232C2semidirect
22. C61×#563C4semidirect
23. C61×#562C4semidirect
271. #563Dic61semidirect
272. #562Dic61semidirect
300. C61#8513semidirect

Maximal subgroups

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

subgroupindex
#214467772
#214467392
#214445142
#214444212
#214443442
#214442322
C61×#35592
#851361