Finite Group Databasebeta

← #29648965

Subgroups — #29648965

Order 15616.

924 conjugacy classes of subgroups.

The subgroup lattice has 522 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21459732C211yesmaximal, characteristic
2#21454948C211yesmaximal, characteristic, complemented
3#21454945C211yesmaximal, characteristic, complemented
4#21444552C211yesmaximal, characteristic, complemented
5#21444548C212yesmaximal, complemented
6#21444523C212yesmaximal, complemented
7#21444495C212yesmaximal, complemented
8#21444491C211yesmaximal, characteristic, complemented
9#21444479C211yesmaximal, characteristic, complemented
10#21444475C212yesmaximal, complemented
11C61×#1549C211yesmaximal, characteristic, complemented
12#16840486—21no—
13#16840478—21no—
14#16840473—21no—
15#16840460—22no—
16#16840423—22no—
17#16840415—21no—
18#16840386C414yes—
19#16840343C412yes—
20#16840317C2212yes—
21#16840313C2212yes—
22#16840312C2211yescharacteristic
23#16840243C412yes—
24C4×#11758222C2211yescharacteristic
25#16839889—21no—
26C8×Dic122—21no—
27#16839841—21no—
28#16839837—22no—
29#16839831—22no—
30#16839827—21no—
31#16839818C2212yes—
32#16839815C2211yescharacteristic
33#16839814C2212yes—
34#16839786C2211yescharacteristic
35#16839784C2212yes—
36C4⋊C8×D61C2212yes—
37#16839510C2211yescharacteristic
38#16839510C2212yes—
39#16839510C2211yescharacteristic
40#16839508C2212yes—
41#16839508C2212yes—
42#16839505C2211yescharacteristic
43#16839505C2211yescharacteristic
44#16839505C2212yes—
45C2×C4⋊Q8×C61C2211yescharacteristic
46C61×#431C2211yescharacteristic
47C61×#423C2211yescharacteristic
48C61×#332C2212yes—
49C61×#332C2212yes—
50C2×#11758226—22no—
51C2×#11758222C2×C411yescharacteristic
52C2×#11758222C2×C412yes—
53C2×#11758194—21no—
54C2×#11758189—22no—
55C2×C8×D61—22no—
56#14519777C2311yescharacteristic
57#14519774C2311yescharacteristic
58#14519772—24no—
59#14519768—22no—
60#14519766—24no—
61#14519766—22no—
62#14519765C2×C412yes—
63#14519762—24no—
64#14519757—22no—
65#14519757C2×C414yes—
66#14519753—22no—
67#14519753—24no—
68#14519752—22no—
69#14519752—22no—
70#14519752C2×C414yes—
71#14519751—21no—
72#14519751—21no—
73#14519751—24no—
74Q8×Dic61—22no—
75C4×Dic122D411yescharacteristic
76C4×Dic122D411yescharacteristic
77#14519742C2×C414yes—
78C4×D244—21no—
79C4×D244—21no—
80C4×D244D411yescharacteristic
81C4×D244D411yescharacteristic
82#14519739—22no—
83#14519739C2×C414yes—
84C4×C22⋊D61—24no—
85C4×C22⋊D61D412yes—
86C4×C22⋊D61D412yes—
87#14519736C2×C411yescharacteristic
88#14519735—22no—
89#14519735—22no—
90#14519734C2×C412yes—
91C4⋊C4×D61—22no—
92C42×D61C2312yes—
93C4⋊#92418—21no—
94C4⋊#92418C2311yescharacteristic
95C4⋊#92418C2312yes—
96C4⋊#92418C2311yescharacteristic
97#14519706—22no—
98#14519706—22no—
99C22⋊#92418—21no—
100#14519689—22no—
101#14519689—22no—
102#14519687—22no—
103C4×#92418—21no—
104C8×Dic61—22no—
105C22×Q8×C61—21no—
106C2×M16×C61—21no—
107C4⋊Q8×C61—21no—
108C4⋊Q8×C61—21no—
109C4⋊Q8×C61C2×C412yes—
110C4⋊Q8×C61C2312yes—
111C2×C4⋊C4×C61—21no—
112C2×C4⋊C4×C61C2×C412yes—
113C2×C42×C61C2311yescharacteristic
114C4⋊C8×C61—21no—
115C4⋊C8×C61C2312yes—
116C4⋊C8×C61C2311yescharacteristic
117C4⋊C8×C61C2311yescharacteristic
118C22⋊C8×C61—21no—
119C4×C8×C61—21no—
120C2×Q8×C61—22no—
121C2×Q8×C61—22no—
122C2×Q8×C61—81no—
123C2×Q8×C61—81no—
124C22×C4×C61—22no—
125C22×C4×C61C22×C411yescharacteristic
126C22×C4×C61C2×D411yescharacteristic
127C22×C4×C61C2×D411yescharacteristic
128C2×C22⋊D61—22no—
129C2×C22⋊D61C22⋊C412yes—
130#11758226—82no—
131#11758226—82no—
132#11758222—42no—
133#11758222—44no—
134C2×D244—21no—
135C2×D244—21no—
136C2×D244—24no—
137C2×D244C22⋊C411yescharacteristic
138C2×D244C22×C412yes—
139C2×C4×D61—22no—
140C2×C4×D61—24no—
141C2×C4×D61C22×C412yes—
142C2×C4×D61C2×D412yes—
143C2×C4×D61C2×D412yes—
144C2×Dic122C22⋊C411yescharacteristic
145C2×Dic122C22×C412yes—
146M16×C61—81no—
147C2×C8×C61—21no—
148C2×C8×C61—21no—
149C2×C8×C61—22no—
150C4⋊C4×C61—22no—
151C4⋊C4×C61—22no—
152C4⋊C4×C61—22no—
153C4⋊C4×C61C22×C414yes—
154C42×C61C22×C411yescharacteristic
155C42×C61C2×D411yescharacteristic
156C42×C61C2×D411yescharacteristic
157C42×C61C2411yescharacteristic
158C22⋊Dic61C22⋊C412yes—
159#11758198—24no—
160#11758198—22no—
161#11758198—22no—
162#11758198—24no—
163#11758198—24no—
164#11758198C22⋊C412yes—
165#11758198C22⋊C412yes—
166C4⋊Dic61—21no—
167C4⋊Dic61—21no—
168C4⋊Dic61—24no—
169C4⋊Dic61C22⋊C411yescharacteristic
170C4⋊Dic61C22⋊C411yescharacteristic
171#11758196—24no—
172#11758196C22⋊C412yes—
173#11758196C22⋊C412yes—
174C4×Dic61—24no—
175C4×Dic61C22×C412yes—
176C4×Dic61C2×D412yes—
177C4×Dic61C2×D412yes—
178#11758194—81no—
179C2×#92418—22no—
180C2×#92418—21no—
181C2×#92418—21no—
182#11758189—82no—
183C8×D61—82no—
184C23×C61C2×C22⋊C411yescharacteristic
185C22×D61—22no—
186C22×D61C2×C22⋊C412yes—
187Q8×C61—42no—
188Q8×C61—42no—
189Q8×C61—82no—
190C2×C4×C61—21no—
191C2×C4×C61—21no—
192C2×C4×C61—21no—
193C2×C4×C61—21no—
194C2×C4×C61—22no—
195C2×C4×C61—24no—
196C2×C4×C61—82no—
197C2×C4×C61C2×C22⋊C411yescharacteristic
198C2×C4×C61C2×C22⋊C411yescharacteristic
199C2×C4×C61C2×C22⋊C411yescharacteristic
200C2×C4×C61C23×C411yescharacteristic
201C2×C4×C61C22×D411yescharacteristic
202C2×C4×C61C22×D411yescharacteristic
203C22⋊D61—42no—
204C22⋊D61—44no—
205C2×Dic61—22no—
206C2×Dic61C2×C22⋊C412yes—
207C2×Dic61C2×C22⋊C412yes—
208C2×Dic61C2×C22⋊C412yes—
209D244—22no—
210D244—42no—
211D244—41no—
212D244—41no—
213D244—41no—
214D244—41no—
215D244—84no—
216C4×D61—22no—
217C4×D61—22no—
218C4×D61—22no—
219C4×D61—22no—
220C4×D61—42no—
221C4×D61—44no—
222C4×D61—84no—
223Dic122—22no—
224Dic122—42no—
225C8×C61—41no—
226C8×C61—42no—
227C8×C61—41no—
228#92418—42no—
229#92418—41no—
230#92418—41no—
231#17781—611nomaximal, Sylow
232C22×C61—21no—
233C22×C61—21no—
234C22×C61—21no—
235C22×C61C22×C22⋊C411yescharacteristic
236C2×D61—22no—
237C2×D61—22no—
238C2×D61—22no—
239C2×D61—42no—
240C2×D61—42no—
241C2×D61—42no—
242C4×C61—21no—
243C4×C61—21no—
244C4×C61—21no—
245C4×C61—44no—
246C4×C61#41312yes—
247C4×C61#57611yescharacteristic
248C4×C61#57711yescharacteristic
249C4×C61#57711yescharacteristic
250C4×C61#57811yescharacteristic
251Dic61—22no—
252Dic61—22no—
253Dic61—22no—
254Dic61—42no—
255#3511—611no—
256#2991—611no—
257#2990—611no—
258#1560—611no—
259#1559—612no—
260#1553—612no—
261#1549D6111yescharacteristic, complemented
262#1547—612no—
263#1546—611no—
264#1543—611no—
265#1542—612no—
266C2×C61—41no—
267C2×C61C2×#41311yescharacteristic
268C2×C61#289711yescharacteristic
269C2×C61#289811yescharacteristic
270D61—42no—
271D61—82no—
272#556—1221no—
273#555—1221no—
274#554—1221no—
275#553—1222no—
276#548—1222no—
277#547—1221no—
278#542—614no—
279#537—612no—
280#533—612no—
281#532—612no—
282#532—611no—
283C2×C4⋊Q8C2×D6111yescharacteristic
284#520—612no—
285C4×#41—611no—
286#446—1221no—
287C8×Q8—1221no—
288#436—1221no—
289#436—1222no—
290#435—1222no—
291C8×D4—1221no—
292#432—612no—
293#431—611no—
294#431—612no—
295#431C2×D6111yescharacteristic
296#423—611no—
297#423—612no—
298#423C2×D6111yescharacteristic
299C2×C4⋊C8—612no—
300#333—611no—
301#333—612no—
302#333—611no—
303#332—612no—
304#332—612no—
305#332C2×D6112yes—
306#332C2×D6112yes—
307#331—611no—
308#331—611no—
309#331—612no—
310C61#1778111yescharacteristic, complemented, Sylow
311C2×#41—611no—
312C2×#41—612no—
313C2×#41—1222no—
314C22×Q8—21no—
315C2×M16—21no—
316C2×M16—1222no—
317C2×M16—1221no—
318C22×C8—1222no—
319C4⋊Q8—21no—
320C4⋊Q8—21no—
321C4⋊Q8—611no—
322C4⋊Q8C4×D6112yes—
323C4⋊Q8C22×D6112yes—
324#127—611no—
325#126—1224no—
326#125—1222no—
327#124—612no—
328#124—1222no—
329#124—1224no—
330#123—1224no—
331C22⋊Q8—614no—
332C22⋊Q8—1222no—
333C22⋊Q8—1224no—
334C22⋊Q8—1222no—
335#121—614no—
336#121—1222no—
337#121—1222no—
338#121—1224no—
339#121—1221no—
340#121—1221no—
341C4×Q8—611no—
342C4×Q8—611no—
343C4×Q8—614no—
344C4×Q8—1222no—
345C4×D4—612no—
346C4×D4—612no—
347C4×D4—614no—
348C4×D4—611no—
349C4×D4—611no—
350C4×D4—1222no—
351C4×D4—1224no—
352C4×D4—1221no—
353C4×D4—1221no—
354#117—611no—
355#117—612no—
356#117—1222no—
357#117—1222no—
358C2×C4⋊C4—21no—
359C2×C4⋊C4—1222no—
360C2×C4⋊C4C4×D6112yes—
361C2×C42—612no—
362C2×C42C22×D6111yescharacteristic
363C4⋊C8—21no—
364C4⋊C8—611no—
365C4⋊C8—612no—
366C4⋊C8—611no—
367C4⋊C8—1222no—
368C4⋊C8—1222no—
369C4⋊C8—1221no—
370C4⋊C8C22×D6112yes—
371C4⋊C8C22×D6111yescharacteristic
372C4⋊C8C22×D6111yescharacteristic
373C22⋊C8—21no—
374C22⋊C8—1222no—
375C22⋊C8—1222no—
376C22⋊C8—1221no—
377#97—1222no—
378C4×C8—21no—
379C4×C8—1222no—
380C4×C8—1221no—
381#41—4882no—
382#41—4882no—
383#41—2442no—
384#41—2444no—
385C2×Q8—22no—
386C2×Q8—22no—
387C2×Q8—81no—
388C2×Q8—81no—
389C2×Q8—611no—
390C2×Q8—612no—
391C2×D4—612no—
392C2×D4—612no—
393C2×D4—611no—
394C2×D4—1224no—
395C2×D4—1222no—
396C2×D4—1221no—
397C2×D4—1221no—
398C22×C4—22no—
399C22×C4—612no—
400C22×C4—612no—
401C22×C4—612no—
402C22×C4—1224no—
403C22×C4—1222no—
404C22×C4C2×C4×D6111yescharacteristic
405C22×C4D4×D6111yescharacteristic
406C22×C4D4×D6111yescharacteristic
407M16—81no—
408M16—4882no—
409M16—4881no—
410C2×C8—21no—
411C2×C8—21no—
412C2×C8—22no—
413C2×C8—1221no—
414C2×C8—1221no—
415C2×C8—1222no—
416C2×C8—4882no—
417C4⋊C4—22no—
418C4⋊C4—22no—
419C4⋊C4—22no—
420C4⋊C4—611no—
421C4⋊C4—612no—
422C4⋊C4—612no—
423C4⋊C4—611no—
424C4⋊C4—1224no—
425C4⋊C4—1224no—
426C4⋊C4—1221no—
427C4⋊C4—1221no—
428C4⋊C4C2×C4×D6114yes—
429C22⋊C4—612no—
430C22⋊C4—612no—
431C22⋊C4—612no—
432C22⋊C4—1224no—
433C22⋊C4—1224no—
434C22⋊C4—1224no—
435C22⋊C4—1222no—
436C22⋊C4—1222no—
437C42—612no—
438C42—612no—
439C42—612no—
440C42—1224no—
441C42C2×C4×D6111yescharacteristic
442C42D4×D6111yescharacteristic
443C42D4×D6111yescharacteristic
444C42C23×D6111yescharacteristic
445C23—612no—
446C23—1222no—
447C23C22⋊C4×D6111yescharacteristic
448Q8—42no—
449Q8—42no—
450Q8—82no—
451Q8—1222no—
452Q8—2442no—
453D4—1222no—
454D4—4884no—
455D4—2442no—
456D4—2442no—
457D4—2444no—
458D4—2441no—
459D4—2441no—
460D4—2441no—
461D4—2441no—
462C2×C4—22no—
463C2×C4—21no—
464C2×C4—21no—
465C2×C4—21no—
466C2×C4—21no—
467C2×C4—24no—
468C2×C4—82no—
469C2×C4—612no—
470C2×C4—612no—
471C2×C4—612no—
472C2×C4—1222no—
473C2×C4—1222no—
474C2×C4—1222no—
475C2×C4—1222no—
476C2×C4—1222no—
477C2×C4—4884no—
478C2×C4—2442no—
479C2×C4—2444no—
480C2×C4C22⋊C4×D6111yescharacteristic
481C2×C4C22⋊C4×D6111yescharacteristic
482C2×C4C22⋊C4×D6111yescharacteristic
483C2×C4C22×C4×D6111yescharacteristic
484C2×C4C2×D4×D6111yescharacteristic
485C2×C4C2×D4×D6111yescharacteristic
486C8—41no—
487C8—42no—
488C8—41no—
489C8—2442no—
490C8—2441no—
491C8—2441no—
492C22—21no—
493C22—21no—
494C22—21no—
495C22—1222no—
496C22—1222no—
497C22—1222no—
498C22—2442no—
499C22—2442no—
500C22—2442no—
501C22C2×C22⋊C4×D6111yescharacteristic
502C4—21no—
503C4—21no—
504C4—21no—
505C4—44no—
506C4—1222no—
507C4—1222no—
508C4—1222no—
509C4—2442no—
510C4#1683974512yes—
511C4#1684056611yescharacteristic
512C4#1684056811yescharacteristic
513C4#1684057211yescharacteristic
514C4#1684057511yescharacteristic
515C2—41no—
516C2—4882no—
517C2—2442no—
518C2C2×#1683974511yescharacteristic
519C2#2145434411yescharacteristic
520C2#2145434711yescharacteristic
subgroupG/H
centerC22C2×C22⋊C4×D61
commutatorC2×C4×C61C23×C4
FrattiniC42C23×D61
FittingC61×#1549C2
socleC22×C61C22×C22⋊C4
prime radical#21459732C2
cosocleC42×C61C24
hypercenter#1549D61
hypocenterC61#17781
LCSG ▷ C2×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1549

Complements

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

NH ≅ G/Ntype
2. #21454948C2semidirect
3. #21454945C2semidirect
4. #21444552C2semidirect
5. #21444548C2semidirect
6. #21444523C2semidirect
7. #21444495C2semidirect
8. #21444491C2semidirect
9. #21444479C2semidirect
10. #21444475C2semidirect
11. C61×#1549C2semidirect
261. #1549D61semidirect
310. C61#17781semidirect

Maximal subgroups

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

subgroupindex
#214597322
#214549482
#214549452
#214445522
#214445482
#214445232
#214444952
#214444912
#214444792
#214444752
C61×#15492
#1778161