Finite Group Databasebeta

← #29665437

Subgroups — #29665437

Order 15616.

620 conjugacy classes of subgroups.

The subgroup lattice has 466 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456764C211yesmaximal, characteristic, complemented
2#21455635C211yesmaximal, characteristic
3#21455601C211yesmaximal, characteristic
4#21454928C211yesmaximal, characteristic
5#21445280C212yesmaximal, complemented
6#21445273C212yesmaximal, complemented
7#21445188C212yesmaximal, complemented
8#21445183C212yesmaximal, complemented
9#21444595C211yesmaximal, characteristic, complemented
10C2×#16839509C211yesmaximal, characteristic, complemented
11C61×#1550C211yesmaximal, characteristic, complemented
12C2×#14519777C2211yescharacteristic
13C4×#11758222C2211yescharacteristic
14#16840141C2212yes—
15#16840114C2212yes—
16#16840066—21no—
17#16840059—21no—
18#16840058—21no—
19#16840054C2211yescharacteristic
20#16840053—21no—
21#16840049C2211yescharacteristic
22#16840048C2212yes—
23#16840044C2211yescharacteristic
24#16840043C2212yes—
25#16840039C2211yescharacteristic
26#16839972—21no—
27#16839972—21no—
28#16839917—22no—
29#16839886C2212yes—
30#16839827C2212yes—
31C8×C22⋊D61—22no—
32#16839815C2211yescharacteristic
33#16839814—21no—
34#16839804—21no—
35#16839798—21no—
36C2×C4⋊#92418C2211yescharacteristic
37C4⋊C8×D61—21no—
38#16839764—21no—
39#16839756—21no—
40#16839514C2212yes—
41#16839514C2212yes—
42#16839509C2212yes—
43#16839509C2212yes—
44C61×#533C2211yescharacteristic
45C61×#423C2211yescharacteristic
46C2×C4⋊C8×C61C2211yescharacteristic
47C61×#333C2212yes—
48C61×#332C2212yes—
49C2×#11758222D411yescharacteristic
50C22×Dic122—21no—
51C2×#11758202—22no—
52C2×#11758202—22no—
53C22×#92418—21no—
54C2×C8×D61—22no—
55#14519777—21no—
56#14519777—21no—
57#14519777—21no—
58#14519777C2312yes—
59C4×Dic122D411yescharacteristic
60C4×Dic122C2311yescharacteristic
61C4×D244D411yescharacteristic
62C4×D244C2311yescharacteristic
63C4×C22⋊D61—21no—
64C4×C22⋊D61—21no—
65#14519736D411yescharacteristic
66#14519734—21no—
67C2×C4⋊Dic61—21no—
68C2×C4⋊Dic61—21no—
69C42×D61—21no—
70#14519711—22no—
71#14519710—21no—
72#14519710—21no—
73#14519709—22no—
74#14519708—21no—
75#14519708—21no—
76C4⋊#92418—21no—
77C4⋊#92418D411yescharacteristic
78C4⋊#92418D411yescharacteristic
79C4⋊#92418C2311yescharacteristic
80C4⋊#92418C2311yescharacteristic
81#14519706—22no—
82#14519702—22no—
83#14519700—21no—
84#14519700—21no—
85#14519700—22no—
86#14519700—22no—
87#14519699—22no—
88#14519697—21no—
89#14519697—21no—
90C22⋊#92418D412yes—
91#14519689—22no—
92C4×#92418C2312yes—
93C8×Dic61—22no—
94C2×M16×C61—21no—
95C22×C8×C61—21no—
96C4⋊Q8×C61C2311yescharacteristic
97C4⋊Q8×C61C2311yescharacteristic
98#125×C61—21no—
99C22⋊Q8×C61—21no—
100C22⋊Q8×C61—21no—
101C4×Q8×C61—21no—
102C4×Q8×C61—21no—
103#117×C61—21no—
104C2×C42×C61C2311yescharacteristic
105C4⋊C8×C61—21no—
106C4⋊C8×C61—21no—
107C4⋊C8×C61C2312yes—
108C4⋊C8×C61C2312yes—
109C2×Q8×C61—21no—
110C2×Q8×C61—21no—
111C22×C4×C61C2×D411yescharacteristic
112C22×C4×C61C2×D411yescharacteristic
113C22×C4×C61#4111yescharacteristic
114C2×C22⋊D61—21no—
115C22×Dic61—41no—
116#11758222—22no—
117#11758222Q1614yescomplemented
118C2×D244C2×D411yescharacteristic
119C2×C4×D61—21no—
120C2×C4×D61—21no—
121C2×C4×D61—21no—
122C2×Dic122—22no—
123C2×Dic122—41no—
124C2×Dic122—81no—
125C2×Dic122—81no—
126C2×Dic122C2×D411yescharacteristic
127M16×C61—42no—
128C2×C8×C61—22no—
129C2×C8×C61—22no—
130C2×C8×C61—81no—
131C4⋊C4×C61—21no—
132C4⋊C4×C61—21no—
133C4⋊C4×C61—21no—
134C4⋊C4×C61—21no—
135C4⋊C4×C61—41no—
136C4⋊C4×C61—41no—
137C22⋊C4×C61—41no—
138C42×C61—41no—
139C42×C61C2×D411yescharacteristic
140C42×C61C2×D411yescharacteristic
141C42×C61#4111yescharacteristic
142C42×C61C2411yescharacteristic
143C22⋊Dic61—21no—
144#11758202—82no—
145#11758202—82no—
146#11758198—21no—
147#11758198—21no—
148C4⋊Dic61—22no—
149C4⋊Dic61—22no—
150C4⋊Dic61—41no—
151C4⋊Dic61—41no—
152C4⋊Dic61C2×D411yescharacteristic
153C4⋊Dic61C2×D411yescharacteristic
154#11758196—21no—
155#11758196—21no—
156C4×Dic61—21no—
157C4×Dic61—21no—
158C4×Dic61—21no—
159C2×#92418—21no—
160C2×#92418—21no—
161C2×#92418—42no—
162C2×#92418C2×D412yes—
163C2×#92418C2×D412yes—
164C8×D61—82no—
165C23×C61#12111yescharacteristic
166C22×D61—21no—
167Q8×C61—41no—
168Q8×C61—41no—
169Q8×C61—81no—
170C2×C4×C61—21no—
171C2×C4×C61—21no—
172C2×C4×C61—41no—
173C2×C4×C61—41no—
174C2×C4×C61—41no—
175C2×C4×C61#12111yescharacteristic
176C2×C4×C61#12111yescharacteristic
177C2×C4×C61#12111yescharacteristic
178C2×C4×C61C2×Q1612yes—
179C2×C4×C61#13512yes—
180C2×C4×C61C22×D411yescharacteristic
181C2×C4×C61C22×D411yescharacteristic
182C2×C4×C61C2×#4111yescharacteristic
183C22⋊D61—24no—
184C2×Dic61—21no—
185C2×Dic61—21no—
186C2×Dic61—21no—
187C2×Dic61—42no—
188C2×Dic61—82no—
189D244—21no—
190D244C2×Q1612yes—
191C4×D61—24no—
192C4×D61—41no—
193C4×D61—41no—
194C4×D61—81no—
195Dic122—21no—
196Dic122—42no—
197Dic122—42no—
198Dic122—82no—
199Dic122C2×Q1612yes—
200C8×C61—42no—
201C8×C61—42no—
202#92418—22no—
203#92418—22no—
204#92418—41no—
205#92418—41no—
206#19411—611nomaximal, Sylow
207C22×C61—21no—
208C22×C61—21no—
209C22×C61#46712yes—
210C22×C61C2×#12111yescharacteristic
211C2×D61—22no—
212C2×D61—41no—
213C2×D61—41no—
214C4×C61—21no—
215C4×C61—21no—
216C4×C61—21no—
217C4×C61—81no—
218C4×C61—81no—
219C4×C61#46712yes—
220C4×C61C22×Q1611yescharacteristic
221C4×C61C2×#13511yescharacteristic
222C4×C61#57611yescharacteristic
223C4×C61#57711yescharacteristic
224Dic61—22no—
225Dic61—41no—
226Dic61—41no—
227Dic61—82no—
228#3209—611no—
229#3079—611no—
230#3076—611no—
231#2988—611no—
232#1684—612no—
233#1683—612no—
234#1670—612no—
235#1669—612no—
236#1571—611no—
237#1550D6111yescharacteristic, complemented
238C2×#332—611no—
239C2×C61—22no—
240C2×C61#304511yescharacteristic
241C2×C61C2×#46711yescharacteristic
242C2×C61#306911yescharacteristic
243D61—42no—
244#533C2×D6111yescharacteristic
245C2×C4⋊Q8—611no—
246C4×#41—611no—
247#494—612no—
248#487—612no—
249#479—1221no—
250#478—1221no—
251#477—611no—
252#477—1221no—
253#476—611no—
254#476—1221no—
255#475—611no—
256#475—612no—
257#474—611no—
258#474—612no—
259#462—1221no—
260#462—1221no—
261#451—1222no—
262C8×Q8—612no—
263C8×D4—612no—
264C8×D4—1222no—
265#431—611no—
266#431—1221no—
267#428—1221no—
268#427—1221no—
269#423C2×D6111yescharacteristic
270C2×C4⋊C8—611no—
271C2×C4⋊C8—1221no—
272C2×C4⋊C8C2×D6111yescharacteristic
273#417—1221no—
274#416—1221no—
275#335—612no—
276#335—612no—
277#333C2×D6112yes—
278#332—612no—
279#332—612no—
280#332C2×D6112yes—
281C61#1941111yescharacteristic, complemented, Sylow
282C2×#41—611no—
283C22×Q8—1221no—
284C2×Q16—1222no—
285C2×Q16—1222no—
286C2×M16—21no—
287C22×C8—21no—
288C22×C8—1222no—
289C22×C8—1221no—
290C4⋊Q8—612no—
291C4⋊Q8—1221no—
292C4⋊Q8—1221no—
293C4⋊Q8—1221no—
294C4⋊Q8C22×D6111yescharacteristic
295C4⋊Q8C22×D6111yescharacteristic
296#125—21no—
297C22⋊Q8—21no—
298C22⋊Q8—21no—
299C4×Q8—21no—
300C4×Q8—21no—
301C4×Q8—611no—
302C4×Q8—611no—
303C4×D4—611no—
304C4×D4—611no—
305C4×D4—1221no—
306C4×D4—1221no—
307#117—21no—
308#117—611no—
309#117—1221no—
310C2×C4⋊C4—1221no—
311C2×C4⋊C4—1221no—
312C2×C42—1221no—
313C2×C42C22×D6111yescharacteristic
314#107—1221no—
315#107—1221no—
316#107—1222no—
317#106—1221no—
318#106—1221no—
319#106—1222no—
320C4⋊C8—21no—
321C4⋊C8—21no—
322C4⋊C8—611no—
323C4⋊C8—611no—
324C4⋊C8—611no—
325C4⋊C8—611no—
326C4⋊C8—1222no—
327C4⋊C8—1221no—
328C4⋊C8C22×D6112yes—
329C4⋊C8C22×D6112yes—
330#103—1222no—
331#103—1222no—
332#103—1221no—
333#103—1221no—
334#103—1222no—
335#102—1221no—
336#102—1221no—
337#102—1222no—
338C22⋊C8—612no—
339C22⋊C8—1222no—
340C4×C8—612no—
341C4×C8—1222no—
342#41—614no—
343#41—1222no—
344C2×Q8—21no—
345C2×Q8—21no—
346C2×Q8—611no—
347C2×Q8—1222no—
348C2×Q8—4881no—
349C2×Q8—4881no—
350C2×Q8—2441no—
351C2×D4—611no—
352C2×D4—1221no—
353C22×C4—1221no—
354C22×C4—1221no—
355C22×C4—1221no—
356C22×C4—2441no—
357C22×C4#1175822211yescharacteristic
358C22×C4D4×D6111yescharacteristic
359C22×C4D4×D6111yescharacteristic
360Q16—4882no—
361Q16—4882no—
362M16—42no—
363C2×C8—22no—
364C2×C8—22no—
365C2×C8—81no—
366C2×C8—612no—
367C2×C8—612no—
368C2×C8—1221no—
369C2×C8—1221no—
370C2×C8—4882no—
371C2×C8—2442no—
372C4⋊C4—21no—
373C4⋊C4—21no—
374C4⋊C4—21no—
375C4⋊C4—21no—
376C4⋊C4—41no—
377C4⋊C4—41no—
378C4⋊C4—611no—
379C4⋊C4—611no—
380C4⋊C4—1221no—
381C4⋊C4—1221no—
382C4⋊C4—1222no—
383C4⋊C4—1222no—
384C4⋊C4—2441no—
385C4⋊C4—2441no—
386C22⋊C4—41no—
387C22⋊C4—1221no—
388C22⋊C4—1221no—
389C22⋊C4—1221no—
390C42—41no—
391C42—1221no—
392C42—1221no—
393C42—1221no—
394C42#1175822211yescharacteristic
395C42D4×D6111yescharacteristic
396C42D4×D6111yescharacteristic
397C42C23×D6111yescharacteristic
398C23—1221no—
399C23#1451975011yescharacteristic
400Q8—41no—
401Q8—41no—
402Q8—81no—
403Q8—612no—
404Q8—1221no—
405Q8—4882no—
406Q8—2442no—
407Q8—2442no—
408D4—612no—
409D4—1224no—
410D4—1221no—
411C2×C4—21no—
412C2×C4—21no—
413C2×C4—41no—
414C2×C4—41no—
415C2×C4—41no—
416C2×C4—1221no—
417C2×C4—1224no—
418C2×C4—1221no—
419C2×C4—1221no—
420C2×C4—4881no—
421C2×C4—4882no—
422C2×C4—2442no—
423C2×C4—2441no—
424C2×C4—2441no—
425C2×C4#1451975011yescharacteristic
426C2×C4#1451975011yescharacteristic
427C2×C4#1451975011yescharacteristic
428C2×C4Q16×D6112yes—
429C2×C4#1451979812yes—
430C2×C4C2×D4×D6111yescharacteristic
431C2×C4C2×D4×D6111yescharacteristic
432C2×C4C2×#1175822211yescharacteristic
433C8—42no—
434C8—42no—
435C8—1222no—
436C8—1222no—
437C8—2441no—
438C8—2441no—
439C22—21no—
440C22—21no—
441C22—1222no—
442C22—2441no—
443C22—2441no—
444C22#1684000212yes—
445C22C2×#1451975011yescharacteristic
446C4—21no—
447C4—21no—
448C4—21no—
449C4—81no—
450C4—81no—
451C4—1222no—
452C4—4882no—
453C4—2441no—
454C4—2441no—
455C4#1684000212yes—
456C4C2×Q16×D6111yescharacteristic
457C4C2×#1451979811yescharacteristic
458C4#1684056611yescharacteristic
459C4#1684057111yescharacteristic
460C2—22no—
461C2—2442no—
462C2#2145535711yescharacteristic
463C2C2×#1684000211yescharacteristic
464C2#2145555911yescharacteristic
subgroupG/H
centerC22C2×#14519750
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#1550C2
socleC22×C61C2×#121
prime radicalC2×#11758222D4
hypercenter#1550D61
hypocenterC61#19411
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1550

Complements

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

NH ≅ G/Ntype
1. #21456764C2semidirect
5. #21445280C2semidirect
6. #21445273C2semidirect
7. #21445188C2semidirect
8. #21445183C2semidirect
9. #21444595C2semidirect
10. C2×#16839509C2semidirect
11. C61×#1550C2semidirect
117. #11758222Q16semidirect
237. #1550D61semidirect
281. C61#19411semidirect

Maximal subgroups

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

subgroupindex
#214567642
#214556352
#214556012
#214549282
#214452802
#214452732
#214451882
#214451832
#214445952
C2×#168395092
C61×#15502
#1941161