Finite Group Databasebeta

← #30600759

Subgroups — #30600759

Order 15616.

410 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 410 conjugacy classes of subgroups — too many to draw. Only its 86 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21462612C211yesmaximal, characteristic
2#21462003C211yesmaximal, characteristic
3#21461333C211yesmaximal, characteristic
4#21461253C211yesmaximal, characteristic
5#21460673C211yesmaximal, characteristic
6#21460644C211yesmaximal, characteristic
7#21456966C211yesmaximal, characteristic
8#16840814—22no—
9#16840813—22no—
10#16840780C2211yescharacteristic
11#16840769—21no—
12#16840767C2211yescharacteristic
13#16840727—21no—
14#16840721C2211yescharacteristic
15#16840706C2211yescharacteristic
16#16840635—21no—
17#16840635—21no—
18#16840622—22no—
19#16840616—22no—
20#16840317C2211yescharacteristic
21C4⋊Q8×D61C2211yescharacteristic
22#16840167C411yescharacteristic, complemented
23#16840165C411yescharacteristic, complemented
24#16840112C411yescharacteristic, complemented
25#16840096—22no—
26#16840055—22no—
27#16839813C2211yescharacteristic
28C61×#487C411yescharacteristic, complemented
29C2×#11758213—21no—
30C2×D61⋊C8D411yescharacteristic
31C2×D61⋊C8D411yescharacteristic
32C2×C4⋊#6923—21no—
33#14519844—22no—
34#14519844—21no—
35#14519844—21no—
36#14519844D412yes—
37C4⋊#92420—22no—
38#14519834—21no—
39#14519832—22no—
40#14519832—21no—
41#14519832—21no—
42C4×#92420—22no—
43C4×#92420—21no—
44(C2×#6923)⋊C4—21no—
45(C2×#6923)⋊C4—21no—
46C2×Q8×D61—21no—
47#14519777C2×C411yescharacteristic
48#14519777C2×C411yescharacteristic
49#14519776—21no—
50#14519776—21no—
51#14519775—21no—
52#14519768—21no—
53#14519757—21no—
54#14519753D412yes—
55Q8×Dic61D412yes—
56#14519742—21no—
57#14519735—21no—
58C4⋊C4×D61—21no—
59C4⋊C4×D61—21no—
60C42×D61C2311yescharacteristic
61#14519711—22no—
62#14519710—22no—
63#14519706—21no—
64#14519701—22no—
65#14519689—21no—
66C4×#92418C2×C411yescharacteristic
67C4⋊Q8×C61C2×C411yescharacteristic
68C4⋊Q8×C61C2×C411yescharacteristic
69#103×C61—22no—
70C4×C8×C61C2×C411yescharacteristic
71C22×#6923—41no—
72C2×Q8×C61—21no—
73C2×Q8×C61C22⋊C412yes—
74Q8×D61—82no—
75C2×C4×D61—41no—
76C2×C4×D61C2×D411yescharacteristic
77C2×C4×D61C2×D411yescharacteristic
78C2×C4×D61#4111yescharacteristic
79C2×Dic122—21no—
80C2×Dic122—21no—
81C2×Dic122—41no—
82C2×#92420—21no—
83C2×#92420—22no—
84C2×#92420—21no—
85C2×#92420—41no—
86C4⋊#6923—41no—
87C4⋊#6923—41no—
88#11758213—41no—
89#11758213—41no—
90D61⋊C8—21no—
91D61⋊C8—21no—
92D61⋊C8—21no—
93D61⋊C8Q1612yes—
94C2×C8×C61—21no—
95C4⋊C4×C61—21no—
96C4⋊C4×C61—21no—
97C4⋊C4×C61—21no—
98C4⋊C4×C61—21no—
99C42×C61C22×C411yescharacteristic
100#11758198—22no—
101C4⋊Dic61—21no—
102C4⋊Dic61—22no—
103C4⋊Dic61—21no—
104C4⋊Dic61—21no—
105C4⋊Dic61C22⋊C412yes—
106#11758196—22no—
107#11758196—41no—
108#11758196—41no—
109C4×Dic61—22no—
110C4×Dic61C2×D411yescharacteristic
111C4×Dic61C2×D411yescharacteristic
112C4×Dic61#4111yescharacteristic
113C2×#92418—21no—
114C22×D61#12111yescharacteristic
115C2×#6923—81no—
116C2×#6923—81no—
117C2×#6923—81no—
118Q8×C61—42no—
119Q8×C61—81no—
120C2×C4×C61—22no—
121C2×C4×C61—41no—
122C2×C4×C61C2×C22⋊C411yescharacteristic
123C2×C4×C61C4×D411yescharacteristic
124C2×C4×C61C4×D411yescharacteristic
125C2×Dic61—22no—
126C2×Dic61—41no—
127C2×Dic61#12011yescharacteristic
128C2×Dic61#12111yescharacteristic
129C2×Dic61#12311yescharacteristic
130C4×D61—41no—
131C4×D61—41no—
132C4×D61—82no—
133C4×D61C2×Q1611yescharacteristic
134C4×D61#13511yescharacteristic
135C4×D61#13711yescharacteristic
136C4×D61#13711yescharacteristic
137Dic122—82no—
138Dic122—81no—
139Dic122—81no—
140#92420—22no—
141#92420—41no—
142#92420—42no—
143#92420—81no—
144C8×C61—41no—
145#92418—41no—
146#9288—611nomaximal, Sylow
147C22×C61#38611yescharacteristic
148C2×D61—21no—
149C2×D61—21no—
150C2×D61#46211yescharacteristic
151C2×D61#46411yescharacteristic
152#6923—161no—
153C4×C61—21no—
154C4×C61—21no—
155C4×C61—42no—
156C4×C61—81no—
157C4×C61C4×Q1611yescharacteristic
158C4×C61#44111yescharacteristic
159Dic61—21no—
160Dic61—21no—
161Dic61—42no—
162Dic61—81no—
163Dic61#45111yescharacteristic
164Dic61#45211yescharacteristic
165#3230—611no—
166#2961—611no—
167#1993—611no—
168#1548—611no—
169#1518—611no—
170#1286—611no—
171#1275—611no—
172C2×C61#187811yescharacteristic
173C2×C61#190311yescharacteristic
174C2×C61#190911yescharacteristic
175D61—21no—
176D61—21no—
177#533—611no—
178C2×C4⋊Q8—611no—
179#500—611no—
180#500—611no—
181#487—611no—
182#487#692311yescharacteristic, complemented
183#484—1222no—
184#477—1222no—
185#446—1222no—
186#446—1222no—
187#431—611no—
188#431—611no—
189#424—1221no—
190#423—611no—
191#405—1221no—
192C2×#97—611no—
193#382—611no—
194#340—1221no—
195#340—1221no—
196#333—1222no—
197#328—1222no—
198C61#928811yescharacteristic, complemented, Sylow
199C22×Q8—1221no—
200C2×M16—1221no—
201C22×C8—611no—
202C22×C8—611no—
203C4⋊Q8—611no—
204C4⋊Q8—611no—
205C4⋊Q8—1221no—
206C4⋊Q8—1221no—
207C4⋊Q8—1221no—
208C4⋊Q8C2×#692311yescharacteristic
209C4⋊Q8C2×#692311yescharacteristic
210#125—1221no—
211C22⋊Q8—612no—
212C22⋊Q8—1221no—
213C4×Q8—612no—
214C4×Q8—1221no—
215#117—1221no—
216C2×C4⋊C4—1221no—
217C2×C4⋊C4—1221no—
218C2×C4⋊C4—1221no—
219C2×C42—611no—
220#107—1222no—
221#107—1222no—
222C4⋊C8—612no—
223C4⋊C8—1221no—
224C4⋊C8—1221no—
225C4⋊C8—1221no—
226C4⋊C8—1222no—
227C4⋊C8—1222no—
228#103—22no—
229#103—1222no—
230C22⋊C8—1221no—
231C22⋊C8—1221no—
232#97—1221no—
233#97—1221no—
234#97—1222no—
235C4×C8—611no—
236C4×C8—1221no—
237C4×C8—1222no—
238C4×C8C2×#692311yescharacteristic
239#95—1221no—
240#95—1221no—
241C2×Q8—21no—
242C2×Q8—1221no—
243C2×Q8—1221no—
244C2×Q8—4882no—
245C2×Q8—2441no—
246C2×Q8C22⋊#692312yes—
247C22×C4—611no—
248C22×C4—611no—
249C22×C4—611no—
250C22×C4—2441no—
251C22×C4—2441no—
252M16—2441no—
253M16—2441no—
254C2×C8—21no—
255C2×C8—612no—
256C2×C8—1221no—
257C2×C8—1221no—
258C2×C8—1222no—
259C2×C8—1221no—
260C2×C8—1221no—
261C2×C8—1221no—
262C2×C8—1221no—
263C2×C8—2441no—
264C4⋊C4—21no—
265C4⋊C4—21no—
266C4⋊C4—21no—
267C4⋊C4—21no—
268C4⋊C4—612no—
269C4⋊C4—1222no—
270C4⋊C4—1222no—
271C4⋊C4—1221no—
272C4⋊C4—1221no—
273C4⋊C4—1221no—
274C4⋊C4—2441no—
275C4⋊C4—2441no—
276C4⋊C4—2441no—
277C4⋊C4—2441no—
278C22⋊C4—1222no—
279C42—611no—
280C42—611no—
281C42—611no—
282C42—1222no—
283C42C22×#692311yescharacteristic
284C23—611no—
285Q8—42no—
286Q8—81no—
287Q8—4881no—
288Q8—4881no—
289Q8—4882no—
290C2×C4—22no—
291C2×C4—41no—
292C2×C4—611no—
293C2×C4—611no—
294C2×C4—611no—
295C2×C4—611no—
296C2×C4—611no—
297C2×C4—611no—
298C2×C4—611no—
299C2×C4—1222no—
300C2×C4—4882no—
301C2×C4—4881no—
302C2×C4—4881no—
303C2×C4—4881no—
304C2×C4—2441no—
305C2×C4—2441no—
306C2×C4—2441no—
307C2×C4C2×C22⋊#692311yescharacteristic
308C2×C4D4×#692311yescharacteristic
309C2×C4D4×#692311yescharacteristic
310C8—41no—
311C8—1222no—
312C8—4881no—
313C8—2441no—
314C8—2441no—
315C8—2442no—
316C22—611no—
317C22—611no—
318C22—1221no—
319C22—1221no—
320C22#1684071011yescharacteristic
321C4—21no—
322C4—21no—
323C4—42no—
324C4—81no—
325C4—611no—
326C4—611no—
327C4—1221no—
328C4—1221no—
329C4—9761no—
330C4—4881no—
331C4—2442no—
332C4Q16×#692311yescharacteristic
333C4#1684080211yescharacteristic
334C2—1221no—
335C2—1221no—
336C2#2146181311yescharacteristic
337C2#2146185111yescharacteristic
338C2#2146186011yescharacteristic
subgroupG/H
centerC22#16840710
commutatorC42×C61C22×C4
FrattiniC42C22×#6923
FittingC61×#487C4
socleC22×C61#386
prime radicalC22×D61#121
cosocleC42×D61C23
hypercenter#487#6923
hypocenterC61#9288
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ #487

Complements

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

NH ≅ G/Ntype
22. #16840167C4semidirect
23. #16840165C4semidirect
24. #16840112C4semidirect
28. C61×#487C4semidirect
182. #487#6923semidirect
198. C61#9288semidirect

Maximal subgroups

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

subgroupindex
#214626122
#214620032
#214613332
#214612532
#214606732
#214606442
#214569662
#928861