Finite Group Databasebeta

← #29576724

Subgroups — #29576724

Order 15616.

454 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 454 conjugacy classes of subgroups — too many to draw. Only its 113 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#21456767C211yesmaximal, characteristic
2#21448568C211yesmaximal, characteristic
3#21446799C211yesmaximal, characteristic
4#21444384C211yesmaximal, characteristic
5#21443679C211yesmaximal, characteristic
6#21443647C211yesmaximal, characteristic
7C61×#1548C211yesmaximal, characteristic
8#16840319C2211yescharacteristic
9#16840141C412yescomplemented
10#16840112C412yescomplemented
11#16839969—22no—
12#16839918—22no—
13#16839815C2211yescharacteristic
14#16839669—22no—
15#16839661—21no—
16#16839648—22no—
17#16839628C2211yescharacteristic
18C4⋊C4×Dic61—21no—
19#16839608C2211yescharacteristic
20#16839523—21no—
21#16839523—21no—
22#16839500—22no—
23C2×C4⋊Q8×C61C2211yescharacteristic
24C61×#431C2211yescharacteristic
25C61×#423C2211yescharacteristic
26C61×#333—22no—
27C2×#11758202—24no—
28#14519777C2×C411yescharacteristic
29#14519777C2×C411yescharacteristic
30#14519769—21no—
31#14519756—21no—
32#14519756—21no—
33C4×Dic122—21no—
34C4×Dic122—21no—
35#14519736—21no—
36C2×C4⋊Dic61—21no—
37C2×C4⋊Dic61—22no—
38C2×C4×Dic61—22no—
39C2×C4×Dic61D411yescharacteristic
40C2×C4×Dic61D411yescharacteristic
41C4⋊#92418—21no—
42#14519701—24no—
43C22⋊#92418—21no—
44C4×#92418C2×C412yes—
45#14519684—22no—
46#14519684—22no—
47#14519684—21no—
48#14519684—21no—
49#14519684—21no—
50#14519684D412yes—
51C22×Q8×C61D412yes—
52C2×M16×C61—21no—
53C4⋊Q8×C61—21no—
54C4⋊Q8×C61—21no—
55C4⋊Q8×C61C2×C412yes—
56C4⋊Q8×C61D412yes—
57C2×C4⋊C4×C61—21no—
58C2×C4⋊C4×C61—21no—
59C2×C42×C61C2311yescharacteristic
60C4⋊C8×C61—21no—
61C4⋊C8×C61—21no—
62C4⋊C8×C61—21no—
63C22⋊C8×C61—21no—
64C4×C8×C61—21no—
65C2×Q8×C61—22no—
66C2×Q8×C61—82no—
67C2×Q8×C61C22⋊C414yes—
68C22×C4×C61—22no—
69C22×C4×C61C2×D411yescharacteristic
70C22×C4×C61C2×D411yescharacteristic
71C22×C4×C61C2×D411yescharacteristic
72C22×Dic61—21no—
73C22×Dic61—21no—
74C22×Dic61—22no—
75C2×Dic122—21no—
76C2×Dic122—21no—
77M16×C61—41no—
78M16×C61—41no—
79C2×C8×C61—41no—
80C2×C8×C61—41no—
81C4⋊C4×C61—22no—
82C4⋊C4×C61—22no—
83C4⋊C4×C61—22no—
84C4⋊C4×C61—22no—
85C42×C61C22×C411yescharacteristic
86C42×C61C2×D411yescharacteristic
87C42×C61#4111yescharacteristic
88C42×C61#4111yescharacteristic
89C22⋊Dic61—41no—
90#11758202—84no—
91C4⋊Dic61—21no—
92C4⋊Dic61—21no—
93C4⋊Dic61—21no—
94C4⋊Dic61—21no—
95C4⋊Dic61—41no—
96C4⋊Dic61—41no—
97C4⋊Dic61—82no—
98#11758196—41no—
99#11758196—41no—
100C4×Dic61—21no—
101C4×Dic61—21no—
102C4×Dic61—21no—
103C4×Dic61—41no—
104C4×Dic61—82no—
105C4×Dic61Q1612yescomplemented
106C2×#92418—22no—
107C23×C61#12011yescharacteristic
108Q8×C61—44no—
109Q8×C61—82no—
110C2×C4×C61—24no—
111C2×C4×C61—41no—
112C2×C4×C61—41no—
113C2×C4×C61—42no—
114C2×C4×C61—42no—
115C2×C4×C61C2×C22⋊C411yescharacteristic
116C2×C4×C61C4×D411yescharacteristic
117C2×C4×C61C4×D411yescharacteristic
118C2×C4×C61#12111yescharacteristic
119C2×C4×C61#12111yescharacteristic
120C2×C4×C61#12311yescharacteristic
121C2×C4×C61C2×Q1611yescharacteristic
122C2×C4×C61#13511yescharacteristic
123C2×C4×C61#13711yescharacteristic
124C2×C4×C61#13711yescharacteristic
125C2×Dic61—22no—
126C2×Dic61—41no—
127C2×Dic61—41no—
128C2×Dic61—42no—
129C2×Dic61—41no—
130C2×Dic61—41no—
131C2×Dic61—42no—
132C2×Dic61—41no—
133C2×Dic61—41no—
134C2×Dic61—41no—
135C2×Dic61—42no—
136Dic122—42no—
137Dic122—81no—
138C8×C61—81no—
139C8×C61—81no—
140#92418—42no—
141#9297—611nomaximal, Sylow
142C22×C61—21no—
143C22×C61—21no—
144C22×C61#38611yescharacteristic
145C22×C61#45111yescharacteristic
146C22×C61#45211yescharacteristic
147C4×C61—21no—
148C4×C61—21no—
149C4×C61—21no—
150C4×C61—21no—
151C4×C61—44no—
152C4×C61C4×Q1611yescharacteristic
153C4×C61#44111yescharacteristic
154C4×C61#46211yescharacteristic
155C4×C61#46411yescharacteristic
156Dic61—42no—
157Dic61—82no—
158Dic61—81no—
159Dic61—81no—
160Dic61—81no—
161#3209—611no—
162#2331—611no—
163#1995—611no—
164#1548D6111yescharacteristic
165#1518—611no—
166#1286—611no—
167#1274—611no—
168C2×C61—21no—
169C2×C61—21no—
170C2×C61#187811yescharacteristic
171C2×C61#190511yescharacteristic
172C2×C61#191111yescharacteristic
173#533—611no—
174C2×C4⋊Q8C2×D6111yescharacteristic
175#494—612no—
176#487—612no—
177#462—1222no—
178#451—1222no—
179#431—611no—
180#431C2×D6111yescharacteristic
181#423C2×D6111yescharacteristic
182#391—1222no—
183#389—1221no—
184#387—1222no—
185#382—611no—
186C4×C4⋊C4—1221no—
187#376—611no—
188#339—1221no—
189#339—1221no—
190#333—22no—
191#328—1222no—
192C61#929711yescharacteristic, complemented, Sylow
193C22×Q8D24412yes—
194C2×Q16—1224no—
195C2×M16—21no—
196C4⋊Q8—21no—
197C4⋊Q8—21no—
198C4⋊Q8—611no—
199C4⋊Q8—611no—
200C4⋊Q8C4×D6112yes—
201C4⋊Q8C22⋊D6112yes—
202#125—1221no—
203C22⋊Q8—1221no—
204C22⋊Q8—1221no—
205C4×Q8—1221no—
206C4×Q8—1221no—
207#117—1221no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—1222no—
211C2×C4⋊C4—1221no—
212C2×C42—611no—
213C2×C42—611no—
214C2×C42—1222no—
215C2×C42C22×D6111yescharacteristic
216C4⋊C8—21no—
217C4⋊C8—21no—
218C4⋊C8—21no—
219C4⋊C8—1221no—
220#103—1224no—
221C22⋊C8—21no—
222C22⋊C8—1221no—
223C4×C8—21no—
224C4×C8—612no—
225#95—612no—
226#95—1222no—
227#95—1222no—
228#95—1221no—
229#95—1221no—
230#95—1221no—
231C2×Q8—22no—
232C2×Q8—82no—
233C2×Q8—1221no—
234C2×Q8—1221no—
235C2×Q8#1175819814yes—
236C22×C4—22no—
237C22×C4—1221no—
238C22×C4—1222no—
239C22×C4—1221no—
240C22×C4C2×D24411yescharacteristic
241C22×C4D4×D6111yescharacteristic
242C22×C4D4×D6111yescharacteristic
243Q16—4884no—
244M16—41no—
245M16—41no—
246C2×C8—41no—
247C2×C8—41no—
248C2×C8—1222no—
249C4⋊C4—22no—
250C4⋊C4—22no—
251C4⋊C4—22no—
252C4⋊C4—22no—
253C4⋊C4—1221no—
254C4⋊C4—1221no—
255C4⋊C4—1221no—
256C4⋊C4—1221no—
257C4⋊C4—4882no—
258C4⋊C4—2441no—
259C4⋊C4—2441no—
260C4⋊C4—2441no—
261C4⋊C4—2441no—
262C22⋊C4—2441no—
263C42—612no—
264C42—1221no—
265C42—1221no—
266C42—1221no—
267C42—4882no—
268C42—2441no—
269C42C2×C4×D6111yescharacteristic
270C42#1175822411yescharacteristic
271C42#1175822411yescharacteristic
272C42C2×C22⋊D6111yescharacteristic
273C23#1451974711yescharacteristic
274Q8—44no—
275Q8—82no—
276Q8—4881no—
277Q8—2442no—
278C2×C4—24no—
279C2×C4—41no—
280C2×C4—41no—
281C2×C4—42no—
282C2×C4—42no—
283C2×C4—1222no—
284C2×C4—2442no—
285C2×C4—2441no—
286C2×C4—2442no—
287C2×C4—2441no—
288C2×C4—2441no—
289C2×C4—2441no—
290C2×C4—2441no—
291C2×C4—2442no—
292C2×C4—2441no—
293C2×C4—2441no—
294C2×C4C2×#1175819811yescharacteristic
295C2×C4#1451973711yescharacteristic
296C2×C4#1451973711yescharacteristic
297C2×C4#1451974911yescharacteristic
298C2×C4#1451974911yescharacteristic
299C2×C4#1451976111yescharacteristic
300C2×C4Q16×D6111yescharacteristic
301C2×C4#1451979811yescharacteristic
302C2×C4#1451980611yescharacteristic
303C2×C4#1451980911yescharacteristic
304C8—81no—
305C8—81no—
306C8—2442no—
307C22—21no—
308C22—21no—
309C22#1683963911yescharacteristic
310C22#1683991911yescharacteristic
311C22#1683992711yescharacteristic
312C4—21no—
313C4—21no—
314C4—21no—
315C4—21no—
316C4—44no—
317C4—4882no—
318C4—4881no—
319C4—4881no—
320C4—4881no—
321C4—2442no—
322C4#1683985411yescharacteristic
323C4#1683986611yescharacteristic
324C4#1683997011yescharacteristic
325C4#1683998211yescharacteristic
326C2—21no—
327C2—21no—
328C2#2144615811yescharacteristic
329C2#2144631111yescharacteristic
330C2#2144634911yescharacteristic
subgroupG/H
centerC22#16839639
commutatorC42×C61C22×C4
FrattiniC2×C42C22×D61
FittingC61×#1548C2
socleC22×C61#386
prime radicalC23×C61#120
cosocleC2×C42×C61C23
hypercenter#1548D61
hypocenterC61#9297
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1548

Complements

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

NH ≅ G/Ntype
9. #16840141C4semidirect
10. #16840112C4semidirect
105. C4×Dic61Q16semidirect
192. C61#9297semidirect

Maximal subgroups

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

subgroupindex
#214567672
#214485682
#214467992
#214443842
#214436792
#214436472
C61×#15482
#929761