Finite Group Databasebeta

← #30599713

Subgroups — #30599713

Order 15616.

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

normal subgroups only

This group has 372 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#21462753C211yesmaximal, characteristic
2#21461743C211yesmaximal, characteristic, complemented
3#21461397C211yesmaximal, characteristic, complemented
4#21461157C211yesmaximal, characteristic
5#21460705C211yesmaximal, characteristic, complemented
6#21460651C211yesmaximal, characteristic, complemented
7#21444200C211yesmaximal, characteristic
8Q8×#92420—21no—
9#16840796—21no—
10#16840796—22no—
11#16840781—21no—
12#16840781C2211yescharacteristic
13#16840766—21no—
14#16840766C2211yescharacteristic
15#16840766C2211yescharacteristic
16C4×#11758213C2211yescharacteristic
17#16840638—22no—
18#16840625—21no—
19#16840624—21no—
20Q8⋊#92420—21no—
21#16840610—21no—
22C4×#11758226C2211yescharacteristic
23#16839814C2211yescharacteristic
24C4×C8×D61C2211yescharacteristic
25Q8⋊#92418C411yescharacteristic
26#16839491C411yescharacteristic
27#16839489C411yescharacteristic
28C61×Q8⋊C8C411yescharacteristic
29C2×#11758213—21no—
30C2×#11758213—21no—
31C2×#11758213—22no—
32C2×#11758213D412yes—
33C4⋊#92420—22no—
34C4⋊#92420—21no—
35C4⋊#92420—21no—
36C4⋊#92420—21no—
37C4⋊#92420—21no—
38C4⋊#92420D411yescharacteristic
39C4⋊#92420D411yescharacteristic
40#14519834—22no—
41#14519834—21no—
42#14519832—21no—
43C4×#92420—22no—
44C4×#92420—21no—
45C4×#92420—21no—
46C2×#11758226Q811yescharacteristic
47C2×C8×D61—21no—
48Q8×Dic61D411yescharacteristic
49C4×D244—21no—
50C4×D244C2×C411yescharacteristic
51#14519739—21no—
52#14519739D411yescharacteristic
53#14519735—21no—
54#14519735Q811yescharacteristic
55C42×D61—21no—
56C42×D61C2311yescharacteristic
57C4⋊#92418C2×C411yescharacteristic
58#14519689—21no—
59C4×#92418C2×C411yescharacteristic
60C8×Dic61—21no—
61C8×Dic61—21no—
62C4×Q8×C61C2×C411yescharacteristic
63C4⋊C8×C61C2×C411yescharacteristic
64C4×C8×C61C2×C411yescharacteristic
65C2×Q8×C61C4⋊C411yescharacteristic
66#11758226—22no—
67#11758226—41no—
68C2×D244—21no—
69C2×D244C4⋊C411yescharacteristic
70C2×C4×D61—21no—
71C2×C4×D61—21no—
72C2×C4×D61—21no—
73C2×C4×D61C2×D411yescharacteristic
74C2×C4×D61C2×Q811yescharacteristic
75C2×C4×D61#4111yescharacteristic
76C2×#92420—21no—
77C2×#92420—22no—
78C2×#92420—21no—
79C2×#92420—41no—
80C2×#92420—41no—
81#11758213—22no—
82#11758213—41no—
83#11758213—42no—
84#11758213—42no—
85#11758213—82no—
86C2×C8×C61—21no—
87C2×C8×C61—21no—
88C4⋊C4×C61—21no—
89C4⋊C4×C61C4⋊C411yescharacteristic
90C42×C61—21no—
91C42×C61C22×C411yescharacteristic
92#11758198—21no—
93#11758198—21no—
94#11758198—21no—
95C4⋊Dic61—21no—
96C4⋊Dic61C4⋊C411yescharacteristic
97C4×Dic61—21no—
98C4×Dic61—21no—
99C4×Dic61C2×D411yescharacteristic
100C4×Dic61C2×D411yescharacteristic
101C4×Dic61#4111yescharacteristic
102C2×#92418—21no—
103C2×#92418—21no—
104C8×D61—41no—
105C8×D61—41no—
106C22×D61—21no—
107C22×D61C22⋊Q811yescharacteristic
108Q8×C61—21no—
109Q8×C61#10812yes—
110C2×C4×C61—21no—
111C2×C4×C61—21no—
112C2×C4×C61C2×C4⋊C411yescharacteristic
113C2×C4×C61C4×D411yescharacteristic
114C2×C4×C61C4×D411yescharacteristic
115C2×Dic61—21no—
116C2×Dic61#12011yescharacteristic
117C2×Dic61C22⋊Q811yescharacteristic
118C2×Dic61#12311yescharacteristic
119D244—21no—
120D244—21no—
121D244—42no—
122C4×D61—21no—
123C4×D61—21no—
124C4×D61—21no—
125C4×D61—21no—
126C4×D61—21no—
127C4×D61—21no—
128C4×D61—42no—
129C4×D61—41no—
130C4×D61—41no—
131C4×D61—41no—
132C4×D61—41no—
133#92420—42no—
134#92420—41no—
135#92420—42no—
136#92420—41no—
137#92420—81no—
138C8×C61—41no—
139C8×C61—41no—
140#92418—41no—
141#92418—41no—
142#8901—611nomaximal, Sylow
143C22×C61#38511yescharacteristic
144C2×D61—21no—
145C2×D61—21no—
146C2×D61—21no—
147C2×D61—41no—
148C2×D61—41no—
149C2×D61—41no—
150C4×C61—22no—
151C4×C61—21no—
152C4×C61—41no—
153C4×C61C2×#10811yescharacteristic
154C4×C61#43011yescharacteristic
155C4×C61#44311yescharacteristic
156C4×C61#44411yescharacteristic
157Dic61—21no—
158Dic61—21no—
159Dic61—41no—
160Dic61#45411yescharacteristic
161Dic61#45511yescharacteristic
162#2991—611no—
163#1837—611no—
164#1574—611no—
165#1489—611no—
166#1476—611no—
167#1299—611no—
168#1278—611no—
169C2×C61#185511yescharacteristic
170C2×C61#186011yescharacteristic
171C2×C61#186611yescharacteristic
172D61—41no—
173D61—81no—
174C4×#41—611no—
175C8×Q8—1221no—
176#436—1221no—
177#436—1222no—
178#431—611no—
179#431—611no—
180#431—1221no—
181#423—611no—
182#423—611no—
183#423—1221no—
184C4×M16—611no—
185C2×C4×C8—611no—
186#341—1222no—
187#335—1221no—
188#334—1221no—
189Q8⋊C8—611no—
190Q8⋊C8—1221no—
191Q8⋊C8#692311yescharacteristic
192D4⋊C8—611no—
193D4⋊C8—611no—
194D4⋊C8—1221no—
195C61#890111yescharacteristic, complemented, Sylow
196C2×#41—611no—
197C2×M16—612no—
198C2×M16—1222no—
199C2×M16—1221no—
200C2×M16—1221no—
201C22×C8—1221no—
202C4×Q8—611no—
203C4×Q8C2×#692311yescharacteristic
204C4×D4—611no—
205C4×D4—611no—
206C4×D4—1221no—
207C4×D4—1221no—
208#117—611no—
209#117—1221no—
210C2×C42—611no—
211C2×C42—1221no—
212C4⋊C8—611no—
213C4⋊C8—611no—
214C4⋊C8—611no—
215C4⋊C8—1221no—
216C4⋊C8—1221no—
217C4⋊C8—1221no—
218C4⋊C8—1221no—
219C4⋊C8—1222no—
220C4⋊C8C2×#692311yescharacteristic
221C22⋊C8—1221no—
222C22⋊C8—1221no—
223C22⋊C8—1222no—
224#97—1221no—
225C4×C8—611no—
226C4×C8—1221no—
227C4×C8—1221no—
228C4×C8—1221no—
229C4×C8—1221no—
230C4×C8—1222no—
231C4×C8C2×#692311yescharacteristic
232#41—1222no—
233#41—2441no—
234C2×Q8C4⋊#692311yescharacteristic
235C2×D4—611no—
236C2×D4—1221no—
237C22×C4—611no—
238C22×C4—611no—
239C22×C4—611no—
240C22×C4—1221no—
241C22×C4—1221no—
242C22×C4—1221no—
243M16—1222no—
244M16—4882no—
245M16—2442no—
246M16—2441no—
247M16—2442no—
248C2×C8—21no—
249C2×C8—21no—
250C2×C8—1221no—
251C2×C8—1221no—
252C2×C8—1222no—
253C2×C8—1221no—
254C2×C8—1221no—
255C2×C8—2441no—
256C2×C8—2441no—
257C2×C8—2441no—
258C2×C8—2441no—
259C4⋊C4—21no—
260C4⋊C4—611no—
261C4⋊C4—1221no—
262C4⋊C4C4⋊#692311yescharacteristic
263C22⋊C4—1221no—
264C22⋊C4—1221no—
265C22⋊C4—1221no—
266C42—21no—
267C42—611no—
268C42—611no—
269C42—611no—
270C42—1221no—
271C42—1221no—
272C42C22×#692311yescharacteristic
273C23—611no—
274C23—1221no—
275Q8—21no—
276Q8#1451984812yes—
277D4—1221no—
278D4—1221no—
279D4—2442no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—611no—
283C2×C4—611no—
284C2×C4—611no—
285C2×C4—1221no—
286C2×C4—1221no—
287C2×C4—1221no—
288C2×C4—1221no—
289C2×C4—1221no—
290C2×C4—1221no—
291C2×C4—1221no—
292C2×C4—2442no—
293C2×C4—2441no—
294C2×C4—2441no—
295C2×C4—2441no—
296C2×C4—2441no—
297C2×C4C2×C4⋊#692311yescharacteristic
298C2×C4D4×#692311yescharacteristic
299C2×C4D4×#692311yescharacteristic
300C8—41no—
301C8—41no—
302C8—4881no—
303C8—2441no—
304C8—2441no—
305C8—2441no—
306C8—2442no—
307C8—2442no—
308C8—2441no—
309C22—1221no—
310C22—1221no—
311C22—1221no—
312C22—2441no—
313C22—2441no—
314C22—2441no—
315C22#1684070911yescharacteristic
316C4—22no—
317C4—21no—
318C4—41no—
319C4—611no—
320C4—611no—
321C4—1221no—
322C4—1221no—
323C4—2441no—
324C4C2×#1451984811yescharacteristic
325C4#1684077711yescharacteristic
326C4#1684080511yescharacteristic
327C4#1684080811yescharacteristic
328C2—4881no—
329C2—2441no—
330C2#2146177111yescharacteristic
331C2#2146177911yescharacteristic
332C2#2146178911yescharacteristic
subgroupG/H
centerC22#16840709
commutatorC42×C61C22×C4
FrattiniC42C22×#6923
FittingC61×Q8⋊C8C4
socleC22×C61#385
prime radicalC2×#11758226Q8
cosocleC42×D61C23
hypercenterQ8⋊C8#6923
hypocenterC61#8901
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ Q8⋊C8

Complements

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

NH ≅ G/Ntype
2. #21461743C2semidirect
3. #21461397C2semidirect
5. #21460705C2semidirect
6. #21460651C2semidirect
195. C61#8901semidirect

Maximal subgroups

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

subgroupindex
#214627532
#214617432
#214613972
#214611572
#214607052
#214606512
#214442002
#890161