Finite Group Databasebeta

← #30607278

Subgroups — #30607278

Order 15616.

426 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 426 conjugacy classes of subgroups — too many to draw. Only its 98 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
1D61⋊(C22⋊C16)C211yesmaximal, characteristic
2#21461696C211yesmaximal, characteristic
3#21460876C212yesmaximal, complemented
4#21460876C212yesmaximal, complemented
5#427×D61C211yesmaximal, characteristic
6C2×D61⋊C16C2212yes—
7C2×#14519846C2212yes—
8#16840704—21no—
9#16840657—21no—
10C22⋊#11758187—21no—
11#16840636—21no—
12C22×C8×D61C2211yescharacteristic
13#117×D61—21no—
14#16839800C411yescharacteristic, complemented
15#16839799C411yescharacteristic, complemented
16#16839799C411yescharacteristic, complemented
17#16839798—21no—
18#16839797—21no—
19#107×D61C2211yescharacteristic
20#107×D61C2211yescharacteristic
21#106×D61—21no—
22C61×#427C411yescharacteristic, complemented
23C2×C4⋊#6923—22no—
24C2×C22⋊#6923—41no—
25D61⋊C16—24no—
26C2×#11758187—22no—
27#14519846—24no—
28(C2×#6923)⋊C4—41no—
29C22×C4×D61D411yescharacteristic
30C22×#92418D411yescharacteristic
31C2×C8×D61—24no—
32C2×C8×D61C2×C411yescharacteristic
33C2×C8×D61D411yescharacteristic
34C2×C8×D61D411yescharacteristic
35C2×C8×D61C2311yescharacteristic
36#14519736—21no—
37#14519735—21no—
38#14519735—21no—
39#14519734—41no—
40#14519733—41no—
41C4⋊C4×D61—21no—
42C4⋊C4×D61—21no—
43C22⋊C4×D61—41no—
44C42×D61—41no—
45#14519711C2×C411yescharacteristic
46#14519711C2×C411yescharacteristic
47#14519710—21no—
48#14519710—21no—
49#14519709—21no—
50#14519708—21no—
51#14519708—21no—
52C22×C8×C61C2×C411yescharacteristic
53#117×C61—21no—
54#107×C61C2×C411yescharacteristic
55#107×C61C2×C411yescharacteristic
56#106×C61—21no—
57C23×D61D811yescharacteristic
58C22×#6923—42no—
59C22×C4×C61C22⋊C411yescharacteristic
60C22×Dic61SD1611yescharacteristic
61C2×C4×D61—22no—
62C2×C4×D61—22no—
63C2×C4×D61—41no—
64C2×C4×D61—41no—
65C2×C4×D61C22⋊C411yescharacteristic
66C2×C4×D61D811yescharacteristic
67C2×C4×D61SD1611yescharacteristic
68C2×C4×D61C2×D411yescharacteristic
69C22⋊#6923—82no—
70C4⋊#6923—42no—
71C4⋊#6923—42no—
72C2×C8×C61—22no—
73C2×C8×C61C22⋊C411yescharacteristic
74C2×C8×C61C22×C411yescharacteristic
75C4⋊C4×C61—21no—
76C4⋊C4×C61—21no—
77C22⋊C4×C61—41no—
78C42×C61—41no—
79C22⋊Dic61—41no—
80#11758198—41no—
81#11758198—41no—
82C4⋊Dic61—21no—
83C4⋊Dic61—21no—
84#11758196—41no—
85#11758196—41no—
86C4×Dic61—41no—
87C4×Dic61—41no—
88C4×Dic61—41no—
89C2×#92418—22no—
90C2×#92418C22⋊C411yescharacteristic
91C2×#92418C2×D411yescharacteristic
92C8×D61—22no—
93C8×D61—22no—
94C8×D61—22no—
95C8×D61#4114yes—
96#11758187—42no—
97C23×C61#10211yescharacteristic
98C22×D61—22no—
99C22×D61—22no—
100C22×D61—21no—
101C22×D61—21no—
102C22×D61#10211yescharacteristic
103C22×D61C2×D811yescharacteristic
104C2×#6923—82no—
105C2×#6923—82no—
106C2×#6923—82no—
107C2×C4×C61—21no—
108C2×C4×C61—21no—
109C2×C4×C61—41no—
110C2×C4×C61—41no—
111C2×C4×C61#10211yescharacteristic
112C2×C4×C61C2×C22⋊C411yescharacteristic
113C2×Dic61—22no—
114C2×Dic61—41no—
115C2×Dic61—41no—
116C2×Dic61#10211yescharacteristic
117C2×Dic61C2×SD1611yescharacteristic
118C4×D61—22no—
119C4×D61—21no—
120C4×D61—21no—
121C4×D61—22no—
122C4×D61—82no—
123C4×D61—81no—
124C4×D61—81no—
125C4×D61#12312yes—
126C4×D61#13712yes—
127C8×C61—21no—
128C8×C61#11712yes—
129#92418—21no—
130#92418#12312yes—
131#12868—611nomaximal, Sylow
132C22×C61—21no—
133C22×C61—21no—
134C22×C61—21no—
135C22×C61C2×#10211yescharacteristic
136C2×D61—21no—
137C2×D61—21no—
138C2×D61—21no—
139C2×D61—42no—
140C2×D61—42no—
141C2×D61#48012yes—
142C2×D61#50812yes—
143C2×D61#50911yescharacteristic
144C2×D61#51011yescharacteristic
145#6923—162no—
146C4×C61—21no—
147C4×C61—81no—
148C4×C61—81no—
149C4×C61#38111yescharacteristic
150C4×C61#41911yescharacteristic
151Dic61—21no—
152Dic61—81no—
153Dic61—81no—
154Dic61#48312yes—
155C2×#427—611no—
156C2×C22⋊C16—611no—
157#1808—611no—
158#1380—612no—
159#1380—612no—
160C2×C61—41no—
161C2×C61#182311yescharacteristic
162C2×C61#213811yescharacteristic
163C2×C61#213911yescharacteristic
164D61—41no—
165D61#223412yes—
166D61#223712yes—
167C23×C8—611no—
168C2×#117—1221no—
169C22×C16—612no—
170#427—611no—
171#427—611no—
172#427—611no—
173#427—1221no—
174#427—1221no—
175#427#692311yescharacteristic, complemented
176C2×#107—611no—
177C2×#107—611no—
178C2×#107—612no—
179C2×#106—1221no—
180#380—1221no—
181C22⋊C16—1221no—
182C22⋊C16—1221no—
183#340—1221no—
184C61#1286811yescharacteristic, complemented, Sylow
185C23×C4—611no—
186C22×C8—611no—
187C22×C8—611no—
188C22×C8—611no—
189C22×C8—611no—
190C22×C8—611no—
191C22×C8—1224no—
192C22×C8C2×#692311yescharacteristic
193#117—21no—
194#117—1221no—
195#117—1221no—
196#117—1221no—
197#117—2441no—
198#117—2441no—
199C2×C4⋊C4—1221no—
200C2×C4⋊C4—1221no—
201C2×C4⋊C4—1222no—
202C2×C22⋊C4—2441no—
203C2×C22⋊C4—2441no—
204C2×C42—2441no—
205C2×C16—1222no—
206C2×C16—1224no—
207#107—611no—
208#107—611no—
209#107—1221no—
210#107—1221no—
211#107—1224no—
212#107C2×#692311yescharacteristic
213#107C2×#692311yescharacteristic
214#106—21no—
215#106—1221no—
216#106—1221no—
217#106—1221no—
218#95—2441no—
219C24—611no—
220C22×C4—611no—
221C22×C4—611no—
222C22×C4—611no—
223C22×C4—611no—
224C22×C4—611no—
225C22×C4—1222no—
226C22×C4—1222no—
227C22×C4—2441no—
228C22×C4—2441no—
229C22×C4—2442no—
230C22×C4C22⋊#692311yescharacteristic
231C2×C8—22no—
232C2×C8—611no—
233C2×C8—614no—
234C2×C8—611no—
235C2×C8—1222no—
236C2×C8—1222no—
237C2×C8—1222no—
238C2×C8—1222no—
239C2×C8C22⋊#692311yescharacteristic
240C2×C8C22×#692311yescharacteristic
241C4⋊C4—21no—
242C4⋊C4—21no—
243C4⋊C4—1221no—
244C4⋊C4—1221no—
245C4⋊C4—2441no—
246C4⋊C4—2441no—
247C4⋊C4—2442no—
248C4⋊C4—2442no—
249C22⋊C4—41no—
250C22⋊C4—4882no—
251C22⋊C4—2441no—
252C22⋊C4—2441no—
253C22⋊C4—2441no—
254C42—41no—
255C42—2441no—
256C42—2441no—
257C42—2441no—
258C16—2442no—
259C23—611no—
260C23—611no—
261C23—1221no—
262C23—1222no—
263C23—1222no—
264C23—1221no—
265C23#1451983911yescharacteristic
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—41no—
269C2×C4—41no—
270C2×C4—611no—
271C2×C4—612no—
272C2×C4—612no—
273C2×C4—611no—
274C2×C4—1222no—
275C2×C4—1222no—
276C2×C4—1222no—
277C2×C4—1221no—
278C2×C4—1221no—
279C2×C4—4882no—
280C2×C4—4881no—
281C2×C4—4881no—
282C2×C4—4882no—
283C2×C4—4882no—
284C2×C4—4882no—
285C2×C4—2441no—
286C2×C4—2441no—
287C2×C4#1451983911yescharacteristic
288C2×C4C2×C22⋊#692311yescharacteristic
289C8—21no—
290C8—612no—
291C8—1221no—
292C8#1451985612yes—
293C22—21no—
294C22—21no—
295C22—21no—
296C22—612no—
297C22—611no—
298C22—611no—
299C22—612no—
300C22—1221no—
301C22—1221no—
302C22—1221no—
303C22—2442no—
304C22—2442no—
305C22C2×#1451983911yescharacteristic
306C4—21no—
307C4—81no—
308C4—81no—
309C4—612no—
310C4—1221no—
311C4—9762no—
312C4—4881no—
313C4—4881no—
314C4#1684070511yescharacteristic
315C4#1684075511yescharacteristic
316C2—41no—
317C2—612no—
318C2—612no—
319C2—2441no—
320C2#2146172211yescharacteristic
321C2#2146212611yescharacteristic
322C2#2146212711yescharacteristic
subgroupG/H
centerC22C2×#14519839
commutatorC2×C8×C61C22×C4
FrattiniC2×C8C22×#6923
FittingC61×#427C4
socleC22×C61C2×#102
prime radicalC23×D61D8
cosocleC2×C8×D61C23
hypercenter#427#6923
hypocenterC61#12868
LCSG ▷ C2×C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ #427

Complements

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

NH ≅ G/Ntype
3. #21460876C2semidirect
4. #21460876C2semidirect
14. #16839800C4semidirect
15. #16839799C4semidirect
16. #16839799C4semidirect
22. C61×#427C4semidirect
175. #427#6923semidirect
184. C61#12868semidirect

Maximal subgroups

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

subgroupindex
D61⋊(C22⋊C16)2
#214616962
#214608762
#427×D612
#1286861