Finite Group Databasebeta

← #30601624

Subgroups — #30601624

Order 15616.

698 conjugacy classes of subgroups; only the 134 normal subgroups are shown.

normal subgroups only

This group has 698 conjugacy classes of subgroups — too many to draw. Only its 134 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#21461736C211yesmaximal, characteristic, complemented
2#21461533C211yesmaximal, characteristic, complemented
3C2×#16840610C212yesmaximal, complemented
4C2×D4⋊#92420C212yesmaximal, complemented
5#21448383C211yesmaximal, characteristic
6C2×C4⋊#92420C2212yes—
7C2×C4×#92420C2212yes—
8#16840628—22no—
9#16840628—22no—
10#16840610—24no—
11D4⋊#92420—24no—
12C2×#14519773C411yescharacteristic
13C2×D4×Dic61C2211yescharacteristic
14C2×#14519739C2211yescharacteristic
15C2×#14519737—22no—
16#16839664C411yescharacteristic
17#16839662—22no—
18#16839631C411yescharacteristic
19C42×Dic61C2211yescharacteristic
20C61×#390C411yescharacteristic
21C22×#92420—22no—
22C22×#92420—22no—
23C4⋊#92420—22no—
24C4⋊#92420—22no—
25C4×#92420—22no—
26C4×#92420D414yescomplemented
27C22×D244C2×C411yescharacteristic
28C22×C22⋊D61—22no—
29C23×Dic61—21no—
30C22×C4×D61—21no—
31#14519773—22no—
32#14519773C814yescomplemented
33D4×Dic61—22no—
34D4×Dic61—22no—
35#14519739—22no—
36#14519739—22no—
37#14519737—48no—
38C2×C4⋊Dic61C2×C411yescharacteristic
39C2×#11758196—22no—
40C2×C22⋊Dic61—21no—
41C2×#11758198—22no—
42C2×#11758198—21no—
43C2×C4×Dic61C2×C411yescharacteristic
44C2×C4×Dic61D412yes—
45C2×C4×Dic61D412yes—
46C2×C4×Dic61C2311yescharacteristic
47C22×D4×C61C2×C411yescharacteristic
48C2×C4⋊C4×C61C2×C411yescharacteristic
49C2×C22⋊C4×C61—22no—
50C2×C42×C61C2×C411yescharacteristic
51C24×C61—21no—
52C23×D61—21no—
53C2×D4×C61—22no—
54C2×D4×C61—21no—
55C2×D4×C61—22no—
56C2×D4×C61C2×C812yes—
57C22×C4×C61—21no—
58C22×C4×C61C22⋊C412yes—
59C22×C4×C61C22×C411yescharacteristic
60C2×C22⋊D61—28no—
61C2×C22⋊D61—44no—
62C2×C22⋊D61—44no—
63C22×Dic61—21no—
64C22×Dic61—41no—
65C22×Dic61—41no—
66C22×Dic61—44no—
67C22×Dic61C22⋊C412yes—
68C22×Dic61C2×D411yescharacteristic
69C22×Dic61C2×D411yescharacteristic
70C2×D244—22no—
71C2×D244—22no—
72C2×D244—21no—
73C2×D244C2×C812yes—
74C2×C4×D61—41no—
75C2×C4×D61—41no—
76C2×C4×D61—44no—
77C2×#92420—24no—
78C2×#92420—44no—
79C2×#92420—42no—
80C2×#92420—44no—
81C4⋊C4×C61—22no—
82C22⋊C4×C61—44no—
83C42×C61—22no—
84C22⋊Dic61—42no—
85#11758198—44no—
86#11758198—42no—
87C4⋊Dic61—22no—
88#11758196—44no—
89C4×Dic61—24no—
90C4×Dic61—42no—
91C4×Dic61C2×C812yes—
92C4×Dic61M1612yes—
93C4×Dic61C2×D412yes—
94C4×Dic61#4112yes—
95C23×C61—22no—
96C23×C61—41no—
97C23×C61—41no—
98C23×C61—42no—
99C23×C61—42no—
100C23×C61C2×C22⋊C411yescharacteristic
101C22×D61—22no—
102C22×D61—41no—
103C22×D61—41no—
104C22×D61—42no—
105C22×D61—42no—
106D4×C61—24no—
107D4×C61—42no—
108C2×C4×C61—22no—
109C2×C4×C61—22no—
110C2×C4×C61—22no—
111C2×C4×C61—41no—
112C2×C4×C61—42no—
113C2×C4×C61C4×D412yes—
114C2×C4×C61C22×C811yescharacteristic
115C2×C4×C61C2×M1611yescharacteristic
116C2×C4×C61#13112yes—
117C22⋊D61—416no—
118C2×Dic61—22no—
119C2×Dic61—22no—
120C2×Dic61—22no—
121C2×Dic61—41no—
122C2×Dic61—42no—
123C2×Dic61—82no—
124C2×Dic61—82no—
125C2×Dic61C22⋊C814yes—
126C2×Dic61#12112yes—
127C2×Dic61#12411yescharacteristic
128C2×Dic61#12711yescharacteristic
129C2×Dic61AGL(1,ℤ/8)14yes—
130D244—24no—
131D244—42no—
132C4×D61—82no—
133C4×D61—82no—
134#92420—44no—
135#92420—82no—
136#9642—611nomaximal, Sylow
137C22×C61—42no—
138C22×C61—44no—
139C22×C61—81no—
140C22×C61—81no—
141C22×C61—82no—
142C22×C61#39011yescharacteristic
143C22×C61C2×C22⋊C811yescharacteristic
144C22×C61#40811yescharacteristic
145C22×C61#41812yes—
146C22×C61#42112yes—
147C2×D61—42no—
148C2×D61—44no—
149C2×D61—81no—
150C2×D61—81no—
151C2×D61—82no—
152C4×C61—42no—
153C4×C61—81no—
154C4×C61C8×D412yes—
155C4×C61#43612yes—
156Dic61—22no—
157Dic61—24no—
158Dic61—81no—
159Dic61#49012yes—
160Dic61#49612yes—
161C4×#127—611no—
162#1831—611no—
163#1744—611no—
164C2×D4⋊C8—612no—
165C2×D4⋊C8—612no—
166C2×C61—82no—
167C2×C61#147812yes—
168C2×C61#149612yes—
169C2×C61#195411yescharacteristic
170C2×C61#196111yescharacteristic
171C2×C61#196311yescharacteristic
172D61—82no—
173C2×#127—611no—
174C2×C4×D4—611no—
175C2×C4×D4—611no—
176C2×C4×D4—1222no—
177C2×C4⋊C8—612no—
178C2×C4×C8—612no—
179#390—611no—
180#390—1222no—
181#390#692311yescharacteristic
182#384—611no—
183C43—611no—
184#336—1222no—
185#336—1222no—
186D4⋊C8—1224no—
187D4⋊C8—1224no—
188C61#964211yescharacteristic, complemented, Sylow
189C22×D4—611no—
190C22×D4—1222no—
191C22×D4C2×#692311yescharacteristic
192C23×C4—1221no—
193C23×C4—1221no—
194C22×C8—1222no—
195C22×C8—1222no—
196#127—614no—
197#127—1222no—
198C4×D4—1222no—
199C4×D4—1222no—
200C4×D4—1222no—
201C4×D4—1222no—
202C4×D4—2448no—
203C2×C4⋊C4—611no—
204C2×C4⋊C4—1222no—
205C2×C4⋊C4C2×#692311yescharacteristic
206C2×C22⋊C4—22no—
207C2×C22⋊C4—1221no—
208C2×C22⋊C4—1221no—
209C2×C22⋊C4—1222no—
210C2×C42—611no—
211C2×C42—611no—
212C2×C42—612no—
213C2×C42—612no—
214C2×C42C2×#692311yescharacteristic
215C4⋊C8—1222no—
216C4⋊C8—1222no—
217C4×C8—614no—
218C4×C8—1222no—
219C24—21no—
220C24—1221no—
221C2×D4—22no—
222C2×D4—21no—
223C2×D4—22no—
224C2×D4—612no—
225C2×D4—1228no—
226C2×D4—1222no—
227C2×D4—1221no—
228C2×D4—1222no—
229C2×D4—2444no—
230C2×D4—2444no—
231C2×D4D61⋊C812yes—
232C22×C4—21no—
233C22×C4—611no—
234C22×C4—611no—
235C22×C4—612no—
236C22×C4—1221no—
237C22×C4—2441no—
238C22×C4—2441no—
239C22×C4—2444no—
240C22×C4—2441no—
241C22×C4—2441no—
242C22×C4—2444no—
243C22×C4C22⋊#692312yes—
244C22×C4C22×#692311yescharacteristic
245C2×C8—1224no—
246C2×C8—2442no—
247C2×C8—2444no—
248C2×C8—2444no—
249C4⋊C4—22no—
250C4⋊C4—1222no—
251C4⋊C4—2444no—
252C22⋊C4—44no—
253C22⋊C4—2444no—
254C22⋊C4—2442no—
255C22⋊C4—2442no—
256C42—22no—
257C42—612no—
258C42—612no—
259C42—612no—
260C42—612no—
261C42—1224no—
262C42—2442no—
263C23—22no—
264C23—41no—
265C23—41no—
266C23—42no—
267C23—42no—
268C23—1222no—
269C23—2441no—
270C23—2441no—
271C23—2442no—
272C23—2442no—
273C23C2×C22⋊#692311yescharacteristic
274D4—24no—
275D4—42no—
276D4—1224no—
277D4—24416no—
278D4—2442no—
279C2×C4—22no—
280C2×C4—22no—
281C2×C4—22no—
282C2×C4—41no—
283C2×C4—42no—
284C2×C4—611no—
285C2×C4—611no—
286C2×C4—614no—
287C2×C4—612no—
288C2×C4—614no—
289C2×C4—1222no—
290C2×C4—1222no—
291C2×C4—1222no—
292C2×C4—4882no—
293C2×C4—4882no—
294C2×C4—4882no—
295C2×C4—4882no—
296C2×C4—2441no—
297C2×C4—2442no—
298C2×C4D4×#692312yes—
299C2×C4C2×D61⋊C811yescharacteristic
300C2×C4D61⋊M1611yescharacteristic
301C2×C4#1451986412yes—
302C8—4882no—
303C8—2444no—
304C22—44no—
305C22—42no—
306C22—82no—
307C22—81no—
308C22—81no—
309C22—4882no—
310C22—4881no—
311C22—4881no—
312C22—2444no—
313C22—2442no—
314C22#1684071411yescharacteristic
315C22D61⋊(C22⋊C8)11yescharacteristic
316C22#1684073511yescharacteristic
317C22#1684075412yes—
318C22#1684075912yes—
319C4—42no—
320C4—81no—
321C4—612no—
322C4—612no—
323C4—1224no—
324C4—1222no—
325C4—4881no—
326C4#1684078912yes—
327C4#1684079512yes—
328C2—82no—
329C2—4882no—
330C2#2146111912yes—
331C2#2146117812yes—
332C2#2146194211yescharacteristic
333C2#2146195111yescharacteristic
334C2#2146195611yescharacteristic
subgroupG/H
centerC23C2×C22⋊#6923
commutatorC2×C4×C61C22×C8
FrattiniC22×C4C22×#6923
FittingC61×#390C4
socleC23×C61C2×C22⋊C4
prime radicalC2×#14519773C4
cosocleC2×C4×Dic61C23
hypercenter#390#6923
hypocenterC61#9642
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #390

Complements

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

NH ≅ G/Ntype
1. #21461736C2semidirect
2. #21461533C2semidirect
3. C2×#16840610C2semidirect
4. C2×D4⋊#92420C2semidirect
26. C4×#92420D4semidirect
32. #14519773C8semidirect
188. C61#9642semidirect

Maximal subgroups

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

subgroupindex
#214617362
#214615332
C2×#168406102
C2×D4⋊#924202
#214483832
#964261