Finite Group Databasebeta

← #30601643

Subgroups — #30601643

Order 15616.

726 conjugacy classes of subgroups; only the 126 normal subgroups are shown.

normal subgroups only

This group has 726 conjugacy classes of subgroups — too many to draw. Only its 126 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#21461734C211yesmaximal, characteristic, complemented
2C4×#14519832C211yesmaximal, characteristic, complemented
3C2×#16840610C212yesmaximal, complemented
4C2×D4⋊#92420C212yesmaximal, complemented
5#21448383C211yesmaximal, characteristic
6C2×C4⋊#92420C2212yes—
7C2×#14519832—22no—
8C2×C4×#92420C2212yes—
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—
25#14519832—28no—
26C4×#92420—22no—
27C4×#92420D414yescomplemented
28C22×D244C2×C411yescharacteristic
29C22×C22⋊D61—22no—
30C23×Dic61—21no—
31C22×C4×D61—21no—
32#14519773—22no—
33#14519773—22no—
34D4×Dic61—22no—
35D4×Dic61—22no—
36#14519739—22no—
37#14519739—22no—
38#14519737—48no—
39C2×C4⋊Dic61C2×C411yescharacteristic
40C2×#11758196—22no—
41C2×C22⋊Dic61—21no—
42C2×#11758198—22no—
43C2×#11758198—21no—
44C2×C4×Dic61C2×C411yescharacteristic
45C2×C4×Dic61D412yes—
46C2×C4×Dic61D412yes—
47C2×C4×Dic61C2311yescharacteristic
48C22×D4×C61C2×C411yescharacteristic
49C2×C4⋊C4×C61C2×C411yescharacteristic
50C2×C22⋊C4×C61—22no—
51C2×C42×C61C2×C411yescharacteristic
52C24×C61—21no—
53C23×D61—21no—
54C2×D4×C61—22no—
55C2×D4×C61—21no—
56C2×D4×C61—22no—
57C2×D4×C61M1612yes—
58C22×C4×C61—21no—
59C22×C4×C61C22⋊C412yes—
60C22×C4×C61C22×C411yescharacteristic
61C2×C22⋊D61—28no—
62C2×C22⋊D61—44no—
63C2×C22⋊D61—44no—
64C22×Dic61—21no—
65C22×Dic61—41no—
66C22×Dic61—41no—
67C22×Dic61—44no—
68C22×Dic61C22⋊C412yes—
69C22×Dic61C2×D411yescharacteristic
70C22×Dic61C2×D411yescharacteristic
71C2×D244—22no—
72C2×D244—22no—
73C2×D244—21no—
74C2×D244M1612yes—
75C2×C4×D61—41no—
76C2×C4×D61—41no—
77C2×C4×D61—44no—
78C2×#92420—28no—
79C2×#92420—24no—
80C2×#92420—42no—
81C2×#92420—44no—
82C4⋊C4×C61—22no—
83C22⋊C4×C61—44no—
84C42×C61—22no—
85C22⋊Dic61—42no—
86#11758198—44no—
87#11758198—42no—
88C4⋊Dic61—22no—
89#11758196—44no—
90C4×Dic61—21no—
91C4×Dic61—21no—
92C4×Dic61—24no—
93C4×Dic61—24no—
94C4×Dic61C2×D412yes—
95C4×Dic61#4112yes—
96C23×C61—22no—
97C23×C61—41no—
98C23×C61—41no—
99C23×C61—42no—
100C23×C61—42no—
101C23×C61C2×C22⋊C411yescharacteristic
102C22×D61—22no—
103C22×D61—41no—
104C22×D61—41no—
105C22×D61—42no—
106C22×D61—42no—
107D4×C61—24no—
108D4×C61—42no—
109C2×C4×C61—22no—
110C2×C4×C61—22no—
111C2×C4×C61—22no—
112C2×C4×C61—41no—
113C2×C4×C61—42no—
114C2×C4×C61C4×D412yes—
115C2×C4×C61C2×M1611yescharacteristic
116C2×C4×C61C2×M1611yescharacteristic
117C2×C4×C61#13112yes—
118C22⋊D61—416no—
119C2×Dic61—22no—
120C2×Dic61—22no—
121C2×Dic61—22no—
122C2×Dic61—41no—
123C2×Dic61—42no—
124C2×Dic61—82no—
125C2×Dic61—82no—
126C2×Dic61#10414yes—
127C2×Dic61#12112yes—
128C2×Dic61#12411yescharacteristic
129C2×Dic61#12711yescharacteristic
130C2×Dic61AGL(1,ℤ/8)14yes—
131D244—24no—
132D244—42no—
133C4×D61—82no—
134C4×D61—82no—
135#92420—28no—
136#92420—82no—
137#9648—611nomaximal, Sylow
138C22×C61—42no—
139C22×C61—44no—
140C22×C61—81no—
141C22×C61—81no—
142C22×C61—82no—
143C22×C61#39011yescharacteristic
144C22×C61#40711yescharacteristic
145C22×C61#40811yescharacteristic
146C22×C61#41812yes—
147C22×C61C2×#10412yes—
148C2×D61—42no—
149C2×D61—44no—
150C2×D61—81no—
151C2×D61—81no—
152C2×D61—82no—
153C4×C61—42no—
154C4×C61—81no—
155C4×C61#43512yes—
156C4×C61#43512yes—
157Dic61—22no—
158Dic61—24no—
159Dic61—81no—
160Dic61#49012yes—
161Dic61#49612yes—
162C4×#127—611no—
163#1831—611no—
164C4×#97—611no—
165C2×D4⋊C8—612no—
166C2×D4⋊C8—612no—
167C2×C61—82no—
168C2×C61#148412yes—
169C2×C61#149012yes—
170C2×C61#195511yescharacteristic
171C2×C61#196111yescharacteristic
172C2×C61#196211yescharacteristic
173D61—82no—
174C2×#127—611no—
175C2×C4×D4—611no—
176C2×C4×D4—611no—
177C2×C4×D4—1222no—
178C2×C4⋊C8—612no—
179C2×#97—1222no—
180C2×C4×C8—612no—
181#390—611no—
182#390—1222no—
183#390#692311yescharacteristic
184#384—611no—
185C43—611no—
186#336—1222no—
187D4⋊C8—1224no—
188D4⋊C8—1224no—
189C61#964811yescharacteristic, complemented, Sylow
190C22×D4—611no—
191C22×D4—1222no—
192C22×D4C2×#692311yescharacteristic
193C23×C4—1221no—
194C23×C4—1221no—
195C22×C8—1222no—
196C22×C8—1222no—
197#127—1222no—
198#127—1222no—
199C4×D4—1222no—
200C4×D4—1222no—
201C4×D4—1222no—
202C4×D4—1222no—
203C4×D4—2448no—
204C2×C4⋊C4—611no—
205C2×C4⋊C4—1222no—
206C2×C4⋊C4C2×#692311yescharacteristic
207C2×C22⋊C4—22no—
208C2×C22⋊C4—1221no—
209C2×C22⋊C4—1221no—
210C2×C22⋊C4—1222no—
211C2×C42—611no—
212C2×C42—612no—
213C2×C42—612no—
214C2×C42—611no—
215C2×C42C2×#692311yescharacteristic
216C4⋊C8—1222no—
217C4⋊C8—1222no—
218#97—1228no—
219C4×C8—614no—
220C4×C8—1222no—
221C24—21no—
222C24—1221no—
223C2×D4—22no—
224C2×D4—21no—
225C2×D4—22no—
226C2×D4—612no—
227C2×D4—1228no—
228C2×D4—1222no—
229C2×D4—1221no—
230C2×D4—1222no—
231C2×D4—2444no—
232C2×D4—2444no—
233C2×D4#1175821312yes—
234C22×C4—21no—
235C22×C4—611no—
236C22×C4—611no—
237C22×C4—612no—
238C22×C4—1221no—
239C22×C4—2441no—
240C22×C4—2441no—
241C22×C4—2444no—
242C22×C4—2441no—
243C22×C4—2441no—
244C22×C4—2444no—
245C22×C4C22⋊#692312yes—
246C22×C4C22×#692311yescharacteristic
247C2×C8—1224no—
248C2×C8—1228no—
249C2×C8—2442no—
250C2×C8—2444no—
251C4⋊C4—22no—
252C4⋊C4—1222no—
253C4⋊C4—2444no—
254C22⋊C4—44no—
255C22⋊C4—2444no—
256C22⋊C4—2442no—
257C22⋊C4—2442no—
258C42—22no—
259C42—612no—
260C42—612no—
261C42—1224no—
262C42—1224no—
263C42—1221no—
264C42—1221no—
265C23—22no—
266C23—41no—
267C23—41no—
268C23—42no—
269C23—42no—
270C23—1222no—
271C23—2441no—
272C23—2441no—
273C23—2442no—
274C23—2442no—
275C23C2×C22⋊#692311yescharacteristic
276D4—24no—
277D4—42no—
278D4—1224no—
279D4—24416no—
280D4—2442no—
281C2×C4—22no—
282C2×C4—22no—
283C2×C4—22no—
284C2×C4—41no—
285C2×C4—42no—
286C2×C4—611no—
287C2×C4—611no—
288C2×C4—614no—
289C2×C4—612no—
290C2×C4—614no—
291C2×C4—1222no—
292C2×C4—1222no—
293C2×C4—1222no—
294C2×C4—4882no—
295C2×C4—4882no—
296C2×C4—4882no—
297C2×C4—4882no—
298C2×C4—2441no—
299C2×C4—2442no—
300C2×C4D4×#692312yes—
301C2×C4D61⋊M1611yescharacteristic
302C2×C4C2×#1175821311yescharacteristic
303C2×C4#1451986412yes—
304C8—1228no—
305C8—4882no—
306C22—44no—
307C22—42no—
308C22—82no—
309C22—81no—
310C22—81no—
311C22—4882no—
312C22—4881no—
313C22—4881no—
314C22—2444no—
315C22—2442no—
316C22#1684071411yescharacteristic
317C22#1684073311yescharacteristic
318C22#1684073511yescharacteristic
319C22#1684075412yes—
320C22D61⋊#10412yes—
321C4—42no—
322C4—81no—
323C4—612no—
324C4—612no—
325C4—1224no—
326C4—1222no—
327C4—4881no—
328C4#1684079112yes—
329C4#1684079412yes—
330C2—82no—
331C2—4882no—
332C2#2146113812yes—
333C2#2146115912yes—
334C2#2146194411yescharacteristic
335C2#2146195111yescharacteristic
336C2#2146195411yescharacteristic
subgroupG/H
centerC23C2×C22⋊#6923
commutatorC22×C4×C61C22×C4
FrattiniC22×C4C22×#6923
FittingC61×#390C4
socleC23×C61C2×C22⋊C4
prime radicalC2×#14519773C4
cosocleC2×C4×Dic61C23
hypercenter#390#6923
hypocenterC61#9648
LCSG ▷ C22×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. #21461734C2semidirect
2. C4×#14519832C2semidirect
3. C2×#16840610C2semidirect
4. C2×D4⋊#92420C2semidirect
27. C4×#92420D4semidirect
189. C61#9648semidirect

Maximal subgroups

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

subgroupindex
#214617342
C4×#145198322
C2×#168406102
C2×D4⋊#924202
#214483832
#964861