Finite Group Databasebeta

← #30599896

Subgroups — #30599896

Order 15616.

722 conjugacy classes of subgroups; only the 112 normal subgroups are shown.

normal subgroups only

This group has 722 conjugacy classes of subgroups — too many to draw. Only its 112 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#21462669C211yesmaximal, characteristic, complemented
2#21461120C212yesmaximal, complemented
3#21461060C211yesmaximal, characteristic, complemented
4#21460647C212yesmaximal, complemented
5#21448381C211yesmaximal, characteristic
6#16840790—24no—
7#16840778C2212yes—
8#16840778C2212yes—
9D61⋊(C22⋊C8)—22no—
10#16840610—24no—
11#16840606—22no—
12C2×#14519774C411yescharacteristic
13C2×#14519739—22no—
14C2×#14519739C2211yescharacteristic
15C2×#14519739C2211yescharacteristic
16#16839663—22no—
17#16839663C411yescharacteristic
18#16839663C411yescharacteristic
19C42×Dic61C2211yescharacteristic
20C61×#384C411yescharacteristic
21C2×D61⋊C8—24no—
22C4⋊#92420—22no—
23C4⋊#92420D414yescomplemented
24#14519834—24no—
25C22⋊#92420—22no—
26C22⋊#92420—22no—
27C4×#92420—22no—
28C4×#92420—22no—
29C22×D244—22no—
30C22×D244C2×C411yescharacteristic
31C22×D244C2×C411yescharacteristic
32C22×C4×D61—21no—
33C22×C4×D61D412yes—
34#14519774—22no—
35#14519774C814yescomplemented
36#14519739—22no—
37#14519739—22no—
38#14519739—22no—
39#14519739—22no—
40#14519739—48no—
41C2×#11758198—22no—
42C2×#11758198—22no—
43C2×#11758198—21no—
44C2×#11758198D412yes—
45C2×C4×Dic61—21no—
46C2×C4×Dic61—21no—
47C2×C4×Dic61C2×C411yescharacteristic
48C2×C4×Dic61C2311yescharacteristic
49C2×C4⋊C4×C61—22no—
50C2×C4⋊C4×C61C2×C411yescharacteristic
51C2×C4⋊C4×C61C2×C411yescharacteristic
52C2×C42×C61C2×C411yescharacteristic
53C23×D61—21no—
54C23×D61C22⋊C412yes—
55C22×C4×C61—21no—
56C22×C4×C61—21no—
57C22×C4×C61C22⋊C412yes—
58C22×C4×C61C22×C411yescharacteristic
59C22×Dic61—21no—
60C22×Dic61C2×D411yescharacteristic
61C22×Dic61C2×D411yescharacteristic
62C2×D244—22no—
63C2×D244—22no—
64C2×D244—22no—
65C2×D244—22no—
66C2×D244—28no—
67C2×D244—21no—
68C2×D244—44no—
69C2×D244—44no—
70C2×D244C2×C812yes—
71C2×D244C2×C812yescomplemented
72C2×D244M1612yes—
73C2×C4×D61—22no—
74C2×C4×D61—22no—
75C2×C4×D61—44no—
76C2×C4×D61—44no—
77C2×C4×D61—41no—
78C2×C4×D61—41no—
79C2×#92420—24no—
80C2×#92420—42no—
81D61⋊C8—84no—
82C4⋊C4×C61—22no—
83C4⋊C4×C61—22no—
84C4⋊C4×C61—44no—
85C42×C61—21no—
86C42×C61C2×C812yescomplemented
87#11758198—24no—
88#11758198—44no—
89#11758198—44no—
90#11758198—42no—
91C4×Dic61—24no—
92C4×Dic61—22no—
93C4×Dic61—42no—
94C4×Dic61C2×D412yes—
95C4×Dic61#4112yes—
96C23×C61C2×C22⋊C411yescharacteristic
97C22×D61—22no—
98C22×D61—22no—
99C22×D61—22no—
100C22×D61—42no—
101C22×D61—42no—
102C22×D61—42no—
103C22×D61—42no—
104C22×D61—41no—
105C22×D61—41no—
106C22×D61C22⋊C814yes—
107C2×C4×C61—22no—
108C2×C4×C61—22no—
109C2×C4×C61—22no—
110C2×C4×C61—22no—
111C2×C4×C61—42no—
112C2×C4×C61—42no—
113C2×C4×C61—41no—
114C2×C4×C61C4×D412yes—
115C2×C4×C61C22×C811yescharacteristic
116C2×C4×C61C2×M1611yescharacteristic
117C2×C4×C61#13112yes—
118C2×Dic61—22no—
119C2×Dic61—22no—
120C2×Dic61—41no—
121C2×Dic61—41no—
122C2×Dic61—41no—
123C2×Dic61#12011yescharacteristic
124C2×Dic61#12112yes—
125C2×Dic61#12311yescharacteristic
126D244—24no—
127D244—24no—
128D244—24no—
129D244—416no—
130D244—42no—
131C4×D61—44no—
132C4×D61—82no—
133C4×D61—82no—
134C4×D61—82no—
135#92420—44no—
136#92420—82no—
137#8959—611nomaximal, Sylow
138C22×C61—21no—
139C22×C61—21no—
140C22×C61#38611yescharacteristic
141C22×C61C2×C22⋊C811yescharacteristic
142C22×C61#40811yescharacteristic
143C2×D61—24no—
144C2×D61—42no—
145C2×D61—42no—
146C2×D61—44no—
147C2×D61—44no—
148C2×D61—42no—
149C2×D61—82no—
150C2×D61—82no—
151C2×D61—81no—
152C2×D61—81no—
153C4×C61—24no—
154C4×C61—42no—
155C4×C61—81no—
156C4×C61C8×D412yes—
157C4×C61#43512yes—
158Dic61—22no—
159Dic61—21no—
160Dic61—42no—
161Dic61#45312yes—
162#2972—611no—
163C4×#127—611no—
164#1478—612no—
165#1460—611no—
166#1276—612no—
167C2×C61—21no—
168C2×C61—21no—
169C2×C61#186911yescharacteristic
170C2×C61#188411yescharacteristic
171C2×C61#188711yescharacteristic
172D61—44no—
173D61—82no—
174C2×#127—611no—
175C2×C4×D4—611no—
176C2×C4×D4—611no—
177C2×C4×D4—1222no—
178C8×D4—1224no—
179#431—612no—
180#431—612no—
181C2×C22⋊C8—1222no—
182#390—611no—
183#390—611no—
184#390—1222no—
185#384#692311yescharacteristic
186C43—611no—
187D4⋊C8—1224no—
188#323—1222no—
189C61#895911yescharacteristic, complemented, Sylow
190C22×D4—611no—
191C22×D4—611no—
192C22×D4—1222no—
193C23×C4—612no—
194C23×C4—1221no—
195C22×C8—1224no—
196#127—614no—
197#127—1222no—
198C4×D4—1222no—
199C4×D4—1222no—
200C4×D4—1222no—
201C4×D4—1222no—
202C4×D4—2448no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4C2×#692311yescharacteristic
205C2×C4⋊C4C2×#692311yescharacteristic
206C2×C22⋊C4—612no—
207C2×C22⋊C4—1221no—
208C2×C22⋊C4—1222no—
209C2×C22⋊C4—1222no—
210C2×C42—611no—
211C2×C42—611no—
212C2×C42—1221no—
213C2×C42—1221no—
214C2×C42C2×#692311yescharacteristic
215C4⋊C8—614no—
216C4⋊C8—1222no—
217C22⋊C8—1222no—
218C22⋊C8—1224no—
219C22⋊C8—1222no—
220C4×C8—1222no—
221C4×C8—1222no—
222C24—612no—
223C24—1221no—
224C2×D4—612no—
225C2×D4—612no—
226C2×D4—612no—
227C2×D4—1228no—
228C2×D4—1222no—
229C2×D4—1222no—
230C2×D4—1221no—
231C2×D4—1222no—
232C2×D4—1222no—
233C2×D4—2444no—
234C2×D4—2444no—
235C22×C4—21no—
236C22×C4—21no—
237C22×C4—611no—
238C22×C4—611no—
239C22×C4—1222no—
240C22×C4—1222no—
241C22×C4—1221no—
242C22×C4—2441no—
243C22×C4—2441no—
244C22×C4—2444no—
245C22×C4—2444no—
246C22×C4C22⋊#692312yes—
247C22×C4C22×#692311yescharacteristic
248C2×C8—1224no—
249C2×C8—4884no—
250C2×C8—2442no—
251C4⋊C4—22no—
252C4⋊C4—22no—
253C4⋊C4—44no—
254C22⋊C4—1224no—
255C22⋊C4—2442no—
256C22⋊C4—2444no—
257C22⋊C4—2444no—
258C42—21no—
259C42—612no—
260C42—612no—
261C42—1224no—
262C42—1222no—
263C42—2442no—
264C42D61⋊C812yescomplemented
265C23—614no—
266C23—1222no—
267C23—1222no—
268C23—1222no—
269C23—2442no—
270C23—2442no—
271C23—2442no—
272C23—2442no—
273C23—2441no—
274C23—2441no—
275C23C2×C22⋊#692311yescharacteristic
276D4—1224no—
277D4—1224no—
278D4—1224no—
279D4—24416no—
280D4—2442no—
281C2×C4—22no—
282C2×C4—22no—
283C2×C4—22no—
284C2×C4—22no—
285C2×C4—42no—
286C2×C4—42no—
287C2×C4—41no—
288C2×C4—611no—
289C2×C4—612no—
290C2×C4—611no—
291C2×C4—1222no—
292C2×C4—1222no—
293C2×C4—4882no—
294C2×C4—4882no—
295C2×C4—4882no—
296C2×C4—2444no—
297C2×C4—2441no—
298C2×C4—2441no—
299C2×C4—2441no—
300C2×C4D4×#692312yes—
301C2×C4C2×D61⋊C811yescharacteristic
302C2×C4D61⋊M1611yescharacteristic
303C2×C4#1451986512yes—
304C8—4882no—
305C8—2444no—
306C22—21no—
307C22—21no—
308C22—1224no—
309C22—4882no—
310C22—4882no—
311C22—4881no—
312C22—4881no—
313C22—2444no—
314C22—2444no—
315C22—2442no—
316C22—2442no—
317C22—2442no—
318C22#1684071011yescharacteristic
319C22D61⋊(C22⋊C8)11yescharacteristic
320C22#1684073711yescharacteristic
321C4—24no—
322C4—42no—
323C4—81no—
324C4—612no—
325C4—1222no—
326C4—1221no—
327C4—2442no—
328C4#1684079012yes—
329C4#1684079212yes—
330C2—21no—
331C2—21no—
332C2—4882no—
333C2—2444no—
334C2#2146179511yescharacteristic
335C2#2146182111yescharacteristic
336C2#2146182611yescharacteristic
subgroupG/H
centerC22#16840710
commutatorC2×C4×C61C22×C8
FrattiniC22×C4C22×#6923
FittingC61×#384C4
socleC22×C61#386
prime radicalC2×#14519774C4
cosocleC2×C4×Dic61C23
hypercenter#384#6923
hypocenterC61#8959
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ #384

Complements

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

NH ≅ G/Ntype
1. #21462669C2semidirect
2. #21461120C2semidirect
3. #21461060C2semidirect
4. #21460647C2semidirect
23. C4⋊#92420D4semidirect
35. #14519774C8semidirect
71. C2×D244C2×C8semidirect
86. C42×C61C2×C8semidirect
189. C61#8959semidirect
264. C42D61⋊C8semidirect

Maximal subgroups

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

subgroupindex
#214626692
#214611202
#214610602
#214606472
#214483812
#895961