Finite Group Databasebeta

← #30599893

Subgroups — #30599893

Order 15616.

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

normal subgroups only

This group has 722 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#21462666C211yesmaximal, characteristic, complemented
2#21461118C212yesmaximal, complemented
3#21461059C211yesmaximal, characteristic, complemented
4#21460647C212yesmaximal, complemented
5#127×Dic61C211yesmaximal, characteristic
6D4×#92420—24no—
7#16840778C2212yes—
8#16840778C2212yes—
9C2×C22⋊#92420—22no—
10D4⋊#92420—24no—
11#16840605—22no—
12C2×D4×Dic61—22no—
13C2×D4×Dic61C2211yescharacteristic
14C2×D4×Dic61C2211yescharacteristic
15#16839665—22no—
16#16839665C411yescharacteristic
17#16839665C411yescharacteristic
18#16839632C411yescharacteristic
19C42×Dic61C2211yescharacteristic
20C2×#127×C61C411yescharacteristic
21C22×#92420—24no—
22C4⋊#92420—22no—
23C4⋊#92420D414yescomplemented
24C22⋊#92420—24no—
25C22⋊#92420—22no—
26C22⋊#92420—22no—
27C4×#92420—22no—
28C4×#92420—22no—
29C23×Dic61—21no—
30C23×Dic61D412yes—
31D4×Dic61—22no—
32D4×Dic61—22no—
33D4×Dic61—22no—
34D4×Dic61—22no—
35D4×Dic61—48no—
36C2×C4⋊Dic61—22no—
37C2×C4⋊Dic61C2×C411yescharacteristic
38C2×C4⋊Dic61C2×C411yescharacteristic
39C2×C22⋊Dic61—22no—
40C2×C22⋊Dic61—22no—
41C2×C22⋊Dic61—21no—
42C2×C22⋊Dic61D412yes—
43C2×C4×Dic61—21no—
44C2×C4×Dic61—21no—
45C2×C4×Dic61C2×C411yescharacteristic
46C2×C4×Dic61C2311yescharacteristic
47C22×D4×C61—22no—
48C22×D4×C61C2×C411yescharacteristic
49C22×D4×C61C2×C411yescharacteristic
50#127×C61—22no—
51#127×C61C814yescomplemented
52C2×C42×C61C2×C411yescharacteristic
53C24×C61—21no—
54C24×C61C22⋊C412yes—
55C2×D4×C61—28no—
56C2×D4×C61—22no—
57C2×D4×C61—22no—
58C2×D4×C61—21no—
59C2×D4×C61—22no—
60C2×D4×C61—22no—
61C2×D4×C61—44no—
62C2×D4×C61—44no—
63C2×D4×C61C2×C812yes—
64C2×D4×C61C2×C812yescomplemented
65C2×D4×C61M1612yes—
66C22×C4×C61—21no—
67C22×C4×C61C22×C411yescharacteristic
68C22×Dic61—21no—
69C22×Dic61—22no—
70C22×Dic61—22no—
71C22×Dic61—21no—
72C22×Dic61—44no—
73C22×Dic61—44no—
74C22×Dic61—41no—
75C22×Dic61—41no—
76C22×Dic61C22⋊C412yes—
77C22×Dic61C2×D411yescharacteristic
78C22×Dic61C2×D411yescharacteristic
79C2×#92420—24no—
80C2×#92420—42no—
81C2×#92420—84no—
82C42×C61—21no—
83C42×C61C2×C812yescomplemented
84C22⋊Dic61—24no—
85C22⋊Dic61—44no—
86C22⋊Dic61—44no—
87C22⋊Dic61—42no—
88C4⋊Dic61—22no—
89C4⋊Dic61—22no—
90C4⋊Dic61—44no—
91C4×Dic61—24no—
92C4×Dic61—22no—
93C4×Dic61—42no—
94C4×Dic61C2×D412yes—
95C4×Dic61#4112yes—
96C23×C61—22no—
97C23×C61—22no—
98C23×C61—22no—
99C23×C61—42no—
100C23×C61—42no—
101C23×C61—42no—
102C23×C61—42no—
103C23×C61—41no—
104C23×C61—41no—
105C23×C61C22⋊C814yes—
106C23×C61C2×C22⋊C411yescharacteristic
107D4×C61—24no—
108D4×C61—24no—
109D4×C61—24no—
110D4×C61—416no—
111D4×C61—42no—
112C2×C4×C61—22no—
113C2×C4×C61—22no—
114C2×C4×C61—22no—
115C2×C4×C61C4×D412yes—
116C2×C4×C61C22×C811yescharacteristic
117C2×C4×C61C2×M1611yescharacteristic
118C2×C4×C61#13112yes—
119C2×Dic61—22no—
120C2×Dic61—22no—
121C2×Dic61—22no—
122C2×Dic61—41no—
123C2×Dic61—41no—
124C2×Dic61—41no—
125C2×Dic61—44no—
126C2×Dic61—42no—
127C2×Dic61—42no—
128C2×Dic61—41no—
129C2×Dic61—82no—
130C2×Dic61—82no—
131C2×Dic61—82no—
132C2×Dic61#12011yescharacteristic
133C2×Dic61#12112yes—
134C2×Dic61#12311yescharacteristic
135#92420—44no—
136#92420—82no—
137#8959—611nomaximal, Sylow
138C22×C61—21no—
139C22×C61—21no—
140C22×C61—24no—
141C22×C61—44no—
142C22×C61—44no—
143C22×C61—42no—
144C22×C61—42no—
145C22×C61—42no—
146C22×C61—82no—
147C22×C61—82no—
148C22×C61—81no—
149C22×C61—81no—
150C22×C61#38611yescharacteristic
151C22×C61C2×C22⋊C811yescharacteristic
152C22×C61#40811yescharacteristic
153C4×C61—24no—
154C4×C61C8×D412yes—
155C4×C61#43512yes—
156Dic61—22no—
157Dic61—21no—
158Dic61—42no—
159Dic61—42no—
160Dic61—81no—
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—44no—
170C2×C61—82no—
171C2×C61#186911yescharacteristic
172C2×C61#188411yescharacteristic
173C2×C61#188711yescharacteristic
174C2×#127#692311yescharacteristic
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—611no—
186C43—611no—
187D4⋊C8—1224no—
188#323—1222no—
189C61#895911yescharacteristic, complemented, Sylow
190C22×D4—22no—
191C22×D4C2×#692311yescharacteristic
192C22×D4C2×#692311yescharacteristic
193C23×C4—612no—
194C23×C4—1221no—
195C22×C8—1224no—
196#127—22no—
197#127#9242014yescomplemented
198C4×D4—1222no—
199C4×D4—1222no—
200C4×D4—1222no—
201C4×D4—1222no—
202C4×D4—2448no—
203C2×C4⋊C4—611no—
204C2×C4⋊C4—611no—
205C2×C4⋊C4—1222no—
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—21no—
223C24C22⋊#692312yes—
224C2×D4—28no—
225C2×D4—22no—
226C2×D4—22no—
227C2×D4—21no—
228C2×D4—22no—
229C2×D4—22no—
230C2×D4—44no—
231C2×D4—44no—
232C2×D4C2×#9242012yes—
233C2×D4C2×#9242012yescomplemented
234C2×D4#1175821712yes—
235C22×C4—21no—
236C22×C4—611no—
237C22×C4—611no—
238C22×C4—612no—
239C22×C4—1222no—
240C22×C4—1222no—
241C22×C4—1221no—
242C22×C4—1221no—
243C22×C4—2441no—
244C22×C4—2441no—
245C22×C4—2444no—
246C22×C4—2444no—
247C22×C4C22×#692311yescharacteristic
248C2×C8—1224no—
249C2×C8—4884no—
250C2×C8—2442no—
251C4⋊C4—1222no—
252C4⋊C4—1222no—
253C4⋊C4—2444no—
254C22⋊C4—1224no—
255C22⋊C4—2442no—
256C22⋊C4—2444no—
257C22⋊C4—2444no—
258C42—21no—
259C42—612no—
260C42—612no—
261C42—1224no—
262C42—1222no—
263C42—2442no—
264C42C2×#9242012yescomplemented
265C23—22no—
266C23—22no—
267C23—22no—
268C23—42no—
269C23—42no—
270C23—42no—
271C23—42no—
272C23—41no—
273C23—41no—
274C23C22⋊#9242014yes—
275C23C2×C22⋊#692311yescharacteristic
276D4—24no—
277D4—24no—
278D4—24no—
279D4—416no—
280D4—42no—
281C2×C4—22no—
282C2×C4—22no—
283C2×C4—22no—
284C2×C4—611no—
285C2×C4—612no—
286C2×C4—611no—
287C2×C4—1222no—
288C2×C4—1222no—
289C2×C4—1222no—
290C2×C4—4882no—
291C2×C4—4882no—
292C2×C4—4882no—
293C2×C4—2444no—
294C2×C4—2441no—
295C2×C4—2442no—
296C2×C4—2441no—
297C2×C4—2441no—
298C2×C4—2441no—
299C2×C4—2442no—
300C2×C4D4×#692312yes—
301C2×C4C22×#9242011yescharacteristic
302C2×C4C2×#1175821711yescharacteristic
303C2×C4#1451986412yes—
304C8—4882no—
305C8—2444no—
306C22—21no—
307C22—21no—
308C22—24no—
309C22—44no—
310C22—44no—
311C22—42no—
312C22—42no—
313C22—42no—
314C22—82no—
315C22—82no—
316C22—81no—
317C22—81no—
318C22#1684071011yescharacteristic
319C22C2×C22⋊#9242011yescharacteristic
320C22#1684073511yescharacteristic
321C4—24no—
322C4—612no—
323C4—1222no—
324C4—1221no—
325C4—4881no—
326C4—2442no—
327C4—2442no—
328C4D4×#9242012yes—
329C4#1684079312yes—
330C2—21no—
331C2—21no—
332C2—44no—
333C2—82no—
334C2#2146179211yescharacteristic
335C2#2146182211yescharacteristic
336C2#2146182611yescharacteristic
subgroupG/H
centerC22#16840710
commutatorC2×C4×C61C22×C8
FrattiniC22×C4C22×#6923
Fitting, prime radicalC2×#127×C61C4
socleC22×C61#386
cosocleC2×C4×Dic61C23
hypercenterC2×#127#6923
hypocenterC61#8959
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#127

Complements

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

NH ≅ G/Ntype
1. #21462666C2semidirect
2. #21461118C2semidirect
3. #21461059C2semidirect
4. #21460647C2semidirect
23. C4⋊#92420D4semidirect
51. #127×C61C8semidirect
64. C2×D4×C61C2×C8semidirect
83. C42×C61C2×C8semidirect
189. C61#8959semidirect
197. #127#92420semidirect
233. C2×D4C2×#92420semidirect
264. C42C2×#92420semidirect

Maximal subgroups

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

subgroupindex
#214626662
#214611182
#214610592
#214606472
#127×Dic612
#895961