Finite Group Databasebeta

← #30595885

Subgroups — #30595885

Order 15616.

414 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 414 conjugacy classes of subgroups — too many to draw. Only its 86 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#21461746C211yesmaximal, characteristic, complemented
2#21461735C211yesmaximal, characteristic, complemented
3#21460640C212yesmaximal, complemented
4#21460635C212yesmaximal
5#21454190C211yesmaximal, characteristic
6C2×#14519844—21no—
7#16840628—21no—
8#16840628—21no—
9#16840628—21no—
10#16840628C2212yes—
11#16840628C2212yes—
12C2×#14519768C411yescharacteristic
13C2×#14519764C411yescharacteristic
14#16839668C411yescharacteristic
15#16839667C2211yescharacteristic
16#16839665C2211yescharacteristic
17#16839655—21no—
18#16839655—21no—
19#16839650—21no—
20#16839650—21no—
21C42×Dic61C2211yescharacteristic
22C61×#390C411yescharacteristic
23C22×#92420—22no—
24C22×#92420—22no—
25#14519844—42no—
26C22×Dic122C2×C411yescharacteristic
27#14519768—42no—
28#14519764—42no—
29C2×C4⋊Dic61—21no—
30C2×C4⋊Dic61C2×C411yescharacteristic
31C2×#11758196—21no—
32C2×C22⋊Dic61—21no—
33C2×C22⋊Dic61—21no—
34C2×C22⋊Dic61—21no—
35C2×C22⋊Dic61—21no—
36C2×C4×Dic61C2×C411yescharacteristic
37C2×C4×Dic61D411yescharacteristic
38C2×C4×Dic61D411yescharacteristic
39C2×C4×Dic61D411yescharacteristic
40C2×C4×Dic61D411yescharacteristic
41C2×C4×Dic61C2311yescharacteristic
42#14519684—21no—
43#14519684—21no—
44#14519684—21no—
45#14519684—21no—
46C22×D4×C61C2×C411yescharacteristic
47C2×C4⋊C4×C61C2×C411yescharacteristic
48C2×C22⋊C4×C61—21no—
49C2×C22⋊C4×C61—21no—
50C2×C42×C61C2×C411yescharacteristic
51C24×C61—21no—
52C2×D4×C61—22no—
53C2×D4×C61—41no—
54C2×D4×C61—41no—
55C22×C4×C61—21no—
56C22×C4×C61C22⋊C411yescharacteristic
57C22×C4×C61C22⋊C411yescharacteristic
58C22×C4×C61C22×C411yescharacteristic
59C22×Dic61—21no—
60C22×Dic61—21no—
61C22×Dic61C22⋊C411yescharacteristic
62C22×Dic61C22⋊C411yescharacteristic
63C22×Dic61C2×D411yescharacteristic
64C22×Dic61C2×D411yescharacteristic
65C2×Dic122—22no—
66C2×Dic122—41no—
67C2×Dic122—41no—
68C2×#92420—42no—
69C2×#92420—42no—
70C2×#92420—42no—
71C2×#92420—42no—
72C2×#92420—42no—
73C2×#92420—42no—
74C4⋊C4×C61—41no—
75C22⋊C4×C61—42no—
76C22⋊C4×C61—42no—
77C42×C61—22no—
78C22⋊Dic61—42no—
79C22⋊Dic61—42no—
80C22⋊Dic61—42no—
81C22⋊Dic61—41no—
82C22⋊Dic61—41no—
83C4⋊Dic61—42no—
84C4⋊Dic61—41no—
85#11758196—42no—
86C4×Dic61—22no—
87C4×Dic61—22no—
88C4×Dic61—21no—
89C4×Dic61—21no—
90C4×Dic61—41no—
91C4×Dic61—41no—
92C4×Dic61—41no—
93C23×C61—41no—
94C23×C61—41no—
95C23×C61—41no—
96C23×C61—41no—
97C23×C61—41no—
98C23×C61—41no—
99C23×C61—41no—
100C23×C61C2×C22⋊C411yescharacteristic
101D4×C61—82no—
102C2×C4×C61—21no—
103C2×C4×C61—21no—
104C2×C4×C61—21no—
105C2×C4×C61—21no—
106C2×C4×C61—21no—
107C2×C4×C61—21no—
108C2×C4×C61—21no—
109C2×C4×C61—21no—
110C2×C4×C61—41no—
111C2×C4×C61—41no—
112C2×C4×C61—41no—
113C2×C4×C61#13112yes—
114C2×Dic61—21no—
115C2×Dic61—21no—
116C2×Dic61—21no—
117C2×Dic61—21no—
118C2×Dic61—21no—
119C2×Dic61—21no—
120C2×Dic61—21no—
121C2×Dic61—21no—
122C2×Dic61—21no—
123C2×Dic61—41no—
124C2×Dic61—41no—
125C2×Dic61—41no—
126C2×Dic61—41no—
127C2×Dic61—41no—
128C2×Dic61—41no—
129C2×Dic61#10012yes—
130C2×Dic61#13512yes—
131C2×Dic61AGL(1,ℤ/8)11yescharacteristic
132C2×Dic61#13711yescharacteristic
133Dic122—82no—
134#92420—82no—
135#92420—82no—
136#7716—611nomaximal, Sylow
137C22×C61—81no—
138C22×C61—81no—
139C22×C61—81no—
140C22×C61—81no—
141C22×C61—81no—
142C22×C61—81no—
143C22×C61—81no—
144C22×C61#40811yescharacteristic
145C22×C61#41011yescharacteristic
146C22×C61C2×#10011yescharacteristic
147C22×C61#41611yescharacteristic
148C22×C61#41711yescharacteristic
149C22×C61#42111yescharacteristic
150C22×C61#42111yescharacteristic
151C4×C61—22no—
152C4×C61—41no—
153C4×C61—41no—
154C4×C61—81no—
155Dic61—22no—
156Dic61—21no—
157Dic61—21no—
158Dic61—22no—
159Dic61—41no—
160Dic61—81no—
161Dic61—81no—
162#2861—611no—
163#1839—611no—
164#1831—611no—
165#1272—612no—
166#1270—612no—
167C2×C61—161no—
168C2×C61#146911yescharacteristic
169C2×C61#148211yescharacteristic
170C2×C61#149511yescharacteristic
171C2×C61#151211yescharacteristic
172C2×C61#152111yescharacteristic
173C2×C61#153611yescharacteristic
174C2×C61#155111yescharacteristic
175C2×#125—611no—
176C2×#124—611no—
177C2×C4⋊C8—1221no—
178#391—611no—
179#391—611no—
180#390—611no—
181#390#692311yescharacteristic
182#388—1221no—
183#388—1221no—
184#388—1221no—
185#388—1221no—
186C43—611no—
187#336—612no—
188#336—612no—
189#336—1221no—
190#336—1221no—
191#336—1221no—
192C61#771611yescharacteristic, complemented, Sylow
193C22×Q8—611no—
194C22×D4C2×#692311yescharacteristic
195C22×C8—1222no—
196C22×C8—1222no—
197#125—2442no—
198#124—2442no—
199C2×C4⋊C4—611no—
200C2×C4⋊C4—1221no—
201C2×C4⋊C4—1221no—
202C2×C4⋊C4C2×#692311yescharacteristic
203C2×C22⋊C4—21no—
204C2×C22⋊C4—21no—
205C2×C22⋊C4—1221no—
206C2×C22⋊C4—1221no—
207C2×C22⋊C4—1221no—
208C2×C22⋊C4—1221no—
209C2×C42—611no—
210C2×C42—611no—
211C2×C42—611no—
212C2×C42—611no—
213C2×C42—611no—
214C2×C42—611no—
215C2×C42C2×#692311yescharacteristic
216C4⋊C8—2442no—
217#95—1221no—
218#95—1221no—
219#95—1221no—
220#95—1221no—
221C24—21no—
222C2×Q8—1222no—
223C2×Q8—2441no—
224C2×Q8—2441no—
225C2×D4—22no—
226C2×D4—41no—
227C2×D4—41no—
228C22×C4—21no—
229C22×C4—611no—
230C22×C4—611no—
231C22×C4—611no—
232C22×C4—611no—
233C22×C4—1221no—
234C22×C4—1221no—
235C22×C4C22⋊#692311yescharacteristic
236C22×C4C22⋊#692311yescharacteristic
237C22×C4C22×#692311yescharacteristic
238C2×C8—2442no—
239C2×C8—2442no—
240C2×C8—2442no—
241C2×C8—2442no—
242C2×C8—2442no—
243C2×C8—2442no—
244C4⋊C4—41no—
245C4⋊C4—2441no—
246C4⋊C4—2442no—
247C4⋊C4—2442no—
248C22⋊C4—42no—
249C22⋊C4—42no—
250C22⋊C4—2442no—
251C22⋊C4—2441no—
252C22⋊C4—2441no—
253C22⋊C4—2442no—
254C22⋊C4—2442no—
255C42—22no—
256C42—1221no—
257C42—1222no—
258C42—1221no—
259C42—1222no—
260C42—2441no—
261C42—2441no—
262C42—2441no—
263C23—41no—
264C23—41no—
265C23—41no—
266C23—41no—
267C23—41no—
268C23—41no—
269C23—41no—
270C23C2×C22⋊#692311yescharacteristic
271Q8—4882no—
272D4—82no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—41no—
282C2×C4—41no—
283C2×C4—41no—
284C2×C4—612no—
285C2×C4—612no—
286C2×C4—611no—
287C2×C4—611no—
288C2×C4—1221no—
289C2×C4—1221no—
290C2×C4—1221no—
291C2×C4—1221no—
292C2×C4—1221no—
293C2×C4—1221no—
294C2×C4—1221no—
295C2×C4—1221no—
296C2×C4—1221no—
297C2×C4—2441no—
298C2×C4—2441no—
299C2×C4—2441no—
300C2×C4—2441no—
301C2×C4—2441no—
302C2×C4—2441no—
303C2×C4#1451986412yes—
304C8—4882no—
305C8—4882no—
306C22—81no—
307C22—81no—
308C22—81no—
309C22—81no—
310C22—81no—
311C22—81no—
312C22—81no—
313C22#1684073511yescharacteristic
314C22#1684074111yescharacteristic
315C22D61⋊#10011yescharacteristic
316C22#1684075211yescharacteristic
317C22#1684075311yescharacteristic
318C22#1684076011yescharacteristic
319C22#1684076011yescharacteristic
320C4—22no—
321C4—41no—
322C4—41no—
323C4—81no—
324C4—1222no—
325C4—1222no—
326C4—1221no—
327C4—1221no—
328C4—4881no—
329C4—4881no—
330C4—2441no—
331C2—161no—
332C2#2146108811yescharacteristic
333C2#2146113011yescharacteristic
334C2#2146117411yescharacteristic
335C2#2146123011yescharacteristic
336C2#2146126111yescharacteristic
337C2#2146130811yescharacteristic
338C2#2146133911yescharacteristic
subgroupG/H
centerC23C2×C22⋊#6923
commutatorC22×C4×C61C22×C4
FrattiniC22×C4C22×#6923
FittingC61×#390C4
socleC23×C61C2×C22⋊C4
prime radicalC22×D4×C61C2×C4
cosocleC2×C4×Dic61C23
hypercenter#390#6923
hypocenterC61#7716
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #390

Complements

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

NH ≅ G/Ntype
1. #21461746C2semidirect
2. #21461735C2semidirect
3. #21460640C2semidirect
192. C61#7716semidirect

Maximal subgroups

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

subgroupindex
#214617462
#214617352
#214606402
#214606352
#214541902
#771661