Finite Group Databasebeta

← #30595797

Subgroups — #30595797

Order 15616.

444 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 444 conjugacy classes of subgroups — too many to draw. Only its 90 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#21461536C211yesmaximal, characteristic, complemented
2#21461534C211yesmaximal, characteristic, complemented
3#21460637C212yesmaximal, complemented
4#21460635C212yesmaximal
5#21454183C211yesmaximal, characteristic
6C2×#14519832—21no—
7C2×C4×#92420—21no—
8#16840628—21no—
9#16840628—21no—
10#16840628C2212yes—
11#16840628C2212yes—
12C2×#14519770—21no—
13#16839665C411yescharacteristic
14#16839665C2211yescharacteristic
15#16839661C2211yescharacteristic
16#16839660C411yescharacteristic
17#16839658C411yescharacteristic
18#16839656—21no—
19#16839655—21no—
20#16839650—21no—
21C42×Dic61C2211yescharacteristic
22C61×#390C411yescharacteristic
23C22×#92420—22no—
24C22×#92420—22no—
25#14519832—24no—
26C4×#92420—24no—
27#14519770—44no—
28C2×C4⋊Dic61—21no—
29C2×C4⋊Dic61C2×C411yescharacteristic
30C2×C4⋊Dic61C2×C411yescharacteristic
31C2×#11758196—21no—
32C2×#11758196—21no—
33C2×C22⋊Dic61—21no—
34C2×C22⋊Dic61—21no—
35C2×C22⋊Dic61—21no—
36C2×C22⋊Dic61—21no—
37C2×C4×Dic61C2×C411yescharacteristic
38C2×C4×Dic61D411yescharacteristic
39C2×C4×Dic61D411yescharacteristic
40C2×C4×Dic61D411yescharacteristic
41C2×C4×Dic61D411yescharacteristic
42C2×C4×Dic61C2311yescharacteristic
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×#92420—24no—
66C2×#92420—24no—
67C2×#92420—42no—
68C2×#92420—42no—
69C2×#92420—42no—
70C2×#92420—42no—
71C4⋊C4×C61—22no—
72C22⋊C4×C61—42no—
73C22⋊C4×C61—42no—
74C42×C61—41no—
75C22⋊Dic61—42no—
76C22⋊Dic61—42no—
77C22⋊Dic61—42no—
78C22⋊Dic61—42no—
79C4⋊Dic61—42no—
80C4⋊Dic61—41no—
81C4⋊Dic61—41no—
82#11758196—42no—
83#11758196—42no—
84C4×Dic61—22no—
85C4×Dic61—22no—
86C4×Dic61—22no—
87C4×Dic61—22no—
88C4×Dic61—22no—
89C4×Dic61—41no—
90C23×C61—41no—
91C23×C61—41no—
92C23×C61—41no—
93C23×C61—41no—
94C23×C61—41no—
95C23×C61—41no—
96C23×C61—41no—
97C23×C61C2×C22⋊C411yescharacteristic
98D4×C61—82no—
99C2×C4×C61—21no—
100C2×C4×C61—21no—
101C2×C4×C61—21no—
102C2×C4×C61—21no—
103C2×C4×C61—21no—
104C2×C4×C61—21no—
105C2×C4×C61—21no—
106C2×C4×C61—41no—
107C2×C4×C61—41no—
108C2×C4×C61—41no—
109C2×C4×C61#10412yes—
110C2×C4×C61#13112yes—
111C2×Dic61—21no—
112C2×Dic61—21no—
113C2×Dic61—21no—
114C2×Dic61—21no—
115C2×Dic61—21no—
116C2×Dic61—21no—
117C2×Dic61—21no—
118C2×Dic61—21no—
119C2×Dic61—21no—
120C2×Dic61—41no—
121C2×Dic61—41no—
122C2×Dic61—41no—
123C2×Dic61—41no—
124C2×Dic61—41no—
125C2×Dic61—41no—
126C2×Dic61#10412yes—
127C2×Dic61#13512yes—
128C2×Dic61AGL(1,ℤ/8)12yes—
129#92420—44no—
130#92420—44no—
131#7692—611nomaximal, Sylow
132C22×C61—81no—
133C22×C61—81no—
134C22×C61—81no—
135C22×C61—81no—
136C22×C61—81no—
137C22×C61—81no—
138C22×C61—81no—
139C22×C61#40811yescharacteristic
140C22×C61#41011yescharacteristic
141C22×C61#41311yescharacteristic
142C22×C61#41611yescharacteristic
143C22×C61#41811yescharacteristic
144C22×C61C2×#10411yescharacteristic
145C22×C61C2×#10411yescharacteristic
146C4×C61—22no—
147C4×C61—22no—
148C4×C61—41no—
149C4×C61—81no—
150Dic61—22no—
151Dic61—22no—
152Dic61—22no—
153Dic61—41no—
154Dic61—81no—
155Dic61—81no—
156#2860—611no—
157#1745—611no—
158#1744—611no—
159#1271—612no—
160#1270—612no—
161C2×C61—161no—
162C2×C61#147311yescharacteristic
163C2×C61#148411yescharacteristic
164C2×C61#149411yescharacteristic
165C2×C61#150011yescharacteristic
166C2×C61#150511yescharacteristic
167C2×C61#152411yescharacteristic
168C2×C61#155311yescharacteristic
169C2×#126—1221no—
170C2×#97—1221no—
171C2×C4×C8—1221no—
172#390—611no—
173#390—611no—
174#390#692311yescharacteristic
175#389—611no—
176#389—611no—
177#389—611no—
178#388—1221no—
179#388—1221no—
180#388—1221no—
181C43—611no—
182#336—612no—
183#336—612no—
184#336—1221no—
185#336—1221no—
186C61#769211yescharacteristic, complemented, Sylow
187C22×D4C2×#692311yescharacteristic
188C22×C8—1222no—
189C22×C8—1222no—
190#126—2444no—
191C2×C4⋊C4—611no—
192C2×C4⋊C4—611no—
193C2×C4⋊C4—1221no—
194C2×C4⋊C4—1221no—
195C2×C4⋊C4—1221no—
196C2×C4⋊C4C2×#692311yescharacteristic
197C2×C22⋊C4—21no—
198C2×C22⋊C4—21no—
199C2×C22⋊C4—1221no—
200C2×C22⋊C4—1221no—
201C2×C22⋊C4—1221no—
202C2×C22⋊C4—1221no—
203C2×C42—611no—
204C2×C42—611no—
205C2×C42—611no—
206C2×C42—611no—
207C2×C42—611no—
208C2×C42—611no—
209C2×C42C2×#692311yescharacteristic
210#97—1224no—
211C4×C8—1224no—
212#95—1221no—
213#95—1221no—
214#95—1221no—
215C24—21no—
216C2×D4—22no—
217C2×D4—41no—
218C2×D4—41no—
219C22×C4—21no—
220C22×C4—611no—
221C22×C4—611no—
222C22×C4—611no—
223C22×C4—611no—
224C22×C4—1221no—
225C22×C4—1221no—
226C22×C4C22⋊#692311yescharacteristic
227C22×C4C22⋊#692311yescharacteristic
228C22×C4C22×#692311yescharacteristic
229C2×C8—1224no—
230C2×C8—1224no—
231C2×C8—2442no—
232C2×C8—2442no—
233C2×C8—2442no—
234C2×C8—2442no—
235C4⋊C4—22no—
236C4⋊C4—2441no—
237C4⋊C4—2441no—
238C4⋊C4—2442no—
239C4⋊C4—2442no—
240C4⋊C4—2442no—
241C22⋊C4—42no—
242C22⋊C4—42no—
243C22⋊C4—2442no—
244C22⋊C4—2442no—
245C22⋊C4—2442no—
246C22⋊C4—2442no—
247C42—41no—
248C42—1222no—
249C42—1222no—
250C42—1222no—
251C42—1222no—
252C42—1222no—
253C42—2441no—
254C23—41no—
255C23—41no—
256C23—41no—
257C23—41no—
258C23—41no—
259C23—41no—
260C23—41no—
261C23C2×C22⋊#692311yescharacteristic
262D4—82no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—41no—
271C2×C4—41no—
272C2×C4—41no—
273C2×C4—612no—
274C2×C4—612no—
275C2×C4—612no—
276C2×C4—1221no—
277C2×C4—1221no—
278C2×C4—1221no—
279C2×C4—1221no—
280C2×C4—1221no—
281C2×C4—1221no—
282C2×C4—1221no—
283C2×C4—1221no—
284C2×C4—1221no—
285C2×C4—2441no—
286C2×C4—2441no—
287C2×C4—2441no—
288C2×C4—2441no—
289C2×C4—2441no—
290C2×C4—2441no—
291C2×C4#1451984112yes—
292C2×C4#1451986412yes—
293C8—2444no—
294C8—2444no—
295C22—81no—
296C22—81no—
297C22—81no—
298C22—81no—
299C22—81no—
300C22—81no—
301C22—81no—
302C22#1684073611yescharacteristic
303C22#1684074111yescharacteristic
304C22#1684074911yescharacteristic
305C22#1684075211yescharacteristic
306C22#1684075411yescharacteristic
307C22C2×#1451984111yescharacteristic
308C22D61⋊#10411yescharacteristic
309C4—22no—
310C4—22no—
311C4—41no—
312C4—81no—
313C4—1222no—
314C4—1222no—
315C4—1222no—
316C4—4881no—
317C4—4881no—
318C4—2441no—
319C2—161no—
320C2#2146110011yescharacteristic
321C2#2146113911yescharacteristic
322C2#2146117111yescharacteristic
323C2#2146119011yescharacteristic
324C2#2146120911yescharacteristic
325C2#2146127011yescharacteristic
326C2#2146134311yescharacteristic
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#7692
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. #21461536C2semidirect
2. #21461534C2semidirect
3. #21460637C2semidirect
186. C61#7692semidirect

Maximal subgroups

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

subgroupindex
#214615362
#214615342
#214606372
#214606352
#214541832
#769261