Finite Group Databasebeta

← #30597813

Subgroups — #30597813

Order 15616.

516 conjugacy classes of subgroups; only the 132 normal subgroups are shown.

normal subgroups only

This group has 516 conjugacy classes of subgroups — too many to draw. Only its 132 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#21461604C211yesmaximal, characteristic
2#21461604C211yesmaximal, characteristic
3C2×#16840621C212yesmaximal
4#21460697C212yesmaximal
5#21448390C211yesmaximal, characteristic
6C2×#14519844C2212yes—
7C2×#14519844C2212yes—
8C2×#14519832—21no—
9C2×#14519832—21no—
10#16840628—21no—
11#16840628—21no—
12#16840621C418yescomplemented
13C2×#14519776C2211yescharacteristic
14C2×#14519742—21no—
15#16839666—21no—
16#16839660C411yescharacteristic
17#16839657—21no—
18#16839631C2211yescharacteristic
19C4⋊C4×Dic61C411yescharacteristic
20C4⋊C4×Dic61C411yescharacteristic
21#16839613—21no—
22C42×Dic61C2211yescharacteristic
23C61×#389C411yescharacteristic
24C22×#92420—22no—
25C22×#92420—22no—
26#14519844—22no—
27#14519844—22no—
28#14519844C2×C414yes—
29#14519844C2×C414yes—
30#14519832—42no—
31#14519832—42no—
32C22×Dic122—21no—
33#14519776—22no—
34#14519776C2×C414yes—
35#14519742—44no—
36C2×C4⋊Dic61C2×C411yescharacteristic
37C2×C4⋊Dic61C2×C411yescharacteristic
38C2×#11758196—21no—
39C2×#11758196—21no—
40C2×#11758196—21no—
41C2×#11758196—21no—
42C2×C4×Dic61—21no—
43C2×C4×Dic61—21no—
44C2×C4×Dic61C2×C411yescharacteristic
45C2×C4×Dic61D411yescharacteristic
46C2×C4×Dic61D411yescharacteristic
47C2×C4×Dic61D411yescharacteristic
48C2×C4×Dic61D411yescharacteristic
49C2×C4×Dic61C2311yescharacteristic
50#14519684—21no—
51#14519684—21no—
52#14519684—21no—
53C2×C4⋊C4×C61—21no—
54C2×C4⋊C4×C61C2×C411yescharacteristic
55C2×C4⋊C4×C61C2×C411yescharacteristic
56C2×C42×C61C2×C411yescharacteristic
57#95×C61—21no—
58C22×C4×C61—21no—
59C22×C4×C61—21no—
60C22×C4×C61C22⋊C411yescharacteristic
61C22×C4×C61C22⋊C411yescharacteristic
62C22×C4×C61C22×C411yescharacteristic
63C22×Dic61—21no—
64C22×Dic61—21no—
65C22×Dic61C22⋊C411yescharacteristic
66C22×Dic61C22⋊C411yescharacteristic
67C22×Dic61C2×D411yescharacteristic
68C22×Dic61C2×D411yescharacteristic
69C2×Dic122—24no—
70C2×Dic122—41no—
71C2×Dic122—42no—
72C2×Dic122—41no—
73C2×#92420—24no—
74C2×#92420—24no—
75C2×#92420—42no—
76C2×#92420—42no—
77C2×#92420—42no—
78C2×#92420—42no—
79C4⋊C4×C61—21no—
80C4⋊C4×C61—42no—
81C4⋊C4×C61—41no—
82C4⋊C4×C61C4212yes—
83C42×C61—41no—
84C4⋊Dic61—21no—
85C4⋊Dic61—41no—
86C4⋊Dic61C4212yes—
87#11758196—24no—
88#11758196—42no—
89#11758196—41no—
90#11758196—42no—
91#11758196—41no—
92C4×Dic61—22no—
93C4×Dic61—21no—
94C4×Dic61—21no—
95C4×Dic61—22no—
96C4×Dic61—22no—
97C4×Dic61—21no—
98C4×Dic61—21no—
99C4×Dic61—42no—
100C4×Dic61—42no—
101C4×Dic61C22×C412yes—
102C4×Dic61#4112yes—
103C23×C61C2×C22⋊C411yescharacteristic
104C2×C4×C61—21no—
105C2×C4×C61—21no—
106C2×C4×C61—22no—
107C2×C4×C61—21no—
108C2×C4×C61—21no—
109C2×C4×C61—21no—
110C2×C4×C61—21no—
111C2×C4×C61—21no—
112C2×C4×C61—21no—
113C2×C4×C61—41no—
114C2×C4×C61—41no—
115C2×C4×C61—41no—
116C2×C4×C61—41no—
117C2×C4×C61—41no—
118C2×C4×C61C2×C4211yescharacteristic
119C2×C4×C61#11711yescharacteristic
120C2×Dic61—21no—
121C2×Dic61—21no—
122C2×Dic61—22no—
123C2×Dic61—21no—
124C2×Dic61—21no—
125C2×Dic61—21no—
126C2×Dic61—41no—
127C2×Dic61—41no—
128C2×Dic61—41no—
129C2×Dic61—41no—
130C2×Dic61—41no—
131C2×Dic61#10112yes—
132C2×Dic61C4×D412yes—
133C2×Dic61C4×D412yes—
134C2×Dic61AGL(1,ℤ/8)11yescharacteristic
135C2×Dic61AGL(1,ℤ/8)12yes—
136C2×Dic61AGL(1,ℤ/8)11yescharacteristic
137C2×Dic61#13711yescharacteristic
138C2×Dic61#13712yes—
139C2×Dic61#13711yescharacteristic
140Dic122—44no—
141Dic122—44no—
142#92420—44no—
143#92420—44no—
144#8233—611nomaximal, Sylow
145C22×C61C4×C22⋊C411yescharacteristic
146C22×C61C2×#10111yescharacteristic
147C22×C61#41311yescharacteristic
148C22×C61#41711yescharacteristic
149C22×C61#41711yescharacteristic
150C22×C61#41811yescharacteristic
151C22×C61#41911yescharacteristic
152C4×C61—21no—
153C4×C61—21no—
154C4×C61—41no—
155C4×C61—41no—
156C4×C61—42no—
157C4×C61—81no—
158Dic61—22no—
159Dic61—41no—
160Dic61—42no—
161Dic61—81no—
162Dic61#44012yes—
163Dic61#44012yes—
164Dic61#44112yes—
165Dic61#44212yes—
166C4×C4⋊Q8—611no—
167#1767—611no—
168#1767—611no—
169C2×#332—612no—
170#1296—612no—
171C2×C61#155611yescharacteristic
172C2×C61#155711yescharacteristic
173C2×C61#155911yescharacteristic
174C2×C61#156111yescharacteristic
175C2×C61C4×#10111yescharacteristic
176C2×C61#176211yescharacteristic
177C2×C61#176311yescharacteristic
178C2×C4⋊Q8—611no—
179C2×C4×Q8—1221no—
180C2×C4⋊C8—612no—
181C2×C4⋊C8—612no—
182C2×#97—1221no—
183C2×#97—1221no—
184#391—1221no—
185#389—611no—
186#389—1221no—
187#389#692311yescharacteristic
188#384—611no—
189C4×C4⋊C4—611no—
190C4×C4⋊C4—611no—
191C4×C4⋊C4—1221no—
192C43—611no—
193#336—1221no—
194#336—1221no—
195#332—618no—
196C61#823311yescharacteristic, complemented, Sylow
197C22×Q8—1221no—
198C22×C8—1222no—
199C22×C8—1222no—
200C4⋊Q8—614no—
201C4⋊Q8—1222no—
202C4×Q8—2444no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—611no—
205C2×C4⋊C4—611no—
206C2×C4⋊C4—1221no—
207C2×C4⋊C4—1221no—
208C2×C4⋊C4—1221no—
209C2×C4⋊C4—1221no—
210C2×C4⋊C4C2×#692311yescharacteristic
211C2×C4⋊C4C2×#692311yescharacteristic
212C2×C42—611no—
213C2×C42—611no—
214C2×C42—611no—
215C2×C42—611no—
216C2×C42—611no—
217C2×C42—611no—
218C2×C42—1221no—
219C2×C42—1221no—
220C2×C42C2×#692311yescharacteristic
221C4⋊C8—614no—
222C4⋊C8—614no—
223C4⋊C8—1222no—
224C4⋊C8—1222no—
225#97—2442no—
226#97—2442no—
227#95—21no—
228#95—1221no—
229#95—1221no—
230#95—1221no—
231C2×Q8—1224no—
232C2×Q8—2442no—
233C2×Q8—2441no—
234C2×Q8—2441no—
235C22×C4—21no—
236C22×C4—21no—
237C22×C4—611no—
238C22×C4—611no—
239C22×C4—611no—
240C22×C4—611no—
241C22×C4—1221no—
242C22×C4—1221no—
243C22×C4C22⋊#692311yescharacteristic
244C22×C4C22⋊#692311yescharacteristic
245C22×C4C22×#692311yescharacteristic
246C2×C8—1224no—
247C2×C8—1224no—
248C2×C8—2442no—
249C2×C8—2442no—
250C2×C8—2442no—
251C2×C8—2442no—
252C4⋊C4—21no—
253C4⋊C4—41no—
254C4⋊C4—42no—
255C4⋊C4—612no—
256C4⋊C4—1224no—
257C4⋊C4—1221no—
258C4⋊C4—2441no—
259C4⋊C4—2442no—
260C4⋊C4—2441no—
261C4⋊C4—2441no—
262C4⋊C4—2442no—
263C4⋊C4C4×#692312yes—
264C42—41no—
265C42—612no—
266C42—612no—
267C42—1221no—
268C42—1221no—
269C42—1222no—
270C42—1221no—
271C42—1221no—
272C42—1222no—
273C42—1222no—
274C42—2442no—
275C42—2442no—
276C23C2×C22⋊#692311yescharacteristic
277Q8—2444no—
278Q8—2444no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—22no—
288C2×C4—41no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—41no—
293C2×C4—611no—
294C2×C4—611no—
295C2×C4—611no—
296C2×C4—611no—
297C2×C4—612no—
298C2×C4—612no—
299C2×C4—612no—
300C2×C4—612no—
301C2×C4—612no—
302C2×C4—1221no—
303C2×C4—1221no—
304C2×C4—1221no—
305C2×C4—1221no—
306C2×C4—1221no—
307C2×C4—1222no—
308C2×C4—2441no—
309C2×C4—2441no—
310C2×C4—2441no—
311C2×C4—2441no—
312C2×C4—2441no—
313C2×C4C2×C4×#692311yescharacteristic
314C2×C4#1451985611yescharacteristic
315C8—2444no—
316C8—2444no—
317C22C4×C22⋊#692311yescharacteristic
318C22D61⋊#10111yescharacteristic
319C22#1684074911yescharacteristic
320C22#1684075311yescharacteristic
321C22#1684075311yescharacteristic
322C22#1684075411yescharacteristic
323C22#1684075511yescharacteristic
324C4—21no—
325C4—21no—
326C4—42no—
327C4—41no—
328C4—41no—
329C4—81no—
330C4—612no—
331C4—612no—
332C4—612no—
333C4—612no—
334C4—1222no—
335C4—4881no—
336C4—2442no—
337C4—2441no—
338C2#2146134911yescharacteristic
339C2#2146135111yescharacteristic
340C2#2146135511yescharacteristic
341C2#2146135911yescharacteristic
342C2#2146157211yescharacteristic
343C2#2146158911yescharacteristic
344C2#2146159111yescharacteristic
subgroupG/H
centerC23C2×C22⋊#6923
commutatorC2×C4×C61C2×C42
FrattiniC22×C4C22×#6923
FittingC61×#389C4
socle, prime radicalC23×C61C2×C22⋊C4
cosocleC2×C4×Dic61C23
hypercenter#389#6923
hypocenterC61#8233
LCSG ▷ C2×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #389

Complements

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

NH ≅ G/Ntype
12. #16840621C4semidirect
196. C61#8233semidirect

Maximal subgroups

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

subgroupindex
#214616042
C2×#168406212
#214606972
#214483902
#823361