Finite Group Databasebeta

← #29595189

Subgroups — #29595189

Order 15616.

628 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 628 conjugacy classes of subgroups — too many to draw. Only its 125 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#21448545C211yesmaximal, characteristic
2C2×#16839555C211yesmaximal, characteristic
3#99×Dic61C212yesmaximal, complemented
4#21443943C212yesmaximal, complemented
5C2×C61×#353C211yesmaximal, characteristic
6C2×D4×Dic61C411yescharacteristic, complemented
7C2×D4×Dic61C2211yescharacteristic
8C2×#14519692C2212yes—
9#16839665C411yescharacteristic, complemented
10#16839665C2211yescharacteristic
11#16839642—22no—
12#16839642—22no—
13#16839628C411yescharacteristic, complemented
14C22⋊C4×Dic61—22no—
15#16839608C411yescharacteristic, complemented
16#16839555C414yescomplemented
17#16839528—22no—
18C2×#127×C61C2211yescharacteristic
19C2×#99×C61C2212yes—
20C61×#353C414yescomplemented
21C23×Dic61—21no—
22C23×Dic61D412yes—
23D4×Dic61—42no—
24D4×Dic61—42no—
25C2×C4⋊Dic61C2×C411yescharacteristic
26C2×C4⋊Dic61C2×C411yescharacteristic
27C2×C22⋊Dic61—22no—
28C2×C22⋊Dic61—21no—
29C2×C22⋊Dic61—22no—
30C2×C22⋊Dic61—22no—
31C2×C22⋊Dic61D412yes—
32C2×C4×Dic61—42no—
33C2×C4×Dic61C2×C411yescharacteristic
34C2×C4×Dic61C2×C411yescharacteristic
35#14519692—22no—
36#14519692C2×C414yescomplemented
37#14519684—21no—
38#14519684—21no—
39#14519684—22no—
40#14519684—42no—
41C22×D4×C61—22no—
42C22×D4×C61C2×C411yescharacteristic
43C22×D4×C61C2311yescharacteristic
44#127×C61—22no—
45#127×C61C2×C412yes—
46#127×C61C2×C412yes—
47C2×C22⋊C4×C61—22no—
48C2×C42×C61C2×C411yescharacteristic
49#99×C61—22no—
50#99×C61C2×C414yescomplemented
51C24×C61—21no—
52C24×C61C2×D412yes—
53C2×D4×C61—24no—
54C2×D4×C61—24no—
55C2×D4×C61—21no—
56C2×D4×C61—41no—
57C2×D4×C61—41no—
58C2×D4×C61—41no—
59C2×D4×C61—41no—
60C2×D4×C61—82no—
61C2×D4×C61—82no—
62C2×D4×C61C4212yescomplemented
63C2×D4×C61C22×C411yescharacteristic
64C2×D4×C61C22×C411yescharacteristic
65C2×D4×C61#4112yes—
66C22×C4×C61—21no—
67C22×C4×C61—42no—
68C22×C4×C61C22×C411yescharacteristic
69C22×Dic61—22no—
70C22×Dic61—22no—
71C22×Dic61—21no—
72C22×Dic61—21no—
73C22×Dic61—42no—
74C22×Dic61—42no—
75C22×Dic61—82no—
76C22×Dic61—82no—
77C22×Dic61C22⋊C411yescharacteristic
78C22×Dic61C22⋊C412yes—
79C22×Dic61C22⋊C411yescharacteristic
80C22⋊C4×C61—24no—
81C22⋊C4×C61—42no—
82C42×C61—21no—
83C42×C61C4212yescomplemented
84C22⋊Dic61—24no—
85C22⋊Dic61—42no—
86C22⋊Dic61—42no—
87C22⋊Dic61—42no—
88C22⋊Dic61—82no—
89C22⋊Dic61—82no—
90C4⋊Dic61—41no—
91C4⋊Dic61—41no—
92C4×Dic61—22no—
93C4×Dic61—41no—
94C4×Dic61—84no—
95C23×C61—22no—
96C23×C61—22no—
97C23×C61—22no—
98C23×C61—42no—
99C23×C61—42no—
100C23×C61—42no—
101C23×C61—42no—
102C23×C61—41no—
103C23×C61—41no—
104C23×C61C2×C22⋊C411yescharacteristic
105C23×C61C4×D412yes—
106C23×C61C4×D412yes—
107D4×C61—41no—
108D4×C61—41no—
109D4×C61—42no—
110D4×C61—42no—
111D4×C61—42no—
112D4×C61—84no—
113D4×C61—84no—
114C2×C4×C61—21no—
115C2×C4×C61—21no—
116C2×C4×C61—22no—
117C2×C4×C61—41no—
118C2×C4×C61—41no—
119C2×C4×C61—44no—
120C2×C4×C61—82no—
121C2×C4×C61—82no—
122C2×C4×C61C2×C4211yescharacteristic
123C2×C4×C61#11711yescharacteristic
124C2×Dic61—21no—
125C2×Dic61—22no—
126C2×Dic61—41no—
127C2×Dic61—44no—
128C2×Dic61—42no—
129C2×Dic61—41no—
130C2×Dic61—44no—
131C2×Dic61—41no—
132C2×Dic61—41no—
133C2×Dic61—42no—
134C2×Dic61—41no—
135C2×Dic61—41no—
136C2×Dic61—82no—
137C2×Dic61—84no—
138C2×Dic61—82no—
139C2×Dic61—82no—
140C2×Dic61#9912yes—
141#12628—611nomaximal, Sylow
142C22×C61—21no—
143C22×C61—21no—
144C22×C61—22no—
145C22×C61—22no—
146C22×C61—42no—
147C22×C61—42no—
148C22×C61—44no—
149C22×C61—42no—
150C22×C61—42no—
151C22×C61—42no—
152C22×C61—82no—
153C22×C61—82no—
154C22×C61—81no—
155C22×C61—81no—
156C22×C61C4×C22⋊C411yescharacteristic
157C22×C61C2×#9911yescharacteristic
158C22×C61#41011yescharacteristic
159C4×C61—21no—
160C4×C61—21no—
161C4×C61—42no—
162C4×C61—84no—
163Dic61—22no—
164Dic61—42no—
165Dic61—41no—
166Dic61—81no—
167Dic61—84no—
168Dic61—81no—
169#2328—611no—
170C2×#353—611no—
171C2×#353D6111yescharacteristic
172C4×#99—612no—
173#1393—612no—
174C2×C61—21no—
175C2×C61—21no—
176C2×C61—42no—
177C2×C61—42no—
178C2×C61—82no—
179C2×C61C4×#9911yescharacteristic
180C2×C61#212711yescharacteristic
181C2×C61#212811yescharacteristic
182C2×#127C2×D6111yescharacteristic
183C2×C4×D4—611no—
184C2×C4×D4—611no—
185C2×#99—612no—
186C2×#99C2×D6112yes—
187#390—611no—
188#390—611no—
189#386—1222no—
190#386—1222no—
191#382—611no—
192C4×C22⋊C4—1222no—
193#376—611no—
194#353—614no—
195#353Dic6114yescomplemented
196#342—1222no—
197C61#1262811yescharacteristic, complemented, Sylow
198C22×D4—22no—
199C22×D4C4×D6111yescharacteristic
200C22×D4C22×D6111yescharacteristic
201C23×C4—612no—
202C23×C4—1221no—
203#127—22no—
204#127C4×D6112yes—
205#127C2×Dic6112yes—
206C4×D4—2442no—
207C4×D4—2442no—
208C2×C4⋊C4—611no—
209C2×C4⋊C4—611no—
210C2×C22⋊C4—22no—
211C2×C22⋊C4—612no—
212C2×C22⋊C4—1222no—
213C2×C22⋊C4—1222no—
214C2×C22⋊C4—1222no—
215C2×C22⋊C4—1221no—
216C2×C42—611no—
217C2×C42—611no—
218C2×C42—2442no—
219C2×C42C4×D6111yescharacteristic
220#99—22no—
221#99—614no—
222#99—1222no—
223#99C2×Dic6114yescomplemented
224#95—1221no—
225#95—1221no—
226#95—1222no—
227#95—2442no—
228C24—21no—
229C24D4×D6112yes—
230C2×D4—24no—
231C2×D4—24no—
232C2×D4—21no—
233C2×D4—41no—
234C2×D4—41no—
235C2×D4—41no—
236C2×D4—41no—
237C2×D4—82no—
238C2×D4—82no—
239C2×D4C4×Dic6112yescomplemented
240C2×D4C2×C4×D6111yescharacteristic
241C2×D4#1175822412yes—
242C2×D4C22×Dic6111yescharacteristic
243C22×C4—21no—
244C22×C4—42no—
245C22×C4—611no—
246C22×C4—611no—
247C22×C4—612no—
248C22×C4—1222no—
249C22×C4—1222no—
250C22×C4—1221no—
251C22×C4—1221no—
252C22×C4—4882no—
253C22×C4—4882no—
254C22×C4—2442no—
255C22×C4—2442no—
256C22×C4C2×C4×D6111yescharacteristic
257C4⋊C4—2441no—
258C4⋊C4—2441no—
259C22⋊C4—24no—
260C22⋊C4—42no—
261C22⋊C4—1224no—
262C22⋊C4—4882no—
263C22⋊C4—4882no—
264C22⋊C4—2442no—
265C22⋊C4—2442no—
266C22⋊C4—2442no—
267C42—21no—
268C42—1222no—
269C42—4884no—
270C42—2441no—
271C42C4×Dic6112yescomplemented
272C23—22no—
273C23—22no—
274C23—22no—
275C23—42no—
276C23—42no—
277C23—42no—
278C23—42no—
279C23—41no—
280C23—41no—
281C23C22⋊C4×D6111yescharacteristic
282C23#1451973712yes—
283C23D4×Dic6112yes—
284D4—41no—
285D4—42no—
286D4—41no—
287D4—42no—
288D4—42no—
289D4—84no—
290D4—84no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—22no—
294C2×C4—41no—
295C2×C4—41no—
296C2×C4—44no—
297C2×C4—82no—
298C2×C4—82no—
299C2×C4—612no—
300C2×C4—1221no—
301C2×C4—1222no—
302C2×C4—4882no—
303C2×C4—4882no—
304C2×C4—4882no—
305C2×C4—4884no—
306C2×C4—2441no—
307C2×C4—2444no—
308C2×C4—2444no—
309C2×C4—2441no—
310C2×C4—2441no—
311C2×C4—2442no—
312C2×C4—2441no—
313C2×C4—2441no—
314C2×C4—2442no—
315C2×C4—2441no—
316C2×C4C2×C4×Dic6111yescharacteristic
317C2×C4#1451973311yescharacteristic
318C22—21no—
319C22—21no—
320C22—22no—
321C22—22no—
322C22—42no—
323C22—42no—
324C22—44no—
325C22—42no—
326C22—42no—
327C22—42no—
328C22—82no—
329C22—82no—
330C22—81no—
331C22—81no—
332C22C22⋊C4×Dic6111yescharacteristic
333C22#99×D6111yescharacteristic
334C22#1683973411yescharacteristic
335C4—21no—
336C4—21no—
337C4—42no—
338C4—84no—
339C4—1222no—
340C4—4881no—
341C4—4884no—
342C4—4881no—
343C4—2442no—
344C4—2441no—
345C2—21no—
346C2—21no—
347C2—42no—
348C2—42no—
349C2—82no—
350C2#99×Dic6111yescharacteristic
351C2#2144748511yescharacteristic
352C2#2144748911yescharacteristic
subgroupG/H
centerC22#99×D61
commutatorC2×C4×C61C2×C42
FrattiniC22×D4C22×D61
FittingC2×C61×#353C2
socleC22×C61C2×#99
prime radicalC2×#127×C61C22
cosocleC22×D4×C61C23
hypercenterC2×#353D61
hypocenterC61#12628
LCSG ▷ C2×C4×C61 ▷ C22×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23 ◁ C22×D4 ◁ C2×#353

Complements

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

NH ≅ G/Ntype
3. #99×Dic61C2semidirect
4. #21443943C2semidirect
6. C2×D4×Dic61C4semidirect
9. #16839665C4semidirect
13. #16839628C4semidirect
15. #16839608C4semidirect
16. #16839555C4semidirect
20. C61×#353C4semidirect
36. #14519692C2×C4semidirect
50. #99×C61C2×C4semidirect
62. C2×D4×C61C42semidirect
83. C42×C61C42semidirect
195. #353Dic61semidirect
197. C61#12628semidirect
223. #99C2×Dic61semidirect
239. C2×D4C4×Dic61semidirect
271. C42C4×Dic61semidirect

Maximal subgroups

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

subgroupindex
#214485452
C2×#168395552
#99×Dic612
#214439432
C2×C61×#3532
#1262861