Finite Group Databasebeta

← #29596809

Subgroups — #29596809

Order 15616.

672 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 672 conjugacy classes of subgroups — too many to draw. Only its 103 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#21448884C211yesmaximal, characteristic
2#21446218C211yesmaximal, characteristic
3#21446102C212yesmaximal, complemented
4#21443936C212yesmaximal, complemented
5C2×C61×#353C211yesmaximal, characteristic
6#16840237C2211yescharacteristic
7C2×D4×Dic61—22no—
8#16839767—21no—
9C2×#14519694C2211yescharacteristic
10#16839665—22no—
11#16839642—21no—
12#16839642C2212yes—
13#16839637—22no—
14C4×C4⋊Dic61—21no—
15#16839528—22no—
16#16839522—21no—
17C2×#127×C61C2211yescharacteristic
18C2×#99×C61C2212yes—
19C61×#353C414yescomplemented
20C23×Dic61—21no—
21C23×Dic61D412yes—
22C2×#11758194—21no—
23D4×Dic61—41no—
24D4×Dic61—41no—
25D4×Dic61—48no—
26#14519736—21no—
27C2×C4⋊Dic61—21no—
28C2×C4⋊Dic61—21no—
29C2×C4⋊Dic61—42no—
30C2×C22⋊Dic61—22no—
31C2×C22⋊Dic61—22no—
32C2×C22⋊Dic61—22no—
33C2×C22⋊Dic61—21no—
34C2×C22⋊Dic61D412yes—
35C2×C4×Dic61—22no—
36#14519698—41no—
37#14519698—41no—
38#14519694—22no—
39#14519684—22no—
40#14519684—42no—
41C22×D4×C61—22no—
42C22×D4×C61D411yescharacteristic
43C22×D4×C61C2311yescharacteristic
44#127×C61—22no—
45#127×C61C2×C412yes—
46#127×C61Q812yes—
47C2×C22⋊C4×C61—22no—
48C2×C42×C61D411yescharacteristic
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—44no—
59C2×D4×C61—44no—
60C2×D4×C61—41no—
61C2×D4×C61—41no—
62C2×D4×C61C4⋊C412yescomplemented
63C2×D4×C61C22×C411yescharacteristic
64C2×D4×C61C2×Q811yescharacteristic
65C2×D4×C61#4112yes—
66C22×C4×C61—21no—
67C22×C4×C61—42no—
68C22×C4×C61C2×D411yescharacteristic
69C22×Dic61—22no—
70C22×Dic61—22no—
71C22×Dic61—22no—
72C22×Dic61—22no—
73C22×Dic61—44no—
74C22×Dic61—44no—
75C22×Dic61—42no—
76C22×Dic61—81no—
77C22⋊C4×C61—24no—
78C22⋊C4×C61—42no—
79C42×C61—21no—
80C42×C61C4⋊C412yescomplemented
81C22⋊Dic61—44no—
82C22⋊Dic61—44no—
83C22⋊Dic61—42no—
84C22⋊Dic61—41no—
85C22⋊Dic61—42no—
86C22⋊Dic61—81no—
87C4⋊Dic61—24no—
88C4⋊Dic61—21no—
89C4⋊Dic61—21no—
90C4⋊Dic61—42no—
91C4⋊Dic61—84no—
92C4×Dic61—44no—
93C4×Dic61—41no—
94#11758194—41no—
95#11758194—41no—
96C2×#92418—41no—
97C23×C61—22no—
98C23×C61—22no—
99C23×C61—22no—
100C23×C61—42no—
101C23×C61—42no—
102C23×C61—42no—
103C23×C61—42no—
104C23×C61—41no—
105C23×C61—41no—
106C23×C61C4×D412yes—
107C23×C61#12011yescharacteristic
108C23×C61C22⋊Q812yes—
109D4×C61—41no—
110D4×C61—41no—
111D4×C61—42no—
112D4×C61—42no—
113D4×C61—48no—
114D4×C61—48no—
115D4×C61—81no—
116C2×C4×C61—22no—
117C2×C4×C61—22no—
118C2×C4×C61—22no—
119C2×C4×C61—21no—
120C2×C4×C61—21no—
121C2×C4×C61—44no—
122C2×C4×C61—82no—
123C2×C4×C61—82no—
124C2×C4×C61C2×C4⋊C411yescharacteristic
125C2×C4×C61#12311yescharacteristic
126C2×Dic61—24no—
127C2×Dic61—42no—
128C2×Dic61—42no—
129C2×Dic61—44no—
130C2×Dic61—42no—
131C2×Dic61—42no—
132C2×Dic61—41no—
133C2×Dic61—41no—
134C2×Dic61—42no—
135C2×Dic61—82no—
136C2×Dic61—84no—
137C2×Dic61—82no—
138C2×Dic61—82no—
139#92418—81no—
140#12964—611nomaximal, Sylow
141C22×C61—21no—
142C22×C61—21no—
143C22×C61—22no—
144C22×C61—22no—
145C22×C61—42no—
146C22×C61—42no—
147C22×C61—44no—
148C22×C61—42no—
149C22×C61—42no—
150C22×C61—42no—
151C22×C61—82no—
152C22×C61—82no—
153C22×C61—81no—
154C22×C61—81no—
155C22×C61#38511yescharacteristic
156C22×C61B(4,𝔽2)11yescharacteristic
157C22×C61#45811yescharacteristic
158C4×C61—24no—
159C4×C61—21no—
160C4×C61—21no—
161C4×C61—84no—
162Dic61—44no—
163Dic61—82no—
164Dic61—81no—
165Dic61—81no—
166#2366—611no—
167C2×#353D6111yescharacteristic
168#1887—611no—
169#1867—612no—
170#1390—612no—
171C2×C61—21no—
172C2×C61—21no—
173C2×C61—42no—
174C2×C61—42no—
175C2×C61—82no—
176C2×C61#186711yescharacteristic
177C2×C61#219611yescharacteristic
178C2×C61#219811yescharacteristic
179C2×#127C2×D6111yescharacteristic
180#518—611no—
181C2×C4×D4—1222no—
182#418—1221no—
183C2×#100—611no—
184C2×#99C2×D6112yes—
185#390—1222no—
186#386—612no—
187#386—1221no—
188#385—1222no—
189C4×C4⋊C4—1221no—
190#353Dic6114yescomplemented
191#342—1222no—
192#339—1221no—
193C61#1296411yescharacteristic, complemented, Sylow
194C22×D4—22no—
195C22×D4D24411yescharacteristic
196C22×D4C22×D6111yescharacteristic
197C23×C4—612no—
198C23×C4—1221no—
199C2×M16—1221no—
200#127—22no—
201#127Dic12212yes—
202#127C2×Dic6112yes—
203C4×D4—2441no—
204C4×D4—2441no—
205C4×D4—2448no—
206#117—1221no—
207C2×C4⋊C4—1221no—
208C2×C4⋊C4—1221no—
209C2×C4⋊C4—2442no—
210C2×C22⋊C4—22no—
211C2×C22⋊C4—612no—
212C2×C22⋊C4—1221no—
213C2×C22⋊C4—1222no—
214C2×C22⋊C4—1222no—
215C2×C22⋊C4—1222no—
216C2×C42—1222no—
217C2×C42D24411yescharacteristic
218#102—2441no—
219#102—2441no—
220#100—1222no—
221#99—22no—
222#99C2×Dic6114yescomplemented
223#95—1222no—
224#95—2442no—
225C24—21no—
226C24D4×D6112yes—
227C2×D4—24no—
228C2×D4—24no—
229C2×D4—21no—
230C2×D4—41no—
231C2×D4—41no—
232C2×D4—44no—
233C2×D4—44no—
234C2×D4—41no—
235C2×D4—41no—
236C2×D4C4⋊Dic6112yescomplemented
237C2×D4C2×Dic12211yescharacteristic
238C2×D4#1175822412yes—
239C2×D4C22×Dic6111yescharacteristic
240C22×C4—21no—
241C22×C4—42no—
242C22×C4—1222no—
243C22×C4—1222no—
244C22×C4—1222no—
245C22×C4—1222no—
246C22×C4—4881no—
247C22×C4—2444no—
248C22×C4—2444no—
249C22×C4—2442no—
250C22×C4C2×D24411yescharacteristic
251M16—2441no—
252M16—2441no—
253C2×C8—2441no—
254C4⋊C4—1224no—
255C4⋊C4—1221no—
256C4⋊C4—1221no—
257C4⋊C4—4884no—
258C4⋊C4—2442no—
259C22⋊C4—24no—
260C22⋊C4—42no—
261C22⋊C4—4881no—
262C22⋊C4—2444no—
263C22⋊C4—2444no—
264C22⋊C4—2442no—
265C22⋊C4—2441no—
266C22⋊C4—2442no—
267C42—21no—
268C42—2444no—
269C42—2441no—
270C42C4⋊Dic6112yescomplemented
271C23—22no—
272C23—22no—
273C23—22no—
274C23—42no—
275C23—42no—
276C23—42no—
277C23—42no—
278C23—41no—
279C23—41no—
280C23D4×Dic6112yes—
281C23#1451974711yescharacteristic
282C23#1451975412yes—
283D4—41no—
284D4—42no—
285D4—41no—
286D4—42no—
287D4—48no—
288D4—48no—
289D4—81no—
290C2×C4—22no—
291C2×C4—22no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—22no—
295C2×C4—44no—
296C2×C4—82no—
297C2×C4—82no—
298C2×C4—1224no—
299C2×C4—4882no—
300C2×C4—4882no—
301C2×C4—4882no—
302C2×C4—4884no—
303C2×C4—2442no—
304C2×C4—2442no—
305C2×C4—2441no—
306C2×C4—2441no—
307C2×C4—2442no—
308C2×C4—2442no—
309C2×C4—2442no—
310C2×C4—2444no—
311C2×C4C2×C4⋊Dic6111yescharacteristic
312C2×C4#1451976111yescharacteristic
313C8—4881no—
314C22—21no—
315C22—21no—
316C22—22no—
317C22—22no—
318C22—42no—
319C22—42no—
320C22—44no—
321C22—42no—
322C22—42no—
323C22—42no—
324C22—82no—
325C22—82no—
326C22—81no—
327C22—81no—
328C22#1683963711yescharacteristic
329C22#1683995011yescharacteristic
330C22#1683995311yescharacteristic
331C4—24no—
332C4—21no—
333C4—21no—
334C4—84no—
335C4—4882no—
336C4—4881no—
337C4—4881no—
338C4—2444no—
339C2—21no—
340C2—21no—
341C2—42no—
342C2—42no—
343C2—82no—
344C2#2144610211yescharacteristic
345C2#2144780511yescharacteristic
346C2#2144781711yescharacteristic
subgroupG/H
centerC22#16839950
commutatorC2×D4×C61C22×C4
FrattiniC22×D4C22×D61
FittingC2×C61×#353C2
socleC22×C61B(4,𝔽2)
prime radicalC2×#127×C61C22
cosocleC22×D4×C61C23
hypercenterC2×#353D61
hypocenterC61#12964
LCSG ▷ C2×D4×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. #21446102C2semidirect
4. #21443936C2semidirect
19. C61×#353C4semidirect
50. #99×C61C2×C4semidirect
62. C2×D4×C61C4⋊C4semidirect
80. C42×C61C4⋊C4semidirect
190. #353Dic61semidirect
193. C61#12964semidirect
222. #99C2×Dic61semidirect
236. C2×D4C4⋊Dic61semidirect
270. C42C4⋊Dic61semidirect

Maximal subgroups

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

subgroupindex
#214488842
#214462182
#214461022
#214439362
C2×C61×#3532
#1296461