Finite Group Databasebeta

← #29573554

Subgroups — #29573554

Order 15616.

474 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 474 conjugacy classes of subgroups — too many to draw. Only its 83 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#21448416C211yesmaximal, characteristic
2#21444145C211yesmaximal, characteristic
3#21444145C211yesmaximal, characteristic
4#21443726C211yesmaximal, characteristic, complemented
5#21443725C211yesmaximal, characteristic, complemented
6#21443724C211yesmaximal, characteristic, complemented
7C61×#1305C211yesmaximal, characteristic, complemented
8C22×#11758198C2211yescharacteristic
9#16839726C2211yescharacteristic
10#16839726C2211yescharacteristic
11#16839654—22no—
12#16839652—22no—
13#16839622C2211yescharacteristic
14#16839617C412yes—
15C22⋊C4×Dic61C412yes—
16#16839485—22no—
17#16839485—22no—
18C61×#407C2211yescharacteristic
19C61×#407C2211yescharacteristic
20C61×#380C2211yescharacteristic
21C24×D61C2×C411yescharacteristic
22C23×Dic61C2×C411yescharacteristic
23C2×#11758194—21no—
24C2×#11758194—21no—
25C2×C4⋊Dic61—21no—
26C2×C22⋊Dic61C2×C412yes—
27C2×#11758198—21no—
28C2×#11758198—21no—
29C2×#11758198—21no—
30C2×#11758198—21no—
31C2×#11758198—22no—
32C2×#11758198—22no—
33C2×#11758198D412yes—
34C2×#11758198D412yes—
35C2×C4×Dic61—22no—
36C22⋊#92418—21no—
37C22⋊#92418—21no—
38C22⋊#92418—21no—
39C22⋊#92418—21no—
40#14519684—21no—
41#14519684—22no—
42C23×C4×C61C2311yescharacteristic
43C2×M16×C61—21no—
44C2×M16×C61—21no—
45C2×C4⋊C4×C61—21no—
46C2×C22⋊C4×C61C2×C412yes—
47C22⋊C8×C61—21no—
48C22⋊C8×C61—21no—
49C22⋊C8×C61—21no—
50C22⋊C8×C61—21no—
51#95×C61—21no—
52C24×C61C22×C411yescharacteristic
53C23×D61—21no—
54C23×D61—21no—
55C23×D61—41no—
56C23×D61—41no—
57C23×D61—81no—
58C23×D61—81no—
59C23×D61C22⋊C412yes—
60C22×C4×C61—22no—
61C22×C4×C61—81no—
62C22×C4×C61C2×D411yescharacteristic
63C22×C4×C61C2×D411yescharacteristic
64C22×C4×C61#4111yescharacteristic
65C22×C4×C61#4111yescharacteristic
66C22×C4×C61#4111yescharacteristic
67C22×C4×C61#4111yescharacteristic
68C22×Dic61—21no—
69C22×Dic61—21no—
70C22×Dic61—22no—
71C22×Dic61—81no—
72C22×Dic61C22⋊C412yes—
73M16×C61—81no—
74M16×C61—81no—
75C2×C8×C61—41no—
76C2×C8×C61—41no—
77C4⋊C4×C61—81no—
78C22⋊C4×C61—42no—
79C22⋊Dic61—42no—
80#11758198—22no—
81#11758198—22no—
82#11758198—22no—
83#11758198—22no—
84#11758198—82no—
85#11758198—82no—
86C4⋊Dic61—81no—
87C4×Dic61—82no—
88#11758194—81no—
89#11758194—81no—
90C2×#92418—41no—
91C2×#92418—41no—
92C23×C61—21no—
93C23×C61—21no—
94C23×C61—41no—
95C23×C61—41no—
96C23×C61C2×C22⋊C411yescharacteristic
97C23×C61#11711yescharacteristic
98C23×C61#11711yescharacteristic
99C22×D61—22no—
100C22×D61—22no—
101C22×D61—22no—
102C22×D61—42no—
103C22×D61—42no—
104C22×D61—42no—
105C22×D61—42no—
106C22×D61—81no—
107C22×D61—81no—
108C22×D61—81no—
109C22×D61—81no—
110C22×D61—84no—
111C22×D61—84no—
112C2×C4×C61—41no—
113C2×C4×C61—41no—
114C2×C4×C61—41no—
115C2×C4×C61—41no—
116C2×C4×C61—41no—
117C2×C4×C61—41no—
118C2×C4×C61—42no—
119C2×C4×C61—42no—
120C2×C4×C61—42no—
121C2×C4×C61#12311yescharacteristic
122C2×C4×C61#12311yescharacteristic
123C2×C4×C61#12311yescharacteristic
124C2×C4×C61#12311yescharacteristic
125C2×Dic61—22no—
126C2×Dic61—42no—
127C2×Dic61—42no—
128C2×Dic61—42no—
129C2×Dic61—42no—
130C2×Dic61—81no—
131C2×Dic61—81no—
132C8×C61—81no—
133C8×C61—81no—
134#92418—81no—
135#92418—81no—
136#8759—611nomaximal, Sylow
137C22×C61—21no—
138C22×C61—21no—
139C22×C61—21no—
140C22×C61—21no—
141C22×C61—21no—
142C22×C61—21no—
143C22×C61—21no—
144C22×C61—21no—
145C22×C61—42no—
146C22×C61—42no—
147C22×C61#38111yescharacteristic
148C22×C61#41311yescharacteristic
149C22×C61#41311yescharacteristic
150C2×D61—42no—
151C2×D61—42no—
152C2×D61—42no—
153C2×D61—84no—
154C2×D61—84no—
155C2×D61—82no—
156C2×D61—82no—
157C4×C61—21no—
158C4×C61—21no—
159C4×C61—21no—
160C4×C61—21no—
161C4×C61—82no—
162Dic61—42no—
163Dic61—82no—
164#2312—611no—
165#1462—611no—
166#1462—611no—
167#1305—611no—
168#1305—611no—
169#1305—611no—
170#1305D6111yescharacteristic, complemented
171C2×C61—21no—
172C2×C61—21no—
173C2×C61—41no—
174C2×C61—41no—
175C2×C61#182211yescharacteristic
176C2×C61#182811yescharacteristic
177C2×C61#182911yescharacteristic
178D61—82no—
179C22×C22⋊C4—611no—
180#407—611no—
181#407—611no—
182#407C2×D6111yescharacteristic
183#407C2×D6111yescharacteristic
184#388—1222no—
185#388—1222no—
186#380—611no—
187#380C2×D6111yescharacteristic
188#379—612no—
189C4×C22⋊C4—612no—
190#323—1222no—
191#323—1222no—
192C61#875911yescharacteristic, complemented, Sylow
193C25—611no—
194C23×C4—611no—
195C23×C4C22×D6111yescharacteristic
196C2×M16—21no—
197C2×M16—21no—
198C2×M16—1221no—
199C2×M16—1221no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—1221no—
202C2×C22⋊C4—612no—
203C2×C22⋊C4—612no—
204C2×C22⋊C4—612no—
205C2×C22⋊C4—1221no—
206C2×C22⋊C4—1221no—
207C2×C22⋊C4—1221no—
208C2×C22⋊C4—1221no—
209C2×C22⋊C4—1222no—
210C2×C22⋊C4—1222no—
211C2×C22⋊C4C4×D6112yes—
212C2×C42—1222no—
213C22⋊C8—21no—
214C22⋊C8—21no—
215C22⋊C8—21no—
216C22⋊C8—21no—
217C22⋊C8—1221no—
218C22⋊C8—1221no—
219C22⋊C8—1221no—
220C22⋊C8—1221no—
221#95—21no—
222#95—1222no—
223#95—1221no—
224C24—612no—
225C24—1221no—
226C24—1221no—
227C24—4881no—
228C24—4881no—
229C24—2441no—
230C24—2441no—
231C24C2×C4×D6111yescharacteristic
232C22×C4—22no—
233C22×C4—81no—
234C22×C4—612no—
235C22×C4—1221no—
236C22×C4—1221no—
237C22×C4—1222no—
238C22×C4—4881no—
239C22×C4#1175822211yescharacteristic
240C22×C4#1175822211yescharacteristic
241C22×C4D4×D6111yescharacteristic
242C22×C4D4×D6111yescharacteristic
243C22×C4#1175822411yescharacteristic
244C22×C4#1175822611yescharacteristic
245M16—81no—
246M16—81no—
247M16—4881no—
248M16—4881no—
249C2×C8—41no—
250C2×C8—41no—
251C2×C8—2441no—
252C2×C8—2441no—
253C4⋊C4—81no—
254C4⋊C4—4881no—
255C22⋊C4—42no—
256C22⋊C4—1222no—
257C22⋊C4—1222no—
258C22⋊C4—1222no—
259C22⋊C4—1222no—
260C22⋊C4—4882no—
261C22⋊C4—4882no—
262C22⋊C4—2442no—
263C42—4882no—
264C23—21no—
265C23—21no—
266C23—41no—
267C23—41no—
268C23—1222no—
269C23—1222no—
270C23—1222no—
271C23—4881no—
272C23—4881no—
273C23—4881no—
274C23—4881no—
275C23—4884no—
276C23—4884no—
277C23—2442no—
278C23—2442no—
279C23—2442no—
280C23—2442no—
281C23C22⋊C4×D6111yescharacteristic
282C23#1451973411yescharacteristic
283C23#1451973511yescharacteristic
284C2×C4—41no—
285C2×C4—41no—
286C2×C4—41no—
287C2×C4—41no—
288C2×C4—41no—
289C2×C4—41no—
290C2×C4—42no—
291C2×C4—42no—
292C2×C4—42no—
293C2×C4—1222no—
294C2×C4—4881no—
295C2×C4—4881no—
296C2×C4—2442no—
297C2×C4—2442no—
298C2×C4—2442no—
299C2×C4—2442no—
300C2×C4#1451975911yescharacteristic
301C2×C4#1451975911yescharacteristic
302C2×C4#1451976211yescharacteristic
303C2×C4#1451976211yescharacteristic
304C8—81no—
305C8—81no—
306C8—4881no—
307C8—4881no—
308C22—21no—
309C22—21no—
310C22—21no—
311C22—21no—
312C22—21no—
313C22—21no—
314C22—21no—
315C22—21no—
316C22—42no—
317C22—42no—
318C22—4882no—
319C22—4882no—
320C22—4884no—
321C22—4884no—
322C22—2442no—
323C22—2442no—
324C22—2442no—
325C22#1683962411yescharacteristic
326C22#1683974311yescharacteristic
327C22#1683974511yescharacteristic
328C4—21no—
329C4—21no—
330C4—21no—
331C4—21no—
332C4—82no—
333C4—4882no—
334C4—2442no—
335C2—21no—
336C2—21no—
337C2—41no—
338C2—41no—
339C2—4882no—
340C2#2144590111yescharacteristic
341C2#2144592611yescharacteristic
342C2#2144592911yescharacteristic
subgroupG/H
centerC22#16839624
commutatorC24×C61C22×C4
FrattiniC23×C4C22×D61
FittingC61×#1305C2
socleC22×C61#381
prime radicalC24×D61C2×C4
cosocleC23×C4×C61C23
hypercenter#1305D61
hypocenterC61#8759
LCSG ▷ C24×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1305

Complements

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

NH ≅ G/Ntype
4. #21443726C2semidirect
5. #21443725C2semidirect
6. #21443724C2semidirect
7. C61×#1305C2semidirect
170. #1305D61semidirect
192. C61#8759semidirect

Maximal subgroups

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

subgroupindex
#214484162
#214441452
#214437262
#214437252
#214437242
C61×#13052
#875961