Finite Group Databasebeta

← #30604994

Subgroups — #30604994

Order 15616.

446 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 446 conjugacy classes of subgroups — too many to draw. Only its 70 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#21461522C211yesmaximal, characteristic
2#21461068C211yesmaximal, characteristic
3#21460817C212yesmaximal, complemented
4#21460803C212yesmaximal, complemented
5#21450101C211yesmaximal, characteristic
6C2×D61⋊M16—21no—
7#16840733—21no—
8#16840642—21no—
9#16840642C2212yes—
10#16840608C2212yes—
11#16840607—21no—
12C2×#14519735C411yescharacteristic
13C2×C4⋊C4×D61C2211yescharacteristic
14#16839702C2211yescharacteristic
15#16839701C411yescharacteristic
16#16839693C2211yescharacteristic
17#16839648—21no—
18#16839644—21no—
19#16839633—21no—
20#16839624—21no—
21#16839624C411yescharacteristic
22C61×#398C411yescharacteristic
23C2×#11758213—41no—
24C2×#11758213—41no—
25D61⋊M16—24no—
26D61⋊M16—42no—
27C2×#11758217—22no—
28C2×D61⋊C8—41no—
29#14519834—41no—
30C22⋊#92420—22no—
31C22×C4×D61C2×C411yescharacteristic
32C22×C4×D61D411yescharacteristic
33C22×C4×D61D411yescharacteristic
34#14519735—41no—
35#14519735—41no—
36C2×C4⋊Dic61C2×C411yescharacteristic
37C2×C4⋊Dic61D411yescharacteristic
38C2×#11758196—21no—
39C2×#11758196D411yescharacteristic
40C2×#11758196C2311yescharacteristic
41C4⋊C4×D61—41no—
42C4⋊C4×D61—41no—
43C2×#11758198—21no—
44C2×#11758198—21no—
45C2×C4×Dic61—21no—
46#14519684—21no—
47#14519684—21no—
48#14519684—21no—
49#14519684—21no—
50#14519684—21no—
51#14519684—21no—
52#14519684C2×C411yescharacteristic
53C2×C4⋊C4×C61—21no—
54C2×C4⋊C4×C61C2×C411yescharacteristic
55C2×C4⋊C4×C61C2×C411yescharacteristic
56#95×C61—21no—
57#95×C61C2×C411yescharacteristic
58C23×D61C22⋊C411yescharacteristic
59C22×C4×C61—21no—
60C22×C4×C61—21no—
61C22×C4×C61C22⋊C411yescharacteristic
62C22×C4×C61C22⋊C411yescharacteristic
63C22×C4×C61C22×C411yescharacteristic
64C22×Dic61—21no—
65C22×Dic61—21no—
66C22×Dic61C22⋊C411yescharacteristic
67C22×Dic61C2×D411yescharacteristic
68C22×Dic61C2×D411yescharacteristic
69C2×C4×D61—21no—
70C2×C4×D61—21no—
71C2×C4×D61—22no—
72C2×C4×D61—21no—
73C2×C4×D61—21no—
74C2×C4×D61—21no—
75C2×C4×D61—21no—
76C2×C4×D61—21no—
77C2×C4×D61—21no—
78C2×C4×D61—42no—
79C2×C4×D61—42no—
80#11758217—24no—
81#11758217—42no—
82C2×#92420—42no—
83C2×#92420—42no—
84#11758213—44no—
85#11758213—82no—
86D61⋊C8—44no—
87C4⋊C4×C61—41no—
88C4⋊C4×C61—41no—
89C4⋊C4×C61—42no—
90#11758198—42no—
91#11758198—41no—
92#11758198—41no—
93C4⋊Dic61—41no—
94C4⋊Dic61—41no—
95#11758196—41no—
96#11758196—41no—
97#11758196—41no—
98#11758196—41no—
99C4×Dic61—42no—
100C23×C61C2×C22⋊C411yescharacteristic
101C22×D61—21no—
102C22×D61—21no—
103C22×D61—22no—
104C22×D61—41no—
105C22×D61#9911yescharacteristic
106C22×D61#9911yescharacteristic
107C2×C4×C61—21no—
108C2×C4×C61—21no—
109C2×C4×C61—21no—
110C2×C4×C61—21no—
111C2×C4×C61—41no—
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—41no—
119C2×C4×C61#10412yes—
120C2×Dic61—21no—
121C2×Dic61—21no—
122C2×Dic61—21no—
123C2×Dic61—21no—
124C2×Dic61—41no—
125C2×Dic61—41no—
126C2×Dic61—41no—
127C2×Dic61—41no—
128C2×Dic61—42no—
129C2×Dic61—41no—
130C2×Dic61—41no—
131C2×Dic61AGL(1,ℤ/8)11yescharacteristic
132C2×Dic61#13711yescharacteristic
133C4×D61—22no—
134C4×D61—22no—
135C4×D61—22no—
136C4×D61—22no—
137C4×D61—42no—
138C4×D61—42no—
139C4×D61—42no—
140C4×D61—81no—
141#92420—44no—
142#92420—82no—
143#11764—611nomaximal, Sylow
144C22×C61—21no—
145C22×C61—21no—
146C22×C61C2×#9911yescharacteristic
147C22×C61#41711yescharacteristic
148C22×C61C2×#10411yescharacteristic
149C2×D61—21no—
150C2×D61—21no—
151C2×D61—21no—
152C2×D61—21no—
153C2×D61—21no—
154C2×D61—42no—
155C2×D61—42no—
156C2×D61#35212yes—
157C4×C61—22no—
158C4×C61—41no—
159C4×C61—41no—
160C4×C61—81no—
161C4×C61—81no—
162Dic61—21no—
163Dic61—21no—
164Dic61—41no—
165Dic61—41no—
166Dic61—81no—
167Dic61—81no—
168#2484—611no—
169C2×#343—611no—
170#1463—611no—
171#1353—612no—
172#1346—612no—
173C2×C61—21no—
174C2×C61—21no—
175C2×C61#150411yescharacteristic
176C2×C61C2×#35211yescharacteristic
177C2×C61#213311yescharacteristic
178D61—22no—
179D61—41no—
180C22×M16—1221no—
181C2×#117—611no—
182C22×C4⋊C4—611no—
183#407—1221no—
184#400—611no—
185#400—611no—
186#398—611no—
187#398#692311yescharacteristic
188#387—1221no—
189#387—1221no—
190#385—1221no—
191#381—611no—
192#381—1221no—
193#343—612no—
194#343—1221no—
195#324—612no—
196#324—1221no—
197C61#1176411yescharacteristic, complemented, Sylow
198C23×C4—611no—
199C23×C4—611no—
200C23×C4—611no—
201C2×M16—1222no—
202C2×M16—1224no—
203C2×M16—2441no—
204C2×M16—2442no—
205C2×M16—2441no—
206C22×C8—2441no—
207#117—2441no—
208#117—2441no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—611no—
211C2×C4⋊C4—611no—
212C2×C4⋊C4—611no—
213C2×C4⋊C4—611no—
214C2×C4⋊C4—1221no—
215C2×C4⋊C4—2441no—
216C2×C4⋊C4—2441no—
217C2×C4⋊C4C2×#692311yescharacteristic
218C2×C4⋊C4C2×#692311yescharacteristic
219C2×C22⋊C4—1221no—
220C2×C22⋊C4—1221no—
221C2×C42—1221no—
222C22⋊C8—1222no—
223C22⋊C8—2441no—
224#95—21no—
225#95—611no—
226#95—1221no—
227#95—1221no—
228#95—1221no—
229#95—1221no—
230#95—1221no—
231#95—1221no—
232#95C2×#692311yescharacteristic
233C24—611no—
234C22×C4—21no—
235C22×C4—21no—
236C22×C4—611no—
237C22×C4—611no—
238C22×C4—611no—
239C22×C4—1221no—
240C22×C4—1221no—
241C22×C4—1221no—
242C22×C4—1221no—
243C22×C4—1221no—
244C22×C4—1221no—
245C22×C4—1221no—
246C22×C4—1221no—
247C22×C4—1222no—
248C22×C4—1221no—
249C22×C4—1221no—
250C22×C4—2442no—
251C22×C4—2442no—
252C22×C4C22⋊#692311yescharacteristic
253C22×C4C22⋊#692311yescharacteristic
254C22×C4C22×#692311yescharacteristic
255M16—1224no—
256M16—4882no—
257M16—2442no—
258M16—2444no—
259C2×C8—2442no—
260C2×C8—2444no—
261C2×C8—2442no—
262C4⋊C4—41no—
263C4⋊C4—41no—
264C4⋊C4—42no—
265C4⋊C4—2441no—
266C4⋊C4—2441no—
267C4⋊C4—2441no—
268C4⋊C4—2441no—
269C4⋊C4—2441no—
270C4⋊C4—2441no—
271C22⋊C4—2441no—
272C22⋊C4—2441no—
273C22⋊C4—2442no—
274C42—2442no—
275C23—611no—
276C23—611no—
277C23—1222no—
278C23—1221no—
279C23—1221no—
280C23—2441no—
281C23C2×C22⋊#692311yescharacteristic
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—41no—
287C2×C4—41no—
288C2×C4—41no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—611no—
295C2×C4—611no—
296C2×C4—1222no—
297C2×C4—1222no—
298C2×C4—1222no—
299C2×C4—1222no—
300C2×C4—1221no—
301C2×C4—1221no—
302C2×C4—1221no—
303C2×C4—1221no—
304C2×C4—4881no—
305C2×C4—2442no—
306C2×C4—2442no—
307C2×C4—2442no—
308C2×C4—2441no—
309C2×C4—2441no—
310C2×C4—2442no—
311C2×C4—2441no—
312C2×C4—2441no—
313C2×C4—2441no—
314C2×C4—2441no—
315C2×C4#1451984112yes—
316C8—4882no—
317C8—2444no—
318C22—21no—
319C22—21no—
320C22—612no—
321C22—1221no—
322C22—1221no—
323C22—1221no—
324C22—1221no—
325C22—1221no—
326C22—2442no—
327C22—2442no—
328C22D61⋊#9911yescharacteristic
329C22#1684075311yescharacteristic
330C22C2×#1451984111yescharacteristic
331C4—22no—
332C4—41no—
333C4—41no—
334C4—81no—
335C4—81no—
336C4—1221no—
337C4—1221no—
338C4—4881no—
339C4—4881no—
340C4—2441no—
341C4—2441no—
342C2—21no—
343C2—21no—
344C2—1222no—
345C2—2441no—
346C2#2146120411yescharacteristic
347C2D61⋊#35211yescharacteristic
348C2#2146211711yescharacteristic
subgroupG/H
centerC22D61⋊#99
commutatorC22×C4×C61C22×C4
FrattiniC22×C4C22×#6923
FittingC61×#398C4
socleC22×C61C2×#99
prime radicalC23×D61C22⋊C4
cosocleC2×#11758196C23
hypercenter#398#6923
hypocenterC61#11764
LCSG ▷ C22×C4×C61 ▷ C23×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #398

Complements

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

NH ≅ G/Ntype
3. #21460817C2semidirect
4. #21460803C2semidirect
197. C61#11764semidirect

Maximal subgroups

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

subgroupindex
#214615222
#214610682
#214608172
#214608032
#214501012
#1176461