Finite Group Databasebeta

← #30605471

Subgroups — #30605471

Order 15616.

452 conjugacy classes of subgroups; only the 80 normal subgroups are shown.

normal subgroups only

This group has 452 conjugacy classes of subgroups — too many to draw. Only its 80 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#21461679C211yesmaximal, characteristic
2#21461058C211yesmaximal, characteristic
3#21460816C214yesmaximal, complemented
4#21450130C211yesmaximal, characteristic
5#16840779—21no—
6C2×#14519835—21no—
7#16840704—21no—
8#16840640C2212yes—
9#16840606C2212yes—
10#16840605—21no—
11C2×C4×C22⋊D61C2211yescharacteristic
12#16839700C412yescomplemented
13#16839697C2212yes—
14#16839658—21no—
15#16839641—21no—
16#16839635—21no—
17#16839622C411yescharacteristic, complemented
18#16839619—21no—
19C22×C4⋊C4×C61C411yescharacteristic, complemented
20C2×C4⋊#6923—41no—
21C2×C22⋊#6923—22no—
22#14519844—41no—
23#14519835—42no—
24C22⋊#92420—22no—
25C4×#92420—41no—
26(C2×#6923)⋊C4—41no—
27C22×C22⋊D61D411yescharacteristic
28C22×C4×D61D411yescharacteristic
29C4×C22⋊D61—41no—
30C4×C22⋊D61—41no—
31C2×C4⋊Dic61—21no—
32C2×#11758196—22no—
33C2×#11758196C2×C411yescharacteristic
34C2×C22⋊Dic61—21no—
35C2×C22⋊Dic61D411yescharacteristic
36C2×#11758198—22no—
37C2×#11758198C2311yescharacteristic
38C2×C4×Dic61D411yescharacteristic
39#14519684—22no—
40#14519684—22no—
41#14519684C2×C412yes—
42C23×C4×C61—21no—
43C23×C4×C61C2×C411yescharacteristic
44C2×C4⋊C4×C61—21no—
45C2×C4⋊C4×C61—41no—
46C2×C4⋊C4×C61—41no—
47C2×C4⋊C4×C61C2×C412yes—
48C24×C61C22⋊C411yescharacteristic
49C23×D61C2×D411yescharacteristic
50C22×#6923—42no—
51C22×C4×C61—21no—
52C22×C4×C61—21no—
53C22×C4×C61—22no—
54C22×C4×C61—42no—
55C22×C4×C61—42no—
56C22×C4×C61—82no—
57C22×C4×C61C22⋊C411yescharacteristic
58C22×C4×C61C22×C411yescharacteristic
59C2×C22⋊D61—41no—
60C2×C22⋊D61—41no—
61C2×C22⋊D61SD1612yes—
62C2×C22⋊D61Q1612yes—
63C22×Dic61—22no—
64C22×Dic61C22⋊C411yescharacteristic
65C22×Dic61C22⋊C411yescharacteristic
66C22×Dic61C2×D411yescharacteristic
67C2×C4×D61—21no—
68C2×C4×D61—21no—
69C2×C4×D61—42no—
70C22⋊#6923—42no—
71C22⋊#6923—42no—
72C2×#92420—42no—
73C4⋊#6923—82no—
74C4⋊C4×C61—42no—
75C4⋊C4×C61—41no—
76C4⋊C4×C61—41no—
77C22⋊Dic61—41no—
78C22⋊Dic61—42no—
79#11758198—41no—
80#11758198—82no—
81C4⋊Dic61—42no—
82#11758196—41no—
83#11758196—82no—
84C4×Dic61—21no—
85C4×Dic61—21no—
86C23×C61—21no—
87C23×C61—21no—
88C23×C61—41no—
89C23×C61—41no—
90C23×C61#10312yes—
91C23×C61C2×C22⋊C411yescharacteristic
92C22×D61—21no—
93C22×D61—21no—
94C22×D61—41no—
95C22×D61—41no—
96C22×D61C2×SD1611yescharacteristic
97C22×D61C2×Q1611yescharacteristic
98C2×#6923—82no—
99C2×#6923—82no—
100C2×#6923—82no—
101C2×C4×C61—22no—
102C2×C4×C61—21no—
103C2×C4×C61—42no—
104C2×C4×C61—42no—
105C2×C4×C61—41no—
106C2×C4×C61—42no—
107C2×C4×C61—42no—
108C2×C4×C61—42no—
109C2×C4×C61—84no—
110C2×C4×C61#9911yescharacteristic
111C2×C4×C61#9911yescharacteristic
112C22⋊D61—42no—
113C22⋊D61—42no—
114C2×Dic61—21no—
115C2×Dic61—21no—
116C2×Dic61—21no—
117C2×Dic61—21no—
118C2×Dic61—41no—
119C2×Dic61—41no—
120C2×Dic61—42no—
121C2×Dic61—42no—
122C2×Dic61—42no—
123C2×Dic61#10312yes—
124C4×D61—42no—
125C4×D61—81no—
126#92420—82no—
127#11899—611nomaximal, Sylow
128C22×C61—21no—
129C22×C61—21no—
130C22×C61—22no—
131C22×C61—42no—
132C22×C61—42no—
133C22×C61—41no—
134C22×C61—41no—
135C22×C61C2×#9911yescharacteristic
136C22×C61C2×#10311yescharacteristic
137C22×C61#42111yescharacteristic
138C2×D61—21no—
139C2×D61—21no—
140C2×D61—41no—
141C2×D61—41no—
142C2×D61—41no—
143C2×D61—41no—
144C2×D61—81no—
145#6923—162no—
146C4×C61—21no—
147C4×C61—41no—
148C4×C61—82no—
149C4×C61#35212yes—
150Dic61—22no—
151Dic61—41no—
152Dic61—41no—
153Dic61—82no—
154#2487—611no—
155#1799—611no—
156#1460—611no—
157#1352—614no—
158C2×C61—21no—
159C2×C61—21no—
160C2×C61—42no—
161C2×C61#152011yescharacteristic
162C2×C61C2×#35211yescharacteristic
163C2×C61#212811yescharacteristic
164D61—41no—
165D61—41no—
166C2×C4×D4—611no—
167C22×C4⋊C4#692311yescharacteristic, complemented
168#431—1221no—
169C2×#99—1221no—
170#400—612no—
171#399—612no—
172#389—1221no—
173#386—1221no—
174#385—1221no—
175#380—611no—
176#380—1221no—
177#380—1221no—
178#342—612no—
179#323—612no—
180#323—1221no—
181C61#1189911yescharacteristic, complemented, Sylow
182C22×D4—611no—
183C23×C4—21no—
184C23×C4—611no—
185C23×C4C2×#692311yescharacteristic
186C4×D4—2441no—
187C4×D4—2441no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—41no—
190C2×C4⋊C4—41no—
191C2×C4⋊C4—611no—
192C2×C4⋊C4—1222no—
193C2×C4⋊C4—1221no—
194C2×C4⋊C4—2441no—
195C2×C4⋊C4C2×#692312yes—
196C2×C22⋊C4—611no—
197C2×C22⋊C4—611no—
198C2×C22⋊C4—1222no—
199C2×C22⋊C4—1221no—
200C2×C22⋊C4—1222no—
201C2×C42—611no—
202C4⋊C8—2441no—
203#99—2442no—
204C22⋊C8—1222no—
205C4×C8—2441no—
206#95—612no—
207#95—1222no—
208#95—1222no—
209#95—2441no—
210C24—611no—
211C24C22⋊#692311yescharacteristic
212C2×D4—612no—
213C2×D4—612no—
214C2×D4—2441no—
215C2×D4—2441no—
216C22×C4—21no—
217C22×C4—21no—
218C22×C4—22no—
219C22×C4—42no—
220C22×C4—42no—
221C22×C4—82no—
222C22×C4—611no—
223C22×C4—611no—
224C22×C4—611no—
225C22×C4—1221no—
226C22×C4—1221no—
227C22×C4—1222no—
228C22×C4—2442no—
229C22×C4—2442no—
230C22×C4C22⋊#692311yescharacteristic
231C22×C4C22×#692311yescharacteristic
232C2×C8—2442no—
233C4⋊C4—42no—
234C4⋊C4—41no—
235C4⋊C4—41no—
236C4⋊C4—4882no—
237C4⋊C4—4882no—
238C4⋊C4—2441no—
239C4⋊C4—2442no—
240C22⋊C4—4882no—
241C22⋊C4—2441no—
242C22⋊C4—2441no—
243C22⋊C4—2442no—
244C22⋊C4—2442no—
245C22⋊C4—2442no—
246C42—1221no—
247C42—1221no—
248C23—21no—
249C23—21no—
250C23—41no—
251C23—41no—
252C23—611no—
253C23—611no—
254C23—1221no—
255C23—1221no—
256C23—2441no—
257C23—2441no—
258C23#1451984012yes—
259C23C2×C22⋊#692311yescharacteristic
260D4—2442no—
261D4—2442no—
262C2×C4—22no—
263C2×C4—21no—
264C2×C4—42no—
265C2×C4—42no—
266C2×C4—41no—
267C2×C4—42no—
268C2×C4—42no—
269C2×C4—42no—
270C2×C4—84no—
271C2×C4—612no—
272C2×C4—1221no—
273C2×C4—1221no—
274C2×C4—1221no—
275C2×C4—1221no—
276C2×C4—4881no—
277C2×C4—4882no—
278C2×C4—4882no—
279C2×C4—4882no—
280C2×C4—2442no—
281C2×C4—2442no—
282C2×C4—2442no—
283C2×C4—2442no—
284C2×C4—2441no—
285C2×C4—2441no—
286C2×C4#1451983511yescharacteristic
287C2×C4#1451983511yescharacteristic
288C8—4882no—
289C22—21no—
290C22—21no—
291C22—22no—
292C22—42no—
293C22—42no—
294C22—41no—
295C22—41no—
296C22—1221no—
297C22—1221no—
298C22—4881no—
299C22—2441no—
300C22—2441no—
301C22—2441no—
302C22—2441no—
303C22C2×#1451983511yescharacteristic
304C22C2×#1451984011yescharacteristic
305C22#1684076111yescharacteristic
306C4—21no—
307C4—41no—
308C4—82no—
309C4—1222no—
310C4—9762no—
311C4—4882no—
312C4—2441no—
313C4—2441no—
314C4#1684066412yes—
315C2—21no—
316C2—21no—
317C2—42no—
318C2—2441no—
319C2—2441no—
320C2#2146125911yescharacteristic
321C2C2×#1684066411yescharacteristic
322C2#2146209811yescharacteristic
subgroupG/H
centerC22C2×#14519835
commutatorC22×C4×C61C22×C4
FrattiniC22×C4C22×#6923
FittingC22×C4⋊C4×C61C4
socleC22×C61C2×#99
prime radicalC22×C22⋊D61D4
cosocleC2×#11758198C23
hypercenterC22×C4⋊C4#6923
hypocenterC61#11899
LCSG ▷ C22×C4×C61 ▷ C23×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ C23×C4 ◁ C22×C4⋊C4

Complements

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

NH ≅ G/Ntype
3. #21460816C2semidirect
12. #16839700C4semidirect
17. #16839622C4semidirect
19. C22×C4⋊C4×C61C4semidirect
167. C22×C4⋊C4#6923semidirect
181. C61#11899semidirect

Maximal subgroups

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

subgroupindex
#214616792
#214610582
#214608162
#214501302
#1189961