Finite Group Databasebeta

← #30594651

Subgroups — #30594651

Order 15616.

402 conjugacy classes of subgroups; only the 56 normal subgroups are shown.

normal subgroups only

This group has 402 conjugacy classes of subgroups — too many to draw. Only its 56 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#21461944C211yesmaximal, characteristic
2#21461566C211yesmaximal, characteristic
3#21445606C211yesmaximal, characteristic
4C2×C4⋊#92420C411yescharacteristic, complemented
5C2×#14519835—21no—
6C2×#14519834—21no—
7C2×C22⋊#92420—21no—
8C2×#14519832C411yescharacteristic, complemented
9C4×C22⋊#6923—21no—
10#16840640C412yescomplemented
11C2×#14519737C2211yescharacteristic
12C2×#14519691—21no—
13C22⋊C4×Dic61—21no—
14#16839529C411yescharacteristic, complemented
15C61×#342C411yescharacteristic, complemented
16C22×#92420—21no—
17C22×#92420—21no—
18C2×C22⋊#6923—22no—
19C2×C4×#6923—41no—
20C4⋊#92420—41no—
21#14519835—22no—
22#14519835—22no—
23#14519834—42no—
24C22⋊#92420—42no—
25#14519832—41no—
26(C2×#6923)⋊C4—41no—
27C22×C22⋊D61D411yescharacteristic
28C23×Dic61D411yescharacteristic
29C22×C4×D61D411yescharacteristic
30#14519737—41no—
31#14519737—41no—
32C2×#11758196Q811yescharacteristic
33C2×C22⋊Dic61—21no—
34C2×#11758198C2×C411yescharacteristic
35C2×C4×Dic61—41no—
36C2×C4×Dic61C2×C411yescharacteristic
37#14519691—24no—
38#14519684—41no—
39C2×C22⋊C4×C61—21no—
40C2×C22⋊C4×C61C2×C411yescharacteristic
41C24×C61C22⋊C411yescharacteristic
42C23×D61C22⋊C411yescharacteristic
43C22×#6923—42no—
44C22×C4×C61—41no—
45C22×C4×C61C4211yescharacteristic
46C22×C4×C61C4⋊C411yescharacteristic
47C2×C22⋊D61—21no—
48C2×C22⋊D61—21no—
49C2×C22⋊D61—22no—
50C2×C22⋊D61—41no—
51C22×Dic61—21no—
52C22×Dic61—21no—
53C22×Dic61—41no—
54C22×Dic61—41no—
55C22×Dic61—41no—
56C22×Dic61C22⋊C411yescharacteristic
57C22×Dic61C4⋊C411yescharacteristic
58C22×Dic61C4⋊C411yescharacteristic
59C2×C4×D61—21no—
60C2×C4×D61—21no—
61C2×C4×D61—41no—
62C2×C4×D61—41no—
63C22⋊#6923—24no—
64C22⋊#6923—42no—
65C2×#92420—22no—
66C2×#92420—41no—
67C2×#92420—41no—
68C2×#92420—41no—
69C2×#92420—41no—
70C2×#92420—41no—
71C4×#6923—81no—
72C4×#6923—81no—
73C22⋊C4×C61—22no—
74C22⋊C4×C61—41no—
75C22⋊C4×C61—41no—
76C22⋊Dic61—22no—
77C22⋊Dic61—41no—
78#11758198—41no—
79#11758196—41no—
80C4×Dic61—21no—
81C4×Dic61—21no—
82C4×Dic61—81no—
83C4×Dic61—81no—
84C23×C61—21no—
85C23×C61—21no—
86C23×C61—41no—
87C23×C61—41no—
88C23×C61#9511yescharacteristic
89C23×C61#10411yescharacteristic
90C23×C61#10411yescharacteristic
91C22×D61—21no—
92C22×D61—21no—
93C22×D61—41no—
94C22×D61—41no—
95C22×D61#10412yes—
96C2×#6923—44no—
97C2×#6923—82no—
98C2×#6923—82no—
99C2×C4×C61—21no—
100C2×C4×C61—21no—
101C2×C4×C61—41no—
102C2×C4×C61—41no—
103C2×C4×C61—42no—
104C2×C4×C61—81no—
105C2×C4×C61—81no—
106C22⋊D61—42no—
107C22⋊D61—42no—
108C2×Dic61—21no—
109C2×Dic61—21no—
110C2×Dic61—21no—
111C2×Dic61—21no—
112C2×Dic61—21no—
113C2×Dic61—21no—
114C2×Dic61—41no—
115C2×Dic61—41no—
116C2×Dic61—41no—
117C2×Dic61—42no—
118C2×Dic61—81no—
119C2×Dic61—81no—
120C2×Dic61—81no—
121C2×Dic61#9911yescharacteristic
122C2×Dic61#9911yescharacteristic
123C4×D61—41no—
124C4×D61—41no—
125C4×D61—81no—
126#92420—42no—
127#92420—81no—
128#7283—611nomaximal, Sylow
129C22×C61—21no—
130C22×C61—21no—
131C22×C61—21no—
132C22×C61—21no—
133C22×C61—41no—
134C22×C61—41no—
135C22×C61—41no—
136C22×C61—41no—
137C22×C61—81no—
138C22×C61#33911yescharacteristic
139C22×C61#33911yescharacteristic
140C22×C61#34211yescharacteristic
141C2×D61—22no—
142C2×D61—41no—
143C2×D61—41no—
144C2×D61—42no—
145C2×D61—81no—
146#6923—84no—
147C4×C61—41no—
148C4×C61—41no—
149C4×C61—82no—
150Dic61—21no—
151Dic61—41no—
152Dic61—41no—
153Dic61—82no—
154Dic61#35311yescharacteristic
155Dic61#35411yescharacteristic
156#1955—611no—
157C4×#99—611no—
158C4×#99—611no—
159C2×C61—21no—
160C2×C61—21no—
161C2×C61—41no—
162C2×C61—41no—
163C2×C61#129711yescharacteristic
164C2×C61#139111yescharacteristic
165C2×C61#139311yescharacteristic
166D61—42no—
167C2×C4×D4—611no—
168C2×C4⋊C8—611no—
169C2×#99—1221no—
170C2×#99—1221no—
171C2×C22⋊C8—1221no—
172C2×C22⋊C8—1221no—
173C2×#97—611no—
174C4×C22⋊C4—1221no—
175C4×C22⋊C4—1221no—
176#342—612no—
177#342—611no—
178#342#692311yescharacteristic, complemented
179C61#728311yescharacteristic, complemented, Sylow
180C22×D4—611no—
181C23×C4—611no—
182C23×C4—611no—
183C22×C8—1221no—
184C22×C8—1221no—
185C4×D4—2441no—
186C4×D4—2441no—
187C2×C4⋊C4—611no—
188C2×C22⋊C4—21no—
189C2×C22⋊C4—611no—
190C2×C22⋊C4—1222no—
191C2×C22⋊C4—1221no—
192C2×C22⋊C4C2×#692311yescharacteristic
193C2×C42—611no—
194C2×C42—2441no—
195C2×C42—2441no—
196C4⋊C8—2441no—
197#99—1224no—
198#99—1222no—
199#99—1222no—
200C22⋊C8—2442no—
201C22⋊C8—2442no—
202#97—2441no—
203#95—2441no—
204#95—2441no—
205C24—611no—
206C24C22⋊#692311yescharacteristic
207C2×D4—1222no—
208C2×D4—1221no—
209C2×D4—1221no—
210C2×D4—2441no—
211C22×C4—41no—
212C22×C4—611no—
213C22×C4—611no—
214C22×C4—611no—
215C22×C4—1221no—
216C22×C4—1221no—
217C22×C4—1221no—
218C22×C4—1221no—
219C22×C4—2441no—
220C22×C4—2441no—
221C22×C4—2441no—
222C22×C4—2441no—
223C22×C4—2441no—
224C22×C4—2442no—
225C22×C4C4×#692311yescharacteristic
226C22×C4C4⋊#692311yescharacteristic
227C2×C8—1222no—
228C2×C8—2441no—
229C2×C8—2441no—
230C2×C8—2441no—
231C2×C8—2441no—
232C2×C8—2441no—
233C4⋊C4—2441no—
234C22⋊C4—22no—
235C22⋊C4—41no—
236C22⋊C4—41no—
237C22⋊C4—1222no—
238C22⋊C4—1224no—
239C22⋊C4—2441no—
240C22⋊C4—2442no—
241C22⋊C4—2441no—
242C42—1221no—
243C42—1221no—
244C42—4881no—
245C42—4881no—
246C42—4881no—
247C42—4881no—
248C23—21no—
249C23—21no—
250C23—41no—
251C23—41no—
252C23—612no—
253C23—1221no—
254C23—1221no—
255C23—2441no—
256C23—2441no—
257C23(C2×#6923)⋊C411yescharacteristic
258C23#1451984111yescharacteristic
259C23#1451984211yescharacteristic
260D4—2442no—
261D4—2442no—
262C2×C4—21no—
263C2×C4—21no—
264C2×C4—41no—
265C2×C4—41no—
266C2×C4—42no—
267C2×C4—81no—
268C2×C4—81no—
269C2×C4—611no—
270C2×C4—611no—
271C2×C4—1221no—
272C2×C4—1221no—
273C2×C4—1221no—
274C2×C4—1221no—
275C2×C4—1221no—
276C2×C4—1221no—
277C2×C4—4881no—
278C2×C4—4881no—
279C2×C4—4881no—
280C2×C4—4881no—
281C2×C4—4882no—
282C2×C4—4882no—
283C2×C4—2441no—
284C2×C4—2441no—
285C2×C4—2441no—
286C2×C4—2441no—
287C2×C4—2442no—
288C2×C4—2444no—
289C2×C4—2441no—
290C8—4881no—
291C8—2442no—
292C22—21no—
293C22—21no—
294C22—21no—
295C22—21no—
296C22—41no—
297C22—41no—
298C22—41no—
299C22—41no—
300C22—81no—
301C22—1222no—
302C22—4881no—
303C22—2442no—
304C22—2441no—
305C22—2441no—
306C22#1684063211yescharacteristic
307C22#1684063311yescharacteristic
308C22#1684063911yescharacteristic
309C4—41no—
310C4—41no—
311C4—82no—
312C4—611no—
313C4—611no—
314C4—1221no—
315C4—4882no—
316C4—4884no—
317C4—2441no—
318C4—2441no—
319C2—21no—
320C2—21no—
321C2—41no—
322C2—41no—
323C2—2442no—
324C2#2146070011yescharacteristic
325C2#2146090611yescharacteristic
326C2#2146091211yescharacteristic
subgroupG/H
centerC22#16840639
commutatorC22×C4×C61C42
FrattiniC2×C22⋊C4C2×#6923
FittingC61×#342C4
socleC22×C61#342
prime radicalC22×C22⋊D61D4
cosocleC2×#14519737C22
hypercenter#342#6923
hypocenterC61#7283
LCSG ▷ C22×C4×C61 ▷ C23×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23 ◁ C2×C22⋊C4 ◁ #342

Complements

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

NH ≅ G/Ntype
4. C2×C4⋊#92420C4semidirect
8. C2×#14519832C4semidirect
10. #16840640C4semidirect
14. #16839529C4semidirect
15. C61×#342C4semidirect
178. #342#6923semidirect
179. C61#7283semidirect

Maximal subgroups

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

subgroupindex
#214619442
#214615662
#214456062
#728361