Finite Group Databasebeta

← #30594649

Subgroups — #30594649

Order 15616.

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

normal subgroups only

This group has 388 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#21461799C211yesmaximal, characteristic
2#21461564C211yesmaximal, characteristic
3#21445610C211yesmaximal, characteristic
4C22×#11758213C411yescharacteristic, complemented
5C2×#14519836—21no—
6C2×#14519834—21no—
7C2×#14519834C411yescharacteristic, complemented
8C4×C22⋊#6923—21no—
9#16840640C412yes—
10#16840628—21no—
11C2×#14519739C2211yescharacteristic
12C2×#14519693—21no—
13M16×Dic61—21no—
14#16839531C411yescharacteristic, complemented
15C61×#343C411yescharacteristic, complemented
16C2×#11758213—41no—
17C2×#11758213—42no—
18C22×#92420—21no—
19C22×#92420—21no—
20C2×C22⋊#6923—22no—
21C2×C4×#6923—41no—
22#14519836—22no—
23#14519836—22no—
24#14519834—42no—
25#14519834—41no—
26(C2×#6923)⋊C4—41no—
27C22×D244D411yescharacteristic
28C22×C4×D61C2×C411yescharacteristic
29C22×C4×D61D411yescharacteristic
30C2×#11758194—21no—
31#14519739—41no—
32#14519739—41no—
33C2×#11758198C2×C411yescharacteristic
34C2×#11758198Q811yescharacteristic
35C2×C4×Dic61D411yescharacteristic
36#14519693—24no—
37#14519687—41no—
38C8×Dic61—41no—
39C2×M16×C61—21no—
40C2×C4⋊C4×C61C2×C411yescharacteristic
41C23×D61C22⋊C411yescharacteristic
42C23×D61C4⋊C411yescharacteristic
43C22×#6923—42no—
44C22×C4×C61C4211yescharacteristic
45C22×C4×C61C22⋊C411yescharacteristic
46C22×C4×C61C4⋊C411yescharacteristic
47C22×Dic61C22⋊C411yescharacteristic
48C22×Dic61C4⋊C411yescharacteristic
49C2×D244—22no—
50C2×D244—21no—
51C2×D244—21no—
52C2×D244—41no—
53C2×C4×D61—21no—
54C2×C4×D61—21no—
55C2×C4×D61—21no—
56C2×C4×D61—21no—
57C2×C4×D61—41no—
58C2×C4×D61—41no—
59C2×C4×D61—41no—
60C2×C4×D61—41no—
61C22⋊#6923—24no—
62C22⋊#6923—42no—
63C2×#92420—22no—
64C2×#92420—41no—
65C2×#92420—41no—
66C2×#92420—41no—
67C2×#92420—41no—
68C2×#92420—41no—
69C4×#6923—81no—
70C4×#6923—81no—
71#11758213—42no—
72#11758213—42no—
73M16×C61—22no—
74M16×C61—41no—
75C2×C8×C61—41no—
76C4⋊C4×C61—41no—
77#11758198—41no—
78#11758198—41no—
79C4×Dic61—21no—
80C4×Dic61—21no—
81#11758194—22no—
82#11758194—41no—
83C2×#92418—41no—
84C23×C61#9511yescharacteristic
85C22×D61—21no—
86C22×D61—21no—
87C22×D61—21no—
88C22×D61—21no—
89C22×D61—21no—
90C22×D61—41no—
91C22×D61—41no—
92C22×D61—41no—
93C22×D61—41no—
94C22×D61#10412yes—
95C2×#6923—44no—
96C2×#6923—82no—
97C2×#6923—82no—
98C2×C4×C61—21no—
99C2×C4×C61—21no—
100C2×C4×C61—21no—
101C2×C4×C61—21no—
102C2×C4×C61—41no—
103C2×C4×C61—41no—
104C2×C4×C61#10411yescharacteristic
105C2×C4×C61#10411yescharacteristic
106C2×Dic61—21no—
107C2×Dic61—21no—
108C2×Dic61—21no—
109C2×Dic61—41no—
110C2×Dic61#9911yescharacteristic
111C2×Dic61#9911yescharacteristic
112D244—42no—
113D244—42no—
114C4×D61—42no—
115C4×D61—41no—
116C4×D61—41no—
117C4×D61—41no—
118C4×D61—41no—
119C4×D61—81no—
120#92420—42no—
121#92420—81no—
122C8×C61—42no—
123#92418—42no—
124#7282—611nomaximal, Sylow
125C22×C61—21no—
126C22×C61—21no—
127C22×C61#33911yescharacteristic
128C22×C61#33911yescharacteristic
129C22×C61#34211yescharacteristic
130C2×D61—22no—
131C2×D61—41no—
132C2×D61—41no—
133C2×D61—42no—
134C2×D61—41no—
135C2×D61—41no—
136C2×D61—41no—
137C2×D61—81no—
138C2×D61—81no—
139C2×D61—81no—
140#6923—84no—
141C4×C61—21no—
142C4×C61—21no—
143C4×C61—41no—
144C4×C61—41no—
145Dic61—21no—
146Dic61—41no—
147Dic61AΣL(1,𝔽16)11yescharacteristic
148Dic61#35211yescharacteristic
149#1870—611no—
150C4×#100—611no—
151C4×#99—611no—
152C2×C61—21no—
153C2×C61—21no—
154C2×C61#129711yescharacteristic
155C2×C61#139011yescharacteristic
156C2×C61#139511yescharacteristic
157D61—42no—
158D61—81no—
159C22×M16—611no—
160C2×C4×D4—611no—
161C2×#100—1221no—
162C2×#99—1221no—
163C2×C22⋊C8—611no—
164C2×C22⋊C8—1221no—
165C4×M16—1221no—
166C4×C22⋊C4—1221no—
167#343—611no—
168#343#692311yescharacteristic, complemented
169#342—612no—
170#336—1221no—
171C61#728211yescharacteristic, complemented, Sylow
172C22×D4—611no—
173C23×C4—611no—
174C23×C4—611no—
175C2×M16—21no—
176C2×M16—1221no—
177C2×M16—2442no—
178C2×M16—2441no—
179C22×C8—1221no—
180C22×C8—1221no—
181C4×D4—2441no—
182C4×D4—2441no—
183C2×C4⋊C4C2×#692311yescharacteristic
184C2×C22⋊C4—611no—
185C2×C22⋊C4—611no—
186C2×C22⋊C4—1222no—
187C2×C42—611no—
188C2×C42—2441no—
189#100—1224no—
190#99—1222no—
191#99—1222no—
192C22⋊C8—2442no—
193C22⋊C8—2441no—
194#97—2441no—
195C4×C8—2441no—
196#95—2441no—
197C24—611no—
198C24—611no—
199C2×D4—1222no—
200C2×D4—1221no—
201C2×D4—1221no—
202C2×D4—2441no—
203C22×C4—611no—
204C22×C4—611no—
205C22×C4—1221no—
206C22×C4—1221no—
207C22×C4—1221no—
208C22×C4—1221no—
209C22×C4—2441no—
210C22×C4—2441no—
211C22×C4—2441no—
212C22×C4—2441no—
213C22×C4—2442no—
214C22×C4C4×#692311yescharacteristic
215C22×C4C4⋊#692311yescharacteristic
216C22×C4C22⋊#692311yescharacteristic
217M16—22no—
218M16—41no—
219M16—1222no—
220M16—2442no—
221M16—2441no—
222M16—2442no—
223C2×C8—41no—
224C2×C8—1222no—
225C2×C8—2441no—
226C2×C8—2441no—
227C2×C8—2441no—
228C2×C8—2441no—
229C2×C8—2441no—
230C2×C8—2441no—
231C4⋊C4—41no—
232C22⋊C4—1224no—
233C22⋊C4—2441no—
234C22⋊C4—2441no—
235C22⋊C4—2442no—
236C42—1221no—
237C42—1221no—
238C42—4881no—
239C42—4881no—
240C23—612no—
241C23—1221no—
242C23—1221no—
243C23—1221no—
244C23—1221no—
245C23—1221no—
246C23—2441no—
247C23—2441no—
248C23—2441no—
249C23—2441no—
250C23(C2×#6923)⋊C411yescharacteristic
251D4—2442no—
252D4—2442no—
253C2×C4—21no—
254C2×C4—21no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—41no—
258C2×C4—41no—
259C2×C4—611no—
260C2×C4—611no—
261C2×C4—1221no—
262C2×C4—1221no—
263C2×C4—1221no—
264C2×C4—4881no—
265C2×C4—4882no—
266C2×C4—4882no—
267C2×C4—2441no—
268C2×C4—2441no—
269C2×C4—2441no—
270C2×C4—2441no—
271C2×C4—2442no—
272C2×C4—2444no—
273C2×C4—2441no—
274C2×C4#1451984111yescharacteristic
275C2×C4#1451984211yescharacteristic
276C8—42no—
277C8—4881no—
278C8—2442no—
279C8—2442no—
280C22—21no—
281C22—21no—
282C22—1222no—
283C22—4881no—
284C22—4881no—
285C22—4881no—
286C22—2442no—
287C22—2441no—
288C22—2441no—
289C22—2441no—
290C22—2441no—
291C22—2441no—
292C22#1684063211yescharacteristic
293C22#1684063311yescharacteristic
294C22#1684063911yescharacteristic
295C4—21no—
296C4—21no—
297C4—41no—
298C4—41no—
299C4—611no—
300C4—611no—
301C4—1221no—
302C4—4884no—
303C4—2441no—
304C2—21no—
305C2—21no—
306C2—4881no—
307C2—2442no—
308C2#2146070011yescharacteristic
309C2#2146090511yescharacteristic
310C2#2146092111yescharacteristic
subgroupG/H
centerC22#16840639
commutatorC22×C4×C61C42
FrattiniC2×C4⋊C4C2×#6923
FittingC61×#343C4
socleC22×C61#342
prime radicalC22×D244D4
cosocleC2×#14519739C22
hypercenter#343#6923
hypocenterC61#7282
LCSG ▷ C22×C4×C61 ▷ C23×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #343

Complements

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

NH ≅ G/Ntype
4. C22×#11758213C4semidirect
7. C2×#14519834C4semidirect
14. #16839531C4semidirect
15. C61×#343C4semidirect
168. #343#6923semidirect
171. C61#7282semidirect

Maximal subgroups

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

subgroupindex
#214617992
#214615642
#214456102
#728261