Finite Group Databasebeta

← #30605487

Subgroups — #30605487

Order 15616.

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

normal subgroups only

This group has 568 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#21461683C211yesmaximal, characteristic
2#21461054C211yesmaximal, characteristic
3#21460820C214yesmaximal, complemented
4#21449178C211yesmaximal, characteristic
5#16840765—21no—
6C2×#14519837—21no—
7D61⋊(C22⋊C8)—21no—
8#16840642C2212yes—
9#16840608C2212yes—
10#16840605—21no—
11C2×#14519753—21no—
12C2×C4×C22⋊D61C2211yescharacteristic
13#16839681C2212yes—
14#16839673C412yes—
15#16839669—21no—
16#16839641—21no—
17#16839635—21no—
18#16839622C411yescharacteristic
19C23×Q8×C61C411yescharacteristic
20C2×#11758217—22no—
21C2×D61⋊C8—41no—
22#14519844—41no—
23C4⋊#92420—41no—
24#14519837—42no—
25#14519834—41no—
26C22⋊#92420—22no—
27C22×C22⋊D61D411yescharacteristic
28C22×C4×D61D411yescharacteristic
29#14519753—44no—
30C4×C22⋊D61—41no—
31C4×C22⋊D61—41no—
32C2×C4⋊Dic61—21no—
33C2×#11758196—22no—
34C2×#11758196C2311yescharacteristic
35C2×C22⋊Dic61—21no—
36C2×C22⋊Dic61D411yescharacteristic
37C2×#11758198—22no—
38C2×#11758198C2×C411yescharacteristic
39C2×C4×Dic61D411yescharacteristic
40#14519684—22no—
41#14519684—22no—
42#14519684C2×C412yes—
43C22×Q8×C61—21no—
44C22×Q8×C61—41no—
45C22×Q8×C61—41no—
46C22×Q8×C61—44no—
47C22×Q8×C61C2×C412yes—
48C23×C4×C61—21no—
49C23×C4×C61C2×C411yescharacteristic
50C24×C61C22⋊C411yescharacteristic
51C23×D61C22⋊C411yescharacteristic
52C2×Q8×C61—24no—
53C2×Q8×C61—42no—
54C2×Q8×C61—44no—
55C2×Q8×C61—42no—
56C2×Q8×C61—48no—
57C2×Q8×C61—44no—
58C2×Q8×C61—44no—
59C2×Q8×C61—41no—
60C2×Q8×C61—41no—
61C22×C4×C61—22no—
62C22×C4×C61—21no—
63C22×C4×C61—21no—
64C22×C4×C61—44no—
65C22×C4×C61—42no—
66C22×C4×C61—42no—
67C22×C4×C61C22⋊C411yescharacteristic
68C22×C4×C61C22×C411yescharacteristic
69C2×C22⋊D61—41no—
70C2×C22⋊D61—41no—
71C2×C22⋊D61SD1612yes—
72C2×C22⋊D61Q1612yes—
73C22×Dic61—22no—
74C22×Dic61C22⋊C411yescharacteristic
75C22×Dic61C2×D411yescharacteristic
76C22×Dic61C2×D411yescharacteristic
77C2×C4×D61—21no—
78C2×C4×D61—21no—
79C2×C4×D61—42no—
80#11758217—42no—
81#11758217—42no—
82C2×#92420—42no—
83C2×#92420—42no—
84D61⋊C8—82no—
85C22⋊Dic61—41no—
86C22⋊Dic61—42no—
87#11758198—41no—
88#11758198—44no—
89C4⋊Dic61—42no—
90#11758196—41no—
91#11758196—44no—
92C4×Dic61—21no—
93C4×Dic61—21no—
94C23×C61—21no—
95C23×C61—21no—
96C23×C61—41no—
97C23×C61—41no—
98C23×C61#10312yes—
99C23×C61C2×C22⋊C411yescharacteristic
100C22×D61—21no—
101C22×D61—21no—
102C22×D61—41no—
103C22×D61—41no—
104C22×D61#10312yes—
105Q8×C61—44no—
106Q8×C61—44no—
107Q8×C61—44no—
108Q8×C61—44no—
109C2×C4×C61—22no—
110C2×C4×C61—24no—
111C2×C4×C61—21no—
112C2×C4×C61—42no—
113C2×C4×C61—42no—
114C2×C4×C61—48no—
115C2×C4×C61—42no—
116C2×C4×C61—42no—
117C2×C4×C61—41no—
118C2×C4×C61#9911yescharacteristic
119C2×C4×C61#9911yescharacteristic
120C22⋊D61—42no—
121C22⋊D61—42no—
122C2×Dic61—21no—
123C2×Dic61—21no—
124C2×Dic61—21no—
125C2×Dic61—21no—
126C2×Dic61—41no—
127C2×Dic61—41no—
128C2×Dic61—42no—
129C2×Dic61—42no—
130C2×Dic61—42no—
131C2×Dic61C2×SD1611yescharacteristic
132C2×Dic61C2×Q1611yescharacteristic
133C4×D61—42no—
134C4×D61—81no—
135#92420—82no—
136#92420—82no—
137#11904—611nomaximal, Sylow
138C22×C61—21no—
139C22×C61—21no—
140C22×C61—22no—
141C22×C61—42no—
142C22×C61—42no—
143C22×C61—41no—
144C22×C61—41no—
145C22×C61C2×#9911yescharacteristic
146C22×C61C2×#10311yescharacteristic
147C22×C61#42111yescharacteristic
148C2×D61—22no—
149C2×D61—41no—
150C2×D61—41no—
151C2×D61—42no—
152C2×D61—81no—
153C4×C61—21no—
154C4×C61—44no—
155C4×C61—41no—
156C4×C61AΣL(1,𝔽16)12yes—
157Dic61—21no—
158Dic61—21no—
159Dic61—41no—
160Dic61—41no—
161Dic61—82no—
162#2409—611no—
163#1800—611no—
164#1459—611no—
165#1354—614no—
166C2×C61—21no—
167C2×C61—21no—
168C2×C61—42no—
169C2×C61#152011yescharacteristic
170C2×C61C2×AΣL(1,𝔽16)11yescharacteristic
171C2×C61#213511yescharacteristic
172D61—42no—
173C23×Q8#692311yescharacteristic
174C2×C22⋊Q8—1221no—
175C2×C4×D4—611no—
176#423—1221no—
177C2×#100—1221no—
178C2×C22⋊C8—1221no—
179#395—612no—
180#393—612no—
181#391—1221no—
182#386—1221no—
183#385—1221no—
184#380—611no—
185#343—612no—
186#324—612no—
187#323—1221no—
188C61#1190411yescharacteristic, complemented, Sylow
189C22×Q8—21no—
190C22×Q8—41no—
191C22×Q8—41no—
192C22×Q8—44no—
193C22×Q8C2×#692312yes—
194C22×D4—611no—
195C23×C4—21no—
196C23×C4—611no—
197C23×C4C2×#692311yescharacteristic
198C2×M16—1222no—
199C22×C8—2441no—
200C22⋊Q8—2444no—
201C4×D4—2441no—
202C4×D4—2441no—
203C2×C4⋊C4—611no—
204C2×C4⋊C4—1222no—
205C2×C4⋊C4—1221no—
206C2×C22⋊C4—611no—
207C2×C22⋊C4—611no—
208C2×C22⋊C4—1222no—
209C2×C22⋊C4—1221no—
210C2×C42—611no—
211C4⋊C8—2441no—
212C4⋊C8—2441no—
213#100—2442no—
214C22⋊C8—1222no—
215C22⋊C8—2441no—
216#95—612no—
217#95—1222no—
218#95—1222no—
219C24—611no—
220C24C22⋊#692311yescharacteristic
221C2×Q8—24no—
222C2×Q8—44no—
223C2×Q8—42no—
224C2×Q8—42no—
225C2×Q8—44no—
226C2×Q8—48no—
227C2×Q8—44no—
228C2×Q8—41no—
229C2×Q8—41no—
230C2×D4—612no—
231C2×D4—612no—
232C2×D4—2441no—
233C2×D4—2441no—
234C22×C4—22no—
235C22×C4—21no—
236C22×C4—21no—
237C22×C4—44no—
238C22×C4—42no—
239C22×C4—42no—
240C22×C4—611no—
241C22×C4—611no—
242C22×C4—611no—
243C22×C4—1221no—
244C22×C4—1221no—
245C22×C4—1222no—
246C22×C4—2442no—
247C22×C4C22⋊#692311yescharacteristic
248C22×C4C22×#692311yescharacteristic
249M16—2442no—
250M16—2442no—
251C2×C8—4882no—
252C2×C8—2442no—
253C2×C8—2442no—
254C4⋊C4—2441no—
255C4⋊C4—2444no—
256C4⋊C4—2442no—
257C22⋊C4—2441no—
258C22⋊C4—2444no—
259C22⋊C4—2441no—
260C22⋊C4—2442no—
261C42—1221no—
262C42—1221no—
263C23—21no—
264C23—21no—
265C23—41no—
266C23—41no—
267C23—612no—
268C23—1221no—
269C23—1221no—
270C23—2441no—
271C23—2441no—
272C23#1451984012yes—
273C23C2×C22⋊#692311yescharacteristic
274Q8—44no—
275Q8—44no—
276Q8—44no—
277Q8—44no—
278D4—2442no—
279D4—2442no—
280C2×C4—22no—
281C2×C4—24no—
282C2×C4—21no—
283C2×C4—42no—
284C2×C4—42no—
285C2×C4—48no—
286C2×C4—42no—
287C2×C4—42no—
288C2×C4—41no—
289C2×C4—611no—
290C2×C4—611no—
291C2×C4—1221no—
292C2×C4—1221no—
293C2×C4—1221no—
294C2×C4—1221no—
295C2×C4—4881no—
296C2×C4—2442no—
297C2×C4—2442no—
298C2×C4—2442no—
299C2×C4—2442no—
300C2×C4—2441no—
301C2×C4—2441no—
302C2×C4#1451983511yescharacteristic
303C2×C4#1451983511yescharacteristic
304C8—4882no—
305C8—4882no—
306C22—21no—
307C22—21no—
308C22—22no—
309C22—42no—
310C22—42no—
311C22—41no—
312C22—41no—
313C22—1222no—
314C22—4881no—
315C22—2442no—
316C22—2441no—
317C22—2441no—
318C22C2×#1451983511yescharacteristic
319C22C2×#1451984011yescharacteristic
320C22#1684076111yescharacteristic
321C4—21no—
322C4—44no—
323C4—41no—
324C4—1221no—
325C4—1221no—
326C4—4882no—
327C4—2441no—
328C4—2441no—
329C4#1684066212yes—
330C2—21no—
331C2—21no—
332C2—42no—
333C2—2442no—
334C2#2146125911yescharacteristic
335C2C2×#1684066211yescharacteristic
336C2#2146212111yescharacteristic
subgroupG/H
centerC22C2×#14519835
commutatorC22×C4×C61C22×C4
FrattiniC22×C4C22×#6923
FittingC23×Q8×C61C4
socleC22×C61C2×#99
prime radicalC22×C22⋊D61D4
cosocleC2×#11758196C23
hypercenterC23×Q8#6923
hypocenterC61#11904
LCSG ▷ C22×C4×C61 ▷ C23×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ C23×C4 ◁ C23×Q8

Complements

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

NH ≅ G/Ntype
3. #21460820C2semidirect
188. C61#11904semidirect

Maximal subgroups

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

subgroupindex
#214616832
#214610542
#214608202
#214491782
#1190461