Finite Group Databasebeta

← #30605468

Subgroups — #30605468

Order 15616.

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

normal subgroups only

This group has 580 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#21460814C214yesmaximal, complemented
4#21449178C211yesmaximal, characteristic
5#16840779—21no—
6C2×#14519835—21no—
7#16840704—21no—
8#16840640C2212yes—
9#16840606C2212yes—
10#16840605—21no—
11C2×#14519753—21no—
12C2×C4×C22⋊D61C2211yescharacteristic
13#16839681C412yescomplemented
14#16839673C2212yes—
15#16839669—21no—
16#16839641—21no—
17#16839635—21no—
18#16839622C411yescharacteristic, complemented
19C23×Q8×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
29#14519753—44no—
30C4×C22⋊D61—41no—
31C4×C22⋊D61—41no—
32C2×C4⋊Dic61—21no—
33C2×#11758196—22no—
34C2×#11758196C2×C411yescharacteristic
35C2×C22⋊Dic61—21no—
36C2×C22⋊Dic61D411yescharacteristic
37C2×#11758198—22no—
38C2×#11758198C2311yescharacteristic
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×D61C2×D411yescharacteristic
52C22×#6923—42no—
53C2×Q8×C61—24no—
54C2×Q8×C61—42no—
55C2×Q8×C61—44no—
56C2×Q8×C61—42no—
57C2×Q8×C61—48no—
58C2×Q8×C61—44no—
59C2×Q8×C61—44no—
60C2×Q8×C61—41no—
61C2×Q8×C61—41no—
62C22×C4×C61—22no—
63C22×C4×C61—21no—
64C22×C4×C61—21no—
65C22×C4×C61—44no—
66C22×C4×C61—42no—
67C22×C4×C61—42no—
68C22×C4×C61C22⋊C411yescharacteristic
69C22×C4×C61C22×C411yescharacteristic
70C2×C22⋊D61—41no—
71C2×C22⋊D61—41no—
72C2×C22⋊D61SD1612yes—
73C2×C22⋊D61Q1612yes—
74C22×Dic61—22no—
75C22×Dic61C22⋊C411yescharacteristic
76C22×Dic61C22⋊C411yescharacteristic
77C22×Dic61C2×D411yescharacteristic
78C2×C4×D61—21no—
79C2×C4×D61—21no—
80C2×C4×D61—42no—
81C22⋊#6923—42no—
82C22⋊#6923—42no—
83C2×#92420—42no—
84C4⋊#6923—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×D61C2×SD1611yescharacteristic
105C22×D61C2×Q1611yescharacteristic
106C2×#6923—82no—
107C2×#6923—82no—
108C2×#6923—82no—
109Q8×C61—44no—
110Q8×C61—44no—
111Q8×C61—44no—
112Q8×C61—44no—
113C2×C4×C61—22no—
114C2×C4×C61—24no—
115C2×C4×C61—21no—
116C2×C4×C61—42no—
117C2×C4×C61—42no—
118C2×C4×C61—48no—
119C2×C4×C61—42no—
120C2×C4×C61—42no—
121C2×C4×C61—41no—
122C2×C4×C61#9911yescharacteristic
123C2×C4×C61#9911yescharacteristic
124C22⋊D61—42no—
125C22⋊D61—42no—
126C2×Dic61—21no—
127C2×Dic61—21no—
128C2×Dic61—21no—
129C2×Dic61—21no—
130C2×Dic61—41no—
131C2×Dic61—41no—
132C2×Dic61—42no—
133C2×Dic61—42no—
134C2×Dic61—42no—
135C2×Dic61#10312yes—
136C4×D61—42no—
137C4×D61—81no—
138#92420—82no—
139#11898—611nomaximal, Sylow
140C22×C61—21no—
141C22×C61—21no—
142C22×C61—22no—
143C22×C61—42no—
144C22×C61—42no—
145C22×C61—41no—
146C22×C61—41no—
147C22×C61C2×#9911yescharacteristic
148C22×C61C2×#10311yescharacteristic
149C22×C61#42111yescharacteristic
150C2×D61—21no—
151C2×D61—21no—
152C2×D61—41no—
153C2×D61—41no—
154C2×D61—41no—
155C2×D61—41no—
156C2×D61—81no—
157#6923—162no—
158C4×C61—21no—
159C4×C61—44no—
160C4×C61—41no—
161C4×C61AΣL(1,𝔽16)12yes—
162Dic61—22no—
163Dic61—41no—
164Dic61—41no—
165Dic61—82no—
166#2409—611no—
167#1799—611no—
168#1460—611no—
169#1351—614no—
170C2×C61—21no—
171C2×C61—21no—
172C2×C61—42no—
173C2×C61#152011yescharacteristic
174C2×C61C2×AΣL(1,𝔽16)11yescharacteristic
175C2×C61#212911yescharacteristic
176D61—41no—
177D61—41no—
178C23×Q8#692311yescharacteristic, complemented
179C2×C22⋊Q8—1221no—
180C2×C4×D4—611no—
181#431—1221no—
182C2×#99—1221no—
183#395—612no—
184#393—612no—
185#391—1221no—
186#386—1221no—
187#385—1221no—
188#380—611no—
189#380—1221no—
190#342—612no—
191#323—612no—
192#323—1221no—
193C61#1189811yescharacteristic, complemented, Sylow
194C22×Q8—21no—
195C22×Q8—41no—
196C22×Q8—41no—
197C22×Q8—44no—
198C22×Q8C2×#692312yes—
199C22×D4—611no—
200C23×C4—21no—
201C23×C4—611no—
202C23×C4C2×#692311yescharacteristic
203C22⋊Q8—2444no—
204C4×D4—2441no—
205C4×D4—2441no—
206C2×C4⋊C4—611no—
207C2×C4⋊C4—1222no—
208C2×C4⋊C4—1221no—
209C2×C4⋊C4—2441no—
210C2×C22⋊C4—611no—
211C2×C22⋊C4—611no—
212C2×C22⋊C4—1222no—
213C2×C22⋊C4—1221no—
214C2×C22⋊C4—1222no—
215C2×C42—611no—
216C4⋊C8—2441no—
217#99—2442no—
218C22⋊C8—1222no—
219C4×C8—2441no—
220#95—612no—
221#95—1222no—
222#95—1222no—
223#95—2441no—
224C24—611no—
225C24C22⋊#692311yescharacteristic
226C2×Q8—24no—
227C2×Q8—44no—
228C2×Q8—42no—
229C2×Q8—42no—
230C2×Q8—44no—
231C2×Q8—48no—
232C2×Q8—44no—
233C2×Q8—41no—
234C2×Q8—41no—
235C2×D4—612no—
236C2×D4—612no—
237C2×D4—2441no—
238C2×D4—2441no—
239C22×C4—22no—
240C22×C4—21no—
241C22×C4—21no—
242C22×C4—44no—
243C22×C4—42no—
244C22×C4—42no—
245C22×C4—611no—
246C22×C4—611no—
247C22×C4—611no—
248C22×C4—1221no—
249C22×C4—1221no—
250C22×C4—1222no—
251C22×C4—2442no—
252C22×C4—2442no—
253C22×C4C22⋊#692311yescharacteristic
254C22×C4C22×#692311yescharacteristic
255C2×C8—2442no—
256C4⋊C4—4882no—
257C4⋊C4—2441no—
258C4⋊C4—2444no—
259C4⋊C4—2442no—
260C22⋊C4—2441no—
261C22⋊C4—2444no—
262C22⋊C4—2441no—
263C22⋊C4—2442no—
264C22⋊C4—2442no—
265C22⋊C4—2442no—
266C42—1221no—
267C42—1221no—
268C23—21no—
269C23—21no—
270C23—41no—
271C23—41no—
272C23—611no—
273C23—611no—
274C23—1221no—
275C23—1221no—
276C23—2441no—
277C23—2441no—
278C23#1451984012yes—
279C23C2×C22⋊#692311yescharacteristic
280Q8—44no—
281Q8—44no—
282Q8—44no—
283Q8—44no—
284D4—2442no—
285D4—2442no—
286C2×C4—22no—
287C2×C4—24no—
288C2×C4—21no—
289C2×C4—42no—
290C2×C4—42no—
291C2×C4—48no—
292C2×C4—42no—
293C2×C4—42no—
294C2×C4—41no—
295C2×C4—612no—
296C2×C4—1221no—
297C2×C4—1221no—
298C2×C4—1221no—
299C2×C4—1221no—
300C2×C4—4881no—
301C2×C4—4882no—
302C2×C4—4882no—
303C2×C4—4882no—
304C2×C4—2442no—
305C2×C4—2442no—
306C2×C4—2442no—
307C2×C4—2442no—
308C2×C4—2441no—
309C2×C4—2441no—
310C2×C4#1451983511yescharacteristic
311C2×C4#1451983511yescharacteristic
312C8—4882no—
313C22—21no—
314C22—21no—
315C22—22no—
316C22—42no—
317C22—42no—
318C22—41no—
319C22—41no—
320C22—1221no—
321C22—1221no—
322C22—4881no—
323C22—2441no—
324C22—2441no—
325C22—2441no—
326C22—2441no—
327C22C2×#1451983511yescharacteristic
328C22C2×#1451984011yescharacteristic
329C22#1684076111yescharacteristic
330C4—21no—
331C4—44no—
332C4—41no—
333C4—1222no—
334C4—9762no—
335C4—4882no—
336C4—2441no—
337C4—2441no—
338C4#1684066212yes—
339C2—21no—
340C2—21no—
341C2—42no—
342C2—2441no—
343C2—2441no—
344C2#2146125911yescharacteristic
345C2C2×#1684066211yescharacteristic
346C2#2146210111yescharacteristic
subgroupG/H
centerC22C2×#14519835
commutatorC22×C4×C61C22×C4
FrattiniC22×C4C22×#6923
FittingC23×Q8×C61C4
socleC22×C61C2×#99
prime radicalC22×C22⋊D61D4
cosocleC2×#11758198C23
hypercenterC23×Q8#6923
hypocenterC61#11898
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. #21460814C2semidirect
13. #16839681C4semidirect
18. #16839622C4semidirect
19. C23×Q8×C61C4semidirect
178. C23×Q8#6923semidirect
193. C61#11898semidirect

Maximal subgroups

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

subgroupindex
#214616792
#214610582
#214608142
#214491782
#1189861