Finite Group Databasebeta

← #30606307

Subgroups — #30606307

Order 15616.

826 conjugacy classes of subgroups; only the 144 normal subgroups are shown.

normal subgroups only

This group has 826 conjugacy classes of subgroups — too many to draw. Only its 144 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
1C2×#16840743C212yesmaximal
2#21460914C212yesmaximal, complemented
3#21460913C212yesmaximal, complemented
4#21448410C211yesmaximal, characteristic
5C2×#14519856—22no—
6#16840743—24no—
7#16840743C418yescomplemented
8C2×#14519836—22no—
9C2×#14519836C2214yes—
10C22×#11758226C2211yescharacteristic
11C2×#14519734C411yescharacteristic, complemented
12C2×C4⋊C4×D61C411yescharacteristic, complemented
13#16839669C411yescharacteristic, complemented
14#16839664C2211yescharacteristic
15#16839663C2211yescharacteristic
16#16839640—22no—
17#16839621—22no—
18C61×#391C411yescharacteristic, complemented
19#14519856—28no—
20#14519856—44no—
21C2×C4⋊#6923—42no—
22C2×C22⋊#6923—24no—
23C2×C4×#6923—42no—
24#14519836—28no—
25#14519836—24no—
26#14519836C2×C418yescomplemented
27C2×#11758226—42no—
28C2×#11758226—42no—
29C2×#11758226C2×C414yes—
30C2×#11758226D412yes—
31C2×#11758226Q812yes—
32C22×D244D412yes—
33C22×D244C2311yescharacteristic
34C22×C4×D61—21no—
35C22×C4×D61C2×C411yescharacteristic
36C22×C4×D61D412yes—
37#14519734—42no—
38C2×C4⋊Dic61C2×C411yescharacteristic
39C2×#11758196—21no—
40C2×#11758196—21no—
41C4⋊C4×D61—42no—
42C2×#11758198—21no—
43C2×#11758198—22no—
44C2×#11758198—22no—
45C2×C4×Dic61C2×C411yescharacteristic
46#14519684—22no—
47C22×Q8×C61C2×C411yescharacteristic
48C2×C4⋊C4×C61C2×C411yescharacteristic
49C2×C42×C61C2×C411yescharacteristic
50#95×C61—22no—
51C23×D61C22⋊C411yescharacteristic
52C23×D61C2×D412yes—
53C22×#6923—44no—
54C2×Q8×C61—41no—
55C2×Q8×C61—41no—
56C2×Q8×C61C4212yes—
57C2×Q8×C61C4⋊C412yes—
58C22×C4×C61—21no—
59C22×C4×C61—21no—
60C22×C4×C61C22⋊C412yes—
61C22×C4×C61C22×C411yescharacteristic
62C22×Dic61—21no—
63C22×Dic61—21no—
64C22×Dic61C22⋊C411yescharacteristic
65#11758226—44no—
66#11758226—44no—
67#11758226—44no—
68#11758226—44no—
69C2×D244—42no—
70C2×D244—42no—
71C2×D244—41no—
72C2×D244—41no—
73C2×D244C22⋊C414yes—
74C2×D244C4⋊C414yes—
75C2×D244C22×C412yes—
76C2×D244C2×D411yescharacteristic
77C2×D244C2×Q811yescharacteristic
78C2×C4×D61—22no—
79C2×C4×D61—22no—
80C2×C4×D61—22no—
81C2×C4×D61—22no—
82C2×C4×D61—24no—
83C2×C4×D61—42no—
84C2×C4×D61—82no—
85C2×C4×D61C4212yes—
86C2×C4×D61C22⋊C414yes—
87C2×C4×D61C4⋊C414yes—
88C2×C4×D61C4⋊C412yes—
89C22⋊#6923—28no—
90C22⋊#6923—44no—
91C4⋊#6923—48no—
92C4×#6923—48no—
93C4⋊C4×C61—41no—
94C42×C61—41no—
95#11758198—42no—
96#11758198—82no—
97#11758198—82no—
98C4⋊Dic61—41no—
99#11758196—42no—
100#11758196—42no—
101C4×Dic61—41no—
102C23×C61C2×C22⋊C411yescharacteristic
103C22×D61—21no—
104C22×D61—21no—
105C22×D61—22no—
106C22×D61—22no—
107C22×D61—41no—
108C22×D61—41no—
109C22×D61—42no—
110C22×D61—42no—
111C22×D61#9512yes—
112C22×D61C2×C22⋊C412yes—
113C22×D61C2×C4⋊C412yes—
114C2×#6923—48no—
115C2×#6923—84no—
116C2×#6923—84no—
117Q8×C61—42no—
118Q8×C61—42no—
119C2×C4×C61—21no—
120C2×C4×C61—21no—
121C2×C4×C61—22no—
122C2×C4×C61—22no—
123C2×C4×C61—41no—
124C2×C4×C61—42no—
125C2×C4×C61—41no—
126C2×C4×C61—41no—
127C2×C4×C61—41no—
128C2×C4×C61#9514yes—
129C2×C4×C61C2×C4211yescharacteristic
130C2×C4×C61C2×C4⋊C411yescharacteristic
131C2×Dic61—41no—
132C2×Dic61—42no—
133C2×Dic61—41no—
134C2×Dic61—41no—
135C2×Dic61—41no—
136C2×Dic61#9512yes—
137C2×Dic61#9912yes—
138C2×Dic61#9912yes—
139D244—44no—
140D244—44no—
141D244—42no—
142D244—41no—
143D244—41no—
144C4×D61—24no—
145C4×D61—24no—
146C4×D61—24no—
147C4×D61—24no—
148C4×D61—24no—
149C4×D61—24no—
150C4×D61—84no—
151#12583—611nomaximal, Sylow
152C22×C61—21no—
153C22×C61—21no—
154C22×C61C2×#9511yescharacteristic
155C22×C61C2×#9911yescharacteristic
156C22×C61C2×#9911yescharacteristic
157C2×D61—22no—
158C2×D61—22no—
159C2×D61—22no—
160C2×D61—42no—
161C2×D61—41no—
162C2×D61—42no—
163C2×D61—42no—
164C2×D61—41no—
165C2×D61—42no—
166C2×D61—42no—
167C2×D61—42no—
168C2×D61—82no—
169#6923—88no—
170C4×C61—24no—
171C4×C61—21no—
172C4×C61—21no—
173C4×C61—81no—
174C4×C61—81no—
175Dic61—81no—
176Dic61—81no—
177Dic61#34214yes—
178C2×#410—612no—
179#2311—611no—
180#1393—612no—
181#1393—612no—
182C2×C61—21no—
183C2×C61—21no—
184C2×C61C2×#34211yescharacteristic
185C2×C61#212811yescharacteristic
186C2×C61#212911yescharacteristic
187D61—42no—
188D61—42no—
189D61—42no—
190C22×#41—611no—
191C2×#117—611no—
192C2×#117—1222no—
193C22×C4⋊C4—611no—
194#410—618no—
195#410—1224no—
196C2×#99—614no—
197C2×#99—1222no—
198#391—611no—
199#391#692311yescharacteristic, complemented
200#390—611no—
201#390—611no—
202#386—1222no—
203#380—1222no—
204C61#1258311yescharacteristic, complemented, Sylow
205C2×#41—614no—
206C2×#41—612no—
207C2×#41—612no—
208C2×#41—2442no—
209C2×#41—2442no—
210C22×Q8C2×#692311yescharacteristic
211C22×D4—612no—
212C22×D4—611no—
213C23×C4—612no—
214C23×C4—611no—
215C23×C4—1221no—
216#117—1228no—
217#117—2442no—
218#117—2444no—
219C2×C4⋊C4—611no—
220C2×C4⋊C4—1221no—
221C2×C4⋊C4—1221no—
222C2×C4⋊C4—2442no—
223C2×C4⋊C4—2442no—
224C2×C4⋊C4C2×#692311yescharacteristic
225C2×C22⋊C4—1224no—
226C2×C22⋊C4—1222no—
227C2×C22⋊C4—1222no—
228C2×C22⋊C4—1221no—
229C2×C42—611no—
230C2×C42—2442no—
231C2×C42C2×#692311yescharacteristic
232#99—618no—
233#99—1224no—
234#99—1228no—
235#95—22no—
236#95—1222no—
237C24—612no—
238C24—611no—
239#41—2444no—
240#41—2444no—
241#41—2444no—
242#41—2444no—
243C2×Q8—41no—
244C2×Q8—41no—
245C2×Q8C4×#692312yes—
246C2×Q8C4⋊#692312yes—
247C2×D4—614no—
248C2×D4—614no—
249C2×D4—612no—
250C2×D4—611no—
251C2×D4—611no—
252C2×D4—2441no—
253C2×D4—2442no—
254C2×D4—2441no—
255C2×D4—2442no—
256C22×C4—21no—
257C22×C4—21no—
258C22×C4—611no—
259C22×C4—614no—
260C22×C4—614no—
261C22×C4—612no—
262C22×C4—612no—
263C22×C4—1222no—
264C22×C4—1222no—
265C22×C4—1222no—
266C22×C4—1222no—
267C22×C4—1224no—
268C22×C4—1221no—
269C22×C4—1221no—
270C22×C4—4882no—
271C22×C4—2442no—
272C22×C4—2444no—
273C22×C4C22⋊#692312yes—
274C22×C4C22×#692311yescharacteristic
275C4⋊C4—41no—
276C4⋊C4—2441no—
277C4⋊C4—2442no—
278C4⋊C4—2442no—
279C4⋊C4—2448no—
280C22⋊C4—1228no—
281C22⋊C4—4882no—
282C22⋊C4—4882no—
283C22⋊C4—2442no—
284C22⋊C4—2444no—
285C42—41no—
286C42—2441no—
287C42—2448no—
288C23—612no—
289C23—612no—
290C23—612no—
291C23—1222no—
292C23—1221no—
293C23—1222no—
294C23—1221no—
295C23—2442no—
296C23—2442no—
297C23—2441no—
298C23—2441no—
299C23C2×C22⋊#692311yescharacteristic
300Q8—42no—
301Q8—42no—
302D4—2444no—
303D4—2444no—
304D4—2442no—
305D4—2441no—
306D4—2441no—
307C2×C4—21no—
308C2×C4—22no—
309C2×C4—21no—
310C2×C4—22no—
311C2×C4—41no—
312C2×C4—42no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—612no—
317C2×C4—612no—
318C2×C4—612no—
319C2×C4—1224no—
320C2×C4—1224no—
321C2×C4—1224no—
322C2×C4—1224no—
323C2×C4—1224no—
324C2×C4—1224no—
325C2×C4—4884no—
326C2×C4—4884no—
327C2×C4—4884no—
328C2×C4—2441no—
329C2×C4—2442no—
330C2×C4—2441no—
331C2×C4—2448no—
332C2×C4—2441no—
333C2×C4—2441no—
334C2×C4(C2×#6923)⋊C414yes—
335C2×C4C2×C4×#692311yescharacteristic
336C2×C4C2×C4⋊#692311yescharacteristic
337C22—21no—
338C22—21no—
339C22—1222no—
340C22—1222no—
341C22—1222no—
342C22—4882no—
343C22—2442no—
344C22—2442no—
345C22—2442no—
346C22—2442no—
347C22—2441no—
348C22—2442no—
349C22—2442no—
350C22—2441no—
351C22C2×(C2×#6923)⋊C411yescharacteristic
352C22D61⋊#9911yescharacteristic
353C22D61⋊#9911yescharacteristic
354C4—24no—
355C4—21no—
356C4—21no—
357C4—81no—
358C4—81no—
359C4—614no—
360C4—4881no—
361C4—4888no—
362C4—4881no—
363C2—21no—
364C2—21no—
365C2—2442no—
366C2—2442no—
367C2—2442no—
368C2#2146151611yescharacteristic
369C2#2146209911yescharacteristic
370C2#2146210211yescharacteristic
subgroupG/H
centerC22D61⋊#99
commutatorC2×C4×C61C2×C42
FrattiniC22×C4C22×#6923
FittingC61×#391C4
socleC22×C61C2×#99
prime radicalC22×#11758226C22
cosocleC22×D244C23
hypercenter#391#6923
hypocenterC61#12583
LCSG ▷ C2×C4×C61 ▷ C22×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23 ◁ C22×Q8 ◁ #391

Complements

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

NH ≅ G/Ntype
2. #21460914C2semidirect
3. #21460913C2semidirect
7. #16840743C4semidirect
11. C2×#14519734C4semidirect
12. C2×C4⋊C4×D61C4semidirect
13. #16839669C4semidirect
18. C61×#391C4semidirect
26. #14519836C2×C4semidirect
199. #391#6923semidirect
204. C61#12583semidirect

Maximal subgroups

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

subgroupindex
C2×#168407432
#214609142
#214609132
#214484102
#1258361