Finite Group Databasebeta

← #30596653

Subgroups — #30596653

Order 15616.

986 conjugacy classes of subgroups; only the 158 normal subgroups are shown.

normal subgroups only

This group has 986 conjugacy classes of subgroups — too many to draw. Only its 158 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#21462263C212yesmaximal
2#21460682C214yesmaximal, complemented
3C2×C22⋊C8×D61C211yesmaximal, characteristic
4C2×#14519856—22no—
5#16840705C414yescomplemented
6#16840704—22no—
7#16840704C2214yes—
8#16840703—22no—
9C4×C22⋊#6923—22no—
10C4×C22⋊#6923C414yescomplemented
11C23×C4×D61C2211yescharacteristic
12C22×C8×D61C412yescomplemented
13C2×C22⋊#92418C411yescharacteristic, complemented
14C2×#14519689C2212yes—
15C22⋊C8×D61—22no—
16C22⋊C8×D61—22no—
17C2×C22⋊C8×C61C411yescharacteristic, complemented
18#14519856—48no—
19C2×C4⋊#6923—24no—
20C2×C22⋊#6923—24no—
21C2×C22⋊#6923—24no—
22C2×C22⋊#6923—24no—
23C2×C4×#6923—24no—
24(C2×#6923)⋊C4—24no—
25(C2×#6923)⋊C4C2×C418yes—
26C24×D61D411yescharacteristic
27C23×Dic61Q811yescharacteristic
28C22×C4×D61—44no—
29C22×C4×D61C2×C411yescharacteristic
30C22×C4×D61C2×C412yes—
31C22×C4×D61C2×C412yes—
32C22×C4×D61D412yes—
33C22×C4×D61D411yescharacteristic
34C22×C4×D61D412yes—
35C22×C4×D61Q811yescharacteristic
36C22×C4×D61C2311yescharacteristic
37C22×#92418C2×C412yes—
38C2×C8×D61—22no—
39C2×C8×D61—22no—
40C2×C8×D61—44no—
41C22⋊#92418—22no—
42#14519689—24no—
43C23×C4×C61C2×C411yescharacteristic
44C22×C8×C61C2×C412yes—
45C22⋊C8×C61—22no—
46C24×C61C4⋊C411yescharacteristic
47C23×D61—24no—
48C23×D61—42no—
49C23×D61—42no—
50C23×D61C22⋊C412yes—
51C23×D61C22⋊C412yes—
52C23×D61C4⋊C411yescharacteristic
53C23×D61C2×D411yescharacteristic
54C22×#6923—28no—
55C22×C4×C61—42no—
56C22×C4×C61C4212yes—
57C22×C4×C61C22⋊C412yes—
58C22×C4×C61C4⋊C411yescharacteristic
59C22×C4×C61C22×C411yescharacteristic
60C22×Dic61—42no—
61C22×Dic61C4⋊C411yescharacteristic
62C22×Dic61C4⋊C412yes—
63C22×Dic61C4⋊C412yes—
64C22×Dic61C2×Q811yescharacteristic
65C2×C4×D61—24no—
66C2×C4×D61—28no—
67C2×C4×D61—44no—
68C2×C4×D61—42no—
69C2×C4×D61—44no—
70C2×C4×D61—42no—
71C2×C4×D61—44no—
72C2×C4×D61—42no—
73C2×C4×D61C4211yescharacteristic
74C2×C4×D61C4211yescharacteristic
75C2×C4×D61C22⋊C412yes—
76C2×C4×D61C22⋊C411yescharacteristic
77C2×C4×D61C22⋊C412yes—
78C2×C4×D61C22⋊C411yescharacteristic
79C2×C4×D61C4⋊C412yes—
80C2×C4×D61C4⋊C412yes—
81C2×C4×D61C22×C412yes—
82C2×C4×D61C2×D412yes—
83C22⋊#6923—48no—
84C22⋊#6923—48no—
85C22⋊#6923—48no—
86C4⋊#6923—48no—
87C4×#6923—48no—
88C2×C8×C61—22no—
89C2×C8×C61—42no—
90C2×#92418—22no—
91C2×#92418—42no—
92C8×D61—44no—
93C8×D61—44no—
94C23×C61—22no—
95C23×C61—41no—
96C23×C61—41no—
97C23×C61#9512yes—
98C23×C61C2×C4⋊C411yescharacteristic
99C22×D61—22no—
100C22×D61—22no—
101C22×D61—22no—
102C22×D61—24no—
103C22×D61—24no—
104C22×D61—24no—
105C22×D61—24no—
106C22×D61—42no—
107C22×D61—41no—
108C22×D61—42no—
109C22×D61—41no—
110C22×D61—416no—
111C22×D61#9511yescharacteristic
112C22×D61#9511yescharacteristic
113C22×D61C2×C22⋊C412yes—
114C22×D61#13514yes—
115C2×#6923—48no—
116C2×#6923—48no—
117C2×#6923—48no—
118C2×C4×C61—22no—
119C2×C4×C61—24no—
120C2×C4×C61—42no—
121C2×C4×C61—41no—
122C2×C4×C61#9512yes—
123C2×C4×C61C2×C4211yescharacteristic
124C2×C4×C61C2×C22⋊C411yescharacteristic
125C2×Dic61—22no—
126C2×Dic61—24no—
127C2×Dic61—42no—
128C2×Dic61—41no—
129C2×Dic61#9511yescharacteristic
130C2×Dic61#9511yescharacteristic
131C2×Dic61C2×C4⋊C412yes—
132C4×D61—22no—
133C4×D61—24no—
134C4×D61—22no—
135C4×D61—24no—
136C4×D61—24no—
137C4×D61—22no—
138C4×D61—42no—
139C4×D61—42no—
140C4×D61—42no—
141C4×D61#9914yes—
142C8×C61—42no—
143#92418—42no—
144#7982—611nomaximal, Sylow
145C22×C61—21no—
146C22×C61—21no—
147C22×C61—22no—
148C22×C61—22no—
149C22×C61—22no—
150C22×C61—44no—
151C22×C61C2×#9511yescharacteristic
152C22×C61#42712yes—
153C2×D61—28no—
154C2×D61—22no—
155C2×D61—21no—
156C2×D61—21no—
157C2×D61—21no—
158C2×D61—22no—
159C2×D61—21no—
160C2×D61—44no—
161C2×D61—48no—
162C2×D61—48no—
163C2×D61H(3,ℤ/4)12yes—
164C2×D61#34212yes—
165C2×D61#41614yes—
166#6923—88no—
167C4×C61—22no—
168C4×C61—21no—
169C4×C61C2×#9912yes—
170Dic61—21no—
171Dic61—22no—
172Dic61#34212yes—
173C22×C22⋊C8—611no—
174#2280—612no—
175#1289—614no—
176C2×C61—22no—
177C2×C61—42no—
178C2×C61C2×H(3,ℤ/4)11yescharacteristic
179C2×C61#173511yescharacteristic
180C2×C61C2×#34211yescharacteristic
181D61—24no—
182D61—22no—
183D61—22no—
184C24×C4—611no—
185C23×C8—612no—
186C2×#117—1222no—
187C2×C22⋊C8—611no—
188C2×C22⋊C8—612no—
189C2×C22⋊C8—1222no—
190C2×C22⋊C8—1222no—
191C2×C22⋊C8#692311yescharacteristic, complemented
192#381—614no—
193#380—614no—
194#380—1222no—
195#379—1222no—
196C4×C22⋊C4—614no—
197C4×C22⋊C4—1222no—
198C61#798211yescharacteristic, complemented, Sylow
199C25—611no—
200C23×C4—611no—
201C23×C4—612no—
202C23×C4—611no—
203C23×C4—612no—
204C23×C4—612no—
205C23×C4—611no—
206C23×C4—611no—
207C23×C4—612no—
208C23×C4—611no—
209C23×C4—2444no—
210C23×C4C2×#692311yescharacteristic
211C22×C8—612no—
212C22×C8—1222no—
213C22×C8—1222no—
214C22×C8—2444no—
215C22×C8C2×#692312yes—
216#117—2448no—
217C2×C4⋊C4—1224no—
218C2×C22⋊C4—1224no—
219C2×C22⋊C4—1224no—
220C2×C22⋊C4—1224no—
221C2×C42—1224no—
222C22⋊C8—22no—
223C22⋊C8—1224no—
224C22⋊C8—1222no—
225#95—618no—
226#95—1224no—
227C24—611no—
228C24—611no—
229C24—612no—
230C24—612no—
231C24—1224no—
232C24—2442no—
233C24—2442no—
234C24C4⋊#692311yescharacteristic
235C22×C4—42no—
236C22×C4—611no—
237C22×C4—612no—
238C22×C4—611no—
239C22×C4—612no—
240C22×C4—611no—
241C22×C4—612no—
242C22×C4—612no—
243C22×C4—611no—
244C22×C4—611no—
245C22×C4—612no—
246C22×C4—611no—
247C22×C4—612no—
248C22×C4—612no—
249C22×C4—612no—
250C22×C4—1228no—
251C22×C4—1224no—
252C22×C4—1228no—
253C22×C4—2442no—
254C22×C4—2444no—
255C22×C4—2444no—
256C22×C4—2442no—
257C22×C4—2444no—
258C22×C4—2442no—
259C22×C4—2442no—
260C22×C4C4×#692312yes—
261C22×C4C4⋊#692311yescharacteristic
262C22×C4C22⋊#692312yes—
263C22×C4C22×#692311yescharacteristic
264C2×C8—22no—
265C2×C8—42no—
266C2×C8—1222no—
267C2×C8—2444no—
268C2×C8—2444no—
269C2×C8—2442no—
270C4⋊C4—2448no—
271C22⋊C4—2448no—
272C22⋊C4—2448no—
273C22⋊C4—2448no—
274C42—2448no—
275C23—22no—
276C23—41no—
277C23—41no—
278C23—611no—
279C23—611no—
280C23—612no—
281C23—614no—
282C23—1222no—
283C23—1222no—
284C23—1224no—
285C23—1224no—
286C23—1224no—
287C23—1224no—
288C23—1222no—
289C23—2441no—
290C23—2442no—
291C23—2441no—
292C23—2442no—
293C23—24416no—
294C23(C2×#6923)⋊C412yes—
295C23C2×C4⋊#692311yescharacteristic
296C2×C4—22no—
297C2×C4—24no—
298C2×C4—42no—
299C2×C4—41no—
300C2×C4—611no—
301C2×C4—614no—
302C2×C4—611no—
303C2×C4—612no—
304C2×C4—1224no—
305C2×C4—1222no—
306C2×C4—1224no—
307C2×C4—1222no—
308C2×C4—1222no—
309C2×C4—1224no—
310C2×C4—1224no—
311C2×C4—1222no—
312C2×C4—2442no—
313C2×C4—2442no—
314C2×C4—2442no—
315C2×C4—2441no—
316C2×C4—2442no—
317C2×C4—2448no—
318C2×C4—2448no—
319C2×C4—2448no—
320C2×C4(C2×#6923)⋊C412yes—
321C2×C4C2×C4×#692311yescharacteristic
322C2×C4C2×C22⋊#692311yescharacteristic
323C8—42no—
324C8—2442no—
325C22—21no—
326C22—21no—
327C22—22no—
328C22—22no—
329C22—22no—
330C22—44no—
331C22—614no—
332C22—612no—
333C22—612no—
334C22—1228no—
335C22—1221no—
336C22—1221no—
337C22—1222no—
338C22—1221no—
339C22—1222no—
340C22—1221no—
341C22—2444no—
342C22—2448no—
343C22—2448no—
344C22C2×(C2×#6923)⋊C411yescharacteristic
345C22#1684077212yes—
346C4—22no—
347C4—21no—
348C4—612no—
349C4—1221no—
350C4—1222no—
351C4—4888no—
352C4D61⋊#9912yes—
353C2—22no—
354C2—42no—
355C2—1224no—
356C2—1222no—
357C2—1222no—
358C2#2146149111yescharacteristic
359C2#2146150511yescharacteristic
360C2#2146151611yescharacteristic
subgroupG/H
centerC22C2×(C2×#6923)⋊C4
commutatorC2×C4×C61C2×C42
FrattiniC22×C4C22×#6923
FittingC2×C22⋊C8×C61C4
socleC22×C61C2×#95
prime radicalC24×D61D4
cosocleC22×C4×D61C23
hypercenterC2×C22⋊C8#6923
hypocenterC61#7982
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×C22⋊C8

Complements

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

NH ≅ G/Ntype
2. #21460682C2semidirect
5. #16840705C4semidirect
10. C4×C22⋊#6923C4semidirect
12. C22×C8×D61C4semidirect
13. C2×C22⋊#92418C4semidirect
17. C2×C22⋊C8×C61C4semidirect
191. C2×C22⋊C8#6923semidirect
198. C61#7982semidirect

Maximal subgroups

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

subgroupindex
#214622632
#214606822
C2×C22⋊C8×D612
#798261