Finite Group Databasebeta

← #30595845

Subgroups — #30595845

Order 15616.

500 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 500 conjugacy classes of subgroups — too many to draw. Only its 92 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#21461709C211yesmaximal, characteristic
2#21461545C211yesmaximal, characteristic
3#21460638C212yesmaximal, complemented
4#21460637C212yesmaximal, complemented
5#21454169C211yesmaximal, characteristic
6C2×C22⋊#92420—21no—
7C2×C22⋊#92420—21no—
8C2×#14519832—21no—
9#16840628—21no—
10#16840628C2212yes—
11#16840628C2212yes—
12C2×#14519776C411yescharacteristic
13C2×#14519754—21no—
14C22×C4×Dic61C2211yescharacteristic
15#16839666—21no—
16#16839661C2211yescharacteristic
17#16839659—21no—
18#16839658C2211yescharacteristic
19#16839631C411yescharacteristic
20#16839622C411yescharacteristic
21#16839620—21no—
22C61×#380C411yescharacteristic
23C22×#92420—22no—
24C22×#92420—22no—
25C22⋊#92420—24no—
26C22⋊#92420—42no—
27#14519832—42no—
28C22×Dic122—21no—
29C23×Dic61D411yescharacteristic
30C23×Dic61D411yescharacteristic
31#14519776—42no—
32#14519754—44no—
33C2×C4⋊Dic61—21no—
34C2×C4⋊Dic61C2×C411yescharacteristic
35C2×C4⋊Dic61C2×C411yescharacteristic
36C2×#11758196—21no—
37C2×#11758196—21no—
38C2×#11758196—21no—
39C2×#11758196—21no—
40C2×C22⋊Dic61—21no—
41C2×C4×Dic61—42no—
42C2×C4×Dic61C2×C411yescharacteristic
43C2×C4×Dic61D411yescharacteristic
44C2×C4×Dic61D411yescharacteristic
45C2×C4×Dic61C2311yescharacteristic
46#14519684—21no—
47#14519684—21no—
48#14519684—21no—
49C23×C4×C61C2×C411yescharacteristic
50C2×C4⋊C4×C61C2×C411yescharacteristic
51C2×C4⋊C4×C61C2×C411yescharacteristic
52C2×C22⋊C4×C61—21no—
53#95×C61—21no—
54C24×C61C22⋊C411yescharacteristic
55C22×C4×C61—22no—
56C22×C4×C61—22no—
57C22×C4×C61—21no—
58C22×C4×C61—21no—
59C22×C4×C61—41no—
60C22×C4×C61C22⋊C411yescharacteristic
61C22×C4×C61C22×C411yescharacteristic
62C22×Dic61—22no—
63C22×Dic61—22no—
64C22×Dic61—22no—
65C22×Dic61—22no—
66C22×Dic61—21no—
67C22×Dic61—21no—
68C22×Dic61—41no—
69C22×Dic61—41no—
70C22×Dic61C22⋊C411yescharacteristic
71C22×Dic61C22⋊C411yescharacteristic
72C22×Dic61C2×D411yescharacteristic
73C22×Dic61C2×D411yescharacteristic
74C2×Dic122—42no—
75C2×Dic122—42no—
76C2×Dic122—42no—
77C2×#92420—24no—
78C2×#92420—42no—
79C2×#92420—42no—
80C2×#92420—42no—
81C2×#92420—42no—
82C2×#92420—42no—
83C4⋊C4×C61—22no—
84C4⋊C4×C61—41no—
85C22⋊C4×C61—42no—
86C22⋊Dic61—42no—
87C4⋊Dic61—22no—
88C4⋊Dic61—42no—
89C4⋊Dic61—41no—
90#11758196—42no—
91#11758196—42no—
92#11758196—42no—
93#11758196—42no—
94C4×Dic61—22no—
95C4×Dic61—22no—
96C4×Dic61—41no—
97C4×Dic61—41no—
98C23×C61—21no—
99C23×C61—21no—
100C23×C61—21no—
101C23×C61—21no—
102C23×C61—21no—
103C23×C61—21no—
104C23×C61#10112yes—
105C23×C61C2×C22⋊C411yescharacteristic
106C2×C4×C61—21no—
107C2×C4×C61—21no—
108C2×C4×C61—21no—
109C2×C4×C61—21no—
110C2×C4×C61—21no—
111C2×C4×C61—44no—
112C2×C4×C61—41no—
113C2×C4×C61—41no—
114C2×C4×C61—41no—
115C2×C4×C61—41no—
116C2×C4×C61—41no—
117C2×C4×C61—41no—
118C2×C4×C61#13112yes—
119C2×Dic61—21no—
120C2×Dic61—21no—
121C2×Dic61—21no—
122C2×Dic61—21no—
123C2×Dic61—21no—
124C2×Dic61—21no—
125C2×Dic61—21no—
126C2×Dic61—21no—
127C2×Dic61—44no—
128C2×Dic61—44no—
129C2×Dic61—41no—
130C2×Dic61—41no—
131C2×Dic61—41no—
132C2×Dic61—41no—
133C2×Dic61—41no—
134C2×Dic61—41no—
135C2×Dic61#9912yes—
136C2×Dic61#10412yes—
137C2×Dic61AGL(1,ℤ/8)12yes—
138C2×Dic61#13712yes—
139Dic122—84no—
140#92420—44no—
141#92420—82no—
142#7705—611nomaximal, Sylow
143C22×C61—22no—
144C22×C61—22no—
145C22×C61—22no—
146C22×C61—41no—
147C22×C61—41no—
148C22×C61—41no—
149C22×C61—41no—
150C22×C61#40811yescharacteristic
151C22×C61C2×#9911yescharacteristic
152C22×C61C2×#10111yescharacteristic
153C22×C61#41811yescharacteristic
154C22×C61#41911yescharacteristic
155C22×C61C2×#10411yescharacteristic
156C22×C61#42111yescharacteristic
157C4×C61—22no—
158C4×C61—41no—
159C4×C61—81no—
160C4×C61—81no—
161Dic61—22no—
162Dic61—22no—
163Dic61—22no—
164Dic61—22no—
165Dic61—81no—
166Dic61—81no—
167#2857—611no—
168#1817—611no—
169#1749—611no—
170#1271—612no—
171#1271—612no—
172C2×C61—42no—
173C2×C61#147211yescharacteristic
174C2×C61#148511yescharacteristic
175C2×C61#149611yescharacteristic
176C2×C61#150711yescharacteristic
177C2×C61#151111yescharacteristic
178C2×C61#153111yescharacteristic
179C2×C61#155611yescharacteristic
180C2×C4⋊Q8—611no—
181C2×C22⋊Q8—1221no—
182C22×C42—611no—
183C2×C22⋊C8—1221no—
184C2×C22⋊C8—1221no—
185C2×#97—1221no—
186#391—1221no—
187#389—611no—
188#389—611no—
189#389—1221no—
190#384—611no—
191#380—611no—
192#380—1221no—
193#380#692311yescharacteristic
194#336—612no—
195#336—612no—
196#336—1221no—
197C61#770511yescharacteristic, complemented, Sylow
198C22×Q8—1221no—
199C23×C4—611no—
200C23×C4—611no—
201C23×C4C2×#692311yescharacteristic
202C22×C8—1222no—
203C22×C8—1222no—
204C4⋊Q8—2442no—
205C22⋊Q8—2444no—
206C2×C4⋊C4—611no—
207C2×C4⋊C4—611no—
208C2×C4⋊C4—1221no—
209C2×C4⋊C4—1221no—
210C2×C4⋊C4—1221no—
211C2×C4⋊C4—1221no—
212C2×C4⋊C4—1221no—
213C2×C4⋊C4C2×#692311yescharacteristic
214C2×C4⋊C4C2×#692311yescharacteristic
215C2×C22⋊C4—21no—
216C2×C22⋊C4—1221no—
217C2×C42—611no—
218C2×C42—611no—
219C2×C42—611no—
220C2×C42—611no—
221C2×C42—2442no—
222C22⋊C8—1224no—
223C22⋊C8—2442no—
224#97—2442no—
225#95—21no—
226#95—1221no—
227#95—1221no—
228#95—1221no—
229C24C22⋊#692311yescharacteristic
230C2×Q8—2442no—
231C2×Q8—2442no—
232C2×Q8—2442no—
233C22×C4—22no—
234C22×C4—22no—
235C22×C4—21no—
236C22×C4—21no—
237C22×C4—41no—
238C22×C4—611no—
239C22×C4—611no—
240C22×C4—611no—
241C22×C4—611no—
242C22×C4—1222no—
243C22×C4—1222no—
244C22×C4—1222no—
245C22×C4—1222no—
246C22×C4—1221no—
247C22×C4—1221no—
248C22×C4—2441no—
249C22×C4—2441no—
250C22×C4C22⋊#692311yescharacteristic
251C22×C4C22×#692311yescharacteristic
252C2×C8—1224no—
253C2×C8—2442no—
254C2×C8—2442no—
255C2×C8—2442no—
256C2×C8—2442no—
257C2×C8—2442no—
258C4⋊C4—22no—
259C4⋊C4—41no—
260C4⋊C4—1222no—
261C4⋊C4—2441no—
262C4⋊C4—2442no—
263C4⋊C4—2442no—
264C4⋊C4—2442no—
265C4⋊C4—2442no—
266C4⋊C4—2442no—
267C22⋊C4—42no—
268C22⋊C4—2442no—
269C42—1222no—
270C42—1222no—
271C42—2441no—
272C42—2441no—
273C23—21no—
274C23—21no—
275C23—21no—
276C23—21no—
277C23—21no—
278C23—21no—
279C23#1451983812yes—
280C23C2×C22⋊#692311yescharacteristic
281Q8—4884no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—44no—
288C2×C4—41no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—612no—
295C2×C4—612no—
296C2×C4—612no—
297C2×C4—612no—
298C2×C4—1221no—
299C2×C4—1221no—
300C2×C4—1221no—
301C2×C4—1221no—
302C2×C4—1221no—
303C2×C4—1221no—
304C2×C4—1221no—
305C2×C4—1221no—
306C2×C4—2444no—
307C2×C4—2444no—
308C2×C4—2441no—
309C2×C4—2441no—
310C2×C4—2441no—
311C2×C4—2441no—
312C2×C4—2441no—
313C2×C4—2441no—
314C2×C4#1451986412yes—
315C8—4882no—
316C8—2444no—
317C22—22no—
318C22—22no—
319C22—22no—
320C22—41no—
321C22—41no—
322C22—41no—
323C22—41no—
324C22#1684073511yescharacteristic
325C22D61⋊#9911yescharacteristic
326C22C2×#1451983811yescharacteristic
327C22#1684075411yescharacteristic
328C22#1684075511yescharacteristic
329C22D61⋊#10411yescharacteristic
330C22#1684075911yescharacteristic
331C4—22no—
332C4—41no—
333C4—81no—
334C4—81no—
335C4—1222no—
336C4—1222no—
337C4—1222no—
338C4—1222no—
339C4—4881no—
340C4—4881no—
341C2—42no—
342C2#2146109811yescharacteristic
343C2#2146114111yescharacteristic
344C2#2146117811yescharacteristic
345C2#2146121411yescharacteristic
346C2#2146122811yescharacteristic
347C2#2146129111yescharacteristic
348C2#2146134811yescharacteristic
subgroupG/H
centerC23C2×C22⋊#6923
commutatorC22×C4×C61C22×C4
FrattiniC22×C4C22×#6923
FittingC61×#380C4
socleC23×C61C2×C22⋊C4
prime radicalC24×C61C22⋊C4
cosocleC2×C4×Dic61C23
hypercenter#380#6923
hypocenterC61#7705
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ #380

Complements

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

NH ≅ G/Ntype
3. #21460638C2semidirect
4. #21460637C2semidirect
197. C61#7705semidirect

Maximal subgroups

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

subgroupindex
#214617092
#214615452
#214606382
#214606372
#214541692
#770561