Finite Group Databasebeta

← #30604317

Subgroups — #30604317

Order 15616.

412 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 412 conjugacy classes of subgroups — too many to draw. Only its 70 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#21461082C211yesmaximal, characteristic, complemented
2#21461062C211yesmaximal, characteristic, complemented
3#21460749C212yesmaximal, complemented
4#21460748C212yesmaximal
5#21450796C211yesmaximal, characteristic
6#16840782—21no—
7#16840778—21no—
8#16840607—21no—
9#16840607—21no—
10#16840607C2212yes—
11#16840607C2212yes—
12C2×#14519766C411yescharacteristic
13C2×#14519766C411yescharacteristic
14C2×Q8×Dic61C2211yescharacteristic
15#16839684—21no—
16#16839684—21no—
17#16839669C2211yescharacteristic
18#16839664C2211yescharacteristic
19#16839652—21no—
20#16839652—21no—
21C4×#11758196C411yescharacteristic
22C2×C4⋊Q8×C61C411yescharacteristic
23#14519844—41no—
24C4⋊#92420—41no—
25C22⋊#92420—22no—
26C22⋊#92420—22no—
27#14519832—41no—
28C4×#92420—41no—
29C22×D244C2×C411yescharacteristic
30#14519766—41no—
31#14519766—41no—
32#14519766—42no—
33Q8×Dic61—41no—
34Q8×Dic61—41no—
35C2×C4⋊Dic61D411yescharacteristic
36C2×C4⋊Dic61D411yescharacteristic
37C2×C4⋊Dic61C2311yescharacteristic
38C2×#11758196—21no—
39C2×#11758198—21no—
40C2×#11758198—21no—
41C2×#11758198—21no—
42C2×#11758198—21no—
43C2×#11758198—21no—
44C2×#11758198—21no—
45C2×C4×Dic61C2×C411yescharacteristic
46C2×C4×Dic61C2×C411yescharacteristic
47C2×C4×Dic61D411yescharacteristic
48C2×C4×Dic61D411yescharacteristic
49#14519684—21no—
50#14519684—21no—
51#14519684—21no—
52C22×Q8×C61C2×C411yescharacteristic
53C22×Q8×C61C2×C411yescharacteristic
54C4⋊Q8×C61—42no—
55C2×C4⋊C4×C61—21no—
56C2×C4⋊C4×C61—21no—
57C2×C42×C61C2×C411yescharacteristic
58C23×D61—21no—
59C2×Q8×C61—22no—
60C2×Q8×C61—22no—
61C2×Q8×C61—22no—
62C2×Q8×C61—41no—
63C2×Q8×C61—41no—
64C2×Q8×C61—41no—
65C22×C4×C61—21no—
66C22×C4×C61—21no—
67C22×C4×C61C22⋊C411yescharacteristic
68C22×C4×C61C22⋊C411yescharacteristic
69C22×C4×C61C22×C411yescharacteristic
70C22×Dic61—21no—
71C22×Dic61C22⋊C411yescharacteristic
72C22×Dic61C22⋊C411yescharacteristic
73C22×Dic61C2×D411yescharacteristic
74C22×Dic61C2×D411yescharacteristic
75C2×D244—22no—
76C2×D244—41no—
77C2×D244—41no—
78C2×#92420—42no—
79C2×#92420—42no—
80C4⋊C4×C61—42no—
81C4⋊C4×C61—42no—
82C42×C61—41no—
83#11758198—42no—
84#11758198—42no—
85#11758198—42no—
86#11758198—42no—
87#11758198—42no—
88#11758198—41no—
89#11758198—41no—
90C4⋊Dic61—41no—
91C4⋊Dic61—41no—
92C4⋊Dic61—41no—
93#11758196—42no—
94C4×Dic61—21no—
95C4×Dic61—21no—
96C4×Dic61—21no—
97C4×Dic61—21no—
98C4×Dic61—21no—
99C4×Dic61—21no—
100C4×Dic61—41no—
101C23×C61C2×C22⋊C411yescharacteristic
102C22×D61—41no—
103C22×D61—41no—
104C22×D61—41no—
105C22×D61—41no—
106C22×D61—41no—
107C22×D61—41no—
108C22×D61—41no—
109Q8×C61—44no—
110Q8×C61—82no—
111C2×C4×C61—21no—
112C2×C4×C61—21no—
113C2×C4×C61—21no—
114C2×C4×C61—21no—
115C2×C4×C61—22no—
116C2×C4×C61—21no—
117C2×C4×C61—21no—
118C2×C4×C61—21no—
119C2×C4×C61—41no—
120C2×C4×C61—41no—
121C2×C4×C61—41no—
122C2×C4×C61—41no—
123C2×C4×C61—41no—
124C2×C4×C61#10012yes—
125C2×C4×C61#13112yes—
126C2×Dic61—21no—
127C2×Dic61—21no—
128C2×Dic61—21no—
129C2×Dic61—41no—
130C2×Dic61—41no—
131C2×Dic61—41no—
132C2×Dic61—41no—
133C2×Dic61—41no—
134C2×Dic61—41no—
135C2×Dic61—41no—
136C2×Dic61#9911yescharacteristic
137C2×Dic61#9911yescharacteristic
138D244—82no—
139#92420—82no—
140#92420—82no—
141#11565—611nomaximal, Sylow
142C22×C61—21no—
143C22×C61—21no—
144C22×C61#40811yescharacteristic
145C22×C61C2×#9911yescharacteristic
146C22×C61C2×#10011yescharacteristic
147C2×D61—81no—
148C2×D61—81no—
149C2×D61—81no—
150C2×D61—81no—
151C2×D61—81no—
152C2×D61—81no—
153C2×D61—81no—
154C4×C61—22no—
155C4×C61—22no—
156C4×C61—42no—
157C4×C61—41no—
158C4×C61—81no—
159Dic61—21no—
160Dic61—21no—
161Dic61—41no—
162Dic61—41no—
163Dic61—41no—
164Dic61—81no—
165#2545—611no—
166#1467—611no—
167#1461—611no—
168#1319—612no—
169#1318—612no—
170C2×C61—21no—
171C2×C61—21no—
172C2×C61#147111yescharacteristic
173C2×C61#212611yescharacteristic
174C2×C61#213311yescharacteristic
175D61—161no—
176C2×C4⋊Q8#692311yescharacteristic
177C2×#124—611no—
178C2×#124—611no—
179C2×C4×Q8—611no—
180#432—1221no—
181#431—1221no—
182#396—1221no—
183#396—1221no—
184#391—611no—
185#390—611no—
186#388—1221no—
187#388—1221no—
188C4×C4⋊C4—611no—
189#324—612no—
190#324—612no—
191#324—1221no—
192#324—1221no—
193C61#1156511yescharacteristic, complemented, Sylow
194C22×Q8C2×#692311yescharacteristic
195C22×Q8C2×#692311yescharacteristic
196C22×D4—611no—
197C4⋊Q8—42no—
198#124—2441no—
199#124—2441no—
200#124—2442no—
201C4×Q8—2441no—
202C4×Q8—2441no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—611no—
206C2×C4⋊C4—611no—
207C2×C4⋊C4—611no—
208C2×C4⋊C4—1221no—
209C2×C22⋊C4—1221no—
210C2×C22⋊C4—1221no—
211C2×C22⋊C4—1221no—
212C2×C22⋊C4—1221no—
213C2×C22⋊C4—1221no—
214C2×C22⋊C4—1221no—
215C2×C42—611no—
216C2×C42—611no—
217C2×C42—611no—
218C2×C42—611no—
219C2×C42C2×#692311yescharacteristic
220C4⋊C8—2441no—
221C4⋊C8—2441no—
222C22⋊C8—1222no—
223C22⋊C8—1222no—
224#97—2441no—
225C4×C8—2441no—
226#95—1221no—
227#95—1221no—
228#95—1221no—
229C24—1221no—
230C2×Q8—22no—
231C2×Q8—22no—
232C2×Q8—22no—
233C2×Q8—41no—
234C2×Q8—41no—
235C2×Q8—41no—
236C2×D4—1222no—
237C2×D4—2441no—
238C2×D4—2441no—
239C22×C4—21no—
240C22×C4—21no—
241C22×C4—611no—
242C22×C4—611no—
243C22×C4—611no—
244C22×C4—611no—
245C22×C4—1221no—
246C22×C4C22⋊#692311yescharacteristic
247C22×C4C22⋊#692311yescharacteristic
248C22×C4C22×#692311yescharacteristic
249C2×C8—2442no—
250C2×C8—2442no—
251C4⋊C4—42no—
252C4⋊C4—42no—
253C4⋊C4—2441no—
254C4⋊C4—2441no—
255C4⋊C4—2441no—
256C4⋊C4—2442no—
257C22⋊C4—2442no—
258C22⋊C4—2442no—
259C22⋊C4—2442no—
260C22⋊C4—2441no—
261C22⋊C4—2441no—
262C22⋊C4—2442no—
263C22⋊C4—2442no—
264C42—41no—
265C42—1221no—
266C42—1221no—
267C42—1221no—
268C42—1221no—
269C42—1221no—
270C42—1221no—
271C42—2441no—
272C23—2441no—
273C23—2441no—
274C23—2441no—
275C23—2441no—
276C23—2441no—
277C23—2441no—
278C23—2441no—
279C23C2×C22⋊#692311yescharacteristic
280Q8—44no—
281Q8—82no—
282D4—4882no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—22no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—41no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—41no—
295C2×C4—41no—
296C2×C4—611no—
297C2×C4—611no—
298C2×C4—1221no—
299C2×C4—1221no—
300C2×C4—1221no—
301C2×C4—2441no—
302C2×C4—2441no—
303C2×C4—2441no—
304C2×C4—2441no—
305C2×C4—2441no—
306C2×C4—2441no—
307C2×C4—2441no—
308C2×C4#1451983712yes—
309C2×C4#1451986412yes—
310C8—4882no—
311C8—4882no—
312C22—21no—
313C22—21no—
314C22—4881no—
315C22—4881no—
316C22—4881no—
317C22—4881no—
318C22—4881no—
319C22—4881no—
320C22—4881no—
321C22#1684073511yescharacteristic
322C22D61⋊#9911yescharacteristic
323C22C2×#1451983711yescharacteristic
324C4—22no—
325C4—22no—
326C4—42no—
327C4—41no—
328C4—81no—
329C4—1221no—
330C4—1221no—
331C4—4881no—
332C4—2441no—
333C4—2441no—
334C4—2441no—
335C2—21no—
336C2—21no—
337C2—9761no—
338C2#2146109511yescharacteristic
339C2#2146209311yescharacteristic
340C2#2146211711yescharacteristic
subgroupG/H
centerC22D61⋊#99
commutatorC22×C4×C61C22×C4
FrattiniC22×C4C22×#6923
FittingC2×C4⋊Q8×C61C4
socleC22×C61C2×#99
prime radicalC22×D244C2×C4
cosocleC2×C4⋊Dic61C23
hypercenterC2×C4⋊Q8#6923
hypocenterC61#11565
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23 ◁ C22×Q8 ◁ C2×C4⋊Q8

Complements

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

NH ≅ G/Ntype
1. #21461082C2semidirect
2. #21461062C2semidirect
3. #21460749C2semidirect
193. C61#11565semidirect

Maximal subgroups

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

subgroupindex
#214610822
#214610622
#214607492
#214607482
#214507962
#1156561