Finite Group Databasebeta

← #30601016

Subgroups — #30601016

Order 15616.

506 conjugacy classes of subgroups; only the 96 normal subgroups are shown.

normal subgroups only

This group has 506 conjugacy classes of subgroups — too many to draw. Only its 96 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#21462714C211yesmaximal, characteristic
2#21461686C211yesmaximal, characteristic
3#21461328C212yesmaximal
4#21460717C212yesmaximal
5#21450803C211yesmaximal, characteristic
6#16840811—24no—
7#16840779—21no—
8#16840779C2212yes—
9#16840778—21no—
10#16840765C2212yes—
11C2×#14519838—21no—
12#16840642—21no—
13#16840621—24no—
14C2×#14519776—21no—
15C2×#14519776C2211yescharacteristic
16C2×#14519742C2211yescharacteristic
17#16839694C411yescharacteristic
18#16839693C411yescharacteristic
19#16839692C411yescharacteristic
20#16839666—21no—
21#16839659—21no—
22#16839657—21no—
23#16839613C2211yescharacteristic
24C61×#398C411yescharacteristic
25C2×#11758217—22no—
26#14519844—22no—
27#14519844—22no—
28#14519844D414yes—
29C4⋊#92420—22no—
30C4⋊#92420—22no—
31#14519838—42no—
32C22⋊#92420—22no—
33C4×#92420—22no—
34C4×#92420—22no—
35C4×#92420—22no—
36C22×Dic122—21no—
37C22×Dic122D411yescharacteristic
38#14519776—22no—
39#14519776—22no—
40#14519776—42no—
41#14519776—42no—
42#14519742—22no—
43#14519742—22no—
44C2×C4⋊Dic61—21no—
45C2×C4⋊Dic61C2×C411yescharacteristic
46C2×#11758196—21no—
47C2×#11758196—21no—
48C2×#11758196—21no—
49C2×#11758196—21no—
50C2×#11758196C2×C411yescharacteristic
51C2×#11758196D411yescharacteristic
52C2×C4×Dic61—21no—
53C2×C4×Dic61D411yescharacteristic
54C2×C4×Dic61D411yescharacteristic
55C2×C4×Dic61C2311yescharacteristic
56#14519684—21no—
57#14519684—21no—
58#14519684—21no—
59#14519684C2×C411yescharacteristic
60C2×C4⋊C4×C61—21no—
61C2×C4⋊C4×C61C2×C411yescharacteristic
62C2×C4⋊C4×C61C2×C411yescharacteristic
63#95×C61—21no—
64#95×C61C2×C411yescharacteristic
65C22×C4×C61—21no—
66C22×C4×C61—21no—
67C22×C4×C61C22⋊C411yescharacteristic
68C22×C4×C61C22⋊C411yescharacteristic
69C22×C4×C61C22×C411yescharacteristic
70C22×Dic61—21no—
71C22×Dic61—21no—
72C22×Dic61C22⋊C411yescharacteristic
73C22×Dic61C22⋊C411yescharacteristic
74C22×Dic61C2×D411yescharacteristic
75C22×Dic61C2×D411yescharacteristic
76C2×Dic122—22no—
77C2×Dic122—22no—
78C2×Dic122—42no—
79C2×Dic122—41no—
80C2×Dic122—41no—
81C2×Dic122—42no—
82C2×Dic122D812yes—
83C2×Dic122SD1612yes—
84#11758217—42no—
85#11758217—42no—
86C2×#92420—24no—
87C2×#92420—42no—
88C4⋊C4×C61—22no—
89C4⋊C4×C61—22no—
90C4⋊C4×C61—42no—
91C4⋊Dic61—22no—
92C4⋊Dic61—42no—
93#11758196—24no—
94#11758196—22no—
95#11758196—22no—
96#11758196—42no—
97#11758196—42no—
98#11758196—41no—
99#11758196—41no—
100C4×Dic61—21no—
101C4×Dic61—21no—
102C4×Dic61—21no—
103C4×Dic61—21no—
104C4×Dic61—41no—
105C4×Dic61—41no—
106C4×Dic61C2×D412yes—
107C4×Dic61#4112yes—
108C23×C61C2×C22⋊C411yescharacteristic
109C2×C4×C61—21no—
110C2×C4×C61—21no—
111C2×C4×C61—22no—
112C2×C4×C61—21no—
113C2×C4×C61—41no—
114C2×C4×C61—41no—
115C2×C4×C61—41no—
116C2×C4×C61—41no—
117C2×C4×C61—41no—
118C2×C4×C61—41no—
119C2×C4×C61—41no—
120C2×C4×C61#10212yes—
121C2×C4×C61C4×D412yes—
122C2×Dic61—21no—
123C2×Dic61—21no—
124C2×Dic61—22no—
125C2×Dic61—22no—
126C2×Dic61—21no—
127C2×Dic61—41no—
128C2×Dic61—41no—
129C2×Dic61—41no—
130C2×Dic61—41no—
131C2×Dic61—41no—
132C2×Dic61—41no—
133C2×Dic61#10212yes—
134C2×Dic61#12011yescharacteristic
135C2×Dic61#12112yes—
136C2×Dic61#12311yescharacteristic
137C2×Dic61C2×D811yescharacteristic
138C2×Dic61C2×SD1611yescharacteristic
139C2×Dic61#13512yes—
140Dic122—24no—
141Dic122—24no—
142Dic122—82no—
143Dic122—82no—
144#92420—44no—
145#92420—82no—
146#9394—611nomaximal, Sylow
147C22×C61—21no—
148C22×C61—21no—
149C22×C61#38611yescharacteristic
150C22×C61C2×#10211yescharacteristic
151C22×C61#41611yescharacteristic
152C4×C61—22no—
153C4×C61—22no—
154C4×C61—42no—
155C4×C61—41no—
156C4×C61—81no—
157Dic61—21no—
158Dic61—22no—
159Dic61—42no—
160Dic61—41no—
161Dic61—81no—
162Dic61#45612yes—
163Dic61#46512yes—
164Dic61#46612yes—
165#2982—611no—
166#2546—611no—
167#1801—611no—
168#1546—612no—
169#1303—612no—
170C2×C61—21no—
171C2×C61—21no—
172C2×C61#188611yescharacteristic
173C2×C61#189311yescharacteristic
174C2×C61#191511yescharacteristic
175C2×C4⋊Q8—611no—
176C2×C4⋊Q8—1221no—
177C2×C4×Q8—611no—
178C8×Q8—1224no—
179#431—612no—
180#431—1221no—
181#431—1221no—
182#423—612no—
183C2×#101—1221no—
184#398—611no—
185#398—611no—
186#398—611no—
187#398#692311yescharacteristic
188#391—1221no—
189#389—1221no—
190#389—1221no—
191C4×C4⋊C4—611no—
192#343—1221no—
193#332—1224no—
194C61#939411yescharacteristic, complemented, Sylow
195C22×Q8—611no—
196C22×Q8—1221no—
197C2×M16—1222no—
198C4⋊Q8—1222no—
199C4⋊Q8—1222no—
200C4⋊Q8—2442no—
201C4⋊Q8—2442no—
202C4×Q8—1222no—
203C4×Q8—1222no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—611no—
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×C42—611no—
216C2×C42—611no—
217C2×C42—611no—
218C2×C42—1221no—
219C4⋊C8—614no—
220C4⋊C8—1222no—
221C4⋊C8—1222no—
222C4⋊C8—1222no—
223C4⋊C8—1222no—
224#101—2442no—
225C22⋊C8—1222no—
226C4×C8—1222no—
227C4×C8—1222no—
228C4×C8—1222no—
229#95—21no—
230#95—611no—
231#95—1221no—
232#95—1221no—
233#95—1221no—
234#95C2×#692311yescharacteristic
235C2×Q8—612no—
236C2×Q8—612no—
237C2×Q8—1222no—
238C2×Q8—1222no—
239C2×Q8—2442no—
240C2×Q8—2441no—
241C2×Q8—2441no—
242C2×Q8—2442no—
243C22×C4—21no—
244C22×C4—21no—
245C22×C4—611no—
246C22×C4—611no—
247C22×C4—611no—
248C22×C4—611no—
249C22×C4—1221no—
250C22×C4—1221no—
251C22×C4C22⋊#692311yescharacteristic
252C22×C4C22⋊#692311yescharacteristic
253C22×C4C22×#692311yescharacteristic
254M16—2442no—
255M16—2442no—
256C2×C8—1224no—
257C2×C8—2442no—
258C4⋊C4—22no—
259C4⋊C4—22no—
260C4⋊C4—42no—
261C4⋊C4—1222no—
262C4⋊C4—1224no—
263C4⋊C4—1222no—
264C4⋊C4—1222no—
265C4⋊C4—2441no—
266C4⋊C4—2441no—
267C4⋊C4—2442no—
268C4⋊C4—2442no—
269C4⋊C4—2442no—
270C42—612no—
271C42—612no—
272C42—1221no—
273C42—1221no—
274C42—1221no—
275C42—1221no—
276C42—2441no—
277C42—2441no—
278C23C2×C22⋊#692311yescharacteristic
279Q8—1224no—
280Q8—1224no—
281Q8—4882no—
282Q8—4882no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—22no—
286C2×C4—21no—
287C2×C4—41no—
288C2×C4—41no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—611no—
295C2×C4—612no—
296C2×C4—611no—
297C2×C4—611no—
298C2×C4—612no—
299C2×C4—611no—
300C2×C4—612no—
301C2×C4—1222no—
302C2×C4—1221no—
303C2×C4—1221no—
304C2×C4—1222no—
305C2×C4—1221no—
306C2×C4—2441no—
307C2×C4—2441no—
308C2×C4—2441no—
309C2×C4—2441no—
310C2×C4—2441no—
311C2×C4—2441no—
312C2×C4#1451983912yes—
313C2×C4D4×#692312yes—
314C8—4882no—
315C8—2444no—
316C22—21no—
317C22—21no—
318C22#1684071011yescharacteristic
319C22C2×#1451983911yescharacteristic
320C22#1684075211yescharacteristic
321C4—22no—
322C4—22no—
323C4—42no—
324C4—41no—
325C4—81no—
326C4—612no—
327C4—612no—
328C4—612no—
329C4—1221no—
330C4—1222no—
331C4—4881no—
332C4—2442no—
333C4—2441no—
334C2—21no—
335C2—21no—
336C2#2146182511yescharacteristic
337C2#2146183311yescharacteristic
338C2#2146187111yescharacteristic
subgroupG/H
centerC22#16840710
commutatorC22×C4×C61C22×C4
FrattiniC22×C4C22×#6923
FittingC61×#398C4
socleC22×C61#386
prime radicalC23×C61C2×C22⋊C4
cosocleC2×C4×Dic61C23
hypercenter#398#6923
hypocenterC61#9394
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ #398

Complements

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

NH ≅ G/Ntype
194. C61#9394semidirect

Maximal subgroups

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

subgroupindex
#214627142
#214616862
#214613282
#214607172
#214508032
#939461