Finite Group Databasebeta

← #29577508

Subgroups — #29577508

Order 15616.

402 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 402 conjugacy classes of subgroups — too many to draw. Only its 85 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#21456537C211yesmaximal, characteristic
2#21454924C211yesmaximal, characteristic, complemented
3#21444453C211yesmaximal, characteristic
4#21444369C211yesmaximal, characteristic, complemented
5#21444365C211yesmaximal, characteristic
6#21443691C211yesmaximal, characteristic, complemented
7C61×#1288C211yesmaximal, characteristic, complemented
8#16840319C2211yescharacteristic
9#16840312C2211yescharacteristic
10#16840167C411yescharacteristic, complemented
11#16840135C411yescharacteristic, complemented
12#16840113C411yescharacteristic, complemented
13#16840109C411yescharacteristic, complemented
14#16840008—22no—
15#16839891—22no—
16#16839890—22no—
17#16839864—22no—
18#16839820—22no—
19#16839819C2211yescharacteristic
20#16839819C2211yescharacteristic
21#16839501—22no—
22#16839498—22no—
23#16839495—22no—
24C61×#432C2211yescharacteristic
25C4×M16×C61C2211yescharacteristic
26C61×#383C2211yescharacteristic
27C2×#11758222—21no—
28C2×#11758190—22no—
29#14519777C2×C411yescharacteristic
30#14519777C2×C411yescharacteristic
31#14519777C2×C411yescharacteristic
32#14519774C2×C411yescharacteristic
33#14519769—21no—
34#14519765—21no—
35#14519756—21no—
36#14519756D412yes—
37#14519751—21no—
38#14519751—21no—
39C4×Dic122—21no—
40C4×Dic122D412yes—
41C4×D244—21no—
42C4×D244—21no—
43#14519736—21no—
44#14519711—22no—
45C4⋊#92418—21no—
46C4⋊#92418—21no—
47#14519706—24no—
48#14519702—22no—
49#14519699—22no—
50C22⋊#92418—21no—
51C22⋊#92418D412yes—
52#14519688—21no—
53#14519688—21no—
54#14519687—22no—
55C8×Dic61—22no—
56C2×M16×C61—21no—
57C2×C42×C61C2311yescharacteristic
58C4⋊C8×C61—21no—
59C22⋊C8×C61—21no—
60#97×C61—21no—
61#97×C61D412yes—
62C4×C8×C61C2×C411yescharacteristic
63C4×C8×C61C2×C411yescharacteristic
64#95×C61—21no—
65#95×C61—21no—
66#95×C61—21no—
67C22×C4×C61—41no—
68C22×C4×C61C2×D411yescharacteristic
69C22×C4×C61C2×D411yescharacteristic
70C22×C4×C61#4111yescharacteristic
71C2×C22⋊D61—41no—
72#11758222—82no—
73C2×D244—21no—
74C2×D244—21no—
75C2×D244—21no—
76C2×C4×D61—41no—
77C2×Dic122—21no—
78C2×Dic122—21no—
79C2×Dic122C22⋊C412yes—
80M16×C61—42no—
81C2×C8×C61—21no—
82C2×C8×C61—21no—
83C2×C8×C61—22no—
84C42×C61C22×C411yescharacteristic
85C42×C61C2×D411yescharacteristic
86C42×C61C2×D411yescharacteristic
87C42×C61#4111yescharacteristic
88C22⋊Dic61—22no—
89#11758198—41no—
90#11758198—41no—
91C4⋊Dic61—21no—
92C4⋊Dic61—21no—
93C4⋊Dic61—21no—
94C4⋊Dic61—21no—
95C4⋊Dic61—21no—
96C4⋊Dic61C22⋊C412yes—
97#11758196—22no—
98#11758196—22no—
99C4×Dic61—22no—
100C2×#92418—22no—
101C2×#92418—41no—
102#11758190—44no—
103C23×C61#12111yescharacteristic
104C22×D61—41no—
105C2×C4×C61—41no—
106C2×C4×C61—41no—
107C2×C4×C61—41no—
108C2×C4×C61—81no—
109C2×C4×C61—81no—
110C2×C4×C61—81no—
111C2×C4×C61C2×C22⋊C411yescharacteristic
112C2×C4×C61C4×D411yescharacteristic
113C2×C4×C61C4×D411yescharacteristic
114C2×C4×C61#12011yescharacteristic
115C2×C4×C61#12111yescharacteristic
116C2×C4×C61#12311yescharacteristic
117C22⋊D61—82no—
118C2×Dic61—22no—
119C2×Dic61—22no—
120C2×Dic61—41no—
121D244—42no—
122D244—81no—
123D244—81no—
124C4×D61—82no—
125Dic122—42no—
126Dic122—81no—
127Dic122—81no—
128C8×C61—22no—
129C8×C61—42no—
130C8×C61—41no—
131#92418—42no—
132#92418—81no—
133#9418—611nomaximal, Sylow
134C22×C61—21no—
135C22×C61—21no—
136C22×C61—21no—
137C22×C61#38611yescharacteristic
138C2×D61—81no—
139C2×D61—81no—
140C2×D61—81no—
141C4×C61—21no—
142C4×C61—21no—
143C4×C61—21no—
144C4×C61—21no—
145C4×C61—21no—
146C4×C61—21no—
147C4×C61—161no—
148Dic61—42no—
149Dic61—42no—
150Dic61—81no—
151#3182—611no—
152#2987—611no—
153#1536—611no—
154#1515—611no—
155#1514—611no—
156#1291—611no—
157#1288D6111yescharacteristic, complemented
158C2×C61—41no—
159C2×C61#188411yescharacteristic
160C2×C61#190311yescharacteristic
161C2×C61#190511yescharacteristic
162D61—161no—
163#533—611no—
164#532—611no—
165#500—611no—
166#492—611no—
167#487—611no—
168#486—611no—
169#468—1222no—
170#446—1222no—
171#446—1222no—
172#440—1222no—
173#433—1222no—
174#432—611no—
175#432—611no—
176#432C2×D6111yescharacteristic
177C4×M16C2×D6111yescharacteristic
178#383C2×D6111yescharacteristic
179#329—1222no—
180#328—1222no—
181#327—1222no—
182C61#941811yescharacteristic, complemented, Sylow
183C2×#41—1221no—
184C2×SD16—1222no—
185C2×M16—21no—
186C4⋊Q8—611no—
187C4⋊Q8—611no—
188C4⋊Q8—611no—
189#127—611no—
190#125—1221no—
191#124—1221no—
192C22⋊Q8—612no—
193C22⋊Q8—1221no—
194#121—1221no—
195#121—1221no—
196C4×Q8—612no—
197C4×Q8—1221no—
198C4×D4—1221no—
199C4×D4—1221no—
200#117—1221no—
201C2×C42C22×D6111yescharacteristic
202#107—1222no—
203C4⋊C8—21no—
204C4⋊C8—1224no—
205C4⋊C8—1221no—
206C4⋊C8—1221no—
207#103—1222no—
208#102—1222no—
209C22⋊C8—21no—
210C22⋊C8—612no—
211C22⋊C8—1221no—
212#97—21no—
213#97—1222no—
214#97—1221no—
215#97—1221no—
216#97D24412yes—
217C4×C8—1222no—
218C4×C8C4×D6111yescharacteristic
219C4×C8C4×D6111yescharacteristic
220#95—21no—
221#95—21no—
222#95—21no—
223#41—4882no—
224C2×Q8—612no—
225C2×Q8—1221no—
226C2×Q8—1221no—
227C2×D4—1221no—
228C2×D4—1221no—
229C2×D4—1221no—
230C2×D4—2441no—
231C22×C4—41no—
232C22×C4—2441no—
233C22×C4#1175822211yescharacteristic
234C22×C4D4×D6111yescharacteristic
235C22×C4D4×D6111yescharacteristic
236SD16—2444no—
237M16—42no—
238C2×C8—21no—
239C2×C8—21no—
240C2×C8—22no—
241C2×C8—1222no—
242C2×C8—2441no—
243C4⋊C4—612no—
244C4⋊C4—1221no—
245C4⋊C4—1221no—
246C4⋊C4—1221no—
247C4⋊C4—1222no—
248C4⋊C4—1222no—
249C4⋊C4—1221no—
250C4⋊C4—1221no—
251C22⋊C4—1222no—
252C22⋊C4—2441no—
253C22⋊C4—2441no—
254C42—1222no—
255C42C2×C4×D6111yescharacteristic
256C42C2×D24411yescharacteristic
257C42D4×D6111yescharacteristic
258C42#1175822611yescharacteristic
259C23—2441no—
260C23#1451975011yescharacteristic
261Q8—4881no—
262Q8—4881no—
263Q8—2442no—
264D4—4881no—
265D4—4881no—
266D4—4882no—
267D4—2442no—
268C2×C4—41no—
269C2×C4—41no—
270C2×C4—41no—
271C2×C4—81no—
272C2×C4—81no—
273C2×C4—81no—
274C2×C4—1222no—
275C2×C4—1222no—
276C2×C4—4882no—
277C2×C4—2441no—
278C2×C4C22⋊C4×D6111yescharacteristic
279C2×C4#1451973911yescharacteristic
280C2×C4C4×D24411yescharacteristic
281C2×C4#1451974711yescharacteristic
282C2×C4#1451975211yescharacteristic
283C2×C4#1451976211yescharacteristic
284C8—22no—
285C8—42no—
286C8—41no—
287C8—4881no—
288C8—2442no—
289C22—21no—
290C22—21no—
291C22—21no—
292C22—4881no—
293C22—4881no—
294C22—4881no—
295C22#1683964011yescharacteristic
296C4—21no—
297C4—21no—
298C4—21no—
299C4—21no—
300C4—21no—
301C4—21no—
302C4—161no—
303C4—4881no—
304C4—2442no—
305C4—2442no—
306C2—41no—
307C2—9761no—
308C2#2144620111yescharacteristic
309C2#2144629811yescharacteristic
310C2#2144631411yescharacteristic
subgroupG/H
centerC22#16839640
commutatorC42×C61C22×C4
FrattiniC2×C42C22×D61
FittingC61×#1288C2
socleC22×C61#386
prime radical#16840312C22
cosocleC2×C42×C61C23
hypercenter#1288D61
hypocenterC61#9418
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1288

Complements

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

NH ≅ G/Ntype
2. #21454924C2semidirect
4. #21444369C2semidirect
6. #21443691C2semidirect
7. C61×#1288C2semidirect
10. #16840167C4semidirect
11. #16840135C4semidirect
12. #16840113C4semidirect
13. #16840109C4semidirect
157. #1288D61semidirect
182. C61#9418semidirect

Maximal subgroups

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

subgroupindex
#214565372
#214549242
#214444532
#214443692
#214443652
#214436912
C61×#12882
#941861