Finite Group Databasebeta

← #29577577

Subgroups — #29577577

Order 15616.

444 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 444 conjugacy classes of subgroups — too many to draw. Only its 99 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#21456215C211yesmaximal, characteristic
2#21444536C211yesmaximal, characteristic, complemented
3#21444365C211yesmaximal, characteristic
4#21444365C211yesmaximal, characteristic
5#21443683C211yesmaximal, characteristic, complemented
6#21443683C211yesmaximal, characteristic, complemented
7C61×#2381C211yesmaximal, characteristic, complemented
8#16840312C2211yescharacteristic
9#16840124—22no—
10#16840114C412yescomplemented
11#16840110C412yescomplemented
12#16839819C2211yescharacteristic
13#16839819C2211yescharacteristic
14#16839761—22no—
15C4×#11758194C2211yescharacteristic
16#16839508—22no—
17#16839495—22no—
18#16839495—22no—
19C2×C4⋊Q8×C61C2211yescharacteristic
20C61×#391—22no—
21C61×#387—22no—
22C61×#387—22no—
23C61×#383C2211yescharacteristic
24C61×#383C2211yescharacteristic
25C2×#11758222—21no—
26C2×#11758194—21no—
27#14519777C2×C411yescharacteristic
28#14519774C2×C411yescharacteristic
29#14519765—21no—
30#14519751—21no—
31#14519751—21no—
32C4×D244—21no—
33C4×D244—21no—
34C4⋊#92418—21no—
35C4⋊#92418—21no—
36#14519700—24no—
37#14519697—24no—
38C22⋊#92418—21no—
39C22⋊#92418—21no—
40#14519688—21no—
41#14519688—21no—
42#14519688—21no—
43C4×#92418C2×C412yes—
44C22×Q8×C61—21no—
45C4⋊Q8×C61—21no—
46C4⋊Q8×C61—21no—
47C4⋊Q8×C61C2×C412yes—
48C4⋊Q8×C61D412yes—
49C2×C4⋊C4×C61—21no—
50C2×C4⋊C4×C61—22no—
51C2×C4⋊C4×C61D412yes—
52C2×C42×C61—22no—
53C2×C42×C61C2311yescharacteristic
54#95×C61—24no—
55#95×C61—21no—
56#95×C61—21no—
57#95×C61—21no—
58#95×C61—21no—
59#95×C61D412yes—
60#95×C61D412yes—
61C2×Q8×C61—22no—
62C2×Q8×C61—22no—
63C2×Q8×C61—81no—
64C2×Q8×C61—81no—
65C22×C4×C61—22no—
66C22×C4×C61—22no—
67C22×C4×C61—22no—
68C22×C4×C61C2×D411yescharacteristic
69C22×C4×C61C2×D411yescharacteristic
70C22×C4×C61C2×D411yescharacteristic
71C2×C22⋊D61—41no—
72#11758222—44no—
73C2×D244—21no—
74C2×D244—21no—
75C2×D244—21no—
76C2×C4×D61—41no—
77C2×Dic122—21no—
78C4⋊C4×C61—22no—
79C4⋊C4×C61—22no—
80C4⋊C4×C61—22no—
81C4⋊C4×C61—82no—
82C4⋊C4×C61C22⋊C414yes—
83C42×C61—82no—
84C42×C61C22×C411yescharacteristic
85C42×C61C2×D411yescharacteristic
86C42×C61#4111yescharacteristic
87C42×C61#4111yescharacteristic
88#11758198—41no—
89#11758198—41no—
90C4⋊Dic61—21no—
91C4⋊Dic61—21no—
92#11758194—42no—
93C2×#92418—22no—
94C2×#92418—41no—
95C2×#92418—41no—
96C23×C61#12011yescharacteristic
97C22×D61—41no—
98Q8×C61—82no—
99Q8×C61—82no—
100C2×C4×C61—22no—
101C2×C4×C61—24no—
102C2×C4×C61—41no—
103C2×C4×C61—41no—
104C2×C4×C61—42no—
105C2×C4×C61—42no—
106C2×C4×C61—42no—
107C2×C4×C61—42no—
108C2×C4×C61—42no—
109C2×C4×C61—42no—
110C2×C4×C61—42no—
111C2×C4×C61—42no—
112C2×C4×C61C2×C22⋊C411yescharacteristic
113C2×C4×C61C4×D411yescharacteristic
114C2×C4×C61C4×D411yescharacteristic
115C2×C4×C61#12111yescharacteristic
116C2×C4×C61#12111yescharacteristic
117C2×C4×C61#12311yescharacteristic
118C22⋊D61—82no—
119C2×Dic61—41no—
120D244—42no—
121D244—81no—
122D244—81no—
123C4×D61—82no—
124Dic122—42no—
125#92418—42no—
126#92418—81no—
127#92418—81no—
128#9430—611nomaximal, Sylow
129C22×C61—21no—
130C22×C61—21no—
131C22×C61#38611yescharacteristic
132C22×C61#45412yes—
133C2×D61—81no—
134C2×D61—81no—
135C2×D61—81no—
136C4×C61—21no—
137C4×C61—21no—
138C4×C61—21no—
139C4×C61—21no—
140C4×C61—21no—
141C4×C61—21no—
142C4×C61—44no—
143C4×C61—82no—
144C4×C61—82no—
145Dic61—81no—
146#3141—611no—
147#2381D6111yescharacteristic, complemented
148#1556—611no—
149#1514—611no—
150#1514—611no—
151#1288—611no—
152#1288—611no—
153C2×C61—22no—
154C2×C61#188811yescharacteristic
155C2×C61#190411yescharacteristic
156C2×C61#190511yescharacteristic
157D61—161no—
158#532—611no—
159C2×C4⋊Q8C2×D6111yescharacteristic
160#489—1222no—
161#487—612no—
162#486—612no—
163#432—611no—
164#432—611no—
165#417—1222no—
166C4×M16—611no—
167#391—22no—
168#387—22no—
169#387—22no—
170#383C2×D6111yescharacteristic
171#383C2×D6111yescharacteristic
172#332—1222no—
173#327—1222no—
174#327—1222no—
175C61#943011yescharacteristic, complemented, Sylow
176C2×#41—1221no—
177C22×Q8—21no—
178C2×M16—1221no—
179C4⋊Q8—21no—
180C4⋊Q8—21no—
181C4⋊Q8—611no—
182C4⋊Q8C4×D6112yes—
183C4⋊Q8D24412yes—
184#127—611no—
185#124—1221no—
186#121—1221no—
187#121—1221no—
188C4×D4—1221no—
189C4×D4—1221no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4C22⋊D6112yes—
193C2×C42—22no—
194C2×C42C22×D6111yescharacteristic
195C4⋊C8—1221no—
196C4⋊C8—1221no—
197#103—1224no—
198#102—1224no—
199C22⋊C8—1221no—
200C22⋊C8—1221no—
201#97—1221no—
202#97—1221no—
203#97—1221no—
204C4×C8—612no—
205#95—24no—
206#95—21no—
207#95—21no—
208#95—21no—
209#95—21no—
210#95C22⋊D6112yes—
211#95C22⋊D6112yes—
212#41—2444no—
213C2×Q8—22no—
214C2×Q8—22no—
215C2×Q8—81no—
216C2×Q8—81no—
217C2×Q8—1221no—
218C2×D4—1221no—
219C2×D4—1221no—
220C2×D4—1221no—
221C2×D4—2441no—
222C22×C4—22no—
223C22×C4—22no—
224C22×C4—22no—
225C22×C4—2441no—
226C22×C4C2×C22⋊D6111yescharacteristic
227C22×C4C2×C22⋊D6111yescharacteristic
228C22×C4C2×C22⋊D6111yescharacteristic
229M16—2442no—
230C2×C8—1222no—
231C2×C8—2441no—
232C2×C8—2441no—
233C4⋊C4—22no—
234C4⋊C4—22no—
235C4⋊C4—22no—
236C4⋊C4—82no—
237C4⋊C4—1221no—
238C4⋊C4—1221no—
239C4⋊C4#1175819814yes—
240C22⋊C4—2441no—
241C22⋊C4—2441no—
242C42—82no—
243C42C2×C4×D6111yescharacteristic
244C42C2×D24411yescharacteristic
245C42#1175822211yescharacteristic
246C42#1175822211yescharacteristic
247C23—2441no—
248C23#1451974611yescharacteristic
249Q8—82no—
250Q8—82no—
251Q8—2442no—
252D4—4881no—
253D4—4881no—
254D4—4882no—
255D4—2442no—
256C2×C4—22no—
257C2×C4—24no—
258C2×C4—41no—
259C2×C4—41no—
260C2×C4—42no—
261C2×C4—42no—
262C2×C4—42no—
263C2×C4—42no—
264C2×C4—42no—
265C2×C4—42no—
266C2×C4—42no—
267C2×C4—42no—
268C2×C4—4882no—
269C2×C4—2441no—
270C2×C4C2×#1175819811yescharacteristic
271C2×C4C4×C22⋊D6111yescharacteristic
272C2×C4C4×C22⋊D6111yescharacteristic
273C2×C4#1451975111yescharacteristic
274C2×C4#1451975111yescharacteristic
275C2×C4#1451976011yescharacteristic
276C8—4881no—
277C8—4881no—
278C8—2442no—
279C22—21no—
280C22—21no—
281C22—4881no—
282C22—4881no—
283C22—4881no—
284C22#1683964111yescharacteristic
285C22#1683993512yes—
286C4—21no—
287C4—21no—
288C4—21no—
289C4—21no—
290C4—21no—
291C4—21no—
292C4—44no—
293C4—82no—
294C4—82no—
295C4—4881no—
296C2—22no—
297C2—9761no—
298C2#2144622211yescharacteristic
299C2#2144630611yescharacteristic
300C2#2144631311yescharacteristic
subgroupG/H
centerC22#16839641
commutatorC42×C61C22×C4
FrattiniC2×C42C22×D61
FittingC61×#2381C2
socleC22×C61#386
prime radical#16840312C22
cosocleC2×C42×C61C23
hypercenter#2381D61
hypocenterC61#9430
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2381

Complements

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

NH ≅ G/Ntype
2. #21444536C2semidirect
5. #21443683C2semidirect
6. #21443683C2semidirect
7. C61×#2381C2semidirect
10. #16840114C4semidirect
11. #16840110C4semidirect
147. #2381D61semidirect
175. C61#9430semidirect

Maximal subgroups

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

subgroupindex
#214562152
#214445362
#214443652
#214436832
C61×#23812
#943061