Finite Group Databasebeta

← #30606520

Subgroups — #30606520

Order 15616.

470 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 470 conjugacy classes of subgroups — too many to draw. Only its 98 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#21462072C212yesmaximal
2#21461574C212yesmaximal, complemented
3#21460911C211yesmaximal, characteristic, complemented
4#21460908C211yesmaximal, characteristic, complemented
5#21454515C211yesmaximal, characteristic
6#16840748C2212yes—
7#16840741C2212yes—
8#16840725—22no—
9#16840665C414yescomplemented
10#16840663C414yescomplemented
11#16840644—22no—
12C2×#14519783C2211yescharacteristic
13C22×C8×D61C411yescharacteristic, complemented
14#16840274C2211yescharacteristic
15#16839835—22no—
16C8×D244—22no—
17#16839789C411yescharacteristic, complemented
18#16839730C2211yescharacteristic
19#16839725C411yescharacteristic, complemented
20C61×#405C411yescharacteristic, complemented
21D61⋊M16—22no—
22#14519856—22no—
23#14519838—22no—
24#14519837C2×C414yes—
25#14519836—22no—
26#14519835C2×C414yes—
27#14519831—42no—
28C8×#6923—42no—
29C2×#11758222C2311yescharacteristic
30C22×C4×D61C2×C411yescharacteristic
31#14519783—41no—
32#14519783—41no—
33C2×#11758194C2×C411yescharacteristic
34C2×#11758189D412yes—
35C22×#92418C2×C411yescharacteristic
36C2×C8×D61—21no—
37C2×C8×D61—21no—
38C2×C8×D61—41no—
39C2×C8×D61—41no—
40C2×C8×D61D412yes—
41#14519761C2×C412yes—
42#14519757—21no—
43#14519752—21no—
44#14519747C2×C412yes—
45C4×D244—22no—
46C4⋊#92418—22no—
47#14519689—22no—
48#14519689—22no—
49C2×M16×C61C2×C411yescharacteristic
50C22×C8×C61C2×C411yescharacteristic
51#117×C61C2×C411yescharacteristic
52#97×C61—21no—
53C4×C8×C61—21no—
54C23×D61C4211yescharacteristic
55C22×C4×C61C22×C411yescharacteristic
56C2×C22⋊D61C22×C412yes—
57C22×Dic61C4211yescharacteristic
58#11758222—41no—
59#11758222—41no—
60C2×D244—22no—
61C2×D244#4111yescharacteristic
62C2×C4×D61—21no—
63C2×C4×D61—21no—
64C2×C4×D61—41no—
65C2×C4×D61—41no—
66C2×C4×D61C2×D412yes—
67C2×Dic122#4111yescharacteristic
68C22⋊#6923—24no—
69#11758217—24no—
70C4⋊#6923—42no—
71C4×#6923—42no—
72#11758213—42no—
73D61⋊C8—42no—
74M16×C61—41no—
75C2×C8×C61—21no—
76C2×C8×C61—21no—
77C2×C8×C61—21no—
78C2×C8×C61C22⋊C411yescharacteristic
79C2×C8×C61C22⋊C411yescharacteristic
80C4⋊C4×C61—21no—
81C22⋊C4×C61C4212yes—
82C42×C61—21no—
83#11758198—22no—
84#11758198—22no—
85C4⋊Dic61—22no—
86#11758194—41no—
87C2×#92418—21no—
88C2×#92418—41no—
89C2×#92418C22⋊C412yes—
90#11758189—42no—
91C8×D61—22no—
92C8×D61—22no—
93C8×D61—42no—
94C8×D61—41no—
95C8×D61—41no—
96C23×C61C2×C4211yescharacteristic
97C22×D61—21no—
98C22×D61—42no—
99C22×D61—41no—
100C22×D61C4×D412yes—
101C2×#6923—44no—
102C2×C4×C61—21no—
103C2×C4×C61—21no—
104C2×C4×C61—22no—
105C2×C4×C61C2×C22⋊C411yescharacteristic
106C2×C4×C61#11711yescharacteristic
107C22⋊D61—42no—
108C2×Dic61—21no—
109C2×Dic61—41no—
110C2×Dic61C4×D412yes—
111D244—41no—
112D244—44no—
113C4×D61—22no—
114C4×D61—22no—
115C4×D61—42no—
116C4×D61—41no—
117C4×D61—41no—
118Dic122—41no—
119#92420—44no—
120C8×C61—21no—
121C8×C61—41no—
122C8×C61#9911yescharacteristic
123C8×C61#9911yescharacteristic
124#92418—22no—
125#92418—41no—
126#12623—611nomaximal, Sylow
127C22×C61—21no—
128C22×C61—41no—
129C22×C61C4×C22⋊C411yescharacteristic
130C2×D61—22no—
131C2×D61—22no—
132C2×D61—41no—
133C2×D61—42no—
134C2×D61—41no—
135C2×D61—42no—
136C2×D61—42no—
137#6923—84no—
138C4×C61—21no—
139C4×C61—42no—
140C4×C61C2×#9911yescharacteristic
141C4×C61#41011yescharacteristic
142Dic61—22no—
143Dic61—41no—
144#2929—611no—
145#2120—612no—
146#1757—612no—
147#1392—611no—
148#1391—611no—
149C2×C61—21no—
150C2×C61—41no—
151C2×C61C4×#9911yescharacteristic
152D61—42no—
153D61—42no—
154C2×#131—611no—
155C23×C8—611no—
156#525—611no—
157#435—1222no—
158C8×D4—1222no—
159#424—611no—
160#413—612no—
161#410—612no—
162#408—611no—
163#407—611no—
164#405—1222no—
165#405#692311yescharacteristic, complemented
166#352—614no—
167AΣL(1,𝔽16)—614no—
168#344—1222no—
169C61#1262311yescharacteristic, complemented, Sylow
170C2×#41—611no—
171C23×C4—611no—
172#131—2441no—
173#131—2441no—
174C2×M16—611no—
175C2×M16—612no—
176C2×M16—1222no—
177C2×M16C2×#692311yescharacteristic
178C22×C8—612no—
179C22×C8—611no—
180C22×C8—1221no—
181C22×C8—1221no—
182C22×C8—2441no—
183C22×C8—2441no—
184C22×C8C2×#692311yescharacteristic
185#123—612no—
186C22⋊Q8—1221no—
187#121—1221no—
188#120—612no—
189C4×D4—1222no—
190#117—1222no—
191#117C2×#692311yescharacteristic
192C4⋊C8—1222no—
193#101—1222no—
194#100—614no—
195#99—614no—
196#99—1222no—
197C22⋊C8—1222no—
198C22⋊C8—1222no—
199#97—21no—
200#97—2442no—
201C4×C8—21no—
202C4×C8—2442no—
203C24—611no—
204#41—2441no—
205#41—2441no—
206C2×Q8—611no—
207C2×D4—612no—
208C2×D4—611no—
209C2×D4—1222no—
210C22×C4—612no—
211C22×C4—611no—
212C22×C4—1221no—
213C22×C4—1221no—
214C22×C4—2441no—
215C22×C4—2441no—
216C22×C4C22×#692311yescharacteristic
217M16—41no—
218M16—1224no—
219M16—2442no—
220M16—2441no—
221M16—2442no—
222C2×C8—21no—
223C2×C8—21no—
224C2×C8—21no—
225C2×C8—612no—
226C2×C8—1222no—
227C2×C8—1222no—
228C2×C8—1221no—
229C2×C8—2442no—
230C2×C8—2442no—
231C2×C8—2441no—
232C2×C8—2441no—
233C2×C8—2441no—
234C2×C8C22⋊#692311yescharacteristic
235C2×C8C22⋊#692311yescharacteristic
236C4⋊C4—21no—
237C4⋊C4—1222no—
238C4⋊C4—2442no—
239C22⋊C4—1222no—
240C22⋊C4—1222no—
241C22⋊C4—1224no—
242C22⋊C4C4×#692312yes—
243C42—21no—
244C42—2442no—
245C23—612no—
246C23—1221no—
247C23—2441no—
248C23—2442no—
249C23C2×C4×#692311yescharacteristic
250Q8—2441no—
251D4—2441no—
252D4—2444no—
253D4—2442no—
254C2×C4—21no—
255C2×C4—21no—
256C2×C4—22no—
257C2×C4—612no—
258C2×C4—1222no—
259C2×C4—1222no—
260C2×C4—1221no—
261C2×C4—2442no—
262C2×C4—2444no—
263C2×C4—2441no—
264C2×C4—2441no—
265C2×C4—2441no—
266C2×C4C2×C22⋊#692311yescharacteristic
267C2×C4#1451985611yescharacteristic
268C8—21no—
269C8—41no—
270C8—1222no—
271C8—2444no—
272C8—2441no—
273C8#1451983611yescharacteristic
274C8#1451983611yescharacteristic
275C22—21no—
276C22—41no—
277C22—1222no—
278C22—1222no—
279C22—2442no—
280C22—2441no—
281C22—2442no—
282C22—2442no—
283C22—2441no—
284C22C4×C22⋊#692311yescharacteristic
285C4—21no—
286C4—42no—
287C4—1222no—
288C4—4884no—
289C4—2441no—
290C4C2×#1451983611yescharacteristic
291C4#1684074111yescharacteristic
292C2—21no—
293C2—41no—
294C2—2442no—
295C2—2442no—
296C2C4×#1451983611yescharacteristic
subgroupG/H
centerC8#14519836
commutatorC23×C61C2×C42
FrattiniC22×C4C22×#6923
FittingC61×#405C4
socleC2×C61C4×#99
prime radical#16840274C22
cosocleC2×#11758222C23
hypercenter#405#6923
hypocenterC61#12623
LCSG ▷ C23×C61 ▷ C22×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C8 ◁ C2×C8 ◁ C22×C8 ◁ #405

Complements

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

NH ≅ G/Ntype
2. #21461574C2semidirect
3. #21460911C2semidirect
4. #21460908C2semidirect
9. #16840665C4semidirect
10. #16840663C4semidirect
13. C22×C8×D61C4semidirect
17. #16839789C4semidirect
19. #16839725C4semidirect
20. C61×#405C4semidirect
165. #405#6923semidirect
169. C61#12623semidirect

Maximal subgroups

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

subgroupindex
#214620722
#214615742
#214609112
#214609082
#214545152
#1262361