Finite Group Databasebeta

← #30599639

Subgroups — #30599639

Order 15616.

458 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 458 conjugacy classes of subgroups — too many to draw. Only its 82 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#21462684C211yesmaximal, characteristic
2#21461799C211yesmaximal, characteristic
3#21461772C212yesmaximal, complemented
4#21460702C212yesmaximal, complemented
5#21444136C211yesmaximal, characteristic
6C22×#11758213C2211yescharacteristic
7#16840796—22no—
8#16840796—22no—
9#16840778—21no—
10C2×#14519847—22no—
11#16840733—21no—
12#16840733C2212yes—
13C2×#14519834—21no—
14C2×#14519834C2211yescharacteristic
15#16840638—22no—
16#16840628—21no—
17C2×#14519739C2211yescharacteristic
18C22⋊C8×D61C2212yes—
19#16839488C411yescharacteristic
20#16839485C412yes—
21C61×#324C411yescharacteristic
22C2×#11758213—22no—
23C2×#11758213—21no—
24C2×#11758213—21no—
25C2×#11758213—42no—
26C2×#11758213D412yes—
27C2×#11758213D412yes—
28C22×#92420—21no—
29C22×#92420—21no—
30#14519847—44no—
31C4⋊#92420—21no—
32C4⋊#92420—21no—
33#14519834—22no—
34#14519834—22no—
35#14519834—22no—
36#14519834—42no—
37C22⋊#92420—22no—
38C4×#92420—22no—
39C22×D244Q811yescharacteristic
40C22×C4×D61D411yescharacteristic
41C22×C4×D61C2311yescharacteristic
42C2×C8×D61—22no—
43#14519739—22no—
44#14519739—22no—
45C2×#11758198C2×C411yescharacteristic
46C2×#11758198Q811yescharacteristic
47C2×C4×Dic61D411yescharacteristic
48C22⋊#92418C2×C412yes—
49#14519689—22no—
50C2×C4⋊C4×C61C2×C411yescharacteristic
51C22⋊C8×C61C2×C412yes—
52C23×D61C4⋊C411yescharacteristic
53C23×D61C2×Q811yescharacteristic
54C22×C4×C61C4⋊C411yescharacteristic
55C22×C4×C61C4⋊C411yescharacteristic
56C22×C4×C61C22×C411yescharacteristic
57C22×Dic61C4⋊C411yescharacteristic
58C22×Dic61C2×D411yescharacteristic
59C2×D244—22no—
60C2×D244—22no—
61C2×D244—41no—
62C2×C4×D61—21no—
63C2×C4×D61—21no—
64C2×C4×D61—41no—
65C2×C4×D61—41no—
66C2×C4×D61—41no—
67C2×C4×D61—41no—
68C2×C4×D61C2×D412yes—
69C2×C4×D61#4112yes—
70C2×#92420—22no—
71C2×#92420—22no—
72C2×#92420—22no—
73C2×#92420—41no—
74C2×#92420—41no—
75C2×#92420—41no—
76C2×#92420—41no—
77C2×#92420—41no—
78#11758213—24no—
79#11758213—42no—
80#11758213—42no—
81#11758213—42no—
82#11758213—81no—
83#11758213—81no—
84C2×C8×C61—22no—
85C4⋊C4×C61—22no—
86#11758198—22no—
87#11758198—22no—
88C4×Dic61—21no—
89C4×Dic61—21no—
90C2×#92418—22no—
91C8×D61—42no—
92C8×D61—42no—
93C23×C61C2×C4⋊C411yescharacteristic
94C22×D61—21no—
95C22×D61—21no—
96C22×D61—21no—
97C22×D61—21no—
98C22×D61—21no—
99C22×D61—41no—
100C22×D61—41no—
101C22×D61—41no—
102C22×D61—41no—
103C22×D61C22⋊Q812yes—
104C2×C4×C61—21no—
105C2×C4×C61—21no—
106C2×C4×C61—21no—
107C2×C4×C61—41no—
108C2×C4×C61—41no—
109C2×C4×C61#10812yescomplemented
110C2×C4×C61C4×D412yes—
111C2×Dic61—21no—
112C2×Dic61—21no—
113C2×Dic61—21no—
114C2×Dic61—41no—
115C2×Dic61#12011yescharacteristic
116C2×Dic61#12311yescharacteristic
117D244—44no—
118C4×D61—22no—
119C4×D61—22no—
120C4×D61—42no—
121C4×D61—41no—
122C4×D61—41no—
123C4×D61—81no—
124#92420—42no—
125#92420—42no—
126#92420—42no—
127#92420—81no—
128C8×C61—42no—
129#92418—42no—
130#8860—611nomaximal, Sylow
131C22×C61—21no—
132C22×C61—21no—
133C22×C61#38511yescharacteristic
134C22×C61C2×#10811yescharacteristic
135C22×C61#43011yescharacteristic
136C2×D61—22no—
137C2×D61—22no—
138C2×D61—22no—
139C2×D61—42no—
140C2×D61—41no—
141C2×D61—41no—
142C2×D61—41no—
143C2×D61—81no—
144C2×D61—81no—
145C2×D61—81no—
146C4×C61—22no—
147C4×C61—22no—
148C4×C61—41no—
149Dic61—21no—
150Dic61—41no—
151Dic61B(4,𝔽2)11yescharacteristic
152Dic61#45811yescharacteristic
153#2975—611no—
154#1870—611no—
155#1855—612no—
156#1459—611no—
157#1298—612no—
158C2×C61—21no—
159C2×C61—21no—
160C2×C61#185511yescharacteristic
161C2×C61#185811yescharacteristic
162C2×C61#186711yescharacteristic
163D61—42no—
164D61—81no—
165C22×M16—611no—
166C2×C4×D4—611no—
167#436—1222no—
168#436—1222no—
169#431—1221no—
170C2×#108—1222no—
171#407—612no—
172#407—1221no—
173C2×C22⋊C8—612no—
174C2×C22⋊C8—611no—
175C2×C22⋊C8—1221no—
176#341—1222no—
177#336—1221no—
178#324—611no—
179#324#692311yescharacteristic
180#323—612no—
181C61#886011yescharacteristic, complemented, Sylow
182C22×D4—611no—
183C23×C4—611no—
184C23×C4—611no—
185C2×M16—612no—
186C2×M16—612no—
187C2×M16—1221no—
188C2×M16—1221no—
189C2×M16—1222no—
190C2×M16—2442no—
191C22×C8—1222no—
192C22×C8—1221no—
193C22×C8—1221no—
194C4×D4—1222no—
195C4×D4—1222no—
196C2×C4⋊C4C2×#692311yescharacteristic
197C2×C22⋊C4—611no—
198C2×C22⋊C4—611no—
199C2×C42—611no—
200#108—2444no—
201C4⋊C8—1221no—
202C4⋊C8—1221no—
203C22⋊C8—612no—
204C22⋊C8—1222no—
205C22⋊C8—1222no—
206C22⋊C8—1222no—
207C22⋊C8—1222no—
208C22⋊C8—1222no—
209C22⋊C8—2442no—
210C22⋊C8C2×#692312yes—
211C4×C8—1222no—
212C24—611no—
213C24—611no—
214C2×D4—1222no—
215C2×D4—1222no—
216C2×D4—2441no—
217C22×C4—611no—
218C22×C4—612no—
219C22×C4—612no—
220C22×C4—611no—
221C22×C4—1221no—
222C22×C4—1221no—
223C22×C4—2441no—
224C22×C4—2441no—
225C22×C4—2441no—
226C22×C4—2441no—
227C22×C4C4⋊#692311yescharacteristic
228C22×C4C4⋊#692311yescharacteristic
229C22×C4C22×#692311yescharacteristic
230M16—1224no—
231M16—4881no—
232M16—4881no—
233M16—2442no—
234M16—2442no—
235M16—2442no—
236C2×C8—22no—
237C2×C8—1222no—
238C2×C8—1222no—
239C2×C8—1222no—
240C2×C8—1222no—
241C2×C8—2442no—
242C2×C8—2442no—
243C2×C8—2441no—
244C2×C8—2441no—
245C2×C8—2441no—
246C2×C8—2441no—
247C2×C8—2441no—
248C4⋊C4—22no—
249C22⋊C4—1222no—
250C22⋊C4—1222no—
251C42—1221no—
252C42—1221no—
253C23—612no—
254C23—1221no—
255C23—1221no—
256C23—1221no—
257C23—1221no—
258C23—1221no—
259C23—2441no—
260C23—2441no—
261C23—2441no—
262C23—2441no—
263C23C2×C4⋊#692311yescharacteristic
264D4—2444no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—41no—
269C2×C4—41no—
270C2×C4—611no—
271C2×C4—611no—
272C2×C4—1222no—
273C2×C4—1222no—
274C2×C4—1221no—
275C2×C4—1221no—
276C2×C4—1221no—
277C2×C4—4881no—
278C2×C4—2442no—
279C2×C4—2441no—
280C2×C4—2441no—
281C2×C4—2441no—
282C2×C4#1451984712yescomplemented
283C2×C4D4×#692312yes—
284C8—42no—
285C8—4881no—
286C8—2442no—
287C8—2442no—
288C8—2442no—
289C8—2442no—
290C22—21no—
291C22—21no—
292C22—1222no—
293C22—1222no—
294C22—1222no—
295C22—4881no—
296C22—4881no—
297C22—4881no—
298C22—2442no—
299C22—2441no—
300C22—2441no—
301C22—2441no—
302C22#1684070911yescharacteristic
303C22C2×#1451984711yescharacteristic
304C22#1684077711yescharacteristic
305C4—22no—
306C4—22no—
307C4—41no—
308C4—611no—
309C4—611no—
310C4—1221no—
311C4—2441no—
312C2—21no—
313C2—21no—
314C2—4881no—
315C2—2442no—
316C2#2146177211yescharacteristic
317C2#2146177711yescharacteristic
318C2#2146179011yescharacteristic
subgroupG/H
centerC22#16840709
commutatorC22×C4×C61C22×C4
FrattiniC22×C4C22×#6923
FittingC61×#324C4
socleC22×C61#385
prime radicalC22×D244Q8
cosocleC22×C4×D61C23
hypercenter#324#6923
hypocenterC61#8860
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ #324

Complements

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

NH ≅ G/Ntype
3. #21461772C2semidirect
4. #21460702C2semidirect
109. C2×C4×C61#108semidirect
181. C61#8860semidirect
282. C2×C4#14519847semidirect

Maximal subgroups

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

subgroupindex
#214626842
#214617992
#214617722
#214607022
#214441362
#886061