Finite Group Databasebeta

← #30599643

Subgroups — #30599643

Order 15616.

420 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 420 conjugacy classes of subgroups — too many to draw. Only its 94 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#21462461C211yesmaximal, characteristic
2#21461677C211yesmaximal, characteristic
3#21461386C212yesmaximal, complemented
4#21460710C212yesmaximal, complemented
5#21444130C211yesmaximal, characteristic
6C22×#11758217—21no—
7#16840768—22no—
8#16840768—22no—
9#16840765—21no—
10#16840765C2212yes—
11C2×#14519844C2211yescharacteristic
12C2×C4⋊#92420C2211yescharacteristic
13C2×C22⋊#92420—21no—
14#16840628—21no—
15#16840624—22no—
16#16840623—22no—
17C2×#14519733C2211yescharacteristic
18#16839812C2212yes—
19#16839487C412yes—
20#16839486C411yescharacteristic
21C61×#323C411yescharacteristic
22C2×#11758217—22no—
23C2×#11758217—22no—
24C2×#11758217—42no—
25C2×#11758217—42no—
26C22×#92420—21no—
27C22×#92420—21no—
28C22×#92420—21no—
29C22×#92420—21no—
30#14519844—22no—
31#14519844—22no—
32#14519844—22no—
33C4⋊#92420—21no—
34C4⋊#92420—21no—
35C4⋊#92420—22no—
36C4⋊#92420D412yes—
37C4⋊#92420D412yes—
38C22⋊#92420—42no—
39C23×Dic61D411yescharacteristic
40#14519733—22no—
41#14519733—22no—
42C2×#11758196C2×C411yescharacteristic
43C2×#11758196Q811yescharacteristic
44C2×C22⋊Dic61Q811yescharacteristic
45C2×C4×Dic61D411yescharacteristic
46C2×C4×Dic61C2311yescharacteristic
47#14519706—22no—
48C22⋊#92418C2×C412yes—
49C8×Dic61—22no—
50C2×C22⋊C4×C61C2×C411yescharacteristic
51C22⋊C8×C61C2×C412yes—
52C24×C61C4⋊C411yescharacteristic
53C22×C4×C61C4⋊C411yescharacteristic
54C22×C4×C61C22×C411yescharacteristic
55C22×Dic61—21no—
56C22×Dic61—21no—
57C22×Dic61—21no—
58C22×Dic61—21no—
59C22×Dic61C4⋊C411yescharacteristic
60C22×Dic61C4⋊C411yescharacteristic
61C22×Dic61SD1612yes—
62C22×Dic61SD1612yes—
63C22×Dic61C2×D411yescharacteristic
64C22×Dic61C2×Q811yescharacteristic
65#11758217—44no—
66#11758217—42no—
67#11758217—42no—
68C2×#92420—22no—
69C2×#92420—22no—
70C2×#92420—42no—
71C2×#92420—42no—
72C2×#92420—41no—
73C2×#92420—41no—
74C2×#92420—41no—
75C2×#92420—41no—
76C2×#92420—41no—
77C2×#92420—41no—
78C2×C8×C61—22no—
79C22⋊C4×C61—22no—
80C22⋊Dic61—22no—
81#11758196—22no—
82#11758196—22no—
83C4×Dic61—21no—
84C4×Dic61—21no—
85C4×Dic61C2×D412yes—
86C4×Dic61#4112yes—
87C2×#92418—22no—
88C23×C61—21no—
89C23×C61—21no—
90C23×C61—41no—
91C23×C61—41no—
92C23×C61#10612yes—
93C23×C61C2×C4⋊C411yescharacteristic
94C2×C4×C61—21no—
95C2×C4×C61—41no—
96C2×C4×C61—41no—
97C2×C4×C61C4×D412yes—
98C2×Dic61—21no—
99C2×Dic61—21no—
100C2×Dic61—22no—
101C2×Dic61—22no—
102C2×Dic61—22no—
103C2×Dic61—22no—
104C2×Dic61—21no—
105C2×Dic61—41no—
106C2×Dic61—41no—
107C2×Dic61#10612yes—
108C2×Dic61#12011yescharacteristic
109C2×Dic61C22⋊Q812yes—
110C2×Dic61#12311yescharacteristic
111C2×Dic61C2×SD1611yescharacteristic
112C2×Dic61C2×SD1611yescharacteristic
113C2×Dic61AGL(1,ℤ/8)11yescharacteristic
114C2×Dic61#13711yescharacteristic
115#92420—42no—
116#92420—42no—
117#92420—81no—
118#92420—81no—
119C8×C61—42no—
120#92418—42no—
121#8862—611nomaximal, Sylow
122C22×C61—21no—
123C22×C61—21no—
124C22×C61—22no—
125C22×C61—42no—
126C22×C61—41no—
127C22×C61—41no—
128C22×C61—81no—
129C22×C61#38511yescharacteristic
130C22×C61C2×#10611yescharacteristic
131C22×C61#42811yescharacteristic
132C4×C61—22no—
133C4×C61—41no—
134Dic61—22no—
135Dic61—22no—
136Dic61—41no—
137Dic61#44811yescharacteristic
138Dic61#45011yescharacteristic
139Dic61#48111yescharacteristic
140Dic61#48311yescharacteristic
141#2905—611no—
142#1798—611no—
143#1570—612no—
144#1458—611no—
145#1301—612no—
146C2×C61—21no—
147C2×C61—21no—
148C2×C61—42no—
149C2×C61#185311yescharacteristic
150C2×C61#185811yescharacteristic
151C2×C61#186811yescharacteristic
152C22×M16—1221no—
153C2×#117—611no—
154#431—612no—
155#424—1222no—
156#424—1222no—
157#423—612no—
158#423—1221no—
159C2×C4⋊C8—611no—
160C2×C4⋊C8—611no—
161C2×C22⋊C8—1221no—
162#336—1221no—
163#334—1222no—
164#334—1222no—
165#324—612no—
166#323—611no—
167#323#692311yescharacteristic
168C61#886211yescharacteristic, complemented, Sylow
169C23×C4—611no—
170C2×M16—1222no—
171C2×M16—1222no—
172C2×M16—2442no—
173C2×M16—2442no—
174C22×C8—1221no—
175C22×C8—1221no—
176C22×C8—1221no—
177C22×C8—1221no—
178#117—1222no—
179#117—1222no—
180C2×C4⋊C4—611no—
181C2×C4⋊C4—611no—
182C2×C22⋊C4—611no—
183C2×C22⋊C4C2×#692311yescharacteristic
184C2×C42—611no—
185C2×C42—611no—
186C4⋊C8—612no—
187C4⋊C8—612no—
188C4⋊C8—1222no—
189C4⋊C8—1222no—
190C4⋊C8—1222no—
191C4⋊C8—1222no—
192C4⋊C8—1221no—
193C4⋊C8—1221no—
194C4⋊C8—1222no—
195C22⋊C8—612no—
196C22⋊C8—2442no—
197C22⋊C8C2×#692312yes—
198C4×C8—1222no—
199C24C4⋊#692311yescharacteristic
200C22×C4—611no—
201C22×C4—612no—
202C22×C4—612no—
203C22×C4—611no—
204C22×C4—611no—
205C22×C4—611no—
206C22×C4—1221no—
207C22×C4—1221no—
208C22×C4—1221no—
209C22×C4—1221no—
210C22×C4C4⋊#692311yescharacteristic
211C22×C4C22×#692311yescharacteristic
212M16—2442no—
213M16—2442no—
214M16—2444no—
215C2×C8—22no—
216C2×C8—1222no—
217C2×C8—1222no—
218C2×C8—1222no—
219C2×C8—2441no—
220C2×C8—2441no—
221C2×C8—2441no—
222C2×C8—2441no—
223C2×C8—2441no—
224C2×C8—2441no—
225C2×C8—2442no—
226C2×C8—2442no—
227C4⋊C4—1222no—
228C4⋊C4—1222no—
229C22⋊C4—22no—
230C22⋊C4—1222no—
231C42—612no—
232C42—612no—
233C42—1221no—
234C42—1221no—
235C23—21no—
236C23—21no—
237C23—41no—
238C23—41no—
239C23#1451984512yes—
240C23C2×C4⋊#692311yescharacteristic
241C2×C4—21no—
242C2×C4—41no—
243C2×C4—41no—
244C2×C4—611no—
245C2×C4—611no—
246C2×C4—611no—
247C2×C4—611no—
248C2×C4—611no—
249C2×C4—611no—
250C2×C4—612no—
251C2×C4—612no—
252C2×C4—1222no—
253C2×C4—1222no—
254C2×C4—1222no—
255C2×C4—1222no—
256C2×C4—1221no—
257C2×C4—1221no—
258C2×C4—1221no—
259C2×C4—2441no—
260C2×C4—2441no—
261C2×C4D4×#692312yes—
262C8—42no—
263C8—4881no—
264C8—4881no—
265C8—2442no—
266C8—2442no—
267C8—2442no—
268C22—21no—
269C22—21no—
270C22—22no—
271C22—42no—
272C22—41no—
273C22—41no—
274C22—81no—
275C22#1684070911yescharacteristic
276C22C2×#1451984511yescharacteristic
277C22#1684077311yescharacteristic
278C4—22no—
279C4—41no—
280C4—611no—
281C4—611no—
282C4—611no—
283C4—611no—
284C4—1222no—
285C4—1222no—
286C4—2441no—
287C2—21no—
288C2—21no—
289C2—42no—
290C2#2146176911yescharacteristic
291C2#2146177611yescharacteristic
292C2#2146179111yescharacteristic
subgroupG/H
centerC22#16840709
commutatorC22×C4×C61C22×C4
FrattiniC22×C4C22×#6923
FittingC61×#323C4
socleC22×C61#385
prime radicalC24×C61C4⋊C4
cosocleC2×C4×Dic61C23
hypercenter#323#6923
hypocenterC61#8862
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ #323

Complements

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

NH ≅ G/Ntype
3. #21461386C2semidirect
4. #21460710C2semidirect
168. C61#8862semidirect

Maximal subgroups

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

subgroupindex
#214624612
#214616772
#214613862
#214607102
#214441302
#886261