Finite Group Databasebeta

← #30599647

Subgroups — #30599647

Order 15616.

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

normal subgroups only

This group has 418 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#21462672C211yesmaximal, characteristic
2#21461944C211yesmaximal, characteristic
3#21461394C212yesmaximal, complemented
4#21460714C212yesmaximal, complemented
5#21444138C211yesmaximal, characteristic
6#16840791—22no—
7#16840791—22no—
8#16840783C2212yes—
9C2×C4⋊#92420C2211yescharacteristic
10#16840733—21no—
11C2×#14519834—21no—
12C2×C22⋊#92420—21no—
13C2×C22⋊#92420—21no—
14C2×#14519832C2211yescharacteristic
15#16840625—22no—
16#16840624—22no—
17C2×#14519737C2211yescharacteristic
18#16839812C2212yes—
19#16839486C411yescharacteristic
20#16839485C412yes—
21C61×#323C411yescharacteristic
22C2×#11758213—21no—
23C2×#11758213—21no—
24C22×#92420—21no—
25C22×#92420—21no—
26C22×#92420—21no—
27C22×#92420—21no—
28C4⋊#92420—22no—
29C4⋊#92420D412yes—
30C4⋊#92420D412yes—
31#14519834—22no—
32#14519834—42no—
33C22⋊#92420—22no—
34C22⋊#92420—22no—
35C22⋊#92420—42no—
36#14519832—22no—
37#14519832—22no—
38C22×C22⋊D61Q811yescharacteristic
39C23×Dic61D411yescharacteristic
40C22×C4×D61D411yescharacteristic
41#14519737—22no—
42#14519737—22no—
43C2×#11758196Q811yescharacteristic
44C2×#11758198C2×C411yescharacteristic
45C2×C4×Dic61C2311yescharacteristic
46#14519706—22no—
47C22⋊#92418C2×C412yes—
48C8×Dic61—22no—
49C2×C22⋊C4×C61C2×C411yescharacteristic
50C22⋊C8×C61C2×C412yes—
51C24×C61C4⋊C411yescharacteristic
52C23×D61C4⋊C411yescharacteristic
53C22×C4×C61C4⋊C411yescharacteristic
54C22×C4×C61C22×C411yescharacteristic
55C2×C22⋊D61—22no—
56C2×C22⋊D61—22no—
57C2×C22⋊D61—41no—
58C22×Dic61—21no—
59C22×Dic61—21no—
60C22×Dic61—41no—
61C22×Dic61—41no—
62C22×Dic61C4⋊C411yescharacteristic
63C22×Dic61C2×D411yescharacteristic
64C22×Dic61C2×Q811yescharacteristic
65C2×C4×D61—21no—
66C2×C4×D61—21no—
67C2×C4×D61—41no—
68C2×C4×D61—41no—
69C2×#92420—22no—
70C2×#92420—22no—
71C2×#92420—22no—
72C2×#92420—41no—
73C2×#92420—41no—
74C2×#92420—41no—
75C2×#92420—41no—
76C2×#92420—41no—
77C2×#92420—81no—
78C2×#92420—81no—
79#11758213—81no—
80#11758213—81no—
81C2×C8×C61—22no—
82C22⋊C4×C61—22no—
83#11758198—22no—
84#11758196—22no—
85C4×Dic61C2×D412yes—
86C4×Dic61#4112yes—
87C2×#92418—22no—
88C23×C61—21no—
89C23×C61—21no—
90C23×C61—41no—
91C23×C61—41no—
92C23×C61#10812yes—
93C23×C61C2×C4⋊C411yescharacteristic
94C22×D61—21no—
95C22×D61—21no—
96C22×D61—21no—
97C22×D61—41no—
98C22×D61—41no—
99C2×C4×C61—21no—
100C2×C4×C61—41no—
101C2×C4×C61—41no—
102C2×C4×C61C4×D412yes—
103C22⋊D61—44no—
104C2×Dic61—21no—
105C2×Dic61—21no—
106C2×Dic61—21no—
107C2×Dic61—21no—
108C2×Dic61—21no—
109C2×Dic61—41no—
110C2×Dic61—41no—
111C2×Dic61—41no—
112C2×Dic61—81no—
113C2×Dic61#12011yescharacteristic
114C2×Dic61C22⋊Q812yes—
115C2×Dic61#12311yescharacteristic
116C4×D61—41no—
117C4×D61—41no—
118C4×D61—81no—
119#92420—42no—
120#92420—42no—
121#92420—42no—
122#92420—81no—
123C8×C61—42no—
124#92418—42no—
125#8864—611nomaximal, Sylow
126C22×C61—21no—
127C22×C61—21no—
128C22×C61—22no—
129C22×C61—42no—
130C22×C61—41no—
131C22×C61—41no—
132C22×C61—81no—
133C22×C61#38511yescharacteristic
134C22×C61C2×#10811yescharacteristic
135C22×C61#43011yescharacteristic
136C2×D61—42no—
137C2×D61—41no—
138C2×D61—41no—
139C2×D61—41no—
140C2×D61—81no—
141C4×C61—22no—
142C4×C61—41no—
143Dic61—22no—
144Dic61—21no—
145Dic61—41no—
146Dic61#45311yescharacteristic
147Dic61#45511yescharacteristic
148#2973—611no—
149#1955—611no—
150#1573—612no—
151#1460—611no—
152#1302—612no—
153C2×C61—21no—
154C2×C61—21no—
155C2×C61—42no—
156C2×C61#185511yescharacteristic
157C2×C61#185611yescharacteristic
158C2×C61#186811yescharacteristic
159D61—81no—
160C2×C4×D4—611no—
161#435—1222no—
162#435—1222no—
163#432—612no—
164#431—612no—
165C2×C4⋊C8—611no—
166#407—1221no—
167C2×C22⋊C8—1221no—
168C2×C22⋊C8—1221no—
169C2×C22⋊C8—1221no—
170C2×#97—611no—
171#335—1222no—
172#334—1222no—
173#323—611no—
174#323—612no—
175#323#692311yescharacteristic
176C61#886411yescharacteristic, complemented, Sylow
177C22×D4—611no—
178C23×C4—611no—
179C23×C4—611no—
180C2×M16—1221no—
181C2×M16—1221no—
182C22×C8—1221no—
183C22×C8—1221no—
184C22×C8—1221no—
185C22×C8—1221no—
186C4×D4—1222no—
187C4×D4—1222no—
188C2×C4⋊C4—611no—
189C2×C22⋊C4—611no—
190C2×C22⋊C4C2×#692311yescharacteristic
191C2×C42—611no—
192C4⋊C8—612no—
193C4⋊C8—612no—
194C4⋊C8—1222no—
195C4⋊C8—1222no—
196C22⋊C8—612no—
197C22⋊C8—1222no—
198C22⋊C8—1222no—
199C22⋊C8—1222no—
200C22⋊C8—2442no—
201C22⋊C8—2442no—
202C22⋊C8C2×#692312yes—
203#97—1222no—
204#97—1222no—
205C4×C8—1222no—
206C24—611no—
207C24C4⋊#692311yescharacteristic
208C2×D4—1222no—
209C2×D4—1222no—
210C2×D4—2441no—
211C22×C4—611no—
212C22×C4—611no—
213C22×C4—611no—
214C22×C4—1221no—
215C22×C4—1221no—
216C22×C4—1221no—
217C22×C4—1221no—
218C22×C4—2441no—
219C22×C4—2441no—
220C22×C4—2441no—
221C22×C4—2441no—
222C22×C4C4⋊#692311yescharacteristic
223C22×C4C22×#692311yescharacteristic
224M16—4881no—
225M16—4881no—
226C2×C8—22no—
227C2×C8—1222no—
228C2×C8—1222no—
229C2×C8—1222no—
230C2×C8—1222no—
231C2×C8—4881no—
232C2×C8—4881no—
233C2×C8—2441no—
234C2×C8—2441no—
235C2×C8—2441no—
236C2×C8—2441no—
237C2×C8—2441no—
238C4⋊C4—1222no—
239C22⋊C4—22no—
240C22⋊C4—1222no—
241C42—612no—
242C42—612no—
243C23—21no—
244C23—21no—
245C23—41no—
246C23—41no—
247C23—1221no—
248C23—1221no—
249C23—1221no—
250C23—2441no—
251C23—2441no—
252C23#1451984812yes—
253C23C2×C4⋊#692311yescharacteristic
254D4—2444no—
255C2×C4—21no—
256C2×C4—41no—
257C2×C4—41no—
258C2×C4—611no—
259C2×C4—612no—
260C2×C4—611no—
261C2×C4—1221no—
262C2×C4—1221no—
263C2×C4—1221no—
264C2×C4—1221no—
265C2×C4—1221no—
266C2×C4—4881no—
267C2×C4—4881no—
268C2×C4—2441no—
269C2×C4—2441no—
270C2×C4—2441no—
271C2×C4—2441no—
272C2×C4—2441no—
273C2×C4D4×#692312yes—
274C8—42no—
275C8—4881no—
276C8—2442no—
277C8—2442no—
278C8—2442no—
279C8—2442no—
280C22—21no—
281C22—21no—
282C22—22no—
283C22—42no—
284C22—41no—
285C22—41no—
286C22—81no—
287C22—4881no—
288C22—2442no—
289C22—2441no—
290C22—2441no—
291C22—2441no—
292C22#1684070911yescharacteristic
293C22C2×#1451984811yescharacteristic
294C22#1684077711yescharacteristic
295C4—22no—
296C4—41no—
297C4—611no—
298C4—611no—
299C4—1222no—
300C4—1221no—
301C4—2441no—
302C2—21no—
303C2—21no—
304C2—42no—
305C2—4881no—
306C2#2146177111yescharacteristic
307C2#2146177411yescharacteristic
308C2#2146179111yescharacteristic
subgroupG/H
centerC22#16840709
commutatorC22×C4×C61C22×C4
FrattiniC22×C4C22×#6923
FittingC61×#323C4
socleC22×C61#385
prime radicalC22×C22⋊D61Q8
cosocleC2×C4×Dic61C23
hypercenter#323#6923
hypocenterC61#8864
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. #21461394C2semidirect
4. #21460714C2semidirect
176. C61#8864semidirect

Maximal subgroups

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

subgroupindex
#214626722
#214619442
#214613942
#214607142
#214441382
#886461