Finite Group Databasebeta

← #30603878

Subgroups — #30603878

Order 15616.

388 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 388 conjugacy classes of subgroups — too many to draw. Only its 92 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#21462047C211yesmaximal, characteristic
2D61⋊(C22⋊C16)C211yesmaximal, characteristic
3#21460758C212yesmaximal, complemented
4#21460757C212yesmaximal, complemented
5#424×D61C211yesmaximal, characteristic
6C2×#14519852—21no—
7C2×D61⋊C16—21no—
8#16840657—21no—
9#16840657—21no—
10C22⋊#11758187C2212yes—
11C22⋊#11758187C2212yes—
12C2×M16×D61C411yescharacteristic
13C22×C8×D61C2211yescharacteristic
14#117×D61C411yescharacteristic
15#16839790C2211yescharacteristic
16#16839789C2211yescharacteristic
17#16839788C411yescharacteristic
18#16839787—21no—
19#16839787—21no—
20C4⋊C8×D61—21no—
21C4⋊C8×D61—21no—
22C61×#424C411yescharacteristic
23#14519852—42no—
24D61⋊C16—42no—
25C2×#11758187—22no—
26C2×#11758187—22no—
27C22×C4×D61C2×C411yescharacteristic
28C2×#11758194C2×C411yescharacteristic
29C2×#11758189—21no—
30M16×D61—42no—
31C22×#92418C2311yescharacteristic
32C2×C8×D61—21no—
33C2×C8×D61—42no—
34C2×C8×D61D411yescharacteristic
35C2×C8×D61D411yescharacteristic
36C2×C8×D61D411yescharacteristic
37C2×C8×D61D411yescharacteristic
38#14519736C2×C411yescharacteristic
39#14519735C811yescharacteristic
40#14519735C811yescharacteristic
41#14519734—21no—
42#14519733—21no—
43C4⋊C4×D61C811yescharacteristic
44C4⋊C4×D61C811yescharacteristic
45C22⋊C4×D61—21no—
46C42×D61—21no—
47C4⋊#92418—21no—
48C4⋊#92418—21no—
49#14519706—21no—
50#14519706—21no—
51#14519706—21no—
52#14519706—21no—
53C2×M16×C61C2×C411yescharacteristic
54C22×C8×C61C2×C411yescharacteristic
55#117×C61C2×C411yescharacteristic
56C4⋊C8×C61—21no—
57C4⋊C8×C61—21no—
58C23×D61M1611yescharacteristic
59C22×C4×C61C22×C411yescharacteristic
60C22×Dic61M1611yescharacteristic
61C2×C4×D61—22no—
62C2×C4×D61—22no—
63C2×C4×D61—21no—
64C2×C4×D61—21no—
65C2×C4×D61C22⋊C411yescharacteristic
66C2×C4×D61C22⋊C411yescharacteristic
67C2×C4×D61C2×C811yescharacteristic
68C2×C4×D61C2×C811yescharacteristic
69M16×C61—41no—
70C2×C8×C61—21no—
71C2×C8×C61—41no—
72C2×C8×C61C22⋊C411yescharacteristic
73C2×C8×C61C22⋊C411yescharacteristic
74C4⋊C4×C61C2×C811yescharacteristic
75C4⋊C4×C61C2×C811yescharacteristic
76C22⋊C4×C61—21no—
77C42×C61—21no—
78C22⋊Dic61—21no—
79#11758198—21no—
80#11758198—21no—
81C4⋊Dic61C2×C811yescharacteristic
82C4⋊Dic61C2×C811yescharacteristic
83#11758196—21no—
84#11758196—21no—
85C4×Dic61—21no—
86C4×Dic61—21no—
87C4×Dic61—21no—
88#11758194—41no—
89C2×#92418—21no—
90C2×#92418—41no—
91C2×#92418C2×D411yescharacteristic
92C2×#92418C2×D411yescharacteristic
93#11758189—42no—
94C8×D61—22no—
95C8×D61—22no—
96C8×D61—41no—
97C8×D61—41no—
98C8×D61—42no—
99#11758187—42no—
100#11758187—42no—
101C23×C61C2×M1611yescharacteristic
102C22×D61—22no—
103C22×D61—22no—
104C22×D61—41no—
105C22×D61C22⋊C811yescharacteristic
106C22×D61C22⋊C811yescharacteristic
107C2×C4×C61—21no—
108C2×C4×C61—21no—
109C2×C4×C61—21no—
110C2×C4×C61—21no—
111C2×C4×C61C2×C22⋊C411yescharacteristic
112C2×C4×C61C22×C811yescharacteristic
113C2×Dic61—22no—
114C2×Dic61—21no—
115C2×Dic61—21no—
116C2×Dic61C22⋊C811yescharacteristic
117C2×Dic61C22⋊C811yescharacteristic
118C4×D61—22no—
119C4×D61—21no—
120C4×D61—21no—
121C4×D61—22no—
122C4×D61—42no—
123C4×D61—42no—
124C4×D61#9912yes—
125C4×D61#10112yes—
126C8×C61—21no—
127C8×C61—21no—
128C8×C61—41no—
129#92418—21no—
130#92418—21no—
131#92418—41no—
132#11444—611nomaximal, Sylow
133C22×C61—21no—
134C22×C61—21no—
135C22×C61—21no—
136C22×C61C2×C22⋊C811yescharacteristic
137C2×D61—21no—
138C2×D61—21no—
139C2×D61—21no—
140C2×D61—21no—
141C2×D61—44no—
142C2×D61#32412yes—
143C2×D61#35012yes—
144C4×C61—21no—
145C4×C61—41no—
146C4×C61—41no—
147C4×C61C2×#9911yescharacteristic
148C4×C61C2×#10111yescharacteristic
149Dic61—21no—
150Dic61—41no—
151Dic61—41no—
152Dic61#32412yes—
153C2×#424—611no—
154#2112—611no—
155C2×C22⋊C16—611no—
156#1323—612no—
157#1323—612no—
158C2×C61—41no—
159C2×C61C2×#32411yescharacteristic
160C2×C61C2×#35011yescharacteristic
161C2×C61#211711yescharacteristic
162D61—22no—
163D61—22no—
164C22×M16—611no—
165C23×C8—611no—
166C2×#117—611no—
167C2×M32—1221no—
168C22×C16—1221no—
169#424—611no—
170#424—611no—
171#424—611no—
172#424—1221no—
173#424—1221no—
174#424#692311yescharacteristic
175C2×C4⋊C8—1221no—
176C2×C4⋊C8—1221no—
177C22⋊C16—612no—
178C22⋊C16—612no—
179C22⋊C16—1221no—
180C22⋊C16—1221no—
181C61#1144411yescharacteristic, complemented, Sylow
182C23×C4—611no—
183C2×M16—611no—
184C2×M16—1221no—
185C2×M16—2442no—
186C2×M16C2×#692311yescharacteristic
187C22×C8—611no—
188C22×C8—611no—
189C22×C8—611no—
190C22×C8—611no—
191C22×C8—611no—
192C22×C8—1221no—
193C22×C8—2442no—
194C22×C8C2×#692311yescharacteristic
195#117—611no—
196#117—611no—
197#117—611no—
198#117—1221no—
199#117—1221no—
200#117C2×#692311yescharacteristic
201C2×C4⋊C4—611no—
202C2×C4⋊C4—611no—
203C2×C22⋊C4—1221no—
204C2×C42—1221no—
205M32—2442no—
206C2×C16—1222no—
207C2×C16—1222no—
208C2×C16—2442no—
209C4⋊C8—21no—
210C4⋊C8—21no—
211C4⋊C8—1221no—
212C4⋊C8—1221no—
213C4⋊C8—1221no—
214C4⋊C8—1221no—
215C4⋊C8—1221no—
216C4⋊C8—1221no—
217C24—611no—
218C22×C4—611no—
219C22×C4—611no—
220C22×C4—611no—
221C22×C4—611no—
222C22×C4—611no—
223C22×C4—1222no—
224C22×C4—1222no—
225C22×C4—1221no—
226C22×C4—1221no—
227C22×C4C22×#692311yescharacteristic
228M16—41no—
229M16—2442no—
230M16—2441no—
231C2×C8—21no—
232C2×C8—41no—
233C2×C8—611no—
234C2×C8—611no—
235C2×C8—1222no—
236C2×C8—1222no—
237C2×C8—1221no—
238C2×C8—2441no—
239C2×C8—2441no—
240C2×C8—2442no—
241C2×C8—2441no—
242C2×C8C22⋊#692311yescharacteristic
243C2×C8C22⋊#692311yescharacteristic
244C4⋊C4—611no—
245C4⋊C4—611no—
246C4⋊C4—1221no—
247C4⋊C4—1221no—
248C4⋊C4D61⋊C811yescharacteristic
249C4⋊C4D61⋊C811yescharacteristic
250C22⋊C4—21no—
251C22⋊C4—1221no—
252C22⋊C4—1221no—
253C22⋊C4—1221no—
254C42—21no—
255C42—1221no—
256C42—1221no—
257C42—1221no—
258C16—2442no—
259C16—2442no—
260C23—611no—
261C23—611no—
262C23—1222no—
263C23—1222no—
264C23—2441no—
265C23D61⋊M1611yescharacteristic
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—611no—
271C2×C4—612no—
272C2×C4—612no—
273C2×C4—611no—
274C2×C4—1222no—
275C2×C4—1222no—
276C2×C4—1222no—
277C2×C4—1221no—
278C2×C4—1221no—
279C2×C4—1221no—
280C2×C4—1221no—
281C2×C4—2442no—
282C2×C4—2442no—
283C2×C4C2×C22⋊#692311yescharacteristic
284C2×C4C2×D61⋊C811yescharacteristic
285C8—21no—
286C8—21no—
287C8—41no—
288C8—1221no—
289C8—1221no—
290C8—2441no—
291C22—21no—
292C22—21no—
293C22—21no—
294C22—612no—
295C22—612no—
296C22—1221no—
297C22—1221no—
298C22—1221no—
299C22—1221no—
300C22—2444no—
301C22D61⋊(C22⋊C8)11yescharacteristic
302C4—21no—
303C4—41no—
304C4—41no—
305C4—612no—
306C4—1221no—
307C4—2441no—
308C4—2441no—
309C4D61⋊#9911yescharacteristic
310C4D61⋊#10111yescharacteristic
311C2—41no—
312C2—1222no—
313C2—1222no—
314C2D61⋊#32411yescharacteristic
315C2D61⋊#35011yescharacteristic
316C2#2146206211yescharacteristic
subgroupG/H
centerC2×C4C2×C22⋊#6923
commutatorC2×C4×C61C22×C8
FrattiniC22×C4C22×#6923
FittingC61×#424C4
socleC22×C61C2×C22⋊C8
prime radicalC23×D61M16
cosocleC22×#92418C23
hypercenter#424#6923
hypocenterC61#11444
LCSG ▷ C2×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ C22×C8 ◁ #424

Complements

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

NH ≅ G/Ntype
3. #21460758C2semidirect
4. #21460757C2semidirect
181. C61#11444semidirect

Maximal subgroups

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

subgroupindex
#214620472
D61⋊(C22⋊C16)2
#214607582
#214607572
#424×D612
#1144461