Finite Group Databasebeta

← #30599230

Subgroups — #30599230

Order 15616.

680 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21461673C211yesmaximal, characteristic
2C4×C22⋊#92420C211yesmaximal, characteristic
3#21460655C214yesmaximal, complemented
4C2×C4⋊C4×Dic61C211yesmaximal, characteristic
5C2×C4⋊#92420C2212yes—
6C2×C22⋊#92420—21no—
7C2×C22⋊#92420—21no—
8C2×C4×#92420C2212yes—
9#16840628—21no—
10#16840628—21no—
11C22×C4⋊Dic61C411yescharacteristic
12C22×C4×Dic61—21no—
13C22×C4×Dic61C2211yescharacteristic
14C4⋊C4×Dic61—21no—
15C4⋊C4×Dic61—21no—
16C4⋊C4×Dic61C412yes—
17C4⋊C4×Dic61C2212yes—
18C2×#14519684—21no—
19C22×C4⋊C4×C61C411yescharacteristic
20C22×#92420—22no—
21C22×#92420—22no—
22C4⋊#92420—24no—
23C22⋊#92420—24no—
24C22⋊#92420—42no—
25C4×#92420—24no—
26C23×Dic61—21no—
27C23×Dic61D411yescharacteristic
28C23×Dic61D411yescharacteristic
29C2×C4⋊Dic61—21no—
30C2×C4⋊Dic61—24no—
31C2×C4⋊Dic61C2×C412yes—
32C2×C4×Dic61—24no—
33C2×C4×Dic61—22no—
34C2×C4×Dic61—22no—
35C2×C4×Dic61—44no—
36C2×C4×Dic61C2×C411yescharacteristic
37C2×C4×Dic61D411yescharacteristic
38C2×C4×Dic61D411yescharacteristic
39C2×C4×Dic61C2311yescharacteristic
40#14519684—22no—
41#14519684—22no—
42C23×C4×C61—21no—
43C23×C4×C61C2×C411yescharacteristic
44C2×C4⋊C4×C61—21no—
45C2×C4⋊C4×C61—22no—
46C2×C4⋊C4×C61C814yescomplemented
47C2×C4⋊C4×C61C2×C412yes—
48C24×C61C22⋊C411yescharacteristic
49C22×C4×C61—22no—
50C22×C4×C61—22no—
51C22×C4×C61—24no—
52C22×C4×C61—22no—
53C22×C4×C61—42no—
54C22×C4×C61—42no—
55C22×C4×C61C22⋊C411yescharacteristic
56C22×C4×C61C2×C812yes—
57C22×C4×C61M1612yes—
58C22×C4×C61C22×C411yescharacteristic
59C22×Dic61—22no—
60C22×Dic61—22no—
61C22×Dic61—24no—
62C22×Dic61—22no—
63C22×Dic61—22no—
64C22×Dic61—22no—
65C22×Dic61—41no—
66C22×Dic61—42no—
67C22×Dic61—42no—
68C22×Dic61C22⋊C411yescharacteristic
69C22×Dic61C22⋊C411yescharacteristic
70C22×Dic61SD1612yes—
71C22×Dic61Q1612yes—
72C22×Dic61C2×D411yescharacteristic
73C22×Dic61C2×D411yescharacteristic
74C2×#92420—24no—
75C2×#92420—42no—
76C2×#92420—42no—
77C2×#92420—42no—
78C2×#92420—42no—
79C2×#92420—42no—
80C4⋊C4×C61—24no—
81C4⋊C4×C61—22no—
82C4⋊C4×C61C2×C814yes—
83C4⋊Dic61—24no—
84C4⋊Dic61—22no—
85C4⋊Dic61M1614yes—
86C4×Dic61—22no—
87C4×Dic61—22no—
88C4×Dic61—22no—
89C4×Dic61—44no—
90C4×Dic61—44no—
91C4×Dic61#4114yes—
92C23×C61—21no—
93C23×C61—21no—
94C23×C61—21no—
95C23×C61—21no—
96C23×C61C22⋊C812yes—
97C23×C61#10312yes—
98C23×C61#10412yes—
99C23×C61C2×C22⋊C411yescharacteristic
100C2×C4×C61—21no—
101C2×C4×C61—21no—
102C2×C4×C61—24no—
103C2×C4×C61—24no—
104C2×C4×C61—24no—
105C2×C4×C61—48no—
106C2×C4×C61—42no—
107C2×C4×C61—42no—
108C2×C4×C61C22⋊C812yes—
109C2×C4×C61#11712yes—
110C2×C4×C61C22×C811yescharacteristic
111C2×C4×C61C2×M1612yes—
112C2×C4×C61C2×M1611yescharacteristic
113C2×Dic61—21no—
114C2×Dic61—21no—
115C2×Dic61—21no—
116C2×Dic61—21no—
117C2×Dic61—24no—
118C2×Dic61—24no—
119C2×Dic61—21no—
120C2×Dic61—24no—
121C2×Dic61—21no—
122C2×Dic61—21no—
123C2×Dic61—44no—
124C2×Dic61—48no—
125C2×Dic61—42no—
126C2×Dic61—42no—
127C2×Dic61#10312yes—
128C2×Dic61#12312yes—
129C2×Dic61#12312yes—
130C2×Dic61C2×SD1611yescharacteristic
131C2×Dic61C2×Q1611yescharacteristic
132C2×Dic61AGL(1,ℤ/8)12yes—
133#92420—44no—
134#92420—82no—
135#8707—611nomaximal, Sylow
136C22×C61—22no—
137C22×C61—22no—
138C22×C61—22no—
139C22×C61—22no—
140C22×C61—22no—
141C22×C61—22no—
142C22×C61Q8⋊C814yes—
143C22×C61#38111yescharacteristic
144C22×C61C2×C22⋊C811yescharacteristic
145C22×C61#40711yescharacteristic
146C22×C61C2×#10311yescharacteristic
147C22×C61#41811yescharacteristic
148C22×C61C2×#10411yescharacteristic
149C22×C61#42111yescharacteristic
150C4×C61—22no—
151C4×C61—44no—
152C4×C61#43112yes—
153C4×C61#43212yes—
154Dic61—22no—
155Dic61—22no—
156Dic61—41no—
157Dic61—44no—
158Dic61#48112yes—
159Dic61#48412yes—
160C2×C4×C4⋊C4—611no—
161#1797—611no—
162C4×C22⋊C8—611no—
163#1279—614no—
164C2×C61—24no—
165C2×C61C2×Q8⋊C811yescharacteristic
166C2×C61#147811yescharacteristic
167C2×C61#148711yescharacteristic
168C2×C61#149011yescharacteristic
169C2×C61#181711yescharacteristic
170C2×C61#182411yescharacteristic
171C2×C61#182611yescharacteristic
172C22×C4⋊C4—611no—
173C22×C4⋊C4#692311yescharacteristic
174C22×C42—611no—
175C22×C42—1221no—
176C2×C4⋊C8—612no—
177C2×C22⋊C8—1221no—
178C2×C22⋊C8—1221no—
179C2×C4×C8—612no—
180C4×C4⋊C4—612no—
181C4×C4⋊C4—612no—
182C4×C4⋊C4—1221no—
183C4×C4⋊C4—1221no—
184C2×#95—1221no—
185#336—1221no—
186#336—1221no—
187C61#870711yescharacteristic, complemented, Sylow
188C23×C4—21no—
189C23×C4—611no—
190C23×C4—611no—
191C23×C4—1221no—
192C23×C4C2×#692311yescharacteristic
193C22×C8—1222no—
194C22×C8—1222no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—612no—
198C2×C4⋊C4—1224no—
199C2×C4⋊C4—1221no—
200C2×C4⋊C4#9242014yescomplemented
201C2×C4⋊C4C2×#692312yes—
202C2×C42—611no—
203C2×C42—611no—
204C2×C42—611no—
205C2×C42—611no—
206C2×C42—1224no—
207C2×C42—1222no—
208C2×C42—1222no—
209C2×C42—2444no—
210C4⋊C8—1224no—
211C22⋊C8—1224no—
212C22⋊C8—2442no—
213C4×C8—1224no—
214#95—1222no—
215#95—1222no—
216C24C22⋊#692311yescharacteristic
217C22×C4—22no—
218C22×C4—22no—
219C22×C4—24no—
220C22×C4—22no—
221C22×C4—42no—
222C22×C4—42no—
223C22×C4—611no—
224C22×C4—612no—
225C22×C4—612no—
226C22×C4—611no—
227C22×C4—611no—
228C22×C4—611no—
229C22×C4—1222no—
230C22×C4—1222no—
231C22×C4—1222no—
232C22×C4—1224no—
233C22×C4—1222no—
234C22×C4—1222no—
235C22×C4—2441no—
236C22×C4—2442no—
237C22×C4—2442no—
238C22×C4C2×#9242012yes—
239C22×C4#1175821712yes—
240C22×C4C22⋊#692311yescharacteristic
241C22×C4C22×#692311yescharacteristic
242C2×C8—1224no—
243C2×C8—2442no—
244C2×C8—2442no—
245C2×C8—2442no—
246C2×C8—2442no—
247C2×C8—2442no—
248C4⋊C4—24no—
249C4⋊C4—22no—
250C4⋊C4—614no—
251C4⋊C4—1224no—
252C4⋊C4—1222no—
253C4⋊C4C2×#9242014yes—
254C42—614no—
255C42—1222no—
256C42—1222no—
257C42—1222no—
258C42—2444no—
259C42—2444no—
260C23—21no—
261C23—21no—
262C23—21no—
263C23—21no—
264C23C22⋊#9242012yes—
265C23#1451984012yes—
266C23#1451984212yes—
267C23C2×C22⋊#692311yescharacteristic
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—24no—
271C2×C4—24no—
272C2×C4—24no—
273C2×C4—48no—
274C2×C4—42no—
275C2×C4—42no—
276C2×C4—612no—
277C2×C4—611no—
278C2×C4—611no—
279C2×C4—612no—
280C2×C4—612no—
281C2×C4—612no—
282C2×C4—1224no—
283C2×C4—1224no—
284C2×C4—1221no—
285C2×C4—1221no—
286C2×C4—1221no—
287C2×C4—1221no—
288C2×C4—1221no—
289C2×C4—1224no—
290C2×C4—1221no—
291C2×C4—1221no—
292C2×C4—2444no—
293C2×C4—2448no—
294C2×C4—2442no—
295C2×C4—2442no—
296C2×C4C22⋊#9242012yes—
297C2×C4#1451985612yes—
298C2×C4C22×#9242011yescharacteristic
299C2×C4C2×#1175821711yescharacteristic
300C2×C4D61⋊M1612yes—
301C8—4882no—
302C8—2444no—
303C22—22no—
304C22—22no—
305C22—22no—
306C22—22no—
307C22—22no—
308C22—22no—
309C22Q8⋊#9242014yes—
310C22#1684070511yescharacteristic
311C22C2×C22⋊#9242011yescharacteristic
312C22#1684073411yescharacteristic
313C22C2×#1451984011yescharacteristic
314C22#1684075411yescharacteristic
315C22C2×#1451984211yescharacteristic
316C22#1684076011yescharacteristic
317C4—22no—
318C4—44no—
319C4—612no—
320C4—612no—
321C4—1222no—
322C4—1222no—
323C4—2444no—
324C4—2441no—
325C4#1684077812yes—
326C4#1684078212yes—
327C2—24no—
328C2C2×Q8⋊#9242011yescharacteristic
329C2#2146111811yescharacteristic
330C2#2146114911yescharacteristic
331C2#2146116011yescharacteristic
332C2#2146170911yescharacteristic
333C2#2146172311yescharacteristic
334C2#2146172611yescharacteristic
subgroupG/H
centerC23C2×C22⋊#6923
commutatorC2×C4×C61C22×C8
FrattiniC22×C4C22×#6923
FittingC22×C4⋊C4×C61C4
socleC23×C61C2×C22⋊C4
prime radicalC24×C61C22⋊C4
cosocleC2×C4×Dic61C23
hypercenterC22×C4⋊C4#6923
hypocenterC61#8707
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ C22×C4⋊C4

Complements

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

NH ≅ G/Ntype
3. #21460655C2semidirect
46. C2×C4⋊C4×C61C8semidirect
187. C61#8707semidirect
200. C2×C4⋊C4#92420semidirect

Maximal subgroups

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

subgroupindex
#214616732
C4×C22⋊#924202
#214606552
C2×C4⋊C4×Dic612
#870761