Finite Group Databasebeta

← #29680604

Subgroups — #29680604

Order 15616.

898 conjugacy classes of subgroups.

The subgroup lattice has 346 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21448226C211yesmaximal, characteristic
2#21448217C211yesmaximal, characteristic
3#21445987C212yesmaximal, complemented
4#21445953C212yesmaximal, complemented
5#21445848C212yesmaximal
6#21445844C212yesmaximal
7#21445716C214yesmaximal, complemented
8C22×#97×C61C211yesmaximal, characteristic
9C2×#14519736—21no—
10C2×#14519711C2214yes—
11C2×#14519709C2214yes—
12#16839632C2212yes—
13#16839630—21no—
14#16839628—21no—
15#16839628C2212yes—
16#16839625—21no—
17#16839622—21no—
18#16839622C2211yescharacteristic
19#16839622C2211yescharacteristic
20#16839622C2212yes—
21C4×C4⋊Dic61—21no—
22C4×C4⋊Dic61C2212yes—
23C4×C4⋊Dic61C2212yes—
24C4×C22⋊Dic61—21no—
25#16839608—21no—
26#16839525C2214yes—
27#16839525C2214yes—
28C23×C8×C61C2212yes—
29C22×C42×C61C2211yescharacteristic
30C2×#97×C61C418yescomplemented
31C2×#97×C61C2211yescharacteristic
32C2×#97×C61C2212yes—
33C2×#97×C61C2211yescharacteristic
34#14519736—44no—
35C2×C4⋊Dic61—22no—
36C2×C4⋊Dic61—22no—
37C2×C4⋊Dic61C2312yes—
38C2×C4⋊Dic61C2314yes—
39C2×C4⋊Dic61C2312yes—
40C2×C22⋊Dic61—22no—
41C2×C22⋊Dic61—21no—
42C2×C22⋊Dic61—21no—
43C2×C4×Dic61—22no—
44C2×C4×Dic61—22no—
45#14519711—28no—
46#14519709—28no—
47#14519684—21no—
48#14519684—21no—
49#14519684—22no—
50#14519684—22no—
51#14519684—22no—
52#14519684—22no—
53#14519684—22no—
54C23×C4×C61D411yescharacteristic
55C23×C4×C61D411yescharacteristic
56C23×C4×C61C2311yescharacteristic
57C22×C8×C61—28no—
58C22×C8×C61C2×C418yes—
59C22×C8×C61C2312yes—
60C22×C8×C61C2312yes—
61C2×C42×C61C2×C414yes—
62C2×C42×C61D411yescharacteristic
63C2×C42×C61D411yescharacteristic
64C2×C42×C61Q814yes—
65C2×C42×C61C2311yescharacteristic
66C2×C42×C61C2311yescharacteristic
67#97×C61C2×C414yes—
68#97×C61C2×C418yes—
69#97×C61C2×C414yes—
70C24×C61C2×D411yescharacteristic
71C22×C4×C61—22no—
72C22×C4×C61—22no—
73C22×C4×C61—24no—
74C22×C4×C61—42no—
75C22×C4×C61C4⋊C414yes—
76C22×C4×C61C4⋊C414yes—
77C22×C4×C61C22×C412yes—
78C22×C4×C61C2×D411yescharacteristic
79C22×C4×C61C2×D411yescharacteristic
80C22×C4×C61C2×D411yescharacteristic
81C22×C4×C61C2×D411yescharacteristic
82C22×C4×C61C2×D411yescharacteristic
83C22×C4×C61C2×Q812yes—
84C22×C4×C61C2411yescharacteristic
85C22×Dic61—22no—
86C22×Dic61—22no—
87C22×Dic61—24no—
88C2×C8×C61—24no—
89C2×C8×C61—216no—
90C2×C8×C61C22×C414yes—
91C2×C8×C61C22×C414yes—
92C2×C8×C61#4118yes—
93C42×C61C4⋊C414yes—
94C42×C61C4⋊C414yes—
95C42×C61C22×C412yes—
96C42×C61C22×C412yes—
97C42×C61C2×Q812yes—
98C42×C61C2×Q812yes—
99C22⋊Dic61—42no—
100C22⋊Dic61—44no—
101C22⋊Dic61—42no—
102C4⋊Dic61—24no—
103C4⋊Dic61—24no—
104C4⋊Dic61—24no—
105C4⋊Dic61—24no—
106C4⋊Dic61—44no—
107C4⋊Dic61—44no—
108C4×Dic61—44no—
109C4×Dic61—44no—
110C23×C61—22no—
111C23×C61—22no—
112C23×C61C2×C4⋊C412yes—
113C23×C61AGL(1,ℤ/8)12yes—
114C23×C61#13712yes—
115C23×C61C22×D411yescharacteristic
116C2×C4×C61—22no—
117C2×C4×C61—21no—
118C2×C4×C61—21no—
119C2×C4×C61—21no—
120C2×C4×C61—21no—
121C2×C4×C61—22no—
122C2×C4×C61—22no—
123C2×C4×C61—28no—
124C2×C4×C61—24no—
125C2×C4×C61—24no—
126C2×C4×C61—44no—
127C2×C4×C61C2×C4⋊C412yes—
128C2×C4×C61C2×C4⋊C412yes—
129C2×C4×C61C2×C4⋊C412yes—
130C2×C4×C61C2×C4⋊C412yes—
131C2×C4×C61C2×C4⋊C412yes—
132C2×C4×C61C23×C411yescharacteristic
133C2×C4×C61C22×Q811yescharacteristic
134C2×C4×C61C2×#4114yes—
135C2×Dic61—44no—
136C2×Dic61—42no—
137C2×Dic61—42no—
138C2×Dic61—44no—
139C2×Dic61—42no—
140C2×Dic61—42no—
141C2×Dic61—42no—
142C2×Dic61—42no—
143C2×Dic61—44no—
144C8×C61—24no—
145C8×C61#11718yescomplemented
146#20823—611nomaximal, Sylow
147C22×C61—22no—
148C22×C61—24no—
149C22×C61—24no—
150C22×C61—22no—
151C22×C61#42814yes—
152C22×C61C22×C4⋊C411yescharacteristic
153C22×C61#53212yes—
154C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
155C22×C61C2×#13711yescharacteristic
156C22×C61#57512yes—
157C4×C61—22no—
158C4×C61—22no—
159C4×C61—22no—
160C4×C61—22no—
161C4×C61—22no—
162C4×C61C2×#11712yes—
163C4×C61#53312yes—
164Dic61—84no—
165Dic61—82no—
166Dic61—82no—
167C22×#97D6111yescharacteristic
168#2285—611no—
169#2283—611no—
170#1847—612no—
171#1836—612no—
172#1808—612no—
173#1807—612no—
174#1776—614no—
175C2×C61—24no—
176C2×C61#228511yescharacteristic
177C2×C61C2×#42811yescharacteristic
178C2×C61#291111yescharacteristic
179C2×C61#318012yes—
180C2×C61#318112yes—
181C23×C8C2×D6112yes—
182C2×#117—1221no—
183C22×C42C2×D6111yescharacteristic
184C2×#107—614no—
185C2×#106—614no—
186C2×#97Dic6118yescomplemented
187C2×#97C2×D6111yescharacteristic
188C2×#97C2×D6111yescharacteristic
189C2×#97C2×D6112yes—
190#384—612no—
191#383—1221no—
192#382—612no—
193#382—1221no—
194#381—1221no—
195#380—611no—
196#380—612no—
197#380—611no—
198#380—1221no—
199C4×C4⋊C4—612no—
200C4×C4⋊C4—612no—
201C4×C4⋊C4—1221no—
202C4×C22⋊C4—1221no—
203#376—1221no—
204#340—614no—
205#340—614no—
206C61#2082311yescharacteristic, complemented, Sylow
207C23×C4D24411yescharacteristic
208C23×C4D24411yescharacteristic
209C23×C4C22×D6111yescharacteristic
210C22×C8—28no—
211C22×C8C2×Dic6118yes—
212C22×C8C22×D6112yes—
213C22×C8C22×D6112yes—
214#117—2444no—
215C2×C4⋊C4—612no—
216C2×C4⋊C4—612no—
217C2×C4⋊C4—614no—
218C2×C4⋊C4—1222no—
219C2×C4⋊C4—1222no—
220C2×C22⋊C4—1222no—
221C2×C22⋊C4—1221no—
222C2×C22⋊C4—1221no—
223C2×C42—1222no—
224C2×C42—1222no—
225C2×C42Dic12214yes—
226C2×C42D24411yescharacteristic
227C2×C42D24411yescharacteristic
228C2×C42C2×Dic6114yes—
229C2×C42C22×D6111yescharacteristic
230C2×C42C22×D6111yescharacteristic
231#107—1228no—
232#106—1228no—
233#97C2×Dic6114yes—
234#97C2×Dic6118yes—
235#97C2×Dic6114yes—
236#95—1221no—
237#95—1221no—
238#95—1222no—
239#95—1222no—
240#95—1222no—
241#95—1222no—
242#95—1222no—
243C24C2×D24411yescharacteristic
244C22×C4—22no—
245C22×C4—22no—
246C22×C4—24no—
247C22×C4—42no—
248C22×C4—1222no—
249C22×C4—1222no—
250C22×C4—1224no—
251C22×C4C4⋊Dic6114yes—
252C22×C4C4⋊Dic6114yes—
253C22×C4C2×Dic12212yes—
254C22×C4C2×D24411yescharacteristic
255C22×C4C2×D24411yescharacteristic
256C22×C4C2×D24411yescharacteristic
257C22×C4C2×D24411yescharacteristic
258C22×C4C2×D24411yescharacteristic
259C22×C4C22×Dic6112yes—
260C22×C4C23×D6111yescharacteristic
261C2×C8—24no—
262C2×C8—216no—
263C2×C8#1175822218yes—
264C2×C8C22×Dic6114yes—
265C2×C8C22×Dic6114yes—
266C4⋊C4—1224no—
267C4⋊C4—1224no—
268C4⋊C4—1224no—
269C4⋊C4—1224no—
270C4⋊C4—2444no—
271C4⋊C4—2444no—
272C22⋊C4—2442no—
273C22⋊C4—2444no—
274C22⋊C4—2442no—
275C42—2444no—
276C42—2444no—
277C42C4⋊Dic6114yes—
278C42C4⋊Dic6114yes—
279C42C2×Dic12212yes—
280C42C2×Dic12212yes—
281C42C22×Dic6112yes—
282C42C22×Dic6112yes—
283C23—22no—
284C23—22no—
285C23C2×C4⋊Dic6112yes—
286C23#1451980512yes—
287C23#1451981012yes—
288C23C22×D24411yescharacteristic
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—22no—
294C2×C4—22no—
295C2×C4—22no—
296C2×C4—28no—
297C2×C4—24no—
298C2×C4—24no—
299C2×C4—44no—
300C2×C4—2444no—
301C2×C4—2442no—
302C2×C4—2442no—
303C2×C4—2442no—
304C2×C4—2442no—
305C2×C4—2444no—
306C2×C4—2444no—
307C2×C4—2442no—
308C2×C4—2442no—
309C2×C4C2×C4⋊Dic6112yes—
310C2×C4C2×C4⋊Dic6112yes—
311C2×C4C2×C4⋊Dic6112yes—
312C2×C4C2×C4⋊Dic6112yes—
313C2×C4C2×C4⋊Dic6112yes—
314C2×C4C23×Dic6111yescharacteristic
315C2×C4C22×Dic12211yescharacteristic
316C2×C4C2×#1175822214yes—
317C8—24no—
318C8#1451973618yescomplemented
319C22—22no—
320C22—22no—
321C22—24no—
322C22—24no—
323C22#1683980414yes—
324C22C22×C4⋊Dic6111yescharacteristic
325C22#1684031212yes—
326C22C2×#1451980511yescharacteristic
327C22C2×#1451981011yescharacteristic
328C22#1684056212yes—
329C4—22no—
330C4—22no—
331C4—22no—
332C4—22no—
333C4—22no—
334C4—4884no—
335C4—4882no—
336C4—4882no—
337C4C2×#1451973612yes—
338C4#1684031912yes—
339C2—24no—
340C2#2144822611yescharacteristic
341C2C2×#1683980411yescharacteristic
342C2#2145440911yescharacteristic
343C2#2145651712yes—
344C2#2145652812yes—
subgroupG/H
centerC23C22×D244
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC22×#97×C61C2
socleC23×C61C22×D4
prime radicalC24×C61C2×D4
cosocleC22×C4×C61C24
hypercenterC22×#97D61
hypocenterC61#20823
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C22×C42 ◁ C22×#97

Complements

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

NH ≅ G/Ntype
3. #21445987C2semidirect
4. #21445953C2semidirect
7. #21445716C2semidirect
30. C2×#97×C61C4semidirect
145. C8×C61#117semidirect
186. C2×#97Dic61semidirect
206. C61#20823semidirect
318. C8#14519736semidirect

Maximal subgroups

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

subgroupindex
#214482262
#214482172
#214459872
#214459532
#214458482
#214458442
#214457162
C22×#97×C612
#2082361