Finite Group Databasebeta

← #108277

Subgroups — #108277

Order 512.

712 conjugacy classes of subgroups.

The subgroup lattice has 381 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60250C211yesmaximal, characteristic
2#21826C211yesmaximal, characteristic
3#17385C211yesmaximal, characteristic
4#7976C211yesmaximal, characteristic
5#7958C211yesmaximal, characteristic
6#7894C211yesmaximal, characteristic
7#7858C211yesmaximal, characteristic
8C2×#533C2211yescharacteristic
9#2986—22no—
10#2985—22no—
11#2984—22no—
12#2983C412yescomplemented
13#2982C412yescomplemented
14#2962—22no—
15#2960—22no—
16#2924—22no—
17#2920C2211yescharacteristic
18#2920C2211yescharacteristic
19C2×#431C2211yescharacteristic
20C2×#431C2211yescharacteristic
21#2283C2211yescharacteristic
22#2283C2211yescharacteristic
23#1996—22no—
24#1996C414yes—
25#1820—22no—
26#1472—22no—
27#1470C414yescomplemented
28#1461—22no—
29#1288—22no—
30#1286—22no—
31#1279—22no—
32#1274—22no—
33#533—21no—
34#533—21no—
35#533—21no—
36#533—21no—
37#533—21no—
38#533—21no—
39#533—21no—
40#533—21no—
41C2×C4⋊Q8Q811yescharacteristic
42C2×C4⋊Q8Q811yescharacteristic
43C2×#125C2×C412yes—
44C2×C22⋊Q8C2×C412yes—
45C2×C22⋊Q8D412yes—
46C2×C4×Q8C2×C412yes—
47C2×C4×Q8D412yes—
48C2×#117D412yes—
49C22×C42C2311yescharacteristic
50#446—24no—
51#446—28no—
52#446—24no—
53#446—24no—
54C8×Q8—24no—
55C8×Q8—24no—
56C8×Q8—24no—
57#433—28no—
58#432—24no—
59#432—21no—
60#432—21no—
61#432—21no—
62#432C2×C412yes—
63#431—24no—
64#431—21no—
65#431—21no—
66#431—21no—
67#431—21no—
68#431—21no—
69#431—21no—
70#431—21no—
71#431C2×C412yes—
72#431C2×C411yescharacteristic
73#431C2×C411yescharacteristic
74#431C2×C412yes—
75#431D412yes—
76#431D412yes—
77#424—24no—
78#423—21no—
79#423—21no—
80C2×C4⋊C8—21no—
81C2×C4⋊C8—21no—
82#407—21no—
83#407—21no—
84C2×C22⋊C8—21no—
85C2×C22⋊C8—21no—
86#405—24no—
87C4×M16—21no—
88C4×M16D412yes—
89C2×C4×C8—21no—
90C2×C4×C8D412yes—
91#383—21no—
92#383—21no—
93#382—21no—
94#382—21no—
95#381—21no—
96#381—21no—
97#380—21no—
98#380C2×C412yes—
99C4×C4⋊C4—21no—
100C4×C4⋊C4—21no—
101C4×C22⋊C4—21no—
102C4×C22⋊C4C2×C412yes—
103#376—21no—
104#376—21no—
105#342—28no—
106#324—28no—
107C22×Q8C4⋊C412yes—
108C22×Q8C4⋊C412yes—
109C23×C4C22×C411yescharacteristic
110C23×C4C2×D411yescharacteristic
111C23×C4C2×D411yescharacteristic
112C2×M16—22no—
113C2×M16—41no—
114C22×C8—22no—
115C22×C8—41no—
116C4⋊Q8—21no—
117C4⋊Q8—21no—
118C4⋊Q8—21no—
119C4⋊Q8—21no—
120C4⋊Q8—21no—
121C4⋊Q8—21no—
122C4⋊Q8—21no—
123C4⋊Q8—21no—
124#125—22no—
125#125—22no—
126#125—22no—
127#125—22no—
128C22⋊Q8—22no—
129C22⋊Q8—22no—
130C22⋊Q8—22no—
131C22⋊Q8—22no—
132C22⋊Q8—22no—
133C22⋊Q8—22no—
134C22⋊Q8—22no—
135C22⋊Q8—22no—
136C4×Q8—22no—
137C4×Q8—22no—
138C4×Q8—22no—
139C4×Q8—22no—
140C4×Q8—22no—
141C4×Q8—22no—
142C4×Q8—22no—
143C4×Q8—22no—
144#117—22no—
145#117—22no—
146#117—22no—
147#117—22no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—41no—
150C2×C4⋊C4C4214yes—
151C2×C4⋊C4C22⋊C414yes—
152C2×C4⋊C4C4⋊C412yes—
153C2×C4⋊C4C4⋊C412yes—
154C2×C4⋊C4C4⋊C412yes—
155C2×C4⋊C4C4⋊C412yes—
156C2×C22⋊C4—22no—
157C2×C22⋊C4—22no—
158C2×C22⋊C4C22⋊C414yes—
159C2×C42—22no—
160C2×C42—41no—
161C2×C42C22⋊C414yes—
162C2×C42C22×C411yescharacteristic
163C2×C42C22×C411yescharacteristic
164C2×C42C2×D411yescharacteristic
165C2×C42C2×D411yescharacteristic
166C2×C42C2×D411yescharacteristic
167C2×C42C2×D411yescharacteristic
168C2×C42C2×D411yescharacteristic
169C2×C42C2×Q811yescharacteristic
170C2×C42#4111yescharacteristic
171C2×C42#4111yescharacteristic
172C2×C42#4111yescharacteristic
173C2×C42#4111yescharacteristic
174C4⋊C8—28no—
175C4⋊C8—28no—
176C4⋊C8—21no—
177C4⋊C8—21no—
178C4⋊C8—22no—
179C4⋊C8—22no—
180C4⋊C8—22no—
181C4⋊C8—22no—
182C4⋊C8—22no—
183C4⋊C8—41no—
184C4⋊C8—41no—
185C22⋊C8—21no—
186C22⋊C8—21no—
187C22⋊C8—22no—
188C22⋊C8—22no—
189C22⋊C8—22no—
190C22⋊C8—22no—
191C22⋊C8—22no—
192C22⋊C8—41no—
193C22⋊C8—41no—
194#97—28no—
195#97—22no—
196#97—22no—
197C4×C8—28no—
198C4×C8—22no—
199C4×C8—22no—
200C4×C8—41no—
201C4×C8—41no—
202C4×C8C22⋊C412yes—
203C4×C8C22⋊C412yes—
204C4×C8C22⋊C414yes—
205#95—22no—
206#95—22no—
207#95—41no—
208#95—41no—
209#95—41no—
210#95—41no—
211#95—41no—
212#95—41no—
213C24C2×C22⋊C411yescharacteristic
214C2×Q8—22no—
215C2×Q8—22no—
216C2×Q8—22no—
217C2×Q8—22no—
218C2×Q8—44no—
219C2×Q8—44no—
220C22×C4—42no—
221C22×C4—42no—
222C22×C4—42no—
223C22×C4—42no—
224C22×C4—42no—
225C22×C4#9514yes—
226C22×C4#9514yes—
227C22×C4C2×C4211yescharacteristic
228C22×C4C2×C22⋊C411yescharacteristic
229C22×C4C2×C22⋊C411yescharacteristic
230C22×C4C2×C4⋊C411yescharacteristic
231C22×C4C2×C4⋊C411yescharacteristic
232C22×C4C2×C4⋊C411yescharacteristic
233C22×C4C4×D411yescharacteristic
234C22×C4C4×D411yescharacteristic
235C22×C4C4×D411yescharacteristic
236C22×C4C4×D411yescharacteristic
237C22×C4#12011yescharacteristic
238C22×C4#12011yescharacteristic
239C22×C4#12111yescharacteristic
240C22×C4#12111yescharacteristic
241C22×C4#12111yescharacteristic
242C22×C4#12111yescharacteristic
243C22×C4#12311yescharacteristic
244C22×C4#12311yescharacteristic
245M16—44no—
246M16—82no—
247C2×C8—22no—
248C2×C8—24no—
249C2×C8—22no—
250C2×C8—44no—
251C2×C8—42no—
252C2×C8—42no—
253C2×C8—82no—
254C4⋊C4—24no—
255C4⋊C4—24no—
256C4⋊C4—22no—
257C4⋊C4—22no—
258C4⋊C4—22no—
259C4⋊C4—22no—
260C4⋊C4—22no—
261C4⋊C4—22no—
262C4⋊C4—22no—
263C4⋊C4—22no—
264C4⋊C4—24no—
265C4⋊C4—24no—
266C4⋊C4—44no—
267C4⋊C4—82no—
268C22⋊C4—24no—
269C22⋊C4—24no—
270C22⋊C4—44no—
271C22⋊C4—44no—
272C42—24no—
273C42—24no—
274C42—44no—
275C42—82no—
276C42C2×C22⋊C411yescharacteristic
277C42C2×C22⋊C411yescharacteristic
278C42#11711yescharacteristic
279C42#11711yescharacteristic
280C42C4×D411yescharacteristic
281C42C4×D411yescharacteristic
282C42C4×D411yescharacteristic
283C42C4×D411yescharacteristic
284C42#12011yescharacteristic
285C42#12011yescharacteristic
286C42C22⋊Q811yescharacteristic
287C42C22⋊Q811yescharacteristic
288C42C22⋊Q811yescharacteristic
289C42C22⋊Q811yescharacteristic
290C42#12311yescharacteristic
291C42#12311yescharacteristic
292C23—22no—
293C23—41no—
294C23—41no—
295C23C2×#9511yescharacteristic
296C23#38611yescharacteristic
297C23#38611yescharacteristic
298Q8—44no—
299Q8—44no—
300Q8—44no—
301Q8—82no—
302C2×C4—22no—
303C2×C4—22no—
304C2×C4—22no—
305C2×C4—22no—
306C2×C4—22no—
307C2×C4—22no—
308C2×C4—22no—
309C2×C4—22no—
310C2×C4—24no—
311C2×C4—24no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—44no—
323C2×C4—44no—
324C2×C4—82no—
325C2×C4—82no—
326C2×C4—82no—
327C2×C4—82no—
328C2×C4—82no—
329C2×C4—82no—
330C2×C4C4×C22⋊C411yescharacteristic
331C2×C4C4×C22⋊C411yescharacteristic
332C2×C4#37911yescharacteristic
333C2×C4#38011yescharacteristic
334C2×C4#38011yescharacteristic
335C2×C4#38111yescharacteristic
336C2×C4#38511yescharacteristic
337C2×C4#38511yescharacteristic
338C2×C4#38511yescharacteristic
339C2×C4#38511yescharacteristic
340C2×C4#38611yescharacteristic
341C2×C4#38611yescharacteristic
342C8—42no—
343C8—42no—
344C8—44no—
345C8—82no—
346C8—82no—
347C22—21no—
348C22—21no—
349C22—21no—
350C22—21no—
351C22—21no—
352C22—21no—
353C22—21no—
354C22—21no—
355C22—82no—
356C22C22⋊#9511yescharacteristic
357C22#191212yes—
358C4—21no—
359C4—21no—
360C4—21no—
361C4—21no—
362C4—21no—
363C4—21no—
364C4—21no—
365C4—21no—
366C4—21no—
367C4—21no—
368C4—44no—
369C4—44no—
370C4—162no—
371C4—162no—
372C4#181212yes—
373C4#186812yes—
374C2—22no—
375C2—41no—
376C2—41no—
377C2#1074111yescharacteristic
378C2#1074811yescharacteristic
379C2#1075911yescharacteristic
subgroupG/H
center, socleC22C22⋊#95
commutatorC22×C4C2×C42
Frattini, cosocleC22×C42C23
prime radicalC24C2×C22⋊C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
12. #2983C4semidirect
13. #2982C4semidirect
27. #1470C4semidirect

Maximal subgroups

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

subgroupindex
#602502
#218262
#173852
#79762
#79582
#78942
#78582