Finite Group Databasebeta

← #117953

Subgroups — #117953

Order 512.

554 conjugacy classes of subgroups.

The subgroup lattice has 368 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20831C211yesmaximal, characteristic
2#17672C211yesmaximal, characteristic
3#17590C211yesmaximal, characteristic
4C2×#1485C211yesmaximal, characteristic
5#7972C211yesmaximal, characteristic
6#7922C211yesmaximal, characteristic
7#7858C211yesmaximal, characteristic
8C2×#533C2211yescharacteristic
9#2920C2211yescharacteristic
10#2920C2211yescharacteristic
11C2×#432C2211yescharacteristic
12C2×#431C2211yescharacteristic
13C2×C4×M16C2211yescharacteristic
14#2283C2211yescharacteristic
15#1848—21no—
16#1837—21no—
17#1811—22no—
18#1778—22no—
19#1537—22no—
20#1531—22no—
21#1528—22no—
22#1518—22no—
23#1511—22no—
24#1506—22no—
25#1485—22no—
26#1485C414yescomplemented
27C2×Q8⋊C8—21no—
28C2×Q8⋊C8—21no—
29#1288—22no—
30#1281—22no—
31#1274—22no—
32C22×M16—21no—
33#533—21no—
34#533—21no—
35#533—21no—
36#533—21no—
37#533—21no—
38#533—21no—
39#533C2×C412yes—
40#533D412yes—
41C2×C4⋊Q8Q811yescharacteristic
42C2×C4⋊Q8Q811yescharacteristic
43C2×#125—21no—
44C2×C22⋊Q8—21no—
45C2×C22⋊Q8—21no—
46C2×C4×Q8—21no—
47C2×C4×Q8—21no—
48C2×#117—21no—
49C22×C42C2311yescharacteristic
50#432—21no—
51#432—21no—
52#432—21no—
53#432—21no—
54#432—21no—
55#432—21no—
56#432—21no—
57#432—21no—
58#431—21no—
59#431—21no—
60#431—21no—
61#431—21no—
62#431—21no—
63#431—21no—
64#431—21no—
65#431C2×C412yes—
66C2×#108—24no—
67#423—21no—
68#423—21no—
69C2×C4⋊C8—21no—
70C2×C4⋊C8—21no—
71#407—21no—
72#407D412yes—
73C2×C22⋊C8—21no—
74C2×C22⋊C8—21no—
75C4×M16—21no—
76C4×M16—21no—
77C4×M16—21no—
78C4×M16C2×C412yes—
79C4×M16D412yes—
80C4×M16D412yes—
81C2×#97—21no—
82C2×#97D412yes—
83C2×C4×C8—21no—
84C2×C4×C8D411yescharacteristic
85C2×C4×C8D411yescharacteristic
86#383—21no—
87#382—21no—
88#381—21no—
89#380—21no—
90C4×C4⋊C4—21no—
91C4×C22⋊C4—21no—
92#376D412yes—
93#341—24no—
94#330—28no—
95#328—28no—
96Q8⋊C8—24no—
97Q8⋊C8—24no—
98C22×Q8—21no—
99C22×Q8—21no—
100C23×C4C2×D411yescharacteristic
101C23×C4#4111yescharacteristic
102C23×C4#4111yescharacteristic
103C2×M16—22no—
104C2×M16—22no—
105C2×M16—22no—
106C2×M16—22no—
107C2×M16—44no—
108C22×C8—21no—
109C22×C8—21no—
110C22×C8—22no—
111C22×C8—41no—
112C4⋊Q8—21no—
113C4⋊Q8—21no—
114C4⋊Q8—21no—
115C4⋊Q8—21no—
116C4⋊Q8—21no—
117C4⋊Q8—21no—
118C4⋊Q8C4⋊C412yes—
119C4⋊Q8C4⋊C412yes—
120#125—22no—
121#125—22no—
122#125—41no—
123#125—41no—
124C22⋊Q8—22no—
125C22⋊Q8—22no—
126C22⋊Q8—22no—
127C22⋊Q8—22no—
128C22⋊Q8—41no—
129C22⋊Q8—41no—
130C22⋊Q8—41no—
131C22⋊Q8—41no—
132C4×Q8—22no—
133C4×Q8—22no—
134C4×Q8—22no—
135C4×Q8—22no—
136C4×Q8—41no—
137C4×Q8—41no—
138C4×Q8—41no—
139C4×Q8—41no—
140#117—22no—
141#117—22no—
142#117—41no—
143#117—41no—
144C2×C4⋊C4—21no—
145C2×C4⋊C4—21no—
146C2×C4⋊C4—21no—
147C2×C4⋊C4—21no—
148C2×C4⋊C4—41no—
149C2×C4⋊C4—41no—
150C2×C4⋊C4—41no—
151C2×C22⋊C4—41no—
152C2×C22⋊C4—41no—
153C2×C42—22no—
154C2×C42—41no—
155C2×C42C22×C411yescharacteristic
156C2×C42C2×D411yescharacteristic
157C2×C42C2×D411yescharacteristic
158C2×C42C2×D411yescharacteristic
159C2×C42C2×D411yescharacteristic
160C2×C42C2×D411yescharacteristic
161C2×C42C2×D411yescharacteristic
162C2×C42C2×Q811yescharacteristic
163C2×C42#4111yescharacteristic
164C2×C42#4111yescharacteristic
165C2×C42#4111yescharacteristic
166C2×C42#4111yescharacteristic
167#108—48no—
168C4⋊C8—22no—
169C4⋊C8—41no—
170C4⋊C8—41no—
171C4⋊C8—41no—
172C4⋊C8—41no—
173C4⋊C8—41no—
174C4⋊C8—41no—
175C4⋊C8—41no—
176C22⋊C8—22no—
177C22⋊C8—22no—
178C22⋊C8—22no—
179C22⋊C8—41no—
180C22⋊C8—41no—
181C22⋊C8—41no—
182C22⋊C8—41no—
183C22⋊C8—41no—
184#97—22no—
185#97—22no—
186#97—22no—
187#97—22no—
188#97—22no—
189#97—41no—
190C4×C8—22no—
191C4×C8—22no—
192C4×C8—21no—
193C4×C8—21no—
194C4×C8—41no—
195C4×C8—41no—
196C4×C8C22⋊C412yes—
197C4×C8C22⋊C412yes—
198#95—22no—
199#95—22no—
200#95—41no—
201#95—41no—
202C24#12411yescharacteristic
203C2×Q8—22no—
204C2×Q8—22no—
205C2×Q8—44no—
206C2×Q8—41no—
207C2×Q8—41no—
208C2×Q8—82no—
209C22×C4—42no—
210C22×C4—42no—
211C22×C4—42no—
212C22×C4—41no—
213C22×C4—41no—
214C22×C4—42no—
215C22×C4#11711yescharacteristic
216C22×C4C4×D411yescharacteristic
217C22×C4C4×D411yescharacteristic
218C22×C4#12011yescharacteristic
219C22×C4#12011yescharacteristic
220C22×C4#12111yescharacteristic
221C22×C4#12111yescharacteristic
222C22×C4#12111yescharacteristic
223C22×C4#12111yescharacteristic
224C22×C4C22⋊Q811yescharacteristic
225C22×C4C22⋊Q811yescharacteristic
226C22×C4C22⋊Q811yescharacteristic
227C22×C4#12311yescharacteristic
228C22×C4#12311yescharacteristic
229C22×C4#12311yescharacteristic
230C22×C4#12411yescharacteristic
231C22×C4#12411yescharacteristic
232C22×C4#12611yescharacteristic
233M16—44no—
234M16—44no—
235M16—44no—
236M16—44no—
237C2×C8—24no—
238C2×C8—22no—
239C2×C8—22no—
240C2×C8—42no—
241C2×C8—44no—
242C2×C8—42no—
243C2×C8—42no—
244C2×C8—42no—
245C2×C8—42no—
246C2×C8—82no—
247C4⋊C4—22no—
248C4⋊C4—22no—
249C4⋊C4—22no—
250C4⋊C4—22no—
251C4⋊C4—42no—
252C4⋊C4—42no—
253C4⋊C4—42no—
254C4⋊C4—42no—
255C4⋊C4—41no—
256C4⋊C4—41no—
257C4⋊C4—41no—
258C4⋊C4—41no—
259C4⋊C4—82no—
260C22⋊C4—42no—
261C22⋊C4—42no—
262C22⋊C4—82no—
263C42—44no—
264C42—42no—
265C42—42no—
266C42C2×C22⋊C411yescharacteristic
267C42C2×C4⋊C411yescharacteristic
268C42C4×D411yescharacteristic
269C42C4×D411yescharacteristic
270C42#12011yescharacteristic
271C42#12111yescharacteristic
272C42#12111yescharacteristic
273C42#12111yescharacteristic
274C42#12111yescharacteristic
275C42C22⋊Q811yescharacteristic
276C42C22⋊Q811yescharacteristic
277C42C22⋊Q811yescharacteristic
278C42#12311yescharacteristic
279C42#12311yescharacteristic
280C42#12611yescharacteristic
281C42#12711yescharacteristic
282C23—22no—
283C23—41no—
284C23—41no—
285C23#38811yescharacteristic
286C23#39311yescharacteristic
287C23#39411yescharacteristic
288Q8—44no—
289Q8—82no—
290Q8—82no—
291Q8—82no—
292C2×C4—22no—
293C2×C4—22no—
294C2×C4—22no—
295C2×C4—22no—
296C2×C4—22no—
297C2×C4—22no—
298C2×C4—22no—
299C2×C4—22no—
300C2×C4—41no—
301C2×C4—41no—
302C2×C4—41no—
303C2×C4—41no—
304C2×C4—41no—
305C2×C4—41no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—42no—
311C2×C4—42no—
312C2×C4—82no—
313C2×C4—82no—
314C2×C4—82no—
315C2×C4—82no—
316C2×C4—82no—
317C2×C4#38011yescharacteristic
318C2×C4#38511yescharacteristic
319C2×C4#38511yescharacteristic
320C2×C4#38611yescharacteristic
321C2×C4#38811yescharacteristic
322C2×C4#39011yescharacteristic
323C2×C4#39211yescharacteristic
324C2×C4#39411yescharacteristic
325C2×C4#39611yescharacteristic
326C2×C4#39611yescharacteristic
327C2×C4#39711yescharacteristic
328C2×C4#39911yescharacteristic
329C8—24no—
330C8—42no—
331C8—82no—
332C8—82no—
333C8—82no—
334C8—82no—
335C22—21no—
336C22—21no—
337C22—21no—
338C22—21no—
339C22—21no—
340C22—21no—
341C22—21no—
342C22—21no—
343C22—82no—
344C22#143911yescharacteristic
345C22#203112yes—
346C4—21no—
347C4—21no—
348C4—21no—
349C4—21no—
350C4—21no—
351C4—21no—
352C4—21no—
353C4—21no—
354C4—21no—
355C4—21no—
356C4—82no—
357C4—82no—
358C4—162no—
359C4#208212yes—
360C4#209812yes—
361C2—22no—
362C2—41no—
363C2—41no—
364C2#1108111yescharacteristic
365C2#1111211yescharacteristic
366C2#1113411yescharacteristic
subgroupG/H
center, socleC22#1439
commutatorC2×C42C22×C4
Frattini, cosocleC22×C42C23
prime radicalC24#124
LCSG ▷ C2×C42 ▷ 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
26. #1485C4semidirect

Maximal subgroups

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

subgroupindex
#208312
#176722
#175902
C2×#14852
#79722
#79222
#78582