Finite Group Databasebeta

← #242965

Subgroups — #242965

Order 512.

648 conjugacy classes of subgroups.

The subgroup lattice has 379 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1993C211yesmaximal, characteristic
2#20830C211yesmaximal, characteristic
3#17746C211yesmaximal, characteristic
4#17742C211yesmaximal, characteristic
5#10636C214yesmaximal, complemented
6#10633C214yesmaximal, complemented
7#7959C211yesmaximal, characteristic
8#7959C211yesmaximal, characteristic
9#7919C211yesmaximal, characteristic
10C22×C4⋊Q8C2211yescharacteristic
11#2920C2211yescharacteristic
12#2920C2211yescharacteristic
13#2920C2211yescharacteristic
14#2920C2211yescharacteristic
15C22×#97C2211yescharacteristic
16C2×#382C2211yescharacteristic
17#1993—22no—
18#1993C2212yes—
19#1993C2212yes—
20#1849C2212yes—
21#1838C2212yes—
22#1810—22no—
23#1778—22no—
24C2×#340—22no—
25#1558—22no—
26#1556—22no—
27#1549C2214yes—
28#1548—22no—
29#1548C2214yes—
30#1546—22no—
31#1286C2214yes—
32#1286C2214yes—
33#1281C2212yes—
34#1281C2212yes—
35C23×Q8—21no—
36C23×C8D412yes—
37C2×C4⋊Q8—21no—
38C2×C4⋊Q8—21no—
39C2×C4⋊Q8—21no—
40C2×C4⋊Q8—21no—
41C2×C4⋊Q8—21no—
42C2×C4⋊Q8—21no—
43C2×C4⋊Q8—21no—
44C2×C4⋊Q8—21no—
45C2×C4⋊Q8—21no—
46C2×C4⋊Q8D412yes—
47C2×C4⋊Q8D412yes—
48C2×C4⋊Q8C2312yes—
49C22×C4⋊C4—21no—
50C22×C4⋊C4—21no—
51C22×C4⋊C4—21no—
52C22×C42C2311yescharacteristic
53#432—21no—
54#432—21no—
55#432—21no—
56#432—21no—
57#432—21no—
58#432—21no—
59#432—21no—
60#432—21no—
61#431—21no—
62#431—21no—
63#431—21no—
64#431—21no—
65#431—21no—
66#431—21no—
67#431C2312yes—
68#431C2312yes—
69C2×#108—22no—
70C2×#108—22no—
71#423—21no—
72#423—21no—
73#423C2312yes—
74#423C2312yes—
75#407—21no—
76#407—21no—
77#407—21no—
78#407—21no—
79C4×M16—21no—
80C4×M16—21no—
81C4×M16—21no—
82C4×M16—21no—
83C2×#97—21no—
84C2×#97—21no—
85C2×#97—21no—
86C2×#97—21no—
87C2×#97D412yes—
88C2×#97C2311yescharacteristic
89C2×#97C2311yescharacteristic
90#382—21no—
91#382—21no—
92#382—21no—
93#382C2312yes—
94C2×#95—21no—
95C2×#95—21no—
96#341—24no—
97#340—24no—
98#340—24no—
99#333—28no—
100#332—28no—
101C22×Q8—22no—
102C22×Q8—22no—
103C22×Q8—22no—
104C22×Q8—41no—
105C22×Q8—41no—
106C22×Q8—41no—
107C22×Q8—84no—
108C23×C4—41no—
109C23×C4—41no—
110C23×C4C2×D411yescharacteristic
111C23×C4#4111yescharacteristic
112C23×C4#4111yescharacteristic
113C2×M16—22no—
114C2×M16—22no—
115C2×M16—41no—
116C2×M16—41no—
117C22×C8—22no—
118C22×C8—22no—
119C22×C8—24no—
120C22×C8—42no—
121C22×C8C2×D412yes—
122C22×C8C2×D412yes—
123C4⋊Q8—22no—
124C4⋊Q8—22no—
125C4⋊Q8—22no—
126C4⋊Q8—22no—
127C4⋊Q8—22no—
128C4⋊Q8—22no—
129C4⋊Q8—22no—
130C4⋊Q8—22no—
131C4⋊Q8—22no—
132C4⋊Q8—41no—
133C4⋊Q8—41no—
134C4⋊Q8—41no—
135C4⋊Q8—41no—
136C4⋊Q8—41no—
137C4⋊Q8—41no—
138C4⋊Q8C2×D414yes—
139C2×C4⋊C4—22no—
140C2×C4⋊C4—22no—
141C2×C4⋊C4—22no—
142C2×C4⋊C4—22no—
143C2×C4⋊C4—22no—
144C2×C4⋊C4—22no—
145C2×C4⋊C4—22no—
146C2×C4⋊C4—41no—
147C2×C4⋊C4—41no—
148C2×C4⋊C4—41no—
149C2×C4⋊C4—41no—
150C2×C4⋊C4—41no—
151C2×C4⋊C4—41no—
152C2×C4⋊C4—44no—
153C2×C4⋊C4—41no—
154C2×C42C2×D411yescharacteristic
155C2×C42C2×D411yescharacteristic
156C2×C42C2×D411yescharacteristic
157C2×C42#4111yescharacteristic
158C2×C42#4111yescharacteristic
159C2×C42#4111yescharacteristic
160C2×C42#4111yescharacteristic
161C2×C42#4111yescharacteristic
162C2×C42#4111yescharacteristic
163C2×C42#4111yescharacteristic
164C2×C42#4111yescharacteristic
165C2×C42C2411yescharacteristic
166#108—44no—
167#108—44no—
168C4⋊C8—22no—
169C4⋊C8—22no—
170C4⋊C8—22no—
171C4⋊C8—22no—
172C4⋊C8—22no—
173C4⋊C8—22no—
174C4⋊C8—41no—
175C4⋊C8—41no—
176C22⋊C8—22no—
177C22⋊C8—22no—
178C22⋊C8—41no—
179C22⋊C8—41no—
180C22⋊C8—41no—
181C22⋊C8—41no—
182C22⋊C8—41no—
183C22⋊C8—41no—
184#97—22no—
185#97—22no—
186#97—21no—
187#97—21no—
188#97—21no—
189#97—21no—
190#97—41no—
191#97—41no—
192#97—41no—
193#97—41no—
194C4×C8—22no—
195C4×C8—22no—
196C4×C8—41no—
197C4×C8—41no—
198#95—22no—
199#95—22no—
200#95—41no—
201#95—41no—
202C24#12411yescharacteristic
203C2×Q8—24no—
204C2×Q8—42no—
205C2×Q8—42no—
206C2×Q8—42no—
207C2×Q8—84no—
208C2×Q8—84no—
209C2×Q8—84no—
210C2×Q8—84no—
211C2×Q8—84no—
212C2×Q8—84no—
213C22×C4—22no—
214C22×C4—22no—
215C22×C4—42no—
216C22×C4—42no—
217C22×C4—42no—
218C22×C4—42no—
219C22×C4—42no—
220C22×C4—42no—
221C22×C4—84no—
222C22×C4—82no—
223C22×C4—82no—
224C22×C4—82no—
225C22×C4#12111yescharacteristic
226C22×C4#12111yescharacteristic
227C22×C4#12311yescharacteristic
228C22×C4#12311yescharacteristic
229C22×C4#12311yescharacteristic
230C22×C4#12311yescharacteristic
231C22×C4#12311yescharacteristic
232C22×C4#12311yescharacteristic
233C22×C4#12411yescharacteristic
234C22×C4#12411yescharacteristic
235C22×C4#12411yescharacteristic
236C22×C4#12611yescharacteristic
237C22×C4#12611yescharacteristic
238C22×C4#12611yescharacteristic
239C22×C4#12611yescharacteristic
240C22×C4C22×D411yescharacteristic
241C22×C4C2×#4111yescharacteristic
242C22×C4C2×#4111yescharacteristic
243M16—42no—
244M16—42no—
245M16—42no—
246M16—42no—
247M16—82no—
248M16—82no—
249C2×C8—22no—
250C2×C8—22no—
251C2×C8—22no—
252C2×C8—22no—
253C2×C8—22no—
254C2×C8—44no—
255C2×C8—44no—
256C2×C8—42no—
257C2×C8—42no—
258C2×C8—42no—
259C2×C8—42no—
260C2×C8#13714yes—
261C4⋊C4—24no—
262C4⋊C4—24no—
263C4⋊C4—42no—
264C4⋊C4—42no—
265C4⋊C4—42no—
266C4⋊C4—42no—
267C4⋊C4—42no—
268C4⋊C4—42no—
269C4⋊C4—44no—
270C4⋊C4—82no—
271C42#12111yescharacteristic
272C42#12111yescharacteristic
273C42#12311yescharacteristic
274C42#12311yescharacteristic
275C42#12311yescharacteristic
276C42#12311yescharacteristic
277C42#12311yescharacteristic
278C42#12311yescharacteristic
279C42#12611yescharacteristic
280C42#12611yescharacteristic
281C42#12611yescharacteristic
282C42#12611yescharacteristic
283C42C22×D411yescharacteristic
284C42C2×#4111yescharacteristic
285C42C2×#4111yescharacteristic
286C42C2×#4111yescharacteristic
287C23—22no—
288C23—22no—
289C23—21no—
290C23—21no—
291C23#39711yescharacteristic
292C23#39711yescharacteristic
293C23C2×#12411yescharacteristic
294Q8—84no—
295Q8—84no—
296Q8—84no—
297Q8—84no—
298C2×C4—22no—
299C2×C4—22no—
300C2×C4—22no—
301C2×C4—22no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—22no—
305C2×C4—22no—
306C2×C4—22no—
307C2×C4—22no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—44no—
317C2×C4—84no—
318C2×C4—84no—
319C2×C4—84no—
320C2×C4—82no—
321C2×C4—82no—
322C2×C4—82no—
323C2×C4—164no—
324C2×C4#39711yescharacteristic
325C2×C4#39711yescharacteristic
326C2×C4#39711yescharacteristic
327C2×C4#39711yescharacteristic
328C2×C4#39711yescharacteristic
329C2×C4#39711yescharacteristic
330C2×C4C2×#12111yescharacteristic
331C2×C4C2×#12311yescharacteristic
332C2×C4C2×#12311yescharacteristic
333C2×C4C2×#12311yescharacteristic
334C2×C4C2×#12611yescharacteristic
335C2×C4C2×#12611yescharacteristic
336C2×C4C2×#13712yes—
337C2×C4#57512yes—
338C8—24no—
339C8—42no—
340C8—82no—
341C8—82no—
342C8—82no—
343C8—82no—
344C22—24no—
345C22—24no—
346C22—21no—
347C22—21no—
348C22—21no—
349C22—21no—
350C22—21no—
351C22—21no—
352C22#208112yes—
353C22#208212yes—
354C22C2×#39711yescharacteristic
355C22#313911yescharacteristic
356C22#314011yescharacteristic
357C4—21no—
358C4—21no—
359C4—21no—
360C4—21no—
361C4—21no—
362C4—21no—
363C4—21no—
364C4—21no—
365C4—21no—
366C4—21no—
367C4—84no—
368C4—162no—
369C4#303712yes—
370C4#309512yes—
371C2—21no—
372C2—22no—
373C2—22no—
374C2—21no—
375C2#2362911yescharacteristic
376C2C2×#208111yescharacteristic
377C2C2×#208211yescharacteristic
subgroupG/H
center, socleC22C2×#397
commutator, Frattini, cosocleC2×C42C24
prime radicalC24#124
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
5. #10636C2semidirect
6. #10633C2semidirect

Maximal subgroups

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

subgroupindex
C2×#19932
#208302
#177462
#177422
#106362
#106332
#79592
#79192