Finite Group Databasebeta

← #107458

Subgroups — #107458

Order 512.

712 conjugacy classes of subgroups.

The subgroup lattice has 384 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2987C211yesmaximal, characteristic
2#21866C211yesmaximal, characteristic
3C2×#1554C211yesmaximal, characteristic
4C2×#1308C211yesmaximal, characteristic
5#8030C211yesmaximal, characteristic
6C2×#1287C211yesmaximal, characteristic
7#7914C211yesmaximal, characteristic
8C2×#533C2211yescharacteristic
9#2987—22no—
10#2987—22no—
11#2987—22no—
12#2987C414yescomplemented
13C2×#446—22no—
14C2×#446—22no—
15C2×#433—22no—
16C2×#432C2211yescharacteristic
17C2×#432C2211yescharacteristic
18C2×#432C2211yescharacteristic
19C2×#432C2211yescharacteristic
20C2×#431C2211yescharacteristic
21C2×#382C2211yescharacteristic
22#1993—21no—
23#1960—21no—
24#1872C414yes—
25#1805—22no—
26#1554—22no—
27#1554C414yescomplemented
28C2×#333—21no—
29C2×#332—21no—
30#1308—22no—
31#1295—22no—
32#1287—22no—
33#1280—22no—
34#533—21no—
35#533—21no—
36#533—21no—
37#533—21no—
38#533C2×C412yes—
39#533C2×C412yes—
40#533D412yes—
41#533Q812yes—
42C2×C4⋊Q8D411yescharacteristic
43C2×C4⋊Q8D411yescharacteristic
44C2×#125—21no—
45C2×C22⋊Q8—21no—
46C2×C22⋊Q8C2×C412yes—
47C2×C4×Q8—21no—
48C2×C4×Q8D412yes—
49C2×#117—21no—
50C22×C4⋊C4C2×C412yes—
51C22×C42C2311yescharacteristic
52#446—28no—
53#446—28no—
54#446—28no—
55#446—28no—
56#433—28no—
57#433—28no—
58#432—21no—
59#432—21no—
60#432—21no—
61#432—21no—
62#432—21no—
63#432—21no—
64#432—21no—
65#432—21no—
66#432—21no—
67#432—21no—
68#432—21no—
69#432C2×C412yes—
70#432C2×C412yes—
71#432C2×C412yes—
72#432D412yes—
73#432D412yes—
74#431—21no—
75#431—21no—
76#431—21no—
77#431C2×C412yes—
78C2×C4⋊C8—24no—
79C2×C4⋊C8—21no—
80C2×C4⋊C8—21no—
81C2×C4⋊C8—21no—
82C2×C4⋊C8—21no—
83C2×C4⋊C8—21no—
84C2×#103—24no—
85C2×C22⋊C8—21no—
86C2×C22⋊C8—21no—
87C2×C22⋊C8—21no—
88C2×C22⋊C8C2×C412yes—
89C2×C22⋊C8D412yes—
90C2×#97—22no—
91C2×#97—21no—
92C2×#97—21no—
93C2×#97—21no—
94C2×#97D412yes—
95C2×C4×C8—22no—
96C2×C4×C8—21no—
97#382—21no—
98#382—21no—
99#382—21no—
100#382—21no—
101C2×#95—21no—
102C2×#95—21no—
103#340—24no—
104#333—24no—
105#332—24no—
106C22×Q8—21no—
107C22×Q8C22⋊C412yes—
108C23×C4—22no—
109C23×C4C22×C411yescharacteristic
110C23×C4C2×D411yescharacteristic
111C23×C4#4111yescharacteristic
112C22×C8—22no—
113C22×C8—22no—
114C22×C8—22no—
115C22×C8—41no—
116C22×C8—41no—
117C4⋊Q8—21no—
118C4⋊Q8—21no—
119C4⋊Q8—21no—
120C4⋊Q8—21no—
121C4⋊Q8C22⋊C412yes—
122C4⋊Q8C22⋊C412yes—
123C4⋊Q8C4⋊C412yes—
124C4⋊Q8C4⋊C412yes—
125#125—22no—
126#125—22no—
127#125—22no—
128#125—22no—
129C22⋊Q8—22no—
130C22⋊Q8—22no—
131C22⋊Q8—22no—
132C22⋊Q8—22no—
133C22⋊Q8—22no—
134C22⋊Q8—22no—
135C22⋊Q8C4214yes—
136C22⋊Q8C4⋊C414yes—
137C4×Q8—22no—
138C4×Q8—22no—
139C4×Q8—22no—
140C4×Q8—22no—
141C4×Q8—22no—
142C4×Q8—22no—
143C4×Q8C22⋊C414yes—
144C4×Q8C4⋊C414yes—
145#117—22no—
146#117—22no—
147#117—22no—
148#117—22no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—21no—
151C2×C4⋊C4—21no—
152C2×C4⋊C4—21no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—82no—
157C2×C4⋊C4C22⋊C412yes—
158C2×C22⋊C4—22no—
159C2×C42—22no—
160C2×C42C22×C411yescharacteristic
161C2×C42C22×C411yescharacteristic
162C2×C42C2×D411yescharacteristic
163C2×C42C2×D411yescharacteristic
164C2×C42C2×D411yescharacteristic
165C2×C42C2×D411yescharacteristic
166C2×C42C2×D411yescharacteristic
167C2×C42C2×D411yescharacteristic
168C2×C42C2×Q811yescharacteristic
169C2×C42#4111yescharacteristic
170C2×C42#4111yescharacteristic
171C2×C42#4111yescharacteristic
172C4⋊C8—216no—
173C4⋊C8—22no—
174C4⋊C8—22no—
175C4⋊C8—22no—
176C4⋊C8—22no—
177C4⋊C8—22no—
178C4⋊C8—22no—
179C4⋊C8—41no—
180C4⋊C8—41no—
181C4⋊C8—41no—
182C4⋊C8—41no—
183#103—84no—
184C22⋊C8—22no—
185C22⋊C8—22no—
186C22⋊C8—22no—
187C22⋊C8—22no—
188C22⋊C8—22no—
189C22⋊C8—22no—
190C22⋊C8—22no—
191C22⋊C8—41no—
192C22⋊C8—41no—
193C22⋊C8C22⋊C414yes—
194#97—28no—
195#97—22no—
196#97—22no—
197#97—22no—
198#97—22no—
199#97—41no—
200#97—41no—
201#97—41no—
202#97C22⋊C414yes—
203C4×C8—28no—
204C4×C8—22no—
205C4×C8—41no—
206#95—41no—
207#95—41no—
208#95—41no—
209#95—41no—
210C24C4×D411yescharacteristic
211C2×Q8—22no—
212C2×Q8—22no—
213C2×Q8—22no—
214C2×Q8—42no—
215C2×Q8—42no—
216C2×Q8—82no—
217C2×Q8#9514yes—
218C22×C4—22no—
219C22×C4—22no—
220C22×C4—41no—
221C22×C4—41no—
222C22×C4—42no—
223C22×C4—42no—
224C22×C4—42no—
225C22×C4—44no—
226C22×C4—82no—
227C22×C4—82no—
228C22×C4C2×C4211yescharacteristic
229C22×C4C2×C22⋊C411yescharacteristic
230C22×C4C2×C22⋊C411yescharacteristic
231C22×C4C2×C4⋊C411yescharacteristic
232C22×C4#11711yescharacteristic
233C22×C4C4×D411yescharacteristic
234C22×C4C4×D411yescharacteristic
235C22×C4C4×D411yescharacteristic
236C22×C4C4×D411yescharacteristic
237C22×C4C4×D411yescharacteristic
238C22×C4#12011yescharacteristic
239C22×C4#12011yescharacteristic
240C22×C4#12111yescharacteristic
241C22×C4#12111yescharacteristic
242C22×C4C22⋊Q811yescharacteristic
243C22×C4C22⋊Q811yescharacteristic
244C22×C4#12311yescharacteristic
245C22×C4#12311yescharacteristic
246C2×C8—24no—
247C2×C8—24no—
248C2×C8—44no—
249C2×C8—44no—
250C2×C8—42no—
251C2×C8—42no—
252C2×C8—42no—
253C2×C8—42no—
254C2×C8—42no—
255C2×C8—82no—
256C2×C8—82no—
257C4⋊C4—24no—
258C4⋊C4—22no—
259C4⋊C4—22no—
260C4⋊C4—22no—
261C4⋊C4—24no—
262C4⋊C4—24no—
263C4⋊C4—24no—
264C4⋊C4—22no—
265C4⋊C4—22no—
266C4⋊C4—22no—
267C4⋊C4—22no—
268C4⋊C4—44no—
269C4⋊C4—82no—
270C4⋊C4#9514yes—
271C22⋊C4—24no—
272C22⋊C4—24no—
273C42—24no—
274C42—24no—
275C42C2×C22⋊C411yescharacteristic
276C42C2×C22⋊C411yescharacteristic
277C42C2×C22⋊C411yescharacteristic
278C42C2×C4⋊C411yescharacteristic
279C42C2×C4⋊C411yescharacteristic
280C42#11711yescharacteristic
281C42C4×D411yescharacteristic
282C42C4×D411yescharacteristic
283C42#12011yescharacteristic
284C42#12011yescharacteristic
285C42#12111yescharacteristic
286C42#12111yescharacteristic
287C42C22⋊Q811yescharacteristic
288C42C22⋊Q811yescharacteristic
289C42#12311yescharacteristic
290C42#12311yescharacteristic
291C23—22no—
292C23—22no—
293C23—21no—
294C23—21no—
295C23C4×C22⋊C411yescharacteristic
296C23#38511yescharacteristic
297C23#38611yescharacteristic
298Q8—44no—
299Q8—82no—
300Q8—82no—
301Q8—82no—
302C2×C4—22no—
303C2×C4—22no—
304C2×C4—22no—
305C2×C4—22no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
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—41no—
323C2×C4—41no—
324C2×C4—44no—
325C2×C4—42no—
326C2×C4—42no—
327C2×C4—82no—
328C2×C4—82no—
329C2×C4—82no—
330C2×C4—164no—
331C2×C4C2×#9511yescharacteristic
332C2×C4C4×C22⋊C411yescharacteristic
333C2×C4#37911yescharacteristic
334C2×C4#38011yescharacteristic
335C2×C4#38011yescharacteristic
336C2×C4#38111yescharacteristic
337C2×C4#38511yescharacteristic
338C2×C4#38511yescharacteristic
339C2×C4#38511yescharacteristic
340C2×C4#38611yescharacteristic
341C2×C4#38611yescharacteristic
342C2×C4#38611yescharacteristic
343C8—44no—
344C8—44no—
345C8—82no—
346C8—82no—
347C8—82no—
348C22—21no—
349C22—21no—
350C22—21no—
351C22—21no—
352C22—21no—
353C22—21no—
354C22—42no—
355C22—42no—
356C22C22⋊#9511yescharacteristic
357C22#188112yes—
358C22#190512yes—
359C22#191411yescharacteristic
360C22#191511yescharacteristic
361C4—21no—
362C4—21no—
363C4—21no—
364C4—21no—
365C4—21no—
366C4—21no—
367C4—21no—
368C4—21no—
369C4—21no—
370C4—21no—
371C4—21no—
372C4—21no—
373C4—44no—
374C4—44no—
375C4—162no—
376C2—41no—
377C2—41no—
378C2#933412yes—
379C2#933612yes—
380C2#1075511yescharacteristic
381C2#1078211yescharacteristic
382C2#1078611yescharacteristic
subgroupG/H
center, socleC23#386
commutatorC22×C4C2×C42
Frattini, cosocleC22×C42C23
prime radicalC24C4×D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
12. #2987C4semidirect
27. #1554C4semidirect

Maximal subgroups

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

subgroupindex
C2×#29872
#218662
C2×#15542
C2×#13082
#80302
C2×#12872
#79142