Finite Group Databasebeta

← #117382

Subgroups — #117382

Order 512.

557 conjugacy classes of subgroups.

The subgroup lattice has 370 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21844C211yesmaximal, characteristic
2#21366C211yesmaximal, characteristic
3#17748C211yesmaximal, characteristic
4#17596C211yesmaximal, characteristic
5#8113C211yesmaximal, characteristic
6#8085C211yesmaximal, characteristic
7#8071C211yesmaximal, characteristic
8C2×#533C2211yescharacteristic
9#2920C2211yescharacteristic
10#2920C2211yescharacteristic
11C2×#432C2211yescharacteristic
12C2×#432C2211yescharacteristic
13#2283C2211yescharacteristic
14C4×#117C2211yescharacteristic
15#1994—22no—
16#1986—22no—
17#1949—22no—
18#1854—22no—
19#1826—22no—
20#1776—22no—
21#1558C412yescomplemented
22#1556C412yescomplemented
23#1554—22no—
24#1550—22no—
25#1520—22no—
26#1515—22no—
27#1506—22no—
28#1306—22no—
29#1302—22no—
30#1300—22no—
31#533—21no—
32#533—21no—
33#533—21no—
34#533—21no—
35#533—21no—
36#533—21no—
37#533D412yes—
38#533Q812yes—
39C2×C4⋊Q8C2×C411yescharacteristic
40C2×C4⋊Q8C2×C411yescharacteristic
41C2×#125—21no—
42C2×C22⋊Q8—21no—
43C2×C22⋊Q8—21no—
44C2×C4×Q8—21no—
45C2×C4×Q8—21no—
46C2×#117—21no—
47C2×#117—21no—
48C22×C42C2311yescharacteristic
49#432—21no—
50#432—21no—
51#432—21no—
52#432—21no—
53#432—21no—
54#432—21no—
55#432—21no—
56#432—21no—
57#432C2×C412yes—
58#432C2×C412yes—
59#432D412yes—
60#432D412yes—
61#431—21no—
62#431—21no—
63#431—21no—
64#431—21no—
65C2×#107—22no—
66C2×#106—22no—
67#423—21no—
68#423—21no—
69C2×C4⋊C8—21no—
70C2×C4⋊C8D412yes—
71#407—21no—
72#407D412yes—
73C2×C22⋊C8—21no—
74C2×C22⋊C8—21no—
75C4×M16—21no—
76C4×M16—21no—
77C2×#97—21no—
78C2×#97—21no—
79#383—21no—
80#382—21no—
81#381—21no—
82#380D412yes—
83C4×C4⋊C4—21no—
84C4×C4⋊C4—21no—
85C4×C4⋊C4D412yes—
86C4×C22⋊C4—21no—
87C4×C22⋊C4—21no—
88C4×C22⋊C4—21no—
89#376—21no—
90#376—21no—
91C43—21no—
92#340—24no—
93#339—28no—
94#333—24no—
95#332—24no—
96#328—28no—
97C22×Q8—21no—
98C22×Q8—21no—
99C23×C4C2×D411yescharacteristic
100C23×C4C2×D411yescharacteristic
101C23×C4#4111yescharacteristic
102C2×M16—22no—
103C2×M16—41no—
104C22×C8—22no—
105C22×C8—41no—
106C4⋊Q8—21no—
107C4⋊Q8—21no—
108C4⋊Q8—21no—
109C4⋊Q8—21no—
110C4⋊Q8—21no—
111C4⋊Q8—21no—
112C4⋊Q8C4⋊C412yes—
113C4⋊Q8C4⋊C412yes—
114#125—22no—
115#125—22no—
116#125—41no—
117#125—41no—
118C22⋊Q8—22no—
119C22⋊Q8—22no—
120C22⋊Q8—22no—
121C22⋊Q8—22no—
122C22⋊Q8—41no—
123C22⋊Q8—41no—
124C22⋊Q8—41no—
125C22⋊Q8—41no—
126C4×Q8—22no—
127C4×Q8—22no—
128C4×Q8—22no—
129C4×Q8—22no—
130C4×Q8—41no—
131C4×Q8—41no—
132C4×Q8—41no—
133C4×Q8—41no—
134#117—22no—
135#117—22no—
136#117—44no—
137#117—41no—
138#117—41no—
139C2×C4⋊C4—21no—
140C2×C4⋊C4—21no—
141C2×C4⋊C4—21no—
142C2×C4⋊C4—21no—
143C2×C4⋊C4—22no—
144C2×C4⋊C4—22no—
145C2×C4⋊C4—41no—
146C2×C4⋊C4—41no—
147C2×C22⋊C4—22no—
148C2×C22⋊C4—41no—
149C2×C22⋊C4—41no—
150C2×C42—22no—
151C2×C42—41no—
152C2×C42—41no—
153C2×C42—41no—
154C2×C42—41no—
155C2×C42C22×C411yescharacteristic
156C2×C42C2×D411yescharacteristic
157C2×C42C2×D411yescharacteristic
158C2×C42C2×D411yescharacteristic
159C2×C42C2×D411yescharacteristic
160C2×C42C2×D411yescharacteristic
161C2×C42C2×Q811yescharacteristic
162C2×C42#4111yescharacteristic
163C2×C42#4111yescharacteristic
164C2×C42#4111yescharacteristic
165C2×C42#4111yescharacteristic
166C2×C42#4111yescharacteristic
167#107—44no—
168#106—44no—
169C4⋊C8—22no—
170C4⋊C8—22no—
171C4⋊C8—22no—
172C4⋊C8—41no—
173C4⋊C8—41no—
174C4⋊C8—41no—
175C4⋊C8—41no—
176C4⋊C8C22⋊C414yes—
177C22⋊C8—22no—
178C22⋊C8—22no—
179C22⋊C8—22no—
180C22⋊C8—22no—
181C22⋊C8—22no—
182C22⋊C8—22no—
183C22⋊C8—41no—
184C22⋊C8—41no—
185#97—22no—
186#97—22no—
187#97—22no—
188#97—22no—
189#97—41no—
190#97—41no—
191C4×C8—41no—
192C4×C8—41no—
193#95—22no—
194#95—22no—
195#95—41no—
196#95—41no—
197#95—41no—
198#95—41no—
199C24#12111yescharacteristic
200C2×Q8—22no—
201C2×Q8—22no—
202C2×Q8—41no—
203C2×Q8—41no—
204C2×Q8—82no—
205C2×Q8—82no—
206C22×C4—42no—
207C22×C4—42no—
208C22×C4—41no—
209C22×C4—41no—
210C22×C4—42no—
211C22×C4—42no—
212C22×C4—81no—
213C22×C4#11711yescharacteristic
214C22×C4C4×D411yescharacteristic
215C22×C4C4×D411yescharacteristic
216C22×C4#12011yescharacteristic
217C22×C4#12011yescharacteristic
218C22×C4#12011yescharacteristic
219C22×C4#12111yescharacteristic
220C22×C4#12111yescharacteristic
221C22×C4#12111yescharacteristic
222C22×C4#12111yescharacteristic
223C22×C4C22⋊Q811yescharacteristic
224C22×C4C22⋊Q811yescharacteristic
225C22×C4C22⋊Q811yescharacteristic
226C22×C4#12311yescharacteristic
227C22×C4#12411yescharacteristic
228C22×C4#12611yescharacteristic
229C22×C4#12611yescharacteristic
230C22×C4#12711yescharacteristic
231M16—44no—
232M16—82no—
233C2×C8—24no—
234C2×C8—42no—
235C2×C8—42no—
236C2×C8—42no—
237C2×C8—42no—
238C2×C8—42no—
239C2×C8—82no—
240C4⋊C4—24no—
241C4⋊C4—22no—
242C4⋊C4—22no—
243C4⋊C4—22no—
244C4⋊C4—22no—
245C4⋊C4—42no—
246C4⋊C4—42no—
247C4⋊C4—42no—
248C4⋊C4—41no—
249C4⋊C4—41no—
250C4⋊C4—41no—
251C4⋊C4—41no—
252C4⋊C4—42no—
253C4⋊C4—42no—
254C4⋊C4—42no—
255C4⋊C4—42no—
256C22⋊C4—44no—
257C22⋊C4—42no—
258C22⋊C4—42no—
259C22⋊C4—82no—
260C42—44no—
261C42—44no—
262C42—42no—
263C42—42no—
264C42—82no—
265C42—82no—
266C42C2×C22⋊C411yescharacteristic
267C42C2×C4⋊C411yescharacteristic
268C42C4×D411yescharacteristic
269C42C4×D411yescharacteristic
270C42#12111yescharacteristic
271C42#12111yescharacteristic
272C42#12111yescharacteristic
273C42C22⋊Q811yescharacteristic
274C42C22⋊Q811yescharacteristic
275C42C22⋊Q811yescharacteristic
276C42#12311yescharacteristic
277C42#12311yescharacteristic
278C42#12311yescharacteristic
279C42#12311yescharacteristic
280C42#12411yescharacteristic
281C42#12411yescharacteristic
282C23—22no—
283C23—41no—
284C23—41no—
285C23#38811yescharacteristic
286C23#39211yescharacteristic
287C23#39611yescharacteristic
288Q8—82no—
289Q8—82no—
290Q8—82no—
291Q8—82no—
292C2×C4—22no—
293C2×C4—22no—
294C2×C4—22no—
295C2×C4—22no—
296C2×C4—22no—
297C2×C4—41no—
298C2×C4—41no—
299C2×C4—41no—
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—44no—
311C2×C4—42no—
312C2×C4—42no—
313C2×C4—82no—
314C2×C4—82no—
315C2×C4—82no—
316C2×C4—82no—
317C2×C4—82no—
318C2×C4—82no—
319C2×C4—82no—
320C2×C4#38011yescharacteristic
321C2×C4#38511yescharacteristic
322C2×C4#38511yescharacteristic
323C2×C4#38611yescharacteristic
324C2×C4#38811yescharacteristic
325C2×C4#39011yescharacteristic
326C2×C4#39311yescharacteristic
327C2×C4#39411yescharacteristic
328C2×C4#39411yescharacteristic
329C2×C4#39611yescharacteristic
330C2×C4#39711yescharacteristic
331C2×C4#39911yescharacteristic
332C8—44no—
333C8—82no—
334C8—82no—
335C8—82no—
336C22—21no—
337C22—21no—
338C22—21no—
339C22—21no—
340C22—21no—
341C22—21no—
342C22—21no—
343C22—21no—
344C22—82no—
345C22#143911yescharacteristic
346C22#200912yes—
347C4—21no—
348C4—21no—
349C4—21no—
350C4—21no—
351C4—21no—
352C4—21no—
353C4—21no—
354C4—21no—
355C4—21no—
356C4—21no—
357C4—21no—
358C4—84no—
359C4—82no—
360C4—82no—
361C4—162no—
362C4#190612yes—
363C2—22no—
364C2—41no—
365C2—41no—
366C2#1106011yescharacteristic
367C2#1113011yescharacteristic
368C2#1114111yescharacteristic
subgroupG/H
center, socleC22#1439
commutatorC2×C42C22×C4
Frattini, cosocleC22×C42C23
prime radicalC24#121
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
21. #1558C4semidirect
22. #1556C4semidirect

Maximal subgroups

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

subgroupindex
#218442
#213662
#177482
#175962
#81132
#80852
#80712