Finite Group Databasebeta

← #110909

Subgroups — #110909

Order 512.

603 conjugacy classes of subgroups.

The subgroup lattice has 391 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20479C211yesmaximal, characteristic
2#17748C211yesmaximal, characteristic
3C2×#1531C211yesmaximal, characteristic
4#17545C211yesmaximal, characteristic
5#8088C211yesmaximal, characteristic
6#8071C211yesmaximal, characteristic
7#7995C211yesmaximal, characteristic
8C2×#533C2211yescharacteristic
9#2920C2211yescharacteristic
10#2920C2211yescharacteristic
11C2×#432C2211yescharacteristic
12C2×#432C2211yescharacteristic
13C22×#97C2211yescharacteristic
14C4×#117C2211yescharacteristic
15#1771—22no—
16#1752—21no—
17C8×C22⋊C4—21no—
18#1745—21no—
19C4×#97—21no—
20#1558C412yescomplemented
21#1556C412yescomplemented
22#1554—22no—
23#1550—22no—
24#1531—21no—
25#1531—21no—
26#1531C414yescomplemented
27C2×#330—22no—
28#1508—21no—
29#1508—21no—
30#1506—22no—
31#1503C414yes—
32#1302—22no—
33#1300—22no—
34#1291—22no—
35C23×C8—21no—
36#533—21no—
37#533—21no—
38#533—21no—
39#533—21no—
40#533—21no—
41#533—21no—
42#533C2×C412yes—
43#533C2×C412yes—
44C2×C4⋊Q8C2×C411yescharacteristic
45C2×C4⋊Q8C2×C411yescharacteristic
46C2×#125—21no—
47C2×C22⋊Q8—21no—
48C2×C22⋊Q8—21no—
49C2×C4×Q8—21no—
50C2×C4×Q8—21no—
51C2×#117—21no—
52C2×#117—21no—
53C22×C42C2311yescharacteristic
54#432—21no—
55#432—21no—
56#432—21no—
57#432—21no—
58#432—21no—
59#432—21no—
60#432—21no—
61#432—21no—
62#432C2×C412yes—
63#432C2×C412yes—
64#432C2×C412yes—
65#432D412yes—
66#431—21no—
67#431—21no—
68#431—21no—
69#431C2×C412yes—
70#423—21no—
71#423—21no—
72C2×C4⋊C8—21no—
73C2×C4⋊C8D412yes—
74#407—21no—
75#407—21no—
76C2×C22⋊C8—21no—
77C2×C22⋊C8—21no—
78C4×M16—21no—
79C4×M16—21no—
80C2×#97—21no—
81C2×#97—21no—
82C2×#97—21no—
83C2×#97—21no—
84C2×#97—21no—
85C2×#97—22no—
86C2×#97C2×C412yes—
87C2×#97D412yes—
88C2×#97D411yescharacteristic
89C2×#97D411yescharacteristic
90C2×C4×C8—22no—
91C4×C4⋊C4—21no—
92C4×C4⋊C4C2×C412yes—
93C4×C22⋊C4—21no—
94C4×C22⋊C4D411yescharacteristic
95C4×C22⋊C4D411yescharacteristic
96#376D412yes—
97C43—21no—
98#336—22no—
99#336—22no—
100#333—24no—
101#332—24no—
102#330—24no—
103#330—24no—
104#328—24no—
105#328—24no—
106C22×Q8—21no—
107C22×Q8—21no—
108C23×C4C2×D411yescharacteristic
109C23×C4#4111yescharacteristic
110C23×C4#4111yescharacteristic
111C2×M16—41no—
112C2×M16—41no—
113C22×C8—21no—
114C22×C8—22no—
115C22×C8—21no—
116C22×C8—22no—
117C22×C8—22no—
118C22×C8—44no—
119C22×C8—41no—
120C4⋊Q8—21no—
121C4⋊Q8—21no—
122C4⋊Q8—21no—
123C4⋊Q8—21no—
124C4⋊Q8—21no—
125C4⋊Q8—21no—
126C4⋊Q8C4212yes—
127C4⋊Q8C4212yes—
128#125—22no—
129#125—22no—
130#125—41no—
131#125—41no—
132C22⋊Q8—22no—
133C22⋊Q8—22no—
134C22⋊Q8—22no—
135C22⋊Q8—22no—
136C22⋊Q8—41no—
137C22⋊Q8—41no—
138C22⋊Q8—41no—
139C22⋊Q8—41no—
140C4×Q8—22no—
141C4×Q8—22no—
142C4×Q8—22no—
143C4×Q8—22no—
144C4×Q8—41no—
145C4×Q8—41no—
146C4×Q8—41no—
147C4×Q8—41no—
148#117—22no—
149#117—22no—
150#117—44no—
151#117—41no—
152#117—41no—
153C2×C4⋊C4—21no—
154C2×C4⋊C4—21no—
155C2×C4⋊C4—21no—
156C2×C4⋊C4—21no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—41no—
159C2×C4⋊C4—41no—
160C2×C22⋊C4—21no—
161C2×C22⋊C4—21no—
162C2×C22⋊C4—41no—
163C2×C42—21no—
164C2×C42—21no—
165C2×C42—22no—
166C2×C42—22no—
167C2×C42—41no—
168C2×C42C22×C411yescharacteristic
169C2×C42C22×C411yescharacteristic
170C2×C42C22×C411yescharacteristic
171C2×C42C2×D411yescharacteristic
172C2×C42C2×D411yescharacteristic
173C2×C42C2×D411yescharacteristic
174C2×C42C2×D411yescharacteristic
175C2×C42C2×D411yescharacteristic
176C2×C42#4111yescharacteristic
177C2×C42#4111yescharacteristic
178C2×C42#4111yescharacteristic
179C2×C42#4111yescharacteristic
180C4⋊C8—22no—
181C4⋊C8—22no—
182C4⋊C8—22no—
183C4⋊C8—22no—
184C4⋊C8—41no—
185C4⋊C8—41no—
186C4⋊C8—41no—
187C4⋊C8C22⋊C414yes—
188C22⋊C8—22no—
189C22⋊C8—22no—
190C22⋊C8—22no—
191C22⋊C8—22no—
192C22⋊C8—22no—
193C22⋊C8—41no—
194C22⋊C8—41no—
195C22⋊C8—41no—
196#97—22no—
197#97—22no—
198#97—22no—
199#97—22no—
200#97—22no—
201#97—22no—
202#97—21no—
203#97—21no—
204#97—44no—
205#97—41no—
206#97—41no—
207#97C22⋊C412yes—
208#97C22⋊C412yes—
209C4×C8—22no—
210C4×C8—44no—
211C4×C8—41no—
212#95—22no—
213#95C22⋊C412yes—
214#95C22⋊C412yes—
215C24#12311yescharacteristic
216C2×Q8—22no—
217C2×Q8—22no—
218C2×Q8—41no—
219C2×Q8—41no—
220C2×Q8—82no—
221C2×Q8—82no—
222C22×C4—22no—
223C22×C4—22no—
224C22×C4—42no—
225C22×C4—42no—
226C22×C4—42no—
227C22×C4—41no—
228C22×C4—41no—
229C22×C4C2×C22⋊C411yescharacteristic
230C22×C4#11711yescharacteristic
231C22×C4#11711yescharacteristic
232C22×C4C4×D411yescharacteristic
233C22×C4C4×D411yescharacteristic
234C22×C4C4×D411yescharacteristic
235C22×C4C4×D411yescharacteristic
236C22×C4C4×D411yescharacteristic
237C22×C4C4×D411yescharacteristic
238C22×C4#12011yescharacteristic
239C22×C4#12111yescharacteristic
240C22×C4#12111yescharacteristic
241C22×C4#12111yescharacteristic
242C22×C4#12111yescharacteristic
243C22×C4#12411yescharacteristic
244C22×C4#12411yescharacteristic
245C22×C4#12611yescharacteristic
246C22×C4#12611yescharacteristic
247M16—82no—
248M16—82no—
249C2×C8—24no—
250C2×C8—22no—
251C2×C8—22no—
252C2×C8—24no—
253C2×C8—44no—
254C2×C8—44no—
255C2×C8—42no—
256C2×C8—42no—
257C2×C8—42no—
258C2×C8—42no—
259C2×C8—42no—
260C2×C8—42no—
261C2×C8—82no—
262C4⋊C4—22no—
263C4⋊C4—22no—
264C4⋊C4—22no—
265C4⋊C4—22no—
266C4⋊C4—44no—
267C4⋊C4—41no—
268C4⋊C4—42no—
269C4⋊C4—41no—
270C4⋊C4—42no—
271C4⋊C4—41no—
272C4⋊C4—41no—
273C4⋊C4—42no—
274C4⋊C4—42no—
275C22⋊C4—24no—
276C22⋊C4—42no—
277C22⋊C4—42no—
278C22⋊C4—42no—
279C42—42no—
280C42—42no—
281C42—44no—
282C42—42no—
283C42—42no—
284C42—82no—
285C42C2×C4211yescharacteristic
286C42C2×C22⋊C411yescharacteristic
287C42C2×C22⋊C411yescharacteristic
288C42#11711yescharacteristic
289C42C4×D411yescharacteristic
290C42C4×D411yescharacteristic
291C42C4×D411yescharacteristic
292C42C4×D411yescharacteristic
293C42C4×D411yescharacteristic
294C42C4×D411yescharacteristic
295C42#12111yescharacteristic
296C42#12111yescharacteristic
297C42#12311yescharacteristic
298C42#12311yescharacteristic
299C42#12411yescharacteristic
300C42#12711yescharacteristic
301C23—22no—
302C23—22no—
303C23—41no—
304C23#38611yescharacteristic
305C23#38811yescharacteristic
306C23#38811yescharacteristic
307Q8—82no—
308Q8—82no—
309Q8—82no—
310Q8—82no—
311C2×C4—22no—
312C2×C4—22no—
313C2×C4—22no—
314C2×C4—22no—
315C2×C4—22no—
316C2×C4—22no—
317C2×C4—22no—
318C2×C4—24no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—41no—
327C2×C4—41no—
328C2×C4—41no—
329C2×C4—41no—
330C2×C4—42no—
331C2×C4—42no—
332C2×C4—42no—
333C2×C4—42no—
334C2×C4—42no—
335C2×C4—42no—
336C2×C4—42no—
337C2×C4—82no—
338C2×C4—82no—
339C2×C4C4×C22⋊C411yescharacteristic
340C2×C4C4×C22⋊C411yescharacteristic
341C2×C4C4×C22⋊C411yescharacteristic
342C2×C4#38611yescharacteristic
343C2×C4#38611yescharacteristic
344C2×C4#38811yescharacteristic
345C2×C4#38811yescharacteristic
346C2×C4#38811yescharacteristic
347C2×C4#38811yescharacteristic
348C2×C4#39011yescharacteristic
349C2×C4#39011yescharacteristic
350C2×C4#39011yescharacteristic
351C8—24no—
352C8—42no—
353C8—44no—
354C8—82no—
355C8—82no—
356C8—82no—
357C22—21no—
358C22—21no—
359C22—21no—
360C22—21no—
361C22—21no—
362C22—21no—
363C22—21no—
364C22—44no—
365C22#143711yescharacteristic
366C22#190212yes—
367C22#193212yes—
368C4—21no—
369C4—21no—
370C4—21no—
371C4—21no—
372C4—21no—
373C4—21no—
374C4—21no—
375C4—21no—
376C4—21no—
377C4—21no—
378C4—21no—
379C4—44no—
380C4—82no—
381C4—82no—
382C4—82no—
383C4#190612yes—
384C2—22no—
385C2—22no—
386C2—41no—
387C2#1094511yescharacteristic
388C2#1094611yescharacteristic
389C2#1095211yescharacteristic
subgroupG/H
center, socleC22#1437
commutatorC42C2×C42
Frattini, cosocleC22×C42C23
prime radicalC24#123
LCSG ▷ 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
20. #1558C4semidirect
21. #1556C4semidirect
26. #1531C4semidirect

Maximal subgroups

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

subgroupindex
#204792
#177482
C2×#15312
#175452
#80882
#80712
#79952