Finite Group Databasebeta

← #116690

Subgroups — #116690

Order 512.

519 conjugacy classes of subgroups.

The subgroup lattice has 354 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#17821C211yesmaximal, characteristic
2C2×#1537C211yesmaximal, characteristic
3#17584C211yesmaximal, characteristic
4#17479C211yesmaximal, characteristic
5#7951C211yesmaximal, characteristic
6#7928C211yesmaximal, characteristic
7#7863C211yesmaximal, characteristic
8C2×#533C2211yescharacteristic
9#2920C2211yescharacteristic
10#2920C2211yescharacteristic
11C2×#432C2211yescharacteristic
12C2×#432C2211yescharacteristic
13C2×#431C2211yescharacteristic
14#2283C2211yescharacteristic
15#1573—22no—
16#1572—22no—
17#1569—22no—
18#1537—22no—
19#1537—22no—
20C2×#330—22no—
21#1518—22no—
22#1515—22no—
23#1508—22no—
24#1497—22no—
25#1493C414yescomplemented
26#1479—22no—
27#1285—22no—
28#1282—22no—
29#1275—22no—
30#533—21no—
31#533—21no—
32#533—21no—
33#533—21no—
34#533—21no—
35#533—21no—
36#533C2×C412yes—
37#533D412yes—
38C2×C4⋊Q8Q811yescharacteristic
39C2×C4⋊Q8Q811yescharacteristic
40C2×#125—21no—
41C2×C22⋊Q8—21no—
42C2×C22⋊Q8—21no—
43C2×C4×Q8—21no—
44C2×C4×Q8—21no—
45C2×#117—21no—
46C22×C42C2311yescharacteristic
47#432—21no—
48#432—21no—
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#432D412yes—
58#432D412yes—
59#431—21no—
60#431—21no—
61#431—21no—
62#431—21no—
63#431—21no—
64#431—21no—
65#431D412yes—
66#431D412yes—
67#423—21no—
68#423C2×C412yes—
69C2×C4⋊C8—21no—
70C2×C4⋊C8—21no—
71C2×C4⋊C8—21no—
72#407—21no—
73#407—21no—
74C2×C22⋊C8—21no—
75C2×C22⋊C8—21no—
76C2×C22⋊C8—21no—
77C4×M16—21no—
78C4×M16—21no—
79C2×#97—21no—
80C2×#97D412yes—
81C2×C4×C8C2×C412yes—
82#383—21no—
83#382—21no—
84#381—21no—
85#380—21no—
86C4×C4⋊C4—21no—
87C4×C22⋊C4D412yes—
88#376—21no—
89#335—24no—
90#334—24no—
91#330—28no—
92#328—28no—
93Q8⋊C8—28no—
94C22×Q8—21no—
95C22×Q8—21no—
96C23×C4C2×D411yescharacteristic
97C23×C4#4111yescharacteristic
98C23×C4#4111yescharacteristic
99C2×M16—22no—
100C2×M16—41no—
101C22×C8—22no—
102C22×C8—22no—
103C22×C8—41no—
104C4⋊Q8—21no—
105C4⋊Q8—21no—
106C4⋊Q8—21no—
107C4⋊Q8—21no—
108C4⋊Q8—21no—
109C4⋊Q8—21no—
110C4⋊Q8C4⋊C412yes—
111C4⋊Q8C4⋊C412yes—
112#125—22no—
113#125—22no—
114#125—41no—
115#125—41no—
116C22⋊Q8—22no—
117C22⋊Q8—22no—
118C22⋊Q8—22no—
119C22⋊Q8—22no—
120C22⋊Q8—41no—
121C22⋊Q8—41no—
122C22⋊Q8—41no—
123C22⋊Q8—41no—
124C4×Q8—22no—
125C4×Q8—22no—
126C4×Q8—22no—
127C4×Q8—22no—
128C4×Q8—41no—
129C4×Q8—41no—
130C4×Q8—41no—
131C4×Q8—41no—
132#117—22no—
133#117—22no—
134#117—41no—
135#117—41no—
136C2×C4⋊C4—21no—
137C2×C4⋊C4—21no—
138C2×C4⋊C4—21no—
139C2×C4⋊C4—21no—
140C2×C4⋊C4—41no—
141C2×C4⋊C4—41no—
142C2×C4⋊C4—41no—
143C2×C22⋊C4—22no—
144C2×C22⋊C4—41no—
145C2×C42—22no—
146C2×C42—41no—
147C2×C42C22×C411yescharacteristic
148C2×C42C2×D411yescharacteristic
149C2×C42C2×D411yescharacteristic
150C2×C42C2×D411yescharacteristic
151C2×C42C2×D411yescharacteristic
152C2×C42C2×D411yescharacteristic
153C2×C42C2×D411yescharacteristic
154C2×C42C2×Q811yescharacteristic
155C2×C42#4111yescharacteristic
156C2×C42#4111yescharacteristic
157C2×C42#4111yescharacteristic
158C2×C42#4111yescharacteristic
159C4⋊C8—22no—
160C4⋊C8—22no—
161C4⋊C8—22no—
162C4⋊C8—41no—
163C4⋊C8—41no—
164C4⋊C8—41no—
165C4⋊C8—41no—
166C4⋊C8—41no—
167C4⋊C8—41no—
168C4⋊C8C22⋊C414yes—
169C22⋊C8—22no—
170C22⋊C8—22no—
171C22⋊C8—22no—
172C22⋊C8—22no—
173C22⋊C8—41no—
174C22⋊C8—41no—
175C22⋊C8—41no—
176C22⋊C8—41no—
177C22⋊C8—41no—
178C22⋊C8—41no—
179#97—22no—
180#97—22no—
181#97—22no—
182#97—22no—
183#97—41no—
184#97—41no—
185C4×C8—22no—
186C4×C8—22no—
187C4×C8—22no—
188C4×C8—22no—
189#95—22no—
190#95—41no—
191#95—41no—
192#95—41no—
193C24#12311yescharacteristic
194C2×Q8—22no—
195C2×Q8—22no—
196C2×Q8—44no—
197C2×Q8—41no—
198C2×Q8—41no—
199C2×Q8—82no—
200C22×C4—42no—
201C22×C4—42no—
202C22×C4—41no—
203C22×C4—41no—
204C22×C4—42no—
205C22×C4—81no—
206C22×C4#11711yescharacteristic
207C22×C4C4×D411yescharacteristic
208C22×C4C4×D411yescharacteristic
209C22×C4#12011yescharacteristic
210C22×C4#12111yescharacteristic
211C22×C4#12111yescharacteristic
212C22×C4#12111yescharacteristic
213C22×C4#12111yescharacteristic
214C22×C4C22⋊Q811yescharacteristic
215C22×C4C22⋊Q811yescharacteristic
216C22×C4C22⋊Q811yescharacteristic
217C22×C4#12311yescharacteristic
218C22×C4#12311yescharacteristic
219C22×C4#12311yescharacteristic
220C22×C4#12411yescharacteristic
221C22×C4#12411yescharacteristic
222C22×C4#12611yescharacteristic
223C22×C4#12711yescharacteristic
224M16—82no—
225M16—82no—
226C2×C8—24no—
227C2×C8—44no—
228C2×C8—42no—
229C2×C8—42no—
230C2×C8—42no—
231C2×C8—42no—
232C2×C8—82no—
233C2×C8—82no—
234C4⋊C4—22no—
235C4⋊C4—22no—
236C4⋊C4—22no—
237C4⋊C4—22no—
238C4⋊C4—42no—
239C4⋊C4—42no—
240C4⋊C4—42no—
241C4⋊C4—41no—
242C4⋊C4—42no—
243C4⋊C4—41no—
244C4⋊C4—41no—
245C4⋊C4—41no—
246C4⋊C4—82no—
247C22⋊C4—42no—
248C22⋊C4—42no—
249C22⋊C4—44no—
250C42—42no—
251C42—42no—
252C42—82no—
253C42C2×C22⋊C411yescharacteristic
254C42C2×C4⋊C411yescharacteristic
255C42C4×D411yescharacteristic
256C42C4×D411yescharacteristic
257C42#12011yescharacteristic
258C42#12011yescharacteristic
259C42#12111yescharacteristic
260C42#12111yescharacteristic
261C42#12111yescharacteristic
262C42#12111yescharacteristic
263C42C22⋊Q811yescharacteristic
264C42C22⋊Q811yescharacteristic
265C42C22⋊Q811yescharacteristic
266C42#12311yescharacteristic
267C42#12411yescharacteristic
268C42#12611yescharacteristic
269C23—22no—
270C23—41no—
271C23—41no—
272C23#38811yescharacteristic
273C23#39411yescharacteristic
274C23#39911yescharacteristic
275Q8—44no—
276Q8—82no—
277Q8—82no—
278Q8—82no—
279C2×C4—22no—
280C2×C4—22no—
281C2×C4—22no—
282C2×C4—22no—
283C2×C4—22no—
284C2×C4—41no—
285C2×C4—41no—
286C2×C4—41no—
287C2×C4—41no—
288C2×C4—41no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—41no—
295C2×C4—41no—
296C2×C4—41no—
297C2×C4—42no—
298C2×C4—42no—
299C2×C4—82no—
300C2×C4—82no—
301C2×C4—82no—
302C2×C4—82no—
303C2×C4—82no—
304C2×C4#38011yescharacteristic
305C2×C4#38511yescharacteristic
306C2×C4#38511yescharacteristic
307C2×C4#38611yescharacteristic
308C2×C4#38811yescharacteristic
309C2×C4#39011yescharacteristic
310C2×C4#39211yescharacteristic
311C2×C4#39311yescharacteristic
312C2×C4#39411yescharacteristic
313C2×C4#39611yescharacteristic
314C2×C4#39611yescharacteristic
315C2×C4#39711yescharacteristic
316C8—44no—
317C8—82no—
318C8—82no—
319C8—82no—
320C8—82no—
321C22—21no—
322C22—21no—
323C22—21no—
324C22—21no—
325C22—21no—
326C22—21no—
327C22—21no—
328C22—21no—
329C22—82no—
330C22#143911yescharacteristic
331C22#209812yes—
332C4—21no—
333C4—21no—
334C4—21no—
335C4—21no—
336C4—21no—
337C4—21no—
338C4—21no—
339C4—21no—
340C4—21no—
341C4—21no—
342C4—21no—
343C4—82no—
344C4—82no—
345C4—162no—
346C4#190212yes—
347C2—22no—
348C2—41no—
349C2—41no—
350C2#1105911yescharacteristic
351C2#1111211yescharacteristic
352C2#1111511yescharacteristic
subgroupG/H
center, socleC22#1439
commutatorC2×C42C22×C4
Frattini, cosocleC22×C42C23
prime radicalC24#123
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
25. #1493C4semidirect

Maximal subgroups

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

subgroupindex
#178212
C2×#15372
#175842
#174792
#79512
#79282
#78632