Finite Group Databasebeta

← #118794

Subgroups — #118794

Order 512.

508 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#17748C211yesmaximal, characteristic
2C2×#1531C211yesmaximal, characteristic
3#17584C211yesmaximal, characteristic
4#8069C211yesmaximal, characteristic
5#8056C211yesmaximal, characteristic
6#8036C211yesmaximal, characteristic
7#7919C211yesmaximal, characteristic
8C2×#533C2211yescharacteristic
9#2920C2211yescharacteristic
10#2920C2211yescharacteristic
11C2×#432C2211yescharacteristic
12C2×#432C2211yescharacteristic
13C22×#97C2211yescharacteristic
14#2283C2211yescharacteristic
15#1558C412yescomplemented
16#1556C412yescomplemented
17#1554—22no—
18#1550—22no—
19#1531—22no—
20#1531—22no—
21C2×#330—22no—
22#1518—22no—
23#1515—22no—
24#1508—22no—
25#1300—22no—
26#1298—22no—
27#1295—22no—
28#1281—22no—
29C23×C8—21no—
30#533—21no—
31#533—21no—
32#533—21no—
33#533—21no—
34#533—21no—
35#533—21no—
36#533D412yes—
37#533D412yes—
38C2×C4⋊Q8C2×C411yescharacteristic
39C2×C4⋊Q8C2×C411yescharacteristic
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#432C2×C412yes—
57#432C2×C412yes—
58#432D412yes—
59#431—21no—
60#431—21no—
61#431—21no—
62#431—21no—
63#423—21no—
64#423—21no—
65C2×C4⋊C8—21no—
66C2×C4⋊C8—21no—
67#407—21no—
68#407D412yes—
69C2×C22⋊C8—21no—
70C2×C22⋊C8—21no—
71C4×M16—21no—
72C4×M16—21no—
73C2×#97—21no—
74C2×#97—21no—
75C2×#97—21no—
76C2×#97—21no—
77C2×#97—21no—
78C2×#97—21no—
79C2×#97—21no—
80C2×#97D412yes—
81#383—21no—
82#382D412yes—
83#381—21no—
84#380—21no—
85C4×C4⋊C4—21no—
86C4×C22⋊C4—21no—
87#376—21no—
88#333—24no—
89#332—24no—
90#330—28no—
91#328—28no—
92C22×Q8—21no—
93C22×Q8—21no—
94C23×C4C2×D411yescharacteristic
95C23×C4#4111yescharacteristic
96C23×C4#4111yescharacteristic
97C2×M16—22no—
98C2×M16—41no—
99C22×C8—22no—
100C22×C8—44no—
101C22×C8—41no—
102C22×C8—41no—
103C22×C8—41no—
104C22×C8—41no—
105C4⋊Q8—21no—
106C4⋊Q8—21no—
107C4⋊Q8—21no—
108C4⋊Q8—21no—
109C4⋊Q8—21no—
110C4⋊Q8—21no—
111C4⋊Q8C22⋊C412yes—
112C4⋊Q8C22⋊C412yes—
113#125—22no—
114#125—22no—
115#125—41no—
116#125—41no—
117C22⋊Q8—22no—
118C22⋊Q8—22no—
119C22⋊Q8—22no—
120C22⋊Q8—22no—
121C22⋊Q8—41no—
122C22⋊Q8—41no—
123C22⋊Q8—41no—
124C22⋊Q8—41no—
125C4×Q8—22no—
126C4×Q8—22no—
127C4×Q8—22no—
128C4×Q8—22no—
129C4×Q8—41no—
130C4×Q8—41no—
131C4×Q8—41no—
132C4×Q8—41no—
133#117—22no—
134#117—22no—
135#117—41no—
136#117—41no—
137C2×C4⋊C4—21no—
138C2×C4⋊C4—21no—
139C2×C4⋊C4—21no—
140C2×C4⋊C4—21no—
141C2×C4⋊C4—22no—
142C2×C4⋊C4—41no—
143C2×C4⋊C4—41no—
144C2×C22⋊C4—41no—
145C2×C22⋊C4—41no—
146C2×C42—41no—
147C2×C42—41no—
148C2×C42C22×C411yescharacteristic
149C2×C42C2×D411yescharacteristic
150C2×C42C2×D411yescharacteristic
151C2×C42C2×D411yescharacteristic
152C2×C42C2×D411yescharacteristic
153C2×C42C2×D411yescharacteristic
154C2×C42#4111yescharacteristic
155C2×C42#4111yescharacteristic
156C2×C42#4111yescharacteristic
157C2×C42#4111yescharacteristic
158C2×C42#4111yescharacteristic
159C2×C42#4111yescharacteristic
160C4⋊C8—22no—
161C4⋊C8—22no—
162C4⋊C8—22no—
163C4⋊C8—41no—
164C4⋊C8—41no—
165C4⋊C8—41no—
166C4⋊C8—41no—
167C4⋊C8—41no—
168C22⋊C8—22no—
169C22⋊C8—22no—
170C22⋊C8—22no—
171C22⋊C8—22no—
172C22⋊C8—22no—
173C22⋊C8—41no—
174C22⋊C8—41no—
175C22⋊C8—41no—
176#97—22no—
177#97—22no—
178#97—22no—
179#97—22no—
180#97—22no—
181#97—41no—
182#97—41no—
183#97—41no—
184#97—41no—
185#97—41no—
186C4×C8—41no—
187C4×C8—41no—
188#95—22no—
189#95—22no—
190#95—41no—
191#95—41no—
192C24#12411yescharacteristic
193C2×Q8—22no—
194C2×Q8—22no—
195C2×Q8—41no—
196C2×Q8—41no—
197C2×Q8—82no—
198C2×Q8—82no—
199C22×C4—42no—
200C22×C4—42no—
201C22×C4—41no—
202C22×C4—41no—
203C22×C4—42no—
204C22×C4—81no—
205C22×C4#11711yescharacteristic
206C22×C4C4×D411yescharacteristic
207C22×C4C4×D411yescharacteristic
208C22×C4#12011yescharacteristic
209C22×C4#12111yescharacteristic
210C22×C4#12111yescharacteristic
211C22×C4#12111yescharacteristic
212C22×C4#12111yescharacteristic
213C22×C4#12111yescharacteristic
214C22×C4#12311yescharacteristic
215C22×C4#12311yescharacteristic
216C22×C4#12311yescharacteristic
217C22×C4#12311yescharacteristic
218C22×C4#12311yescharacteristic
219C22×C4#12311yescharacteristic
220C22×C4#12611yescharacteristic
221C22×C4#12611yescharacteristic
222C22×C4#12611yescharacteristic
223M16—82no—
224M16—82no—
225C2×C8—24no—
226C2×C8—42no—
227C2×C8—44no—
228C2×C8—44no—
229C2×C8—42no—
230C2×C8—42no—
231C2×C8—42no—
232C2×C8—42no—
233C2×C8—42no—
234C2×C8—82no—
235C2×C8—82no—
236C4⋊C4—22no—
237C4⋊C4—22no—
238C4⋊C4—22no—
239C4⋊C4—22no—
240C4⋊C4—41no—
241C4⋊C4—41no—
242C4⋊C4—41no—
243C4⋊C4—41no—
244C4⋊C4—42no—
245C4⋊C4—42no—
246C4⋊C4—42no—
247C4⋊C4—42no—
248C4⋊C4—82no—
249C22⋊C4—42no—
250C22⋊C4—42no—
251C22⋊C4—82no—
252C42—42no—
253C42—42no—
254C42—82no—
255C42C2×C22⋊C411yescharacteristic
256C42#11711yescharacteristic
257C42C4×D411yescharacteristic
258C42C4×D411yescharacteristic
259C42#12011yescharacteristic
260C42#12111yescharacteristic
261C42#12111yescharacteristic
262C42#12111yescharacteristic
263C42#12311yescharacteristic
264C42#12311yescharacteristic
265C42#12311yescharacteristic
266C42#12311yescharacteristic
267C42#12411yescharacteristic
268C42#12411yescharacteristic
269C42#12411yescharacteristic
270C42#12611yescharacteristic
271C23—22no—
272C23—22no—
273C23—41no—
274C23#38811yescharacteristic
275C23#39411yescharacteristic
276C23#39711yescharacteristic
277Q8—82no—
278Q8—82no—
279Q8—82no—
280Q8—82no—
281C2×C4—22no—
282C2×C4—22no—
283C2×C4—22no—
284C2×C4—22no—
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—41no—
298C2×C4—41no—
299C2×C4—42no—
300C2×C4—42no—
301C2×C4—82no—
302C2×C4—82no—
303C2×C4—82no—
304C2×C4—82no—
305C2×C4—82no—
306C2×C4#38111yescharacteristic
307C2×C4#38611yescharacteristic
308C2×C4#38611yescharacteristic
309C2×C4#38811yescharacteristic
310C2×C4#38811yescharacteristic
311C2×C4#38811yescharacteristic
312C2×C4#39411yescharacteristic
313C2×C4#39411yescharacteristic
314C2×C4#39411yescharacteristic
315C2×C4#39711yescharacteristic
316C2×C4#39711yescharacteristic
317C2×C4#39711yescharacteristic
318C8—44no—
319C8—82no—
320C8—82no—
321C8—82no—
322C8—82no—
323C22—21no—
324C22—21no—
325C22—21no—
326C22—21no—
327C22—21no—
328C22—21no—
329C22—21no—
330C22—44no—
331C22#144011yescharacteristic
332C22#193212yes—
333C22#204912yes—
334C4—21no—
335C4—21no—
336C4—21no—
337C4—21no—
338C4—21no—
339C4—21no—
340C4—21no—
341C4—21no—
342C4—21no—
343C4—21no—
344C4—21no—
345C4—21no—
346C4—82no—
347C4—82no—
348C4—162no—
349C2—22no—
350C2—22no—
351C2—41no—
352C2#1116511yescharacteristic
353C2#1118611yescharacteristic
354C2#1118811yescharacteristic
subgroupG/H
center, socleC22#1440
commutatorC2×C42C22×C4
Frattini, cosocleC22×C42C23
prime radicalC24#124
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
15. #1558C4semidirect
16. #1556C4semidirect

Maximal subgroups

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

subgroupindex
#177482
C2×#15312
#175842
#80692
#80562
#80362
#79192