Finite Group Databasebeta

← #243044

Subgroups — #243044

Order 512.

591 conjugacy classes of subgroups.

The subgroup lattice has 387 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21889C211yesmaximal, characteristic
2#21879C211yesmaximal, characteristic
3C2×#1571C211yesmaximal, characteristic
4#17479C211yesmaximal, characteristic
5#10640C214yesmaximal, complemented
6#10639C214yesmaximal, complemented
7#8082C211yesmaximal, characteristic
8#7962C211yesmaximal, characteristic
9#7956C211yesmaximal, characteristic
10C2×#533C2211yescharacteristic
11#2920C2211yescharacteristic
12#2920C2211yescharacteristic
13C2×#431C2211yescharacteristic
14C2×#423C2211yescharacteristic
15C22×C4⋊C8C2211yescharacteristic
16#2283C2211yescharacteristic
17#1997—21no—
18#1988—21no—
19#1981—21no—
20#1966—21no—
21#1901—22no—
22#1892—22no—
23#1805C2214yes—
24#1801C2214yes—
25#1571—22no—
26#1571C2212yes—
27#1571C2212yes—
28C2×#335—22no—
29#1497—22no—
30#1493—22no—
31#1479C2212yes—
32#1479C2212yes—
33#1301C2214yes—
34#1286C2212yes—
35#1286C2212yes—
36#1286C2214yes—
37C22×M16—21no—
38C23×C8—21no—
39#533—21no—
40#533—21no—
41#533—21no—
42#533—21no—
43#533—21no—
44#533—21no—
45#533—21no—
46#533—21no—
47C2×C4⋊Q8D411yescharacteristic
48C2×C4⋊Q8D411yescharacteristic
49C2×#125—21no—
50C2×C22⋊Q8—21no—
51C2×C22⋊Q8D412yes—
52C2×C4×Q8—21no—
53C2×C4×Q8C2312yes—
54C2×#117—21no—
55C22×C42C2311yescharacteristic
56#432—21no—
57#432—21no—
58#432—21no—
59#432—21no—
60#431—21no—
61#431—21no—
62#431—21no—
63#431—21no—
64#431—21no—
65#431D412yes—
66#431C2312yes—
67#431C2311yescharacteristic
68#431C2311yescharacteristic
69#423—21no—
70#423—21no—
71#423—21no—
72#423—21no—
73#423C2312yes—
74#423C2312yes—
75C2×C4⋊C8—21no—
76C2×C4⋊C8—21no—
77C2×C4⋊C8—21no—
78C2×C4⋊C8—21no—
79C2×C4⋊C8—21no—
80C2×C4⋊C8—21no—
81C2×C4⋊C8—21no—
82C2×C4⋊C8—21no—
83C2×C4⋊C8C2312yes—
84C2×#103—24no—
85C2×#101—24no—
86#407—21no—
87#407—21no—
88C2×C22⋊C8—21no—
89C4×M16—21no—
90C4×M16—21no—
91C2×C4×C8D412yes—
92#383—21no—
93#382C2312yes—
94#381—21no—
95#380—21no—
96C4×C4⋊C4—21no—
97C4×C22⋊C4—21no—
98#376—21no—
99#343—24no—
100#340—24no—
101#335—24no—
102#335—24no—
103Q8⋊C8—24no—
104Q8⋊C8—24no—
105C22×Q8—21no—
106C22×Q8C2×D412yes—
107C23×C4C2×D411yescharacteristic
108C23×C4#4111yescharacteristic
109C23×C4#4111yescharacteristic
110C2×M16—22no—
111C2×M16—22no—
112C2×M16—41no—
113C2×M16—41no—
114C2×M16—82no—
115C22×C8—22no—
116C22×C8—22no—
117C22×C8—41no—
118C22×C8—41no—
119C22×C8—41no—
120C22×C8—82no—
121C4⋊Q8—21no—
122C4⋊Q8—21no—
123C4⋊Q8—21no—
124C4⋊Q8—21no—
125C4⋊Q8—21no—
126C4⋊Q8—21no—
127C4⋊Q8—21no—
128C4⋊Q8—21no—
129#125—41no—
130#125—41no—
131#125—41no—
132#125—41no—
133C22⋊Q8—22no—
134C22⋊Q8—22no—
135C22⋊Q8—22no—
136C22⋊Q8—22no—
137C22⋊Q8—41no—
138C22⋊Q8—41no—
139C22⋊Q8—41no—
140C22⋊Q8—41no—
141C4×Q8—22no—
142C4×Q8—22no—
143C4×Q8—22no—
144C4×Q8—22no—
145C4×Q8—41no—
146C4×Q8—41no—
147C4×Q8—41no—
148C4×Q8—41no—
149#117—41no—
150#117—41no—
151#117—41no—
152#117—41no—
153C2×C4⋊C4—21no—
154C2×C4⋊C4—21no—
155C2×C4⋊C4—21no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4C2×D412yes—
160C2×C22⋊C4—22no—
161C2×C22⋊C4—41no—
162C2×C42—22no—
163C2×C42—41no—
164C2×C42C2×D411yescharacteristic
165C2×C42C2×D411yescharacteristic
166C2×C42C2×D411yescharacteristic
167C2×C42#4111yescharacteristic
168C2×C42#4111yescharacteristic
169C2×C42#4111yescharacteristic
170C2×C42#4111yescharacteristic
171C2×C42#4111yescharacteristic
172C2×C42#4111yescharacteristic
173C2×C42#4111yescharacteristic
174C2×C42#4111yescharacteristic
175C2×C42C2411yescharacteristic
176C4⋊C8—22no—
177C4⋊C8—22no—
178C4⋊C8—22no—
179C4⋊C8—22no—
180C4⋊C8—21no—
181C4⋊C8—21no—
182C4⋊C8—22no—
183C4⋊C8—22no—
184C4⋊C8—22no—
185C4⋊C8—22no—
186C4⋊C8—41no—
187C4⋊C8—41no—
188C4⋊C8—41no—
189C4⋊C8—41no—
190C4⋊C8—41no—
191#103—84no—
192#101—84no—
193C22⋊C8—22no—
194C22⋊C8—21no—
195C22⋊C8—21no—
196C22⋊C8—22no—
197C22⋊C8—41no—
198C22⋊C8—41no—
199C22⋊C8—41no—
200#97—41no—
201#97—41no—
202C4×C8—22no—
203C4×C8—22no—
204C4×C8—41no—
205C4×C8C2×D412yes—
206C4×C8C2×D412yes—
207#95—22no—
208#95—22no—
209#95—41no—
210#95—41no—
211C24#12311yescharacteristic
212C2×Q8—22no—
213C2×Q8—22no—
214C2×Q8—24no—
215C2×Q8—24no—
216C2×Q8—41no—
217C2×Q8—41no—
218C2×Q8—82no—
219C22×C4—22no—
220C22×C4—22no—
221C22×C4—22no—
222C22×C4—22no—
223C22×C4—42no—
224C22×C4—42no—
225C22×C4—42no—
226C22×C4#12111yescharacteristic
227C22×C4#12111yescharacteristic
228C22×C4#12311yescharacteristic
229C22×C4#12311yescharacteristic
230C22×C4#12311yescharacteristic
231C22×C4#12311yescharacteristic
232C22×C4#12311yescharacteristic
233C22×C4#12311yescharacteristic
234C22×C4#12311yescharacteristic
235C22×C4#12411yescharacteristic
236C22×C4#12411yescharacteristic
237C22×C4#12611yescharacteristic
238C22×C4#12611yescharacteristic
239C22×C4#12611yescharacteristic
240C22×C4#12611yescharacteristic
241C22×C4C22×D411yescharacteristic
242C22×C4C2×#4111yescharacteristic
243C22×C4C2×#4111yescharacteristic
244M16—42no—
245M16—44no—
246M16—42no—
247M16—82no—
248M16—82no—
249C2×C8—22no—
250C2×C8—22no—
251C2×C8—44no—
252C2×C8—42no—
253C2×C8—42no—
254C2×C8—42no—
255C2×C8—42no—
256C2×C8—44no—
257C2×C8—82no—
258C2×C8—82no—
259C2×C8—82no—
260C4⋊C4—22no—
261C4⋊C4—22no—
262C4⋊C4—42no—
263C4⋊C4—42no—
264C4⋊C4—41no—
265C4⋊C4—41no—
266C4⋊C4—41no—
267C4⋊C4—41no—
268C4⋊C4—42no—
269C4⋊C4—42no—
270C4⋊C4—41no—
271C4⋊C4—41no—
272C4⋊C4—42no—
273C4⋊C4—42no—
274C22⋊C4—42no—
275C22⋊C4—42no—
276C22⋊C4—82no—
277C42—42no—
278C42—42no—
279C42—82no—
280C42#12111yescharacteristic
281C42#12111yescharacteristic
282C42#12311yescharacteristic
283C42#12311yescharacteristic
284C42#12311yescharacteristic
285C42#12311yescharacteristic
286C42#12411yescharacteristic
287C42#12411yescharacteristic
288C42#12611yescharacteristic
289C42#12611yescharacteristic
290C42#12611yescharacteristic
291C42#12611yescharacteristic
292C42C22×D411yescharacteristic
293C42C2×#4111yescharacteristic
294C42C2×#4111yescharacteristic
295C42C2×#4111yescharacteristic
296C23—22no—
297C23—22no—
298C23—41no—
299C23#39711yescharacteristic
300C23#39711yescharacteristic
301C23C2×#12311yescharacteristic
302Q8—44no—
303Q8—42no—
304Q8—42no—
305Q8—82no—
306Q8—82no—
307C2×C4—22no—
308C2×C4—22no—
309C2×C4—22no—
310C2×C4—22no—
311C2×C4—22no—
312C2×C4—22no—
313C2×C4—22no—
314C2×C4—22no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—44no—
327C2×C4—42no—
328C2×C4—42no—
329C2×C4—82no—
330C2×C4—82no—
331C2×C4—82no—
332C2×C4—82no—
333C2×C4#39711yescharacteristic
334C2×C4#39711yescharacteristic
335C2×C4#39711yescharacteristic
336C2×C4#39711yescharacteristic
337C2×C4#39711yescharacteristic
338C2×C4#39711yescharacteristic
339C2×C4C2×#12111yescharacteristic
340C2×C4C2×#12311yescharacteristic
341C2×C4C2×#12311yescharacteristic
342C2×C4C2×#12411yescharacteristic
343C2×C4C2×#12611yescharacteristic
344C2×C4C2×#12611yescharacteristic
345C2×C4#57612yes—
346C2×C4#57812yes—
347C8—42no—
348C8—42no—
349C8—82no—
350C8—82no—
351C8—82no—
352C8—82no—
353C22—21no—
354C22—21no—
355C22—21no—
356C22—21no—
357C22—21no—
358C22—21no—
359C22—21no—
360C22—44no—
361C22#208212yes—
362C22C2×#39711yescharacteristic
363C22#309812yes—
364C4—21no—
365C4—21no—
366C4—21no—
367C4—21no—
368C4—21no—
369C4—21no—
370C4—21no—
371C4—21no—
372C4—21no—
373C4—82no—
374C4—82no—
375C4—162no—
376C4#208012yes—
377C4#305811yescharacteristic
378C4#305911yescharacteristic
379C4#309812yes—
380C2—22no—
381C2—22no—
382C2—41no—
383C2#2363911yescharacteristic
384C2C2×#208011yescharacteristic
385C2C2×#208211yescharacteristic
subgroupG/H
center, socleC22C2×#397
commutator, Frattini, cosocleC2×C42C24
prime radicalC24#123
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
5. #10640C2semidirect
6. #10639C2semidirect

Maximal subgroups

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

subgroupindex
#218892
#218792
C2×#15712
#174792
#106402
#106392
#80822
#79622
#79562