Finite Group Databasebeta

← #223250

Subgroups — #223250

Order 512.

993 conjugacy classes of subgroups.

The subgroup lattice has 400 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36187C211yesmaximal, characteristic
2#22750C212yesmaximal, complemented
3#9905C212yesmaximal, complemented
4#9142C212yesmaximal, complemented
5#9133C212yesmaximal, complemented
6#8729C212yesmaximal, complemented
7#8592C211yesmaximal, characteristic, complemented
8#8231C211yesmaximal, characteristic, complemented
9#8044C212yesmaximal, complemented
10C2×#555C412yes—
11C2×#432C2212yes—
12C22×C4⋊C8C2212yes—
13#2868C2211yescharacteristic
14#2856C2212yes—
15#2563C412yes—
16#2537—22no—
17#2500C414yes—
18#2491—22no—
19#2491—22no—
20#2354—22no—
21C4×#127C2211yescharacteristic
22#2285C2212yes—
23C42×D4C2211yescharacteristic
24#1924—22no—
25#1917—22no—
26#1869—22no—
27#1831C2212yes—
28#1797C2212yes—
29#1797C2212yes—
30#1767C2212yes—
31#1752—22no—
32C2×#331C2212yes—
33C2×#331C2211yescharacteristic
34C2×#331C2211yescharacteristic
35#1296C2211yescharacteristic
36#1296C2212yes—
37#1296C2212yes—
38#1296C2212yes—
39#1296C2211yescharacteristic
40#1296C2212yes—
41#1296C2212yes—
42C23×C8—22no—
43#555—48no—
44C2×C4⋊Q8—22no—
45C2×C4⋊Q8C2×C412yes—
46C2×#127C2311yescharacteristic
47C2×#124—24no—
48C2×#121—22no—
49C2×#121—22no—
50C2×#121—22no—
51C2×#121C2×C414yes—
52C2×C4×D4—22no—
53C2×C4×D4C2×C412yes—
54C2×C4×D4C2×C411yescharacteristic
55C2×C4×D4D412yes—
56C2×#117—22no—
57C2×#117C2×C414yes—
58C22×C42C2312yes—
59#432—24no—
60#432—42no—
61C2×C4⋊C8—24no—
62C2×C4⋊C8—22no—
63C2×C4⋊C8—42no—
64C2×C4⋊C8C2312yes—
65C2×C4⋊C8C2312yes—
66C2×C4⋊C8C2312yes—
67C2×C4⋊C8C2312yes—
68C2×C22⋊C8—22no—
69C2×C22⋊C8—22no—
70C2×C22⋊C8—22no—
71C2×#97—22no—
72C2×#97—22no—
73#394—24no—
74#394—24no—
75#393—22no—
76#393—22no—
77#390—22no—
78#390—22no—
79#390—22no—
80#390C2×C412yes—
81#390C2×C414yes—
82#388—24no—
83#388—24no—
84#384—22no—
85#384C2×C412yes—
86#384C2311yescharacteristic
87#384C2312yes—
88#382C2×C414yes—
89#380—22no—
90#380—22no—
91#380—24no—
92C4×C4⋊C4—22no—
93C4×C4⋊C4—22no—
94C4×C4⋊C4C2×C411yescharacteristic
95C4×C4⋊C4D412yes—
96C4×C22⋊C4C2×C412yes—
97C4×C22⋊C4D412yes—
98C4×C22⋊C4D412yes—
99C43C2311yescharacteristic
100#336—22no—
101#336—22no—
102#336—22no—
103#336—22no—
104#336—22no—
105#331—24no—
106#331—41no—
107#331—41no—
108C22×Q8—22no—
109C22×D4—24no—
110C22×D4—22no—
111C22×D4C22⋊C411yescharacteristic
112C22×D4C22×C412yes—
113C23×C4—22no—
114C23×C4C22×C412yes—
115C23×C4C2×D412yes—
116C23×C4C2×D412yes—
117C22×C8—22no—
118C22×C8—22no—
119C22×C8—22no—
120C22×C8—22no—
121C22×C8—44no—
122C22×C8—44no—
123C22×C8—82no—
124C4⋊Q8—44no—
125C4⋊Q8—82no—
126C4⋊Q8—82no—
127#127—22no—
128#127—21no—
129#127—21no—
130#124—416no—
131#121—48no—
132#121—48no—
133#121—82no—
134#121—82no—
135#121—82no—
136#121—82no—
137C4×D4—24no—
138C4×D4—42no—
139C4×D4—42no—
140C4×D4—44no—
141C4×D4—44no—
142C4×D4—44no—
143#117—48no—
144#117—48no—
145C2×C4⋊C4—22no—
146C2×C4⋊C4—22no—
147C2×C4⋊C4—22no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—24no—
150C2×C4⋊C4C22⋊C411yescharacteristic
151C2×C4⋊C4C22⋊C412yes—
152C2×C4⋊C4C22×C412yes—
153C2×C4⋊C4C22×C414yes—
154C2×C22⋊C4—24no—
155C2×C22⋊C4—24no—
156C2×C22⋊C4—24no—
157C2×C22⋊C4—22no—
158C2×C22⋊C4—24no—
159C2×C22⋊C4—24no—
160C2×C22⋊C4C22⋊C412yes—
161C2×C22⋊C4C22⋊C412yes—
162C2×C22⋊C4C22⋊C412yes—
163C2×C42—22no—
164C2×C42—22no—
165C2×C42—22no—
166C2×C42—22no—
167C2×C42—24no—
168C2×C42C22×C411yescharacteristic
169C2×C42C22×C412yes—
170C2×C42C22×C411yescharacteristic
171C2×C42C2×D412yes—
172C2×C42C2×D412yes—
173C2×C42C2×D412yes—
174C2×C42C2×D412yes—
175C2×C42C2411yescharacteristic
176C4⋊C8—22no—
177C4⋊C8—22no—
178C4⋊C8—22no—
179C4⋊C8—22no—
180C4⋊C8—22no—
181C4⋊C8—22no—
182C4⋊C8—22no—
183C4⋊C8—44no—
184C4⋊C8#4114yes—
185C22⋊C8—44no—
186C22⋊C8—44no—
187C22⋊C8—82no—
188#97—44no—
189#97—82no—
190#95—24no—
191#95—24no—
192#95C22⋊C412yes—
193#95C22⋊C412yes—
194#95C22⋊C411yescharacteristic
195#95C22⋊C411yescharacteristic
196C24—22no—
197C24C2×C22⋊C412yes—
198C2×Q8—44no—
199C2×Q8—84no—
200C2×D4—22no—
201C2×D4—22no—
202C2×D4—42no—
203C2×D4—42no—
204C2×D4—42no—
205C2×D4—48no—
206C2×D4—42no—
207C2×D4—42no—
208C2×D4—42no—
209C2×D4—42no—
210C2×D4—42no—
211C2×D4—42no—
212C2×D4—84no—
213C2×D4—84no—
214C2×D4#10014yes—
215C22×C4—24no—
216C22×C4—22no—
217C22×C4—22no—
218C22×C4—22no—
219C22×C4—22no—
220C22×C4—22no—
221C22×C4—22no—
222C22×C4—22no—
223C22×C4—22no—
224C22×C4—22no—
225C22×C4—22no—
226C22×C4—22no—
227C22×C4—22no—
228C22×C4—22no—
229C22×C4—22no—
230C22×C4—22no—
231C22×C4—24no—
232C22×C4—42no—
233C22×C4—48no—
234C22×C4—44no—
235C22×C4C2×C22⋊C412yes—
236C22×C4C2×C22⋊C412yes—
237C22×C4C2×C22⋊C412yes—
238C22×C4C2×C22⋊C412yes—
239C22×C4C2×C22⋊C411yescharacteristic
240C22×C4C2×C22⋊C411yescharacteristic
241C22×C4C23×C411yescharacteristic
242C22×C4C22×D412yes—
243C2×C8—24no—
244C2×C8—42no—
245C2×C8—42no—
246C2×C8—42no—
247C2×C8—42no—
248C2×C8—42no—
249C2×C8—42no—
250C2×C8—42no—
251C2×C8—42no—
252C2×C8—42no—
253C2×C8—42no—
254C2×C8—42no—
255C2×C8—88no—
256C4⋊C4—22no—
257C4⋊C4—42no—
258C4⋊C4—42no—
259C4⋊C4—44no—
260C4⋊C4—42no—
261C4⋊C4—42no—
262C4⋊C4—42no—
263C4⋊C4—42no—
264C4⋊C4—42no—
265C4⋊C4—42no—
266C4⋊C4—44no—
267C4⋊C4—84no—
268C22⋊C4—24no—
269C22⋊C4—42no—
270C22⋊C4—42no—
271C22⋊C4—48no—
272C22⋊C4—48no—
273C22⋊C4—44no—
274C22⋊C4—48no—
275C22⋊C4—84no—
276C22⋊C4—84no—
277C42—22no—
278C42—22no—
279C42—22no—
280C42—22no—
281C42—22no—
282C42—22no—
283C42—22no—
284C42—22no—
285C42—22no—
286C42—22no—
287C42—22no—
288C42—41no—
289C42—48no—
290C42C2×#4112yes—
291C42H(5,𝔽2)11yescharacteristic
292C422−1+411yescharacteristic
293C23—22no—
294C23—22no—
295C23—22no—
296C23—22no—
297C23—22no—
298C23—22no—
299C23—42no—
300C23—42no—
301C23—42no—
302C23—44no—
303C23—44no—
304C23C2×#10014yes—
305C23C22×C22⋊C411yescharacteristic
306Q8—164no—
307D4—44no—
308D4—82no—
309D4—82no—
310D4—82no—
311D4—82no—
312D4—82no—
313D4—82no—
314D4—168no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—22no—
318C2×C4—22no—
319C2×C4—22no—
320C2×C4—22no—
321C2×C4—21no—
322C2×C4—22no—
323C2×C4—22no—
324C2×C4—22no—
325C2×C4—22no—
326C2×C4—22no—
327C2×C4—22no—
328C2×C4—22no—
329C2×C4—22no—
330C2×C4—22no—
331C2×C4—22no—
332C2×C4—44no—
333C2×C4—44no—
334C2×C4—42no—
335C2×C4—42no—
336C2×C4—42no—
337C2×C4—42no—
338C2×C4—42no—
339C2×C4—42no—
340C2×C4—42no—
341C2×C4—42no—
342C2×C4—42no—
343C2×C4—44no—
344C2×C4—44no—
345C2×C4—44no—
346C2×C4—44no—
347C2×C4—88no—
348C2×C4C2×#10012yes—
349C2×C4C4×#4111yescharacteristic
350C2×C4#52011yescharacteristic
351C2×C4#54612yes—
352C2×C4#54712yes—
353C2×C4#57512yes—
354C2×C4#57512yes—
355C2×C4#57612yes—
356C2×C4#57712yes—
357C8—44no—
358C8—82no—
359C8—82no—
360C8—82no—
361C22—24no—
362C22—24no—
363C22—24no—
364C22—42no—
365C22—42no—
366C22—42no—
367C22—42no—
368C22—82no—
369C22—84no—
370C22—82no—
371C22—82no—
372C22—84no—
373C22#233911yescharacteristic
374C22C22×#10011yescharacteristic
375C22#288911yescharacteristic
376C22#289612yes—
377C22#289712yes—
378C4—22no—
379C4—21no—
380C4—21no—
381C4—42no—
382C4—41no—
383C4—42no—
384C4—42no—
385C4—42no—
386C4—82no—
387C4—84no—
388C4#331912yes—
389C4#332112yes—
390C4#334712yes—
391C4#334812yes—
392C2—44no—
393C2—162no—
394C2#1775411yescharacteristic
395C2#1776912yes—
396C2#1777011yescharacteristic
397C2#2247911yescharacteristic
398C2#2249212yes—
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radical#36187C2
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C42×D4 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #22750C2semidirect
3. #9905C2semidirect
4. #9142C2semidirect
5. #9133C2semidirect
6. #8729C2semidirect
7. #8592C2semidirect
8. #8231C2semidirect
9. #8044C2semidirect

Maximal subgroups

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

subgroupindex
#361872
#227502
#99052
#91422
#91332
#87292
#85922
#82312
#80442