Finite Group Databasebeta

← #288066

Subgroups — #288066

Order 512.

644 conjugacy classes of subgroups.

The subgroup lattice has 371 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#28676C211yesmaximal, characteristic
2#25324C212yesmaximal
3#23157C211yesmaximal, characteristic
4C2×#1710C211yesmaximal, characteristic
5C2×#1674C211yesmaximal, characteristic
6#10541C212yesmaximal
7#10065C212yesmaximal
8#9730C211yesmaximal, characteristic
9#9594C211yesmaximal, characteristic
10#9065C212yesmaximal
11C8×#103C211yesmaximal, characteristic
12C2×#500C2211yescharacteristic
13C2×#500C2211yescharacteristic
14C2×#487C2212yes—
15C2×#487C2212yes—
16C2×#462C2212yes—
17C2×C8×Q8C2211yescharacteristic
18#2567C2212yes—
19#2365C2212yes—
20#2087—22no—
21#2072—22no—
22#2056—22no—
23#1988C2212yes—
24#1987C2212yes—
25#1987C2212yes—
26#1942—21no—
27#1942—21no—
28#1942—21no—
29#1939C2212yes—
30#1918—22no—
31#1878—22no—
32#1803—22no—
33C4×#107—21no—
34C4×#107C2211yescharacteristic
35C8×C4⋊C4C2211yescharacteristic
36C4×#103C2211yescharacteristic
37#1710—24no—
38#1674—24no—
39C2×#335C2211yescharacteristic
40C2×#333C2212yes—
41C2×#332C2212yes—
42C2×Q8⋊C8C2211yescharacteristic
43C2×C82C2211yescharacteristic
44#1279C2211yescharacteristic
45#1279C2212yes—
46#1277C2211yescharacteristic
47C22×Q16—22no—
48C2×C4⋊Q8C2312yes—
49C2×C4⋊Q8C2312yes—
50C2×C4×Q8—22no—
51C2×C4×Q8C2311yescharacteristic
52#500—22no—
53#500—22no—
54#500—22no—
55#500D414yes—
56#487—24no—
57#487—24no—
58#487—24no—
59#487—42no—
60#462—24no—
61#462—42no—
62C8×Q8—22no—
63C8×Q8—22no—
64C2×#107—21no—
65C2×#107—21no—
66C2×#107—21no—
67C2×#107—21no—
68C2×#107D411yescharacteristic
69C2×#107D411yescharacteristic
70C2×C4⋊C8—21no—
71C2×C4⋊C8—22no—
72C2×C4⋊C8C2311yescharacteristic
73C2×C4⋊C8C2312yes—
74C2×C4⋊C8C2311yescharacteristic
75C2×#103—22no—
76C2×#103—22no—
77C2×#103—22no—
78C2×#103—42no—
79C2×#103D411yescharacteristic
80C2×#103D411yescharacteristic
81C2×C4×C8—21no—
82C2×C4×C8—21no—
83C2×C4×C8C2311yescharacteristic
84C2×C4×C8C2311yescharacteristic
85C2×C4×C8C2311yescharacteristic
86#395—42no—
87#391—22no—
88#391—22no—
89#389—22no—
90#389—22no—
91#389—22no—
92#387—42no—
93#384—22no—
94C4×C4⋊C4—22no—
95C4×C4⋊C4C2312yes—
96C4×C4⋊C4C2311yescharacteristic
97#340—22no—
98#340—22no—
99#340—22no—
100#340—22no—
101#340—22no—
102#340—22no—
103#340D411yescharacteristic
104#340D411yescharacteristic
105#340D412yes—
106#336—22no—
107#336—21no—
108#335—22no—
109#333—42no—
110#332—24no—
111Q8⋊C8—22no—
112C82—22no—
113C22×Q8—22no—
114C22×Q8—22no—
115C22×Q8C2×D411yescharacteristic
116C2×Q16—44no—
117C2×Q16—44no—
118C2×Q16—82no—
119C22×C8—21no—
120C22×C8—22no—
121C22×C8—21no—
122C22×C8—21no—
123C22×C8C2×D411yescharacteristic
124C22×C8C2×D411yescharacteristic
125C22×C8C2×D411yescharacteristic
126C22×C8C2×D411yescharacteristic
127C4⋊Q8—22no—
128C4⋊Q8—22no—
129C4⋊Q8—22no—
130C4⋊Q8—22no—
131C4⋊Q8—22no—
132C4⋊Q8—22no—
133C4⋊Q8—22no—
134C4⋊Q8C2×D414yes—
135C4×Q8—21no—
136C4×Q8—21no—
137C4×Q8—21no—
138C4×Q8—21no—
139C4×Q8—44no—
140C4×Q8—82no—
141C2×C4⋊C4—22no—
142C2×C4⋊C4—22no—
143C2×C4⋊C4—21no—
144C2×C4⋊C4—22no—
145C2×C4⋊C4—22no—
146C2×C4⋊C4—42no—
147C2×C4⋊C4—42no—
148C2×C4⋊C4—42no—
149C2×C4⋊C4C2×D411yescharacteristic
150C2×C4⋊C4C2×D412yes—
151C2×C4⋊C4C2×D411yescharacteristic
152C2×C4⋊C4C2×D412yes—
153C2×C4⋊C4C2×D411yescharacteristic
154C2×C42—22no—
155C2×C42—21no—
156C2×C42—21no—
157C2×C42—42no—
158C2×C42C2411yescharacteristic
159#107—24no—
160#107—24no—
161#107—22no—
162#107—42no—
163#107—42no—
164#107Q1614yescomplemented
165C4⋊C8—21no—
166C4⋊C8—22no—
167C4⋊C8—21no—
168C4⋊C8—21no—
169C4⋊C8—21no—
170C4⋊C8—22no—
171C4⋊C8—42no—
172C4⋊C8—82no—
173#103—22no—
174#103—22no—
175#103—48no—
176#103—44no—
177#103—44no—
178#103—44no—
179C4×C8—21no—
180C4×C8—21no—
181C4×C8—21no—
182C4×C8—42no—
183C4×C8—42no—
184C4×C8C2×D412yes—
185C4×C8#4112yes—
186C4×C8#4112yes—
187#95—21no—
188#95—22no—
189#95—42no—
190#95—42no—
191#95—42no—
192#95—42no—
193C2×Q8—21no—
194C2×Q8—21no—
195C2×Q8—21no—
196C2×Q8—24no—
197C2×Q8—21no—
198C2×Q8—21no—
199C2×Q8—42no—
200C2×Q8—42no—
201C2×Q8—42no—
202C2×Q8—42no—
203C2×Q8—42no—
204C2×Q8—42no—
205C2×Q8—42no—
206C2×Q8—82no—
207C2×Q8—82no—
208C2×Q8#13512yes—
209C22×C4—22no—
210C22×C4—21no—
211C22×C4—21no—
212C22×C4—21no—
213C22×C4—22no—
214C22×C4—21no—
215C22×C4—42no—
216C22×C4—42no—
217C22×C4C22×D411yescharacteristic
218C22×C4C22×D411yescharacteristic
219C22×C4H(5,𝔽2)11yescharacteristic
220Q16—88no—
221C2×C8—21no—
222C2×C8—21no—
223C2×C8—21no—
224C2×C8—21no—
225C2×C8—22no—
226C2×C8—21no—
227C2×C8—21no—
228C2×C8—21no—
229C2×C8—21no—
230C2×C8—41no—
231C2×C8—41no—
232C2×C8—41no—
233C2×C8—42no—
234C2×C8—41no—
235C2×C8—42no—
236C2×C8—41no—
237C2×C8—42no—
238C2×C8—41no—
239C2×C8—41no—
240C2×C8—41no—
241C2×C8#12112yes—
242C2×C8#12112yes—
243C2×C8C2×Q1612yes—
244C2×C8#13712yes—
245C4⋊C4—21no—
246C4⋊C4—22no—
247C4⋊C4—24no—
248C4⋊C4—21no—
249C4⋊C4—21no—
250C4⋊C4—24no—
251C4⋊C4—21no—
252C4⋊C4—22no—
253C4⋊C4—22no—
254C4⋊C4—21no—
255C4⋊C4—41no—
256C4⋊C4—42no—
257C4⋊C4—41no—
258C4⋊C4—42no—
259C4⋊C4—42no—
260C4⋊C4—42no—
261C4⋊C4—42no—
262C4⋊C4—42no—
263C4⋊C4—84no—
264C4⋊C4—162no—
265C4⋊C4—162no—
266C4⋊C4C2×Q1614yes—
267C4⋊C4#13512yes—
268C42—41no—
269C42—41no—
270C42—42no—
271C42—41no—
272C42—42no—
273C42—41no—
274C42—84no—
275C42C22×D411yescharacteristic
276C42C2×#4111yescharacteristic
277C42C2×#4111yescharacteristic
278C42H(5,𝔽2)11yescharacteristic
279C23D4211yescharacteristic
280Q8—22no—
281Q8—22no—
282Q8—41no—
283Q8—41no—
284Q8—44no—
285Q8—44no—
286Q8—84no—
287Q8—82no—
288Q8—82no—
289Q8—162no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—22no—
293C2×C4—22no—
294C2×C4—24no—
295C2×C4—41no—
296C2×C4—41no—
297C2×C4—42no—
298C2×C4—41no—
299C2×C4—42no—
300C2×C4—41no—
301C2×C4—41no—
302C2×C4—44no—
303C2×C4—41no—
304C2×C4—42no—
305C2×C4—42no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—42no—
310C2×C4—41no—
311C2×C4—82no—
312C2×C4—82no—
313C2×C4—82no—
314C2×C4—82no—
315C2×C4—82no—
316C2×C4C2×#12111yescharacteristic
317C2×C4#53211yescharacteristic
318C2×C4#54011yescharacteristic
319C2×C4D4211yescharacteristic
320C2×C4#54611yescharacteristic
321C2×C4#55011yescharacteristic
322C2×C4C22×Q1611yescharacteristic
323C2×C4C2×#13511yescharacteristic
324C2×C4C2×#13511yescharacteristic
325C2×C4C2×#13711yescharacteristic
326C2×C4#57611yescharacteristic
327C2×C4#57811yescharacteristic
328C2×C4#57811yescharacteristic
329C2×C4#57811yescharacteristic
330C8—22no—
331C8—42no—
332C8—41no—
333C8—41no—
334C8—81no—
335C8—82no—
336C8—81no—
337C8#46214yes—
338C22#242811yescharacteristic
339C22#327611yescharacteristic
340C22#328011yescharacteristic
341C22#328511yescharacteristic
342C22D4×Q1611yescharacteristic
343C22#329011yescharacteristic
344C22#329011yescharacteristic
345C4—21no—
346C4—21no—
347C4—44no—
348C4—42no—
349C4—42no—
350C4—84no—
351C4—81no—
352C4—82no—
353C4—81no—
354C4—162no—
355C4C2×#46211yescharacteristic
356C4#304411yescharacteristic
357C4#316811yescharacteristic
358C4#317511yescharacteristic
359C4#328011yescharacteristic
360C4#328311yescharacteristic
361C4#339511yescharacteristic
362C4#339811yescharacteristic
363C2#1990311yescharacteristic
364C2#1994011yescharacteristic
365C2#1999311yescharacteristic
366C2#2000811yescharacteristic
367C2#2537011yescharacteristic
368C2#2541311yescharacteristic
369C2#2543211yescharacteristic
subgroupG/H
center, socle, prime radicalC23D42
commutator, Frattini, cosocleC2×C42C24
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C2×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
164. #107Q16semidirect

Maximal subgroups

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

subgroupindex
#286762
#253242
#231572
C2×#17102
C2×#16742
#105412
#100652
#97302
#95942
#90652
C8×#1032