Finite Group Databasebeta

← #277314

Subgroups — #277314

Order 512.

612 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#24682C212yesmaximal
3#22727C211yesmaximal, characteristic
4C2×#1676C211yesmaximal, characteristic
5C2×C8×Q16C211yesmaximal, characteristic
6#10601C212yesmaximal
7#10415C212yesmaximal
8#9823C211yesmaximal, characteristic
9#9594C211yesmaximal, characteristic
10#8974C212yesmaximal
11#8645C211yesmaximal, characteristic
12C2×#500C2211yescharacteristic
13C2×#477C2212yes—
14C2×C8×Q8C2211yescharacteristic
15C2×C8×Q8C2211yescharacteristic
16C2×C4×Q16C2211yescharacteristic
17#2567C2212yes—
18#2355C2212yes—
19#2087—22no—
20#2072—22no—
21#2056—22no—
22#1990C2212yes—
23#1987C2212yes—
24#1956C2212yes—
25#1949—22no—
26#1948C2212yes—
27#1947—22no—
28#1942—21no—
29#1942—21no—
30#1939C2212yes—
31#1928—22no—
32#1846—21no—
33#1845C2211yescharacteristic
34#1803C2212yes—
35C4×#107—21no—
36C4×#107C2211yescharacteristic
37#1676—24no—
38C8×Q16—24no—
39C2×#335C2211yescharacteristic
40C2×#335C2211yescharacteristic
41C2×#333C2212yes—
42C2×Q8⋊C8C2211yescharacteristic
43C2×Q8⋊C8C2211yescharacteristic
44C2×C82C2211yescharacteristic
45#1296C2212yes—
46#1279C2212yes—
47C22×Q16D411yescharacteristic
48C2×C4⋊Q8C2312yes—
49C2×C4×Q8C2311yescharacteristic
50C2×C4×Q8C2311yescharacteristic
51#500—22no—
52#500—22no—
53#477—24no—
54#477—24no—
55C8×Q8—22no—
56C8×Q8—22no—
57C8×Q8—22no—
58C8×Q8D414yes—
59C4×Q16—22no—
60C4×Q16—22no—
61C2×#107—21no—
62C2×#107—21no—
63C2×#107—21no—
64C2×#107—22no—
65C2×#107—21no—
66C2×#107D411yescharacteristic
67C2×#107D411yescharacteristic
68C2×#107D411yescharacteristic
69C2×#106—22no—
70C2×C4⋊C8—21no—
71C2×C4⋊C8—22no—
72C2×C4⋊C8—21no—
73C2×C4⋊C8C2312yes—
74C2×C4⋊C8C2311yescharacteristic
75C2×C4⋊C8C2311yescharacteristic
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#387—42no—
92#384—22no—
93#384C2312yes—
94#382—22no—
95C4×C4⋊C4—22no—
96C4×C4⋊C4—22no—
97C4×C4⋊C4C2312yes—
98#340—22no—
99#340—22no—
100#340—22no—
101#340—22no—
102#340—22no—
103#340—22no—
104#340D412yes—
105#336—22no—
106#335—22no—
107#335—22no—
108#333—24no—
109Q8⋊C8—22no—
110Q8⋊C8—22no—
111C82—22no—
112C22×Q8—22no—
113C22×Q8C2×D411yescharacteristic
114C22×Q8C2×D411yescharacteristic
115C2×Q16—22no—
116C2×Q16—22no—
117C2×Q16—22no—
118C22×C8—21no—
119C22×C8—21no—
120C22×C8—22no—
121C22×C8—21no—
122C22×C8C2×D411yescharacteristic
123C22×C8C2×D411yescharacteristic
124C22×C8C2×D411yescharacteristic
125C22×C8C2×D411yescharacteristic
126C4⋊Q8—22no—
127C4⋊Q8—22no—
128C4⋊Q8—22no—
129C4⋊Q8—22no—
130C4×Q8—21no—
131C4×Q8—21no—
132C4×Q8—21no—
133C4×Q8—21no—
134C4×Q8—21no—
135C4×Q8—21no—
136C4×Q8C2×D412yes—
137C4×Q8#4112yes—
138C2×C4⋊C4—21no—
139C2×C4⋊C4—21no—
140C2×C4⋊C4—22no—
141C2×C4⋊C4—22no—
142C2×C4⋊C4—22no—
143C2×C4⋊C4—22no—
144C2×C4⋊C4—22no—
145C2×C4⋊C4—42no—
146C2×C4⋊C4—42no—
147C2×C4⋊C4C2×D411yescharacteristic
148C2×C4⋊C4C2×D412yes—
149C2×C4⋊C4C2×D411yescharacteristic
150C2×C4⋊C4C2×D412yes—
151C2×C42—21no—
152C2×C42—21no—
153C2×C42—22no—
154C2×C42—42no—
155C2×C42C2411yescharacteristic
156#107—22no—
157#107—22no—
158#107—42no—
159#107—42no—
160#107—42no—
161#107—44no—
162#107—81no—
163#107Q1614yescomplemented
164#106—82no—
165C4⋊C8—24no—
166C4⋊C8—21no—
167C4⋊C8—22no—
168C4⋊C8—21no—
169C4⋊C8—21no—
170C4⋊C8—22no—
171C4⋊C8—42no—
172C4⋊C8—44no—
173C4⋊C8C2×D412yes—
174#103—22no—
175#103—22no—
176#103—44no—
177#103—44no—
178#103—84no—
179C4×C8—24no—
180C4×C8—21no—
181C4×C8—21no—
182C4×C8—21no—
183C4×C8—21no—
184C4×C8—42no—
185C4×C8C2×D412yes—
186C4×C8#4112yes—
187#95—22no—
188#95—42no—
189#95—42no—
190#95—42no—
191#95—42no—
192#95—42no—
193C2×Q8—21no—
194C2×Q8—21no—
195C2×Q8—21no—
196C2×Q8—21no—
197C2×Q8—21no—
198C2×Q8—21no—
199C2×Q8—21no—
200C2×Q8—21no—
201C2×Q8—21no—
202C2×Q8—21no—
203C2×Q8—42no—
204C2×Q8—42no—
205C2×Q8—82no—
206C2×Q8—82no—
207C2×Q8#12112yes—
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—28no—
221C2×C8—21no—
222C2×C8—21no—
223C2×C8—21no—
224C2×C8—21no—
225C2×C8—22no—
226C2×C8—21no—
227C2×C8—22no—
228C2×C8—21no—
229C2×C8—21no—
230C2×C8—41no—
231C2×C8—41no—
232C2×C8—41no—
233C2×C8—41no—
234C2×C8—41no—
235C2×C8—42no—
236C2×C8—42no—
237C2×C8—42no—
238C2×C8—41no—
239C2×C8—41no—
240C2×C8#12112yes—
241C2×C8#12112yes—
242C2×C8C2×Q1612yes—
243C2×C8#13512yes—
244C2×C8#13712yes—
245C4⋊C4—22no—
246C4⋊C4—21no—
247C4⋊C4—22no—
248C4⋊C4—22no—
249C4⋊C4—21no—
250C4⋊C4—21no—
251C4⋊C4—22no—
252C4⋊C4—22no—
253C4⋊C4—41no—
254C4⋊C4—41no—
255C4⋊C4—42no—
256C4⋊C4—42no—
257C4⋊C4—42no—
258C4⋊C4—42no—
259C4⋊C4—42no—
260C4⋊C4—42no—
261C4⋊C4—42no—
262C4⋊C4—42no—
263C4⋊C4—82no—
264C4⋊C4—162no—
265C4⋊C4—162no—
266C4⋊C4#12112yes—
267C4⋊C4C2×Q1614yes—
268C42—22no—
269C42—22no—
270C42—41no—
271C42—41no—
272C42—42no—
273C42—42no—
274C42—162no—
275C42C22×D411yescharacteristic
276C42C2×#4111yescharacteristic
277C42C2×#4111yescharacteristic
278C422−1+411yescharacteristic
279C23D4211yescharacteristic
280Q8—22no—
281Q8—22no—
282Q8—22no—
283Q8—22no—
284Q8—41no—
285Q8—41no—
286Q8—41no—
287Q8—41no—
288Q8—84no—
289Q8—162no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—22no—
293C2×C4—22no—
294C2×C4—22no—
295C2×C4—24no—
296C2×C4—41no—
297C2×C4—42no—
298C2×C4—41no—
299C2×C4—41no—
300C2×C4—41no—
301C2×C4—41no—
302C2×C4—41no—
303C2×C4—42no—
304C2×C4—41no—
305C2×C4—42no—
306C2×C4—42no—
307C2×C4—41no—
308C2×C4—42no—
309C2×C4—41no—
310C2×C4—82no—
311C2×C4—82no—
312C2×C4—82no—
313C2×C4—82no—
314C2×C4—82no—
315C2×C4—82no—
316C2×C4C2×#12111yescharacteristic
317C2×C4C2×#12111yescharacteristic
318C2×C4#53711yescharacteristic
319C2×C4#54711yescharacteristic
320C2×C4#54811yescharacteristic
321C2×C4#55311yescharacteristic
322C2×C4C22×Q1611yescharacteristic
323C2×C4C2×#13511yescharacteristic
324C2×C4C2×#13511yescharacteristic
325C2×C4C2×#13711yescharacteristic
326C2×C4#57711yescharacteristic
327C2×C4#57811yescharacteristic
328C2×C4#57811yescharacteristic
329C2×C4#57811yescharacteristic
330C8—22no—
331C8—22no—
332C8—42no—
333C8—42no—
334C8—41no—
335C8—81no—
336C8—82no—
337C8#46214yes—
338C22#241611yescharacteristic
339C22#327811yescharacteristic
340C22#328311yescharacteristic
341C22#328511yescharacteristic
342C22D4×Q1611yescharacteristic
343C22#329011yescharacteristic
344C22#329211yescharacteristic
345C4—21no—
346C4—21no—
347C4—44no—
348C4—42no—
349C4—42no—
350C4—42no—
351C4—81no—
352C4—82no—
353C4—162no—
354C4—162no—
355C4C2×#46211yescharacteristic
356C4#304611yescharacteristic
357C4#305011yescharacteristic
358C4#305911yescharacteristic
359C4#333611yescharacteristic
360C4#334211yescharacteristic
361C4#336311yescharacteristic
362C4#337411yescharacteristic
363C2#1994211yescharacteristic
364C2#1995911yescharacteristic
365C2#2000611yescharacteristic
366C2#2000811yescharacteristic
367C2#2473211yescharacteristic
368C2#2477911yescharacteristic
369C2#2479711yescharacteristic
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
163. #107Q16semidirect

Maximal subgroups

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

subgroupindex
#286762
#246822
#227272
C2×#16762
C2×C8×Q162
#106012
#104152
#98232
#95942
#89742
#86452