Finite Group Databasebeta

← #329983

Subgroups — #329983

Order 512.

594 conjugacy classes of subgroups.

The subgroup lattice has 349 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33114C212yesmaximal
2#29876C211yesmaximal, characteristic
3#21972C211yesmaximal, characteristic
4#10655C212yesmaximal, complemented
5#10602C212yesmaximal, complemented
6#10594C212yesmaximal, complemented
7#10545C212yesmaximal, complemented
8#9129C212yesmaximal
9#8599C211yesmaximal, characteristic
10C2×#500C2211yescharacteristic
11C2×#485—22no—
12C2×#431C2211yescharacteristic
13C22×C4⋊C8C2211yescharacteristic
14#2857C2212yes—
15C4×#117C2211yescharacteristic
16#2086—21no—
17#2086—21no—
18#2075—21no—
19#2075—21no—
20#1990C2212yes—
21#1981C2212yes—
22#1981C2212yes—
23#1948C2212yes—
24#1947C2212yes—
25#1939C2212yes—
26#1939C2212yes—
27#1928—22no—
28#1927—22no—
29#1873—22no—
30#1850—21no—
31#1850—21no—
32#1842—21no—
33#1834C2211yescharacteristic
34#1831—21no—
35#1797C2212yes—
36C2×#335C2211yescharacteristic
37C2×#335C2211yescharacteristic
38C2×#332C2212yes—
39C2×#332C2212yes—
40#1296C2212yes—
41#1296C2212yes—
42#1277C2212yes—
43C23×C8—21no—
44C2×C4⋊Q8C2312yes—
45C2×C22⋊Q8—22no—
46C2×#117D411yescharacteristic
47C2×#117D411yescharacteristic
48C22×C42C2311yescharacteristic
49#500—41no—
50#500—41no—
51#485—48no—
52#431—41no—
53#431—41no—
54C2×#107—21no—
55C2×#107—21no—
56C2×#107—22no—
57C2×#106—22no—
58C2×C4⋊C8—21no—
59C2×C4⋊C8—22no—
60C2×C4⋊C8—22no—
61C2×C4⋊C8—41no—
62C2×C4⋊C8—41no—
63C2×C4⋊C8D411yescharacteristic
64C2×C4⋊C8D411yescharacteristic
65C2×C4⋊C8C2312yes—
66C2×C4⋊C8C2312yes—
67C2×C4⋊C8C2312yes—
68C2×#103—22no—
69C2×#103—22no—
70C2×#103—22no—
71C2×#103—22no—
72C2×C22⋊C8—22no—
73C2×C22⋊C8D411yescharacteristic
74C2×C22⋊C8D411yescharacteristic
75C2×C4×C8C2311yescharacteristic
76C2×C4×C8C2311yescharacteristic
77#391—22no—
78#389—22no—
79#389—22no—
80#384C2312yes—
81#380—22no—
82#380—22no—
83C4×C4⋊C4D412yes—
84C4×C4⋊C4C2312yes—
85C4×C22⋊C4—21no—
86C4×C22⋊C4—21no—
87#376—21no—
88C43—21no—
89#340—21no—
90#340—21no—
91#340—22no—
92#340—22no—
93#340—22no—
94#340—22no—
95#336—22no—
96#336—22no—
97#336—22no—
98#335—41no—
99#335—41no—
100#332—42no—
101#332—42no—
102C22×Q8—22no—
103C23×C4C2×D411yescharacteristic
104C23×C4C2×D411yescharacteristic
105C23×C4#4111yescharacteristic
106C22×C8—22no—
107C22×C8—22no—
108C22×C8—22no—
109C22×C8—42no—
110C22×C8—82no—
111C22×C8C2×D411yescharacteristic
112C22×C8C2×D411yescharacteristic
113C22×C8C2×D411yescharacteristic
114C22×C8C2×D411yescharacteristic
115C4⋊Q8—22no—
116C4⋊Q8—22no—
117C4⋊Q8—22no—
118C4⋊Q8—22no—
119C22⋊Q8—44no—
120C22⋊Q8—44no—
121#117—22no—
122#117—41no—
123#117—41no—
124#117D814yes—
125C2×C4⋊C4—22no—
126C2×C4⋊C4—22no—
127C2×C4⋊C4—22no—
128C2×C4⋊C4—22no—
129C2×C4⋊C4—22no—
130C2×C4⋊C4—42no—
131C2×C4⋊C4C2×D412yes—
132C2×C4⋊C4C2×D412yes—
133C2×C22⋊C4—21no—
134C2×C22⋊C4—21no—
135C2×C22⋊C4—42no—
136C2×C42—21no—
137C2×C42—21no—
138C2×C42—21no—
139C2×C42—21no—
140C2×C42—21no—
141C2×C42—21no—
142C2×C42—22no—
143C2×C42—22no—
144C2×C42C2×D411yescharacteristic
145C2×C42C2×D411yescharacteristic
146C2×C42#4111yescharacteristic
147C2×C42C2411yescharacteristic
148#107—42no—
149#107—42no—
150#107—44no—
151#106—44no—
152C4⋊C8—22no—
153C4⋊C8—22no—
154C4⋊C8—22no—
155C4⋊C8—22no—
156C4⋊C8—22no—
157C4⋊C8—22no—
158C4⋊C8—41no—
159C4⋊C8—41no—
160C4⋊C8—42no—
161C4⋊C8—82no—
162C4⋊C8—82no—
163#103—44no—
164#103—44no—
165#103—44no—
166#103—44no—
167C22⋊C8—41no—
168C22⋊C8—41no—
169C22⋊C8—44no—
170C4×C8—21no—
171C4×C8—21no—
172C4×C8—21no—
173C4×C8—21no—
174#95—22no—
175#95—22no—
176#95—42no—
177#95—42no—
178C24#12111yescharacteristic
179C2×Q8—42no—
180C2×Q8—42no—
181C2×Q8—42no—
182C2×Q8—42no—
183C2×Q8—42no—
184C2×Q8—42no—
185C22×C4—21no—
186C22×C4—21no—
187C22×C4—21no—
188C22×C4—21no—
189C22×C4—22no—
190C22×C4—21no—
191C22×C4—21no—
192C22×C4—22no—
193C22×C4—22no—
194C22×C4—21no—
195C22×C4—21no—
196C22×C4—22no—
197C22×C4—42no—
198C22×C4—42no—
199C22×C4—42no—
200C22×C4#12111yescharacteristic
201C22×C4#12111yescharacteristic
202C22×C4#12111yescharacteristic
203C22×C4C2×D812yes—
204C22×C4#13512yes—
205C22×C4C22×D411yescharacteristic
206C22×C4C22×D411yescharacteristic
207C22×C4C2×#4111yescharacteristic
208C2×C8—21no—
209C2×C8—21no—
210C2×C8—21no—
211C2×C8—21no—
212C2×C8—22no—
213C2×C8—22no—
214C2×C8—22no—
215C2×C8—22no—
216C2×C8—42no—
217C2×C8—42no—
218C2×C8—42no—
219C2×C8—42no—
220C2×C8—42no—
221C2×C8—42no—
222C2×C8—42no—
223C2×C8—42no—
224C2×C8—42no—
225C2×C8—84no—
226C4⋊C4—22no—
227C4⋊C4—22no—
228C4⋊C4—22no—
229C4⋊C4—42no—
230C4⋊C4—42no—
231C4⋊C4—42no—
232C4⋊C4—42no—
233C4⋊C4—42no—
234C4⋊C4—42no—
235C4⋊C4—42no—
236C4⋊C4—42no—
237C4⋊C4—82no—
238C4⋊C4—84no—
239C4⋊C4C2×D814yes—
240C22⋊C4—24no—
241C22⋊C4—42no—
242C22⋊C4—84no—
243C42—21no—
244C42—21no—
245C42—21no—
246C42—21no—
247C42—21no—
248C42—21no—
249C42—24no—
250C42—42no—
251C42—42no—
252C42—42no—
253C42—82no—
254C42H(5,𝔽2)11yescharacteristic
255C42H(5,𝔽2)11yescharacteristic
256C422−1+411yescharacteristic
257C422−1+411yescharacteristic
258C23—21no—
259C23—22no—
260C23—21no—
261C23—22no—
262C23#46612yes—
263C23C2×#12111yescharacteristic
264Q8—82no—
265Q8—82no—
266Q8—82no—
267Q8—82no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—22no—
271C2×C4—22no—
272C2×C4—22no—
273C2×C4—22no—
274C2×C4—22no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—22no—
278C2×C4—24no—
279C2×C4—22no—
280C2×C4—42no—
281C2×C4—42no—
282C2×C4—41no—
283C2×C4—41no—
284C2×C4—42no—
285C2×C4—42no—
286C2×C4—42no—
287C2×C4—42no—
288C2×C4—81no—
289C2×C4—81no—
290C2×C4—82no—
291C2×C4—82no—
292C2×C4—82no—
293C2×C4—82no—
294C2×C4—82no—
295C2×C4—82no—
296C2×C4#46612yes—
297C2×C4#53411yescharacteristic
298C2×C4#53611yescharacteristic
299C2×C4#53711yescharacteristic
300C2×C4#53711yescharacteristic
301C2×C4#53911yescharacteristic
302C2×C4#53911yescharacteristic
303C2×C4C22×D811yescharacteristic
304C2×C4C2×#13511yescharacteristic
305C2×C4#57511yescharacteristic
306C2×C4#57511yescharacteristic
307C2×C4#57712yes—
308C2×C4#57812yes—
309C8—41no—
310C8—41no—
311C8—41no—
312C8—41no—
313C8—82no—
314C8—82no—
315C8—82no—
316C22—22no—
317C22—24no—
318C22—41no—
319C22—41no—
320C22—42no—
321C22#261811yescharacteristic
322C22#304612yes—
323C22C2×#46611yescharacteristic
324C22#305411yescharacteristic
325C22#306912yes—
326C4—22no—
327C4—21no—
328C4—21no—
329C4—41no—
330C4—41no—
331C4—44no—
332C4—82no—
333C4—162no—
334C4—162no—
335C4#320011yescharacteristic
336C4#320211yescharacteristic
337C4#324212yes—
338C4#324511yescharacteristic
339C4#324511yescharacteristic
340C4#325011yescharacteristic
341C4#325011yescharacteristic
342C2—42no—
343C2#1941612yes—
344C2#1945412yes—
345C2#2976611yescharacteristic
346C2#2977011yescharacteristic
347C2#2979211yescharacteristic
subgroupG/H
center, socleC23C2×#121
commutator, Frattini, cosocleC2×C42C24
prime radicalC24#121
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #10655C2semidirect
5. #10602C2semidirect
6. #10594C2semidirect
7. #10545C2semidirect

Maximal subgroups

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

subgroupindex
#331142
#298762
#219722
#106552
#106022
#105942
#105452
#91292
#85992