Finite Group Databasebeta

← #329980

Subgroups — #329980

Order 512.

582 conjugacy classes of subgroups.

The subgroup lattice has 350 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33154C212yesmaximal
2#29808C211yesmaximal, characteristic
3#21972C211yesmaximal, characteristic
4#10656C212yesmaximal, complemented
5#10600C212yesmaximal, complemented
6#10592C212yesmaximal, complemented
7#10538C212yesmaximal, complemented
8#9131C212yesmaximal
9#8599C211yesmaximal, characteristic
10C2×#494C2211yescharacteristic
11C2×#431C2211yescharacteristic
12C22×C4⋊C8C2211yescharacteristic
13#2857C2212yes—
14C4×#117C2211yescharacteristic
15#2056—21no—
16#2056—21no—
17#2001—21no—
18#2001—21no—
19#1988C2212yes—
20#1988C2212yes—
21#1956C2212yes—
22#1948C2212yes—
23#1947C2212yes—
24#1939C2212yes—
25#1939C2212yes—
26#1921—22no—
27#1920—22no—
28#1854—22no—
29#1850—21no—
30#1850—21no—
31#1842—21no—
32#1834C2211yescharacteristic
33#1831—21no—
34#1825—22no—
35#1797C2212yes—
36C2×#335C2211yescharacteristic
37C2×#335C2211yescharacteristic
38C2×#332C2212yes—
39C2×#332C2212yes—
40#1296C2212yes—
41#1296C2212yes—
42#1277C2212yes—
43C22×Q16—21no—
44C22×Q16—21no—
45C23×C8—21no—
46C2×C4⋊Q8C2312yes—
47C2×C22⋊Q8—22no—
48C2×#117D411yescharacteristic
49C2×#117D411yescharacteristic
50C22×C42C2311yescharacteristic
51#494—41no—
52#494—41no—
53#431—41no—
54#431—41no—
55C2×#107—22no—
56C2×#106—22no—
57C2×C4⋊C8—21no—
58C2×C4⋊C8—22no—
59C2×C4⋊C8—22no—
60C2×C4⋊C8—41no—
61C2×C4⋊C8—41no—
62C2×C4⋊C8D411yescharacteristic
63C2×C4⋊C8D411yescharacteristic
64C2×C4⋊C8C2312yes—
65C2×C4⋊C8C2312yes—
66C2×C4⋊C8C2312yes—
67C2×#103—22no—
68C2×#103—22no—
69C2×C22⋊C8—22no—
70C2×C22⋊C8D411yescharacteristic
71C2×C22⋊C8D411yescharacteristic
72C2×C4×C8C2311yescharacteristic
73C2×C4×C8C2311yescharacteristic
74#391—22no—
75#389—22no—
76#389—22no—
77#384C2312yes—
78#380—22no—
79#380—22no—
80C4×C4⋊C4D412yes—
81C4×C4⋊C4C2312yes—
82C4×C22⋊C4—21no—
83C4×C22⋊C4—21no—
84#376—21no—
85C43—21no—
86#340—21no—
87#340—21no—
88#340—22no—
89#340—22no—
90#340—22no—
91#340—22no—
92#340—22no—
93#340—22no—
94#336—22no—
95#336—22no—
96#336—22no—
97#335—41no—
98#335—41no—
99#332—42no—
100#332—42no—
101C22×Q8—22no—
102C23×C4C2×D411yescharacteristic
103C23×C4C2×D411yescharacteristic
104C23×C4#4111yescharacteristic
105C2×Q16—42no—
106C2×Q16—42no—
107C2×Q16—82no—
108C2×Q16—82no—
109C22×C8—22no—
110C22×C8—22no—
111C22×C8—22no—
112C22×C8—42no—
113C22×C8—82no—
114C22×C8C2×D411yescharacteristic
115C22×C8C2×D411yescharacteristic
116C22×C8C2×D411yescharacteristic
117C22×C8C2×D411yescharacteristic
118C4⋊Q8—22no—
119C4⋊Q8—22no—
120C4⋊Q8—22no—
121C4⋊Q8—22no—
122C22⋊Q8—82no—
123C22⋊Q8—82no—
124#117—22no—
125#117—41no—
126#117—41no—
127#117Q1614yescomplemented
128C2×C4⋊C4—22no—
129C2×C4⋊C4—22no—
130C2×C4⋊C4—22no—
131C2×C4⋊C4—22no—
132C2×C4⋊C4—22no—
133C2×C4⋊C4—42no—
134C2×C4⋊C4C2×D412yes—
135C2×C4⋊C4C2×D412yes—
136C2×C22⋊C4—21no—
137C2×C22⋊C4—21no—
138C2×C22⋊C4—42no—
139C2×C42—21no—
140C2×C42—21no—
141C2×C42—21no—
142C2×C42—21no—
143C2×C42—21no—
144C2×C42—21no—
145C2×C42—22no—
146C2×C42—22no—
147C2×C42C2×D411yescharacteristic
148C2×C42C2×D411yescharacteristic
149C2×C42#4111yescharacteristic
150C2×C42C2411yescharacteristic
151#107—44no—
152#106—44no—
153C4⋊C8—22no—
154C4⋊C8—22no—
155C4⋊C8—22no—
156C4⋊C8—22no—
157C4⋊C8—22no—
158C4⋊C8—22no—
159C4⋊C8—41no—
160C4⋊C8—41no—
161C4⋊C8—42no—
162C4⋊C8—82no—
163C4⋊C8—82no—
164#103—82no—
165#103—82no—
166C22⋊C8—41no—
167C22⋊C8—41no—
168C22⋊C8—44no—
169C4×C8—21no—
170C4×C8—21no—
171C4×C8—21no—
172C4×C8—21no—
173#95—22no—
174#95—22no—
175#95—42no—
176#95—42no—
177C24#12111yescharacteristic
178C2×Q8—42no—
179C2×Q8—42no—
180C2×Q8—42no—
181C2×Q8—42no—
182C2×Q8—42no—
183C2×Q8—42no—
184C22×C4—21no—
185C22×C4—21no—
186C22×C4—21no—
187C22×C4—21no—
188C22×C4—22no—
189C22×C4—21no—
190C22×C4—21no—
191C22×C4—22no—
192C22×C4—22no—
193C22×C4—21no—
194C22×C4—21no—
195C22×C4—22no—
196C22×C4—42no—
197C22×C4—42no—
198C22×C4—42no—
199C22×C4#12111yescharacteristic
200C22×C4#12111yescharacteristic
201C22×C4#12111yescharacteristic
202C22×C4C2×Q1612yes—
203C22×C4#13512yes—
204C22×C4C22×D411yescharacteristic
205C22×C4C22×D411yescharacteristic
206C22×C4C2×#4111yescharacteristic
207Q16—84no—
208Q16—84no—
209C2×C8—21no—
210C2×C8—21no—
211C2×C8—21no—
212C2×C8—21no—
213C2×C8—22no—
214C2×C8—22no—
215C2×C8—22no—
216C2×C8—22no—
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—42no—
226C2×C8—84no—
227C4⋊C4—22no—
228C4⋊C4—22no—
229C4⋊C4—22no—
230C4⋊C4—42no—
231C4⋊C4—42no—
232C4⋊C4—42no—
233C4⋊C4—42no—
234C4⋊C4—42no—
235C4⋊C4—42no—
236C4⋊C4—42no—
237C4⋊C4—42no—
238C4⋊C4—82no—
239C4⋊C4—84no—
240C4⋊C4C2×Q1614yes—
241C22⋊C4—24no—
242C22⋊C4—42no—
243C22⋊C4—84no—
244C42—21no—
245C42—21no—
246C42—21no—
247C42—21no—
248C42—21no—
249C42—21no—
250C42—24no—
251C42—42no—
252C42—42no—
253C42—42no—
254C42—82no—
255C42H(5,𝔽2)11yescharacteristic
256C42H(5,𝔽2)11yescharacteristic
257C422−1+411yescharacteristic
258C422−1+411yescharacteristic
259C23—21no—
260C23—22no—
261C23—21no—
262C23—22no—
263C23#46712yes—
264C23C2×#12111yescharacteristic
265Q8—82no—
266Q8—82no—
267Q8—82no—
268Q8—82no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—22no—
272C2×C4—22no—
273C2×C4—22no—
274C2×C4—22no—
275C2×C4—22no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—22no—
279C2×C4—24no—
280C2×C4—22no—
281C2×C4—42no—
282C2×C4—42no—
283C2×C4—41no—
284C2×C4—41no—
285C2×C4—42no—
286C2×C4—42no—
287C2×C4—42no—
288C2×C4—42no—
289C2×C4—81no—
290C2×C4—81no—
291C2×C4—82no—
292C2×C4—82no—
293C2×C4—82no—
294C2×C4—82no—
295C2×C4—82no—
296C2×C4—82no—
297C2×C4#46712yes—
298C2×C4#53411yescharacteristic
299C2×C4#53611yescharacteristic
300C2×C4#53711yescharacteristic
301C2×C4#53711yescharacteristic
302C2×C4#53911yescharacteristic
303C2×C4#53911yescharacteristic
304C2×C4C22×Q1611yescharacteristic
305C2×C4C2×#13511yescharacteristic
306C2×C4#57511yescharacteristic
307C2×C4#57511yescharacteristic
308C2×C4#57612yes—
309C2×C4#57712yes—
310C8—41no—
311C8—41no—
312C8—41no—
313C8—41no—
314C8—82no—
315C8—82no—
316C8—82no—
317C22—22no—
318C22—24no—
319C22—41no—
320C22—41no—
321C22—42no—
322C22#261811yescharacteristic
323C22#304512yes—
324C22C2×#46711yescharacteristic
325C22#305411yescharacteristic
326C22#306912yes—
327C4—22no—
328C4—21no—
329C4—21no—
330C4—41no—
331C4—41no—
332C4—44no—
333C4—82no—
334C4—162no—
335C4—162no—
336C4#319911yescharacteristic
337C4#320011yescharacteristic
338C4#324112yes—
339C4#324911yescharacteristic
340C4#324911yescharacteristic
341C4#325011yescharacteristic
342C4#325011yescharacteristic
343C2—42no—
344C2#1941112yes—
345C2#1944912yes—
346C2#2976511yescharacteristic
347C2#2977111yescharacteristic
348C2#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. #10656C2semidirect
5. #10600C2semidirect
6. #10592C2semidirect
7. #10538C2semidirect
127. #117Q16semidirect

Maximal subgroups

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

subgroupindex
#331542
#298082
#219722
#106562
#106002
#105922
#105382
#91312
#85992