Finite Group Databasebeta

← #330121

Subgroups — #330121

Order 512.

539 conjugacy classes of subgroups.

The subgroup lattice has 337 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#29926C212yesmaximal
2#29878C211yesmaximal, characteristic
3#21973C211yesmaximal, characteristic
4#10658C212yesmaximal, complemented
5#10598C212yesmaximal, complemented
6#10594C212yesmaximal, complemented
7#10545C212yesmaximal, complemented
8#9129C212yesmaximal
9#8600C211yesmaximal, characteristic
10C2×#500C2211yescharacteristic
11C2×#431C2211yescharacteristic
12C22×C4⋊C8C2211yescharacteristic
13#2857C2212yes—
14#2283C2211yescharacteristic
15#2097—22no—
16#2087—21no—
17#2087—21no—
18#2076—21no—
19#2076—21no—
20#2074—22no—
21#2036—22no—
22#2035—22no—
23#1990C2212yes—
24#1981C2212yes—
25#1981C2212yes—
26#1948C2212yes—
27#1947C2212yes—
28#1939C2212yes—
29#1939C2212yes—
30#1850—21no—
31#1850—21no—
32#1843—21no—
33#1834C2211yescharacteristic
34#1832—21no—
35#1797C2212yes—
36C2×#335C2211yescharacteristic
37C2×#334C2211yescharacteristic
38C2×#332C2212yes—
39C2×#332C2212yes—
40#1296C2212yes—
41#1296C2212yes—
42#1277C2212yes—
43C23×C8—21no—
44C2×C4⋊Q8C2312yes—
45C2×C22⋊Q8—22no—
46C22×C42C2311yescharacteristic
47#500—41no—
48#500—41no—
49#431—41no—
50#431—41no—
51C2×#107—21no—
52C2×#107—21no—
53C2×#107—22no—
54C2×#106—22no—
55C2×C4⋊C8—21no—
56C2×C4⋊C8—22no—
57C2×C4⋊C8—22no—
58C2×C4⋊C8—41no—
59C2×C4⋊C8—41no—
60C2×C4⋊C8D411yescharacteristic
61C2×C4⋊C8D411yescharacteristic
62C2×C4⋊C8C2312yes—
63C2×C4⋊C8C2312yes—
64C2×C4⋊C8C2312yes—
65C2×#103—22no—
66C2×#103—22no—
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#383—21no—
79#382D412yes—
80#381—21no—
81#380—22no—
82#380—22no—
83#380D411yescharacteristic
84#380D411yescharacteristic
85C4×C4⋊C4C2312yes—
86C4×C22⋊C4—21no—
87#376—21no—
88#340—21no—
89#340—21no—
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#334—41no—
99#332—42no—
100#332—42no—
101C22×Q8—22no—
102C23×C4C2×D411yescharacteristic
103C23×C4C2×D411yescharacteristic
104C23×C4#4111yescharacteristic
105C22×C8—22no—
106C22×C8—22no—
107C22×C8—22no—
108C22×C8—42no—
109C22×C8—82no—
110C22×C8C2×D411yescharacteristic
111C22×C8C2×D411yescharacteristic
112C22×C8C2×D411yescharacteristic
113C22×C8C2×D411yescharacteristic
114C4⋊Q8—22no—
115C4⋊Q8—22no—
116C4⋊Q8—22no—
117C4⋊Q8—22no—
118C22⋊Q8—82no—
119C22⋊Q8—82no—
120C2×C4⋊C4—22no—
121C2×C4⋊C4—22no—
122C2×C4⋊C4—22no—
123C2×C4⋊C4—22no—
124C2×C4⋊C4—22no—
125C2×C4⋊C4—42no—
126C2×C4⋊C4C2×D412yes—
127C2×C4⋊C4C2×D412yes—
128C2×C22⋊C4—21no—
129C2×C22⋊C4—21no—
130C2×C22⋊C4—42no—
131C2×C42—21no—
132C2×C42—21no—
133C2×C42—21no—
134C2×C42—21no—
135C2×C42—22no—
136C2×C42C2×D411yescharacteristic
137C2×C42C2×D411yescharacteristic
138C2×C42#4111yescharacteristic
139C2×C42C2411yescharacteristic
140#107—42no—
141#107—42no—
142#107—82no—
143#106—82no—
144C4⋊C8—22no—
145C4⋊C8—22no—
146C4⋊C8—22no—
147C4⋊C8—22no—
148C4⋊C8—22no—
149C4⋊C8—22no—
150C4⋊C8—41no—
151C4⋊C8—41no—
152C4⋊C8—42no—
153C4⋊C8—82no—
154C4⋊C8—82no—
155#103—82no—
156#103—82no—
157#103—82no—
158#103—82no—
159C22⋊C8—41no—
160C22⋊C8—41no—
161C22⋊C8—82no—
162C4×C8—21no—
163C4×C8—21no—
164C4×C8—21no—
165C4×C8—21no—
166#95—21no—
167#95—21no—
168#95—22no—
169#95—22no—
170#95—22no—
171#95—42no—
172#95—42no—
173C24#12111yescharacteristic
174C2×Q8—42no—
175C2×Q8—42no—
176C2×Q8—42no—
177C2×Q8—42no—
178C2×Q8—42no—
179C2×Q8—42no—
180C22×C4—22no—
181C22×C4—21no—
182C22×C4—21no—
183C22×C4—21no—
184C22×C4—21no—
185C22×C4—21no—
186C22×C4—21no—
187C22×C4—22no—
188C22×C4—22no—
189C22×C4—22no—
190C22×C4—22no—
191C22×C4—22no—
192C22×C4—22no—
193C22×C4—41no—
194C22×C4—42no—
195C22×C4—42no—
196C22×C4#12111yescharacteristic
197C22×C4#12111yescharacteristic
198C22×C4#12111yescharacteristic
199C22×C4C22×D411yescharacteristic
200C22×C4C22×D411yescharacteristic
201C22×C4C2×#4111yescharacteristic
202C2×C8—21no—
203C2×C8—21no—
204C2×C8—21no—
205C2×C8—21no—
206C2×C8—22no—
207C2×C8—22no—
208C2×C8—22no—
209C2×C8—22no—
210C2×C8—42no—
211C2×C8—42no—
212C2×C8—42no—
213C2×C8—42no—
214C2×C8—42no—
215C2×C8—42no—
216C2×C8—42no—
217C2×C8—42no—
218C2×C8—42no—
219C2×C8—84no—
220C4⋊C4—22no—
221C4⋊C4—22no—
222C4⋊C4—22no—
223C4⋊C4—22no—
224C4⋊C4—42no—
225C4⋊C4—42no—
226C4⋊C4—42no—
227C4⋊C4—42no—
228C4⋊C4—42no—
229C4⋊C4—42no—
230C4⋊C4—42no—
231C4⋊C4—42no—
232C4⋊C4—82no—
233C4⋊C4—84no—
234C22⋊C4—42no—
235C22⋊C4—42no—
236C22⋊C4—84no—
237C42—21no—
238C42—21no—
239C42—21no—
240C42—21no—
241C42—21no—
242C42—21no—
243C42—42no—
244C42—42no—
245C42H(5,𝔽2)11yescharacteristic
246C42H(5,𝔽2)11yescharacteristic
247C422−1+411yescharacteristic
248C422−1+411yescharacteristic
249C23—21no—
250C23—22no—
251C23—22no—
252C23—21no—
253C23#47312yes—
254C23C2×#12111yescharacteristic
255Q8—82no—
256Q8—82no—
257Q8—82no—
258Q8—82no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—21no—
264C2×C4—22no—
265C2×C4—22no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—42no—
269C2×C4—42no—
270C2×C4—41no—
271C2×C4—41no—
272C2×C4—42no—
273C2×C4—42no—
274C2×C4—41no—
275C2×C4—41no—
276C2×C4—42no—
277C2×C4—42no—
278C2×C4—42no—
279C2×C4—42no—
280C2×C4—42no—
281C2×C4—42no—
282C2×C4—82no—
283C2×C4—82no—
284C2×C4—82no—
285C2×C4—82no—
286C2×C4—82no—
287C2×C4—82no—
288C2×C4#53411yescharacteristic
289C2×C4#53611yescharacteristic
290C2×C4#53711yescharacteristic
291C2×C4#53711yescharacteristic
292C2×C4#53911yescharacteristic
293C2×C4#53911yescharacteristic
294C2×C4#57612yes—
295C2×C4#57712yes—
296C2×C4#57712yes—
297C2×C4#57812yes—
298C8—41no—
299C8—41no—
300C8—41no—
301C8—41no—
302C8—82no—
303C8—82no—
304C8—82no—
305C22—22no—
306C22—24no—
307C22—41no—
308C22—41no—
309C22—42no—
310C22#261811yescharacteristic
311C22#304612yes—
312C22#306212yes—
313C22C2×#47311yescharacteristic
314C22#306711yescharacteristic
315C4—21no—
316C4—21no—
317C4—41no—
318C4—41no—
319C4—41no—
320C4—82no—
321C4—82no—
322C4—162no—
323C4—162no—
324C4#320011yescharacteristic
325C4#320211yescharacteristic
326C4#324212yes—
327C4#325712yes—
328C4#325911yescharacteristic
329C4#326311yescharacteristic
330C2—42no—
331C2#1958412yes—
332C2#1959012yes—
333C2#2976611yescharacteristic
334C2#2978511yescharacteristic
335C2#2978911yescharacteristic
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. #10658C2semidirect
5. #10598C2semidirect
6. #10594C2semidirect
7. #10545C2semidirect

Maximal subgroups

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

subgroupindex
#299262
#298782
#219732
#106582
#105982
#105942
#105452
#91292
#86002