Finite Group Databasebeta

← #330120

Subgroups — #330120

Order 512.

523 conjugacy classes of subgroups.

The subgroup lattice has 328 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#29942C212yesmaximal
2#29876C211yesmaximal, characteristic
3#21973C211yesmaximal, characteristic
4#10658C212yesmaximal, complemented
5#10601C212yesmaximal, complemented
6#10589C212yesmaximal, complemented
7#10543C212yesmaximal, complemented
8#9127C212yesmaximal
9#8600C211yesmaximal, characteristic
10C2×#500C2211yescharacteristic
11C2×#431C2211yescharacteristic
12C22×C4⋊C8C2211yescharacteristic
13#2857C2212yes—
14#2283C2211yescharacteristic
15#2101—22no—
16#2086—21no—
17#2086—21no—
18#2085—22no—
19#2077—22no—
20#2075—21no—
21#2075—21no—
22#2075—22no—
23#1990C2212yes—
24#1987C2212yes—
25#1987C2212yes—
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×#333C2212yes—
39C2×#333C2212yes—
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×C22⋊C8—22no—
66C2×C22⋊C8D411yescharacteristic
67C2×C22⋊C8D411yescharacteristic
68C2×C4×C8C2311yescharacteristic
69C2×C4×C8C2311yescharacteristic
70#391—22no—
71#389—22no—
72#389—22no—
73#384C2312yes—
74#383—21no—
75#382D412yes—
76#381—21no—
77#380—22no—
78#380—22no—
79#380D411yescharacteristic
80#380D411yescharacteristic
81C4×C4⋊C4C2312yes—
82C4×C22⋊C4—21no—
83#376—21no—
84#340—21no—
85#340—21no—
86#340—22no—
87#340—22no—
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#334—41no—
99#333—42no—
100#333—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—84no—
119C2×C4⋊C4—22no—
120C2×C4⋊C4—22no—
121C2×C4⋊C4—22no—
122C2×C4⋊C4—22no—
123C2×C4⋊C4—22no—
124C2×C4⋊C4—42no—
125C2×C4⋊C4C2×D412yes—
126C2×C4⋊C4C2×D412yes—
127C2×C22⋊C4—21no—
128C2×C22⋊C4—21no—
129C2×C22⋊C4—42no—
130C2×C42—21no—
131C2×C42—21no—
132C2×C42—21no—
133C2×C42—21no—
134C2×C42—22no—
135C2×C42C2×D411yescharacteristic
136C2×C42C2×D411yescharacteristic
137C2×C42#4111yescharacteristic
138C2×C42C2411yescharacteristic
139#107—42no—
140#107—42no—
141#107—82no—
142#106—82no—
143C4⋊C8—22no—
144C4⋊C8—22no—
145C4⋊C8—22no—
146C4⋊C8—22no—
147C4⋊C8—22no—
148C4⋊C8—22no—
149C4⋊C8—41no—
150C4⋊C8—41no—
151C4⋊C8—42no—
152C4⋊C8—82no—
153C4⋊C8—82no—
154C22⋊C8—41no—
155C22⋊C8—41no—
156C22⋊C8—82no—
157C4×C8—21no—
158C4×C8—21no—
159C4×C8—21no—
160C4×C8—21no—
161#95—21no—
162#95—21no—
163#95—22no—
164#95—22no—
165#95—22no—
166#95—42no—
167#95—42no—
168C24#12111yescharacteristic
169C2×Q8—42no—
170C2×Q8—42no—
171C2×Q8—82no—
172C2×Q8—82no—
173C22×C4—22no—
174C22×C4—21no—
175C22×C4—21no—
176C22×C4—21no—
177C22×C4—21no—
178C22×C4—21no—
179C22×C4—21no—
180C22×C4—22no—
181C22×C4—22no—
182C22×C4—22no—
183C22×C4—22no—
184C22×C4—22no—
185C22×C4—22no—
186C22×C4—41no—
187C22×C4—42no—
188C22×C4—42no—
189C22×C4#12111yescharacteristic
190C22×C4#12111yescharacteristic
191C22×C4#12111yescharacteristic
192C22×C4C22×D411yescharacteristic
193C22×C4C22×D411yescharacteristic
194C22×C4C2×#4111yescharacteristic
195C2×C8—21no—
196C2×C8—21no—
197C2×C8—21no—
198C2×C8—21no—
199C2×C8—22no—
200C2×C8—22no—
201C2×C8—22no—
202C2×C8—22no—
203C2×C8—42no—
204C2×C8—42no—
205C2×C8—42no—
206C2×C8—42no—
207C2×C8—42no—
208C2×C8—42no—
209C2×C8—42no—
210C2×C8—42no—
211C2×C8—42no—
212C2×C8—84no—
213C4⋊C4—22no—
214C4⋊C4—22no—
215C4⋊C4—22no—
216C4⋊C4—22no—
217C4⋊C4—42no—
218C4⋊C4—42no—
219C4⋊C4—42no—
220C4⋊C4—42no—
221C4⋊C4—42no—
222C4⋊C4—42no—
223C4⋊C4—42no—
224C4⋊C4—42no—
225C4⋊C4—82no—
226C4⋊C4—84no—
227C22⋊C4—42no—
228C22⋊C4—42no—
229C22⋊C4—84no—
230C42—21no—
231C42—21no—
232C42—21no—
233C42—21no—
234C42—21no—
235C42—21no—
236C42—42no—
237C42—42no—
238C42H(5,𝔽2)11yescharacteristic
239C42H(5,𝔽2)11yescharacteristic
240C422−1+411yescharacteristic
241C422−1+411yescharacteristic
242C23—21no—
243C23—22no—
244C23—22no—
245C23—21no—
246C23#47212yes—
247C23C2×#12111yescharacteristic
248Q8—162no—
249Q8—162no—
250C2×C4—21no—
251C2×C4—21no—
252C2×C4—21no—
253C2×C4—21no—
254C2×C4—21no—
255C2×C4—22no—
256C2×C4—22no—
257C2×C4—21no—
258C2×C4—21no—
259C2×C4—42no—
260C2×C4—42no—
261C2×C4—41no—
262C2×C4—41no—
263C2×C4—42no—
264C2×C4—42no—
265C2×C4—41no—
266C2×C4—41no—
267C2×C4—42no—
268C2×C4—42no—
269C2×C4—42no—
270C2×C4—42no—
271C2×C4—42no—
272C2×C4—42no—
273C2×C4—82no—
274C2×C4—82no—
275C2×C4—82no—
276C2×C4—82no—
277C2×C4—82no—
278C2×C4—82no—
279C2×C4#53411yescharacteristic
280C2×C4#53611yescharacteristic
281C2×C4#53711yescharacteristic
282C2×C4#53711yescharacteristic
283C2×C4#53911yescharacteristic
284C2×C4#53911yescharacteristic
285C2×C4#57712yes—
286C2×C4#57712yes—
287C2×C4#57812yes—
288C2×C4#57812yes—
289C8—41no—
290C8—41no—
291C8—41no—
292C8—41no—
293C8—82no—
294C8—82no—
295C8—82no—
296C22—22no—
297C22—24no—
298C22—41no—
299C22—41no—
300C22—42no—
301C22#261811yescharacteristic
302C22#304612yes—
303C22#306312yes—
304C22C2×#47211yescharacteristic
305C22#306711yescharacteristic
306C4—21no—
307C4—21no—
308C4—41no—
309C4—41no—
310C4—41no—
311C4—82no—
312C4—82no—
313C4—162no—
314C4—162no—
315C4#320011yescharacteristic
316C4#320211yescharacteristic
317C4#324212yes—
318C4#325812yes—
319C4#326111yescharacteristic
320C4#326311yescharacteristic
321C2—42no—
322C2#1958712yes—
323C2#1959112yes—
324C2#2976611yescharacteristic
325C2#2978611yescharacteristic
326C2#2978811yescharacteristic
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. #10601C2semidirect
6. #10589C2semidirect
7. #10543C2semidirect

Maximal subgroups

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

subgroupindex
#299422
#298762
#219732
#106582
#106012
#105892
#105432
#91272
#86002