Finite Group Databasebeta

← #330397

Subgroups — #330397

Order 512.

634 conjugacy classes of subgroups.

The subgroup lattice has 353 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33289C211yesmaximal, characteristic
2#33198C211yesmaximal, characteristic
3#33163C211yesmaximal, characteristic
4#21339C211yesmaximal, characteristic
5#10594C214yesmaximal
6#10478C214yesmaximal
7#9896C211yesmaximal, characteristic
8#9692C211yesmaximal, characteristic
9#8806C211yesmaximal, characteristic
10C2×#477—21no—
11C2×#462—21no—
12C2×#462—21no—
13C2×#462C2212yes—
14#2868C2211yescharacteristic
15#2868C2211yescharacteristic
16C42×Q8C2211yescharacteristic
17#1990—22no—
18#1990C2214yes—
19#1987—21no—
20#1984C2212yes—
21#1981—21no—
22#1981—21no—
23#1981C2212yes—
24#1947—21no—
25#1947C2212yes—
26#1939C2212yes—
27#1916—22no—
28#1842C2211yescharacteristic
29#1842C2211yescharacteristic
30#1834—21no—
31#1834—21no—
32#1803—21no—
33#1803C2212yes—
34C4×C4⋊C8—21no—
35C4×C4⋊C8C2211yescharacteristic
36#1744—21no—
37C42×C8C2211yescharacteristic
38C2×#335C2214yes—
39C2×#332C2214yes—
40#1296C2214yes—
41C22×Q16—22no—
42C2×C4⋊Q8—21no—
43C2×C4⋊Q8—21no—
44C2×C4⋊Q8—21no—
45C2×C4⋊Q8—21no—
46C2×C4⋊Q8—21no—
47C2×C4⋊Q8C2312yes—
48C2×C4×Q8D411yescharacteristic
49C2×C4×Q8D411yescharacteristic
50C2×C4×Q8C2311yescharacteristic
51#477—44no—
52#462—44no—
53#462—44no—
54#462—44no—
55C2×#107—22no—
56C2×C4⋊C8—22no—
57C2×C4⋊C8—21no—
58C2×C4⋊C8—21no—
59C2×C4⋊C8—21no—
60C2×C4⋊C8—21no—
61C2×C4⋊C8C2312yes—
62C2×C4⋊C8C2312yes—
63C2×C4⋊C8C2312yes—
64C2×#103—22no—
65C2×#103—22no—
66C2×#103—22no—
67C2×#103—22no—
68C2×C4×C8—21no—
69C2×C4×C8D411yescharacteristic
70C2×C4×C8D412yes—
71C2×C4×C8D411yescharacteristic
72C2×C4×C8C2312yes—
73#384—21no—
74#384—21no—
75#384—21no—
76#384—21no—
77#384—21no—
78#384—21no—
79#384—21no—
80#384C2312yes—
81C4×C4⋊C4—21no—
82C4×C4⋊C4—21no—
83C4×C4⋊C4—21no—
84C4×C4⋊C4D411yescharacteristic
85C4×C4⋊C4D411yescharacteristic
86C4×C4⋊C4C2311yescharacteristic
87C43—21no—
88C43C2311yescharacteristic
89#340—24no—
90#340—24no—
91#340—22no—
92#340—22no—
93#336—22no—
94#336—22no—
95#336—22no—
96#336—21no—
97#335—44no—
98#332—44no—
99C22×Q8—22no—
100C22×Q8—41no—
101C22×Q8C2×D411yescharacteristic
102C2×Q16—44no—
103C2×Q16—44no—
104C2×Q16—44no—
105C22×C8—22no—
106C22×C8—22no—
107C22×C8—22no—
108C22×C8C2×D412yes—
109C22×C8C2×D412yes—
110C4⋊Q8—24no—
111C4⋊Q8—24no—
112C4⋊Q8—44no—
113C4⋊Q8—42no—
114C4⋊Q8—42no—
115C4⋊Q8—42no—
116C4⋊Q8—42no—
117C4⋊Q8—42no—
118C4⋊Q8—42no—
119C4⋊Q8—82no—
120C4×Q8—22no—
121C4×Q8—22no—
122C4×Q8—22no—
123C4×Q8—22no—
124C4×Q8—22no—
125C4×Q8D814yes—
126C2×C4⋊C4—21no—
127C2×C4⋊C4—21no—
128C2×C4⋊C4—21no—
129C2×C4⋊C4—22no—
130C2×C4⋊C4—22no—
131C2×C4⋊C4—22no—
132C2×C4⋊C4—22no—
133C2×C4⋊C4—22no—
134C2×C4⋊C4—41no—
135C2×C4⋊C4—41no—
136C2×C4⋊C4—41no—
137C2×C4⋊C4—41no—
138C2×C4⋊C4—41no—
139C2×C4⋊C4—41no—
140C2×C4⋊C4—41no—
141C2×C4⋊C4C2×D411yescharacteristic
142C2×C4⋊C4C2×D411yescharacteristic
143C2×C4⋊C4C2×D411yescharacteristic
144C2×C42—21no—
145C2×C42—21no—
146C2×C42—21no—
147C2×C42—21no—
148C2×C42—21no—
149C2×C42—21no—
150C2×C42C2×D411yescharacteristic
151C2×C42C2×D411yescharacteristic
152C2×C42C2×D411yescharacteristic
153C2×C42C2×D411yescharacteristic
154C2×C42#4111yescharacteristic
155C2×C42#4111yescharacteristic
156C2×C42C2411yescharacteristic
157#107—44no—
158C4⋊C8—22no—
159C4⋊C8—24no—
160C4⋊C8—24no—
161C4⋊C8—22no—
162C4⋊C8—44no—
163C4⋊C8—42no—
164C4⋊C8—42no—
165C4⋊C8—42no—
166C4⋊C8—42no—
167#103—44no—
168#103—44no—
169#103—44no—
170#103—44no—
171C4×C8—22no—
172C4×C8—22no—
173C4×C8—22no—
174C4×C8—42no—
175C4×C8—42no—
176C4×C8Q1614yescomplemented
177#95—21no—
178#95—21no—
179#95—21no—
180C2×Q8—21no—
181C2×Q8—21no—
182C2×Q8—21no—
183C2×Q8—21no—
184C2×Q8—21no—
185C2×Q8—42no—
186C2×Q8—42no—
187C2×Q8—42no—
188C2×Q8—42no—
189C2×Q8—42no—
190C2×Q8—42no—
191C2×Q8—82no—
192C2×Q8—82no—
193C2×Q8—82no—
194C2×Q8C2×D812yes—
195C22×C4—21no—
196C22×C4—21no—
197C22×C4—21no—
198C22×C4—21no—
199C22×C4—42no—
200C22×C4—42no—
201C22×C4#12111yescharacteristic
202C22×C4#12111yescharacteristic
203C22×C4#12111yescharacteristic
204C22×C4#12111yescharacteristic
205C22×C4C22×D411yescharacteristic
206C22×C4C22×D411yescharacteristic
207C22×C4C2×#4111yescharacteristic
208Q16—88no—
209C2×C8—22no—
210C2×C8—22no—
211C2×C8—22no—
212C2×C8—22no—
213C2×C8—22no—
214C2×C8—42no—
215C2×C8—42no—
216C2×C8—42no—
217C2×C8—42no—
218C2×C8—42no—
219C2×C8—42no—
220C2×C8—42no—
221C2×C8—42no—
222C2×C8—42no—
223C2×C8C2×Q1614yes—
224C4⋊C4—21no—
225C4⋊C4—24no—
226C4⋊C4—21no—
227C4⋊C4—21no—
228C4⋊C4—21no—
229C4⋊C4—21no—
230C4⋊C4—42no—
231C4⋊C4—42no—
232C4⋊C4—44no—
233C4⋊C4—44no—
234C4⋊C4—42no—
235C4⋊C4—42no—
236C4⋊C4—44no—
237C4⋊C4—82no—
238C4⋊C4—82no—
239C4⋊C4—82no—
240C4⋊C4—82no—
241C4⋊C4—82no—
242C4⋊C4—82no—
243C4⋊C4—82no—
244C4⋊C4—82no—
245C4⋊C4C2×D812yes—
246C42—22no—
247C42—24no—
248C42—22no—
249C42—22no—
250C42—42no—
251C42—42no—
252C42—42no—
253C42—42no—
254C42—42no—
255C42C2×D812yes—
256C42C2×Q1612yes—
257C42#13512yes—
258C42AGL(1,ℤ/8)12yes—
259C42#13712yes—
260C42#13712yes—
261C42H(5,𝔽2)12yes—
262C422−1+412yes—
263C23C2×#12111yescharacteristic
264Q8—22no—
265Q8—22no—
266Q8—41no—
267Q8—41no—
268Q8—82no—
269Q8—82no—
270Q8—82no—
271Q8—82no—
272Q8—164no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—22no—
280C2×C4—22no—
281C2×C4—41no—
282C2×C4—41no—
283C2×C4—41no—
284C2×C4—41no—
285C2×C4—41no—
286C2×C4—41no—
287C2×C4—41no—
288C2×C4—41no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—82no—
292C2×C4—82no—
293C2×C4—82no—
294C2×C4—82no—
295C2×C4—82no—
296C2×C4—82no—
297C2×C4#46212yes—
298C2×C4#46212yes—
299C2×C4#46612yes—
300C2×C4#46612yes—
301C2×C4#46812yes—
302C2×C4#46812yes—
303C2×C4#47112yes—
304C2×C4#47312yes—
305C2×C4#53511yescharacteristic
306C2×C4#53611yescharacteristic
307C2×C4#53712yes—
308C2×C4#54212yes—
309C2×C4C22×D811yescharacteristic
310C2×C4C22×Q1611yescharacteristic
311C2×C4C2×#13511yescharacteristic
312C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
313C2×C4C2×#13711yescharacteristic
314C2×C4C2×#13711yescharacteristic
315C2×C4#57512yes—
316C8—24no—
317C8—42no—
318C8—82no—
319C8—82no—
320C8—82no—
321C22#262011yescharacteristic
322C22C2×#46211yescharacteristic
323C22#304011yescharacteristic
324C22C2×#46611yescharacteristic
325C22C2×#46811yescharacteristic
326C22C2×#47111yescharacteristic
327C22C2×#47311yescharacteristic
328C4—22no—
329C4—22no—
330C4—22no—
331C4—42no—
332C4—42no—
333C4—81no—
334C4—81no—
335C4—162no—
336C4—162no—
337C4#167414yes—
338C4#169214yes—
339C4#320511yescharacteristic
340C4#321011yescharacteristic
341C4#322611yescharacteristic
342C4#322811yescharacteristic
343C4#324912yes—
344C4#325212yes—
345C2C2×#167411yescharacteristic
346C2C2×#169211yescharacteristic
347C2#1927611yescharacteristic
348C2#1930011yescharacteristic
349C2#2983011yescharacteristic
350C2#2984211yescharacteristic
351C2#2985911yescharacteristic
subgroupG/H
center, socle, prime radicalC23C2×#121
commutator, Frattini, cosocleC2×C42C24
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C43 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
176. C4×C8Q16semidirect

Maximal subgroups

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

subgroupindex
#332892
#331982
#331632
#213392
#105942
#104782
#98962
#96922
#88062