Finite Group Databasebeta

← #344992

Subgroups — #344992

Order 512.

628 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26302C212yesmaximal, complemented
2#24812C212yesmaximal
3#10593C212yesmaximal, complemented
4#10481C212yesmaximal, complemented
5#10056C212yesmaximal, complemented
6#9825C211yesmaximal, characteristic, complemented
7#9657C211yesmaximal, characteristic
8#9594C211yesmaximal, characteristic, complemented
9#8882C212yesmaximal, complemented
10C2×#494C2211yescharacteristic
11C2×#476C2212yes—
12C2×C4×D8C2211yescharacteristic
13#2563C2212yes—
14D4×C4⋊C4C2212yes—
15#2085—22no—
16#2076—22no—
17#2076—22no—
18#2065—22no—
19#1990C2212yes—
20#1990C2212yes—
21#1987C2212yes—
22#1954C2212yes—
23#1947C2212yes—
24#1939C2212yes—
25#1879—22no—
26#1852—22no—
27#1845C2211yescharacteristic
28#1824—22no—
29#1804—22no—
30#1802C2212yes—
31C4×#107C2211yescharacteristic
32C2×#335C2211yescharacteristic
33C2×#335C2211yescharacteristic
34C2×#333C2212yes—
35C2×D4⋊C8C2212yes—
36C2×C82C2211yescharacteristic
37#1296C2212yes—
38#1279C2212yes—
39C22×Q16—41no—
40C22×D8D411yescharacteristic
41C2×C4⋊Q8C2312yes—
42C2×C4×D4—22no—
43C2×C4×D4C2312yes—
44C22×C4⋊C4—22no—
45#494—22no—
46#494—41no—
47#476—42no—
48#476—42no—
49C4×D8—41no—
50C4×D8—41no—
51C2×#107—22no—
52C2×#107—41no—
53C2×#107D411yescharacteristic
54C2×#107D411yescharacteristic
55C2×#107D411yescharacteristic
56C2×#106—22no—
57C2×C4⋊C8—22no—
58C2×C4⋊C8C2312yes—
59C2×C4⋊C8C2312yes—
60C2×#102—22no—
61C2×#102—22no—
62C2×#102D412yes—
63C2×C22⋊C8—22no—
64C2×C22⋊C8—22no—
65C2×C4×C8C2311yescharacteristic
66C2×C4×C8C2311yescharacteristic
67C2×C4×C8C2311yescharacteristic
68#393—42no—
69#390—22no—
70#389—22no—
71#385—42no—
72#384C2312yes—
73#382—22no—
74#380—22no—
75#380—22no—
76#380—22no—
77C4×C4⋊C4C2312yes—
78#340—22no—
79#340—22no—
80#340—22no—
81#340—22no—
82#340—22no—
83#340—22no—
84#340—22no—
85#340—22no—
86#340D412yes—
87#336—22no—
88#335—41no—
89#335—41no—
90#333—42no—
91D4⋊C8—24no—
92C82—41no—
93C22×Q8—22no—
94C22×D4C2×D412yes—
95C23×C4—22no—
96C23×C4—42no—
97C2×Q16—44no—
98C2×Q16—82no—
99C2×Q16—82no—
100C2×D8—41no—
101C2×D8—41no—
102C2×D8Q1614yescomplemented
103C22×C8—21no—
104C22×C8—22no—
105C22×C8—22no—
106C22×C8C2×D411yescharacteristic
107C22×C8C2×D411yescharacteristic
108C22×C8C2×D411yescharacteristic
109C22×C8C2×D411yescharacteristic
110C4⋊Q8—22no—
111C4⋊Q8—22no—
112C4⋊Q8—22no—
113C4⋊Q8—22no—
114C4×D4—22no—
115C4×D4—22no—
116C4×D4—22no—
117C4×D4—22no—
118C4×D4—44no—
119C4×D4—82no—
120C2×C4⋊C4—22no—
121C2×C4⋊C4—22no—
122C2×C4⋊C4—22no—
123C2×C4⋊C4—22no—
124C2×C4⋊C4—22no—
125C2×C4⋊C4—44no—
126C2×C4⋊C4—42no—
127C2×C4⋊C4—82no—
128C2×C4⋊C4C2×D412yes—
129C2×C4⋊C4C2×D412yes—
130C2×C4⋊C4C2×D412yes—
131C2×C22⋊C4—22no—
132C2×C22⋊C4—42no—
133C2×C22⋊C4—42no—
134C2×C22⋊C4—42no—
135C2×C42—22no—
136C2×C42C2411yescharacteristic
137#107—22no—
138#107—41no—
139#107—41no—
140#107—82no—
141#107—82no—
142#106—44no—
143C4⋊C8—22no—
144C4⋊C8—22no—
145C4⋊C8—22no—
146C4⋊C8—22no—
147C4⋊C8—82no—
148#102—42no—
149#102—44no—
150#102—82no—
151C22⋊C8—44no—
152C22⋊C8—82no—
153C4×C8—21no—
154C4×C8—21no—
155C4×C8—21no—
156C4×C8—21no—
157C4×C8—21no—
158C4×C8#4112yes—
159#95—22no—
160#95—42no—
161#95—42no—
162#95—42no—
163C24—22no—
164C2×Q8—42no—
165C2×Q8—42no—
166C2×Q8—82no—
167C2×Q8—82no—
168C2×D4—22no—
169C2×D4—22no—
170C2×D4—22no—
171C2×D4—22no—
172C2×D4—22no—
173C2×D4C2×Q1614yes—
174C22×C4—22no—
175C22×C4—22no—
176C22×C4—22no—
177C22×C4—42no—
178C22×C4—42no—
179C22×C4—42no—
180C22×C4—42no—
181C22×C4—42no—
182C22×C4—42no—
183C22×C4—42no—
184C22×C4—42no—
185C22×C4—84no—
186C22×C4—84no—
187C22×C4—162no—
188C22×C4C22×D411yescharacteristic
189C22×C4C22×D411yescharacteristic
190C22×C4H(5,𝔽2)11yescharacteristic
191Q16—88no—
192D8—44no—
193C2×C8—22no—
194C2×C8—22no—
195C2×C8—21no—
196C2×C8—21no—
197C2×C8—22no—
198C2×C8—21no—
199C2×C8—21no—
200C2×C8—21no—
201C2×C8—21no—
202C2×C8—21no—
203C2×C8—21no—
204C2×C8—21no—
205C2×C8—21no—
206C2×C8—42no—
207C2×C8—42no—
208C2×C8—42no—
209C2×C8—42no—
210C2×C8—42no—
211C2×C8—42no—
212C2×C8C2×Q1612yes—
213C2×C8#13512yes—
214C4⋊C4—22no—
215C4⋊C4—22no—
216C4⋊C4—22no—
217C4⋊C4—22no—
218C4⋊C4—22no—
219C4⋊C4—22no—
220C4⋊C4—42no—
221C4⋊C4—42no—
222C4⋊C4—42no—
223C4⋊C4—42no—
224C4⋊C4—42no—
225C4⋊C4—42no—
226C4⋊C4—42no—
227C4⋊C4—42no—
228C4⋊C4—82no—
229C4⋊C4—84no—
230C22⋊C4—42no—
231C22⋊C4—42no—
232C22⋊C4—84no—
233C22⋊C4—162no—
234C22⋊C4—162no—
235C42—42no—
236C42—42no—
237C42C2×#4111yescharacteristic
238C42H(5,𝔽2)11yescharacteristic
239C42H(5,𝔽2)11yescharacteristic
240C422−1+411yescharacteristic
241C23—24no—
242C23—42no—
243C23—42no—
244C23—42no—
245C23—42no—
246C23—42no—
247C23—42no—
248C23D4211yescharacteristic
249Q8—84no—
250Q8—162no—
251D4—24no—
252D4—24no—
253D4—42no—
254D4—42no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—42no—
258C2×C4—42no—
259C2×C4—42no—
260C2×C4—42no—
261C2×C4—42no—
262C2×C4—42no—
263C2×C4—42no—
264C2×C4—42no—
265C2×C4—42no—
266C2×C4—82no—
267C2×C4—82no—
268C2×C4—82no—
269C2×C4—82no—
270C2×C4—82no—
271C2×C4—82no—
272C2×C4—82no—
273C2×C4—82no—
274C2×C4—82no—
275C2×C4—82no—
276C2×C4—168no—
277C2×C4#53411yescharacteristic
278C2×C4#53711yescharacteristic
279C2×C4#53811yescharacteristic
280C2×C4#54711yescharacteristic
281C2×C4#55011yescharacteristic
282C2×C4#56211yescharacteristic
283C2×C4C22×Q1611yescharacteristic
284C2×C4C2×#13511yescharacteristic
285C2×C4#57511yescharacteristic
286C2×C4#57511yescharacteristic
287C2×C4#57611yescharacteristic
288C2×C4#57611yescharacteristic
289C2×C4#57811yescharacteristic
290C2×C4#57811yescharacteristic
291C8—24no—
292C8—22no—
293C8—22no—
294C8—41no—
295C8—41no—
296C8—82no—
297C8—82no—
298C22—44no—
299C22—44no—
300C22—44no—
301C22—82no—
302C22—82no—
303C22—82no—
304C22—82no—
305C22#268211yescharacteristic
306C22#327311yescharacteristic
307C22#327311yescharacteristic
308C22#328311yescharacteristic
309C22D4×Q1611yescharacteristic
310C22#329111yescharacteristic
311C22#329211yescharacteristic
312C4—21no—
313C4—21no—
314C4—82no—
315C4—82no—
316C4—82no—
317C4—162no—
318C4—162no—
319C4#318911yescharacteristic
320C4#320311yescharacteristic
321C4#325011yescharacteristic
322C4#326211yescharacteristic
323C4#333811yescharacteristic
324C4#335211yescharacteristic
325C4#339611yescharacteristic
326C4#340211yescharacteristic
327C2—84no—
328C2#1992711yescharacteristic
329C2#1995411yescharacteristic
330C2#2000611yescharacteristic
331C2#2030211yescharacteristic
332C2#3119011yescharacteristic
333C2#3121011yescharacteristic
334C2#3122811yescharacteristic
subgroupG/H
center, socleC23D42
commutator, Frattini, cosocleC2×C42C24
prime radicalC22×D8D4
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C2×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #26302C2semidirect
3. #10593C2semidirect
4. #10481C2semidirect
5. #10056C2semidirect
6. #9825C2semidirect
8. #9594C2semidirect
9. #8882C2semidirect
102. C2×D8Q16semidirect

Maximal subgroups

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

subgroupindex
#263022
#248122
#105932
#104812
#100562
#98252
#96572
#95942
#88822