Finite Group Databasebeta

← #318744

Subgroups — #318744

Order 512.

538 conjugacy classes of subgroups.

The subgroup lattice has 339 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#29272C211yesmaximal, characteristic
2#29259C211yesmaximal, characteristic
3#22307C211yesmaximal, characteristic
4#22218C211yesmaximal, characteristic
5#10594C212yesmaximal
6#10592C212yesmaximal
7#9736C212yesmaximal
8#9611C212yesmaximal
9#8811C211yesmaximal, characteristic
10#8593C211yesmaximal, characteristic
11#8593C211yesmaximal, characteristic
12C2×#487C2211yescharacteristic
13C4×C4⋊Q8C2211yescharacteristic
14C4×C4⋊Q8C2211yescharacteristic
15C4×C4⋊Q8C2211yescharacteristic
16#2099—21no—
17#2095—21no—
18#2057—21no—
19#2024—21no—
20#1990C2211yescharacteristic
21#1988C2212yes—
22#1984—22no—
23#1981C2212yes—
24#1956C2211yescharacteristic
25#1947C2212yes—
26#1947C2212yes—
27#1939C2212yes—
28#1927—22no—
29#1918—22no—
30#1916—22no—
31#1865—22no—
32#1863—22no—
33#1844C2211yescharacteristic
34#1831C2211yescharacteristic
35#1831C2211yescharacteristic
36#1831C2211yescharacteristic
37C8×C4⋊C4C2212yes—
38#1744C2211yescharacteristic
39C2×#335C2212yes—
40C2×#334C2212yes—
41C2×#332C2212yes—
42C2×#332C2212yes—
43#1296C2212yes—
44#1296C2212yes—
45C2×C4⋊Q8D411yescharacteristic
46C2×C4⋊Q8C2311yescharacteristic
47C2×C4⋊Q8C2311yescharacteristic
48C2×C4×Q8—21no—
49C2×C4×Q8—21no—
50C2×C4×Q8—21no—
51#487—22no—
52#487Q814yescomplemented
53C2×#107—22no—
54C2×#107—22no—
55C2×#106—22no—
56C2×C4⋊C8—22no—
57C2×C4⋊C8C2312yes—
58C2×C4⋊C8C2312yes—
59C2×C4⋊C8C2312yes—
60C2×#103—22no—
61C2×#103—22no—
62C2×#103—22no—
63C2×#103—22no—
64C2×C4×C8—22no—
65C2×C4×C8C2311yescharacteristic
66C2×C4×C8C2311yescharacteristic
67#391—21no—
68#391—21no—
69#391—21no—
70#389—21no—
71#389—21no—
72#389—21no—
73#389—21no—
74#389—21no—
75#389D412yes—
76#384D411yescharacteristic
77#384C2311yescharacteristic
78#384C2311yescharacteristic
79C4×C4⋊C4—21no—
80C4×C4⋊C4—21no—
81C4×C4⋊C4—21no—
82C4×C4⋊C4—21no—
83C4×C4⋊C4—21no—
84C4×C4⋊C4C2312yes—
85C43C2311yescharacteristic
86#340—22no—
87#340—22no—
88#340—22no—
89#340—22no—
90#340—22no—
91#336—22no—
92#336—22no—
93#336—21no—
94#336—21no—
95#336—21no—
96#336—21no—
97#336—21no—
98#336—21no—
99#336D412yes—
100#336D412yes—
101#335—42no—
102#334—24no—
103#332—42no—
104#332—42no—
105C22×Q8—21no—
106C22×Q8—21no—
107C22×Q8—21no—
108C22×C8—22no—
109C22×C8—22no—
110C22×C8—22no—
111C22×C8C2×D412yes—
112C22×C8C2×D412yes—
113C4⋊Q8—24no—
114C4⋊Q8—21no—
115C4⋊Q8—21no—
116C4⋊Q8—21no—
117C4⋊Q8—21no—
118C4⋊Q8—21no—
119C4⋊Q8—21no—
120C4⋊Q8C2×Q812yes—
121C4⋊Q8C2×Q812yes—
122C4×Q8—82no—
123C4×Q8—82no—
124C4×Q8—82no—
125C2×C4⋊C4—21no—
126C2×C4⋊C4—21no—
127C2×C4⋊C4—21no—
128C2×C4⋊C4—21no—
129C2×C4⋊C4—21no—
130C2×C4⋊C4—21no—
131C2×C4⋊C4—21no—
132C2×C4⋊C4—21no—
133C2×C4⋊C4—21no—
134C2×C4⋊C4—21no—
135C2×C4⋊C4—22no—
136C2×C4⋊C4—21no—
137C2×C4⋊C4—21no—
138C2×C4⋊C4—21no—
139C2×C4⋊C4—41no—
140C2×C4⋊C4—41no—
141C2×C4⋊C4C2×D412yes—
142C2×C4⋊C4C2×D412yes—
143C2×C42—22no—
144C2×C42—41no—
145C2×C42—41no—
146C2×C42C2×D411yescharacteristic
147C2×C42C2×D411yescharacteristic
148C2×C42C2×D411yescharacteristic
149C2×C42C2×D411yescharacteristic
150C2×C42#4111yescharacteristic
151C2×C42#4111yescharacteristic
152C2×C42C2411yescharacteristic
153#107—44no—
154#107—82no—
155#106—82no—
156C4⋊C8—22no—
157C4⋊C8—22no—
158C4⋊C8—22no—
159C4⋊C8—22no—
160C4⋊C8—22no—
161C4⋊C8—22no—
162C4⋊C8—82no—
163#103—28no—
164#103—44no—
165#103—82no—
166#103—82no—
167C4×C8—21no—
168C4×C8—21no—
169C4×C8—82no—
170C4×C8C2×Q812yes—
171C4×C8#4112yes—
172#95—22no—
173#95—22no—
174#95—41no—
175#95—41no—
176#95—41no—
177#95—41no—
178C2×Q8—24no—
179C2×Q8—22no—
180C2×Q8—22no—
181C2×Q8—41no—
182C2×Q8—41no—
183C2×Q8—42no—
184C2×Q8—42no—
185C2×Q8—42no—
186C2×Q8—42no—
187C2×Q8—81no—
188C2×Q8—81no—
189C22×C4—22no—
190C22×C4—22no—
191C22×C4—41no—
192C22×C4—41no—
193C22×C4—41no—
194C22×C4—41no—
195C22×C4#12111yescharacteristic
196C22×C4#12111yescharacteristic
197C22×C4#12111yescharacteristic
198C22×C4#12111yescharacteristic
199C22×C4C22×D411yescharacteristic
200C22×C4C22×D411yescharacteristic
201C22×C4C2×#4111yescharacteristic
202C2×C8—22no—
203C2×C8—22no—
204C2×C8—22no—
205C2×C8—22no—
206C2×C8—22no—
207C2×C8—42no—
208C2×C8—42no—
209C2×C8—42no—
210C2×C8—42no—
211C2×C8—42no—
212C2×C8—42no—
213C2×C8—42no—
214C2×C8—42no—
215C2×C8—42no—
216C2×C8C22⋊Q814yes—
217C4⋊C4—24no—
218C4⋊C4—22no—
219C4⋊C4—22no—
220C4⋊C4—22no—
221C4⋊C4—22no—
222C4⋊C4—22no—
223C4⋊C4—22no—
224C4⋊C4—22no—
225C4⋊C4—41no—
226C4⋊C4—41no—
227C4⋊C4—41no—
228C4⋊C4—41no—
229C4⋊C4—42no—
230C4⋊C4—42no—
231C4⋊C4—42no—
232C4⋊C4—42no—
233C4⋊C4—81no—
234C4⋊C4—82no—
235C4⋊C4—81no—
236C4⋊C4—81no—
237C4⋊C4—81no—
238C4⋊C4—82no—
239C4⋊C4—82no—
240C4⋊C4#13514yes—
241C42—22no—
242C42—22no—
243C42—22no—
244C42—22no—
245C42—41no—
246C42—41no—
247C42—42no—
248C42—42no—
249C42—82no—
250C42—82no—
251C42C22×Q811yescharacteristic
252C42C2×#4111yescharacteristic
253C42H(5,𝔽2)11yescharacteristic
254C422−1+411yescharacteristic
255C23C2×#12111yescharacteristic
256Q8—44no—
257Q8—44no—
258Q8—84no—
259Q8—82no—
260Q8—82no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—24no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—42no—
277C2×C4—42no—
278C2×C4—42no—
279C2×C4—42no—
280C2×C4—42no—
281C2×C4—42no—
282C2×C4—42no—
283C2×C4—81no—
284C2×C4—81no—
285C2×C4—81no—
286C2×C4—81no—
287C2×C4—81no—
288C2×C4—81no—
289C2×C4—81no—
290C2×C4—81no—
291C2×C4—81no—
292C2×C4—81no—
293C2×C4C2×C22⋊Q811yescharacteristic
294C2×C4#53711yescharacteristic
295C2×C4D4×Q811yescharacteristic
296C2×C4#55011yescharacteristic
297C2×C4#55411yescharacteristic
298C2×C4#55611yescharacteristic
299C2×C4C2×#13512yes—
300C2×C4#57612yes—
301C2×C4#57712yes—
302C2×C4#57812yes—
303C8—24no—
304C8—42no—
305C8—82no—
306C8—82no—
307C8—82no—
308C22#253811yescharacteristic
309C22#304111yescharacteristic
310C22#304411yescharacteristic
311C22#305011yescharacteristic
312C22#305711yescharacteristic
313C22#306911yescharacteristic
314C22#306911yescharacteristic
315C4—21no—
316C4—21no—
317C4—41no—
318C4—41no—
319C4—41no—
320C4—44no—
321C4—41no—
322C4—82no—
323C4—82no—
324C4—82no—
325C4—161no—
326C4—161no—
327C4#308312yes—
328C4#326212yes—
329C4#338112yes—
330C4#339412yes—
331C2#1938811yescharacteristic
332C2#1940711yescharacteristic
333C2#1950511yescharacteristic
334C2#1952811yescharacteristic
335C2#2826611yescharacteristic
336C2#2828811yescharacteristic
337C2#2830011yescharacteristic
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
52. #487Q8semidirect

Maximal subgroups

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

subgroupindex
#292722
#292592
#223072
#222182
#105942
#105922
#97362
#96112
#88112
#85932