Finite Group Databasebeta

← #325178

Subgroups — #325178

Order 512.

494 conjugacy classes of subgroups.

The subgroup lattice has 309 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#30307C211yesmaximal, characteristic
2#29319C211yesmaximal, characteristic
3#29313C211yesmaximal, characteristic
4#22307C211yesmaximal, characteristic
5#10655C212yesmaximal
6#10652C212yesmaximal
7#9907C211yesmaximal, characteristic
8#9901C211yesmaximal, characteristic
9#9613C212yesmaximal
10#9612C212yesmaximal
11#8277C211yesmaximal, characteristic
12#2859C2211yescharacteristic
13#2859C2211yescharacteristic
14C4×C4⋊Q8C2211yescharacteristic
15#2085—22no—
16#2084—21no—
17#2084—21no—
18#2059—22no—
19#2058—21no—
20#2057—21no—
21#1984—22no—
22#1948C2212yes—
23#1948C2212yes—
24#1948C2212yes—
25#1947C2212yes—
26#1943—22no—
27#1941—22no—
28#1939C2212yes—
29#1916—22no—
30#1845C2211yescharacteristic
31#1844C2211yescharacteristic
32#1834C2211yescharacteristic
33#1834C2211yescharacteristic
34#1831C2211yescharacteristic
35C8×C4⋊C4C2212yes—
36#1767C2211yescharacteristic
37C4×C4⋊C8C2211yescharacteristic
38#1744C2211yescharacteristic
39C2×#334C2212yes—
40#1296C2212yes—
41#1296C2212yes—
42#1296C2212yes—
43#1296C2212yes—
44#1277C2212yes—
45C2×C4⋊Q8D411yescharacteristic
46C2×#125—21no—
47C2×#125—21no—
48C2×C4×Q8—21no—
49C2×#107—22no—
50C2×#107—22no—
51C2×#106—22no—
52C2×#106—22no—
53C2×#106—22no—
54C2×#106—22no—
55C2×C4⋊C8—22no—
56C2×C4⋊C8C2312yes—
57C2×C4⋊C8C2312yes—
58C2×C4⋊C8C2312yes—
59C2×#97—21no—
60C2×C4×C8—22no—
61C2×C4×C8—21no—
62C2×C4×C8C2311yescharacteristic
63C2×C4×C8C2311yescharacteristic
64#391—21no—
65#389—21no—
66#389—21no—
67#389—21no—
68#389—21no—
69#389—21no—
70#389—21no—
71#389—21no—
72#389—21no—
73#389—21no—
74#389D412yes—
75#384D411yescharacteristic
76#384C2311yescharacteristic
77#384C2311yescharacteristic
78#384C2311yescharacteristic
79#384C2311yescharacteristic
80#382—21no—
81#382—21no—
82C4×C4⋊C4—21no—
83C4×C4⋊C4C2312yes—
84C43C2311yescharacteristic
85#340—22no—
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#336D412yes—
98#336D412yes—
99#334Q818yescomplemented
100C22×Q8—21no—
101C22×C8—22no—
102C22×C8—22no—
103C22×C8—22no—
104C22×C8C2×D412yes—
105C22×C8C2×D412yes—
106C4⋊Q8—24no—
107#125—82no—
108#125—82no—
109C4×Q8—82no—
110C2×C4⋊C4—21no—
111C2×C4⋊C4—21no—
112C2×C4⋊C4—21no—
113C2×C4⋊C4—21no—
114C2×C4⋊C4—21no—
115C2×C4⋊C4—21no—
116C2×C4⋊C4—21no—
117C2×C4⋊C4—21no—
118C2×C4⋊C4—21no—
119C2×C4⋊C4—21no—
120C2×C4⋊C4—22no—
121C2×C4⋊C4—21no—
122C2×C4⋊C4—21no—
123C2×C4⋊C4—21no—
124C2×C4⋊C4—21no—
125C2×C4⋊C4—21no—
126C2×C4⋊C4—41no—
127C2×C4⋊C4—41no—
128C2×C4⋊C4—41no—
129C2×C4⋊C4—41no—
130C2×C4⋊C4C2×D412yes—
131C2×C4⋊C4C2×D412yes—
132C2×C42—22no—
133C2×C42C2×D411yescharacteristic
134C2×C42C2×D411yescharacteristic
135C2×C42C2×D411yescharacteristic
136C2×C42C2×D411yescharacteristic
137C2×C42#4111yescharacteristic
138C2×C42#4111yescharacteristic
139C2×C42C2411yescharacteristic
140#107—44no—
141#107—82no—
142#106—44no—
143#106—82no—
144#106—82no—
145#106—82no—
146C4⋊C8—22no—
147C4⋊C8—22no—
148C4⋊C8—24no—
149C4⋊C8—82no—
150C4⋊C8C2×Q814yes—
151C4⋊C8C2×Q814yes—
152#97—81no—
153C4×C8—22no—
154C4×C8—81no—
155C4×C8—82no—
156C4×C8C2×Q814yes—
157#95—22no—
158#95—22no—
159#95—41no—
160#95—41no—
161#95—41no—
162#95—41no—
163C2×Q8—24no—
164C2×Q8—81no—
165C2×Q8—81no—
166C22×C4—22no—
167C22×C4—22no—
168C22×C4—41no—
169C22×C4—41no—
170C22×C4—41no—
171C22×C4—41no—
172C22×C4#12111yescharacteristic
173C22×C4#12111yescharacteristic
174C22×C4#12111yescharacteristic
175C22×C4#12111yescharacteristic
176C22×C4C22×D411yescharacteristic
177C22×C4C22×D411yescharacteristic
178C22×C4C2×#4111yescharacteristic
179C2×C8—22no—
180C2×C8—22no—
181C2×C8—22no—
182C2×C8—24no—
183C2×C8—24no—
184C2×C8—22no—
185C2×C8—22no—
186C2×C8—42no—
187C2×C8—42no—
188C2×C8—42no—
189C2×C8—42no—
190C2×C8—42no—
191C2×C8—42no—
192C2×C8—42no—
193C2×C8C4⋊Q814yes—
194C4⋊C4—24no—
195C4⋊C4—22no—
196C4⋊C4—22no—
197C4⋊C4—22no—
198C4⋊C4—42no—
199C4⋊C4—42no—
200C4⋊C4—41no—
201C4⋊C4—41no—
202C4⋊C4—42no—
203C4⋊C4—42no—
204C4⋊C4—42no—
205C4⋊C4—42no—
206C4⋊C4—41no—
207C4⋊C4—41no—
208C4⋊C4—81no—
209C4⋊C4—82no—
210C4⋊C4—81no—
211C4⋊C4—81no—
212C4⋊C4—81no—
213C4⋊C4—81no—
214C4⋊C4—81no—
215C4⋊C4—82no—
216C4⋊C4—82no—
217C4⋊C4—82no—
218C4⋊C4—82no—
219C4⋊C4#13514yes—
220C42—22no—
221C42—21no—
222C42—21no—
223C42—22no—
224C42—22no—
225C42—41no—
226C42—41no—
227C42—42no—
228C42—42no—
229C42C22×Q812yes—
230C422−1+412yes—
231C23C2×#12111yescharacteristic
232Q8—84no—
233C2×C4—21no—
234C2×C4—21no—
235C2×C4—21no—
236C2×C4—21no—
237C2×C4—21no—
238C2×C4—21no—
239C2×C4—21no—
240C2×C4—21no—
241C2×C4—24no—
242C2×C4—21no—
243C2×C4—21no—
244C2×C4—21no—
245C2×C4—21no—
246C2×C4—21no—
247C2×C4—21no—
248C2×C4—42no—
249C2×C4—42no—
250C2×C4—42no—
251C2×C4—42no—
252C2×C4—42no—
253C2×C4—81no—
254C2×C4—81no—
255C2×C4—81no—
256C2×C4—81no—
257C2×C4—81no—
258C2×C4—81no—
259C2×C4—81no—
260C2×C4—81no—
261C2×C4—81no—
262C2×C4—81no—
263C2×C4—81no—
264C2×C4—81no—
265C2×C4C2×C4⋊Q811yescharacteristic
266C2×C4#53611yescharacteristic
267C2×C4D4×Q812yes—
268C2×C4#55712yes—
269C2×C4C2×#13512yes—
270C2×C4#57711yescharacteristic
271C2×C4#57711yescharacteristic
272C2×C4#57712yes—
273C2×C4#57712yes—
274C8—24no—
275C8—44no—
276C8—44no—
277C8—42no—
278C8—82no—
279C22#254611yescharacteristic
280C22#304211yescharacteristic
281C22#304311yescharacteristic
282C22#305011yescharacteristic
283C22#306011yescharacteristic
284C22#306811yescharacteristic
285C22#306911yescharacteristic
286C4—22no—
287C4—41no—
288C4—41no—
289C4—41no—
290C4—44no—
291C4—41no—
292C4—82no—
293C4—161no—
294C4—161no—
295C4—161no—
296C4—161no—
297C4#316312yes—
298C4#324412yes—
299C4#338112yes—
300C4#338112yes—
301C2#1939411yescharacteristic
302C2#1940011yescharacteristic
303C2#1951711yescharacteristic
304C2#1952511yescharacteristic
305C2#2867811yescharacteristic
306C2#2869611yescharacteristic
307C2#2870311yescharacteristic
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
99. #334Q8semidirect

Maximal subgroups

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

subgroupindex
#303072
#293192
#293132
#223072
#106552
#106522
#99072
#99012
#96132
#96122
#82772