Finite Group Databasebeta

← #324874

Subgroups — #324874

Order 512.

498 conjugacy classes of subgroups.

The subgroup lattice has 320 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#30229C211yesmaximal, characteristic
2#29347C211yesmaximal, characteristic
3#29319C211yesmaximal, characteristic
4#22307C211yesmaximal, characteristic
5#10598C212yesmaximal
6#10589C212yesmaximal
7#9901C211yesmaximal, characteristic
8#9709C211yesmaximal, characteristic
9#9567C212yesmaximal
10#9566C212yesmaximal
11#8277C211yesmaximal, characteristic
12#2862C2211yescharacteristic
13#2859C2211yescharacteristic
14C4×C4⋊Q8C2211yescharacteristic
15#2101—22no—
16#2101—22no—
17#2084—21no—
18#2072—21no—
19#2057—21no—
20#2024—21no—
21#1990C2211yescharacteristic
22#1990C2211yescharacteristic
23#1990C2211yescharacteristic
24#1990C2211yescharacteristic
25#1987C2212yes—
26#1984—22no—
27#1981C2212yes—
28#1948C2212yes—
29#1947C2212yes—
30#1943—22no—
31#1939C2212yes—
32#1933—22no—
33#1916—22no—
34#1831C2211yescharacteristic
35C8×C4⋊C4C2212yes—
36#1767C2211yescharacteristic
37C4×C4⋊C8C2211yescharacteristic
38#1744C2211yescharacteristic
39C2×#334C2212yes—
40C2×#333C2212yes—
41C2×#332C2212yes—
42#1296C2212yes—
43#1296C2212yes—
44#1277C2212yes—
45C2×C4⋊Q8D411yescharacteristic
46C2×C4⋊Q8C2311yescharacteristic
47C2×C4⋊Q8C2311yescharacteristic
48C2×#125—21no—
49C2×C4×Q8—21no—
50C2×#107—22no—
51C2×#106—22no—
52C2×C4⋊C8—22no—
53C2×C4⋊C8C2312yes—
54C2×C4⋊C8C2312yes—
55C2×C4⋊C8C2312yes—
56C2×#103—22no—
57C2×#103—22no—
58C2×#97—21no—
59C2×C4×C8—22no—
60C2×C4×C8—21no—
61C2×C4×C8C2311yescharacteristic
62C2×C4×C8C2311yescharacteristic
63#391—21no—
64#391—21no—
65#391—21no—
66#391—21no—
67#391—21no—
68#389—21no—
69#389—21no—
70#389—21no—
71#389—21no—
72#389—21no—
73#389D412yes—
74#384D411yescharacteristic
75#384C2311yescharacteristic
76#384C2311yescharacteristic
77#382—21no—
78#382—21no—
79#382—21no—
80C4×C4⋊C4—21no—
81C4×C4⋊C4C2312yes—
82C43C2311yescharacteristic
83#340—22no—
84#340—22no—
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#334—24no—
100#333—24no—
101#332—24no—
102C22×Q8—21no—
103C22×Q8—21no—
104C22×Q8—21no—
105C22×C8—22no—
106C22×C8—22no—
107C22×C8—22no—
108C22×C8C2×D412yes—
109C22×C8C2×D412yes—
110C4⋊Q8—24no—
111C4⋊Q8—22no—
112C4⋊Q8—22no—
113C4⋊Q8—22no—
114C4⋊Q8—22no—
115#125—82no—
116C4×Q8—82no—
117C2×C4⋊C4—21no—
118C2×C4⋊C4—21no—
119C2×C4⋊C4—21no—
120C2×C4⋊C4—21no—
121C2×C4⋊C4—21no—
122C2×C4⋊C4—21no—
123C2×C4⋊C4—21no—
124C2×C4⋊C4—21no—
125C2×C4⋊C4—21no—
126C2×C4⋊C4—21no—
127C2×C4⋊C4—22no—
128C2×C4⋊C4—21no—
129C2×C4⋊C4—21no—
130C2×C4⋊C4—21no—
131C2×C4⋊C4—41no—
132C2×C4⋊C4—41no—
133C2×C4⋊C4C2×D412yes—
134C2×C4⋊C4C2×D412yes—
135C2×C42—22no—
136C2×C42C2×D411yescharacteristic
137C2×C42C2×D411yescharacteristic
138C2×C42C2×D411yescharacteristic
139C2×C42C2×D411yescharacteristic
140C2×C42#4111yescharacteristic
141C2×C42#4111yescharacteristic
142C2×C42C2411yescharacteristic
143#107—44no—
144#106—82no—
145C4⋊C8—22no—
146C4⋊C8—22no—
147C4⋊C8—22no—
148C4⋊C8—22no—
149C4⋊C8—22no—
150C4⋊C8—82no—
151C4⋊C8#4114yes—
152#103—44no—
153#103—82no—
154#97—81no—
155C4×C8—22no—
156C4×C8—22no—
157C4×C8—81no—
158C4×C8—82no—
159#95—22no—
160#95—22no—
161#95—41no—
162#95—41no—
163#95—41no—
164#95—41no—
165#95—41no—
166#95—41no—
167C2×Q8—24no—
168C2×Q8—42no—
169C2×Q8—42no—
170C2×Q8—42no—
171C2×Q8—41no—
172C2×Q8—41no—
173C2×Q8—81no—
174C2×Q8—81no—
175C2×Q8—81no—
176C2×Q8—81no—
177C22×C4—22no—
178C22×C4—22no—
179C22×C4—41no—
180C22×C4—41no—
181C22×C4—41no—
182C22×C4—41no—
183C22×C4#12111yescharacteristic
184C22×C4#12111yescharacteristic
185C22×C4#12111yescharacteristic
186C22×C4#12111yescharacteristic
187C22×C4C22×D411yescharacteristic
188C22×C4C22×D411yescharacteristic
189C22×C4C2×#4111yescharacteristic
190C2×C8—22no—
191C2×C8—22no—
192C2×C8—24no—
193C2×C8—22no—
194C2×C8—22no—
195C2×C8—22no—
196C2×C8—22no—
197C2×C8—42no—
198C2×C8—42no—
199C2×C8—42no—
200C2×C8—42no—
201C2×C8—42no—
202C2×C8—42no—
203C2×C8—42no—
204C2×C8—42no—
205C4⋊C4—24no—
206C4⋊C4—22no—
207C4⋊C4—22no—
208C4⋊C4—22no—
209C4⋊C4—41no—
210C4⋊C4—41no—
211C4⋊C4—41no—
212C4⋊C4—41no—
213C4⋊C4—42no—
214C4⋊C4—42no—
215C4⋊C4—42no—
216C4⋊C4—41no—
217C4⋊C4—42no—
218C4⋊C4—41no—
219C4⋊C4—42no—
220C4⋊C4—81no—
221C4⋊C4—82no—
222C4⋊C4—81no—
223C4⋊C4—81no—
224C4⋊C4—81no—
225C4⋊C4—82no—
226C4⋊C4—82no—
227C4⋊C4#13514yes—
228C42—22no—
229C42—21no—
230C42—21no—
231C42—22no—
232C42—22no—
233C42—41no—
234C42—41no—
235C42—42no—
236C42—42no—
237C42C2×#4112yes—
238C422−1+412yes—
239C23C2×#12111yescharacteristic
240Q8—84no—
241Q8—82no—
242Q8—82no—
243Q8—162no—
244C2×C4—21no—
245C2×C4—21no—
246C2×C4—21no—
247C2×C4—21no—
248C2×C4—21no—
249C2×C4—21no—
250C2×C4—21no—
251C2×C4—21no—
252C2×C4—24no—
253C2×C4—21no—
254C2×C4—21no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—21no—
258C2×C4—21no—
259C2×C4—42no—
260C2×C4—42no—
261C2×C4—42no—
262C2×C4—42no—
263C2×C4—42no—
264C2×C4—81no—
265C2×C4—81no—
266C2×C4—81no—
267C2×C4—81no—
268C2×C4—81no—
269C2×C4—81no—
270C2×C4—81no—
271C2×C4—81no—
272C2×C4—81no—
273C2×C4—81no—
274C2×C4—81no—
275C2×C4—81no—
276C2×C4#53211yescharacteristic
277C2×C4#53611yescharacteristic
278C2×C4#54812yes—
279C2×C4#55612yes—
280C2×C4C2×#13512yes—
281C2×C4#57711yescharacteristic
282C2×C4#57711yescharacteristic
283C2×C4#57712yes—
284C2×C4#57812yes—
285C8—44no—
286C8—42no—
287C8—42no—
288C8—82no—
289C8—82no—
290C22#254511yescharacteristic
291C22#304211yescharacteristic
292C22#304411yescharacteristic
293C22#305011yescharacteristic
294C22#306011yescharacteristic
295C22#306811yescharacteristic
296C22#306911yescharacteristic
297C4—22no—
298C4—41no—
299C4—41no—
300C4—41no—
301C4—44no—
302C4—41no—
303C4—82no—
304C4—161no—
305C4—161no—
306C4—161no—
307C4—161no—
308C4#317812yes—
309C4#324212yes—
310C4#336012yes—
311C4#336512yes—
312C2#1939411yescharacteristic
313C2#1940711yescharacteristic
314C2#1951711yescharacteristic
315C2#1953211yescharacteristic
316C2#2864211yescharacteristic
317C2#2866611yescharacteristic
318C2#2867211yescharacteristic
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

Maximal subgroups

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

subgroupindex
#302292
#293472
#293192
#223072
#105982
#105892
#99012
#97092
#95672
#95662
#82772