Finite Group Databasebeta

← #315134

Subgroups — #315134

Order 512.

541 conjugacy classes of subgroups.

The subgroup lattice has 314 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#28811C211yesmaximal, characteristic
2#28682C211yesmaximal, characteristic
3#27729C212yesmaximal
4#10414C212yesmaximal
5#10355C212yesmaximal
6#9836C211yesmaximal, characteristic
7#9730C211yesmaximal, characteristic
8#9568C212yesmaximal
9#8651C211yesmaximal, characteristic
10#8285C211yesmaximal, characteristic
11#8211C211yesmaximal, characteristic
12C2×#500C2211yescharacteristic
13C2×#499—21no—
14C2×#446C2211yescharacteristic
15#2567C2212yes—
16#2552C2212yes—
17#2100—22no—
18#2086—22no—
19#2072—22no—
20#2059—22no—
21#1990C2212yes—
22#1987C2212yes—
23#1987C2212yes—
24#1956C2212yes—
25#1947C2212yes—
26#1944—21no—
27#1944—21no—
28#1941—21no—
29#1933—22no—
30#1844C2211yescharacteristic
31#1834—22no—
32#1803C2212yes—
33C4×#106C2211yescharacteristic
34C8×C4⋊C4C2211yescharacteristic
35C8×C4⋊C4C2212yes—
36C4×#103C2211yescharacteristic
37C2×#335C2211yescharacteristic
38C2×#333C2212yes—
39C2×Q8⋊C8C2211yescharacteristic
40C2×#322C2211yescharacteristic
41#1296C2212yes—
42#1279C2211yescharacteristic
43#1279C2212yes—
44#1277C2211yescharacteristic
45C2×C4⋊Q8C2312yes—
46C2×#125—22no—
47C2×C4×Q8C2311yescharacteristic
48#500—22no—
49#500—22no—
50#499—44no—
51#446—22no—
52#446—22no—
53C2×#107—21no—
54C2×#107—21no—
55C2×#107—22no—
56C2×#106—21no—
57C2×#106—21no—
58C2×#106D411yescharacteristic
59C2×#106D411yescharacteristic
60C2×C4⋊C8—21no—
61C2×C4⋊C8C2312yes—
62C2×C4⋊C8C2311yescharacteristic
63C2×C4⋊C8C2311yescharacteristic
64C2×#103—22no—
65C2×#103—42no—
66C2×#103D412yes—
67C2×#97—21no—
68C2×C4×C8—22no—
69C2×C4×C8—21no—
70C2×C4×C8C2311yescharacteristic
71C2×C4×C8C2311yescharacteristic
72C2×C4×C8C2311yescharacteristic
73#398—42no—
74#395—42no—
75#391—22no—
76#389—22no—
77#389—22no—
78#389—22no—
79#389—22no—
80#384—22no—
81#384—22no—
82#384C2312yes—
83C4×C4⋊C4C2312yes—
84C4×C4⋊C4C2311yescharacteristic
85#340—22no—
86#340—22no—
87#340—22no—
88#340—22no—
89#340—22no—
90#340—22no—
91#340—22no—
92#340—22no—
93#340D412yes—
94#340D412yes—
95#336—21no—
96#336—22no—
97#335—22no—
98#333—24no—
99Q8⋊C8—22no—
100#322—22no—
101C22×Q8—22no—
102C22×Q8C2×D411yescharacteristic
103C22×C8—21no—
104C22×C8—22no—
105C22×C8—21no—
106C22×C8—21no—
107C22×C8C2×D412yes—
108C22×C8C2×D411yescharacteristic
109C22×C8C2×D411yescharacteristic
110C4⋊Q8—22no—
111C4⋊Q8—22no—
112C4⋊Q8—22no—
113C4⋊Q8—22no—
114#125—44no—
115#125—82no—
116C4×Q8—21no—
117C4×Q8—21no—
118C4×Q8—21no—
119C4×Q8—21no—
120C2×C4⋊C4—21no—
121C2×C4⋊C4—22no—
122C2×C4⋊C4—22no—
123C2×C4⋊C4—22no—
124C2×C4⋊C4—22no—
125C2×C4⋊C4—22no—
126C2×C4⋊C4—42no—
127C2×C4⋊C4—42no—
128C2×C4⋊C4—42no—
129C2×C4⋊C4—42no—
130C2×C4⋊C4—42no—
131C2×C4⋊C4C2×D411yescharacteristic
132C2×C4⋊C4C2×D412yes—
133C2×C4⋊C4C2×D412yes—
134C2×C4⋊C4C2×D412yes—
135C2×C42—21no—
136C2×C42—21no—
137C2×C42—22no—
138C2×C42C2411yescharacteristic
139#107—42no—
140#107—42no—
141#107—44no—
142#106—42no—
143#106—81no—
144#106SD1614yes—
145#106Q1614yes—
146C4⋊C8—22no—
147C4⋊C8—22no—
148C4⋊C8—21no—
149C4⋊C8—21no—
150C4⋊C8—21no—
151C4⋊C8—42no—
152C4⋊C8#4112yes—
153#103—42no—
154#103—44no—
155#103—84no—
156#97—42no—
157C4×C8—21no—
158C4×C8—21no—
159C4×C8—21no—
160C4×C8—21no—
161C4×C8—44no—
162C4×C8—42no—
163C4×C8#4112yes—
164C4×C8#4112yes—
165#95—22no—
166#95—21no—
167#95—42no—
168#95—42no—
169#95—42no—
170C2×Q8—21no—
171C2×Q8—21no—
172C2×Q8—22no—
173C2×Q8—22no—
174C2×Q8—42no—
175C2×Q8—42no—
176C2×Q8—82no—
177C2×Q8—82no—
178C22×C4—22no—
179C22×C4—22no—
180C22×C4—22no—
181C22×C4—21no—
182C22×C4—21no—
183C22×C4—42no—
184C22×C4—42no—
185C22×C4C22×D411yescharacteristic
186C22×C4C22×D411yescharacteristic
187C22×C4H(5,𝔽2)11yescharacteristic
188C2×C8—21no—
189C2×C8—21no—
190C2×C8—22no—
191C2×C8—22no—
192C2×C8—22no—
193C2×C8—22no—
194C2×C8—41no—
195C2×C8—41no—
196C2×C8—42no—
197C2×C8—41no—
198C2×C8—42no—
199C2×C8—41no—
200C2×C8—42no—
201C2×C8—42no—
202C2×C8—41no—
203C2×C8—41no—
204C2×C8#12412yes—
205C2×C8#12412yes—
206C2×C8C2×SD1612yes—
207C2×C8C2×Q1612yes—
208C4⋊C4—22no—
209C4⋊C4—22no—
210C4⋊C4—21no—
211C4⋊C4—21no—
212C4⋊C4—22no—
213C4⋊C4—22no—
214C4⋊C4—41no—
215C4⋊C4—41no—
216C4⋊C4—42no—
217C4⋊C4—42no—
218C4⋊C4—42no—
219C4⋊C4—42no—
220C4⋊C4—42no—
221C4⋊C4—42no—
222C4⋊C4—42no—
223C4⋊C4—42no—
224C4⋊C4—84no—
225C4⋊C4—84no—
226C4⋊C4—82no—
227C4⋊C4—162no—
228C4⋊C4—162no—
229C4⋊C4—162no—
230C4⋊C4C2×SD1614yes—
231C4⋊C4C2×Q1614yes—
232C42—41no—
233C42—41no—
234C42—41no—
235C42—42no—
236C42—41no—
237C42—42no—
238C42C2×#4111yescharacteristic
239C42C2×#4111yescharacteristic
240C42C2×#4111yescharacteristic
241C422−1+411yescharacteristic
242C23D4211yescharacteristic
243Q8—42no—
244Q8—41no—
245Q8—41no—
246Q8—84no—
247Q8—162no—
248C2×C4—22no—
249C2×C4—24no—
250C2×C4—24no—
251C2×C4—42no—
252C2×C4—42no—
253C2×C4—42no—
254C2×C4—42no—
255C2×C4—41no—
256C2×C4—41no—
257C2×C4—42no—
258C2×C4—42no—
259C2×C4—42no—
260C2×C4—42no—
261C2×C4—42no—
262C2×C4—82no—
263C2×C4—82no—
264C2×C4—82no—
265C2×C4—82no—
266C2×C4—82no—
267C2×C4—82no—
268C2×C4C2×#12411yescharacteristic
269C2×C4#53211yescharacteristic
270C2×C4#54211yescharacteristic
271C2×C4#54711yescharacteristic
272C2×C4#54811yescharacteristic
273C2×C4#55511yescharacteristic
274C2×C4C22×SD1611yescharacteristic
275C2×C4C22×Q1611yescharacteristic
276C2×C4#57512yes—
277C2×C4#57711yescharacteristic
278C2×C4#57812yes—
279C2×C4#57811yescharacteristic
280C8—22no—
281C8—42no—
282C8—42no—
283C8—81no—
284C8—81no—
285C8—82no—
286C8#48714yes—
287C22#249111yescharacteristic
288C22#326912yes—
289C22D4×SD1611yescharacteristic
290C22D4×Q1611yescharacteristic
291C22#328812yes—
292C4—21no—
293C4—21no—
294C4—44no—
295C4—44no—
296C4—82no—
297C4—81no—
298C4—81no—
299C4—162no—
300C4—162no—
301C4C2×#48711yescharacteristic
302C4#313711yescharacteristic
303C4#317711yescharacteristic
304C4#318311yescharacteristic
305C4#335212yes—
306C4#335911yescharacteristic
307C4#336311yescharacteristic
308C2#1976812yes—
309C2#1981612yes—
310C2#2776911yescharacteristic
311C2#2780611yescharacteristic
312C2#2781111yescharacteristic
subgroupG/H
center, socle, prime radicalC23D42
commutator, Frattini, cosocleC2×C42C24
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C2×C42 ◁ 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
#288112
#286822
#277292
#104142
#103552
#98362
#97302
#95682
#86512
#82852
#82112