Finite Group Databasebeta

← #331563

Subgroups — #331563

Order 512.

614 conjugacy classes of subgroups.

The subgroup lattice has 317 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33297C211yesmaximal, characteristic
2#33294C211yesmaximal, characteristic
3#33256C211yesmaximal, characteristic
4#33225C211yesmaximal, characteristic
5#30304C211yesmaximal, characteristic
6#30230C211yesmaximal, characteristic
7#10686C212yesmaximal
8#10682C212yesmaximal
9#10679C212yesmaximal
10#10675C212yesmaximal
11#8132C211yesmaximal, characteristic
12C2×#487—22no—
13#2868C2211yescharacteristic
14#2868C2211yescharacteristic
15#2859C2211yescharacteristic
16#2859C2211yescharacteristic
17#2100—22no—
18#2100—22no—
19#2083—22no—
20#2057—22no—
21#1993—22no—
22#1992—22no—
23#1992—22no—
24#1990C2212yes—
25#1990C2212yes—
26#1990C2212yes—
27#1990C2212yes—
28#1956—22no—
29#1956C2212yes—
30#1956C2212yes—
31#1941—22no—
32#1941—22no—
33#1845C2212yes—
34#1844C2211yescharacteristic
35#1844C2211yescharacteristic
36#1836—22no—
37#1834C2212yes—
38#1834C2212yes—
39#1834C2212yes—
40#1834C2211yescharacteristic
41#1834C2211yescharacteristic
42#1744C2211yescharacteristic
43C4×#97C2211yescharacteristic
44C42×C8C2211yescharacteristic
45#1273C2212yes—
46#1273C2212yes—
47C2×C4⋊Q8—21no—
48C2×C4⋊Q8—21no—
49C2×C4⋊Q8—21no—
50C2×C4⋊Q8—21no—
51C2×C4⋊Q8—21no—
52C2×C4⋊Q8C2312yes—
53C2×#125—21no—
54C2×#125—21no—
55#487—48no—
56C2×#107—24no—
57C2×#106—22no—
58C2×#106—22no—
59C2×#103—24no—
60C2×#103—24no—
61C2×#97D412yes—
62C2×#97D412yes—
63C2×C4×C8D412yes—
64C2×C4×C8D412yes—
65C2×C4×C8C2311yescharacteristic
66C2×C4×C8C2312yes—
67C2×C4×C8C2312yes—
68C2×C4×C8C2311yescharacteristic
69#389—21no—
70#389—21no—
71#389—21no—
72#389—21no—
73#389—21no—
74#389—21no—
75#389—21no—
76#389—21no—
77#384—21no—
78#384—21no—
79#384—21no—
80#384—21no—
81#384—21no—
82#384—21no—
83#384—21no—
84#384C2311yescharacteristic
85#384C2311yescharacteristic
86#384C2311yescharacteristic
87#384C2312yes—
88#384C2311yescharacteristic
89#382—21no—
90#382—21no—
91C43C2311yescharacteristic
92#340—24no—
93#340—22no—
94#340—22no—
95#340—24no—
96#340—24no—
97#340—24no—
98#340—24no—
99#340—24no—
100#340—24no—
101#336D412yes—
102#336D412yes—
103C22×Q8—22no—
104C22×Q8—41no—
105C22×C8C2×D414yes—
106C22×C8C2×D414yes—
107C22×C8C2×D412yes—
108C22×C8C2×D412yes—
109C4⋊Q8—24no—
110C4⋊Q8—24no—
111C4⋊Q8—44no—
112C4⋊Q8—44no—
113C4⋊Q8—44no—
114C4⋊Q8—44no—
115C4⋊Q8—82no—
116#125—44no—
117#125—44no—
118C2×C4⋊C4—21no—
119C2×C4⋊C4—21no—
120C2×C4⋊C4—21no—
121C2×C4⋊C4—21no—
122C2×C4⋊C4—22no—
123C2×C4⋊C4—22no—
124C2×C4⋊C4—22no—
125C2×C4⋊C4—22no—
126C2×C4⋊C4—22no—
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—41no—
136C2×C4⋊C4—41no—
137C2×C4⋊C4—41no—
138C2×C4⋊C4—41no—
139C2×C4⋊C4—41no—
140C2×C4⋊C4—41no—
141C2×C4⋊C4—41no—
142C2×C4⋊C4—41no—
143C2×C4⋊C4—41no—
144C2×C4⋊C4—41no—
145C2×C4⋊C4—41no—
146C2×C42C2×D411yescharacteristic
147C2×C42C2×D411yescharacteristic
148C2×C42C2×D411yescharacteristic
149C2×C42C2×D411yescharacteristic
150C2×C42C2×D411yescharacteristic
151C2×C42C2×D411yescharacteristic
152C2×C42C2411yescharacteristic
153#107—84no—
154#106—44no—
155#106—44no—
156#103—48no—
157#103—48no—
158#97—24no—
159#97—24no—
160C4×C8—24no—
161C4×C8—22no—
162C4×C8—22no—
163C4×C8—22no—
164C4×C8—22no—
165C4×C8—22no—
166C4×C8—22no—
167C4×C8—42no—
168#95—41no—
169#95—41no—
170#95—41no—
171#95—41no—
172C2×Q8—42no—
173C2×Q8—42no—
174C2×Q8—44no—
175C2×Q8—44no—
176C2×Q8—82no—
177C2×Q8—82no—
178C2×Q8—82no—
179C22×C4—41no—
180C22×C4—41no—
181C22×C4—41no—
182C22×C4—41no—
183C22×C4—42no—
184C22×C4—42no—
185C22×C4#12711yescharacteristic
186C22×C4#12711yescharacteristic
187C22×C4#12711yescharacteristic
188C22×C4#12711yescharacteristic
189C22×C4C22×D411yescharacteristic
190C22×C4C22×D411yescharacteristic
191C22×C4C22×D411yescharacteristic
192C2×C8—22no—
193C2×C8—22no—
194C2×C8—24no—
195C2×C8—22no—
196C2×C8—24no—
197C2×C8—24no—
198C2×C8—22no—
199C2×C8—24no—
200C2×C8—22no—
201C2×C8—24no—
202C2×C8—24no—
203C2×C8AGL(1,ℤ/8)14yes—
204C4⋊C4—44no—
205C4⋊C4—44no—
206C4⋊C4—42no—
207C4⋊C4—42no—
208C4⋊C4—44no—
209C4⋊C4—44no—
210C4⋊C4—42no—
211C4⋊C4—42no—
212C4⋊C4—42no—
213C4⋊C4—42no—
214C4⋊C4—42no—
215C4⋊C4—42no—
216C4⋊C4—42no—
217C4⋊C4—44no—
218C4⋊C4—42no—
219C4⋊C4—42no—
220C4⋊C4—42no—
221C4⋊C4—82no—
222C4⋊C4—82no—
223C4⋊C4—82no—
224C4⋊C4—82no—
225C4⋊C4—82no—
226C4⋊C4—82no—
227C4⋊C4—82no—
228C4⋊C4—82no—
229C4⋊C4—82no—
230C4⋊C4—82no—
231C4⋊C4—82no—
232C42—22no—
233C42—22no—
234C42—22no—
235C42—22no—
236C42#13514yes—
237C42#13714yes—
238C422−1+412yes—
239C422−1+412yes—
240C23C2×#12711yescharacteristic
241Q8—84no—
242Q8—84no—
243Q8—164no—
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—81no—
253C2×C4—81no—
254C2×C4—82no—
255C2×C4—81no—
256C2×C4—81no—
257C2×C4—82no—
258C2×C4—81no—
259C2×C4—81no—
260C2×C4—82no—
261C2×C4—81no—
262C2×C4—81no—
263C2×C4—82no—
264C2×C4—82no—
265C2×C4—81no—
266C2×C4—81no—
267C2×C4—81no—
268C2×C4—81no—
269C2×C4—82no—
270C2×C4#49512yes—
271C2×C4#49512yes—
272C2×C4#49712yes—
273C2×C4#49712yes—
274C2×C4#53612yes—
275C2×C4#53611yescharacteristic
276C2×C4#53611yescharacteristic
277C2×C4#53612yes—
278C2×C4C2×#13512yes—
279C2×C4C2×AGL(1,ℤ/8)12yes—
280C2×C4C2×#13712yes—
281C2×C4#57512yes—
282C2×C4#57712yes—
283C2×C4#57812yes—
284C8—24no—
285C8—44no—
286C8—42no—
287C8—44no—
288C22#265811yescharacteristic
289C22C2×#49511yescharacteristic
290C22C2×#49711yescharacteristic
291C22#315111yescharacteristic
292C22#315311yescharacteristic
293C22#315411yescharacteristic
294C22#315611yescharacteristic
295C4—22no—
296C4—22no—
297C4—22no—
298C4—22no—
299C4—162no—
300C4—161no—
301C4—162no—
302C4—161no—
303C4—161no—
304C4—161no—
305C4#323114yes—
306C4#323412yes—
307C4#323512yes—
308C4#324214yes—
309C2#1962511yescharacteristic
310C2#1962811yescharacteristic
311C2#1968411yescharacteristic
312C2#1968611yescharacteristic
313C2#3028111yescharacteristic
314C2#3028311yescharacteristic
315C2#3028811yescharacteristic
subgroupG/H
center, socle, prime radicalC23C2×#127
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
#332972
#332942
#332562
#332252
#303042
#302302
#106862
#106822
#106792
#106752
#81322