Finite Group Databasebeta

← #315131

Subgroups — #315131

Order 512.

567 conjugacy classes of subgroups.

The subgroup lattice has 319 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#28809C211yesmaximal, characteristic
2#28682C211yesmaximal, characteristic
3#27815C212yesmaximal
4#10415C212yesmaximal
5#10349C212yesmaximal
6#9836C211yesmaximal, characteristic
7#9730C211yesmaximal, characteristic
8#9567C212yesmaximal
9#8651C211yesmaximal, characteristic
10#8285C211yesmaximal, characteristic
11#8211C211yesmaximal, characteristic
12C2×#500C2211yescharacteristic
13C2×#498—21no—
14C2×#446C2211yescharacteristic
15#2567C2212yes—
16#2551C2212yes—
17#2100—22no—
18#2086—22no—
19#2058—22no—
20#2024—22no—
21#1990—22no—
22#1990C2212yes—
23#1987C2212yes—
24#1987C2212yes—
25#1956C2212yes—
26#1948C2212yes—
27#1944—21no—
28#1944—21no—
29#1941—21no—
30#1933—22no—
31#1844C2211yescharacteristic
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⋊Q8—22no—
46C2×C4⋊Q8—22no—
47C2×C4⋊Q8C2312yes—
48C2×C4×Q8C2311yescharacteristic
49#500—22no—
50#500—22no—
51#498—44no—
52#446—22no—
53#446—22no—
54C2×#107—21no—
55C2×#107—21no—
56C2×#106—21no—
57C2×#106—22no—
58C2×#106—21no—
59C2×#106D411yescharacteristic
60C2×#106D411yescharacteristic
61C2×C4⋊C8—21no—
62C2×C4⋊C8C2312yes—
63C2×C4⋊C8C2311yescharacteristic
64C2×C4⋊C8C2311yescharacteristic
65C2×#103—22no—
66C2×#103—42no—
67C2×#103D412yes—
68C2×#97—21no—
69C2×C4×C8—21no—
70C2×C4×C8—22no—
71C2×C4×C8C2311yescharacteristic
72C2×C4×C8C2311yescharacteristic
73C2×C4×C8C2311yescharacteristic
74#395—42no—
75#395—42no—
76#391—22no—
77#389—22no—
78#389—22no—
79#389—22no—
80#389—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×Q8—42no—
103C22×Q8C2×D411yescharacteristic
104C22×C8—21no—
105C22×C8—22no—
106C22×C8—21no—
107C22×C8—21no—
108C22×C8C2×D412yes—
109C22×C8C2×D411yescharacteristic
110C22×C8C2×D411yescharacteristic
111C4⋊Q8—22no—
112C4⋊Q8—22no—
113C4⋊Q8—22no—
114C4⋊Q8—22no—
115C4⋊Q8—44no—
116C4⋊Q8—44no—
117C4⋊Q8—82no—
118C4⋊Q8—82no—
119C4×Q8—21no—
120C4×Q8—21no—
121C4×Q8—21no—
122C4×Q8—21no—
123C2×C4⋊C4—21no—
124C2×C4⋊C4—22no—
125C2×C4⋊C4—22no—
126C2×C4⋊C4—22no—
127C2×C4⋊C4—22no—
128C2×C4⋊C4—22no—
129C2×C4⋊C4—42no—
130C2×C4⋊C4—42no—
131C2×C4⋊C4—42no—
132C2×C4⋊C4—42no—
133C2×C4⋊C4C2×D411yescharacteristic
134C2×C4⋊C4C2×D412yes—
135C2×C4⋊C4C2×D412yes—
136C2×C4⋊C4C2×D412yes—
137C2×C42—21no—
138C2×C42—21no—
139C2×C42—22no—
140C2×C42C2411yescharacteristic
141#107—42no—
142#107—42no—
143#106—42no—
144#106—44no—
145#106—81no—
146#106SD1614yes—
147#106Q1614yes—
148C4⋊C8—22no—
149C4⋊C8—22no—
150C4⋊C8—21no—
151C4⋊C8—21no—
152C4⋊C8—21no—
153C4⋊C8—42no—
154C4⋊C8#4112yes—
155#103—24no—
156#103—82no—
157#103—84no—
158#97—42no—
159C4×C8—21no—
160C4×C8—21no—
161C4×C8—21no—
162C4×C8—21no—
163C4×C8—42no—
164C4×C8—82no—
165C4×C8#4112yes—
166C4×C8#4112yes—
167#95—22no—
168#95—21no—
169#95—42no—
170#95—42no—
171#95—42no—
172C2×Q8—21no—
173C2×Q8—21no—
174C2×Q8—22no—
175C2×Q8—22no—
176C2×Q8—42no—
177C2×Q8—42no—
178C2×Q8—84no—
179C2×Q8—84no—
180C2×Q8—82no—
181C2×Q8—82no—
182C2×Q8—162no—
183C22×C4—22no—
184C22×C4—22no—
185C22×C4—22no—
186C22×C4—21no—
187C22×C4—21no—
188C22×C4—42no—
189C22×C4—42no—
190C22×C4C22×D411yescharacteristic
191C22×C4C22×D411yescharacteristic
192C22×C4H(5,𝔽2)11yescharacteristic
193C2×C8—21no—
194C2×C8—21no—
195C2×C8—22no—
196C2×C8—22no—
197C2×C8—22no—
198C2×C8—22no—
199C2×C8—41no—
200C2×C8—41no—
201C2×C8—42no—
202C2×C8—41no—
203C2×C8—42no—
204C2×C8—41no—
205C2×C8—42no—
206C2×C8—41no—
207C2×C8—42no—
208C2×C8—41no—
209C2×C8#12412yes—
210C2×C8#12412yes—
211C2×C8C2×SD1612yes—
212C2×C8C2×Q1612yes—
213C4⋊C4—22no—
214C4⋊C4—22no—
215C4⋊C4—22no—
216C4⋊C4—21no—
217C4⋊C4—21no—
218C4⋊C4—41no—
219C4⋊C4—41no—
220C4⋊C4—42no—
221C4⋊C4—42no—
222C4⋊C4—42no—
223C4⋊C4—42no—
224C4⋊C4—42no—
225C4⋊C4—42no—
226C4⋊C4—42no—
227C4⋊C4—42no—
228C4⋊C4—84no—
229C4⋊C4—84no—
230C4⋊C4—82no—
231C4⋊C4—162no—
232C4⋊C4—162no—
233C4⋊C4C2×SD1614yes—
234C4⋊C4C2×Q1614yes—
235C4⋊C4#13714yes—
236C42—41no—
237C42—41no—
238C42—41no—
239C42—42no—
240C42—42no—
241C42—41no—
242C42C2×#4111yescharacteristic
243C42C2×#4111yescharacteristic
244C42C2×#4111yescharacteristic
245C422−1+411yescharacteristic
246C23D4211yescharacteristic
247Q8—42no—
248Q8—41no—
249Q8—41no—
250Q8—84no—
251Q8—162no—
252Q8—168no—
253C2×C4—22no—
254C2×C4—24no—
255C2×C4—24no—
256C2×C4—24no—
257C2×C4—42no—
258C2×C4—42no—
259C2×C4—42no—
260C2×C4—42no—
261C2×C4—42no—
262C2×C4—42no—
263C2×C4—41no—
264C2×C4—41no—
265C2×C4—42no—
266C2×C4—42no—
267C2×C4—82no—
268C2×C4—82no—
269C2×C4—82no—
270C2×C4—82no—
271C2×C4—82no—
272C2×C4—82no—
273C2×C4C2×#12411yescharacteristic
274C2×C4#53211yescharacteristic
275C2×C4#54211yescharacteristic
276C2×C4#54711yescharacteristic
277C2×C4#54811yescharacteristic
278C2×C4#55511yescharacteristic
279C2×C4C22×SD1611yescharacteristic
280C2×C4C22×Q1611yescharacteristic
281C2×C4C2×#13712yes—
282C2×C4#57712yes—
283C2×C4#57711yescharacteristic
284C2×C4#57811yescharacteristic
285C8—22no—
286C8—42no—
287C8—42no—
288C8—81no—
289C8—81no—
290C8—82no—
291C8#48714yes—
292C22#249111yescharacteristic
293C22#326612yes—
294C22D4×SD1611yescharacteristic
295C22D4×Q1611yescharacteristic
296C22#328812yes—
297C4—21no—
298C4—21no—
299C4—44no—
300C4—44no—
301C4—44no—
302C4—81no—
303C4—81no—
304C4—162no—
305C4—162no—
306C4C2×#48711yescharacteristic
307C4#313711yescharacteristic
308C4#317311yescharacteristic
309C4#318511yescharacteristic
310C4#335612yes—
311C4#335911yescharacteristic
312C4#336311yescharacteristic
313C2#1976512yes—
314C2#1981312yes—
315C2#2776911yescharacteristic
316C2#2780311yescharacteristic
317C2#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
#288092
#286822
#278152
#104152
#103492
#98362
#97302
#95672
#86512
#82852
#82112