Finite Group Databasebeta

← #325627

Subgroups — #325627

Order 512.

566 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#30280C211yesmaximal, characteristic
2#29222C211yesmaximal, characteristic
3#28755C211yesmaximal, characteristic
4#22299C211yesmaximal, characteristic
5#10544C212yesmaximal
6#10542C212yesmaximal
7#9736C212yesmaximal
8#9729C212yesmaximal
9#9701C211yesmaximal, characteristic
10#9694C211yesmaximal, characteristic
11#8277C211yesmaximal, characteristic
12C2×#498C2211yescharacteristic
13C2×#498C2211yescharacteristic
14C2×#491—21no—
15C2×#487C2211yescharacteristic
16C2×#487C2211yescharacteristic
17#2868C2211yescharacteristic
18#2862C2211yescharacteristic
19C4×#125C2211yescharacteristic
20#2101—22no—
21#2096—22no—
22#2096—22no—
23#2043—22no—
24#1991—21no—
25#1988C2212yes—
26#1987C2212yes—
27#1981C2212yes—
28#1981C2212yes—
29#1960—21no—
30#1939—22no—
31#1939C2212yes—
32#1916—22no—
33#1842C2211yescharacteristic
34#1835—21no—
35C8×C4⋊C4C2212yes—
36#1767C2211yescharacteristic
37C4×C4⋊C8C2211yescharacteristic
38#1744C2211yescharacteristic
39C2×#334C2212yes—
40C2×#333C2212yes—
41C2×#332C2212yes—
42C2×#332C2212yes—
43C2×#332C2212yes—
44#1277C2212yes—
45C2×C4⋊Q8—21no—
46C2×C4⋊Q8—21no—
47C2×C4⋊Q8C2311yescharacteristic
48C2×C4⋊Q8C2311yescharacteristic
49C2×C4⋊Q8C2311yescharacteristic
50C2×C4⋊Q8C2311yescharacteristic
51C2×#125D411yescharacteristic
52#498—24no—
53#498—42no—
54#491—44no—
55#487—24no—
56#487—42no—
57C2×#106—21no—
58C2×#106—21no—
59C2×#106—22no—
60C2×C4⋊C8—22no—
61C2×C4⋊C8—22no—
62C2×C4⋊C8C2312yes—
63C2×C4⋊C8C2312yes—
64C2×C4⋊C8C2312yes—
65C2×#103—21no—
66C2×#103—21no—
67C2×#103—22no—
68C2×#103—22no—
69C2×#103—22no—
70C2×#103—22no—
71C2×#103—22no—
72C2×#97—21no—
73C2×C4×C8—21no—
74C2×C4×C8—22no—
75C2×C4×C8C2311yescharacteristic
76C2×C4×C8C2311yescharacteristic
77#391—22no—
78#391—22no—
79#389D412yes—
80#387—22no—
81#384—22no—
82#384—22no—
83#382—21no—
84#382—21no—
85#382D411yescharacteristic
86C4×C4⋊C4—22no—
87C4×C4⋊C4C2312yes—
88C43C2311yescharacteristic
89#340—22no—
90#340—22no—
91#340—22no—
92#340—22no—
93#336—22no—
94#336—22no—
95#336—21no—
96#336—21no—
97#336D412yes—
98#336D412yes—
99#334—24no—
100#333—42no—
101#332—42no—
102#332—42no—
103#332—42no—
104C22×Q8—21no—
105C22×Q8—21no—
106C22×Q8—21no—
107C22×Q8—21no—
108C22×C8—22no—
109C22×C8—22no—
110C22×C8—22no—
111C22×C8C2×D411yescharacteristic
112C22×C8C2×D411yescharacteristic
113C22×C8C2×D412yes—
114C4⋊Q8—22no—
115C4⋊Q8—22no—
116C4⋊Q8—22no—
117C4⋊Q8—22no—
118C4⋊Q8—22no—
119C4⋊Q8—22no—
120C4⋊Q8—22no—
121C4⋊Q8—22no—
122C4⋊Q8—42no—
123C4⋊Q8—42no—
124C4⋊Q8—82no—
125#125—22no—
126#125SD1614yes—
127C2×C4⋊C4—22no—
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—21no—
136C2×C4⋊C4—22no—
137C2×C4⋊C4—42no—
138C2×C4⋊C4—42no—
139C2×C4⋊C4C2×D412yes—
140C2×C4⋊C4C2×D412yes—
141C2×C42—22no—
142C2×C42C2×D411yescharacteristic
143C2×C42C2×D412yes—
144C2×C42C2×D411yescharacteristic
145C2×C42#4112yes—
146C2×C42C2411yescharacteristic
147#106—42no—
148#106—42no—
149#106—44no—
150C4⋊C8—22no—
151C4⋊C8—22no—
152C4⋊C8—22no—
153C4⋊C8—22no—
154C4⋊C8—22no—
155C4⋊C8—22no—
156C4⋊C8—82no—
157C4⋊C8—82no—
158#103—42no—
159#103—42no—
160#103—44no—
161#103—44no—
162#103—44no—
163#103—82no—
164#103—82no—
165#97—42no—
166C4×C8—22no—
167C4×C8—42no—
168C4×C8—82no—
169C4×C8#4114yes—
170#95—22no—
171#95—22no—
172#95—22no—
173#95—42no—
174#95—42no—
175C2×Q8—41no—
176C2×Q8—41no—
177C2×Q8—42no—
178C2×Q8—42no—
179C2×Q8—42no—
180C2×Q8—41no—
181C2×Q8—41no—
182C2×Q8—41no—
183C2×Q8—41no—
184C2×Q8—42no—
185C2×Q8—42no—
186C2×Q8—42no—
187C2×Q8—42no—
188C2×Q8—82no—
189C22×C4—22no—
190C22×C4—22no—
191C22×C4—41no—
192C22×C4—41no—
193C22×C4—41no—
194C22×C4—41no—
195C22×C4#12112yes—
196C22×C4#12112yes—
197C22×C4C22×D411yescharacteristic
198C22×C4C22×D411yescharacteristic
199C22×C4C2×#4111yescharacteristic
200C2×C8—22no—
201C2×C8—22no—
202C2×C8—21no—
203C2×C8—21no—
204C2×C8—22no—
205C2×C8—22no—
206C2×C8—42no—
207C2×C8—42no—
208C2×C8—42no—
209C2×C8—42no—
210C2×C8—42no—
211C2×C8—42no—
212C2×C8—42no—
213C2×C8—42no—
214C2×C8—42no—
215C2×C8#12414yes—
216C4⋊C4—22no—
217C4⋊C4—28no—
218C4⋊C4—22no—
219C4⋊C4—22no—
220C4⋊C4—41no—
221C4⋊C4—41no—
222C4⋊C4—41no—
223C4⋊C4—41no—
224C4⋊C4—41no—
225C4⋊C4—41no—
226C4⋊C4—42no—
227C4⋊C4—42no—
228C4⋊C4—42no—
229C4⋊C4—42no—
230C4⋊C4—42no—
231C4⋊C4—82no—
232C4⋊C4—84no—
233C4⋊C4—84no—
234C4⋊C4C2×SD1614yes—
235C42—22no—
236C42—22no—
237C42—22no—
238C42—42no—
239C42—42no—
240C42—42no—
241C42C2×SD1612yes—
242C42#13512yes—
243C42C2×#4112yes—
244C42H(5,𝔽2)12yes—
245C23C2×#12111yescharacteristic
246Q8—82no—
247Q8—82no—
248Q8—82no—
249Q8—82no—
250Q8—82no—
251Q8—81no—
252Q8—81no—
253Q8—162no—
254C2×C4—22no—
255C2×C4—22no—
256C2×C4—22no—
257C2×C4—22no—
258C2×C4—24no—
259C2×C4—22no—
260C2×C4—22no—
261C2×C4—42no—
262C2×C4—42no—
263C2×C4—42no—
264C2×C4—42no—
265C2×C4—42no—
266C2×C4—82no—
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—82no—
276C2×C4#46514yes—
277C2×C4C2×#12411yescharacteristic
278C2×C4#53511yescharacteristic
279C2×C4#55012yes—
280C2×C4#55912yes—
281C2×C4C22×SD1611yescharacteristic
282C2×C4C2×#13511yescharacteristic
283C2×C4#57712yes—
284C2×C4#57712yes—
285C2×C4#57812yes—
286C8—24no—
287C8—41no—
288C8—41no—
289C8—82no—
290C8—82no—
291C8—82no—
292C22#254811yescharacteristic
293C22#304612yes—
294C22C2×#46511yescharacteristic
295C22#305711yescharacteristic
296C22#306912yes—
297C4—22no—
298C4—24no—
299C4—42no—
300C4—44no—
301C4—82no—
302C4—161no—
303C4—161no—
304C4—161no—
305C4—161no—
306C4#313712yes—
307C4#322111yescharacteristic
308C4#322311yescharacteristic
309C4#339212yes—
310C4#339412yes—
311C2#1941512yes—
312C2#1954512yes—
313C2#2873811yescharacteristic
314C2#2876011yescharacteristic
315C2#2876711yescharacteristic
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
#302802
#292222
#287552
#222992
#105442
#105422
#97362
#97292
#97012
#96942
#82772