Finite Group Databasebeta

← #471702

Subgroups — #471702

Order 512.

712 conjugacy classes of subgroups.

The subgroup lattice has 325 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23746C211yesmaximal, characteristic
2#23553C211yesmaximal, characteristic
3C2×#1991C211yesmaximal, characteristic
4#21020C211yesmaximal, characteristic
5#11383C211yesmaximal, characteristic
6#11355C211yesmaximal, characteristic
7#11303C212yesmaximal
8#11303C212yesmaximal
9#11290C212yesmaximal
10#11216C212yesmaximal
11#10822C211yesmaximal, characteristic
12C22×#125C2211yescharacteristic
13C2×#428C2212yes—
14C2×#424C2211yescharacteristic
15C22×C4×C8C2211yescharacteristic
16#2296C2211yescharacteristic
17C2×#382C2211yescharacteristic
18#2101—24no—
19#2059—24no—
20#1991—24no—
21#1916—24no—
22C2×#343C2212yes—
23C2×#343C2212yes—
24C2×#343C2212yes—
25C2×#343C2212yes—
26C2×#341C2212yes—
27C2×#341C2212yes—
28#1736C2212yes—
29#1736C2212yes—
30#1736C2212yes—
31C2×#340C2212yes—
32C2×#340C2212yes—
33#1729C2211yescharacteristic
34#1729C2211yescharacteristic
35#1729C2211yescharacteristic
36#1728C2211yescharacteristic
37#1728C2211yescharacteristic
38#1728C2211yescharacteristic
39C22×M16C2311yescharacteristic
40C22×M16C2311yescharacteristic
41C22×M16C2312yes—
42C22×M16C2312yes—
43C23×C8C2311yescharacteristic
44C23×C8C2311yescharacteristic
45C2×#125—24no—
46C2×#125—22no—
47C2×#125—22no—
48C2×#125—42no—
49C2×#117C2311yescharacteristic
50C2×#117C2311yescharacteristic
51C22×C4⋊C4—22no—
52C22×C4⋊C4C2312yes—
53C22×C4⋊C4C2312yes—
54C22×C42C2311yescharacteristic
55#428—24no—
56#428—42no—
57C2×#107—22no—
58C2×#106—22no—
59#424—42no—
60C2×C4⋊C8—22no—
61C2×C4×C8—42no—
62C2×C4×C8D414yes—
63#389D414yes—
64#385—22no—
65#382—22no—
66#382—22no—
67C4×C4⋊C4—22no—
68C2×#95—22no—
69#343—24no—
70#343—24no—
71#343—24no—
72#343—42no—
73#341—24no—
74#341—42no—
75#340—24no—
76#340—24no—
77#340—24no—
78#340—24no—
79#340—24no—
80#336—22no—
81#336—22no—
82#336—22no—
83#336—22no—
84#336—22no—
85#336—22no—
86C23×C4—22no—
87C23×C4—22no—
88C23×C4#4112yes—
89C23×C4C2411yescharacteristic
90C2×M16—22no—
91C2×M16—22no—
92C2×M16—22no—
93C2×M16—22no—
94C2×M16—22no—
95C2×M16—22no—
96C2×M16—22no—
97C2×M16—22no—
98C2×M16—22no—
99C2×M16—22no—
100C2×M16—22no—
101C2×M16—22no—
102C2×M16—21no—
103C2×M16—22no—
104C2×M16—21no—
105C2×M16—21no—
106C2×M16—21no—
107C2×M16—22no—
108C2×M16—22no—
109C2×M16#4114yes—
110C22×C8—22no—
111C22×C8—22no—
112C22×C8—22no—
113C22×C8—22no—
114C22×C8—21no—
115C22×C8—21no—
116C22×C8—21no—
117C22×C8—21no—
118C22×C8—22no—
119C22×C8—21no—
120C22×C8—21no—
121C22×C8—22no—
122C22×C8C2×D412yes—
123C22×C8C2×D412yes—
124#125—28no—
125#125—48no—
126#125—44no—
127#117—22no—
128#117—22no—
129#117—21no—
130#117—21no—
131#117—21no—
132#117—21no—
133C2×C4⋊C4—22no—
134C2×C4⋊C4—22no—
135C2×C4⋊C4—22no—
136C2×C4⋊C4—22no—
137C2×C4⋊C4—24no—
138C2×C4⋊C4—22no—
139C2×C4⋊C4—22no—
140C2×C4⋊C4—22no—
141C2×C4⋊C4—22no—
142C2×C4⋊C4—22no—
143C2×C4⋊C4—22no—
144C2×C4⋊C4—22no—
145C2×C4⋊C4—44no—
146C2×C4⋊C4—44no—
147C2×C4⋊C4—44no—
148C2×C4⋊C4C2×D412yes—
149C2×C4⋊C4C2×D412yes—
150C2×C4⋊C4#4114yes—
151C2×C22⋊C4—21no—
152C2×C22⋊C4—21no—
153C2×C42—22no—
154C2×C42—22no—
155C2×C42—21no—
156C2×C42—21no—
157C2×C42C2×D412yes—
158C2×C42C2×D412yes—
159#107—44no—
160#106—44no—
161C4⋊C8—44no—
162C4×C8—44no—
163#95—24no—
164#95—44no—
165C24C2×#4111yescharacteristic
166C22×C4—22no—
167C22×C4—22no—
168C22×C4—22no—
169C22×C4—22no—
170C22×C4—24no—
171C22×C4—44no—
172C22×C4—42no—
173C22×C4—42no—
174C22×C4—42no—
175C22×C4—42no—
176C22×C4—42no—
177C22×C4—84no—
178C22×C4—84no—
179C22×C4#12114yes—
180C22×C4C22×D411yescharacteristic
181C22×C4C22×D411yescharacteristic
182C22×C4C2×#4112yes—
183C22×C4C2×#4112yes—
184C22×C4H(5,𝔽2)12yes—
185C22×C42−1+411yescharacteristic
186C22×C42−1+411yescharacteristic
187C22×C42−1+411yescharacteristic
188C22×C42−1+411yescharacteristic
189C22×C42−1+412yes—
190M16—24no—
191M16—24no—
192M16—42no—
193M16—42no—
194M16—42no—
195M16—42no—
196M16—42no—
197M16—42no—
198M16—42no—
199M16—42no—
200M16—42no—
201M16—42no—
202C2×C8—22no—
203C2×C8—22no—
204C2×C8—22no—
205C2×C8—22no—
206C2×C8—22no—
207C2×C8—22no—
208C2×C8—24no—
209C2×C8—42no—
210C2×C8—42no—
211C2×C8—42no—
212C2×C8—42no—
213C2×C8—42no—
214C2×C8—42no—
215C2×C8—41no—
216C2×C8—41no—
217C2×C8—41no—
218C2×C8—41no—
219C2×C8—41no—
220C2×C8—41no—
221C2×C8—42no—
222C2×C8—42no—
223C2×C8—42no—
224C4⋊C4—24no—
225C4⋊C4—22no—
226C4⋊C4—22no—
227C4⋊C4—22no—
228C4⋊C4—24no—
229C4⋊C4—22no—
230C4⋊C4—48no—
231C4⋊C4—416no—
232C4⋊C4—42no—
233C4⋊C4—42no—
234C4⋊C4—42no—
235C4⋊C4—42no—
236C4⋊C4—42no—
237C4⋊C4—42no—
238C22⋊C4—41no—
239C22⋊C4—41no—
240C22⋊C4—42no—
241C42—22no—
242C42—22no—
243C42—24no—
244C42—41no—
245C42—41no—
246C42—41no—
247C42—41no—
248C23—24no—
249C23C2×#12111yescharacteristic
250C23#54111yescharacteristic
251C23#54111yescharacteristic
252C23#55112yes—
253C23#55612yes—
254C2×C4—24no—
255C2×C4—24no—
256C2×C4—24no—
257C2×C4—44no—
258C2×C4—44no—
259C2×C4—42no—
260C2×C4—42no—
261C2×C4—44no—
262C2×C4—44no—
263C2×C4—44no—
264C2×C4—84no—
265C2×C4—84no—
266C2×C4—84no—
267C2×C4—82no—
268C2×C4—84no—
269C2×C4—162no—
270C2×C4—162no—
271C2×C4#52511yescharacteristic
272C2×C4#53211yescharacteristic
273C2×C4#53411yescharacteristic
274C2×C4#53611yescharacteristic
275C2×C4#53611yescharacteristic
276C2×C4#53611yescharacteristic
277C2×C4#53611yescharacteristic
278C2×C4#53611yescharacteristic
279C2×C4#53912yes—
280C2×C4#54112yes—
281C2×C4#54112yes—
282C2×C4#54212yes—
283C2×C4#54612yes—
284C2×C4#54812yes—
285C2×C4#55212yes—
286C2×C4#55612yes—
287C2×C4#56312yes—
288C2×C4#56312yes—
289C8—42no—
290C8—42no—
291C8—44no—
292C8—81no—
293C8—81no—
294C8—82no—
295C22—22no—
296C22—22no—
297C22—22no—
298C22—22no—
299C22—22no—
300C22—22no—
301C22#253411yescharacteristic
302C22#254511yescharacteristic
303C22#262111yescharacteristic
304C22#275812yes—
305C22#278912yes—
306C22#306714yes—
307C4—44no—
308C4—82no—
309C4—82no—
310C4—84no—
311C4—162no—
312C4#260411yescharacteristic
313C4#263411yescharacteristic
314C4#266311yescharacteristic
315C4#269012yes—
316C4#271612yes—
317C4#281711yescharacteristic
318C2—22no—
319C2—22no—
320C2#1680411yescharacteristic
321C2#2813612yes—
322C2#2867112yes—
323C2#2988812yes—
subgroupG/H
center, socleC23C2×#121
commutator, Frattini, cosocleC23×C4C24
prime radicalC24C2×#41
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Maximal subgroups

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

subgroupindex
#237462
#235532
C2×#19912
#210202
#113832
#113552
#113032
#112902
#112162
#108222