Finite Group Databasebeta

← #475963

Subgroups — #475963

Order 512.

720 conjugacy classes of subgroups.

The subgroup lattice has 327 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23747C212yesmaximal
2C2×#1942C211yesmaximal, characteristic
3#21143C211yesmaximal, characteristic
4#11365C212yesmaximal
5#11270C212yesmaximal
6#11242C214yesmaximal
7#11204C212yesmaximal
8#10864C211yesmaximal, characteristic
9C22×#108C2212yes—
10C22×#107C2211yescharacteristic
11C22×#107C2211yescharacteristic
12C2×#424C2212yes—
13C2×#423C2211yescharacteristic
14C22×C4×C8C2211yescharacteristic
15C2×#389C2212yes—
16#2285C2211yescharacteristic
17#2101—24no—
18#2100—22no—
19#2099—22no—
20#1942—22no—
21#1942—22no—
22#1939—22no—
23#1939—22no—
24C2×#343C2214yes—
25C2×#343C2214yes—
26C2×#341C2212yes—
27#1736C2214yes—
28#1736C2214yes—
29C2×#340C2212yes—
30#1729C2212yes—
31#1728C2212yes—
32C22×M16C2312yes—
33C22×M16C2314yes—
34C23×C8C2311yescharacteristic
35C23×C8C2311yescharacteristic
36C2×#117C2312yes—
37C22×C4⋊C4—22no—
38C22×C4⋊C4C2312yes—
39C22×C4⋊C4C2312yes—
40C22×C42C2311yescharacteristic
41C2×#108—24no—
42C2×#108—42no—
43C2×#108—42no—
44C2×#107—22no—
45C2×#107—22no—
46C2×#107—22no—
47C2×#107—42no—
48C2×#107Q814yes—
49#424—28no—
50#423—42no—
51C2×C4⋊C8—24no—
52C2×C4⋊C8—21no—
53C2×C4⋊C8—21no—
54C2×C4×C8—42no—
55C2×C4×C8D412yes—
56C2×C4×C8D412yes—
57#389—22no—
58#389—22no—
59#389—22no—
60#389—22no—
61#384D411yescharacteristic
62#384D411yescharacteristic
63#382D412yes—
64#380—21no—
65#380—21no—
66C4×C4⋊C4—21no—
67C4×C4⋊C4—21no—
68C2×#95—22no—
69#343—44no—
70#343—44no—
71#341—24no—
72#340—24no—
73#340—24no—
74#340—24no—
75#340—24no—
76#340—24no—
77#336—24no—
78#336—22no—
79#336—22no—
80C23×C4—22no—
81C23×C4—22no—
82C23×C4#4111yescharacteristic
83C23×C4#4111yescharacteristic
84C23×C4C2411yescharacteristic
85C2×M16—24no—
86C2×M16—24no—
87C2×M16—24no—
88C2×M16—22no—
89C2×M16—24no—
90C2×M16—24no—
91C2×M16—24no—
92C2×M16—22no—
93C2×M16—24no—
94C2×M16—24no—
95C22×C8—22no—
96C22×C8—22no—
97C22×C8—21no—
98C22×C8—22no—
99C22×C8—21no—
100C22×C8—24no—
101C22×C8—22no—
102C22×C8C2×D412yes—
103C22×C8C2×D412yes—
104C22×C8C2×Q812yes—
105C22×C8#4112yes—
106C22×C8#4114yes—
107#117—24no—
108#117—24no—
109C2×C4⋊C4—22no—
110C2×C4⋊C4—22no—
111C2×C4⋊C4—22no—
112C2×C4⋊C4—22no—
113C2×C4⋊C4—21no—
114C2×C4⋊C4—21no—
115C2×C4⋊C4—21no—
116C2×C4⋊C4—21no—
117C2×C4⋊C4—22no—
118C2×C4⋊C4—22no—
119C2×C4⋊C4—22no—
120C2×C4⋊C4—24no—
121C2×C4⋊C4—24no—
122C2×C4⋊C4—42no—
123C2×C4⋊C4—42no—
124C2×C4⋊C4—84no—
125C2×C4⋊C4C2×D412yes—
126C2×C4⋊C4C2×D412yes—
127C2×C4⋊C4C2×Q814yes—
128C2×C22⋊C4—22no—
129C2×C42—22no—
130C2×C42—22no—
131C2×C42—22no—
132C2×C42C2×D411yescharacteristic
133C2×C42C2×D411yescharacteristic
134C2×C42C2×D411yescharacteristic
135C2×C42C2×D411yescharacteristic
136#108—48no—
137#107—28no—
138#107—44no—
139C4⋊C8—42no—
140C4⋊C8—42no—
141C4⋊C8—48no—
142C4×C8—42no—
143C4×C8—42no—
144#95—22no—
145#95—22no—
146#95—42no—
147#95—42no—
148C24C2×#4111yescharacteristic
149C22×C4—22no—
150C22×C4—22no—
151C22×C4—22no—
152C22×C4—22no—
153C22×C4—22no—
154C22×C4—22no—
155C22×C4—24no—
156C22×C4—41no—
157C22×C4—41no—
158C22×C4—42no—
159C22×C4—42no—
160C22×C4—42no—
161C22×C4—44no—
162C22×C4—82no—
163C22×C4—82no—
164C22×C4—82no—
165C22×C4—82no—
166C22×C4#12111yescharacteristic
167C22×C4#12111yescharacteristic
168C22×C4#12111yescharacteristic
169C22×C4#12111yescharacteristic
170C22×C4C22×D411yescharacteristic
171C22×C4C22×D411yescharacteristic
172C22×C4C22×Q811yescharacteristic
173C22×C4C2×#4111yescharacteristic
174C22×C4C2×#4112yes—
175C22×C4H(5,𝔽2)12yes—
176C22×C4H(5,𝔽2)11yescharacteristic
177C22×C42−1+412yes—
178C22×C42−1+412yes—
179C22×C42−1+411yescharacteristic
180M16—44no—
181M16—44no—
182M16—44no—
183M16—44no—
184M16—44no—
185M16—44no—
186C2×C8—22no—
187C2×C8—22no—
188C2×C8—22no—
189C2×C8—22no—
190C2×C8—24no—
191C2×C8—24no—
192C2×C8—24no—
193C2×C8—24no—
194C2×C8—42no—
195C2×C8—42no—
196C2×C8—42no—
197C2×C8—44no—
198C2×C8—44no—
199C2×C8—44no—
200C2×C8C22⋊Q814yes—
201C2×C8#12414yes—
202C4⋊C4—24no—
203C4⋊C4—24no—
204C4⋊C4—24no—
205C4⋊C4—24no—
206C4⋊C4—42no—
207C4⋊C4—42no—
208C4⋊C4—42no—
209C4⋊C4—42no—
210C4⋊C4—42no—
211C4⋊C4—42no—
212C4⋊C4—42no—
213C4⋊C4—42no—
214C4⋊C4—44no—
215C4⋊C4—84no—
216C4⋊C4—84no—
217C22⋊C4—44no—
218C42—21no—
219C42—21no—
220C42—21no—
221C42—21no—
222C42—22no—
223C42—22no—
224C42—42no—
225C42—42no—
226C23—21no—
227C23—21no—
228C23—21no—
229C23—21no—
230C23C2×#12111yescharacteristic
231C23#53912yes—
232C23#55411yescharacteristic
233C23#55611yescharacteristic
234C23#55612yes—
235C2×C4—21no—
236C2×C4—21no—
237C2×C4—21no—
238C2×C4—21no—
239C2×C4—21no—
240C2×C4—21no—
241C2×C4—21no—
242C2×C4—21no—
243C2×C4—21no—
244C2×C4—21no—
245C2×C4—44no—
246C2×C4—44no—
247C2×C4—44no—
248C2×C4—42no—
249C2×C4—42no—
250C2×C4—42no—
251C2×C4—42no—
252C2×C4—42no—
253C2×C4—42no—
254C2×C4—84no—
255C2×C4—82no—
256C2×C4—82no—
257C2×C4—84no—
258C2×C4—82no—
259C2×C4—82no—
260C2×C4—82no—
261C2×C4—164no—
262C2×C4#47212yes—
263C2×C4#47312yes—
264C2×C4C2×C22⋊Q811yescharacteristic
265C2×C4C2×#12411yescharacteristic
266C2×C4#53411yescharacteristic
267C2×C4#53611yescharacteristic
268C2×C4#53611yescharacteristic
269C2×C4#53711yescharacteristic
270C2×C4#53712yes—
271C2×C4#53812yes—
272C2×C4#53912yes—
273C2×C4#53912yes—
274C2×C4#54412yes—
275C2×C4#54712yes—
276C2×C4D4×Q811yescharacteristic
277C2×C4#55011yescharacteristic
278C2×C4#55612yes—
279C2×C4#55712yes—
280C2×C4#56312yes—
281C2×C4#56412yes—
282C8—24no—
283C8—24no—
284C8—82no—
285C8—84no—
286C22—22no—
287C22—22no—
288C22—22no—
289C22—22no—
290C22—21no—
291C22—21no—
292C22—21no—
293C22—21no—
294C22#253811yescharacteristic
295C22#254711yescharacteristic
296C22#261811yescharacteristic
297C22#280512yes—
298C22#281012yes—
299C22C2×#47211yescharacteristic
300C22C2×#47311yescharacteristic
301C22#306711yescharacteristic
302C22#306711yescharacteristic
303C4—22no—
304C4—22no—
305C4—41no—
306C4—41no—
307C4—82no—
308C4—82no—
309C4—84no—
310C4—162no—
311C4#256811yescharacteristic
312C4#262711yescharacteristic
313C4#263911yescharacteristic
314C4#272612yes—
315C4#272712yes—
316C4#282211yescharacteristic
317C2—22no—
318C2—22no—
319C2#1707411yescharacteristic
320C2#2829511yescharacteristic
321C2#2829711yescharacteristic
322C2#2873111yescharacteristic
323C2#2873311yescharacteristic
324C2#2978811yescharacteristic
325C2#2978911yescharacteristic
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
#237472
C2×#19422
#211432
#113652
#112702
#112422
#112042
#108642