Finite Group Databasebeta

← #386285

Subgroups — #386285

Order 512.

782 conjugacy classes of subgroups.

The subgroup lattice has 331 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21038C212yesmaximal
2#21037C212yesmaximal
3C2×#1847C211yesmaximal, characteristic
4#11366C212yesmaximal
5#11365C212yesmaximal
6#11211C211yesmaximal, characteristic
7#11211C211yesmaximal, characteristic
8#10891C212yesmaximal
9#10819C212yesmaximal
10C22×#107C2212yes—
11C22×#106C2212yes—
12C22×#97C2211yescharacteristic
13#2316C2211yescharacteristic
14#2316C2211yescharacteristic
15C2×#384C2211yescharacteristic
16C2×#382C2211yescharacteristic
17#2274C2212yes—
18#1921—24no—
19#1920—24no—
20#1919—24no—
21#1918—24no—
22#1847C418yescomplemented
23C2×#343C2211yescharacteristic
24C2×#343C2212yes—
25C2×#343C2211yescharacteristic
26C2×#341C2212yes—
27#1736C2212yes—
28#1736C2212yes—
29#1736C2212yes—
30#1736C2212yes—
31#1735C2212yes—
32C2×#339C2212yes—
33C2×#339C2212yes—
34#1729C2212yes—
35#1729C2212yes—
36C22×M16C2312yes—
37C22×M16C2312yes—
38C23×C8C2312yes—
39C2×#117C2312yes—
40C2×#117C2312yes—
41C22×C4⋊C4—41no—
42C22×C4⋊C4C2311yescharacteristic
43C22×C4⋊C4C2311yescharacteristic
44C22×C4⋊C4C2312yes—
45C22×C42C2311yescharacteristic
46C2×#107—22no—
47C2×#107—22no—
48C2×#107—24no—
49C2×#107C2×C418yes—
50C2×#106—22no—
51C2×#106—22no—
52C2×#106—24no—
53C2×#106C2×C418yes—
54C2×#97—22no—
55C2×#97—22no—
56C2×#97C2×C414yes—
57#389—24no—
58#389—24no—
59#384—41no—
60#384C2×C414yes—
61#382—41no—
62#382C2×C414yes—
63#380—41no—
64#380—41no—
65C4×C4⋊C4—24no—
66C4×C4⋊C4—24no—
67C4×C22⋊C4—42no—
68C2×#95—41no—
69#343—41no—
70#343—41no—
71#343—42no—
72#341—24no—
73#340—24no—
74#340—24no—
75#340—24no—
76#340—24no—
77#340D414yes—
78#340D414yes—
79#339—24no—
80#339—24no—
81#336—22no—
82#336—22no—
83#336—22no—
84#336—22no—
85C23×C4—21no—
86C23×C4—22no—
87C23×C4—21no—
88C23×C4—21no—
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—
102C22×C8—22no—
103C22×C8—22no—
104C22×C8—22no—
105C22×C8—22no—
106C22×C8—22no—
107C22×C8—22no—
108C22×C8C22×C414yes—
109C22×C8C2×D412yes—
110C22×C8C2×D412yes—
111C22×C8#4114yes—
112#117—22no—
113#117—22no—
114#117—22no—
115#117—22no—
116#117—22no—
117#117—22no—
118#117—22no—
119#117—22no—
120C2×C4⋊C4—22no—
121C2×C4⋊C4—22no—
122C2×C4⋊C4—22no—
123C2×C4⋊C4—22no—
124C2×C4⋊C4—22no—
125C2×C4⋊C4—22no—
126C2×C4⋊C4—22no—
127C2×C4⋊C4—22no—
128C2×C4⋊C4—21no—
129C2×C4⋊C4—21no—
130C2×C4⋊C4—22no—
131C2×C4⋊C4—44no—
132C2×C4⋊C4—42no—
133C2×C4⋊C4—42no—
134C2×C4⋊C4—84no—
135C2×C4⋊C4C22×C412yes—
136C2×C4⋊C4C22×C412yes—
137C2×C4⋊C4C22×C414yes—
138C2×C4⋊C4C2×D414yes—
139C2×C4⋊C4C2×D414yes—
140C2×C22⋊C4—22no—
141C2×C22⋊C4—22no—
142C2×C42—21no—
143C2×C42—21no—
144C2×C42—21no—
145C2×C42—22no—
146C2×C42—22no—
147C2×C42—44no—
148C2×C42C22×C412yes—
149C2×C42#4112yes—
150C2×C42#4112yes—
151#107—28no—
152#107—28no—
153#106—28no—
154#106—28no—
155#97—28no—
156#95—44no—
157#95—44no—
158#95—42no—
159#95—42no—
160C24H(5,𝔽2)11yescharacteristic
161C22×C4—22no—
162C22×C4—22no—
163C22×C4—22no—
164C22×C4—24no—
165C22×C4—24no—
166C22×C4—22no—
167C22×C4—22no—
168C22×C4—24no—
169C22×C4—42no—
170C22×C4—42no—
171C22×C4—42no—
172C22×C4—41no—
173C22×C4—41no—
174C22×C4—41no—
175C22×C4—42no—
176C22×C4—41no—
177C22×C4—84no—
178C22×C4—82no—
179C22×C4—82no—
180C22×C4C23×C411yescharacteristic
181C22×C4C22×D412yes—
182C22×C4C2×#4111yescharacteristic
183C22×C4C2×#4111yescharacteristic
184C22×C4C2×#4112yes—
185C22×C4H(5,𝔽2)11yescharacteristic
186C22×C42−1+411yescharacteristic
187C22×C42−1+411yescharacteristic
188C22×C42−1+412yes—
189C22×C42−1+412yes—
190M16—42no—
191M16—42no—
192M16—42no—
193M16—42no—
194M16—42no—
195M16—42no—
196M16—42no—
197M16—42no—
198C2×C8—24no—
199C2×C8—24no—
200C2×C8—22no—
201C2×C8—22no—
202C2×C8—22no—
203C2×C8—22no—
204C2×C8—24no—
205C2×C8—24no—
206C2×C8—42no—
207C2×C8—42no—
208C2×C8—42no—
209C2×C8—42no—
210C2×C8—42no—
211C2×C8—42no—
212C2×C8C4×D414yes—
213C2×C8C4×D414yes—
214C4⋊C4—24no—
215C4⋊C4—24no—
216C4⋊C4—24no—
217C4⋊C4—24no—
218C4⋊C4—24no—
219C4⋊C4—24no—
220C4⋊C4—24no—
221C4⋊C4—24no—
222C4⋊C4—42no—
223C4⋊C4—42no—
224C4⋊C4—42no—
225C4⋊C4—42no—
226C4⋊C4—88no—
227C4⋊C4—162no—
228C4⋊C4—162no—
229C22⋊C4—42no—
230C22⋊C4—42no—
231C22⋊C4—42no—
232C22⋊C4—42no—
233C42—22no—
234C42—22no—
235C42—22no—
236C42—42no—
237C42—42no—
238C42—42no—
239C42—42no—
240C42—164no—
241C42#11714yes—
242C23—22no—
243C23—22no—
244C23#51811yescharacteristic
245C23#53911yescharacteristic
246C23#53911yescharacteristic
247C23D4211yescharacteristic
248C23#55911yescharacteristic
249C23#56311yescharacteristic
250C23#56311yescharacteristic
251C2×C4—24no—
252C2×C4—42no—
253C2×C4—44no—
254C2×C4—44no—
255C2×C4—42no—
256C2×C4—44no—
257C2×C4—42no—
258C2×C4—42no—
259C2×C4—44no—
260C2×C4—44no—
261C2×C4—44no—
262C2×C4—44no—
263C2×C4—44no—
264C2×C4—44no—
265C2×C4—84no—
266C2×C4—82no—
267C2×C4—82no—
268C2×C4—84no—
269C2×C4—82no—
270C2×C4—82no—
271C2×C4—162no—
272C2×C4—161no—
273C2×C4—161no—
274C2×C4C2×#11711yescharacteristic
275C2×C4C2×C4×D412yes—
276C2×C4#51911yescharacteristic
277C2×C4#51912yes—
278C2×C4#53612yes—
279C2×C4#53612yes—
280C2×C4#53911yescharacteristic
281C2×C4#53911yescharacteristic
282C2×C4#54112yes—
283C2×C4#54112yes—
284C2×C4#54712yes—
285C2×C4#54712yes—
286C2×C4#55012yes—
287C2×C4#55212yes—
288C2×C4#55212yes—
289C2×C4#56312yes—
290C8—24no—
291C8—24no—
292C8—82no—
293C8—82no—
294C22—22no—
295C22—22no—
296C22—22no—
297C22—22no—
298C22—21no—
299C22—21no—
300C22—21no—
301C22—21no—
302C22#231311yescharacteristic
303C22#236611yescharacteristic
304C22#238111yescharacteristic
305C22#268011yescharacteristic
306C22#268011yescharacteristic
307C22#279411yescharacteristic
308C22#279411yescharacteristic
309C22#328812yes—
310C22#328912yes—
311C4—44no—
312C4—84no—
313C4—84no—
314C4—84no—
315C4—82no—
316C4—82no—
317C4#232712yes—
318C4#234412yes—
319C4#272112yes—
320C4#272112yes—
321C2—21no—
322C2—21no—
323C2—21no—
324C2—21no—
325C2#1442411yescharacteristic
326C2#2320211yescharacteristic
327C2#2320311yescharacteristic
328C2#3111412yes—
329C2#3111512yes—
subgroupG/H
center, socleC23D42
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC24H(5,𝔽2)
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
22. #1847C4semidirect

Maximal subgroups

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

subgroupindex
#210382
#210372
C2×#18472
#113662
#113652
#112112
#108912
#108192