Finite Group Databasebeta

← #360610

Subgroups — #360610

Order 512.

1366 conjugacy classes of subgroups.

The subgroup lattice has 409 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34179C211yesmaximal, characteristic
2#20661C212yesmaximal
3C2×#1742C211yesmaximal, characteristic
4#13939C211yesmaximal, characteristic
5#11342C212yesmaximal, complemented
6#11296C212yesmaximal, complemented
7#10899C214yesmaximal, complemented
8#10732C212yesmaximal
9C23×M16C2212yes—
10C22×#117C2211yescharacteristic
11C2×#428C2214yes—
12#2347—22no—
13#2347—22no—
14#2346—24no—
15#2316C2212yes—
16#2312—22no—
17#2283—22no—
18#2280C2212yes—
19C2×#380C2211yescharacteristic
20#2274C2212yes—
21C2×C4×C22⋊C4C2211yescharacteristic
22#1809—24no—
23#1808—22no—
24#1807—22no—
25#1742C418yescomplemented
26C2×#343C2211yescharacteristic
27C2×#343C2211yescharacteristic
28C2×#343C2212yes—
29#1736C2214yes—
30C2×#339C2214yes—
31C2×#339C2214yes—
32#1440—22no—
33C4⋊#95C2212yes—
34C4⋊#95C2212yes—
35#1434—22no—
36C24×C4C2311yescharacteristic
37C22×M16—22no—
38C22×M16—24no—
39C22×M16—28no—
40C22×M16—44no—
41C22×M16C2×C418yes—
42C22×M16C2312yes—
43C22×M16C2312yes—
44C23×C8—22no—
45C2×#117—22no—
46C2×#117—22no—
47C2×#117—24no—
48C2×#117C2×C414yes—
49C2×#117Q814yes—
50C2×#117C2314yes—
51C22×C4⋊C4C2312yes—
52C22×C4⋊C4C2311yescharacteristic
53C22×C4⋊C4C2311yescharacteristic
54C22×C22⋊C4—21no—
55C22×C22⋊C4—21no—
56C22×C22⋊C4D412yes—
57C22×C42D412yes—
58C22×C42C2312yes—
59#428—28no—
60#428—44no—
61C2×#107—24no—
62C2×#106—24no—
63#389—24no—
64#389—24no—
65#388—28no—
66#388—28no—
67#383—24no—
68#382—24no—
69#381—22no—
70#380—24no—
71#380—22no—
72#380—22no—
73#380D412yes—
74#380D412yes—
75#380D414yes—
76#379—22no—
77C4×C4⋊C4—22no—
78C4×C4⋊C4—22no—
79C4×C4⋊C4—24no—
80C4×C4⋊C4—24no—
81C4×C22⋊C4—24no—
82C4×C22⋊C4—22no—
83C4×C22⋊C4—22no—
84C4×C22⋊C4—24no—
85#376—24no—
86C2×#95—21no—
87C2×#95—21no—
88C2×#95—24no—
89C2×#95—24no—
90#343C2×C414yes—
91#343C2×C418yes—
92#343C2×C414yes—
93#340—24no—
94#340—24no—
95#339—28no—
96#339—44no—
97C25C2×D411yescharacteristic
98C23×C4—24no—
99C23×C4—22no—
100C23×C4—22no—
101C23×C4—22no—
102C23×C4—22no—
103C23×C4C22×C412yes—
104C23×C4C2×D411yescharacteristic
105C23×C4C2×D412yes—
106C23×C4C2×D412yes—
107C23×C4C2×D412yes—
108C23×C4C2×D412yes—
109C23×C4C2×Q812yes—
110C23×C4#4114yes—
111C23×C4C2411yescharacteristic
112C2×M16—22no—
113C2×M16—24no—
114C2×M16—28no—
115C2×M16—22no—
116C2×M16—24no—
117C2×M16—24no—
118C2×M16—216no—
119C2×M16—48no—
120C2×M16—48no—
121C2×M16—416no—
122C2×M16C22×C414yes—
123C2×M16C22×C414yes—
124C2×M16#4118yes—
125C22×C8—22no—
126C22×C8—28no—
127C22×C8—22no—
128C22×C8—48no—
129#117—28no—
130#117—28no—
131#117—28no—
132#117—28no—
133#117—24no—
134#117—24no—
135#117—24no—
136#117—24no—
137C2×C4⋊C4—22no—
138C2×C4⋊C4—22no—
139C2×C4⋊C4—22no—
140C2×C4⋊C4—22no—
141C2×C4⋊C4—24no—
142C2×C4⋊C4—24no—
143C2×C4⋊C4—22no—
144C2×C4⋊C4—22no—
145C2×C4⋊C4—24no—
146C2×C4⋊C4—24no—
147C2×C4⋊C4C22×C412yes—
148C2×C4⋊C4C22×C412yes—
149C2×C4⋊C4C2×D414yes—
150C2×C4⋊C4C2×D414yes—
151C2×C4⋊C4C2×Q812yes—
152C2×C4⋊C4C2×Q812yes—
153C2×C22⋊C4—24no—
154C2×C22⋊C4—24no—
155C2×C22⋊C4—24no—
156C2×C22⋊C4—22no—
157C2×C22⋊C4—22no—
158C2×C22⋊C4—24no—
159C2×C22⋊C4—44no—
160C2×C22⋊C4—42no—
161C2×C22⋊C4—42no—
162C2×C22⋊C4—82no—
163C2×C22⋊C4C4⋊C418yes—
164C2×C42—22no—
165C2×C42—22no—
166C2×C42—24no—
167C2×C42—24no—
168C2×C42—24no—
169C2×C42—24no—
170C2×C42—22no—
171C2×C42—22no—
172C2×C42—24no—
173C2×C42—24no—
174C2×C42C4⋊C418yes—
175#107—48no—
176#106—48no—
177#95—28no—
178#95—28no—
179#95—24no—
180#95—22no—
181#95—22no—
182#95—48no—
183#95—41no—
184#95—41no—
185#95—42no—
186#95—84no—
187C24—22no—
188C24—22no—
189C24—42no—
190C24—41no—
191C24—41no—
192C24C2×C4⋊C412yes—
193C24#12012yes—
194C24#12312yes—
195C24C22×D411yescharacteristic
196C22×C4—22no—
197C22×C4—22no—
198C22×C4—24no—
199C22×C4—22no—
200C22×C4—22no—
201C22×C4—24no—
202C22×C4—24no—
203C22×C4—24no—
204C22×C4—24no—
205C22×C4—24no—
206C22×C4—24no—
207C22×C4—28no—
208C22×C4—28no—
209C22×C4—22no—
210C22×C4—22no—
211C22×C4—22no—
212C22×C4—22no—
213C22×C4—24no—
214C22×C4—24no—
215C22×C4—24no—
216C22×C4—44no—
217C22×C4—42no—
218C22×C4—42no—
219C22×C4—42no—
220C22×C4—42no—
221C22×C4—42no—
222C22×C4—42no—
223C22×C4—42no—
224C22×C4—42no—
225C22×C4—44no—
226C22×C4—44no—
227C22×C4—42no—
228C22×C4—42no—
229C22×C4—42no—
230C22×C4C2×C4⋊C412yes—
231C22×C4C2×C4⋊C414yes—
232C22×C4C2×C4⋊C414yes—
233C22×C4C4×D414yes—
234C22×C4C4×D414yes—
235C22×C4#12011yescharacteristic
236C22×C4#12011yescharacteristic
237C22×C4C22⋊Q814yes—
238C22×C4C22⋊Q814yes—
239C22×C4#12312yes—
240C22×C4C23×C411yescharacteristic
241C22×C4C22×D412yes—
242C22×C4C22×Q811yescharacteristic
243C22×C4C2×#4114yes—
244C22×C4C2×#4112yes—
245C22×C4H(5,𝔽2)12yes—
246C22×C42−1+412yes—
247M16—24no—
248M16—28no—
249M16—24no—
250M16—24no—
251M16—416no—
252M16—44no—
253M16#11718yes—
254C2×C8—24no—
255C2×C8—28no—
256C2×C8—24no—
257C2×C8—416no—
258C2×C8—44no—
259C4⋊C4—24no—
260C4⋊C4—24no—
261C4⋊C4—24no—
262C4⋊C4—24no—
263C4⋊C4—24no—
264C4⋊C4—24no—
265C4⋊C4—24no—
266C4⋊C4—24no—
267C4⋊C4—44no—
268C4⋊C4—44no—
269C4⋊C4—84no—
270C4⋊C4—84no—
271C22⋊C4—216no—
272C22⋊C4—48no—
273C22⋊C4—44no—
274C22⋊C4—44no—
275C22⋊C4—84no—
276C22⋊C4—82no—
277C22⋊C4—82no—
278C22⋊C4—84no—
279C42—28no—
280C42—28no—
281C42—44no—
282C42—44no—
283C42—44no—
284C42—44no—
285C42—84no—
286C42—84no—
287C23—22no—
288C23—21no—
289C23—21no—
290C23—24no—
291C23—24no—
292C23—22no—
293C23—22no—
294C23—22no—
295C23—24no—
296C23—22no—
297C23—42no—
298C23—42no—
299C23—42no—
300C23—42no—
301C23—82no—
302C23—82no—
303C23—82no—
304C23—82no—
305C23#38514yes—
306C23C22×C4⋊C411yescharacteristic
307C23C2×#12011yescharacteristic
308C23C2×#12311yescharacteristic
309C23#53212yes—
310C23#53712yes—
311C2×C4—22no—
312C2×C4—22no—
313C2×C4—22no—
314C2×C4—24no—
315C2×C4—22no—
316C2×C4—24no—
317C2×C4—24no—
318C2×C4—24no—
319C2×C4—24no—
320C2×C4—28no—
321C2×C4—28no—
322C2×C4—24no—
323C2×C4—24no—
324C2×C4—44no—
325C2×C4—42no—
326C2×C4—42no—
327C2×C4—44no—
328C2×C4—44no—
329C2×C4—44no—
330C2×C4—44no—
331C2×C4—44no—
332C2×C4—44no—
333C2×C4—44no—
334C2×C4—44no—
335C2×C4—44no—
336C2×C4—44no—
337C2×C4—44no—
338C2×C4—82no—
339C2×C4—82no—
340C2×C4—82no—
341C2×C4—82no—
342C2×C4—82no—
343C2×C4—82no—
344C2×C4—84no—
345C2×C4—84no—
346C2×C4—82no—
347C2×C4—82no—
348C2×C4—82no—
349C2×C4#38512yes—
350C2×C4#38512yes—
351C2×C4C2×#11712yes—
352C2×C4C2×C4×D412yes—
353C2×C4#51811yescharacteristic
354C2×C4#51911yescharacteristic
355C2×C4C2×C22⋊Q812yes—
356C2×C4#53312yes—
357C2×C4#54412yes—
358C2×C4#54614yes—
359C2×C4#54714yes—
360C2×C4#55114yes—
361C2×C4#55214yes—
362C8—28no—
363C8—44no—
364C22—21no—
365C22—21no—
366C22—21no—
367C22—21no—
368C22—22no—
369C22—22no—
370C22—22no—
371C22—22no—
372C22—24no—
373C22—24no—
374C22—24no—
375C22—44no—
376C22—44no—
377C22—84no—
378C22—82no—
379C22—82no—
380C22#228511yescharacteristic
381C22C2×#38511yescharacteristic
382C22#231611yescharacteristic
383C22#243012yes—
384C22#249812yes—
385C22#301812yes—
386C22#302111yescharacteristic
387C22#302211yescharacteristic
388C4—22no—
389C4—22no—
390C4—44no—
391C4—44no—
392C4—84no—
393C4—84no—
394C4—84no—
395C4—162no—
396C4#234212yes—
397C4#234412yes—
398C4#243814yes—
399C2—22no—
400C2—21no—
401C2—21no—
402C2—44no—
403C2#1387811yescharacteristic
404C2#2085511yescharacteristic
405C2#2085811yescharacteristic
406C2#2551612yes—
407C2#2551812yes—
subgroupG/H
center, socleC23C2×#120
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C24×C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
5. #11342C2semidirect
6. #11296C2semidirect
7. #10899C2semidirect
25. #1742C4semidirect

Maximal subgroups

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

subgroupindex
#341792
#206612
C2×#17422
#139392
#113422
#112962
#108992
#107322