Finite Group Databasebeta

← #6479794

Subgroups — #6479794

Order 512.

1380 conjugacy classes of subgroups.

The subgroup lattice has 410 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35291C212yesmaximal
2#33207C211yesmaximal, characteristic
3#33172C211yesmaximal, characteristic
4#33049C212yesmaximal
5#29777C214yesmaximal
6#29439C218yesmaximal, complemented
7#29191C212yesmaximal, complemented
8#29179C212yesmaximal, complemented
9#28653C212yesmaximal, complemented
10#28616C212yesmaximal, complemented
11#22192C214yesmaximal
12C4×#424C211yesmaximal, characteristic
13C2×#498C2211yescharacteristic
14C2×#494C2211yescharacteristic
15C2×#487C2212yes—
16C2×#467C2214yes—
17C2×#424C2212yes—
18C2×C4×M16C2211yescharacteristic
19C22×C4×C8C2211yescharacteristic
20#2862C2212yes—
21#2859C2212yes—
22#2857C2212yes—
23#2857C2212yes—
24#2645C2214yes—
25#2625C2214yes—
26#2514—22no—
27#2509—22no—
28#2508—22no—
29#2495—22no—
30#2494—22no—
31#2489—22no—
32#2453—22no—
33#2421—22no—
34#2323C2214yes—
35#2320C2214yes—
36C4×C4⋊Q8C2212yes—
37C4×C4⋊Q8C2212yes—
38C4×#117C2211yescharacteristic
39#2087C2218yes—
40#2066C2218yes—
41#2066C2218yes—
42#2056C2218yes—
43#1990C2212yes—
44#1990C2212yes—
45#1969C2214yes—
46#1956C2212yes—
47#1956C2212yes—
48#1941C2214yes—
49#1935C2218yes—
50#1927C2218yes—
51#1921C2218yes—
52#1894C2218yes—
53#1878C2218yes—
54#1865C2218yes—
55#1845C2211yescharacteristic
56#1844C2211yescharacteristic
57#1834C2212yes—
58#1807C2214yes—
59#1768C2214yes—
60C4×C4⋊C8C2212yes—
61C4×C4⋊C8C2212yes—
62#1729C2212yes—
63C22×Q16C2314yes—
64C22×M16C2312yes—
65C23×C8C2312yes—
66C2×C4⋊Q8C2312yes—
67C2×C4⋊Q8C2312yes—
68C2×#125—22no—
69C2×C22⋊Q8—22no—
70C2×C22⋊Q8C2314yes—
71C2×C4×Q8—22no—
72C2×C4×Q8C2314yes—
73C2×#117C2311yescharacteristic
74C2×#117C2311yescharacteristic
75C22×C42C2311yescharacteristic
76#498—42no—
77#494—42no—
78#487—44no—
79#467—216no—
80C2×#107C2314yes—
81C2×#106C2314yes—
82C2×#106C2314yes—
83#424—28no—
84C2×C4⋊C8C2314yes—
85C2×C4⋊C8C2314yes—
86C2×#103C2318yes—
87C2×#103C2318yes—
88C2×#103C2318yes—
89C4×M16—42no—
90C2×#97C2312yes—
91C2×C4×C8—42no—
92C2×C4×C8C2312yes—
93C2×C4×C8C2311yescharacteristic
94C2×C4×C8C2311yescharacteristic
95C2×C4×C8C2312yes—
96#401—24no—
97#400—24no—
98#400—24no—
99#400—24no—
100#400—24no—
101#398—24no—
102#397—24no—
103#397—24no—
104#391—22no—
105#391—22no—
106#391—22no—
107#391—22no—
108#391C2314yes—
109#391C2314yes—
110#391C2314yes—
111#391C2314yes—
112#389—24no—
113#389—22no—
114#389—22no—
115#389—22no—
116#389—22no—
117#389—22no—
118#389—22no—
119#389—22no—
120#389—22no—
121#389C2314yes—
122#389C2314yes—
123#389C2314yes—
124#389C2314yes—
125#388—24no—
126#388—24no—
127#387—24no—
128#387—24no—
129#387—24no—
130#384C2312yes—
131#384C2312yes—
132#383—24no—
133#382—22no—
134#382—22no—
135#382—22no—
136#382—24no—
137#380—22no—
138#380—22no—
139#380—22no—
140#380C2314yes—
141C4×C4⋊C4—22no—
142C4×C4⋊C4—22no—
143C4×C4⋊C4—22no—
144C4×C4⋊C4D412yes—
145C4×C4⋊C4D412yes—
146C4×C4⋊C4C2314yes—
147C4×C22⋊C4D412yes—
148C4×C22⋊C4D412yes—
149#376C2312yes—
150C43C2312yes—
151#340C2318yes—
152#340C2318yes—
153#340C2318yes—
154#336C2314yes—
155#336C2314yes—
156C22×Q8—22no—
157C22×Q8C2414yes—
158C23×C4C2×D412yes—
159C23×C4C2411yescharacteristic
160C2×Q16—28no—
161C2×Q16—44no—
162C2×Q16—44no—
163C2×M16—24no—
164C2×M16—42no—
165C2×M16—42no—
166C22×C8—42no—
167C22×C8—42no—
168C22×C8#4118yes—
169C22×C8C2414yes—
170C22×C8C2412yes—
171C22×C8C2412yes—
172C4⋊Q8—24no—
173C4⋊Q8—24no—
174C4⋊Q8—24no—
175C4⋊Q8—24no—
176#125—84no—
177C22⋊Q8—216no—
178C22⋊Q8—84no—
179C4×Q8—48no—
180C4×Q8—84no—
181#117—24no—
182#117—42no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—24no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—24no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—24no—
193C2×C4⋊C4—24no—
194C2×C4⋊C4—24no—
195C2×C4⋊C4—24no—
196C2×C4⋊C4C2×D411yescharacteristic
197C2×C4⋊C4C2×D411yescharacteristic
198C2×C4⋊C4C2×D412yes—
199C2×C4⋊C4C2414yes—
200C2×C4⋊C4C2414yes—
201C2×C4⋊C4C2414yes—
202C2×C22⋊C4—24no—
203C2×C22⋊C4—24no—
204C2×C22⋊C4C2×D412yes—
205C2×C22⋊C4C2×D411yescharacteristic
206C2×C22⋊C4C2×D411yescharacteristic
207C2×C42—22no—
208C2×C42—22no—
209C2×C42—24no—
210C2×C42—24no—
211C2×C42C2×D412yes—
212C2×C42C2×D412yes—
213C2×C42C2×D412yes—
214C2×C42C2×D412yes—
215C2×C42C2×D412yes—
216C2×C42C2412yes—
217C2×C42C2412yes—
218C2×C42C2411yescharacteristic
219C2×C42C2411yescharacteristic
220#107—28no—
221#106—44no—
222#106—44no—
223C4⋊C8—28no—
224C4⋊C8—28no—
225#103—216no—
226#103—48no—
227#103—48no—
228#97—42no—
229C4×C8—22no—
230C4×C8—22no—
231C4×C8—24no—
232C4×C8—42no—
233#95—24no—
234#95—24no—
235#95—24no—
236#95—24no—
237#95—24no—
238#95—24no—
239#95—24no—
240#95—24no—
241#95—24no—
242#95—24no—
243#95—24no—
244#95—24no—
245#95—24no—
246#95—24no—
247#95C2×D412yes—
248#95C2×D412yes—
249#95C2×D412yes—
250#95C2×D412yes—
251C24C22×D411yescharacteristic
252C2×Q8—28no—
253C2×Q8—24no—
254C2×Q8—24no—
255C2×Q8—28no—
256C2×Q8—44no—
257C2×Q8—44no—
258C2×Q8—44no—
259C22×C4—21no—
260C22×C4—21no—
261C22×C4—22no—
262C22×C4—24no—
263C22×C4—24no—
264C22×C4—24no—
265C22×C4—24no—
266C22×C4—24no—
267C22×C4—24no—
268C22×C4—24no—
269C22×C4C22×D412yes—
270C22×C4C22×D412yes—
271C22×C4C22×D412yes—
272C22×C4C22×D412yes—
273C22×C4C22×D412yes—
274C22×C4C22×D411yescharacteristic
275C22×C4C22×D412yes—
276C22×C4C2×#4114yes—
277C22×C4C2511yescharacteristic
278Q16—88no—
279M16—48no—
280C2×C8—22no—
281C2×C8—22no—
282C2×C8—22no—
283C2×C8—22no—
284C2×C8—24no—
285C2×C8—24no—
286C2×C8—24no—
287C2×C8—48no—
288C2×C8C2×#4114yes—
289C2×C8C2×#4114yes—
290C4⋊C4—22no—
291C4⋊C4—22no—
292C4⋊C4—28no—
293C4⋊C4—24no—
294C4⋊C4—24no—
295C4⋊C4—28no—
296C4⋊C4—42no—
297C4⋊C4—48no—
298C4⋊C4—48no—
299C4⋊C4—44no—
300C4⋊C4—44no—
301C4⋊C4—44no—
302C4⋊C4—44no—
303C4⋊C4—44no—
304C4⋊C4—44no—
305C4⋊C4—44no—
306C4⋊C4—84no—
307C4⋊C4—84no—
308C4⋊C4—84no—
309C4⋊C4—84no—
310C22⋊C4—24no—
311C22⋊C4—41no—
312C22⋊C4—41no—
313C22⋊C4—48no—
314C22⋊C4—84no—
315C42—24no—
316C42—24no—
317C42—24no—
318C42—24no—
319C42—24no—
320C42—24no—
321C42—21no—
322C42—21no—
323C42—24no—
324C42—48no—
325C42—84no—
326C42H(5,𝔽2)12yes—
327C422−1+412yes—
328C23—21no—
329C23—21no—
330C23—22no—
331C23—22no—
332C23#53212yes—
333C23C23×D411yescharacteristic
334Q8—48no—
335Q8—48no—
336Q8—88no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—24no—
340C2×C4—24no—
341C2×C4—22no—
342C2×C4—22no—
343C2×C4—22no—
344C2×C4—22no—
345C2×C4—22no—
346C2×C4—22no—
347C2×C4—22no—
348C2×C4—22no—
349C2×C4—22no—
350C2×C4—22no—
351C2×C4—22no—
352C2×C4—22no—
353C2×C4—22no—
354C2×C4—22no—
355C2×C4—22no—
356C2×C4—48no—
357C2×C4—44no—
358C2×C4—44no—
359C2×C4—44no—
360C2×C4—44no—
361C2×C4—44no—
362C2×C4—44no—
363C2×C4—44no—
364C2×C4—44no—
365C2×C4—44no—
366C2×C4—44no—
367C2×C4—44no—
368C2×C4—44no—
369C2×C4#53212yes—
370C2×C4#53714yes—
371C2×C4#57714yes—
372C2×C4#57814yes—
373C2×C4C22×#4112yes—
374C2×C4C2×H(5,𝔽2)11yescharacteristic
375C2×C4C2×2−1+411yescharacteristic
376C2×C4#58512yes—
377C8—24no—
378C8—24no—
379C8—44no—
380C22—22no—
381C22—22no—
382C22—22no—
383C22—42no—
384C22—42no—
385C22C2×#53211yescharacteristic
386C22C2×#53711yescharacteristic
387C22#344911yescharacteristic
388C22C2×#57712yes—
389C22C2×#57812yes—
390C4—22no—
391C4—42no—
392C4—42no—
393C4—42no—
394C4—42no—
395C4—42no—
396C4—84no—
397C4—84no—
398C4—84no—
399C4—84no—
400C4#325818yes—
401C4#350512yes—
402C4#350712yes—
403C2—42no—
404C2#3510311yescharacteristic
405C2#6071511yescharacteristic
406C2#6071611yescharacteristic
407C2C2×#325812yes—
408C2#6100812yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC24C22×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
6. #29439C2semidirect
7. #29191C2semidirect
8. #29179C2semidirect
9. #28653C2semidirect
10. #28616C2semidirect

Maximal subgroups

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

subgroupindex
#352912
#332072
#331722
#330492
#297772
#294392
#291912
#291792
#286532
#286162
#221922
C4×#4242