Finite Group Databasebeta

← #6667455

Subgroups — #6667455

Order 512.

1269 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#35283C212yesmaximal
2#32855C214yesmaximal, complemented
3#29947C218yesmaximal
4#29878C211yesmaximal, characteristic
5#29877C212yesmaximal
6#29876C211yesmaximal, characteristic
7#29293C214yesmaximal, complemented
8#29219C212yesmaximal, complemented
9#29218C212yesmaximal, complemented
10#28688C212yesmaximal, complemented
11#28688C212yesmaximal, complemented
12#20809C211yesmaximal, characteristic
13C2×#500C2211yescharacteristic
14C2×#500C2211yescharacteristic
15C2×#498C2212yes—
16C2×#431C2211yescharacteristic
17C2×#431C2211yescharacteristic
18#2868C2212yes—
19#2862C2212yes—
20#2857C2212yes—
21#2857C2212yes—
22#2647C2214yes—
23#2628C2214yes—
24#2511—22no—
25#2511—22no—
26#2492—22no—
27#2492—22no—
28#2451—22no—
29#2451—22no—
30#2423—22no—
31#2423—22no—
32#2331C2214yes—
33#2314C2214yes—
34C4×C4⋊Q8C2212yes—
35C4×C4⋊Q8C2212yes—
36C4×#117C2211yescharacteristic
37#2100C2218yes—
38#2099C2218yes—
39#2088C2218yes—
40#2087C2214yes—
41#2086C2214yes—
42#2085C2218yes—
43#2085C2218yes—
44#2077C2218yes—
45#2076C2214yes—
46#2075C2214yes—
47#1990C2212yes—
48#1990C2212yes—
49#1990C2212yes—
50#1990C2212yes—
51#1988C2218yes—
52#1987C2218yes—
53#1949C2218yes—
54#1949C2218yes—
55#1843C2212yes—
56#1834C2211yescharacteristic
57#1834C2212yes—
58#1834C2211yescharacteristic
59#1832C2212yes—
60#1798C2214yes—
61C4×C4⋊C8C2212yes—
62C4×C4⋊C8C2212yes—
63C2×C4⋊Q8—22no—
64C2×C4⋊Q8—22no—
65C2×C4⋊Q8C2312yes—
66C2×C4⋊Q8C2312yes—
67C2×C4⋊Q8C2314yes—
68C2×C22⋊Q8—22no—
69C2×C22⋊Q8—22no—
70C2×C4×Q8—22no—
71C2×C4×Q8—22no—
72C2×#117C2312yes—
73C22×C42C2311yescharacteristic
74#500—42no—
75#500—42no—
76#498—44no—
77#431—42no—
78#431—42no—
79C2×#107C2314yes—
80C2×#107C2314yes—
81C2×#106C2318yes—
82C2×C4⋊C8C2314yes—
83C2×C4⋊C8C2314yes—
84C2×C4⋊C8C2312yes—
85C2×C4⋊C8C2312yes—
86C2×C22⋊C8C2312yes—
87C2×C22⋊C8C2312yes—
88C2×C4×C8C2312yes—
89C2×C4×C8C2311yescharacteristic
90C2×C4×C8C2311yescharacteristic
91#400—24no—
92#400—24no—
93#400—24no—
94#399—24no—
95#398—24no—
96#398—24no—
97#395—24no—
98#393—24no—
99#391—22no—
100#391—22no—
101#391—22no—
102#391—22no—
103#391—22no—
104#391—22no—
105#391—22no—
106#391—22no—
107#391—24no—
108#391—24no—
109#389—22no—
110#389—22no—
111#389—22no—
112#389—22no—
113#389C2314yes—
114#389C2314yes—
115#389C2314yes—
116#389C2314yes—
117#388—24no—
118#388—24no—
119#387—24no—
120#387—24no—
121#387—24no—
122#387—24no—
123#384—22no—
124#384—22no—
125#384C2312yes—
126#384C2312yes—
127#384C2314yes—
128#382C2314yes—
129#382C2314yes—
130#380—22no—
131#380—22no—
132#380C2314yes—
133#380C2314yes—
134C4×C4⋊C4—22no—
135C4×C4⋊C4—22no—
136C4×C4⋊C4D411yescharacteristic
137C4×C4⋊C4D412yes—
138C4×C4⋊C4D411yescharacteristic
139C4×C4⋊C4C2314yes—
140C4×C4⋊C4C2314yes—
141C4×C22⋊C4D411yescharacteristic
142C4×C22⋊C4D412yes—
143C4×C22⋊C4D411yescharacteristic
144#376C2311yescharacteristic
145#376C2311yescharacteristic
146C43C2312yes—
147#340C2318yes—
148#340C2314yes—
149#340C2318yes—
150#340C2314yes—
151#340C2318yes—
152#340C2318yes—
153#340C2318yes—
154#336C2314yes—
155#336C2314yes—
156C22×Q8—22no—
157C22×Q8—22no—
158C22×Q8—24no—
159C23×C4C2×D411yescharacteristic
160C23×C4C2×D411yescharacteristic
161C23×C4C2411yescharacteristic
162C22×C8C2414yes—
163C22×C8C2412yes—
164C22×C8C2412yes—
165C4⋊Q8—24no—
166C4⋊Q8—24no—
167C4⋊Q8—216no—
168C4⋊Q8—24no—
169C4⋊Q8—24no—
170C4⋊Q8—84no—
171C4⋊Q8—84no—
172C22⋊Q8—84no—
173C22⋊Q8—84no—
174C4×Q8—84no—
175C4×Q8—84no—
176#117—28no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—24no—
184C2×C4⋊C4—24no—
185C2×C4⋊C4—24no—
186C2×C4⋊C4—24no—
187C2×C4⋊C4—24no—
188C2×C4⋊C4—24no—
189C2×C4⋊C4—24no—
190C2×C4⋊C4C2×D412yes—
191C2×C4⋊C4C2×D412yes—
192C2×C4⋊C4C2414yes—
193C2×C4⋊C4C2414yes—
194C2×C4⋊C4C2414yes—
195C2×C4⋊C4C2414yes—
196C2×C22⋊C4—24no—
197C2×C22⋊C4—24no—
198C2×C22⋊C4C2×D412yes—
199C2×C22⋊C4C2×D412yes—
200C2×C42—22no—
201C2×C42—22no—
202C2×C42—24no—
203C2×C42—24no—
204C2×C42C2×D412yes—
205C2×C42C2×D412yes—
206C2×C42C2×D412yes—
207C2×C42C2×D412yes—
208C2×C42C2×D411yescharacteristic
209C2×C42C2×D411yescharacteristic
210C2×C42C2412yes—
211C2×C42C2411yescharacteristic
212C2×C42C2411yescharacteristic
213C2×C42C2412yes—
214#107—44no—
215#107—44no—
216#106—48no—
217C4⋊C8—28no—
218C4⋊C8—28no—
219C4⋊C8—42no—
220C4⋊C8—42no—
221C22⋊C8—42no—
222C22⋊C8—42no—
223C4×C8—24no—
224C4×C8—22no—
225C4×C8—22no—
226#95—24no—
227#95—24no—
228#95—24no—
229#95—24no—
230#95—24no—
231#95—24no—
232#95—24no—
233#95—24no—
234#95—24no—
235#95—24no—
236#95—24no—
237#95—24no—
238#95C2×D411yescharacteristic
239#95C2×D412yes—
240#95C2×D411yescharacteristic
241#95C2×D411yescharacteristic
242#95C2×D412yes—
243#95C2×D411yescharacteristic
244C24C22×D411yescharacteristic
245C2×Q8—48no—
246C2×Q8—44no—
247C2×Q8—44no—
248C2×Q8—44no—
249C2×Q8—44no—
250C2×Q8—44no—
251C2×Q8—44no—
252C2×Q8—84no—
253C2×Q8—84no—
254C22×C4—22no—
255C22×C4—22no—
256C22×C4—22no—
257C22×C4—24no—
258C22×C4—22no—
259C22×C4—21no—
260C22×C4—21no—
261C22×C4—24no—
262C22×C4—24no—
263C22×C4—24no—
264C22×C4—24no—
265C22×C4—24no—
266C22×C4C22×D411yescharacteristic
267C22×C4C22×D412yes—
268C22×C4C22×D411yescharacteristic
269C22×C4C22×D411yescharacteristic
270C22×C4C22×D412yes—
271C22×C4C22×D412yes—
272C22×C4C22×D412yes—
273C22×C4C22×D411yescharacteristic
274C22×C4C22×D411yescharacteristic
275C22×C4C2511yescharacteristic
276C2×C8—24no—
277C2×C8—22no—
278C2×C8—22no—
279C2×C8—24no—
280C2×C8—24no—
281C2×C8—24no—
282C2×C8—24no—
283C4⋊C4—24no—
284C4⋊C4—24no—
285C4⋊C4—24no—
286C4⋊C4—24no—
287C4⋊C4—28no—
288C4⋊C4—28no—
289C4⋊C4—24no—
290C4⋊C4—24no—
291C4⋊C4—48no—
292C4⋊C4—44no—
293C4⋊C4—44no—
294C4⋊C4—44no—
295C4⋊C4—44no—
296C4⋊C4—44no—
297C4⋊C4—44no—
298C4⋊C4—84no—
299C4⋊C4—84no—
300C4⋊C4—84no—
301C4⋊C4—84no—
302C4⋊C4—84no—
303C4⋊C4—84no—
304C22⋊C4—24no—
305C22⋊C4—24no—
306C22⋊C4—84no—
307C22⋊C4—84no—
308C42—22no—
309C42—22no—
310C42—24no—
311C42—24no—
312C42—24no—
313C42—24no—
314C42—21no—
315C42—21no—
316C42—24no—
317C42—84no—
318C42—84no—
319C42H(5,𝔽2)12yes—
320C422−1+418yes—
321C422−1+412yes—
322C23—22no—
323C23—21no—
324C23—21no—
325C23—22no—
326C23—21no—
327C23C23×D411yescharacteristic
328Q8—88no—
329Q8—88no—
330Q8—168no—
331C2×C4—22no—
332C2×C4—22no—
333C2×C4—22no—
334C2×C4—22no—
335C2×C4—22no—
336C2×C4—22no—
337C2×C4—22no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—22no—
342C2×C4—21no—
343C2×C4—22no—
344C2×C4—22no—
345C2×C4—22no—
346C2×C4—22no—
347C2×C4—21no—
348C2×C4—44no—
349C2×C4—44no—
350C2×C4—42no—
351C2×C4—42no—
352C2×C4—44no—
353C2×C4—44no—
354C2×C4—44no—
355C2×C4—44no—
356C2×C4—44no—
357C2×C4—44no—
358C2×C4—44no—
359C2×C4—44no—
360C2×C4—44no—
361C2×C4—44no—
362C2×C4—44no—
363C2×C4—44no—
364C2×C4#53614yes—
365C2×C4#53614yes—
366C2×C4#53712yes—
367C2×C4#53712yes—
368C2×C4#57612yes—
369C2×C4#57714yes—
370C2×C4#57812yes—
371C2×C4C2×H(5,𝔽2)11yescharacteristic
372C2×C4C2×2−1+414yes—
373C2×C4C2×2−1+411yescharacteristic
374C8—44no—
375C8—42no—
376C8—42no—
377C22—41no—
378C22—42no—
379C22—42no—
380C22—41no—
381C22—41no—
382C22C2×#53612yes—
383C22C2×#53711yescharacteristic
384C22C2×#57611yescharacteristic
385C22C2×#57712yes—
386C22C2×#57811yescharacteristic
387C4—24no—
388C4—24no—
389C4—41no—
390C4—42no—
391C4—42no—
392C4—41no—
393C4—41no—
394C4—84no—
395C4—84no—
396C4—84no—
397C4—84no—
398C4#326214yes—
399C4#326314yes—
400C4#355012yes—
401C4#355612yes—
402C2—81no—
403C2#3764111yescharacteristic
404C2#6087712yes—
405C2#6087812yes—
406C2C2×#326211yescharacteristic
407C2C2×#326311yescharacteristic
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
2. #32855C2semidirect
7. #29293C2semidirect
8. #29219C2semidirect
9. #29218C2semidirect
10. #28688C2semidirect
11. #28688C2semidirect

Maximal subgroups

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

subgroupindex
#352832
#328552
#299472
#298782
#298772
#298762
#292932
#292192
#292182
#286882
#208092