Finite Group Databasebeta

← #6849748

Subgroups — #6849748

Order 512.

1300 conjugacy classes of subgroups.

The subgroup lattice has 397 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36209C212yesmaximal
2#33263C211yesmaximal, characteristic, complemented
3#32518C214yesmaximal, complemented
4#32471C214yesmaximal
5#29877C212yesmaximal, complemented
6#29177C212yesmaximal, complemented
7#27949C212yesmaximal, complemented
8#27876C212yesmaximal, complemented
9#27483C218yesmaximal, complemented
10#21984C212yesmaximal, complemented
11#21727C211yesmaximal, characteristic, complemented
12#21265C211yesmaximal, characteristic
13C2×#498C2211yescharacteristic
14C2×#431C2212yes—
15#2859C2212yes—
16#2857C2212yes—
17#2857C2212yes—
18#2536—22no—
19#2508—22no—
20#2508C2214yes—
21#2495—22no—
22#2495C2214yes—
23#2489—22no—
24#2489—22no—
25#2485—22no—
26#2485C2214yes—
27#2356—22no—
28#2346—22no—
29#2346C2214yes—
30C4×C4⋊Q8C2212yes—
31#2285C2212yes—
32#2285C2212yes—
33C42×D4C2211yescharacteristic
34#2088C2218yes—
35#2088C2218yes—
36#2085C2218yes—
37#2077C2218yes—
38#2077C2218yes—
39#1990C2212yes—
40#1990C2212yes—
41#1990C2212yes—
42#1955C2212yes—
43#1949C2218yes—
44#1944C2214yes—
45#1944C2214yes—
46#1943C2214yes—
47#1940C2212yes—
48#1924C2214yes—
49#1919C2218yes—
50#1919C2218yes—
51#1854C2218yes—
52#1854C2218yes—
53#1845C2211yescharacteristic
54#1845C2211yescharacteristic
55#1844C2212yes—
56#1844C2211yescharacteristic
57#1834C2212yes—
58#1834C2212yes—
59#1776C2214yes—
60C4×C4⋊C8C2211yescharacteristic
61C4×C22⋊C8C2212yes—
62C4×#97C2211yescharacteristic
63C2×C4⋊Q8C2312yes—
64C2×#125—22no—
65C2×C22⋊Q8—22no—
66C2×C22⋊Q8—22no—
67C2×C4×Q8—22no—
68C2×C4×D4D412yes—
69C2×C4×D4C2311yescharacteristic
70C2×#117—22no—
71C2×#117C2314yes—
72C22×C42C2312yes—
73#498—24no—
74#431—28no—
75C2×#107C2314yes—
76C2×#107C2314yes—
77C2×#106C2314yes—
78C2×#106C2314yes—
79C2×#106C2318yes—
80C2×C4⋊C8C2314yes—
81C2×C4⋊C8C2312yes—
82C2×C22⋊C8C2314yes—
83C2×C22⋊C8C2314yes—
84C2×#97C2312yes—
85C2×#97C2312yes—
86C2×C4×C8C2312yes—
87C2×C4×C8C2311yescharacteristic
88C2×C4×C8C2311yescharacteristic
89#400—24no—
90#400—24no—
91#398—24no—
92#397—24no—
93#397—24no—
94#394—24no—
95#391—22no—
96#391—22no—
97#391—22no—
98#389—22no—
99#389—22no—
100#389—22no—
101#389—22no—
102#389—22no—
103#389C2314yes—
104#389C2314yes—
105#389C2314yes—
106#389C2314yes—
107#389C2314yes—
108#388—24no—
109#388—24no—
110#388—24no—
111#388—24no—
112#387—24no—
113#387—24no—
114#384C2312yes—
115#384C2312yes—
116#384C2312yes—
117#383—24no—
118#382—22no—
119#382—22no—
120#382—24no—
121#382—24no—
122#382C2314yes—
123#380—22no—
124#380—22no—
125#380—22no—
126#380—22no—
127#380—22no—
128#380C2314yes—
129#380C2314yes—
130#380C2314yes—
131C4×C4⋊C4—22no—
132C4×C4⋊C4—22no—
133C4×C4⋊C4—22no—
134C4×C4⋊C4—22no—
135C4×C4⋊C4D412yes—
136C4×C4⋊C4C2314yes—
137C4×C4⋊C4C2314yes—
138C4×C4⋊C4C2311yescharacteristic
139C4×C22⋊C4D412yes—
140C4×C22⋊C4D412yes—
141C4×C22⋊C4C2312yes—
142#376—24no—
143C43C2311yescharacteristic
144#340C2318yes—
145#340C2318yes—
146#340C2318yes—
147#340C2318yes—
148#340C2318yes—
149#336C2314yes—
150#336C2312yes—
151C22×Q8—22no—
152C22×D4C2×D411yescharacteristic
153C23×C4C2×D412yes—
154C23×C4C2×D412yes—
155C23×C4C2412yes—
156C22×C8C2414yes—
157C22×C8C2412yes—
158C22×C8C2412yes—
159C4⋊Q8—24no—
160C4⋊Q8—24no—
161#125—84no—
162C22⋊Q8—84no—
163C22⋊Q8—84no—
164C4×Q8—48no—
165C4×D4—44no—
166C4×D4—42no—
167#117—48no—
168#117—48no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—24no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—24no—
178C2×C4⋊C4—24no—
179C2×C4⋊C4—24no—
180C2×C4⋊C4—24no—
181C2×C4⋊C4C2×D412yes—
182C2×C4⋊C4C2×D411yescharacteristic
183C2×C4⋊C4C2414yes—
184C2×C4⋊C4C2414yes—
185C2×C4⋊C4C2414yes—
186C2×C4⋊C4C2414yes—
187C2×C22⋊C4—24no—
188C2×C22⋊C4—24no—
189C2×C22⋊C4—24no—
190C2×C22⋊C4—24no—
191C2×C22⋊C4C2×D412yes—
192C2×C22⋊C4C2×D412yes—
193C2×C22⋊C4C2×D412yes—
194C2×C42—24no—
195C2×C42—24no—
196C2×C42—24no—
197C2×C42—24no—
198C2×C42—24no—
199C2×C42C2×D412yes—
200C2×C42C2×D412yes—
201C2×C42C2×D412yes—
202C2×C42C2×D412yes—
203C2×C42C2411yescharacteristic
204C2×C42C2411yescharacteristic
205C2×C42C2412yes—
206C2×C42C2411yescharacteristic
207#107—44no—
208#107—44no—
209#106—28no—
210#106—28no—
211#106—48no—
212C4⋊C8—28no—
213C4⋊C8—42no—
214C22⋊C8—28no—
215C22⋊C8—28no—
216#97—24no—
217#97—42no—
218C4×C8—24no—
219C4×C8—22no—
220C4×C8#4114yes—
221#95—24no—
222#95—24no—
223#95—24no—
224#95—24no—
225#95—24no—
226#95—24no—
227#95—24no—
228#95—24no—
229#95—24no—
230#95—24no—
231#95—24no—
232#95—24no—
233#95C2×D412yes—
234#95C2×D411yescharacteristic
235#95C2×D411yescharacteristic
236#95C2×D412yes—
237C24C22×D412yes—
238C2×Q8—44no—
239C2×Q8—44no—
240C2×Q8—44no—
241C2×D4—41no—
242C2×D4—42no—
243C22×C4—24no—
244C22×C4—24no—
245C22×C4—24no—
246C22×C4—24no—
247C22×C4—24no—
248C22×C4—24no—
249C22×C4—22no—
250C22×C4—24no—
251C22×C4—22no—
252C22×C4—24no—
253C22×C4—24no—
254C22×C4—24no—
255C22×C4—24no—
256C22×C4—24no—
257C22×C4C22×D412yes—
258C22×C4C22×D412yes—
259C22×C4C22×D412yes—
260C22×C4C22×D412yes—
261C22×C4C22×D411yescharacteristic
262C22×C4C22×D411yescharacteristic
263C22×C4C22×D412yes—
264C22×C4C2511yescharacteristic
265C2×C8—22no—
266C2×C8—22no—
267C2×C8—22no—
268C2×C8—22no—
269C2×C8—24no—
270C2×C8—24no—
271C2×C8—24no—
272C2×C8C2×#4114yes—
273C2×C82−1+414yes—
274C4⋊C4—24no—
275C4⋊C4—28no—
276C4⋊C4—24no—
277C4⋊C4—28no—
278C4⋊C4—28no—
279C4⋊C4—41no—
280C4⋊C4—42no—
281C4⋊C4—48no—
282C4⋊C4—44no—
283C4⋊C4—44no—
284C4⋊C4—44no—
285C4⋊C4—44no—
286C4⋊C4—44no—
287C4⋊C4—44no—
288C4⋊C4—44no—
289C4⋊C4—84no—
290C4⋊C4—84no—
291C4⋊C4—84no—
292C4⋊C4—84no—
293C22⋊C4—42no—
294C22⋊C4—42no—
295C22⋊C4—42no—
296C22⋊C4—48no—
297C22⋊C4—48no—
298C22⋊C4—84no—
299C22⋊C4—84no—
300C42—24no—
301C42—24no—
302C42—24no—
303C42—24no—
304C42—22no—
305C42—22no—
306C42—22no—
307C42—22no—
308C42—48no—
309C42—48no—
310C42—48no—
311C42C2×#4112yes—
312C422−1+412yes—
313C23—22no—
314C23—22no—
315C23—22no—
316C23—22no—
317C23—22no—
318C23—22no—
319C23—22no—
320C23C23×D411yescharacteristic
321Q8—88no—
322D4—44no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—22no—
326C2×C4—22no—
327C2×C4—22no—
328C2×C4—22no—
329C2×C4—22no—
330C2×C4—22no—
331C2×C4—22no—
332C2×C4—22no—
333C2×C4—22no—
334C2×C4—22no—
335C2×C4—22no—
336C2×C4—22no—
337C2×C4—22no—
338C2×C4—22no—
339C2×C4—48no—
340C2×C4—48no—
341C2×C4—44no—
342C2×C4—44no—
343C2×C4—44no—
344C2×C4—44no—
345C2×C4—44no—
346C2×C4—44no—
347C2×C4—44no—
348C2×C4—44no—
349C2×C4—44no—
350C2×C4—44no—
351C2×C4—44no—
352C2×C4—44no—
353C2×C4—44no—
354C2×C4—44no—
355C2×C4#54714yes—
356C2×C4#57514yes—
357C2×C4#57714yes—
358C2×C4C22×#4111yescharacteristic
359C2×C4C2×2−1+412yes—
360C2×C4C2×2−1+411yescharacteristic
361C2×C4#58512yes—
362C8—24no—
363C8—24no—
364C8—44no—
365C22—42no—
366C22—42no—
367C22—42no—
368C22—42no—
369C22—42no—
370C22—42no—
371C22—42no—
372C22#344911yescharacteristic
373C22C2×#54711yescharacteristic
374C22#349911yescharacteristic
375C22C2×#57512yes—
376C22C2×#57712yes—
377C4—42no—
378C4—41no—
379C4—42no—
380C4—42no—
381C4—42no—
382C4—41no—
383C4—84no—
384C4—84no—
385C4—84no—
386C4—84no—
387C4#334818yes—
388C4#350712yes—
389C4#356312yes—
390C2—82no—
391C2#3948411yescharacteristic
392C2#6097211yescharacteristic
393C2#6101511yescharacteristic
394C2C2×#334812yes—
395C2#6248312yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×D4C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C42×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #33263C2semidirect
3. #32518C2semidirect
5. #29877C2semidirect
6. #29177C2semidirect
7. #27949C2semidirect
8. #27876C2semidirect
9. #27483C2semidirect
10. #21984C2semidirect
11. #21727C2semidirect

Maximal subgroups

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

subgroupindex
#362092
#332632
#325182
#324712
#298772
#291772
#279492
#278762
#274832
#219842
#217272
#212652