Finite Group Databasebeta

← #6849747

Subgroups — #6849747

Order 512.

1358 conjugacy classes of subgroups.

The subgroup lattice has 404 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36189C212yesmaximal
2#33263C211yesmaximal, characteristic, complemented
3#32514C214yesmaximal, complemented
4#32467C214yesmaximal
5#29877C212yesmaximal, complemented
6#29178C212yesmaximal, complemented
7#27950C212yesmaximal, complemented
8#27876C212yesmaximal, complemented
9#27473C218yesmaximal, complemented
10#21982C212yesmaximal, complemented
11#21727C211yesmaximal, characteristic, complemented
12#21265C211yesmaximal, characteristic
13C2×#498C2211yescharacteristic
14C2×#431C2212yes—
15#2859C2212yes—
16#2857C2212yes—
17#2857C2212yes—
18#2570—22no—
19#2511—22no—
20#2507C2214yes—
21#2497C2214yes—
22#2487C2214yes—
23#2442—22no—
24#2441—22no—
25#2423—22no—
26#2417—22no—
27#2360—22no—
28#2352—22no—
29#2345C2214yes—
30C4×C4⋊Q8C2212yes—
31#2285C2212yes—
32C2×#384C2212yes—
33C42×D4C2211yescharacteristic
34#2088C2218yes—
35#2086C2218yes—
36#2083C2218yes—
37#2077C2218yes—
38#2077C2218yes—
39#1990C2212yes—
40#1990C2212yes—
41#1990C2212yes—
42#1955C2212yes—
43#1947C2218yes—
44#1944C2214yes—
45#1944C2214yes—
46#1943C2214yes—
47#1940C2212yes—
48#1924C2214yes—
49#1919C2218yes—
50#1918C2218yes—
51#1852C2218yes—
52#1852C2218yes—
53#1845C2211yescharacteristic
54#1845C2212yes—
55#1845C2211yescharacteristic
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—
71C22×C4⋊C4—22no—
72C22×C4⋊C4—22no—
73C22×C4⋊C4C2314yes—
74C22×C42C2312yes—
75#498—24no—
76#431—28no—
77C2×#107C2314yes—
78C2×#107C2318yes—
79C2×#107C2314yes—
80C2×#106C2314yes—
81C2×#106C2314yes—
82C2×C4⋊C8C2314yes—
83C2×C4⋊C8C2312yes—
84C2×C22⋊C8C2314yes—
85C2×C22⋊C8C2314yes—
86C2×#97C2312yes—
87C2×#97C2312yes—
88C2×C4×C8C2312yes—
89C2×C4×C8C2311yescharacteristic
90C2×C4×C8C2311yescharacteristic
91#400—24no—
92#400—24no—
93#399—24no—
94#398—24no—
95#398—24no—
96#396—24no—
97#391—22no—
98#391—22no—
99#391—22no—
100#389—22no—
101#389—22no—
102#389—22no—
103#389—22no—
104#389—22no—
105#389—24no—
106#389C2314yes—
107#389C2314yes—
108#389C2314yes—
109#389C2314yes—
110#389C2314yes—
111#388—24no—
112#387—24no—
113#387—24no—
114#387—24no—
115#387—24no—
116#386—24no—
117#385—24no—
118#384—22no—
119#384—22no—
120#384C2312yes—
121#384C2314yes—
122#384C2312yes—
123#384C2312yes—
124#382—22no—
125#382—22no—
126#380—22no—
127#380—22no—
128#380—22no—
129#380—24no—
130#380C2314yes—
131#380C2314yes—
132#380C2314yes—
133C4×C4⋊C4—22no—
134C4×C4⋊C4—22no—
135C4×C4⋊C4D412yes—
136C4×C4⋊C4C2314yes—
137C4×C4⋊C4C2314yes—
138C4×C4⋊C4C2311yescharacteristic
139C4×C22⋊C4—24no—
140C4×C22⋊C4D412yes—
141C4×C22⋊C4D412yes—
142C4×C22⋊C4C2312yes—
143C43C2311yescharacteristic
144#340C2318yes—
145#340C2318yes—
146#340C2318yes—
147#340C2318yes—
148#340C2318yes—
149#336C2314yes—
150#336C2312yes—
151C22×Q8—22no—
152C22×D4C2×D411yescharacteristic
153C23×C4—24no—
154C23×C4C2×D412yes—
155C23×C4C2×D412yes—
156C23×C4C2412yes—
157C22×C8C2414yes—
158C22×C8C2412yes—
159C22×C8C2412yes—
160C4⋊Q8—24no—
161C4⋊Q8—24no—
162#125—84no—
163C22⋊Q8—48no—
164C22⋊Q8—84no—
165C4×Q8—84no—
166C4×D4—28no—
167C4×D4—42no—
168#117—84no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—24no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—24no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—24no—
179C2×C4⋊C4—24no—
180C2×C4⋊C4—24no—
181C2×C4⋊C4—24no—
182C2×C4⋊C4—24no—
183C2×C4⋊C4—24no—
184C2×C4⋊C4—48no—
185C2×C4⋊C4—84no—
186C2×C4⋊C4—84no—
187C2×C4⋊C4C2×D412yes—
188C2×C4⋊C4C2×D411yescharacteristic
189C2×C4⋊C4C2414yes—
190C2×C4⋊C4C2414yes—
191C2×C4⋊C4C2414yes—
192C2×C4⋊C4C2414yes—
193C2×C22⋊C4—24no—
194C2×C22⋊C4—24no—
195C2×C22⋊C4—24no—
196C2×C22⋊C4C2×D412yes—
197C2×C22⋊C4C2×D412yes—
198C2×C22⋊C4C2×D412yes—
199C2×C42—24no—
200C2×C42—24no—
201C2×C42—24no—
202C2×C42—24no—
203C2×C42C2×D412yes—
204C2×C42C2×D412yes—
205C2×C42C2×D412yes—
206C2×C42C2×D412yes—
207C2×C42C2411yescharacteristic
208C2×C42C2411yescharacteristic
209C2×C42C2412yes—
210C2×C42C2411yescharacteristic
211#107—44no—
212#107—44no—
213#107—48no—
214#106—28no—
215#106—28no—
216C4⋊C8—28no—
217C4⋊C8—42no—
218C22⋊C8—28no—
219C22⋊C8—28no—
220#97—24no—
221#97—42no—
222C4×C8—24no—
223C4×C8—22no—
224C4×C8#4114yes—
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#95—24no—
234#95—24no—
235#95C2×D411yescharacteristic
236#95C2×D411yescharacteristic
237#95C2×D412yes—
238#95C2×D412yes—
239C24C22×D412yes—
240C2×Q8—44no—
241C2×Q8—44no—
242C2×Q8—44no—
243C2×D4—24no—
244C2×D4—41no—
245C22×C4—24no—
246C22×C4—24no—
247C22×C4—24no—
248C22×C4—24no—
249C22×C4—24no—
250C22×C4—22no—
251C22×C4—24no—
252C22×C4—22no—
253C22×C4—24no—
254C22×C4—24no—
255C22×C4—24no—
256C22×C4—24no—
257C22×C4—24no—
258C22×C4—48no—
259C22×C4—84no—
260C22×C4—84no—
261C22×C4#13718yes—
262C22×C4C22×D411yescharacteristic
263C22×C4C22×D412yes—
264C22×C4C22×D412yes—
265C22×C4C22×D412yes—
266C22×C4C22×D411yescharacteristic
267C22×C4C22×D412yes—
268C22×C4C22×D412yes—
269C22×C4C2511yescharacteristic
270C2×C8—22no—
271C2×C8—22no—
272C2×C8—22no—
273C2×C8—22no—
274C2×C8—24no—
275C2×C8—24no—
276C2×C8—24no—
277C2×C8C2×#4114yes—
278C2×C82−1+414yes—
279C4⋊C4—24no—
280C4⋊C4—24no—
281C4⋊C4—28no—
282C4⋊C4—24no—
283C4⋊C4—28no—
284C4⋊C4—28no—
285C4⋊C4—41no—
286C4⋊C4—48no—
287C4⋊C4—44no—
288C4⋊C4—44no—
289C4⋊C4—48no—
290C4⋊C4—44no—
291C4⋊C4—44no—
292C4⋊C4—44no—
293C4⋊C4—44no—
294C4⋊C4—44no—
295C4⋊C4—84no—
296C4⋊C4—84no—
297C4⋊C4—84no—
298C4⋊C4—84no—
299C4⋊C4—84no—
300C4⋊C4—84no—
301C22⋊C4—24no—
302C22⋊C4—24no—
303C22⋊C4—42no—
304C22⋊C4—48no—
305C22⋊C4—84no—
306C22⋊C4—84no—
307C42—24no—
308C42—24no—
309C42—24no—
310C42—24no—
311C42—22no—
312C42—22no—
313C42—22no—
314C42—22no—
315C42—84no—
316C42—84no—
317C42C2×#4112yes—
318C422−1+412yes—
319C23—22no—
320C23—22no—
321C23—22no—
322C23—22no—
323C23—22no—
324C23—22no—
325C23C2×#13714yes—
326C23C23×D411yescharacteristic
327Q8—88no—
328D4—44no—
329C2×C4—22no—
330C2×C4—22no—
331C2×C4—216no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—22no—
335C2×C4—22no—
336C2×C4—22no—
337C2×C4—22no—
338C2×C4—22no—
339C2×C4—22no—
340C2×C4—22no—
341C2×C4—22no—
342C2×C4—22no—
343C2×C4—22no—
344C2×C4—22no—
345C2×C4—48no—
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—44no—
356C2×C4—44no—
357C2×C4—44no—
358C2×C4—44no—
359C2×C4—44no—
360C2×C4—168no—
361C2×C4#54714yes—
362C2×C4C2×#13714yes—
363C2×C4C2×#13714yes—
364C2×C4#57614yes—
365C2×C4C22×#4111yescharacteristic
366C2×C4C2×2−1+412yes—
367C2×C4C2×2−1+411yescharacteristic
368C2×C4#58512yes—
369C8—24no—
370C8—24no—
371C8—44no—
372C22—24no—
373C22—24no—
374C22—24no—
375C22—42no—
376C22—42no—
377C22—42no—
378C22—42no—
379C22#344911yescharacteristic
380C22C2×#54711yescharacteristic
381C22#349911yescharacteristic
382C22C22×#13712yes—
383C22C2×#57612yes—
384C4—24no—
385C4—42no—
386C4—41no—
387C4—42no—
388C4—42no—
389C4—41no—
390C4—84no—
391C4—84no—
392C4—84no—
393C4—84no—
394C4#334618yes—
395C4#350712yes—
396C4#356312yes—
397C2—44no—
398C2#3948411yescharacteristic
399C2#6097111yescharacteristic
400C2#6101111yescharacteristic
401C2C2×#334612yes—
402C2#6248012yes—
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. #32514C2semidirect
5. #29877C2semidirect
6. #29178C2semidirect
7. #27950C2semidirect
8. #27876C2semidirect
9. #27473C2semidirect
10. #21982C2semidirect
11. #21727C2semidirect

Maximal subgroups

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

subgroupindex
#361892
#332632
#325142
#324672
#298772
#291782
#279502
#278762
#274732
#219822
#217272
#212652