Finite Group Databasebeta

← #6389765

Subgroups — #6389765

Order 512.

1682 conjugacy classes of subgroups.

The subgroup lattice has 394 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3160C211yesmaximal, characteristic
2#34394C212yesmaximal
3#23163C214yesmaximal, complemented
4#23158C212yesmaximal, complemented
5#23157C212yesmaximal, complemented
6#22616C214yesmaximal
7#22614C212yesmaximal
8#22613C212yesmaximal
9#22252C214yesmaximal, complemented
10#21839C212yesmaximal
11#21723C214yesmaximal, complemented
12#20817C211yesmaximal, characteristic
13#20476C211yesmaximal, characteristic
14C2×#533C2212yes—
15#3160C4116yescomplemented
16C2×#500C2211yescharacteristic
17C2×#500C2211yescharacteristic
18C2×#499C2212yes—
19C2×#499C2212yes—
20C2×#498C2212yes—
21C2×#432C2211yescharacteristic
22C2×#427C2212yes—
23C2×#427C2212yes—
24C22×C4×C8C2211yescharacteristic
25#2378—24no—
26#2369C2214yes—
27#2365C2214yes—
28#2356—22no—
29#2351—22no—
30#2347C2214yes—
31#2346C2214yes—
32#2330C2218yes—
33#2323—24no—
34#2320—24no—
35#2285C2212yes—
36#2283C2211yescharacteristic
37#1992C2218yes—
38#1988C2214yes—
39#1987C2214yes—
40#1944C2218yes—
41#1942C2214yes—
42#1942C2214yes—
43#1921C2218yes—
44#1920C2218yes—
45#1919C2218yes—
46#1918C2218yes—
47#1845C2212yes—
48#1834C2212yes—
49#1825C2214yes—
50#1825C2214yes—
51#1808C2211yescharacteristic
52#1808C2211yescharacteristic
53#1808C2212yes—
54#1776C2218yes—
55C4×#107C2212yes—
56C4×#107C2212yes—
57C4×#107C2214yes—
58C4×#107C2214yes—
59#1770C2212yes—
60C8×C4⋊C4C2212yes—
61#1752C2212yes—
62C8×C22⋊C4C2212yes—
63#1746C2212yes—
64#1726C2212yes—
65C23×C8C2311yescharacteristic
66C23×C8C2311yescharacteristic
67#533—28no—
68#533C2×C4116yes—
69C2×C4⋊Q8C2312yes—
70C2×C4⋊Q8C2312yes—
71C2×#125C2314yes—
72C2×C22⋊Q8—22no—
73C2×C22⋊Q8—22no—
74C2×C4×Q8—24no—
75C2×C4×Q8—22no—
76C2×C4×Q8—22no—
77C2×#117—22no—
78C2×#117C2314yes—
79C22×C42C2311yescharacteristic
80#500C2×C418yes—
81#500C2×C418yes—
82#499C2×C4116yes—
83#499C2×C4116yes—
84#498C2×C4116yes—
85#432—24no—
86#427C2×C4116yes—
87#427C2×C4116yes—
88C2×#107C2312yes—
89C2×#107C2312yes—
90C2×#107C2314yes—
91C2×#107C2312yes—
92C2×#107C2312yes—
93C2×#107C2312yes—
94C2×#107C2312yes—
95C2×#106C2314yes—
96C2×#106C2314yes—
97C2×C4⋊C8C2312yes—
98C2×C22⋊C8C2312yes—
99C2×#97C2312yes—
100C2×#97C2314yes—
101C2×C4×C8C2×C418yes—
102C2×C4×C8C2314yes—
103C2×C4×C8C2311yescharacteristic
104C2×C4×C8C2311yescharacteristic
105C2×C4×C8C2311yescharacteristic
106C2×C4×C8C2311yescharacteristic
107#391—24no—
108#391—28no—
109#389—24no—
110#389—28no—
111#389—28no—
112#389C2318yes—
113#389C2318yes—
114#388—28no—
115#388—28no—
116#387—28no—
117#387—28no—
118#387—28no—
119#384C2314yes—
120#383—24no—
121#383D412yes—
122#382—22no—
123#382—24no—
124#382D411yescharacteristic
125#382D411yescharacteristic
126#381D412yes—
127#380—22no—
128#380D412yes—
129#380C2312yes—
130#380C2312yes—
131C4×C4⋊C4—24no—
132C4×C4⋊C4—22no—
133C4×C4⋊C4—24no—
134C4×C4⋊C4C2318yes—
135C4×C4⋊C4C2318yes—
136C4×C4⋊C4C2314yes—
137C4×C4⋊C4C2312yes—
138C4×C22⋊C4C2312yes—
139#376—24no—
140#376C2312yes—
141#340C2314yes—
142#340C2312yes—
143#340C2312yes—
144#340C2318yes—
145#340C2318yes—
146#340C2318yes—
147#340C2318yes—
148#336C2314yes—
149#336C2314yes—
150C22×Q8—22no—
151C22×Q8—22no—
152C23×C4C2×D411yescharacteristic
153C23×C4C2×D411yescharacteristic
154C23×C4C2411yescharacteristic
155C22×C8—24no—
156C22×C8—24no—
157C22×C8C22×C414yes—
158C22×C8C22×C414yes—
159C22×C8C2414yes—
160C22×C8C2411yescharacteristic
161C22×C8C2411yescharacteristic
162C22×C8C2411yescharacteristic
163C22×C8C2411yescharacteristic
164C4⋊Q8—24no—
165C4⋊Q8—24no—
166C4⋊Q8C22×C418yes—
167C4⋊Q8C22×C418yes—
168#125—28no—
169#125C22×C4116yes—
170C22⋊Q8—216no—
171C22⋊Q8—216no—
172C4×Q8—216no—
173C4×Q8—216no—
174C4×Q8—88no—
175#117—28no—
176#117—84no—
177#117C22×C4116yes—
178C2×C4⋊C4—24no—
179C2×C4⋊C4—24no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—24no—
183C2×C4⋊C4—28no—
184C2×C4⋊C4—24no—
185C2×C4⋊C4—28no—
186C2×C4⋊C4C2×D412yes—
187C2×C4⋊C4C2×D412yes—
188C2×C4⋊C4C2414yes—
189C2×C4⋊C4C2414yes—
190C2×C4⋊C4C2414yes—
191C2×C4⋊C4C2414yes—
192C2×C22⋊C4—24no—
193C2×C22⋊C4—22no—
194C2×C22⋊C4—22no—
195C2×C22⋊C4C2×D414yes—
196C2×C42—24no—
197C2×C42—24no—
198C2×C42—28no—
199C2×C42—28no—
200C2×C42C22×C414yes—
201C2×C42C2×D411yescharacteristic
202C2×C42C2×D411yescharacteristic
203C2×C42#4114yes—
204C2×C42C2414yes—
205C2×C42C2411yescharacteristic
206C2×C42C2411yescharacteristic
207#107—24no—
208#107—24no—
209#107—28no—
210#107C22×C418yes—
211#107C22×C418yes—
212#107C22×C418yes—
213#107C22×C418yes—
214#106C22×C4116yes—
215#106C22×C4116yes—
216C4⋊C8—24no—
217C22⋊C8—24no—
218#97—28no—
219#97—24no—
220C4×C8—28no—
221C4×C8C22×C414yes—
222C4×C8C22×C414yes—
223C4×C8C22×C414yes—
224C4×C8C22×C414yes—
225#95—22no—
226#95—28no—
227#95—22no—
228#95—28no—
229#95—24no—
230#95—28no—
231#95—28no—
232#95C2×D412yes—
233#95C2×D414yes—
234#95C2×D412yes—
235#95C2×D412yes—
236#95C2×D412yes—
237#95C2×D414yes—
238C24C22×D411yescharacteristic
239C2×Q8—28no—
240C2×Q8—28no—
241C2×Q8—44no—
242C2×Q8—44no—
243C2×Q8—84no—
244C22×C4—22no—
245C22×C4—22no—
246C22×C4—22no—
247C22×C4—22no—
248C22×C4—24no—
249C22×C4—24no—
250C22×C4—24no—
251C22×C4—24no—
252C22×C4—42no—
253C22×C4C4×D414yes—
254C22×C4C4×D414yes—
255C22×C4C23×C412yes—
256C22×C4C22×D414yes—
257C22×C4C22×D414yes—
258C22×C4C22×D411yescharacteristic
259C22×C4C22×D411yescharacteristic
260C22×C4C22×D411yescharacteristic
261C22×C4C22×D411yescharacteristic
262C22×C4C22×D411yescharacteristic
263C22×C4C2×#4112yes—
264C22×C4C2511yescharacteristic
265C2×C8—22no—
266C2×C8—22no—
267C2×C8—28no—
268C2×C8—28no—
269C2×C8—24no—
270C2×C8—24no—
271C2×C8—24no—
272C2×C8C23×C412yes—
273C2×C8C23×C412yes—
274C2×C8C23×C412yes—
275C2×C8C23×C412yes—
276C2×C82−1+414yes—
277C2×C82−1+414yes—
278C4⋊C4—216no—
279C4⋊C4—216no—
280C4⋊C4—28no—
281C4⋊C4—28no—
282C4⋊C4—24no—
283C4⋊C4—24no—
284C4⋊C4—24no—
285C4⋊C4—24no—
286C4⋊C4—24no—
287C4⋊C4—24no—
288C4⋊C4—24no—
289C4⋊C4—24no—
290C4⋊C4—84no—
291C4⋊C4—88no—
292C4⋊C4—84no—
293C4⋊C4—88no—
294C4⋊C4C23×C418yes—
295C4⋊C4C23×C418yes—
296C22⋊C4—216no—
297C22⋊C4—44no—
298C22⋊C4—82no—
299C22⋊C4—82no—
300C42—216no—
301C42—24no—
302C42—24no—
303C42—48no—
304C42—88no—
305C42—88no—
306C42C4×D414yes—
307C42C4×D414yes—
308C42C23×C412yes—
309C42C23×C412yes—
310C42C2×#4112yes—
311C42C2×#4112yes—
312C23—22no—
313C23—22no—
314C23—21no—
315C23—21no—
316C23C2×C4×D412yes—
317C23C23×D411yescharacteristic
318Q8—416no—
319Q8—88no—
320C2×C4—22no—
321C2×C4—22no—
322C2×C4—22no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—22no—
326C2×C4—22no—
327C2×C4—28no—
328C2×C4—24no—
329C2×C4—24no—
330C2×C4—24no—
331C2×C4—28no—
332C2×C4—28no—
333C2×C4—24no—
334C2×C4—24no—
335C2×C4—24no—
336C2×C4—24no—
337C2×C4—44no—
338C2×C4—48no—
339C2×C4—44no—
340C2×C4—44no—
341C2×C4—44no—
342C2×C4—44no—
343C2×C4—44no—
344C2×C4—44no—
345C2×C4—44no—
346C2×C4—44no—
347C2×C4C2×C4×D412yes—
348C2×C4C2×C4×D412yes—
349C2×C4C2×C4×D412yes—
350C2×C4C2×C4×D412yes—
351C2×C4C2×C4×D412yes—
352C2×C4#57612yes—
353C2×C4#57812yes—
354C2×C4C24×C411yescharacteristic
355C2×C4C22×#4111yescharacteristic
356C2×C4C2×2−1+412yes—
357C2×C4C2×2−1+412yes—
358C8—22no—
359C8—22no—
360C8—44no—
361C8#51914yes—
362C8#51914yes—
363C22—22no—
364C22—22no—
365C22—22no—
366C22—42no—
367C22—42no—
368C22C22×C4×D411yescharacteristic
369C22#349912yes—
370C22C2×#57611yescharacteristic
371C22C2×#57811yescharacteristic
372C22#358412yes—
373C4—22no—
374C4—22no—
375C4—22no—
376C4—22no—
377C4—48no—
378C4—48no—
379C4—44no—
380C4—44no—
381C4—84no—
382C4—84no—
383C4#295114yes—
384C4C2×#51911yescharacteristic
385C4C2×#51911yescharacteristic
386C4#350912yes—
387C2—42no—
388C2#3420911yescharacteristic
389C2C2×#295111yescharacteristic
390C2#6016711yescharacteristic
391C2#6249512yes—
392C2#6249612yes—
subgroupG/H
center, socleC23C23×D4
commutatorC2×C4C24×C4
Frattini, cosocleC22×C4C25
prime radicalC24C22×D4
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ #2283 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #23163C2semidirect
4. #23158C2semidirect
5. #23157C2semidirect
9. #22252C2semidirect
11. #21723C2semidirect
15. #3160C4semidirect

Maximal subgroups

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

subgroupindex
C2×#31602
#343942
#231632
#231582
#231572
#226162
#226142
#226132
#222522
#218392
#217232
#208172
#204762