Finite Group Databasebeta

← #6389759

Subgroups — #6389759

Order 512.

1706 conjugacy classes of subgroups.

The subgroup lattice has 377 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3160C211yesmaximal, characteristic
2#34231C212yesmaximal
3#23163C214yesmaximal, complemented
4#23162C212yesmaximal, complemented
5#23161C212yesmaximal, complemented
6#22620C212yesmaximal
7#22619C212yesmaximal
8#22616C214yesmaximal
9#22249C214yesmaximal, complemented
10#21837C212yesmaximal
11#21728C214yesmaximal, complemented
12#20818C211yesmaximal, characteristic
13#20473C211yesmaximal, characteristic
14C2×#533C2212yes—
15#3160C4116yescomplemented
16C2×#500C2211yescharacteristic
17C2×#500C2211yescharacteristic
18C2×#499C2212yes—
19C2×#499C2212yes—
20C2×#498C2212yes—
21C2×#431C2211yescharacteristic
22C2×#427C2212yes—
23C2×#427C2212yes—
24C22×C4×C8C2211yescharacteristic
25#2369C2214yes—
26#2365C2214yes—
27#2362—24no—
28#2356—24no—
29#2347C2214yes—
30#2346C2214yes—
31#2332—24no—
32#2320—22no—
33C4×#125C2218yes—
34#2283C2212yes—
35#2276—22no—
36C4×#117C2211yescharacteristic
37#1991C2218yes—
38#1988C2214yes—
39#1987C2214yes—
40#1944C2218yes—
41#1943C2214yes—
42#1943C2214yes—
43#1921C2218yes—
44#1920C2218yes—
45#1919C2218yes—
46#1918C2218yes—
47#1846C2212yes—
48#1835C2212yes—
49#1825C2214yes—
50#1825C2214yes—
51#1808C2212yes—
52#1807C2211yescharacteristic
53#1807C2211yescharacteristic
54#1776C2218yes—
55#1776C2214yes—
56#1776C2214yes—
57C4×#107C2212yes—
58C4×#106C2212yes—
59#1770C2212yes—
60#1770C2212yes—
61#1752C2212yes—
62#1752C2212yes—
63#1744C2212yes—
64C4×#97C2212yes—
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×#117D412yes—
78C2×#117C2314yes—
79C22×C42C2311yescharacteristic
80#500C2×C418yes—
81#500C2×C418yes—
82#499C2×C4116yes—
83#499C2×C4116yes—
84#498C2×C4116yes—
85#431—24no—
86#427C2×C4116yes—
87#427C2×C4116yes—
88C2×#107C2314yes—
89C2×#107C2312yes—
90C2×#107C2312yes—
91C2×#107C2312yes—
92C2×#107C2312yes—
93C2×#106C2312yes—
94C2×#106C2312yes—
95C2×#106C2314yes—
96C2×#106C2314yes—
97C2×C4⋊C8C2312yes—
98C2×C22⋊C8C2312yes—
99C2×#97C2314yes—
100C2×#97C2314yes—
101C2×C4×C8C2×C418yes—
102C2×C4×C8C2312yes—
103C2×C4×C8C2311yescharacteristic
104C2×C4×C8C2311yescharacteristic
105C2×C4×C8C2311yescharacteristic
106C2×C4×C8C2311yescharacteristic
107#391—24no—
108#391—28no—
109#389—24no—
110#389C2318yes—
111#389C2318yes—
112#388—28no—
113#388—28no—
114#387—28no—
115#387—28no—
116#387—28no—
117#383—22no—
118#383—24no—
119#382—24no—
120#382C2314yes—
121#381—22no—
122#380C2312yes—
123#380C2312yes—
124C4×C4⋊C4—24no—
125C4×C4⋊C4—28no—
126C4×C4⋊C4—28no—
127C4×C4⋊C4—24no—
128C4×C4⋊C4D412yes—
129C4×C4⋊C4C2318yes—
130C4×C4⋊C4C2318yes—
131C4×C4⋊C4C2314yes—
132C4×C4⋊C4C2312yes—
133C4×C22⋊C4—22no—
134C4×C22⋊C4D412yes—
135C4×C22⋊C4C2312yes—
136#376—22no—
137#376—24no—
138#376D412yes—
139C43C2312yes—
140#340C2312yes—
141#340C2312yes—
142#340C2314yes—
143#340C2318yes—
144#340C2318yes—
145#340C2318yes—
146#340C2318yes—
147#336C2314yes—
148#336C2314yes—
149C22×Q8—22no—
150C22×Q8—22no—
151C23×C4C2×D411yescharacteristic
152C23×C4C2×D411yescharacteristic
153C23×C4C2411yescharacteristic
154C22×C8—24no—
155C22×C8—24no—
156C22×C8C22×C414yes—
157C22×C8C22×C414yes—
158C22×C8C2414yes—
159C22×C8C2411yescharacteristic
160C22×C8C2411yescharacteristic
161C22×C8C2411yescharacteristic
162C22×C8C2411yescharacteristic
163C4⋊Q8—24no—
164C4⋊Q8—24no—
165C4⋊Q8C22×C418yes—
166C4⋊Q8C22×C418yes—
167#125—28no—
168#125C22×C4116yes—
169C22⋊Q8—216no—
170C22⋊Q8—216no—
171C4×Q8—216no—
172C4×Q8—216no—
173C4×Q8—416no—
174#117—28no—
175#117—44no—
176#117C22×C4116yes—
177C2×C4⋊C4—24no—
178C2×C4⋊C4—24no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—28no—
182C2×C4⋊C4—28no—
183C2×C4⋊C4C2×D414yes—
184C2×C4⋊C4C2414yes—
185C2×C4⋊C4C2414yes—
186C2×C4⋊C4C2414yes—
187C2×C4⋊C4C2414yes—
188C2×C22⋊C4—24no—
189C2×C22⋊C4—22no—
190C2×C22⋊C4—22no—
191C2×C22⋊C4C2×D414yes—
192C2×C42—24no—
193C2×C42—24no—
194C2×C42—28no—
195C2×C42—28no—
196C2×C42C22×C414yes—
197C2×C42C2×D414yes—
198C2×C42C2×D414yes—
199C2×C42C2×D411yescharacteristic
200C2×C42C2×D411yescharacteristic
201C2×C42#4114yes—
202C2×C42C2414yes—
203C2×C42C2411yescharacteristic
204C2×C42C2411yescharacteristic
205#107—28no—
206#107C22×C418yes—
207#107C22×C418yes—
208#107C22×C418yes—
209#107C22×C418yes—
210#106—24no—
211#106—24no—
212#106C22×C4116yes—
213#106C22×C4116yes—
214C4⋊C8—24no—
215C22⋊C8—24no—
216#97—28no—
217#97—28no—
218C4×C8—24no—
219C4×C8C22×C414yes—
220C4×C8C22×C414yes—
221C4×C8C22×C414yes—
222C4×C8C22×C414yes—
223#95—22no—
224#95—22no—
225#95—24no—
226#95—28no—
227#95—24no—
228#95—28no—
229#95—24no—
230#95—28no—
231#95—28no—
232#95C2×D414yes—
233#95C2×D414yes—
234C24C22×D411yescharacteristic
235C2×Q8—28no—
236C2×Q8—28no—
237C2×Q8—48no—
238C2×Q8—82no—
239C2×Q8—82no—
240C22×C4—24no—
241C22×C4—22no—
242C22×C4—22no—
243C22×C4—24no—
244C22×C4—24no—
245C22×C4—24no—
246C22×C4—24no—
247C22×C4—42no—
248C22×C4C4×D414yes—
249C22×C4C4×D414yes—
250C22×C4C23×C412yes—
251C22×C4C22×D418yes—
252C22×C4C22×D411yescharacteristic
253C22×C4C22×D411yescharacteristic
254C22×C4C22×D411yescharacteristic
255C22×C4C22×D411yescharacteristic
256C22×C4C22×D411yescharacteristic
257C22×C4C2×#4112yes—
258C22×C4C2511yescharacteristic
259C2×C8—22no—
260C2×C8—22no—
261C2×C8—28no—
262C2×C8—28no—
263C2×C8—24no—
264C2×C8—24no—
265C2×C8—24no—
266C2×C8C23×C412yes—
267C2×C8C23×C412yes—
268C2×C8C23×C412yes—
269C2×C8C23×C412yes—
270C2×C82−1+414yes—
271C2×C82−1+414yes—
272C4⋊C4—216no—
273C4⋊C4—216no—
274C4⋊C4—28no—
275C4⋊C4—28no—
276C4⋊C4—28no—
277C4⋊C4—24no—
278C4⋊C4—24no—
279C4⋊C4—24no—
280C4⋊C4—24no—
281C4⋊C4—24no—
282C4⋊C4—24no—
283C4⋊C4—416no—
284C4⋊C4—88no—
285C4⋊C4C23×C418yes—
286C4⋊C4C23×C418yes—
287C22⋊C4—216no—
288C22⋊C4—44no—
289C22⋊C4—82no—
290C22⋊C4—82no—
291C42—216no—
292C42—24no—
293C42—24no—
294C42—44no—
295C42—44no—
296C42—416no—
297C42—48no—
298C42—88no—
299C42C4×D414yes—
300C42C4×D414yes—
301C42C23×C412yes—
302C42C23×C412yes—
303C42C2×#4112yes—
304C42C2×#4112yes—
305C23—22no—
306C23—22no—
307C23—22no—
308C23C2×C4×D412yes—
309C23C23×D411yescharacteristic
310Q8—416no—
311Q8—88no—
312C2×C4—22no—
313C2×C4—22no—
314C2×C4—22no—
315C2×C4—22no—
316C2×C4—22no—
317C2×C4—22no—
318C2×C4—28no—
319C2×C4—24no—
320C2×C4—28no—
321C2×C4—28no—
322C2×C4—28no—
323C2×C4—28no—
324C2×C4—28no—
325C2×C4—44no—
326C2×C4—44no—
327C2×C4—44no—
328C2×C4—48no—
329C2×C4—44no—
330C2×C4—44no—
331C2×C4—44no—
332C2×C4—44no—
333C2×C4—44no—
334C2×C4—44no—
335C2×C4C2×C4×D412yes—
336C2×C4C2×C4×D412yes—
337C2×C4C2×C4×D412yes—
338C2×C4C2×C4×D412yes—
339C2×C4C2×C4×D412yes—
340C2×C4#57514yes—
341C2×C4C24×C411yescharacteristic
342C2×C4C22×#4111yescharacteristic
343C2×C4C2×2−1+412yes—
344C2×C4C2×2−1+412yes—
345C8—22no—
346C8—22no—
347C8—44no—
348C8#51914yes—
349C8#51914yes—
350C22—22no—
351C22—22no—
352C22—22no—
353C22—44no—
354C22C22×C4×D411yescharacteristic
355C22#349912yes—
356C22C2×#57512yes—
357C22#358412yes—
358C4—22no—
359C4—22no—
360C4—22no—
361C4—22no—
362C4—48no—
363C4—48no—
364C4—48no—
365C4—84no—
366C4—84no—
367C4#294314yes—
368C4C2×#51911yescharacteristic
369C4C2×#51911yescharacteristic
370C4#350912yes—
371C2—42no—
372C2#3420911yescharacteristic
373C2C2×#294311yescharacteristic
374C2#6016711yescharacteristic
375C2#6248514yes—
subgroupG/H
center, socleC23C23×D4
commutatorC2×C4C24×C4
Frattini, cosocleC22×C4C25
prime radicalC24C22×D4
LCSG ▷ C2×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
3. #23163C2semidirect
4. #23162C2semidirect
5. #23161C2semidirect
9. #22249C2semidirect
11. #21728C2semidirect
15. #3160C4semidirect

Maximal subgroups

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

subgroupindex
C2×#31602
#342312
#231632
#231622
#231612
#226202
#226192
#226162
#222492
#218372
#217282
#208182
#204732