Finite Group Databasebeta

← #6389766

Subgroups — #6389766

Order 512.

1688 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#499C211yesmaximal, characteristic
2#34378C212yesmaximal
3#23164C214yesmaximal, complemented
4#23159C214yesmaximal, complemented
5#22618C212yesmaximal
6#22617C212yesmaximal
7#22609C214yesmaximal
8#22252C214yesmaximal, complemented
9#21838C212yesmaximal
10#21722C214yesmaximal, complemented
11#20816C211yesmaximal, characteristic
12#20476C211yesmaximal, characteristic
13C22×#125C2212yes—
14C2×#499C4116yescomplemented
15C2×#499C2212yes—
16C2×#499C2212yes—
17C2×#499C2212yes—
18C2×#499C2212yes—
19C2×#432C2211yescharacteristic
20C22×#107C2211yescharacteristic
21C22×#107C2211yescharacteristic
22C22×#106C2212yes—
23C22×C4×C8C2211yescharacteristic
24#2382—24no—
25#2370C2218yes—
26#2345C2214yes—
27#2344—24no—
28#2344C2214yes—
29#2336—24no—
30#2330C2218yes—
31#2316—22no—
32#2313—22no—
33#2285C2212yes—
34#2283C2211yescharacteristic
35#1992C2218yes—
36#1989C2218yes—
37#1943C2214yes—
38#1943C2214yes—
39#1941C2218yes—
40#1921C2218yes—
41#1920C2218yes—
42#1919C2218yes—
43#1918C2218yes—
44#1844C2212yes—
45#1834C2212yes—
46#1824C2214yes—
47#1823C2214yes—
48#1807C2212yes—
49#1807C2212yes—
50#1776C2214yes—
51#1776C2214yes—
52C4×#106C2212yes—
53C4×#106C2212yes—
54C4×#106C2218yes—
55#1770C2212yes—
56C8×C4⋊C4C2212yes—
57#1752C2212yes—
58C8×C22⋊C4C2212yes—
59#1746C2212yes—
60#1726C2212yes—
61C23×C8C2311yescharacteristic
62C23×C8C2311yescharacteristic
63C2×#125—22no—
64C2×#125—22no—
65C2×#125—28no—
66C2×#125C2×C4116yes—
67C2×#125C2314yes—
68C2×#125C2314yes—
69C2×#117—22no—
70C2×#117—24no—
71C22×C4⋊C4—22no—
72C22×C4⋊C4—22no—
73C22×C4⋊C4C2312yes—
74C22×C4⋊C4C2312yes—
75C22×C42C2311yescharacteristic
76#499C2×C4116yes—
77#499C2×C4116yes—
78#499C2×C4116yes—
79#499C2×C4116yes—
80#432—24no—
81C2×#107C2×C418yes—
82C2×#107C2×C418yes—
83C2×#107C2312yes—
84C2×#107C2312yes—
85C2×#107C2312yes—
86C2×#107C2312yes—
87C2×#106C2×C4116yes—
88C2×#106C2314yes—
89C2×#106C2314yes—
90C2×#106C2314yes—
91C2×#106C2314yes—
92C2×C4⋊C8C2312yes—
93C2×C22⋊C8C2312yes—
94C2×#97C2312yes—
95C2×#97C2314yes—
96C2×C4×C8C2×C418yes—
97C2×C4×C8C2314yes—
98C2×C4×C8C2311yescharacteristic
99C2×C4×C8C2311yescharacteristic
100C2×C4×C8C2311yescharacteristic
101C2×C4×C8C2311yescharacteristic
102#389—28no—
103#389—28no—
104#389—28no—
105#389C2318yes—
106#389C2318yes—
107#387—28no—
108#387—28no—
109#387—28no—
110#387—28no—
111#387—28no—
112#387—28no—
113#384C2314yes—
114#383D412yes—
115#382—22no—
116#382D412yes—
117#381—24no—
118#381D412yes—
119#380—22no—
120#380—24no—
121#380D412yes—
122#380C2314yes—
123C4×C4⋊C4—22no—
124C4×C4⋊C4—28no—
125C4×C4⋊C4C2318yes—
126C4×C4⋊C4C2318yes—
127C4×C4⋊C4C2314yes—
128C4×C4⋊C4C2312yes—
129C4×C22⋊C4—24no—
130C4×C22⋊C4C2312yes—
131#376C2312yes—
132#340C2314yes—
133#340C2314yes—
134#340C2318yes—
135#340C2318yes—
136#340C2318yes—
137#340C2318yes—
138#336C2314yes—
139#336C2314yes—
140C23×C4—22no—
141C23×C4—22no—
142C23×C4C2×D411yescharacteristic
143C23×C4C2×D411yescharacteristic
144C23×C4C2411yescharacteristic
145C22×C8—24no—
146C22×C8—24no—
147C22×C8C22×C414yes—
148C22×C8C22×C414yes—
149C22×C8C2414yes—
150C22×C8C2411yescharacteristic
151C22×C8C2411yescharacteristic
152C22×C8C2411yescharacteristic
153C22×C8C2411yescharacteristic
154#125—216no—
155#125—216no—
156#125—28no—
157#125—28no—
158#125C22×C4116yes—
159#125C22×C4116yes—
160#117—88no—
161#117—84no—
162C2×C4⋊C4—216no—
163C2×C4⋊C4—216no—
164C2×C4⋊C4—24no—
165C2×C4⋊C4—24no—
166C2×C4⋊C4—24no—
167C2×C4⋊C4—24no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—24no—
171C2×C4⋊C4—24no—
172C2×C4⋊C4—24no—
173C2×C4⋊C4—28no—
174C2×C4⋊C4—24no—
175C2×C4⋊C4—28no—
176C2×C4⋊C4C22×C418yes—
177C2×C4⋊C4C22×C418yes—
178C2×C4⋊C4C2×D414yes—
179C2×C4⋊C4C2418yes—
180C2×C4⋊C4C2414yes—
181C2×C4⋊C4C2414yes—
182C2×C22⋊C4—24no—
183C2×C22⋊C4C2×D414yes—
184C2×C42—24no—
185C2×C42—28no—
186C2×C42—28no—
187C2×C42C22×C414yes—
188C2×C42C2×D411yescharacteristic
189C2×C42C2×D411yescharacteristic
190C2×C42#4114yes—
191C2×C42C2414yes—
192C2×C42C2411yescharacteristic
193C2×C42C2411yescharacteristic
194#107C22×C418yes—
195#107C22×C418yes—
196#107C22×C418yes—
197#107C22×C418yes—
198#106—28no—
199#106—28no—
200#106C22×C4116yes—
201#106C22×C4116yes—
202C4⋊C8—24no—
203C22⋊C8—24no—
204#97—28no—
205#97—24no—
206C4×C8—28no—
207C4×C8C22×C414yes—
208C4×C8C22×C414yes—
209C4×C8C22×C414yes—
210C4×C8C22×C414yes—
211#95—28no—
212#95—24no—
213#95—28no—
214#95—24no—
215#95—28no—
216#95—28no—
217#95C2×D414yes—
218#95C2×D414yes—
219#95C2×D414yes—
220#95C2×D414yes—
221C24C22×D411yescharacteristic
222C22×C4—24no—
223C22×C4—22no—
224C22×C4—22no—
225C22×C4—28no—
226C22×C4—24no—
227C22×C4—24no—
228C22×C4—28no—
229C22×C4—28no—
230C22×C4—42no—
231C22×C4—44no—
232C22×C4—44no—
233C22×C4—84no—
234C22×C4C4×D414yes—
235C22×C4C4×D414yes—
236C22×C4C23×C412yes—
237C22×C4C22×D418yes—
238C22×C4C22×D411yescharacteristic
239C22×C4C22×D411yescharacteristic
240C22×C4C22×D411yescharacteristic
241C22×C4C22×D411yescharacteristic
242C22×C4C22×D411yescharacteristic
243C22×C4C2×#4112yes—
244C22×C4C2511yescharacteristic
245C2×C8—22no—
246C2×C8—22no—
247C2×C8—28no—
248C2×C8—28no—
249C2×C8—24no—
250C2×C8—24no—
251C2×C8—24no—
252C2×C8C23×C412yes—
253C2×C8C23×C412yes—
254C2×C8C23×C412yes—
255C2×C8C23×C412yes—
256C2×C82−1+414yes—
257C2×C82−1+414yes—
258C4⋊C4—216no—
259C4⋊C4—216no—
260C4⋊C4—216no—
261C4⋊C4—216no—
262C4⋊C4—28no—
263C4⋊C4—28no—
264C4⋊C4—24no—
265C4⋊C4—24no—
266C4⋊C4—28no—
267C4⋊C4—28no—
268C4⋊C4—44no—
269C4⋊C4—84no—
270C4⋊C4—88no—
271C4⋊C4—84no—
272C4⋊C4—88no—
273C4⋊C4C23×C418yes—
274C4⋊C4C23×C418yes—
275C22⋊C4—28no—
276C22⋊C4—48no—
277C42—24no—
278C42—24no—
279C42—88no—
280C42—84no—
281C42—88no—
282C42C4×D414yes—
283C42C4×D414yes—
284C42C23×C412yes—
285C42C23×C412yes—
286C42C2×#4112yes—
287C42C2×#4112yes—
288C23—22no—
289C23—22no—
290C23C2×C4×D412yes—
291C23#57714yes—
292C23C23×D411yescharacteristic
293C2×C4—22no—
294C2×C4—22no—
295C2×C4—28no—
296C2×C4—22no—
297C2×C4—22no—
298C2×C4—22no—
299C2×C4—22no—
300C2×C4—22no—
301C2×C4—24no—
302C2×C4—28no—
303C2×C4—28no—
304C2×C4—28no—
305C2×C4—28no—
306C2×C4—28no—
307C2×C4—44no—
308C2×C4—48no—
309C2×C4—416no—
310C2×C4—44no—
311C2×C4—44no—
312C2×C4—48no—
313C2×C4—48no—
314C2×C4—48no—
315C2×C4—88no—
316C2×C4C2×C4×D412yes—
317C2×C4C2×C4×D412yes—
318C2×C4C2×C4×D412yes—
319C2×C4C2×C4×D412yes—
320C2×C4C2×C4×D412yes—
321C2×C4C24×C411yescharacteristic
322C2×C4C22×#4111yescharacteristic
323C2×C4C2×2−1+412yes—
324C2×C4C2×2−1+412yes—
325C8—22no—
326C8—22no—
327C8—44no—
328C8#51914yes—
329C8#51914yes—
330C22—22no—
331C22—28no—
332C22—22no—
333C22#295214yes—
334C22C22×C4×D411yescharacteristic
335C22#349912yes—
336C22C2×#57712yes—
337C22#358412yes—
338C4—22no—
339C4—22no—
340C4—22no—
341C4—22no—
342C4—22no—
343C4—48no—
344C4—48no—
345C4—48no—
346C4—88no—
347C4C2×#51911yescharacteristic
348C4C2×#51911yescharacteristic
349C4#350912yes—
350C2—24no—
351C2#3420911yescharacteristic
352C2C2×#295211yescharacteristic
353C2#6016711yescharacteristic
354C2#6249114yes—
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. #23164C2semidirect
4. #23159C2semidirect
8. #22252C2semidirect
10. #21722C2semidirect
14. C2×#499C4semidirect

Maximal subgroups

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

subgroupindex
C22×#4992
#343782
#231642
#231592
#226182
#226172
#226092
#222522
#218382
#217222
#208162
#204762