Finite Group Databasebeta

← #6667424

Subgroups — #6667424

Order 512.

1336 conjugacy classes of subgroups.

The subgroup lattice has 390 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36640C212yesmaximal
2#32850C211yesmaximal, characteristic, complemented
3#32849C212yesmaximal, complemented
4#32848C211yesmaximal, characteristic, complemented
5#29944C214yesmaximal
6#29943C214yesmaximal
7#29879C214yesmaximal
8#29291C214yesmaximal, complemented
9#29221C214yesmaximal, complemented
10#28690C214yesmaximal, complemented
11#20808C211yesmaximal, characteristic
12C2×#499C2214yes—
13C2×#431C2212yes—
14#2864—22no—
15#2864—22no—
16#2859C2212yes—
17#2859C2212yes—
18#2859C2212yes—
19#2859C2212yes—
20#2851—22no—
21#2851—22no—
22#2851C2214yes—
23#2851C2214yes—
24C4×#125—22no—
25C4×#125—22no—
26C4×#125C2214yes—
27C4×#125C2214yes—
28#2296—22no—
29#2296—22no—
30#2285C2212yes—
31#2285C2212yes—
32C2×C43C2211yescharacteristic
33#2100C2214yes—
34#2100C2214yes—
35#2099C2214yes—
36#2099C2214yes—
37#2088C2218yes—
38#2087C2214yes—
39#2086C2214yes—
40#2084C2214yes—
41#2084C2214yes—
42#2083C2214yes—
43#2083C2214yes—
44#2077C2218yes—
45#2076C2214yes—
46#2075C2214yes—
47#1991C2218yes—
48#1989C2218yes—
49#1989C2218yes—
50#1949C2218yes—
51#1949C2218yes—
52#1842C2212yes—
53#1835C2214yes—
54#1831C2211yescharacteristic
55#1831C2211yescharacteristic
56#1797C2214yes—
57C4×C4⋊C8C2214yes—
58C2×#125—22no—
59C2×#125—22no—
60C2×#125C2314yes—
61C2×#125C2314yes—
62C2×#117—22no—
63C2×#117—22no—
64C22×C42D411yescharacteristic
65C22×C42D412yes—
66C22×C42D411yescharacteristic
67C22×C42C2312yes—
68C22×C42C2311yescharacteristic
69#499—48no—
70#431—44no—
71C2×#107C2314yes—
72C2×#107C2314yes—
73C2×#106C2318yes—
74C2×C4⋊C8C2318yes—
75C2×C4⋊C8C2312yes—
76C2×C4⋊C8C2312yes—
77C2×C22⋊C8C2314yes—
78C2×C4×C8C2314yes—
79#389—22no—
80#389—22no—
81#389—22no—
82#389—22no—
83#389—22no—
84#389—22no—
85#389—22no—
86#389—22no—
87#389—22no—
88#389—22no—
89#389—22no—
90#389—22no—
91#389C2314yes—
92#389C2314yes—
93#389C2314yes—
94#389C2314yes—
95#387—24no—
96#387—24no—
97#387—24no—
98#387—24no—
99#387—24no—
100#387—24no—
101#387—24no—
102#387—24no—
103#385—24no—
104#385—24no—
105#385—24no—
106#385—24no—
107#384C2312yes—
108#384C2312yes—
109#384C2312yes—
110#384C2312yes—
111#382—22no—
112#382—22no—
113#382—22no—
114#382—22no—
115#382C2314yes—
116#382C2314yes—
117#380—22no—
118#380—22no—
119#380C2314yes—
120#380C2314yes—
121C4×C4⋊C4—22no—
122C4×C4⋊C4—22no—
123C4×C4⋊C4—24no—
124C4×C4⋊C4—24no—
125C4×C4⋊C4—24no—
126C4×C4⋊C4—24no—
127C4×C4⋊C4C2314yes—
128C4×C4⋊C4C2314yes—
129C43D411yescharacteristic
130C43D412yes—
131C43D411yescharacteristic
132C43C2311yescharacteristic
133C43C2312yes—
134C43C2311yescharacteristic
135#340C2318yes—
136#340C2314yes—
137#340C2318yes—
138#340C2314yes—
139#340C2318yes—
140#340C2318yes—
141#340C2318yes—
142#336C2314yes—
143#336C2314yes—
144C23×C4C2×D412yes—
145C23×C4C2×D411yescharacteristic
146C23×C4C2×D412yes—
147C23×C4C2×D411yescharacteristic
148C23×C4C2411yescharacteristic
149C22×C8C2414yes—
150C22×C8C2414yes—
151#125—28no—
152#125—216no—
153#125—28no—
154#125—48no—
155#125—48no—
156#117—48no—
157#117—48no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—24no—
167C2×C4⋊C4—24no—
168C2×C4⋊C4—24no—
169C2×C4⋊C4—24no—
170C2×C4⋊C4—24no—
171C2×C4⋊C4—24no—
172C2×C4⋊C4—24no—
173C2×C4⋊C4—24no—
174C2×C4⋊C4C2414yes—
175C2×C4⋊C4C2414yes—
176C2×C4⋊C4C2414yes—
177C2×C4⋊C4C2414yes—
178C2×C22⋊C4—24no—
179C2×C22⋊C4—24no—
180C2×C42—24no—
181C2×C42—22no—
182C2×C42—28no—
183C2×C42—22no—
184C2×C42—24no—
185C2×C42—24no—
186C2×C42—44no—
187C2×C42—42no—
188C2×C42C2×D412yes—
189C2×C42C2×D412yes—
190C2×C42C2×D412yes—
191C2×C42C2×D411yescharacteristic
192C2×C42C2×D411yescharacteristic
193C2×C42C2×D411yescharacteristic
194C2×C42C2×D412yes—
195C2×C42C2×D412yes—
196C2×C42C2×D411yescharacteristic
197C2×C42C2×D412yes—
198C2×C42C2×D412yes—
199C2×C42C2×D412yes—
200C2×C42C2×D411yescharacteristic
201C2×C42C2×D411yescharacteristic
202C2×C42C2412yes—
203C2×C42C2411yescharacteristic
204C2×C42C2411yescharacteristic
205C2×C42C2412yes—
206#107—44no—
207#107—44no—
208#106—48no—
209C4⋊C8—216no—
210C4⋊C8—42no—
211C4⋊C8—42no—
212C22⋊C8—44no—
213C4×C8—28no—
214#95—24no—
215#95—24no—
216#95—24no—
217#95—24no—
218#95—24no—
219#95—24no—
220#95—24no—
221#95—24no—
222#95—24no—
223#95—24no—
224#95—24no—
225#95—24no—
226C24C22×D411yescharacteristic
227C22×C4—24no—
228C22×C4—22no—
229C22×C4—24no—
230C22×C4—22no—
231C22×C4—22no—
232C22×C4—24no—
233C22×C4—24no—
234C22×C4—22no—
235C22×C4—24no—
236C22×C4—21no—
237C22×C4—24no—
238C22×C4—21no—
239C22×C4—24no—
240C22×C4—24no—
241C22×C4—24no—
242C22×C4—24no—
243C22×C4—42no—
244C22×C4—42no—
245C22×C4C22×D411yescharacteristic
246C22×C4C22×D411yescharacteristic
247C22×C4C22×D412yes—
248C22×C4C22×D412yes—
249C22×C4C22×D411yescharacteristic
250C22×C4C22×D411yescharacteristic
251C22×C4C22×D411yescharacteristic
252C22×C4C22×D412yes—
253C22×C4C22×D412yes—
254C22×C4C2511yescharacteristic
255C2×C8—24no—
256C2×C8—24no—
257C2×C8—28no—
258C2×C8—28no—
259C4⋊C4—24no—
260C4⋊C4—24no—
261C4⋊C4—28no—
262C4⋊C4—28no—
263C4⋊C4—24no—
264C4⋊C4—24no—
265C4⋊C4—44no—
266C4⋊C4—44no—
267C4⋊C4—44no—
268C4⋊C4—44no—
269C4⋊C4—48no—
270C4⋊C4—48no—
271C4⋊C4—48no—
272C4⋊C4—48no—
273C4⋊C4—48no—
274C4⋊C4—44no—
275C4⋊C4—44no—
276C4⋊C4—48no—
277C4⋊C4—48no—
278C4⋊C4—44no—
279C4⋊C4—44no—
280C4⋊C4—48no—
281C22⋊C4—48no—
282C22⋊C4—48no—
283C42—24no—
284C42—24no—
285C42—22no—
286C42—22no—
287C42—22no—
288C42—24no—
289C42—24no—
290C42—24no—
291C42—21no—
292C42—24no—
293C42—21no—
294C42—42no—
295C42—42no—
296C42—41no—
297C42—42no—
298C42—41no—
299C42—48no—
300C42—48no—
301C42#13514yes—
302C42H(5,𝔽2)12yes—
303C422−1+418yes—
304C422−1+412yes—
305C23—22no—
306C23—21no—
307C23—21no—
308C23—22no—
309C23#57512yes—
310C23C23×D411yescharacteristic
311C2×C4—216no—
312C2×C4—28no—
313C2×C4—22no—
314C2×C4—22no—
315C2×C4—21no—
316C2×C4—22no—
317C2×C4—21no—
318C2×C4—22no—
319C2×C4—22no—
320C2×C4—22no—
321C2×C4—22no—
322C2×C4—22no—
323C2×C4—22no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—22no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—22no—
330C2×C4—48no—
331C2×C4—44no—
332C2×C4—42no—
333C2×C4—42no—
334C2×C4—44no—
335C2×C4—44no—
336C2×C4—44no—
337C2×C4—44no—
338C2×C4—44no—
339C2×C4—44no—
340C2×C4—44no—
341C2×C4—44no—
342C2×C4—44no—
343C2×C4—44no—
344C2×C4—44no—
345C2×C4—44no—
346C2×C4#53614yes—
347C2×C4#53614yes—
348C2×C4#53712yes—
349C2×C4#53712yes—
350C2×C4C2×#13514yes—
351C2×C4C2×#13512yes—
352C2×C4#57514yes—
353C2×C4C2×H(5,𝔽2)11yescharacteristic
354C2×C4C2×2−1+414yes—
355C2×C4C2×2−1+411yescharacteristic
356C8—44no—
357C8—44no—
358C22—24no—
359C22—22no—
360C22—41no—
361C22—42no—
362C22—41no—
363C22C2×#53612yes—
364C22C2×#53711yescharacteristic
365C22C22×#13511yescharacteristic
366C22C2×#57512yes—
367C22C2×#57511yescharacteristic
368C4—24no—
369C4—24no—
370C4—24no—
371C4—24no—
372C4—22no—
373C4—22no—
374C4—22no—
375C4—41no—
376C4—41no—
377C4—84no—
378C4—84no—
379C4—84no—
380C4—84no—
381C4#355012yes—
382C4#355612yes—
383C2—42no—
384C2#3764111yescharacteristic
385C2#6084612yes—
386C2#6086112yes—
387C2#6090011yescharacteristic
388C2#6090711yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC24C22×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C43 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #32850C2semidirect
3. #32849C2semidirect
4. #32848C2semidirect
8. #29291C2semidirect
9. #29221C2semidirect
10. #28690C2semidirect

Maximal subgroups

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

subgroupindex
#366402
#328502
#328492
#328482
#299442
#299432
#298792
#292912
#292212
#286902
#208082