Finite Group Databasebeta

← #6498036

Subgroups — #6498036

Order 512.

1372 conjugacy classes of subgroups.

The subgroup lattice has 381 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36640C212yesmaximal
2#33315C211yesmaximal, characteristic, complemented
3#33299C211yesmaximal, characteristic, complemented
4#33295C212yesmaximal, complemented
5#29879C214yesmaximal
6#29184C214yesmaximal, complemented
7#29183C214yesmaximal, complemented
8#27881C214yesmaximal
9#27880C214yesmaximal
10#22249C214yesmaximal, complemented
11#20463C211yesmaximal, 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#2088C2218yes—
34#2087C2214yes—
35#2086C2214yes—
36#2077C2218yes—
37#2076C2214yes—
38#2075C2214yes—
39#1992C2214yes—
40#1992C2214yes—
41#1992C2214yes—
42#1992C2214yes—
43#1991C2218yes—
44#1921C2218yes—
45#1920C2218yes—
46#1919C2218yes—
47#1918C2218yes—
48#1847C2214yes—
49#1847C2214yes—
50#1836C2214yes—
51#1836C2214yes—
52#1835C2214yes—
53#1767C2214yes—
54#1749C2214yes—
55#1744C2212yes—
56C4×#97C2211yescharacteristic
57C4×#97C2211yescharacteristic
58C2×#125—22no—
59C2×#125—22no—
60C2×#125C2314yes—
61C2×#125C2314yes—
62C2×#117—22no—
63C2×#117—22no—
64C22×C42D411yescharacteristic
65C22×C42D412yes—
66C22×C42D411yescharacteristic
67C22×C42C2311yescharacteristic
68C22×C42C2312yes—
69#499—216no—
70#431—28no—
71C2×#107C2314yes—
72C2×#107C2314yes—
73C2×#106C2318yes—
74C2×C4⋊C8C2314yes—
75C2×C22⋊C8C2314yes—
76C2×#97C2314yes—
77C2×#97C2314yes—
78C2×C4×C8C2312yes—
79C2×C4×C8C2312yes—
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#389—22no—
92#389C2314yes—
93#389C2314yes—
94#389C2314yes—
95#389C2314yes—
96#387—24no—
97#387—24no—
98#387—24no—
99#387—24no—
100#387—24no—
101#387—24no—
102#387—24no—
103#387—24no—
104#385—24no—
105#385—24no—
106#385—24no—
107#385—24no—
108#384C2312yes—
109#384C2312yes—
110#384C2312yes—
111#384C2312yes—
112#382—22no—
113#382—22no—
114#382—22no—
115#382—22no—
116#382C2314yes—
117#382C2314yes—
118#380—22no—
119#380—22no—
120#380C2314yes—
121#380C2314yes—
122C4×C4⋊C4—22no—
123C4×C4⋊C4—22no—
124C4×C4⋊C4—24no—
125C4×C4⋊C4—24no—
126C4×C4⋊C4—24no—
127C4×C4⋊C4—24no—
128C4×C4⋊C4C2314yes—
129C4×C4⋊C4C2314yes—
130C43D411yescharacteristic
131C43D412yes—
132C43D411yescharacteristic
133C43C2311yescharacteristic
134C43C2312yes—
135C43C2311yescharacteristic
136#340C2314yes—
137#340C2318yes—
138#340C2314yes—
139#340C2318yes—
140#340C2318yes—
141#340C2318yes—
142#340C2318yes—
143#336C2318yes—
144C23×C4C2×D412yes—
145C23×C4C2×D411yescharacteristic
146C23×C4C2×D412yes—
147C23×C4C2×D411yescharacteristic
148C23×C4C2411yescharacteristic
149C22×C8C2414yes—
150C22×C8C2414yes—
151#125—216no—
152#125—216no—
153#125—48no—
154#125—48no—
155#117—48no—
156#117—48no—
157C2×C4⋊C4—22no—
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—24no—
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⋊C4C2414yes—
174C2×C4⋊C4C2414yes—
175C2×C4⋊C4C2414yes—
176C2×C4⋊C4C2414yes—
177C2×C22⋊C4—24no—
178C2×C22⋊C4—24no—
179C2×C42—24no—
180C2×C42—28no—
181C2×C42—24no—
182C2×C42—24no—
183C2×C42—44no—
184C2×C42—42no—
185C2×C42—42no—
186C2×C42C2×D412yes—
187C2×C42C2×D412yes—
188C2×C42C2×D412yes—
189C2×C42C2×D412yes—
190C2×C42C2×D411yescharacteristic
191C2×C42C2×D411yescharacteristic
192C2×C42C2×D412yes—
193C2×C42C2×D411yescharacteristic
194C2×C42C2×D412yes—
195C2×C42C2×D411yescharacteristic
196C2×C42C2×D412yes—
197C2×C42C2×D411yescharacteristic
198C2×C42C2×D412yes—
199C2×C42C2×D411yescharacteristic
200C2×C42C2412yes—
201C2×C42C2411yescharacteristic
202C2×C42C2411yescharacteristic
203C2×C42C2412yes—
204#107—28no—
205#107—28no—
206#106—216no—
207C4⋊C8—28no—
208C22⋊C8—28no—
209#97—28no—
210#97—28no—
211C4×C8—24no—
212C4×C8#4118yes—
213#95—24no—
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—
225C24C22×D411yescharacteristic
226C22×C4—24no—
227C22×C4—22no—
228C22×C4—22no—
229C22×C4—24no—
230C22×C4—24no—
231C22×C4—22no—
232C22×C4—22no—
233C22×C4—24no—
234C22×C4—22no—
235C22×C4—24no—
236C22×C4—24no—
237C22×C4—24no—
238C22×C4—24no—
239C22×C4—24no—
240C22×C4—42no—
241C22×C4—42no—
242C22×C4—42no—
243C22×C4C22×D411yescharacteristic
244C22×C4C22×D412yes—
245C22×C4C22×D411yescharacteristic
246C22×C4C22×D411yescharacteristic
247C22×C4C22×D411yescharacteristic
248C22×C4C22×D412yes—
249C22×C4C22×D412yes—
250C22×C4C22×D411yescharacteristic
251C22×C4C22×D412yes—
252C22×C4C2511yescharacteristic
253C2×C8—28no—
254C2×C8—24no—
255C2×C8—28no—
256C2×C8C2×#4118yes—
257C4⋊C4—28no—
258C4⋊C4—28no—
259C4⋊C4—28no—
260C4⋊C4—28no—
261C4⋊C4—44no—
262C4⋊C4—44no—
263C4⋊C4—44no—
264C4⋊C4—44no—
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—48no—
275C4⋊C4—48no—
276C4⋊C4—48no—
277C22⋊C4—48no—
278C22⋊C4—48no—
279C42—22no—
280C42—22no—
281C42—24no—
282C42—24no—
283C42—24no—
284C42—24no—
285C42—24no—
286C42—22no—
287C42—24no—
288C42—41no—
289C42—42no—
290C42—41no—
291C42—42no—
292C42—41no—
293C42—42no—
294C42—41no—
295C42—48no—
296C42—48no—
297C42C2×#4114yes—
298C422−1+418yes—
299C23—21no—
300C23—22no—
301C23—22no—
302C23—21no—
303C23#57512yes—
304C23C23×D411yescharacteristic
305C2×C4—28no—
306C2×C4—216no—
307C2×C4—21no—
308C2×C4—22no—
309C2×C4—22no—
310C2×C4—22no—
311C2×C4—22no—
312C2×C4—22no—
313C2×C4—22no—
314C2×C4—22no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—22no—
321C2×C4—22no—
322C2×C4—22no—
323C2×C4—22no—
324C2×C4—22no—
325C2×C4—44no—
326C2×C4—48no—
327C2×C4—44no—
328C2×C4—44no—
329C2×C4—44no—
330C2×C4—44no—
331C2×C4—44no—
332C2×C4—44no—
333C2×C4—44no—
334C2×C4—44no—
335C2×C4—44no—
336C2×C4—44no—
337C2×C4—44no—
338C2×C4—44no—
339C2×C4—44no—
340C2×C4#53212yes—
341C2×C4#53212yes—
342C2×C4#53614yes—
343C2×C4#53614yes—
344C2×C4#57514yes—
345C2×C4#57512yes—
346C2×C4C22×#4112yes—
347C2×C4C2×2−1+414yes—
348C8—28no—
349C8—44no—
350C22—22no—
351C22—24no—
352C22—41no—
353C22—42no—
354C22—41no—
355C22C2×#53211yescharacteristic
356C22C2×#53612yes—
357C22C2×#57512yes—
358C22C2×#57511yescharacteristic
359C22C2×#57511yescharacteristic
360C4—22no—
361C4—22no—
362C4—22no—
363C4—24no—
364C4—24no—
365C4—24no—
366C4—41no—
367C4—42no—
368C4—41no—
369C4—84no—
370C4—84no—
371C4—84no—
372C4—84no—
373C4#350914yes—
374C2—42no—
375C2#3532211yescharacteristic
376C2#6067611yescharacteristic
377C2#6068811yescharacteristic
378C2#6086112yes—
379C2#6086312yes—
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. #33315C2semidirect
3. #33299C2semidirect
4. #33295C2semidirect
6. #29184C2semidirect
7. #29183C2semidirect
10. #22249C2semidirect

Maximal subgroups

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

subgroupindex
#366402
#333152
#332992
#332952
#298792
#291842
#291832
#278812
#278802
#222492
#204632