Finite Group Databasebeta

← #6853687

Subgroups — #6853687

Order 512.

1242 conjugacy classes of subgroups.

The subgroup lattice has 369 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36574C212yesmaximal
2#33263C211yesmaximal, characteristic
3#32692C214yesmaximal
4#32519C214yesmaximal
5#30307C212yesmaximal
6#29207C212yesmaximal
7#29177C212yesmaximal
8#29176C212yesmaximal
9#26881C218yesmaximal
10#22246C212yesmaximal
11#21727C211yesmaximal, characteristic
12#21340C211yesmaximal, characteristic
13C2×#498C2211yescharacteristic
14#2859C2212yes—
15#2859C2212yes—
16#2859C2212yes—
17#2584—22no—
18#2584C2214yes—
19#2581—22no—
20#2509—22no—
21#2509C2214yes—
22#2508—22no—
23#2508C2214yes—
24#2453—22no—
25#2453—22no—
26#2370—22no—
27#2370C2214yes—
28#2351—22no—
29C4×C4⋊Q8C2212yes—
30C4×C4⋊Q8C2212yes—
31C4×C4⋊Q8C2212yes—
32C42×Q8C2211yescharacteristic
33#2088C2218yes—
34#2088C2218yes—
35#2085C2218yes—
36#2059C2218yes—
37#2059C2218yes—
38#1990C2212yes—
39#1990C2212yes—
40#1990C2212yes—
41#1985C2212yes—
42#1949C2218yes—
43#1944C2214yes—
44#1944C2214yes—
45#1943C2214yes—
46#1940C2212yes—
47#1920C2218yes—
48#1920C2218yes—
49#1919C2218yes—
50#1919C2218yes—
51#1917C2214yes—
52#1845C2211yescharacteristic
53#1845C2211yescharacteristic
54#1844C2212yes—
55#1844C2211yescharacteristic
56#1834C2212yes—
57#1834C2212yes—
58#1776C2214yes—
59C4×C4⋊C8C2212yes—
60C4×C4⋊C8C2211yescharacteristic
61#1744C2212yes—
62C4×#97C2211yescharacteristic
63C2×C4⋊Q8C2312yes—
64C2×#125—22no—
65C2×#125—22no—
66C2×#125C2314yes—
67C2×C4×Q8—22no—
68C2×C4×Q8—22no—
69C2×C4×Q8—22no—
70C2×C4×Q8D412yes—
71C2×C4×Q8C2311yescharacteristic
72#498Q818yescomplemented
73C2×#107C2314yes—
74C2×#107C2314yes—
75C2×#106C2314yes—
76C2×#106C2318yes—
77C2×#106C2314yes—
78C2×C4⋊C8C2314yes—
79C2×C4⋊C8C2312yes—
80C2×#97C2312yes—
81C2×#97C2312yes—
82C2×C4×C8C2312yes—
83C2×C4×C8C2311yescharacteristic
84C2×C4×C8C2311yescharacteristic
85#400—24no—
86#400—24no—
87#400—24no—
88#398—24no—
89#398—24no—
90#395—24no—
91#391—22no—
92#391—22no—
93#391—22no—
94#389—24no—
95#389—24no—
96#389—24no—
97#389—24no—
98#389—22no—
99#389—22no—
100#389—22no—
101#389—22no—
102#389—22no—
103#389—22no—
104#389—22no—
105#389—22no—
106#389—22no—
107#389—22no—
108#389—22no—
109#389C2314yes—
110#389C2314yes—
111#389C2314yes—
112#389C2314yes—
113#389C2314yes—
114#389C2314yes—
115#389C2314yes—
116#387—24no—
117#387—24no—
118#387—24no—
119#387—24no—
120#384C2312yes—
121#384C2312yes—
122#384C2312yes—
123#382—22no—
124#382—22no—
125#382—24no—
126#382C2314yes—
127C4×C4⋊C4—22no—
128C4×C4⋊C4—22no—
129C4×C4⋊C4—22no—
130C4×C4⋊C4—24no—
131C4×C4⋊C4D412yes—
132C4×C4⋊C4D412yes—
133C4×C4⋊C4D412yes—
134C4×C4⋊C4C2314yes—
135C4×C4⋊C4C2314yes—
136C4×C4⋊C4C2314yes—
137C4×C4⋊C4C2312yes—
138C4×C4⋊C4C2311yescharacteristic
139C43C2311yescharacteristic
140C43C2312yes—
141#340C2318yes—
142#340C2318yes—
143#340C2318yes—
144#340C2318yes—
145#340C2318yes—
146#336C2314yes—
147#336C2314yes—
148#336C2314yes—
149#336C2312yes—
150C22×Q8—22no—
151C22×Q8C2×D411yescharacteristic
152C22×C8C2414yes—
153C22×C8C2412yes—
154C22×C8C2412yes—
155C4⋊Q8—24no—
156C4⋊Q8C2×Q818yes—
157#125—48no—
158#125—48no—
159#125—84no—
160C4×Q8—44no—
161C4×Q8—42no—
162C4×Q8—48no—
163C4×Q8—84no—
164C4×Q8—84no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—24no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—24no—
173C2×C4⋊C4—24no—
174C2×C4⋊C4—24no—
175C2×C4⋊C4—24no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—24no—
178C2×C4⋊C4—24no—
179C2×C4⋊C4—24no—
180C2×C4⋊C4—24no—
181C2×C4⋊C4C2×D412yes—
182C2×C4⋊C4C2×D412yes—
183C2×C4⋊C4C2×D412yes—
184C2×C4⋊C4C2×D411yescharacteristic
185C2×C4⋊C4C2×D412yes—
186C2×C4⋊C4C2414yes—
187C2×C4⋊C4C2414yes—
188C2×C4⋊C4C2414yes—
189C2×C4⋊C4C2414yes—
190C2×C42—24no—
191C2×C42—24no—
192C2×C42—24no—
193C2×C42C2×D412yes—
194C2×C42C2×D412yes—
195C2×C42C2×D412yes—
196C2×C42C2×D412yes—
197C2×C42C2×D412yes—
198C2×C42C2×D412yes—
199C2×C42C2412yes—
200C2×C42C2411yescharacteristic
201C2×C42C2411yescharacteristic
202C2×C42C2412yes—
203C2×C42C2411yescharacteristic
204#107—44no—
205#107—44no—
206#106—28no—
207#106—48no—
208#106C2×Q8116yes—
209C4⋊C8—28no—
210C4⋊C8—42no—
211#97—24no—
212#97—42no—
213C4×C8—22no—
214C4×C8—24no—
215C4×C8C2×Q814yes—
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—
226#95—24no—
227#95—24no—
228#95C2×D412yes—
229#95C2×D412yes—
230#95C2×D411yescharacteristic
231#95C2×D411yescharacteristic
232C2×Q8—28no—
233C2×Q8—41no—
234C2×Q8—42no—
235C2×Q8—44no—
236C2×Q8—44no—
237C22×C4—24no—
238C22×C4—24no—
239C22×C4—24no—
240C22×C4—24no—
241C22×C4C22×D412yes—
242C22×C4C22×D412yes—
243C22×C4C22×D412yes—
244C22×C4C22×D411yescharacteristic
245C22×C4C22×D412yes—
246C22×C4C22×D411yescharacteristic
247C22×C4C22×D412yes—
248C22×C4C22×D412yes—
249C22×C4C2511yescharacteristic
250C2×C8—22no—
251C2×C8—22no—
252C2×C8—22no—
253C2×C8—22no—
254C2×C8—24no—
255C2×C8—24no—
256C2×C8—24no—
257C2×C8C22×Q814yes—
258C2×C82−1+414yes—
259C4⋊C4—28no—
260C4⋊C4—24no—
261C4⋊C4—28no—
262C4⋊C4—28no—
263C4⋊C4—28no—
264C4⋊C4—41no—
265C4⋊C4—42no—
266C4⋊C4—42no—
267C4⋊C4—42no—
268C4⋊C4—42no—
269C4⋊C4—48no—
270C4⋊C4—48no—
271C4⋊C4—48no—
272C4⋊C4—48no—
273C4⋊C4—48no—
274C4⋊C4—44no—
275C4⋊C4—44no—
276C4⋊C4—44no—
277C4⋊C4—44no—
278C4⋊C4—44no—
279C4⋊C4—44no—
280C4⋊C4—84no—
281C4⋊C4—84no—
282C4⋊C4—84no—
283C4⋊C4—84no—
284C4⋊C4C22×Q818yes—
285C42—24no—
286C42—24no—
287C42—24no—
288C42—24no—
289C42—24no—
290C42—24no—
291C42—22no—
292C42—24no—
293C42—22no—
294C42—24no—
295C42—48no—
296C42—84no—
297C42—84no—
298C42C22×Q812yes—
299C422−1+412yes—
300C23C23×D411yescharacteristic
301Q8—44no—
302Q8—416no—
303C2×C4—22no—
304C2×C4—22no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—28no—
308C2×C4—22no—
309C2×C4—22no—
310C2×C4—22no—
311C2×C4—22no—
312C2×C4—22no—
313C2×C4—22no—
314C2×C4—22no—
315C2×C4—22no—
316C2×C4—22no—
317C2×C4—22no—
318C2×C4—22no—
319C2×C4—22no—
320C2×C4—22no—
321C2×C4—22no—
322C2×C4—22no—
323C2×C4—44no—
324C2×C4—44no—
325C2×C4—44no—
326C2×C4—44no—
327C2×C4—44no—
328C2×C4—44no—
329C2×C4—44no—
330C2×C4—44no—
331C2×C4—44no—
332C2×C4—44no—
333C2×C4—44no—
334C2×C4D4×Q814yes—
335C2×C4#57514yes—
336C2×C4#57714yes—
337C2×C4C23×Q811yescharacteristic
338C2×C4C2×2−1+412yes—
339C2×C4C2×2−1+411yescharacteristic
340C2×C4#58512yes—
341C8—24no—
342C8—24no—
343C8—44no—
344C22#344911yescharacteristic
345C22C2×D4×Q811yescharacteristic
346C22#349911yescharacteristic
347C22C2×#57512yes—
348C22C2×#57712yes—
349C4—42no—
350C4—42no—
351C4—42no—
352C4—42no—
353C4—41no—
354C4—42no—
355C4—41no—
356C4—48no—
357C4—84no—
358C4—84no—
359C4—84no—
360C4#338818yes—
361C4#350212yes—
362C4#356312yes—
363C2#3961211yescharacteristic
364C2#6097211yescharacteristic
365C2#6101511yescharacteristic
366C2C2×#338812yes—
367C2#6248312yes—
subgroupG/H
center, socle, prime radicalC23C23×D4
commutator, Frattini, cosocleC22×C4C25
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C42×Q8 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
72. #498Q8semidirect

Maximal subgroups

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

subgroupindex
#365742
#332632
#326922
#325192
#303072
#292072
#291772
#291762
#268812
#222462
#217272
#213402