Finite Group Databasebeta

← #6853652

Subgroups — #6853652

Order 512.

1254 conjugacy classes of subgroups.

The subgroup lattice has 394 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36574C212yesmaximal
2#33258C211yesmaximal, characteristic
3#32686C214yesmaximal
4#32509C214yesmaximal
5#30307C212yesmaximal
6#29207C212yesmaximal
7#29178C212yesmaximal
8#29177C212yesmaximal
9#26870C218yesmaximal
10#22248C212yesmaximal
11C4×#500C211yesmaximal, characteristic
12#21339C211yesmaximal, characteristic
13C2×#500C2211yescharacteristic
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#2086C2218yes—
35#2085C2218yes—
36#2059C2218yes—
37#2059C2218yes—
38#1990C2212yes—
39#1990C2212yes—
40#1990C2212yes—
41#1984C2212yes—
42#1949C2218yes—
43#1942C2214yes—
44#1942C2214yes—
45#1942C2214yes—
46#1939C2212yes—
47#1921C2218yes—
48#1921C2218yes—
49#1919C2218yes—
50#1918C2218yes—
51#1916C2214yes—
52#1845C2211yescharacteristic
53#1845C2211yescharacteristic
54#1845C2211yescharacteristic
55#1844C2212yes—
56#1834C2212yes—
57#1834C2212yes—
58C4×#107C2214yes—
59C4×C4⋊C8C2212yes—
60C4×C4⋊C8C2211yescharacteristic
61#1744C2212yes—
62C42×C8C2211yescharacteristic
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×Q8D411yescharacteristic
71C2×C4×Q8D411yescharacteristic
72C2×C4×Q8C2311yescharacteristic
73#500Q818yescomplemented
74C2×#107C2314yes—
75C2×#107C2314yes—
76C2×#107C2314yes—
77C2×#107C2314yes—
78C2×#106C2318yes—
79C2×C4⋊C8C2314yes—
80C2×C4⋊C8C2312yes—
81C2×C4×C8C2312yes—
82C2×C4×C8C2311yescharacteristic
83C2×C4×C8C2312yes—
84C2×C4×C8C2312yes—
85C2×C4×C8C2311yescharacteristic
86#400—24no—
87#400—24no—
88#400—24no—
89#398—24no—
90#398—24no—
91#395—24no—
92#391—22no—
93#391—22no—
94#391—22no—
95#389—24no—
96#389—24no—
97#389—24no—
98#389—24no—
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#389—22no—
110#389C2314yes—
111#389C2314yes—
112#389C2314yes—
113#389C2314yes—
114#389C2314yes—
115#389C2314yes—
116#389C2314yes—
117#387—24no—
118#387—24no—
119#387—24no—
120#387—24no—
121#384C2312yes—
122#384C2312yes—
123#384C2312yes—
124#382—22no—
125#382—22no—
126#382—24no—
127#382C2314yes—
128C4×C4⋊C4—22no—
129C4×C4⋊C4—22no—
130C4×C4⋊C4—22no—
131C4×C4⋊C4—24no—
132C4×C4⋊C4D412yes—
133C4×C4⋊C4D411yescharacteristic
134C4×C4⋊C4D411yescharacteristic
135C4×C4⋊C4D412yes—
136C4×C4⋊C4C2314yes—
137C4×C4⋊C4C2314yes—
138C4×C4⋊C4C2314yes—
139C4×C4⋊C4C2312yes—
140C4×C4⋊C4C2311yescharacteristic
141C43C2311yescharacteristic
142C43C2312yes—
143#340C2318yes—
144#340C2318yes—
145#340C2318yes—
146#340C2318yes—
147#340C2318yes—
148#336C2314yes—
149#336C2314yes—
150#336C2314yes—
151#336C2312yes—
152C22×Q8—22no—
153C22×Q8C2×D411yescharacteristic
154C22×C8C2414yes—
155C22×C8C2412yes—
156C22×C8C2412yes—
157C4⋊Q8—24no—
158C4⋊Q8C2×Q818yes—
159#125—48no—
160#125—48no—
161#125—84no—
162C4×Q8—24no—
163C4×Q8—42no—
164C4×Q8—42no—
165C4×Q8—48no—
166C4×Q8—84no—
167C4×Q8—84no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—24no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—24no—
176C2×C4⋊C4—24no—
177C2×C4⋊C4—24no—
178C2×C4⋊C4—24no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—24no—
181C2×C4⋊C4—24no—
182C2×C4⋊C4—24no—
183C2×C4⋊C4—24no—
184C2×C4⋊C4C2×D412yes—
185C2×C4⋊C4C2×D411yescharacteristic
186C2×C4⋊C4C2×D411yescharacteristic
187C2×C4⋊C4C2×D412yes—
188C2×C4⋊C4C2×D411yescharacteristic
189C2×C4⋊C4C2×D412yes—
190C2×C4⋊C4C2414yes—
191C2×C4⋊C4C2414yes—
192C2×C4⋊C4C2414yes—
193C2×C4⋊C4C2414yes—
194C2×C42—24no—
195C2×C42—24no—
196C2×C42—24no—
197C2×C42C2×D411yescharacteristic
198C2×C42C2×D412yes—
199C2×C42C2×D412yes—
200C2×C42C2×D412yes—
201C2×C42C2×D411yescharacteristic
202C2×C42C2×D412yes—
203C2×C42C2×D411yescharacteristic
204C2×C42C2×D411yescharacteristic
205C2×C42C2412yes—
206C2×C42C2411yescharacteristic
207C2×C42C2411yescharacteristic
208C2×C42C2412yes—
209C2×C42C2411yescharacteristic
210#107—28no—
211#107—44no—
212#107—44no—
213#107C2×Q8116yes—
214#106—48no—
215C4⋊C8—28no—
216C4⋊C8—42no—
217C4×C8—24no—
218C4×C8—22no—
219C4×C8—24no—
220C4×C8—42no—
221C4×C8C2×Q814yes—
222#95—24no—
223#95—24no—
224#95—24no—
225#95—24no—
226#95—24no—
227#95—24no—
228#95—24no—
229#95—24no—
230#95—24no—
231#95—24no—
232#95—24no—
233#95—24no—
234#95C2×D412yes—
235#95C2×D412yes—
236#95C2×D412yes—
237C2×Q8—28no—
238C2×Q8—22no—
239C2×Q8—41no—
240C2×Q8—41no—
241C2×Q8—44no—
242C2×Q8—44no—
243C22×C4—24no—
244C22×C4—24no—
245C22×C4—24no—
246C22×C4—24no—
247C22×C4C22×D411yescharacteristic
248C22×C4C22×D412yes—
249C22×C4C22×D411yescharacteristic
250C22×C4C22×D412yes—
251C22×C4C22×D412yes—
252C22×C4C22×D412yes—
253C22×C4C22×D411yescharacteristic
254C22×C4C22×D411yescharacteristic
255C22×C4C22×D411yescharacteristic
256C22×C4C22×D411yescharacteristic
257C22×C4C2511yescharacteristic
258C2×C8—22no—
259C2×C8—22no—
260C2×C8—22no—
261C2×C8—22no—
262C2×C8—24no—
263C2×C8—24no—
264C2×C8—24no—
265C2×C8C22×Q814yes—
266C2×C82−1+414yes—
267C4⋊C4—28no—
268C4⋊C4—22no—
269C4⋊C4—24no—
270C4⋊C4—24no—
271C4⋊C4—28no—
272C4⋊C4—28no—
273C4⋊C4—28no—
274C4⋊C4—41no—
275C4⋊C4—41no—
276C4⋊C4—42no—
277C4⋊C4—42no—
278C4⋊C4—48no—
279C4⋊C4—48no—
280C4⋊C4—48no—
281C4⋊C4—48no—
282C4⋊C4—48no—
283C4⋊C4—44no—
284C4⋊C4—44no—
285C4⋊C4—44no—
286C4⋊C4—44no—
287C4⋊C4—44no—
288C4⋊C4—44no—
289C4⋊C4—84no—
290C4⋊C4—84no—
291C4⋊C4—84no—
292C4⋊C4—84no—
293C4⋊C4C22×Q818yes—
294C42—24no—
295C42—22no—
296C42—24no—
297C42—22no—
298C42—24no—
299C42—24no—
300C42—22no—
301C42—22no—
302C42—24no—
303C42—22no—
304C42—24no—
305C42—48no—
306C42—84no—
307C42—84no—
308C42#13514yes—
309C42C22×Q812yes—
310C422−1+412yes—
311C23C23×D411yescharacteristic
312Q8—44no—
313Q8—416no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—22no—
319C2×C4—22no—
320C2×C4—21no—
321C2×C4—28no—
322C2×C4—22no—
323C2×C4—21no—
324C2×C4—22no—
325C2×C4—22no—
326C2×C4—22no—
327C2×C4—22no—
328C2×C4—22no—
329C2×C4—21no—
330C2×C4—22no—
331C2×C4—22no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—22no—
335C2×C4—21no—
336C2×C4—22no—
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—44no—
347C2×C4—44no—
348C2×C4D4×Q812yes—
349C2×C4D4×Q812yes—
350C2×C4C2×#13512yes—
351C2×C4C2×#13512yes—
352C2×C4C2×#13512yes—
353C2×C4#57512yes—
354C2×C4#57612yes—
355C2×C4#57812yes—
356C2×C4C23×Q811yescharacteristic
357C2×C4C2×2−1+412yes—
358C2×C4C2×2−1+411yescharacteristic
359C2×C4#58512yes—
360C8—24no—
361C8—24no—
362C8—44no—
363C22#344911yescharacteristic
364C22C2×D4×Q811yescharacteristic
365C22#349911yescharacteristic
366C22C22×#13511yescharacteristic
367C22C2×#57511yescharacteristic
368C22C2×#57611yescharacteristic
369C22C2×#57811yescharacteristic
370C4—22no—
371C4—22no—
372C4—42no—
373C4—42no—
374C4—42no—
375C4—41no—
376C4—42no—
377C4—41no—
378C4—48no—
379C4—84no—
380C4—84no—
381C4—84no—
382C4#338314yes—
383C4#338914yes—
384C4#350212yes—
385C4#356312yes—
386C2#3961211yescharacteristic
387C2#6094411yescharacteristic
388C2#6101411yescharacteristic
389C2C2×#338311yescharacteristic
390C2C2×#338911yescharacteristic
391C2#6247411yescharacteristic
392C2#6248211yescharacteristic
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
73. #500Q8semidirect

Maximal subgroups

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

subgroupindex
#365742
#332582
#326862
#325092
#303072
#292072
#291782
#291772
#268702
#222482
C4×#5002
#213392