Finite Group Databasebeta

← #6853649

Subgroups — #6853649

Order 512.

1458 conjugacy classes of subgroups.

The subgroup lattice has 406 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36560C212yesmaximal
2#33258C211yesmaximal, characteristic
3#32682C214yesmaximal
4#32504C214yesmaximal
5#30304C212yesmaximal
6#29208C212yesmaximal
7#29178C212yesmaximal
8#29178C212yesmaximal
9#26856C218yesmaximal
10#22247C212yesmaximal
11C4×#500C211yesmaximal, characteristic
12#21339C211yesmaximal, characteristic
13C2×#557—22no—
14C2×#500C2211yescharacteristic
15#2868C2212yes—
16#2859C2212yes—
17#2859C2212yes—
18#2568—22no—
19#2565—22no—
20#2565—22no—
21#2546—22no—
22#2546C2214yes—
23#2506—22no—
24#2506—22no—
25#2506C2214yes—
26#2506C2214yes—
27#2365—22no—
28#2365C2214yes—
29C4×C4⋊Q8C2212yes—
30C4×C4⋊Q8C2212yes—
31C4×C4⋊Q8C2212yes—
32C42×Q8C2211yescharacteristic
33#2086C2218yes—
34#2086C2218yes—
35#2083C2218yes—
36#2057C2218yes—
37#2057C2218yes—
38#1990C2212yes—
39#1990C2212yes—
40#1990C2212yes—
41#1984C2212yes—
42#1947C2218yes—
43#1942C2214yes—
44#1942C2214yes—
45#1942C2214yes—
46#1939C2212yes—
47#1918C2218yes—
48#1918C2218yes—
49#1918C2218yes—
50#1918C2218yes—
51#1916C2214yes—
52#1845C2211yescharacteristic
53#1845C2212yes—
54#1845C2211yescharacteristic
55#1845C2211yescharacteristic
56#1834C2212yes—
57#1834C2212yes—
58C4×#107C2214yes—
59C4×C4⋊C8C2212yes—
60C4×C4⋊C8C2211yescharacteristic
61#1744C2212yes—
62C42×C8C2211yescharacteristic
63#557—416no—
64C2×C4⋊Q8—24no—
65C2×C4⋊Q8—22no—
66C2×C4⋊Q8—22no—
67C2×C4⋊Q8C2312yes—
68C2×C4⋊Q8C2314yes—
69C2×#125—24no—
70C2×#125—24no—
71C2×#125—22no—
72C2×#125—22no—
73C2×C4×Q8—24no—
74C2×C4×Q8—22no—
75C2×C4×Q8—22no—
76C2×C4×Q8—22no—
77C2×C4×Q8D411yescharacteristic
78C2×C4×Q8D411yescharacteristic
79C2×C4×Q8C2311yescharacteristic
80#500Q818yescomplemented
81C2×#107C2314yes—
82C2×#107C2314yes—
83C2×#107C2318yes—
84C2×#107C2314yes—
85C2×#107C2314yes—
86C2×C4⋊C8C2314yes—
87C2×C4⋊C8C2312yes—
88C2×C4×C8C2312yes—
89C2×C4×C8C2311yescharacteristic
90C2×C4×C8C2312yes—
91C2×C4×C8C2312yes—
92C2×C4×C8C2311yescharacteristic
93#400—24no—
94#400—24no—
95#398—24no—
96#398—24no—
97#398—24no—
98#398—24no—
99#391—24no—
100#391—24no—
101#391—22no—
102#391—22no—
103#391—22no—
104#389—22no—
105#389—22no—
106#389—22no—
107#389—22no—
108#389—22no—
109#389—22no—
110#389—22no—
111#389C2314yes—
112#389C2314yes—
113#389C2314yes—
114#389C2314yes—
115#389C2314yes—
116#389C2314yes—
117#389C2314yes—
118#387—24no—
119#387—24no—
120#387—24no—
121#387—24no—
122#384—22no—
123#384—22no—
124#384C2312yes—
125#384C2312yes—
126#384C2312yes—
127#384C2314yes—
128#382—22no—
129#382—22no—
130C4×C4⋊C4—22no—
131C4×C4⋊C4—22no—
132C4×C4⋊C4—22no—
133C4×C4⋊C4D411yescharacteristic
134C4×C4⋊C4D412yes—
135C4×C4⋊C4D412yes—
136C4×C4⋊C4D411yescharacteristic
137C4×C4⋊C4C2314yes—
138C4×C4⋊C4C2314yes—
139C4×C4⋊C4C2314yes—
140C4×C4⋊C4C2312yes—
141C4×C4⋊C4C2311yescharacteristic
142C43C2311yescharacteristic
143C43C2312yes—
144#340C2318yes—
145#340C2318yes—
146#340C2318yes—
147#340C2318yes—
148#340C2318yes—
149#336C2314yes—
150#336C2314yes—
151#336C2314yes—
152#336C2312yes—
153C22×Q8—22no—
154C22×Q8—24no—
155C22×Q8C2×D411yescharacteristic
156C22×C8C2414yes—
157C22×C8C2412yes—
158C22×C8C2412yes—
159C4⋊Q8—24no—
160C4⋊Q8—48no—
161C4⋊Q8—48no—
162C4⋊Q8—416no—
163C4⋊Q8—48no—
164C4⋊Q8C2×Q818yes—
165#125—416no—
166#125—416no—
167#125—48no—
168#125—48no—
169C4×Q8—42no—
170C4×Q8—42no—
171C4×Q8—416no—
172C4×Q8—48no—
173C4×Q8—48no—
174C4×Q8—48no—
175C4×Q8D818yes—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—24no—
179C2×C4⋊C4—24no—
180C2×C4⋊C4—24no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—24no—
184C2×C4⋊C4—24no—
185C2×C4⋊C4—24no—
186C2×C4⋊C4—24no—
187C2×C4⋊C4—24no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—24no—
191C2×C4⋊C4—24no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4C2×D412yes—
194C2×C4⋊C4C2×D411yescharacteristic
195C2×C4⋊C4C2×D412yes—
196C2×C4⋊C4C2×D411yescharacteristic
197C2×C4⋊C4C2×D412yes—
198C2×C4⋊C4C2×D411yescharacteristic
199C2×C4⋊C4C2414yes—
200C2×C4⋊C4C2414yes—
201C2×C4⋊C4C2414yes—
202C2×C4⋊C4C2414yes—
203C2×C42—24no—
204C2×C42—24no—
205C2×C42—24no—
206C2×C42C2×D412yes—
207C2×C42C2×D411yescharacteristic
208C2×C42C2×D411yescharacteristic
209C2×C42C2×D412yes—
210C2×C42C2×D411yescharacteristic
211C2×C42C2×D412yes—
212C2×C42C2×D411yescharacteristic
213C2×C42C2×D412yes—
214C2×C42C2412yes—
215C2×C42C2411yescharacteristic
216C2×C42C2411yescharacteristic
217C2×C42C2412yes—
218C2×C42C2411yescharacteristic
219#107—28no—
220#107—44no—
221#107—44no—
222#107—48no—
223#107C2×Q8116yes—
224C4⋊C8—28no—
225C4⋊C8—42no—
226C4×C8—24no—
227C4×C8—22no—
228C4×C8—24no—
229C4×C8—42no—
230C4×C8C2×Q814yes—
231#95—24no—
232#95—24no—
233#95—24no—
234#95—24no—
235#95—24no—
236#95—24no—
237#95—24no—
238#95—24no—
239#95—24no—
240#95—24no—
241#95C2×D412yes—
242#95C2×D412yes—
243#95C2×D412yes—
244C2×Q8—28no—
245C2×Q8—41no—
246C2×Q8—41no—
247C2×Q8—48no—
248C2×Q8—48no—
249C2×Q8—44no—
250C2×Q8—48no—
251C2×Q8—44no—
252C2×Q8C2×D814yes—
253C22×C4—24no—
254C22×C4—24no—
255C22×C4—24no—
256C22×C4—24no—
257C22×C4C22×D411yescharacteristic
258C22×C4C22×D411yescharacteristic
259C22×C4C22×D412yes—
260C22×C4C22×D412yes—
261C22×C4C22×D411yescharacteristic
262C22×C4C22×D412yes—
263C22×C4C22×D412yes—
264C22×C4C22×D411yescharacteristic
265C22×C4C22×D411yescharacteristic
266C22×C4C22×D411yescharacteristic
267C22×C4C2511yescharacteristic
268C2×C8—22no—
269C2×C8—22no—
270C2×C8—22no—
271C2×C8—22no—
272C2×C8—24no—
273C2×C8—24no—
274C2×C8—24no—
275C2×C8C22×Q814yes—
276C2×C82−1+414yes—
277C4⋊C4—28no—
278C4⋊C4—24no—
279C4⋊C4—28no—
280C4⋊C4—28no—
281C4⋊C4—28no—
282C4⋊C4—41no—
283C4⋊C4—42no—
284C4⋊C4—41no—
285C4⋊C4—42no—
286C4⋊C4—48no—
287C4⋊C4—48no—
288C4⋊C4—48no—
289C4⋊C4—44no—
290C4⋊C4—48no—
291C4⋊C4—48no—
292C4⋊C4—48no—
293C4⋊C4—48no—
294C4⋊C4—44no—
295C4⋊C4—44no—
296C4⋊C4—48no—
297C4⋊C4—48no—
298C4⋊C4—48no—
299C4⋊C4—44no—
300C4⋊C4—44no—
301C4⋊C4—44no—
302C4⋊C4C2×D818yes—
303C4⋊C4C2×D814yes—
304C4⋊C4C22×Q818yes—
305C42—22no—
306C42—24no—
307C42—22no—
308C42—24no—
309C42—22no—
310C42—24no—
311C42—22no—
312C42—24no—
313C42—22no—
314C42—24no—
315C42—48no—
316C42—48no—
317C42—48no—
318C42C2×D814yes—
319C42C2×D818yes—
320C42C22×Q812yes—
321C422−1+412yes—
322C23C23×D411yescharacteristic
323Q8—44no—
324Q8—416no—
325Q8—816no—
326C2×C4—21no—
327C2×C4—22no—
328C2×C4—21no—
329C2×C4—22no—
330C2×C4—21no—
331C2×C4—22no—
332C2×C4—21no—
333C2×C4—22no—
334C2×C4—22no—
335C2×C4—21no—
336C2×C4—28no—
337C2×C4—21no—
338C2×C4—22no—
339C2×C4—22no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—22no—
343C2×C4—21no—
344C2×C4—22no—
345C2×C4—21no—
346C2×C4—22no—
347C2×C4—44no—
348C2×C4—44no—
349C2×C4—44no—
350C2×C4—44no—
351C2×C4—44no—
352C2×C4—44no—
353C2×C4—44no—
354C2×C4—44no—
355C2×C4—44no—
356C2×C4—44no—
357C2×C4—44no—
358C2×C4D4×Q812yes—
359C2×C4D4×Q812yes—
360C2×C4C22×D812yes—
361C2×C4C22×D814yes—
362C2×C4C22×D814yes—
363C2×C4C22×D812yes—
364C2×C4C22×D812yes—
365C2×C4#57512yes—
366C2×C4#57812yes—
367C2×C4#57812yes—
368C2×C4C23×Q811yescharacteristic
369C2×C4C2×2−1+412yes—
370C2×C4C2×2−1+411yescharacteristic
371C2×C4#58512yes—
372C8—24no—
373C8—24no—
374C8—44no—
375C22#344911yescharacteristic
376C22C2×D4×Q811yescharacteristic
377C22#349911yescharacteristic
378C22C23×D811yescharacteristic
379C22C2×#57511yescharacteristic
380C22C2×#57811yescharacteristic
381C22C2×#57811yescharacteristic
382C4—22no—
383C4—24no—
384C4—22no—
385C4—24no—
386C4—42no—
387C4—41no—
388C4—42no—
389C4—41no—
390C4—48no—
391C4—84no—
392C4—84no—
393C4—84no—
394C4Q8×D814yes—
395C4#339014yes—
396C4#350212yes—
397C4#356312yes—
398C2#3961211yescharacteristic
399C2#6094011yescharacteristic
400C2#6101311yescharacteristic
401C2C2×Q8×D811yescharacteristic
402C2C2×#339011yescharacteristic
403C2#6247111yescharacteristic
404C2#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
80. #500Q8semidirect

Maximal subgroups

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

subgroupindex
#365602
#332582
#326822
#325042
#303042
#292082
#291782
#268562
#222472
C4×#5002
#213392