Finite Group Databasebeta

← #6853650

Subgroups — #6853650

Order 512.

1298 conjugacy classes of subgroups.

The subgroup lattice has 401 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36563C212yesmaximal
2#33258C211yesmaximal, characteristic
3#32684C214yesmaximal
4#32506C214yesmaximal
5#30304C212yesmaximal
6#29208C212yesmaximal
7#29179C212yesmaximal
8#29179C212yesmaximal
9#26862C218yesmaximal
10#22248C212yesmaximal
11C4×#500C211yesmaximal, characteristic
12#21339C211yesmaximal, characteristic
13C2×#500C2211yescharacteristic
14#2868C2212yes—
15#2859C2212yes—
16#2859C2212yes—
17#2582—22no—
18#2567C2214yes—
19#2552—22no—
20#2550—22no—
21#2550—22no—
22#2507C2214yes—
23#2507C2214yes—
24#2451—22no—
25#2451—22no—
26#2369C2214yes—
27#2355—22no—
28Q8×C4⋊C4—22no—
29C4×C4⋊Q8C2212yes—
30C4×C4⋊Q8C2212yes—
31C4×C4⋊Q8C2212yes—
32C42×Q8C2211yescharacteristic
33#2087C2218yes—
34#2087C2218yes—
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#1921C2218yes—
48#1921C2218yes—
49#1921C2218yes—
50#1921C2218yes—
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
63C2×C4⋊Q8—22no—
64C2×C4⋊Q8—22no—
65C2×C4⋊Q8C2312yes—
66C2×C4⋊Q8C2314yes—
67C2×#125—22no—
68C2×#125—22no—
69C2×C4×Q8—22no—
70C2×C4×Q8—22no—
71C2×C4×Q8—22no—
72C2×C4×Q8D411yescharacteristic
73C2×C4×Q8D411yescharacteristic
74C2×C4×Q8C2311yescharacteristic
75#500Q818yescomplemented
76C2×#107C2314yes—
77C2×#107C2314yes—
78C2×#107C2318yes—
79C2×#107C2314yes—
80C2×#107C2314yes—
81C2×C4⋊C8C2314yes—
82C2×C4⋊C8C2312yes—
83C2×C4×C8C2312yes—
84C2×C4×C8C2311yescharacteristic
85C2×C4×C8C2312yes—
86C2×C4×C8C2312yes—
87C2×C4×C8C2311yescharacteristic
88#398—24no—
89#398—24no—
90#398—24no—
91#398—24no—
92#395—24no—
93#395—24no—
94#391—24no—
95#391—22no—
96#391—22no—
97#391—22no—
98#391—24no—
99#391—24no—
100#389—24no—
101#389—22no—
102#389—22no—
103#389—22no—
104#389—22no—
105#389—22no—
106#389—22no—
107#389—22no—
108#389—24no—
109#389—24no—
110#389C2314yes—
111#389C2314yes—
112#389C2314yes—
113#389C2314yes—
114#389C2314yes—
115#389C2314yes—
116#389C2314yes—
117#387—24no—
118#387—24no—
119#384—24no—
120#384—22no—
121#384—22no—
122#384C2312yes—
123#384C2312yes—
124#384C2312yes—
125#384C2314yes—
126#382—22no—
127#382—22no—
128C4×C4⋊C4—24no—
129C4×C4⋊C4—22no—
130C4×C4⋊C4—22no—
131C4×C4⋊C4—22no—
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×Q8—24no—
154C22×Q8C2×D411yescharacteristic
155C22×C8C2414yes—
156C22×C8C2412yes—
157C22×C8C2412yes—
158C4⋊Q8—24no—
159C4⋊Q8—48no—
160C4⋊Q8—84no—
161C4⋊Q8—84no—
162C4⋊Q8C2×Q818yes—
163#125—84no—
164#125—84no—
165C4×Q8—42no—
166C4×Q8—42no—
167C4×Q8—48no—
168C4×Q8—84no—
169C4×Q8—84no—
170C4×Q8Q1618yes—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—24no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—24no—
180C2×C4⋊C4—24no—
181C2×C4⋊C4—24no—
182C2×C4⋊C4—24no—
183C2×C4⋊C4—24no—
184C2×C4⋊C4—24no—
185C2×C4⋊C4—24no—
186C2×C4⋊C4—24no—
187C2×C4⋊C4—24no—
188C2×C4⋊C4C2×D412yes—
189C2×C4⋊C4C2×D411yescharacteristic
190C2×C4⋊C4C2×D412yes—
191C2×C4⋊C4C2×D411yescharacteristic
192C2×C4⋊C4C2×D412yes—
193C2×C4⋊C4C2×D411yescharacteristic
194C2×C4⋊C4C2414yes—
195C2×C4⋊C4C2414yes—
196C2×C4⋊C4C2414yes—
197C2×C4⋊C4C2414yes—
198C2×C42—24no—
199C2×C42—24no—
200C2×C42—24no—
201C2×C42C2×D411yescharacteristic
202C2×C42C2×D412yes—
203C2×C42C2×D412yes—
204C2×C42C2×D411yescharacteristic
205C2×C42C2×D411yescharacteristic
206C2×C42C2×D412yes—
207C2×C42C2×D411yescharacteristic
208C2×C42C2×D412yes—
209C2×C42C2412yes—
210C2×C42C2411yescharacteristic
211C2×C42C2411yescharacteristic
212C2×C42C2412yes—
213C2×C42C2411yescharacteristic
214#107—28no—
215#107—44no—
216#107—44no—
217#107—48no—
218#107C2×Q8116yes—
219C4⋊C8—28no—
220C4⋊C8—42no—
221C4×C8—24no—
222C4×C8—22no—
223C4×C8—24no—
224C4×C8—42no—
225C4×C8C2×Q814yes—
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#95—24no—
235#95—24no—
236#95C2×D412yes—
237#95C2×D412yes—
238#95C2×D412yes—
239C2×Q8—28no—
240C2×Q8—41no—
241C2×Q8—41no—
242C2×Q8—48no—
243C2×Q8—44no—
244C2×Q8—44no—
245C2×Q8—84no—
246C2×Q8—84no—
247C2×Q8C2×Q1614yes—
248C22×C4—24no—
249C22×C4—24no—
250C22×C4—24no—
251C22×C4—24no—
252C22×C4C22×D411yescharacteristic
253C22×C4C22×D412yes—
254C22×C4C22×D412yes—
255C22×C4C22×D411yescharacteristic
256C22×C4C22×D411yescharacteristic
257C22×C4C22×D411yescharacteristic
258C22×C4C22×D412yes—
259C22×C4C22×D412yes—
260C22×C4C22×D411yescharacteristic
261C22×C4C22×D411yescharacteristic
262C22×C4C2511yescharacteristic
263C2×C8—22no—
264C2×C8—22no—
265C2×C8—22no—
266C2×C8—22no—
267C2×C8—24no—
268C2×C8—24no—
269C2×C8—24no—
270C2×C8C22×Q814yes—
271C2×C82−1+414yes—
272C4⋊C4—28no—
273C4⋊C4—24no—
274C4⋊C4—28no—
275C4⋊C4—28no—
276C4⋊C4—28no—
277C4⋊C4—42no—
278C4⋊C4—41no—
279C4⋊C4—42no—
280C4⋊C4—41no—
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⋊C4—84no—
294C4⋊C4—84no—
295C4⋊C4—84no—
296C4⋊C4—84no—
297C4⋊C4C2×Q1618yes—
298C4⋊C4C2×Q1614yes—
299C4⋊C4C22×Q818yes—
300C42—22no—
301C42—24no—
302C42—22no—
303C42—24no—
304C42—22no—
305C42—24no—
306C42—22no—
307C42—24no—
308C42—22no—
309C42—24no—
310C42—48no—
311C42—84no—
312C42—84no—
313C42C2×Q1614yes—
314C42C2×Q1618yes—
315C42C22×Q812yes—
316C422−1+412yes—
317C23C23×D411yescharacteristic
318Q8—44no—
319Q8—416no—
320Q8—168no—
321C2×C4—22no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—22no—
325C2×C4—22no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—22no—
329C2×C4—21no—
330C2×C4—22no—
331C2×C4—28no—
332C2×C4—21no—
333C2×C4—22no—
334C2×C4—22no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—22no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—22no—
341C2×C4—22no—
342C2×C4—44no—
343C2×C4—44no—
344C2×C4—44no—
345C2×C4—44no—
346C2×C4—44no—
347C2×C4—44no—
348C2×C4—44no—
349C2×C4—44no—
350C2×C4—44no—
351C2×C4—44no—
352C2×C4—44no—
353C2×C4D4×Q812yes—
354C2×C4D4×Q812yes—
355C2×C4C22×Q1612yes—
356C2×C4C22×Q1614yes—
357C2×C4C22×Q1612yes—
358C2×C4C22×Q1614yes—
359C2×C4C22×Q1612yes—
360C2×C4#57512yes—
361C2×C4#57612yes—
362C2×C4#57612yes—
363C2×C4C23×Q811yescharacteristic
364C2×C4C2×2−1+412yes—
365C2×C4C2×2−1+411yescharacteristic
366C2×C4#58512yes—
367C8—24no—
368C8—24no—
369C8—44no—
370C22#344911yescharacteristic
371C22C2×D4×Q811yescharacteristic
372C22#349911yescharacteristic
373C22C23×Q1611yescharacteristic
374C22C2×#57511yescharacteristic
375C22C2×#57611yescharacteristic
376C22C2×#57611yescharacteristic
377C4—22no—
378C4—24no—
379C4—24no—
380C4—22no—
381C4—42no—
382C4—41no—
383C4—41no—
384C4—42no—
385C4—48no—
386C4—84no—
387C4—84no—
388C4—84no—
389C4Q8×Q1614yes—
390C4#338914yes—
391C4#350212yes—
392C4#356312yes—
393C2#3961211yescharacteristic
394C2#6094311yescharacteristic
395C2#6101111yescharacteristic
396C2C2×Q8×Q1611yescharacteristic
397C2C2×#338911yescharacteristic
398C2#6247211yescharacteristic
399C2#6248111yescharacteristic
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
75. #500Q8semidirect

Maximal subgroups

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

subgroupindex
#365632
#332582
#326842
#325062
#303042
#292082
#291792
#268622
#222482
C4×#5002
#213392