Finite Group Databasebeta

← #6434606

Subgroups — #6434606

Order 512.

1498 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#36635C212yesmaximal
2#30382C214yesmaximal, complemented
3#30344C214yesmaximal, complemented
4#30293C211yesmaximal, characteristic
5#30242C211yesmaximal, characteristic
6#30187C212yesmaximal
7#29773C214yesmaximal
8#23782C214yesmaximal, complemented
9#23685C214yesmaximal, complemented
10#21527C214yesmaximal
11C2×#1767C211yesmaximal, characteristic
12C2×#467C2214yes—
13C2×#441C2214yes—
14C22×C4⋊C8C2212yes—
15C22×#97C2212yes—
16#2864—22no—
17#2863—22no—
18#2863C2214yes—
19#2862C2212yes—
20#2859C2212yes—
21#2857C2212yes—
22#2857C2212yes—
23#2855—22no—
24#2855C2214yes—
25#2851—22no—
26C4×C4⋊Q8C2212yes—
27C4×C4⋊Q8C2212yes—
28C4×#125—22no—
29C4×C22⋊Q8—22no—
30C4×C22⋊Q8C2214yes—
31#2295—22no—
32#2295C2214yes—
33#2283—22no—
34C2×C43C2211yescharacteristic
35#2100C2214yes—
36#2095C2214yes—
37#2086C2218yes—
38#2084C2214yes—
39#2072C2214yes—
40#2066C2218yes—
41#2066C2218yes—
42#2058C2214yes—
43#2056C2218yes—
44#2043C2214yes—
45#2024C2214yes—
46#2016C2214yes—
47#1971C2214yes—
48#1943C2214yes—
49#1935C2218yes—
50#1892C2218yes—
51#1807C2214yes—
52#1776C2214yes—
53#1767C2214yes—
54#1767C2214yes—
55#1763C2218yes—
56C2×#336C2211yescharacteristic
57C2×#336C2211yescharacteristic
58C22×Q16C2314yes—
59C23×C8C2312yes—
60C23×C8C2312yes—
61C2×C4⋊Q8C2312yes—
62C2×C4⋊Q8C2312yes—
63C2×#125—22no—
64C2×C22⋊Q8—22no—
65C2×C22⋊Q8C2314yes—
66C2×C4×Q8—22no—
67C2×C4×Q8C2314yes—
68C22×C42D412yes—
69C22×C42D412yes—
70C22×C42C2311yescharacteristic
71C22×C42C2311yescharacteristic
72C22×C42C2311yescharacteristic
73#467—216no—
74#441—48no—
75C2×#107C2314yes—
76C2×#106C2314yes—
77C2×#106C2314yes—
78C2×C4⋊C8—28no—
79C2×C4⋊C8C2318yes—
80C2×#103C2318yes—
81C2×#103C2318yes—
82C2×#103C2318yes—
83C2×#97—44no—
84C2×#97C2314yes—
85C2×#97C2314yes—
86#391—22no—
87#391—22no—
88#391—22no—
89#391—22no—
90#391C2314yes—
91#391C2314yes—
92#391C2314yes—
93#391C2314yes—
94#389—22no—
95#389—22no—
96#389—22no—
97#389—22no—
98#389—22no—
99#389—22no—
100#389—22no—
101#389—22no—
102#389C2314yes—
103#389C2314yes—
104#389C2314yes—
105#389C2314yes—
106#387—24no—
107#387—24no—
108#387—24no—
109#387—24no—
110#387—24no—
111#387—24no—
112#387—24no—
113#387—24no—
114#385—24no—
115#385—24no—
116#384C2312yes—
117#384C2312yes—
118#383—24no—
119#382—22no—
120#382—22no—
121#382—22no—
122#381—24no—
123#380—22no—
124#380—22no—
125#380—22no—
126#380C2314yes—
127C4×C4⋊C4—22no—
128C4×C4⋊C4—22no—
129C4×C4⋊C4—24no—
130C4×C4⋊C4—24no—
131C4×C4⋊C4—22no—
132C4×C4⋊C4C2314yes—
133C4×C22⋊C4—24no—
134#376—24no—
135C43D412yes—
136C43D412yes—
137C43C2312yes—
138C43C2312yes—
139#340C2318yes—
140#340C2318yes—
141#340C2318yes—
142#336C2314yes—
143#336C2314yes—
144C22×Q8—22no—
145C22×Q8C2414yes—
146C23×C4C2×D411yescharacteristic
147C23×C4C2×D411yescharacteristic
148C23×C4C2×D412yes—
149C23×C4C2×D412yes—
150C23×C4C2411yescharacteristic
151C2×Q16—28no—
152C2×Q16—44no—
153C2×Q16—44no—
154C22×C8—24no—
155C22×C8—42no—
156C22×C8—42no—
157C22×C8—42no—
158C22×C8—42no—
159C22×C8#4118yes—
160C22×C8C2414yes—
161C22×C8C2414yes—
162C4⋊Q8—44no—
163C4⋊Q8—44no—
164#125—48no—
165C22⋊Q8—216no—
166C22⋊Q8—48no—
167C4×Q8—28no—
168C4×Q8—28no—
169C4×Q8—48no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—24no—
172C2×C4⋊C4—24no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—24no—
175C2×C4⋊C4—24no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—24no—
179C2×C4⋊C4—24no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4C2414yes—
184C2×C4⋊C4C2414yes—
185C2×C4⋊C4C2414yes—
186C2×C22⋊C4—24no—
187C2×C22⋊C4—24no—
188C2×C42—28no—
189C2×C42—24no—
190C2×C42—22no—
191C2×C42—22no—
192C2×C42—24no—
193C2×C42—24no—
194C2×C42—44no—
195C2×C42—42no—
196C2×C42C2×D412yes—
197C2×C42C2×D412yes—
198C2×C42C2×D412yes—
199C2×C42C2×D412yes—
200C2×C42C2×D412yes—
201C2×C42C2×D412yes—
202C2×C42C2×D412yes—
203C2×C42C2×D412yes—
204C2×C42C2×D411yescharacteristic
205C2×C42C2×D411yescharacteristic
206C2×C42C2×D412yes—
207C2×C42C2×D412yes—
208C2×C42C2412yes—
209C2×C42C2411yescharacteristic
210C2×C42C2412yes—
211C2×C42C2411yescharacteristic
212#107—28no—
213#106—44no—
214#106—44no—
215C4⋊C8—216no—
216#103—216no—
217#103—48no—
218#103—48no—
219#97—28no—
220#97—44no—
221#95—24no—
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#95—24no—
235C24C22×D411yescharacteristic
236C2×Q8—28no—
237C2×Q8—24no—
238C2×Q8—24no—
239C2×Q8—28no—
240C2×Q8—44no—
241C2×Q8—44no—
242C2×Q8—44no—
243C22×C4—22no—
244C22×C4—24no—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—22no—
248C22×C4—22no—
249C22×C4—24no—
250C22×C4—24no—
251C22×C4—24no—
252C22×C4—24no—
253C22×C4—24no—
254C22×C4—24no—
255C22×C4—24no—
256C22×C4—42no—
257C22×C4—42no—
258C22×C4—42no—
259C22×C4#13718yes—
260C22×C4C22×D412yes—
261C22×C4C22×D412yes—
262C22×C4C22×D411yescharacteristic
263C22×C4C22×D412yes—
264C22×C4C22×D412yes—
265C22×C4C22×D412yes—
266C22×C4C22×D412yes—
267C22×C4C2×#4114yes—
268C22×C4C2511yescharacteristic
269Q16—416no—
270C2×C8—24no—
271C2×C8—24no—
272C2×C8—24no—
273C2×C8—24no—
274C2×C8—24no—
275C2×C8—48no—
276C2×C8—48no—
277C2×C8C2×#4118yes—
278C4⋊C4—28no—
279C4⋊C4—24no—
280C4⋊C4—24no—
281C4⋊C4—28no—
282C4⋊C4—44no—
283C4⋊C4—44no—
284C4⋊C4—44no—
285C4⋊C4—48no—
286C4⋊C4—44no—
287C4⋊C4—44no—
288C4⋊C4—44no—
289C4⋊C4—48no—
290C4⋊C4—48no—
291C4⋊C4—44no—
292C4⋊C4—48no—
293C4⋊C4—48no—
294C4⋊C4—48no—
295C22⋊C4—48no—
296C22⋊C4—48no—
297C42—24no—
298C42—24no—
299C42—24no—
300C42—24no—
301C42—24no—
302C42—24no—
303C42—24no—
304C42—24no—
305C42—24no—
306C42—21no—
307C42—21no—
308C42—24no—
309C42—41no—
310C42—41no—
311C42—42no—
312C42—42no—
313C42—48no—
314C42—48no—
315C42H(5,𝔽2)12yes—
316C422−1+412yes—
317C23—21no—
318C23—21no—
319C23—22no—
320C23#53212yes—
321C23C2×#13714yes—
322C23C23×D411yescharacteristic
323Q8—48no—
324Q8—48no—
325Q8—88no—
326C2×C4—28no—
327C2×C4—22no—
328C2×C4—21no—
329C2×C4—22no—
330C2×C4—22no—
331C2×C4—21no—
332C2×C4—22no—
333C2×C4—22no—
334C2×C4—22no—
335C2×C4—22no—
336C2×C4—22no—
337C2×C4—24no—
338C2×C4—24no—
339C2×C4—22no—
340C2×C4—22no—
341C2×C4—22no—
342C2×C4—22no—
343C2×C4—22no—
344C2×C4—22no—
345C2×C4—48no—
346C2×C4—48no—
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×C4—44no—
359C2×C4—44no—
360C2×C4#53212yes—
361C2×C4#53714yes—
362C2×C4C2×#13714yes—
363C2×C4C2×#13714yes—
364C2×C4#57514yes—
365C2×C4C22×#4112yes—
366C2×C4C2×H(5,𝔽2)11yescharacteristic
367C2×C4C2×2−1+411yescharacteristic
368C2×C4#58512yes—
369C8—28no—
370C8—44no—
371C22—22no—
372C22—24no—
373C22—22no—
374C22—24no—
375C22#318514yes—
376C22C2×#53211yescharacteristic
377C22C2×#53711yescharacteristic
378C22#344911yescharacteristic
379C22C22×#13712yes—
380C22C2×#57512yes—
381C4—24no—
382C4—24no—
383C4—24no—
384C4—22no—
385C4—24no—
386C4—42no—
387C4—42no—
388C4—84no—
389C4—84no—
390C4—84no—
391C4—84no—
392C4#345912yes—
393C4#346112yes—
394C2—24no—
395C2#3459411yescharacteristic
396C2#6068811yescharacteristic
397C2C2×#318511yescharacteristic
398C2#6091812yes—
399C2#6096012yes—
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. #30382C2semidirect
3. #30344C2semidirect
8. #23782C2semidirect
9. #23685C2semidirect

Maximal subgroups

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

subgroupindex
#366352
#303822
#303442
#302932
#302422
#301872
#297732
#237822
#236852
#215272
C2×#17672