Finite Group Databasebeta

← #160748

Subgroups — #160748

Order 512.

1054 conjugacy classes of subgroups.

The subgroup lattice has 381 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34026C211yesmaximal, characteristic
2C4×#432C212yesmaximal, complemented
3#7630C214yesmaximal, complemented
4#7628C214yesmaximal, complemented
5#7564C212yesmaximal, complemented
6#7563C212yesmaximal
7C2×C4×#41C411yescharacteristic
8C2×#432—22no—
9#2298—22no—
10#2297C411yescharacteristic
11#2297C2211yescharacteristic
12#2296C411yescharacteristic
13#2296C2211yescharacteristic
14#2295C412yes—
15#2295C2212yes—
16#2293C412yes—
17#2293C2212yes—
18#2290—22no—
19#2290—22no—
20C4×#117—22no—
21C4×#117C411yescharacteristic
22C2×C43C2211yescharacteristic
23#1767C2214yes—
24C4×C4⋊C8—22no—
25#1749—22no—
26C4×C22⋊C8C2214yes—
27#1745C2214yes—
28C4×#97—22no—
29#1272C2214yes—
30#1271C2218yes—
31#1270C2214yes—
32C22×#41C2×C411yescharacteristic
33C4×#41—24no—
34C4×#41—24no—
35C2×C4×Q8C2×C412yes—
36C2×C4×D4—22no—
37C2×C4×D4C2×C412yes—
38C2×#117—21no—
39C2×#117—22no—
40C2×#117—22no—
41C2×#117C2×C411yescharacteristic
42C2×#117C2×C411yescharacteristic
43C2×#117C2×C411yescharacteristic
44C22×C42—21no—
45C22×C42C2×C411yescharacteristic
46C22×C42C2×C411yescharacteristic
47C22×C42D412yes—
48C22×C42D412yes—
49C22×C42C2311yescharacteristic
50#432—216no—
51C2×C4⋊C8—24no—
52C2×C22⋊C8—24no—
53C2×#97—24no—
54C2×#97—24no—
55C2×C4×C8—24no—
56#391C2×C411yescharacteristic
57#391C2×C412yes—
58#391C2311yescharacteristic
59#390C2×C411yescharacteristic
60#390C2×C412yes—
61#390C2311yescharacteristic
62#389C2×C412yes—
63#389C2×C412yes—
64#389C2×C412yes—
65#389C2312yes—
66#388—22no—
67#388—22no—
68#388—22no—
69#388—22no—
70#387—22no—
71#387—22no—
72#387—22no—
73#387—22no—
74#386—21no—
75#386—21no—
76#386—22no—
77#385—21no—
78#385—21no—
79#385—22no—
80C4×C4⋊C4—22no—
81C4×C4⋊C4—22no—
82C4×C4⋊C4—22no—
83C4×C4⋊C4—22no—
84C4×C4⋊C4C2×C412yes—
85C4×C4⋊C4C2×C412yes—
86C4×C22⋊C4—22no—
87C4×C22⋊C4—22no—
88C4×C22⋊C4—22no—
89C4×C22⋊C4—22no—
90C4×C22⋊C4—22no—
91#376—22no—
92#376—21no—
93#376—22no—
94C43—21no—
95C43C2×C412yes—
96C43D412yes—
97C43D412yes—
98C43C2312yes—
99#336—24no—
100#336C2318yes—
101C2×#41—22no—
102C2×#41—42no—
103C2×#41—42no—
104C2×#41M1614yes—
105C22×Q8C22×C411yescharacteristic
106C22×D4—21no—
107C22×D4C22×C411yescharacteristic
108C23×C4—22no—
109C23×C4—21no—
110C23×C4C22⋊C412yes—
111C23×C4C22⋊C411yescharacteristic
112C23×C4C22⋊C411yescharacteristic
113C23×C4C22×C411yescharacteristic
114C23×C4C2×D412yes—
115C22×C8—28no—
116C4×Q8—24no—
117C4×Q8—24no—
118C4×D4—216no—
119C4×D4—24no—
120C4×D4—24no—
121#117—28no—
122#117—22no—
123#117—48no—
124#117—48no—
125#117—42no—
126#117—42no—
127#117M1614yes—
128C2×C4⋊C4—22no—
129C2×C4⋊C4—22no—
130C2×C4⋊C4—22no—
131C2×C4⋊C4—22no—
132C2×C4⋊C4—22no—
133C2×C4⋊C4C22×C412yes—
134C2×C4⋊C4C22×C411yescharacteristic
135C2×C4⋊C4C22×C411yescharacteristic
136C2×C4⋊C4C22×C412yes—
137C2×C22⋊C4—21no—
138C2×C22⋊C4—22no—
139C2×C22⋊C4—21no—
140C2×C22⋊C4—21no—
141C2×C22⋊C4—22no—
142C2×C22⋊C4—22no—
143C2×C22⋊C4—22no—
144C2×C22⋊C4—22no—
145C2×C42—28no—
146C2×C42—22no—
147C2×C42—21no—
148C2×C42—22no—
149C2×C42—22no—
150C2×C42—28no—
151C2×C42—22no—
152C2×C42—22no—
153C2×C42—21no—
154C2×C42—21no—
155C2×C42—42no—
156C2×C42—44no—
157C2×C42—42no—
158C2×C42C22⋊C412yes—
159C2×C42C22⋊C411yescharacteristic
160C2×C42C22⋊C411yescharacteristic
161C2×C42C22⋊C412yes—
162C2×C42C22⋊C412yes—
163C2×C42C22⋊C412yes—
164C2×C42C22⋊C412yes—
165C2×C42C22×C412yes—
166C2×C42C22×C411yescharacteristic
167C2×C42C22×C412yes—
168C2×C42C2×D412yes—
169C2×C42C2×D412yes—
170C2×C42C2×D412yes—
171C2×C42C2×D412yes—
172C2×C42C2×D412yes—
173C2×C42C2411yescharacteristic
174C4⋊C8—216no—
175C22⋊C8—216no—
176#97—216no—
177#97—48no—
178C4×C8—48no—
179#95—22no—
180#95—22no—
181#95—22no—
182#95—22no—
183#95—22no—
184#95—22no—
185#95—22no—
186#95—22no—
187#95—22no—
188#95—22no—
189C24—21no—
190C24C2×C22⋊C411yescharacteristic
191#41—48no—
192#41—48no—
193C2×Q8—21no—
194C2×Q8—41no—
195C2×Q8—41no—
196C2×Q8C2×M1612yes—
197C2×D4—24no—
198C2×D4—21no—
199C2×D4—41no—
200C2×D4—41no—
201C2×D4—42no—
202C2×D4—42no—
203C2×D4C2×M1612yes—
204C22×C4—22no—
205C22×C4—22no—
206C22×C4—28no—
207C22×C4—21no—
208C22×C4—22no—
209C22×C4—24no—
210C22×C4—24no—
211C22×C4—24no—
212C22×C4—24no—
213C22×C4—24no—
214C22×C4—24no—
215C22×C4—22no—
216C22×C4—21no—
217C22×C4—21no—
218C22×C4—41no—
219C22×C4—42no—
220C22×C4—41no—
221C22×C4—42no—
222C22×C4—42no—
223C22×C4—44no—
224C22×C4—42no—
225C22×C4—42no—
226C22×C4—44no—
227C22×C4C2×C22⋊C412yes—
228C22×C4C2×C22⋊C412yes—
229C22×C4C2×C22⋊C412yes—
230C22×C4C2×C22⋊C412yes—
231C22×C4C2×C22⋊C412yes—
232C22×C4C2×C22⋊C411yescharacteristic
233C22×C4C2×M1612yes—
234C22×C4C2×M1612yes—
235C22×C4C23×C411yescharacteristic
236C22×C4C22×D412yes—
237C2×C8—216no—
238C2×C8—48no—
239C2×C8—48no—
240C4⋊C4—28no—
241C4⋊C4—22no—
242C4⋊C4—42no—
243C4⋊C4—41no—
244C4⋊C4—41no—
245C4⋊C4—44no—
246C4⋊C4—44no—
247C4⋊C4—44no—
248C4⋊C4—44no—
249C4⋊C4C2×M1614yes—
250C22⋊C4—28no—
251C22⋊C4—24no—
252C22⋊C4—44no—
253C22⋊C4—42no—
254C22⋊C4—42no—
255C22⋊C4—44no—
256C22⋊C4—44no—
257C22⋊C4—44no—
258C42—28no—
259C42—22no—
260C42—22no—
261C42—24no—
262C42—24no—
263C42—22no—
264C42—24no—
265C42—22no—
266C42—24no—
267C42—24no—
268C42—24no—
269C42—24no—
270C42—24no—
271C42—42no—
272C42—42no—
273C42—41no—
274C42—42no—
275C42—41no—
276C42—44no—
277C42—42no—
278C42—42no—
279C42—44no—
280C42#10418yes—
281C23—21no—
282C23—21no—
283C23—22no—
284C23—22no—
285C23—22no—
286C23—41no—
287C23—41no—
288C23—42no—
289C23—42no—
290C23#40712yes—
291C23C22×C22⋊C411yescharacteristic
292Q8—42no—
293Q8—42no—
294D4—42no—
295D4—42no—
296D4—48no—
297C2×C4—22no—
298C2×C4—22no—
299C2×C4—22no—
300C2×C4—21no—
301C2×C4—22no—
302C2×C4—21no—
303C2×C4—22no—
304C2×C4—22no—
305C2×C4—22no—
306C2×C4—22no—
307C2×C4—24no—
308C2×C4—24no—
309C2×C4—22no—
310C2×C4—24no—
311C2×C4—22no—
312C2×C4—22no—
313C2×C4—22no—
314C2×C4—22no—
315C2×C4—22no—
316C2×C4—22no—
317C2×C4—48no—
318C2×C4—44no—
319C2×C4—48no—
320C2×C4—416no—
321C2×C4—44no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—42no—
325C2×C4—42no—
326C2×C4—48no—
327C2×C4—42no—
328C2×C4—41no—
329C2×C4—41no—
330C2×C4—42no—
331C2×C4—42no—
332C2×C4—42no—
333C2×C4—42no—
334C2×C4#40712yes—
335C2×C4#41012yes—
336C2×C4#41312yes—
337C2×C4C2×#10414yes—
338C2×C4C2×#10414yes—
339C2×C4C2×#10414yes—
340C2×C4C22×M1611yescharacteristic
341C2×C4C22×M1611yescharacteristic
342C2×C4#57514yes—
343C8—416no—
344C22—24no—
345C22—22no—
346C22—41no—
347C22—42no—
348C22—41no—
349C22—44no—
350C22—42no—
351C22—81no—
352C22—82no—
353C22—81no—
354C22C2×#40711yescharacteristic
355C22C2×#41011yescharacteristic
356C22C2×#41311yescharacteristic
357C22#289612yes—
358C22C22×#10412yes—
359C4—24no—
360C4—24no—
361C4—24no—
362C4—21no—
363C4—21no—
364C4—24no—
365C4—42no—
366C4—44no—
367C4—42no—
368C4—42no—
369C4—81no—
370C4—82no—
371C4—81no—
372C4#146414yes—
373C2—42no—
374C2—82no—
375C2C2×#146411yescharacteristic
376C2#1744712yes—
377C2#1754312yes—
378C2#1763811yescharacteristic
379C2#1774011yescharacteristic
subgroupG/H
centerC22×C4C2×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC23C22×C22⋊C4
prime radicalC22×#41C2×C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C4 ◁ C2×C43 ◁ G
LECSG ▷ C2×C42 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C4×#432C2semidirect
3. #7630C2semidirect
4. #7628C2semidirect
5. #7564C2semidirect

Maximal subgroups

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

subgroupindex
#340262
C4×#4322
#76302
#76282
#75642
#75632