Finite Group Databasebeta

← #496411

Subgroups — #496411

Order 512.

858 conjugacy classes of subgroups.

The subgroup lattice has 366 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35620C211yesmaximal, characteristic
2#20731C212yesmaximal
3#13303C214yesmaximal, complemented
4#13301C214yesmaximal, complemented
5#12863C212yesmaximal
6#12861C212yesmaximal
7C2×#544C411yescharacteristic
8C2×#520C2211yescharacteristic
9#2921—22no—
10#2568C2212yes—
11#2566C412yes—
12#2565C2212yes—
13#2563C412yes—
14#2450—22no—
15#2450—22no—
16#2442—22no—
17#2439—22no—
18#2317C411yescharacteristic
19#2311C2211yescharacteristic
20#2297C2211yescharacteristic
21#2296C411yescharacteristic
22C2×#382C411yescharacteristic
23#1821—22no—
24#1801C2214yes—
25#1800—22no—
26#1785C2214yes—
27C4×#101C2214yes—
28C4×#100—22no—
29#1394C2214yes—
30#1394C2214yes—
31#1394C2218yes—
32C22×#41C2311yescharacteristic
33#544—44no—
34C2×C4⋊Q8—21no—
35C2×C4⋊Q8C2×C412yes—
36C2×#125—21no—
37C2×#125C2×C412yes—
38C2×C22⋊Q8—22no—
39#520—44no—
40C2×C4×Q8C2312yes—
41C2×C4×D4C2×C412yes—
42C2×C4×D4D412yes—
43C2×C4×D4D412yes—
44C2×#117C2×C411yescharacteristic
45C2×#117C2×C411yescharacteristic
46C2×#117C2×C411yescharacteristic
47C2×#117D412yes—
48C22×C4⋊C4C2×C411yescharacteristic
49C22×C4⋊C4C2×C411yescharacteristic
50C22×C4⋊C4D412yes—
51C22×C42C2×C411yescharacteristic
52#433—48no—
53#423—24no—
54C2×#101C2318yes—
55C2×#100—24no—
56#407—24no—
57C4×M16—24no—
58#400—22no—
59#400—22no—
60#398—22no—
61#398—22no—
62#396—22no—
63#396—22no—
64#393—22no—
65#393—22no—
66#391—22no—
67#391—22no—
68#391C2311yescharacteristic
69#391C2312yes—
70#391C2311yescharacteristic
71#390C2×C411yescharacteristic
72#390C2×C412yes—
73#390C2×C411yescharacteristic
74#389—22no—
75#389—21no—
76#389—21no—
77#387—22no—
78#387—22no—
79#386—22no—
80#386—22no—
81#385—22no—
82#385—22no—
83#385—21no—
84#385—21no—
85#385—22no—
86#384C2×C412yes—
87#382—22no—
88#382C2×C412yes—
89#382C2×C412yes—
90#382C2×C412yes—
91#380—22no—
92#380—22no—
93#380—22no—
94C4×C4⋊C4C2×C412yes—
95C4×C4⋊C4C2×C412yes—
96C4×C22⋊C4—22no—
97C2×#95—21no—
98C2×#95—21no—
99#343—24no—
100#343—24no—
101C2×#41—24no—
102C2×#41—42no—
103C2×#41—42no—
104C22×Q8—22no—
105C22×Q8C2411yescharacteristic
106C22×D4C22×C411yescharacteristic
107C22×D4C2×D412yes—
108C23×C4—21no—
109C23×C4—21no—
110C23×C4C22⋊C411yescharacteristic
111C23×C4C22⋊C411yescharacteristic
112C23×C4C22⋊C412yes—
113C23×C4C22⋊C412yes—
114C23×C4C22×C411yescharacteristic
115C23×C4C2×D412yes—
116C2×M16—28no—
117C4⋊Q8—44no—
118C4⋊Q8—44no—
119#125—44no—
120#125—44no—
121C22⋊Q8—48no—
122C4×Q8—44no—
123C4×D4—44no—
124C4×D4—44no—
125C4×D4—44no—
126#117—24no—
127#117—28no—
128#117—42no—
129#117—42no—
130C2×C4⋊C4—21no—
131C2×C4⋊C4—21no—
132C2×C4⋊C4—22no—
133C2×C4⋊C4—22no—
134C2×C4⋊C4—22no—
135C2×C4⋊C4—22no—
136C2×C4⋊C4—22no—
137C2×C4⋊C4—22no—
138C2×C4⋊C4—22no—
139C2×C4⋊C4—42no—
140C2×C4⋊C4—44no—
141C2×C4⋊C4—42no—
142C2×C4⋊C4C22⋊C412yes—
143C2×C4⋊C4C22⋊C411yescharacteristic
144C2×C4⋊C4C22⋊C411yescharacteristic
145C2×C4⋊C4C22×C412yes—
146C2×C4⋊C4C22×C411yescharacteristic
147C2×C4⋊C4C22×C411yescharacteristic
148C2×C4⋊C4C22×C412yes—
149C2×C4⋊C4C2×D412yes—
150C2×C4⋊C4C2×D412yes—
151C2×C22⋊C4—21no—
152C2×C22⋊C4—21no—
153C2×C22⋊C4—22no—
154C2×C22⋊C4—22no—
155C2×C22⋊C4—22no—
156C2×C22⋊C4—22no—
157C2×C22⋊C4C22⋊C412yes—
158C2×C22⋊C4C22⋊C412yes—
159C2×C22⋊C4C22⋊C412yes—
160C2×C42—21no—
161C2×C42—21no—
162C2×C42—42no—
163C2×C42C22×C412yes—
164C2×C42C22×C412yes—
165C2×C42C22×C411yescharacteristic
166C2×C42C22×C411yescharacteristic
167C2×C42C2×D412yes—
168C2×C42C2×D412yes—
169C4⋊C8—48no—
170#101—216no—
171#100—48no—
172C22⋊C8—48no—
173#97—44no—
174C4×C8—44no—
175#95—22no—
176#95—22no—
177#95—22no—
178#95—22no—
179#95—22no—
180#95—22no—
181#95—22no—
182#95—22no—
183#95—22no—
184#95—22no—
185#95—22no—
186#95—22no—
187C24C2×C22⋊C412yes—
188C24C2×C22⋊C411yescharacteristic
189#41—416no—
190C2×Q8—41no—
191C2×Q8—41no—
192C2×Q8—44no—
193C2×Q8—82no—
194C2×Q8—82no—
195C2×Q82−1+414yes—
196C2×D4—24no—
197C2×D4—22no—
198C2×D4—24no—
199C2×D4—41no—
200C2×D4—42no—
201C2×D4—41no—
202C22×C4—22no—
203C22×C4—22no—
204C22×C4—22no—
205C22×C4—24no—
206C22×C4—24no—
207C22×C4—21no—
208C22×C4—21no—
209C22×C4—21no—
210C22×C4—22no—
211C22×C4—22no—
212C22×C4—21no—
213C22×C4—42no—
214C22×C4—42no—
215C22×C4—42no—
216C22×C4—41no—
217C22×C4—41no—
218C22×C4—42no—
219C22×C4—42no—
220C22×C4—81no—
221C22×C4—81no—
222C22×C4—81no—
223C22×C4—81no—
224C22×C4—81no—
225C22×C4—81no—
226C22×C4—82no—
227C22×C4—82no—
228C22×C4C2×C22⋊C411yescharacteristic
229C22×C4C2×C22⋊C412yes—
230C22×C4C2×C22⋊C412yes—
231C22×C4C2×C22⋊C412yes—
232C22×C4C2×C22⋊C412yes—
233C22×C4C23×C411yescharacteristic
234C22×C4C22×D412yes—
235M16—48no—
236M16—48no—
237C2×C8—48no—
238C4⋊C4—24no—
239C4⋊C4—22no—
240C4⋊C4—42no—
241C4⋊C4—42no—
242C4⋊C4—42no—
243C4⋊C4—42no—
244C4⋊C4—41no—
245C4⋊C4—41no—
246C4⋊C4—42no—
247C4⋊C4—44no—
248C4⋊C4—44no—
249C4⋊C4—42no—
250C4⋊C4—44no—
251C4⋊C4—82no—
252C4⋊C4—82no—
253C4⋊C4—82no—
254C4⋊C4—82no—
255C4⋊C4#9914yes—
256C22⋊C4—24no—
257C22⋊C4—24no—
258C22⋊C4—24no—
259C22⋊C4—42no—
260C22⋊C4—42no—
261C22⋊C4—82no—
262C22⋊C4—82no—
263C22⋊C4—82no—
264C22⋊C4—82no—
265C42—24no—
266C42—24no—
267C42—24no—
268C42—41no—
269C42—41no—
270C42—42no—
271C42—42no—
272C42—42no—
273C23—21no—
274C23—21no—
275C23—22no—
276C23—22no—
277C23—22no—
278C23—41no—
279C23—41no—
280C23—42no—
281C23—42no—
282C23#41012yes—
283C23C22×C22⋊C411yescharacteristic
284Q8—44no—
285Q8—164no—
286D4—48no—
287D4—44no—
288C2×C4—28no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—22no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—22no—
295C2×C4—22no—
296C2×C4—22no—
297C2×C4—22no—
298C2×C4—42no—
299C2×C4—48no—
300C2×C4—42no—
301C2×C4—42no—
302C2×C4—42no—
303C2×C4—42no—
304C2×C4—41no—
305C2×C4—41no—
306C2×C4—42no—
307C2×C4—42no—
308C2×C4—42no—
309C2×C4—42no—
310C2×C4—42no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—42no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—42no—
319C2×C4—84no—
320C2×C4—161no—
321C2×C4—161no—
322C2×C4—162no—
323C2×C4—162no—
324C2×C4—162no—
325C2×C4C2×#9914yes—
326C2×C4C2×#9912yes—
327C2×C4#51912yes—
328C2×C4#53614yes—
329C8—88no—
330C22—21no—
331C22—21no—
332C22—22no—
333C22—42no—
334C22—42no—
335C22—44no—
336C22—41no—
337C22—41no—
338C22—42no—
339C22—42no—
340C22—42no—
341C22—82no—
342C22#231011yescharacteristic
343C22C22×#9911yescharacteristic
344C22C2×#41011yescharacteristic
345C4—24no—
346C4—22no—
347C4—22no—
348C4—24no—
349C4—42no—
350C4—42no—
351C4—42no—
352C4—82no—
353C4—81no—
354C4—81no—
355C4—82no—
356C4—81no—
357C4—81no—
358C2—21no—
359C2—21no—
360C2—42no—
361C2—82no—
362C2#2182511yescharacteristic
363C2#3337011yescharacteristic
364C2#3337111yescharacteristic
subgroupG/H
center, socleC22C22×#99
commutatorC22×C4C23×C4
Frattini, cosocleC22×Q8C24
prime radicalC22×#41C23
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C4⋊C4 ◁ C2×#520 ◁ G
LECSG ▷ C22×Q8 ▷ C22 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #13303C2semidirect
4. #13301C2semidirect

Maximal subgroups

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

subgroupindex
#356202
#207312
#133032
#133012
#128632
#128612