Finite Group Databasebeta

← #163269

Subgroups — #163269

Order 512.

863 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36641C211yesmaximal, characteristic
2#20752C212yesmaximal, complemented
3#7733C214yesmaximal, complemented
4#7732C214yesmaximal, complemented
5#7632C212yesmaximal, complemented
6#7631C212yesmaximal
7C2×#423—22no—
8#2863C412yes—
9#2863C2212yes—
10#2860C412yes—
11#2860C2212yes—
12#2852—22no—
13#2852—22no—
14C4×#126—22no—
15C4×#126—22no—
16#2297C411yescharacteristic
17#2297C411yescharacteristic
18#2297C2211yescharacteristic
19#2296C411yescharacteristic
20#2296C411yescharacteristic
21#2296C2211yescharacteristic
22C2×C43C2211yescharacteristic
23#1831—22no—
24#1831—22no—
25#1831C2214yes—
26#1831C2214yes—
27#1728—22no—
28#1728C2214yes—
29#1272C2214yes—
30#1271C2218yes—
31#1270C2214yes—
32C22×#41C2×C411yescharacteristic
33C22×M16—24no—
34C2×#126—22no—
35C2×#126—22no—
36C2×#117C2×C411yescharacteristic
37C2×#117C2×C411yescharacteristic
38C2×#117C2×C411yescharacteristic
39C22×C42C2×C411yescharacteristic
40C22×C42C2×C411yescharacteristic
41C22×C42D412yes—
42C22×C42D412yes—
43C22×C42C2311yescharacteristic
44#423—48no—
45C2×C4⋊C8—24no—
46C2×C4⋊C8—24no—
47#391C2×C411yescharacteristic
48#391C2×C412yes—
49#391C2×C412yes—
50#391C2311yescharacteristic
51#390C2×C411yescharacteristic
52#390C2×C412yes—
53#390C2×C412yes—
54#390C2311yescharacteristic
55#389C2×C412yes—
56#389C2×C412yes—
57#389C2×C412yes—
58#389C2×C412yes—
59#389C2×C412yes—
60#389C2312yes—
61#388—22no—
62#388—22no—
63#388—22no—
64#388—22no—
65#388—22no—
66#388—22no—
67#387—22no—
68#387—22no—
69#387—22no—
70#387—22no—
71#387—22no—
72#387—22no—
73#386—21no—
74#386—22no—
75#386—21no—
76#386—22no—
77#385—21no—
78#385—22no—
79#385—21no—
80#385—22no—
81#383—22no—
82#383—22no—
83C4×C4⋊C4—22no—
84C4×C4⋊C4—21no—
85C4×C4⋊C4—21no—
86C4×C4⋊C4—22no—
87C4×C22⋊C4—22no—
88C4×C22⋊C4—21no—
89C4×C22⋊C4—21no—
90C4×C22⋊C4—22no—
91C43C2×C412yes—
92C43D412yes—
93C43D412yes—
94C43C2312yes—
95#336—24no—
96#336—24no—
97#336—24no—
98#336C2318yes—
99C2×#41—42no—
100C2×#41—42no—
101C2×#41—42no—
102C22×Q8C22×C411yescharacteristic
103C22×D4—21no—
104C22×D4C22×C411yescharacteristic
105C23×C4—21no—
106C23×C4C22⋊C411yescharacteristic
107C23×C4C22⋊C412yes—
108C23×C4C22⋊C411yescharacteristic
109C23×C4C22×C411yescharacteristic
110C23×C4C2×D412yes—
111C2×M16—48no—
112C2×M16—48no—
113C2×M16—48no—
114C22×C8—28no—
115#126—48no—
116#126—44no—
117#126—44no—
118#117—42no—
119#117—42no—
120#117—42no—
121C2×C4⋊C4—22no—
122C2×C4⋊C4—22no—
123C2×C4⋊C4—22no—
124C2×C4⋊C4—22no—
125C2×C4⋊C4—22no—
126C2×C4⋊C4—22no—
127C2×C4⋊C4C22×C412yes—
128C2×C4⋊C4C22×C411yescharacteristic
129C2×C4⋊C4C22×C411yescharacteristic
130C2×C4⋊C4C22×C412yes—
131C2×C22⋊C4—21no—
132C2×C22⋊C4—21no—
133C2×C22⋊C4—21no—
134C2×C22⋊C4—22no—
135C2×C22⋊C4—22no—
136C2×C22⋊C4—22no—
137C2×C22⋊C4—22no—
138C2×C22⋊C4—22no—
139C2×C22⋊C4—22no—
140C2×C42—24no—
141C2×C42—24no—
142C2×C42—21no—
143C2×C42—21no—
144C2×C42—21no—
145C2×C42—42no—
146C2×C42—42no—
147C2×C42—44no—
148C2×C42—42no—
149C2×C42C22⋊C412yes—
150C2×C42C22⋊C412yes—
151C2×C42C22⋊C412yes—
152C2×C42C22⋊C412yes—
153C2×C42C22⋊C412yes—
154C2×C42C22⋊C411yescharacteristic
155C2×C42C22⋊C411yescharacteristic
156C2×C42C22×C412yes—
157C2×C42C22×C411yescharacteristic
158C2×C42C22×C412yes—
159C2×C42C2×D412yes—
160C2×C42C2×D412yes—
161C2×C42C2×D412yes—
162C2×C42C2×D412yes—
163C2×C42C2×D412yes—
164C2×C42C2411yescharacteristic
165C4⋊C8—48no—
166C4⋊C8—48no—
167#95—22no—
168#95—22no—
169#95—22no—
170#95—22no—
171#95—22no—
172#95—22no—
173#95—22no—
174#95—22no—
175#95—22no—
176#95—22no—
177#95—22no—
178#95—22no—
179C24—21no—
180C24C2×C22⋊C411yescharacteristic
181#41—88no—
182C2×Q8—22no—
183C2×Q8—41no—
184C2×Q8—41no—
185C2×D4—22no—
186C2×D4—41no—
187C2×D4—41no—
188C2×D4—42no—
189C2×D4—42no—
190C2×D4—42no—
191C22×C4—22no—
192C22×C4—24no—
193C22×C4—22no—
194C22×C4—22no—
195C22×C4—22no—
196C22×C4—22no—
197C22×C4—24no—
198C22×C4—24no—
199C22×C4—24no—
200C22×C4—21no—
201C22×C4—22no—
202C22×C4—22no—
203C22×C4—21no—
204C22×C4—21no—
205C22×C4—42no—
206C22×C4—41no—
207C22×C4—42no—
208C22×C4—41no—
209C22×C4—42no—
210C22×C4—42no—
211C22×C4—42no—
212C22×C4—42no—
213C22×C4C2×C22⋊C412yes—
214C22×C4C2×C22⋊C412yes—
215C22×C4C2×C22⋊C411yescharacteristic
216C22×C4C2×C22⋊C412yes—
217C22×C4C2×C22⋊C412yes—
218C22×C4C2×C22⋊C412yes—
219C22×C4C23×C411yescharacteristic
220C22×C4C22×D412yes—
221M16—816no—
222C2×C8—48no—
223C2×C8—48no—
224C2×C8—48no—
225C4⋊C4—24no—
226C4⋊C4—42no—
227C4⋊C4—41no—
228C4⋊C4—41no—
229C4⋊C4—44no—
230C4⋊C4—44no—
231C4⋊C4—42no—
232C4⋊C4—42no—
233C4⋊C4—44no—
234C4⋊C4—44no—
235C4⋊C4—44no—
236C22⋊C4—44no—
237C22⋊C4—44no—
238C22⋊C4—42no—
239C22⋊C4—42no—
240C22⋊C4—42no—
241C22⋊C4—42no—
242C22⋊C4—42no—
243C22⋊C4—44no—
244C22⋊C4—44no—
245C22⋊C4—44no—
246C42—24no—
247C42—22no—
248C42—22no—
249C42—24no—
250C42—24no—
251C42—24no—
252C42—24no—
253C42—24no—
254C42—41no—
255C42—41no—
256C42—41no—
257C42—42no—
258C42—42no—
259C42—42no—
260C42—42no—
261C42—41no—
262C42—42no—
263C42—42no—
264C42—42no—
265C42AGL(1,ℤ/8)14yes—
266C42#13714yes—
267C23—21no—
268C23—21no—
269C23—22no—
270C23—22no—
271C23—41no—
272C23—41no—
273C23—41no—
274C23—42no—
275C23—42no—
276C23#41012yes—
277C23C22×C22⋊C411yescharacteristic
278Q8—82no—
279D4—82no—
280D4—84no—
281C2×C4—28no—
282C2×C4—28no—
283C2×C4—28no—
284C2×C4—22no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—22no—
289C2×C4—22no—
290C2×C4—22no—
291C2×C4—21no—
292C2×C4—22no—
293C2×C4—22no—
294C2×C4—22no—
295C2×C4—22no—
296C2×C4—22no—
297C2×C4—22no—
298C2×C4—22no—
299C2×C4—22no—
300C2×C4—22no—
301C2×C4—22no—
302C2×C4—44no—
303C2×C4—48no—
304C2×C4—44no—
305C2×C4—42no—
306C2×C4—41no—
307C2×C4—42no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—42no—
313C2×C4—42no—
314C2×C4—42no—
315C2×C4—42no—
316C2×C4—42no—
317C2×C4—42no—
318C2×C4—84no—
319C2×C4#41312yes—
320C2×C4#41714yes—
321C2×C4#41714yes—
322C2×C4#42114yes—
323C2×C4#56812yes—
324C2×C4C2×AGL(1,ℤ/8)12yes—
325C2×C4C2×#13712yes—
326C8—88no—
327C22—24no—
328C22—22no—
329C22—41no—
330C22—41no—
331C22—42no—
332C22—81no—
333C22—81no—
334C22—82no—
335C22—82no—
336C22—81no—
337C22#287911yescharacteristic
338C22C2×#41011yescharacteristic
339C22C2×#41311yescharacteristic
340C22C2×#41712yes—
341C22C2×#42112yes—
342C4—24no—
343C4—24no—
344C4—22no—
345C4—22no—
346C4—24no—
347C4—22no—
348C4—22no—
349C4—42no—
350C4—42no—
351C4—81no—
352C4—81no—
353C4—82no—
354C4—82no—
355C4—81no—
356C2—42no—
357C2—161no—
358C2#1739811yescharacteristic
359C2#1749812yes—
360C2#1758812yes—
361C2#1766511yescharacteristic
362C2#1775211yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC22×#41C2×C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C2×C43 ◁ G
LECSG ▷ C2×C42 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #20752C2semidirect
3. #7733C2semidirect
4. #7732C2semidirect
5. #7632C2semidirect

Maximal subgroups

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

subgroupindex
#366412
#207522
#77332
#77322
#76322
#76312