Finite Group Databasebeta

← #163268

Subgroups — #163268

Order 512.

815 conjugacy classes of subgroups.

The subgroup lattice has 351 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36641C211yesmaximal, characteristic
2#20772C212yesmaximal, complemented
3#7729C214yesmaximal, complemented
4#7726C214yesmaximal, complemented
5#7632C212yesmaximal, complemented
6#7631C212yesmaximal
7#2863C412yes—
8#2863C2212yes—
9#2860C412yes—
10#2860C2212yes—
11#2852—22no—
12#2852—22no—
13C4×#126—22no—
14C4×#126—22no—
15#2297C411yescharacteristic
16#2297C411yescharacteristic
17#2297C2211yescharacteristic
18#2296C411yescharacteristic
19#2296C411yescharacteristic
20#2296C2211yescharacteristic
21C2×C43C2211yescharacteristic
22#1839—22no—
23#1839—22no—
24#1839C2214yes—
25#1839C2214yes—
26#1728—22no—
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#391C2×C411yescharacteristic
45#391C2×C412yes—
46#391C2×C412yes—
47#391C2311yescharacteristic
48#390C2×C411yescharacteristic
49#390C2×C412yes—
50#390C2×C412yes—
51#390C2311yescharacteristic
52#389C2×C412yes—
53#389C2×C412yes—
54#389C2×C412yes—
55#389C2×C412yes—
56#389C2×C412yes—
57#389C2312yes—
58#388—22no—
59#388—22no—
60#388—22no—
61#388—22no—
62#388—22no—
63#388—22no—
64#387—22no—
65#387—22no—
66#387—22no—
67#387—22no—
68#387—22no—
69#387—22no—
70#386—21no—
71#386—22no—
72#386—21no—
73#386—22no—
74#385—21no—
75#385—22no—
76#385—21no—
77#385—22no—
78#383—22no—
79#383—22no—
80C4×C4⋊C4—22no—
81C4×C4⋊C4—21no—
82C4×C4⋊C4—21no—
83C4×C4⋊C4—22no—
84C4×C22⋊C4—22no—
85C4×C22⋊C4—21no—
86C4×C22⋊C4—21no—
87C4×C22⋊C4—22no—
88C43C2×C412yes—
89C43D412yes—
90C43D412yes—
91C43C2312yes—
92#336—24no—
93#336—24no—
94#336—24no—
95#336—24no—
96#336—24no—
97#336C2318yes—
98C2×#41—42no—
99C2×#41—42no—
100C2×#41—42no—
101C22×Q8C22×C411yescharacteristic
102C22×D4—21no—
103C22×D4C22×C411yescharacteristic
104C23×C4—21no—
105C23×C4C22⋊C412yes—
106C23×C4C22⋊C411yescharacteristic
107C23×C4C22⋊C411yescharacteristic
108C23×C4C22×C411yescharacteristic
109C23×C4C2×D412yes—
110C2×M16—48no—
111C2×M16—48no—
112C2×M16—48no—
113C22×C8—28no—
114#126—48no—
115#126—48no—
116#117—42no—
117#117—42no—
118#117—42no—
119C2×C4⋊C4—22no—
120C2×C4⋊C4—22no—
121C2×C4⋊C4—22no—
122C2×C4⋊C4—22no—
123C2×C4⋊C4—22no—
124C2×C4⋊C4—22no—
125C2×C4⋊C4C22×C412yes—
126C2×C4⋊C4C22×C411yescharacteristic
127C2×C4⋊C4C22×C411yescharacteristic
128C2×C4⋊C4C22×C412yes—
129C2×C22⋊C4—21no—
130C2×C22⋊C4—21no—
131C2×C22⋊C4—21no—
132C2×C22⋊C4—22no—
133C2×C22⋊C4—22no—
134C2×C22⋊C4—22no—
135C2×C22⋊C4—22no—
136C2×C22⋊C4—22no—
137C2×C22⋊C4—22no—
138C2×C42—21no—
139C2×C42—21no—
140C2×C42—21no—
141C2×C42—42no—
142C2×C42—42no—
143C2×C42—44no—
144C2×C42—44no—
145C2×C42—42no—
146C2×C42C22⋊C412yes—
147C2×C42C22⋊C412yes—
148C2×C42C22⋊C412yes—
149C2×C42C22⋊C412yes—
150C2×C42C22⋊C412yes—
151C2×C42C22⋊C411yescharacteristic
152C2×C42C22⋊C411yescharacteristic
153C2×C42C22×C412yes—
154C2×C42C22×C411yescharacteristic
155C2×C42C22×C412yes—
156C2×C42C2×D412yes—
157C2×C42C2×D412yes—
158C2×C42C2×D412yes—
159C2×C42C2×D412yes—
160C2×C42C2×D412yes—
161C2×C42C2411yescharacteristic
162#95—22no—
163#95—22no—
164#95—22no—
165#95—22no—
166#95—22no—
167#95—22no—
168#95—22no—
169#95—22no—
170#95—22no—
171#95—22no—
172#95—22no—
173#95—22no—
174C24—21no—
175C24C2×C22⋊C411yescharacteristic
176#41—88no—
177C2×Q8—22no—
178C2×Q8—41no—
179C2×Q8—41no—
180C2×D4—22no—
181C2×D4—41no—
182C2×D4—41no—
183C2×D4—42no—
184C2×D4—42no—
185C2×D4—42no—
186C22×C4—22no—
187C22×C4—24no—
188C22×C4—22no—
189C22×C4—24no—
190C22×C4—22no—
191C22×C4—24no—
192C22×C4—21no—
193C22×C4—22no—
194C22×C4—22no—
195C22×C4—21no—
196C22×C4—21no—
197C22×C4—42no—
198C22×C4—41no—
199C22×C4—41no—
200C22×C4—42no—
201C22×C4—42no—
202C22×C4—42no—
203C22×C4—42no—
204C22×C4—42no—
205C22×C4—42no—
206C22×C4—42no—
207C22×C4C2×C22⋊C412yes—
208C22×C4C2×C22⋊C412yes—
209C22×C4C2×C22⋊C412yes—
210C22×C4C2×C22⋊C412yes—
211C22×C4C2×C22⋊C411yescharacteristic
212C22×C4C2×C22⋊C412yes—
213C22×C4C23×C411yescharacteristic
214C22×C4C22×D412yes—
215M16—816no—
216C2×C8—48no—
217C2×C8—48no—
218C2×C8—48no—
219C4⋊C4—24no—
220C4⋊C4—42no—
221C4⋊C4—41no—
222C4⋊C4—41no—
223C4⋊C4—44no—
224C4⋊C4—44no—
225C4⋊C4—44no—
226C4⋊C4—44no—
227C4⋊C4—44no—
228C4⋊C4—44no—
229C22⋊C4—44no—
230C22⋊C4—44no—
231C22⋊C4—44no—
232C22⋊C4—44no—
233C22⋊C4—42no—
234C22⋊C4—42no—
235C22⋊C4—42no—
236C22⋊C4—44no—
237C22⋊C4—44no—
238C42—24no—
239C42—24no—
240C42—24no—
241C42—24no—
242C42—24no—
243C42—24no—
244C42—41no—
245C42—41no—
246C42—42no—
247C42—42no—
248C42—41no—
249C42—42no—
250C42—42no—
251C42—42no—
252C42—42no—
253C42—41no—
254C42—42no—
255C42—42no—
256C42—42no—
257C23—21no—
258C23—21no—
259C23—22no—
260C23—22no—
261C23—41no—
262C23—41no—
263C23—41no—
264C23—42no—
265C23—42no—
266C23#41012yes—
267C23C22×C22⋊C411yescharacteristic
268Q8—82no—
269D4—82no—
270D4—84no—
271C2×C4—216no—
272C2×C4—28no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—22no—
277C2×C4—22no—
278C2×C4—21no—
279C2×C4—22no—
280C2×C4—22no—
281C2×C4—22no—
282C2×C4—22no—
283C2×C4—22no—
284C2×C4—22no—
285C2×C4—22no—
286C2×C4—22no—
287C2×C4—22no—
288C2×C4—22no—
289C2×C4—22no—
290C2×C4—22no—
291C2×C4—22no—
292C2×C4—22no—
293C2×C4—44no—
294C2×C4—48no—
295C2×C4—44no—
296C2×C4—42no—
297C2×C4—41no—
298C2×C4—42no—
299C2×C4—41no—
300C2×C4—41no—
301C2×C4—41no—
302C2×C4—41no—
303C2×C4—42no—
304C2×C4—42no—
305C2×C4—42no—
306C2×C4—42no—
307C2×C4—42no—
308C2×C4—42no—
309C2×C4—84no—
310C2×C4#41312yes—
311C2×C4#42114yes—
312C2×C4#56812yes—
313C2×C4#57514yes—
314C8—88no—
315C22—24no—
316C22—22no—
317C22—41no—
318C22—42no—
319C22—41no—
320C22—81no—
321C22—82no—
322C22—81no—
323C22—82no—
324C22—81no—
325C22#287911yescharacteristic
326C22C2×#41011yescharacteristic
327C22C2×#41311yescharacteristic
328C22#289612yes—
329C22C2×#42112yes—
330C4—24no—
331C4—24no—
332C4—22no—
333C4—22no—
334C4—42no—
335C4—42no—
336C4—42no—
337C4—42no—
338C4—81no—
339C4—82no—
340C4—81no—
341C4—82no—
342C4—81no—
343C2—42no—
344C2—161no—
345C2#1739811yescharacteristic
346C2#1749912yes—
347C2#1759312yes—
348C2#1766511yescharacteristic
349C2#1775711yescharacteristic
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. #20772C2semidirect
3. #7729C2semidirect
4. #7726C2semidirect
5. #7632C2semidirect

Maximal subgroups

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

subgroupindex
#366412
#207722
#77292
#77262
#76322
#76312