Finite Group Databasebeta

← #163632

Subgroups — #163632

Order 512.

791 conjugacy classes of subgroups.

The subgroup lattice has 350 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36641C211yesmaximal, characteristic
2#20778C212yesmaximal, complemented
3#7709C214yesmaximal, complemented
4#7706C214yesmaximal, complemented
5#7646C212yesmaximal, complemented
6#7645C212yesmaximal
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#1831C2214yes—
24#1817C2214yes—
25#1797—22no—
26#1767—22no—
27#1749—22no—
28#1745C2214yes—
29#1272C2214yes—
30#1271C2218yes—
31#1270C2214yes—
32C22×#41C2×C411yescharacteristic
33C2×#126—22no—
34C2×#126—22no—
35C2×#117C2×C411yescharacteristic
36C2×#117C2×C411yescharacteristic
37C2×#117C2×C411yescharacteristic
38C22×C42C2×C411yescharacteristic
39C22×C42C2×C411yescharacteristic
40C22×C42D412yes—
41C22×C42D412yes—
42C22×C42C2311yescharacteristic
43C2×C4⋊C8—24no—
44C2×C22⋊C8—24no—
45C2×#97—24no—
46#391C2×C411yescharacteristic
47#391C2×C412yes—
48#391C2×C412yes—
49#391C2311yescharacteristic
50#390C2×C411yescharacteristic
51#390C2×C412yes—
52#390C2×C412yes—
53#390C2311yescharacteristic
54#389C2×C412yes—
55#389C2×C412yes—
56#389C2×C412yes—
57#389C2×C412yes—
58#389C2×C412yes—
59#389C2312yes—
60#388—22no—
61#388—22no—
62#388—22no—
63#388—22no—
64#388—22no—
65#388—22no—
66#387—22no—
67#387—22no—
68#387—22no—
69#387—22no—
70#387—22no—
71#387—22no—
72#386—22no—
73#386—21no—
74#386—21no—
75#386—22no—
76#385—22no—
77#385—21no—
78#385—21no—
79#385—22no—
80#383—22no—
81#383—22no—
82C4×C4⋊C4—22no—
83C4×C4⋊C4—21no—
84C4×C4⋊C4—21no—
85C4×C4⋊C4—22no—
86C4×C22⋊C4—22no—
87C4×C22⋊C4—21no—
88C4×C22⋊C4—21no—
89C4×C22⋊C4—22no—
90C43C2×C412yes—
91C43D412yes—
92C43D412yes—
93C43C2312yes—
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—
110C22×C8—28no—
111#126—48no—
112#126—48no—
113#117—42no—
114#117—42no—
115#117—42no—
116C2×C4⋊C4—22no—
117C2×C4⋊C4—22no—
118C2×C4⋊C4—22no—
119C2×C4⋊C4—22no—
120C2×C4⋊C4—22no—
121C2×C4⋊C4—22no—
122C2×C4⋊C4C22×C412yes—
123C2×C4⋊C4C22×C411yescharacteristic
124C2×C4⋊C4C22×C411yescharacteristic
125C2×C4⋊C4C22×C412yes—
126C2×C22⋊C4—21no—
127C2×C22⋊C4—21no—
128C2×C22⋊C4—21no—
129C2×C22⋊C4—22no—
130C2×C22⋊C4—22no—
131C2×C22⋊C4—22no—
132C2×C22⋊C4—22no—
133C2×C22⋊C4—22no—
134C2×C22⋊C4—22no—
135C2×C42—21no—
136C2×C42—21no—
137C2×C42—21no—
138C2×C42—42no—
139C2×C42—42no—
140C2×C42—44no—
141C2×C42—44no—
142C2×C42—42no—
143C2×C42C22⋊C412yes—
144C2×C42C22⋊C412yes—
145C2×C42C22⋊C412yes—
146C2×C42C22⋊C412yes—
147C2×C42C22⋊C412yes—
148C2×C42C22⋊C411yescharacteristic
149C2×C42C22⋊C411yescharacteristic
150C2×C42C22×C412yes—
151C2×C42C22×C411yescharacteristic
152C2×C42C22×C412yes—
153C2×C42C2×D412yes—
154C2×C42C2×D412yes—
155C2×C42C2×D412yes—
156C2×C42C2×D412yes—
157C2×C42C2×D412yes—
158C2×C42C2411yescharacteristic
159C4⋊C8—48no—
160C22⋊C8—48no—
161#97—48no—
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—22no—
188C22×C4—22no—
189C22×C4—24no—
190C22×C4—24no—
191C22×C4—24no—
192C22×C4—21no—
193C22×C4—22no—
194C22×C4—22no—
195C22×C4—21no—
196C22×C4—21no—
197C22×C4—42no—
198C22×C4—42no—
199C22×C4—41no—
200C22×C4—41no—
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—
215C2×C8—48no—
216C2×C8—48no—
217C2×C8—48no—
218C4⋊C4—24no—
219C4⋊C4—42no—
220C4⋊C4—41no—
221C4⋊C4—41no—
222C4⋊C4—44no—
223C4⋊C4—44no—
224C4⋊C4—44no—
225C4⋊C4—44no—
226C4⋊C4—44no—
227C4⋊C4—44no—
228C22⋊C4—44no—
229C22⋊C4—44no—
230C22⋊C4—44no—
231C22⋊C4—44no—
232C22⋊C4—42no—
233C22⋊C4—42no—
234C22⋊C4—42no—
235C22⋊C4—44no—
236C22⋊C4—44no—
237C42—24no—
238C42—24no—
239C42—24no—
240C42—24no—
241C42—24no—
242C42—24no—
243C42—41no—
244C42—41no—
245C42—42no—
246C42—42no—
247C42—42no—
248C42—41no—
249C42—42no—
250C42—42no—
251C42—42no—
252C42—41no—
253C42—42no—
254C42—42no—
255C42—42no—
256C23—21no—
257C23—21no—
258C23—22no—
259C23—22no—
260C23—41no—
261C23—41no—
262C23—41no—
263C23—42no—
264C23—42no—
265C23#41012yes—
266C23C22×C22⋊C411yescharacteristic
267Q8—82no—
268D4—82no—
269D4—84no—
270C2×C4—28no—
271C2×C4—22no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—22no—
276C2×C4—21no—
277C2×C4—22no—
278C2×C4—22no—
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—48no—
292C2×C4—48no—
293C2×C4—44no—
294C2×C4—44no—
295C2×C4—42no—
296C2×C4—41no—
297C2×C4—42no—
298C2×C4—41no—
299C2×C4—41no—
300C2×C4—41no—
301C2×C4—41no—
302C2×C4—42no—
303C2×C4—42no—
304C2×C4—42no—
305C2×C4—42no—
306C2×C4—42no—
307C2×C4—42no—
308C2×C4—84no—
309C2×C4#41312yes—
310C2×C4#42114yes—
311C2×C4#56812yes—
312C2×C4#57514yes—
313C8—88no—
314C22—24no—
315C22—22no—
316C22—41no—
317C22—41no—
318C22—42no—
319C22—81no—
320C22—81no—
321C22—82no—
322C22—82no—
323C22—81no—
324C22#287911yescharacteristic
325C22C2×#41011yescharacteristic
326C22C2×#41311yescharacteristic
327C22#289612yes—
328C22C2×#42112yes—
329C4—24no—
330C4—24no—
331C4—22no—
332C4—22no—
333C4—42no—
334C4—42no—
335C4—42no—
336C4—42no—
337C4—81no—
338C4—81no—
339C4—82no—
340C4—82no—
341C4—81no—
342C2—42no—
343C2—161no—
344C2#1739811yescharacteristic
345C2#1749412yes—
346C2#1759112yes—
347C2#1767311yescharacteristic
348C2#1774011yescharacteristic
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. #20778C2semidirect
3. #7709C2semidirect
4. #7706C2semidirect
5. #7646C2semidirect

Maximal subgroups

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

subgroupindex
#366412
#207782
#77092
#77062
#76462
#76452