Finite Group Databasebeta

← #164881

Subgroups — #164881

Order 512.

794 conjugacy classes of subgroups.

The subgroup lattice has 339 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36508C211yesmaximal, characteristic
2#22022C212yesmaximal, complemented
3#7716C214yesmaximal
4#7682C212yesmaximal, complemented
5#7653C212yesmaximal, complemented
6#7635C211yesmaximal, characteristic
7#7605C212yesmaximal, complemented
8#7567C211yesmaximal, characteristic
9C2×#557C411yescharacteristic
10C2×#556C411yescharacteristic
11#2862C2211yescharacteristic
12#2861C2212yes—
13#2565C412yes—
14#2546C411yescharacteristic
15#2545C412yes—
16#2542C411yescharacteristic
17#2519—24no—
18#2429—22no—
19#2362—22no—
20C4×C4⋊Q8C2212yes—
21C4×#124C2211yescharacteristic
22C42×Q8C2211yescharacteristic
23#1984—22no—
24#1939—22no—
25#1917—24no—
26#1839C2214yes—
27#1831C2214yes—
28#1767C2212yes—
29#1744C2212yes—
30#1272C2214yes—
31#1272C2214yes—
32#1272C2211yescharacteristic
33#1272C2211yescharacteristic
34#1272C2212yes—
35#1270C2212yes—
36#1270C2212yes—
37#557—44no—
38#556—44no—
39C2×C4⋊Q8—21no—
40C2×C4⋊Q8C2×C412yes—
41C2×C4⋊Q8C2×C411yescharacteristic
42C2×C4⋊Q8C2×C411yescharacteristic
43C2×#126—22no—
44C2×#125—22no—
45C2×#125C2×C412yes—
46C2×#124C2×C411yescharacteristic
47C2×#124C2×C412yes—
48C2×C22⋊Q8—21no—
49C2×C22⋊Q8—22no—
50C2×C4×Q8—21no—
51C2×C4×Q8C2×C412yes—
52C2×C4×Q8C2×C412yes—
53C2×C4×Q8C2×C411yescharacteristic
54C2×C4×Q8C2×C411yescharacteristic
55C2×C4×Q8D412yes—
56C2×C4×D4C2×C411yescharacteristic
57C2×#117—21no—
58C2×C4⋊C8—22no—
59C2×C4⋊C8—24no—
60C2×#97—22no—
61C2×C4×C8—22no—
62#399—22no—
63#398—24no—
64#398—22no—
65#396—24no—
66#391—22no—
67#391C2×C412yes—
68#391C2×C412yes—
69#391C2311yescharacteristic
70#391C2312yes—
71#390C2×C412yes—
72#390C2311yescharacteristic
73#389—22no—
74#389—24no—
75#388—24no—
76#388—24no—
77#388—22no—
78#388—24no—
79#387—24no—
80#384C2×C412yes—
81#382—22no—
82#382C2×C411yescharacteristic
83C4×C4⋊C4—24no—
84C4×C4⋊C4C2×C412yes—
85C4×C4⋊C4C2×C411yescharacteristic
86C4×C4⋊C4D412yes—
87C4×C4⋊C4D414yes—
88C4×C22⋊C4—22no—
89C43C2312yes—
90C43C2311yescharacteristic
91#336—24no—
92#336—24no—
93#336—24no—
94#336—22no—
95#336C2312yes—
96#336C2314yes—
97#336C2312yes—
98C22×Q8—21no—
99C22×Q8C22⋊C411yescharacteristic
100C22×Q8C22×C412yes—
101C22×Q8C22×C411yescharacteristic
102C22×D4C22×C411yescharacteristic
103C23×C4—21no—
104C22×C8—22no—
105C22×C8—22no—
106C22×C8—24no—
107C4⋊Q8—24no—
108C4⋊Q8—42no—
109C4⋊Q8—42no—
110C4⋊Q8—42no—
111C4⋊Q8—44no—
112#126—48no—
113#125—44no—
114#125—48no—
115#124—22no—
116#124—41no—
117#124—44no—
118C22⋊Q8—48no—
119C22⋊Q8—44no—
120C4×Q8—24no—
121C4×Q8—44no—
122C4×Q8—44no—
123C4×Q8—42no—
124C4×Q8—44no—
125C4×Q8—44no—
126C4×D4—42no—
127#117—44no—
128C2×C4⋊C4—21no—
129C2×C4⋊C4—22no—
130C2×C4⋊C4—22no—
131C2×C4⋊C4—21no—
132C2×C4⋊C4—22no—
133C2×C4⋊C4—24no—
134C2×C4⋊C4—21no—
135C2×C4⋊C4—22no—
136C2×C4⋊C4—21no—
137C2×C4⋊C4—24no—
138C2×C4⋊C4C22⋊C412yes—
139C2×C4⋊C4C22⋊C412yes—
140C2×C4⋊C4C22⋊C414yes—
141C2×C4⋊C4C22⋊C411yescharacteristic
142C2×C4⋊C4C22×C411yescharacteristic
143C2×C4⋊C4C22×C411yescharacteristic
144C2×C4⋊C4C22×C412yes—
145C2×C22⋊C4—22no—
146C2×C22⋊C4—21no—
147C2×C22⋊C4—22no—
148C2×C22⋊C4—24no—
149C2×C22⋊C4—22no—
150C2×C22⋊C4—22no—
151C2×C42—22no—
152C2×C42—22no—
153C2×C42—21no—
154C2×C42C22×C411yescharacteristic
155C2×C42C22×C412yes—
156C2×C42C22×C411yescharacteristic
157C2×C42C22×C412yes—
158C2×C42C2×D412yes—
159C2×C42C2×D414yes—
160C2×C42C2×D412yes—
161C2×C42C2×D414yes—
162C2×C42C2411yescharacteristic
163C4⋊C8—44no—
164C4⋊C8—84no—
165#97—82no—
166C4×C8—44no—
167#95—22no—
168#95—24no—
169#95—24no—
170#95—22no—
171#95—24no—
172#95C22⋊C414yes—
173#95C22⋊C412yes—
174C24—21no—
175C2×Q8—21no—
176C2×Q8—22no—
177C2×Q8—24no—
178C2×Q8—22no—
179C2×Q8—22no—
180C2×Q8—41no—
181C2×Q8—41no—
182C2×Q8—42no—
183C2×Q8—42no—
184C2×Q8—42no—
185C2×Q8—44no—
186C2×Q8#13112yes—
187C2×D4—21no—
188C2×D4—21no—
189C2×D4—41no—
190C2×D4—41no—
191C22×C4—21no—
192C22×C4—21no—
193C22×C4—22no—
194C22×C4—21no—
195C22×C4—22no—
196C22×C4—42no—
197C22×C4—44no—
198C22×C4C2×C22⋊C412yes—
199C22×C4C2×C22⋊C414yes—
200C22×C4C2×C22⋊C411yescharacteristic
201C22×C4C2×C22⋊C412yes—
202C22×C4C2×C22⋊C412yes—
203C22×C4C2×C22⋊C411yescharacteristic
204C22×C4C23×C411yescharacteristic
205C22×C4C22×D412yes—
206C2×C8—42no—
207C2×C8—42no—
208C2×C8—44no—
209C2×C8—42no—
210C2×C8—42no—
211C2×C8—44no—
212C2×C8—44no—
213C2×C8—42no—
214C2×C8—42no—
215C4⋊C4—24no—
216C4⋊C4—24no—
217C4⋊C4—44no—
218C4⋊C4—48no—
219C4⋊C4—41no—
220C4⋊C4—41no—
221C4⋊C4—42no—
222C4⋊C4—44no—
223C4⋊C4—42no—
224C4⋊C4—44no—
225C4⋊C4—48no—
226C4⋊C4—42no—
227C4⋊C4—44no—
228C4⋊C4—42no—
229C4⋊C4—42no—
230C4⋊C4—44no—
231C4⋊C4#10014yes—
232C22⋊C4—44no—
233C22⋊C4—42no—
234C22⋊C4—48no—
235C22⋊C4—44no—
236C22⋊C4—42no—
237C22⋊C4—42no—
238C22⋊C4—44no—
239C42—21no—
240C42—22no—
241C42—21no—
242C42—22no—
243C42—22no—
244C42—24no—
245C42—24no—
246C42—22no—
247C42—22no—
248C42—24no—
249C42—24no—
250C42—44no—
251C42—41no—
252C42—44no—
253C42—44no—
254C42—42no—
255C23—41no—
256C23—41no—
257C23—41no—
258C23—42no—
259C23—42no—
260C23C22×C22⋊C411yescharacteristic
261Q8—42no—
262Q8—42no—
263Q8—44no—
264Q8—82no—
265Q8—82no—
266Q8—84no—
267D4—81no—
268D4—81no—
269C2×C4—22no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—24no—
273C2×C4—22no—
274C2×C4—22no—
275C2×C4—24no—
276C2×C4—22no—
277C2×C4—21no—
278C2×C4—22no—
279C2×C4—22no—
280C2×C4—22no—
281C2×C4—24no—
282C2×C4—22no—
283C2×C4—24no—
284C2×C4—24no—
285C2×C4—41no—
286C2×C4—41no—
287C2×C4—42no—
288C2×C4—42no—
289C2×C4—41no—
290C2×C4—42no—
291C2×C4—41no—
292C2×C4—42no—
293C2×C4—42no—
294C2×C4—44no—
295C2×C4—42no—
296C2×C4—84no—
297C2×C4C2×#10014yes—
298C2×C4C2×#10012yes—
299C2×C4#42114yes—
300C2×C4C2×#13111yescharacteristic
301C2×C4#56811yescharacteristic
302C2×C4#57712yes—
303C2×C4#57812yes—
304C8—82no—
305C8—84no—
306C8—82no—
307C22—81no—
308C22—81no—
309C22—82no—
310C22—82no—
311C22—81no—
312C22#288011yescharacteristic
313C22#288411yescharacteristic
314C22C22×#10011yescharacteristic
315C22#289812yes—
316C22C2×#42112yes—
317C4—21no—
318C4—21no—
319C4—24no—
320C4—22no—
321C4—22no—
322C4—42no—
323C4—41no—
324C4—42no—
325C4—44no—
326C4—42no—
327C4—82no—
328C4—81no—
329C4—81no—
330C4—82no—
331C4#153614yes—
332C2—161no—
333C2#1742511yescharacteristic
334C2#1752512yes—
335C2#1760112yes—
336C2C2×#153611yescharacteristic
337C2#1777911yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC22×D4C22×C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C42×Q8 ◁ G
LECSG ▷ C2×C42 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #22022C2semidirect
4. #7682C2semidirect
5. #7653C2semidirect
7. #7605C2semidirect

Maximal subgroups

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

subgroupindex
#365082
#220222
#77162
#76822
#76532
#76352
#76052
#75672