Finite Group Databasebeta

← #195971

Subgroups — #195971

Order 512.

1219 conjugacy classes of subgroups.

The subgroup lattice has 343 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2982C211yesmaximal, characteristic, complemented
2C2×#2637C211yesmaximal, characteristic
3#21822C211yesmaximal, characteristic, complemented
4C2×#1473C212yesmaximal
5#9082C214yesmaximal, complemented
6#9080C214yesmaximal
7#7914C212yesmaximal, complemented
8C22×#126C2211yescharacteristic
9C22×C4×Q8C2211yescharacteristic
10#2982—28no—
11C2×C8×Q8—24no—
12C2×#432C2212yes—
13C2×#431C2214yes—
14C2×#431C2212yes—
15#2637—24no—
16#2637—24no—
17C2×#398C411yescharacteristic
18C2×#398C412yes—
19C2×#397C411yescharacteristic
20C2×#396C412yes—
21C2×#395C411yescharacteristic
22C2×#393C411yescharacteristic
23C2×#388C2212yes—
24C2×#387C2212yes—
25C2×#382C2211yescharacteristic
26#1985—24no—
27#1817C2214yes—
28#1473—28no—
29C2×#324C2214yes—
30C2×#323C2214yes—
31#1280C2218yes—
32C23×Q8C2×C411yescharacteristic
33C2×#126—22no—
34C2×#126—22no—
35C2×#126—24no—
36C2×#126—24no—
37C2×C4×Q8—24no—
38C2×C4×Q8—24no—
39C2×C4×Q8D414yes—
40C2×C4×Q8D414yes—
41C22×C4⋊C4C2×C412yes—
42C22×C4⋊C4C2×C411yescharacteristic
43C22×C4⋊C4C2×C411yescharacteristic
44C22×C4⋊C4C2×C412yes—
45C22×C4⋊C4C2×C411yescharacteristic
46C22×C4⋊C4C2×C412yes—
47C22×C4⋊C4C2×C412yes—
48C22×C4⋊C4C2311yescharacteristic
49C22×C22⋊C4C2×C412yes—
50C22×C22⋊C4C2×C411yescharacteristic
51C22×C22⋊C4C2×C411yescharacteristic
52C22×C22⋊C4C2×C412yes—
53C22×C22⋊C4C2311yescharacteristic
54C22×C42C2312yes—
55C22×C42C2311yescharacteristic
56C8×Q8—232no—
57#432—24no—
58#432—24no—
59#431—24no—
60#431—216no—
61#431D418yes—
62C2×C4⋊C8—24no—
63C2×C4⋊C8—22no—
64C2×C4⋊C8—22no—
65C2×C22⋊C8C2314yes—
66C2×C22⋊C8C2314yes—
67C2×#97—22no—
68C2×C4×C8—24no—
69C2×C4×C8—22no—
70#398—24no—
71#398—28no—
72#397—24no—
73#396—28no—
74#395—24no—
75#393—24no—
76#388—24no—
77#388—24no—
78#387—24no—
79#387—24no—
80#382—22no—
81#382—22no—
82C2×#95C2×C411yescharacteristic
83C2×#95C2×C412yes—
84C2×#95C2×C411yescharacteristic
85C2×#95C2×C412yes—
86C2×#95C2×C412yes—
87C2×#95C2×C412yes—
88C2×#95C2312yes—
89#336—28no—
90#324—28no—
91#323—28no—
92C25C22×C411yescharacteristic
93C22×Q8—24no—
94C22×Q8—28no—
95C22×Q8C22⋊C414yes—
96C23×C4C22×C411yescharacteristic
97C23×C4C22×C411yescharacteristic
98C23×C4C22×C412yes—
99C23×C4C22×C412yes—
100C23×C4C22×C412yes—
101C23×C4C22×C411yescharacteristic
102C23×C4C22×C411yescharacteristic
103C23×C4C22×C412yes—
104C23×C4C22×C411yescharacteristic
105C23×C4C2411yescharacteristic
106C22×C8—24no—
107C22×C8—24no—
108#126—44no—
109#126—44no—
110#126—44no—
111#126—44no—
112C4×Q8—28no—
113C4×Q8—28no—
114C4×Q8—28no—
115C4×Q8—28no—
116C2×C4⋊C4—28no—
117C2×C4⋊C4—24no—
118C2×C4⋊C4—22no—
119C2×C4⋊C4—22no—
120C2×C4⋊C4—24no—
121C2×C4⋊C4—24no—
122C2×C4⋊C4—24no—
123C2×C4⋊C4—28no—
124C2×C4⋊C4—22no—
125C2×C4⋊C4—21no—
126C2×C4⋊C4—21no—
127C2×C4⋊C4—22no—
128C2×C4⋊C4—24no—
129C2×C4⋊C4—24no—
130C2×C4⋊C4—82no—
131C2×C4⋊C4—82no—
132C2×C4⋊C4C22⋊C418yes—
133C2×C4⋊C4C22⋊C414yes—
134C2×C22⋊C4—22no—
135C2×C22⋊C4—22no—
136C2×C22⋊C4—24no—
137C2×C22⋊C4—24no—
138C2×C22⋊C4—24no—
139C2×C22⋊C4—28no—
140C2×C22⋊C4—22no—
141C2×C22⋊C4—21no—
142C2×C22⋊C4—21no—
143C2×C22⋊C4—22no—
144C2×C22⋊C4—24no—
145C2×C22⋊C4—82no—
146C2×C42—24no—
147C2×C42—24no—
148C2×C42C2×D414yes—
149C2×C42C2×D414yes—
150C2×C42C2×D414yes—
151C2×C42C2×D412yes—
152C2×C42C2×D412yes—
153C2×C42#4114yes—
154C4⋊C8—216no—
155C4⋊C8—28no—
156C4⋊C8—44no—
157C22⋊C8—24no—
158C22⋊C8—24no—
159C22⋊C8—24no—
160C22⋊C8C2×D418yes—
161#97—44no—
162C4×C8—216no—
163C4×C8—28no—
164#95—22no—
165#95—22no—
166#95—24no—
167#95—24no—
168#95—24no—
169#95—24no—
170#95—22no—
171#95—22no—
172C24—21no—
173C24—22no—
174C24—24no—
175C24—41no—
176C24—42no—
177C24—41no—
178C24C23×C411yescharacteristic
179C2×Q8—232no—
180C2×Q8—28no—
181C2×Q8—216no—
182C22×C4—24no—
183C22×C4—24no—
184C22×C4—24no—
185C22×C4—24no—
186C22×C4—22no—
187C22×C4—22no—
188C22×C4—21no—
189C22×C4—21no—
190C22×C4—21no—
191C22×C4—22no—
192C22×C4—24no—
193C22×C4—22no—
194C22×C4—22no—
195C22×C4—24no—
196C22×C4—24no—
197C22×C4—41no—
198C22×C4—41no—
199C22×C4—42no—
200C22×C4—42no—
201C22×C4—42no—
202C22×C4—42no—
203C22×C4—42no—
204C22×C4—41no—
205C22×C4—42no—
206C22×C4—42no—
207C22×C4—42no—
208C22×C4—41no—
209C22×C4C2×C22⋊C414yes—
210C22×C4C2×C22⋊C412yes—
211C22×C4C2×C22⋊C412yes—
212C22×C4C2×C22⋊C414yes—
213C22×C4C4×D414yes—
214C22×C4C4×D414yes—
215C22×C4#13118yes—
216C22×C4C22×D411yescharacteristic
217C22×C4C22×D411yescharacteristic
218C22×C4C22×D412yes—
219C22×C4C2×#4112yes—
220C2×C8—28no—
221C2×C8—48no—
222C2×C8—48no—
223C2×C8—44no—
224C4⋊C4—216no—
225C4⋊C4—28no—
226C4⋊C4—41no—
227C4⋊C4—42no—
228C4⋊C4—41no—
229C4⋊C4—44no—
230C4⋊C4—44no—
231C4⋊C4—48no—
232C4⋊C4—84no—
233C4⋊C4—84no—
234C22⋊C4—41no—
235C22⋊C4—42no—
236C22⋊C4—41no—
237C22⋊C4—44no—
238C22⋊C4—44no—
239C22⋊C4—48no—
240C22⋊C4—84no—
241C42—24no—
242C42—28no—
243C42—24no—
244C42#12014yes—
245C42#12114yes—
246C42#12114yes—
247C42#12314yes—
248C23—22no—
249C23—21no—
250C23—21no—
251C23—42no—
252C23—42no—
253C23—41no—
254C23—41no—
255C23—41no—
256C23—44no—
257C23—82no—
258C23—84no—
259C23—84no—
260C23—82no—
261C23C22×C22⋊C411yescharacteristic
262C23C2×C4×D412yes—
263C23C2×#13114yes—
264Q8—232no—
265C2×C4—24no—
266C2×C4—24no—
267C2×C4—22no—
268C2×C4—22no—
269C2×C4—22no—
270C2×C4—22no—
271C2×C4—28no—
272C2×C4—216no—
273C2×C4—24no—
274C2×C4—24no—
275C2×C4—24no—
276C2×C4—24no—
277C2×C4—44no—
278C2×C4—42no—
279C2×C4—42no—
280C2×C4—42no—
281C2×C4—42no—
282C2×C4—42no—
283C2×C4—42no—
284C2×C4—42no—
285C2×C4—42no—
286C2×C4—42no—
287C2×C4—41no—
288C2×C4—42no—
289C2×C4—41no—
290C2×C4—44no—
291C2×C4—41no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—84no—
295C2×C4—82no—
296C2×C4—84no—
297C2×C4—84no—
298C2×C4—82no—
299C2×C4#38614yes—
300C2×C4#38614yes—
301C2×C4#40818yes—
302C2×C4C2×#12011yescharacteristic
303C2×C4C2×#12112yes—
304C2×C4C2×#12311yescharacteristic
305C8—48no—
306C8—84no—
307C22—22no—
308C22—24no—
309C22—24no—
310C22—22no—
311C22—81no—
312C22—81no—
313C22—82no—
314C22—82no—
315C22—81no—
316C22—164no—
317C22C2×#38611yescharacteristic
318C22C2×#40812yes—
319C22#288412yes—
320C22#288811yescharacteristic
321C22#288911yescharacteristic
322C22#292914yes—
323C4—24no—
324C4—24no—
325C4—28no—
326C4—21no—
327C4—22no—
328C4—21no—
329C4—44no—
330C4—81no—
331C4—81no—
332C4—82no—
333C4—82no—
334C4—81no—
335C2—24no—
336C2—161no—
337C2#1739012yes—
338C2#1739112yes—
339C2#2089511yescharacteristic
340C2#2098611yescharacteristic
341C2#2099211yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC24C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25C22×C4
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C4×Q8 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#2982C2semidirect
3. #21822C2semidirect
5. #9082C2semidirect
7. #7914C2semidirect

Maximal subgroups

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

subgroupindex
C2×#29822
C2×#26372
#218222
C2×#14732
#90822
#90802
#79142