Finite Group Databasebeta

← #164893

Subgroups — #164893

Order 512.

854 conjugacy classes of subgroups.

The subgroup lattice has 357 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36534C211yesmaximal, characteristic
2#22828C212yesmaximal, complemented
3#7662C212yesmaximal, complemented
4#7656C214yesmaximal, complemented
5#7651C211yesmaximal, characteristic
6#7649C212yesmaximal, complemented
7#7569C211yesmaximal, characteristic
8#7567C212yesmaximal, complemented
9C2×#556C411yescharacteristic
10C2×#554C411yescharacteristic
11#2862C2211yescharacteristic
12#2861C2212yes—
13#2565C412yes—
14#2563C411yescharacteristic
15#2560—24no—
16#2547C411yescharacteristic
17#2544C412yes—
18#2536—22no—
19#2352—22no—
20C4×#127C2211yescharacteristic
21C4×#124C2212yes—
22C42×Q8C2211yescharacteristic
23#1984—22no—
24#1916—24no—
25#1839C2214yes—
26#1831C2212yes—
27C8×C4⋊C4—22no—
28C4×C4⋊C8C2212yes—
29#1744C2214yes—
30#1272C2214yes—
31#1272C2214yes—
32#1270C2211yescharacteristic
33#1270C2212yes—
34#1270C2211yescharacteristic
35#1270C2212yes—
36#1270C2212yes—
37#556—44no—
38#554—44no—
39C2×C4⋊Q8C2×C411yescharacteristic
40C2×C4⋊Q8C2×C411yescharacteristic
41C2×#127C2×C411yescharacteristic
42C2×#126—22no—
43C2×#125C2×C412yes—
44C2×#124C2×C412yes—
45C2×#124C2×C412yes—
46C2×C22⋊Q8—21no—
47C2×C22⋊Q8—21no—
48C2×C22⋊Q8—22no—
49C2×C4×Q8C2×C412yes—
50C2×C4×Q8C2×C411yescharacteristic
51C2×C4×Q8D412yes—
52C2×C4×D4—22no—
53C2×C4×D4C2×C411yescharacteristic
54C2×C4×D4C2×C411yescharacteristic
55C2×C4×D4C2×C412yes—
56C2×#117—21no—
57C22×C4⋊C4—21no—
58C2×C4⋊C8—22no—
59C2×C4⋊C8—22no—
60C2×C4×C8—24no—
61C2×C4×C8—22no—
62#398—24no—
63#397—22no—
64#394—24no—
65#393—22no—
66#391—22no—
67#391C2×C412yes—
68#391C2×C412yes—
69#391C2312yes—
70#390—22no—
71#390C2×C412yes—
72#390C2311yescharacteristic
73#390C2311yescharacteristic
74#388—24no—
75#388—24no—
76#388—24no—
77#387—24no—
78#386—24no—
79#384C2×C411yescharacteristic
80#382—22no—
81#382C2×C412yes—
82#380—22no—
83#380—22no—
84C4×C4⋊C4C2×C411yescharacteristic
85C4×C4⋊C4C2×C412yes—
86C4×C4⋊C4D414yes—
87C4×C4⋊C4D412yes—
88C4×C22⋊C4—24no—
89C43C2311yescharacteristic
90C43C2312yes—
91#336—22no—
92#336—24no—
93#336—24no—
94#336—24no—
95#336C2314yes—
96#336C2312yes—
97#336C2312yes—
98C22×Q8—21no—
99C22×Q8C22⋊C411yescharacteristic
100C22×Q8C22×C412yes—
101C22×D4—22no—
102C22×D4C22×C411yescharacteristic
103C22×D4C22×C411yescharacteristic
104C23×C4—21no—
105C23×C4—21no—
106C23×C4—22no—
107C22×C8—22no—
108C22×C8—24no—
109C22×C8—22no—
110C4⋊Q8—42no—
111C4⋊Q8—42no—
112#127—22no—
113#127—41no—
114#126—48no—
115#125—44no—
116#124—24no—
117#124—42no—
118#124—44no—
119C22⋊Q8—48no—
120C22⋊Q8—44no—
121C22⋊Q8—44no—
122C4×Q8—24no—
123C4×Q8—24no—
124C4×Q8—24no—
125C4×Q8—44no—
126C4×D4—42no—
127C4×D4—44no—
128C4×D4—42no—
129C4×D4—44no—
130C4×D4—44no—
131#117—44no—
132C2×C4⋊C4—22no—
133C2×C4⋊C4—21no—
134C2×C4⋊C4—21no—
135C2×C4⋊C4—22no—
136C2×C4⋊C4—22no—
137C2×C4⋊C4—22no—
138C2×C4⋊C4—21no—
139C2×C4⋊C4—44no—
140C2×C4⋊C4C22⋊C412yes—
141C2×C4⋊C4C22⋊C412yes—
142C2×C4⋊C4C22⋊C414yes—
143C2×C4⋊C4C22⋊C411yescharacteristic
144C2×C4⋊C4C22×C411yescharacteristic
145C2×C4⋊C4C22×C411yescharacteristic
146C2×C4⋊C4C22×C412yes—
147C2×C22⋊C4—22no—
148C2×C22⋊C4—22no—
149C2×C22⋊C4—21no—
150C2×C22⋊C4—21no—
151C2×C22⋊C4—22no—
152C2×C22⋊C4—24no—
153C2×C22⋊C4—22no—
154C2×C22⋊C4—22no—
155C2×C22⋊C4—24no—
156C2×C42—22no—
157C2×C42C22×C411yescharacteristic
158C2×C42C22×C412yes—
159C2×C42C22×C412yes—
160C2×C42C22×C411yescharacteristic
161C2×C42C2×D414yes—
162C2×C42C2×D414yes—
163C2×C42C2×D412yes—
164C2×C42C2×D412yes—
165C2×C42C2411yescharacteristic
166C4⋊C8—44no—
167C4⋊C8—44no—
168C4×C8—48no—
169C4×C8—44no—
170#95—24no—
171#95—24no—
172#95—24no—
173#95—24no—
174#95C22⋊C414yes—
175#95C22⋊C412yes—
176C24—21no—
177C24—21no—
178C2×Q8—22no—
179C2×Q8—24no—
180C2×Q8—22no—
181C2×Q8—22no—
182C2×Q8—42no—
183C2×Q8—42no—
184C2×Q8—42no—
185C2×Q8—82no—
186C2×D4—21no—
187C2×D4—21no—
188C2×D4—21no—
189C2×D4—41no—
190C2×D4—41no—
191C2×D4—44no—
192C2×D4—41no—
193C2×D4—41no—
194C2×D4—44no—
195C2×D4—82no—
196C2×D4#13112yes—
197C22×C4—22no—
198C22×C4—22no—
199C22×C4—21no—
200C22×C4—21no—
201C22×C4—42no—
202C22×C4—42no—
203C22×C4—44no—
204C22×C4—44no—
205C22×C4—44no—
206C22×C4—84no—
207C22×C4C2×C22⋊C414yes—
208C22×C4C2×C22⋊C411yescharacteristic
209C22×C4C2×C22⋊C412yes—
210C22×C4C2×C22⋊C412yes—
211C22×C4C2×C22⋊C412yes—
212C22×C4C2×C22⋊C411yescharacteristic
213C22×C4C23×C411yescharacteristic
214C22×C4C22×D412yes—
215C2×C8—24no—
216C2×C8—42no—
217C2×C8—42no—
218C2×C8—44no—
219C2×C8—42no—
220C2×C8—42no—
221C2×C8—44no—
222C2×C8—42no—
223C2×C8—44no—
224C4⋊C4—24no—
225C4⋊C4—24no—
226C4⋊C4—28no—
227C4⋊C4—41no—
228C4⋊C4—41no—
229C4⋊C4—42no—
230C4⋊C4—44no—
231C4⋊C4—44no—
232C4⋊C4—42no—
233C4⋊C4—44no—
234C4⋊C4—42no—
235C4⋊C4—42no—
236C4⋊C4—44no—
237C4⋊C4#10014yes—
238C22⋊C4—48no—
239C22⋊C4—42no—
240C22⋊C4—44no—
241C22⋊C4—42no—
242C22⋊C4—44no—
243C22⋊C4—42no—
244C22⋊C4—42no—
245C22⋊C4—42no—
246C22⋊C4—42no—
247C22⋊C4—44no—
248C22⋊C4—84no—
249C42—22no—
250C42—21no—
251C42—22no—
252C42—21no—
253C42—24no—
254C42—24no—
255C42—22no—
256C42—24no—
257C42—44no—
258C42—42no—
259C42—41no—
260C42—44no—
261C42#13514yes—
262C42#13514yes—
263C23—22no—
264C23—41no—
265C23—41no—
266C23—42no—
267C23—42no—
268C23—41no—
269C23—41no—
270C23—41no—
271C23—42no—
272C23—42no—
273C23C22×C22⋊C411yescharacteristic
274Q8—44no—
275Q8—82no—
276Q8—82no—
277Q8—162no—
278D4—42no—
279D4—42no—
280D4—88no—
281D4—81no—
282D4—81no—
283C2×C4—22no—
284C2×C4—22no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—22no—
288C2×C4—24no—
289C2×C4—22no—
290C2×C4—24no—
291C2×C4—21no—
292C2×C4—22no—
293C2×C4—24no—
294C2×C4—24no—
295C2×C4—24no—
296C2×C4—22no—
297C2×C4—42no—
298C2×C4—42no—
299C2×C4—41no—
300C2×C4—42no—
301C2×C4—41no—
302C2×C4—42no—
303C2×C4—42no—
304C2×C4—42no—
305C2×C4—44no—
306C2×C4—84no—
307C2×C4—84no—
308C2×C4—164no—
309C2×C4C2×#10014yes—
310C2×C4C2×#10012yes—
311C2×C4#41614yes—
312C2×C4#41614yes—
313C2×C4C2×#13111yescharacteristic
314C2×C4#56811yescharacteristic
315C2×C4C2×#13512yes—
316C2×C4C2×#13512yes—
317C8—44no—
318C8—82no—
319C8—84no—
320C22—42no—
321C22—44no—
322C22—81no—
323C22—81no—
324C22—82no—
325C22—81no—
326C22—81no—
327C22—81no—
328C22—82no—
329C22—82no—
330C22#288011yescharacteristic
331C22#288411yescharacteristic
332C22C22×#10011yescharacteristic
333C22C2×#41612yes—
334C22#290112yes—
335C4—21no—
336C4—21no—
337C4—24no—
338C4—24no—
339C4—22no—
340C4—22no—
341C4—41no—
342C4—42no—
343C4—44no—
344C4—82no—
345C4—82no—
346C4—81no—
347C4—81no—
348C4#154314yes—
349C2—82no—
350C2—161no—
351C2#1742511yescharacteristic
352C2#1750812yes—
353C2#1761612yes—
354C2#1768511yescharacteristic
355C2C2×#154311yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC2×#127C2×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. #22828C2semidirect
3. #7662C2semidirect
4. #7656C2semidirect
6. #7649C2semidirect
8. #7567C2semidirect

Maximal subgroups

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

subgroupindex
#365342
#228282
#76622
#76562
#76512
#76492
#75692
#75672