Finite Group Databasebeta

← #164367

Subgroups — #164367

Order 512.

890 conjugacy classes of subgroups.

The subgroup lattice has 342 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36508C211yesmaximal, characteristic
2#20573C212yesmaximal, complemented
3#7682C212yesmaximal, complemented
4#7676C214yesmaximal
5#7639C212yesmaximal, complemented
6#7635C211yesmaximal, characteristic
7#7571C212yesmaximal, complemented
8#7567C211yesmaximal, characteristic
9C2×#557C411yescharacteristic
10C2×#556C411yescharacteristic
11C2×#446—22no—
12#2862C2211yescharacteristic
13#2861C2212yes—
14#2565C412yes—
15#2546C411yescharacteristic
16#2545C412yes—
17#2542C411yescharacteristic
18#2519—24no—
19#2429—22no—
20#2362—22no—
21C4×C4⋊Q8C2212yes—
22C4×#124C2211yescharacteristic
23C42×Q8C2211yescharacteristic
24#1771—24no—
25C8×C4⋊C4—22no—
26#1767C2212yes—
27C4×C4⋊C8C2214yes—
28#1745C2214yes—
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—
58#446—216no—
59C2×C4⋊C8—24no—
60C2×C4⋊C8—22no—
61C2×#97—24no—
62C2×#97—22no—
63C2×C4×C8—22no—
64C2×C4×C8—24no—
65#399—22no—
66#398—24no—
67#398—22no—
68#396—24no—
69#391—22no—
70#391C2×C412yes—
71#391C2×C412yes—
72#391C2311yescharacteristic
73#391C2312yes—
74#390C2×C412yes—
75#390C2311yescharacteristic
76#389—22no—
77#389—24no—
78#388—24no—
79#388—24no—
80#388—22no—
81#388—24no—
82#387—24no—
83#384C2×C412yes—
84#382—22no—
85#382C2×C411yescharacteristic
86C4×C4⋊C4—24no—
87C4×C4⋊C4C2×C412yes—
88C4×C4⋊C4C2×C411yescharacteristic
89C4×C4⋊C4D412yes—
90C4×C4⋊C4D414yes—
91C4×C22⋊C4—22no—
92C43C2312yes—
93C43C2311yescharacteristic
94#336—22no—
95#336—24no—
96#336C2312yes—
97#336C2312yes—
98#336C2314yes—
99C22×Q8—21no—
100C22×Q8C22⋊C411yescharacteristic
101C22×Q8C22×C412yes—
102C22×Q8C22×C411yescharacteristic
103C22×D4C22×C411yescharacteristic
104C23×C4—21no—
105C22×C8—22no—
106C22×C8—22no—
107C22×C8—24no—
108C4⋊Q8—24no—
109C4⋊Q8—42no—
110C4⋊Q8—42no—
111C4⋊Q8—42no—
112C4⋊Q8—44no—
113#126—48no—
114#125—44no—
115#125—48no—
116#124—22no—
117#124—41no—
118#124—44no—
119C22⋊Q8—44no—
120C22⋊Q8—48no—
121C4×Q8—24no—
122C4×Q8—28no—
123C4×Q8—44no—
124C4×Q8—42no—
125C4×Q8—44no—
126C4×Q8—44no—
127C4×D4—42no—
128#117—44no—
129C2×C4⋊C4—21no—
130C2×C4⋊C4—22no—
131C2×C4⋊C4—22no—
132C2×C4⋊C4—21no—
133C2×C4⋊C4—22no—
134C2×C4⋊C4—24no—
135C2×C4⋊C4—21no—
136C2×C4⋊C4—22no—
137C2×C4⋊C4—21no—
138C2×C4⋊C4—24no—
139C2×C4⋊C4C22⋊C412yes—
140C2×C4⋊C4C22⋊C414yes—
141C2×C4⋊C4C22⋊C411yescharacteristic
142C2×C4⋊C4C22⋊C412yes—
143C2×C4⋊C4C22×C411yescharacteristic
144C2×C4⋊C4C22×C411yescharacteristic
145C2×C4⋊C4C22×C412yes—
146C2×C22⋊C4—22no—
147C2×C22⋊C4—21no—
148C2×C22⋊C4—22no—
149C2×C22⋊C4—24no—
150C2×C22⋊C4—22no—
151C2×C22⋊C4—22no—
152C2×C42—22no—
153C2×C42—22no—
154C2×C42—21no—
155C2×C42C22×C411yescharacteristic
156C2×C42C22×C412yes—
157C2×C42C22×C411yescharacteristic
158C2×C42C22×C412yes—
159C2×C42C2×D412yes—
160C2×C42C2×D414yes—
161C2×C42C2×D414yes—
162C2×C42C2×D412yes—
163C2×C42C2411yescharacteristic
164C4⋊C8—28no—
165C4⋊C8—216no—
166#97—216no—
167#97—82no—
168C4×C8—28no—
169C4×C8—84no—
170#95—22no—
171#95—24no—
172#95—24no—
173#95—22no—
174#95—24no—
175#95C22⋊C414yes—
176#95C22⋊C412yes—
177C24—21no—
178C2×Q8—21no—
179C2×Q8—22no—
180C2×Q8—24no—
181C2×Q8—22no—
182C2×Q8—22no—
183C2×Q8—41no—
184C2×Q8—41no—
185C2×Q8—42no—
186C2×Q8—42no—
187C2×Q8—42no—
188C2×Q8—44no—
189C2×Q8#13112yes—
190C2×D4—21no—
191C2×D4—21no—
192C2×D4—41no—
193C2×D4—41no—
194C22×C4—21no—
195C22×C4—21no—
196C22×C4—22no—
197C22×C4—21no—
198C22×C4—22no—
199C22×C4—42no—
200C22×C4—44no—
201C22×C4C2×C22⋊C412yes—
202C22×C4C2×C22⋊C411yescharacteristic
203C22×C4C2×C22⋊C412yes—
204C22×C4C2×C22⋊C412yes—
205C22×C4C2×C22⋊C414yes—
206C22×C4C2×C22⋊C411yescharacteristic
207C22×C4C23×C411yescharacteristic
208C22×C4C22×D412yes—
209C2×C8—24no—
210C2×C8—24no—
211C2×C8—28no—
212C2×C8—42no—
213C2×C8—42no—
214C2×C8—44no—
215C2×C8—42no—
216C2×C8—44no—
217C2×C8—42no—
218C4⋊C4—24no—
219C4⋊C4—28no—
220C4⋊C4—48no—
221C4⋊C4—41no—
222C4⋊C4—41no—
223C4⋊C4—42no—
224C4⋊C4—44no—
225C4⋊C4—42no—
226C4⋊C4—44no—
227C4⋊C4—42no—
228C4⋊C4—44no—
229C4⋊C4—42no—
230C4⋊C4—42no—
231C4⋊C4—44no—
232C4⋊C4—48no—
233C4⋊C4#10014yes—
234C4⋊C4#10418yes—
235C22⋊C4—44no—
236C22⋊C4—42no—
237C22⋊C4—44no—
238C22⋊C4—42no—
239C22⋊C4—42no—
240C22⋊C4—44no—
241C22⋊C4—48no—
242C42—21no—
243C42—22no—
244C42—21no—
245C42—22no—
246C42—24no—
247C42—22no—
248C42—24no—
249C42—22no—
250C42—22no—
251C42—24no—
252C42—24no—
253C42—28no—
254C42—41no—
255C42—44no—
256C42—44no—
257C42—42no—
258C23—41no—
259C23—41no—
260C23—41no—
261C23—42no—
262C23—42no—
263C23C22×C22⋊C411yescharacteristic
264Q8—42no—
265Q8—42no—
266Q8—44no—
267Q8—82no—
268Q8—82no—
269Q8—84no—
270D4—81no—
271D4—81no—
272C2×C4—24no—
273C2×C4—22no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—24no—
277C2×C4—24no—
278C2×C4—22no—
279C2×C4—22no—
280C2×C4—22no—
281C2×C4—21no—
282C2×C4—22no—
283C2×C4—22no—
284C2×C4—22no—
285C2×C4—22no—
286C2×C4—24no—
287C2×C4—41no—
288C2×C4—41no—
289C2×C4—42no—
290C2×C4—42no—
291C2×C4—41no—
292C2×C4—42no—
293C2×C4—41no—
294C2×C4—42no—
295C2×C4—42no—
296C2×C4—44no—
297C2×C4—42no—
298C2×C4—84no—
299C2×C4C2×#10014yes—
300C2×C4C2×#10012yes—
301C2×C4C2×#10418yes—
302C2×C4C2×#10414yes—
303C2×C4C2×#13111yescharacteristic
304C2×C4#56811yescharacteristic
305C2×C4#57712yes—
306C2×C4#57812yes—
307C8—44no—
308C8—48no—
309C8—44no—
310C22—81no—
311C22—82no—
312C22—81no—
313C22—82no—
314C22—81no—
315C22#288011yescharacteristic
316C22#288411yescharacteristic
317C22C22×#10011yescharacteristic
318C22#289812yes—
319C22C22×#10412yes—
320C4—21no—
321C4—21no—
322C4—28no—
323C4—24no—
324C4—22no—
325C4—22no—
326C4—42no—
327C4—42no—
328C4—41no—
329C4—42no—
330C4—82no—
331C4—81no—
332C4—81no—
333C4—82no—
334C4#152514yes—
335C2—161no—
336C2#1742511yescharacteristic
337C2#1751912yes—
338C2#1754912yes—
339C2C2×#152511yescharacteristic
340C2#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. #20573C2semidirect
3. #7682C2semidirect
5. #7639C2semidirect
7. #7571C2semidirect

Maximal subgroups

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

subgroupindex
#365082
#205732
#76822
#76762
#76392
#76352
#75712
#75672