Finite Group Databasebeta

← #164821

Subgroups — #164821

Order 512.

891 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36496C211yesmaximal, characteristic
2#22022C212yesmaximal, complemented
3#7716C214yesmaximal, complemented
4#7679C212yesmaximal, complemented
5#7651C212yesmaximal, complemented
6#7635C211yesmaximal, characteristic, complemented
7#7603C212yesmaximal, complemented
8#7567C211yesmaximal, characteristic, complemented
9C2×#559C411yescharacteristic
10C2×#554C411yescharacteristic
11#2861C2212yes—
12#2858C2211yescharacteristic
13#2567C411yescharacteristic
14#2566C412yes—
15#2544C412yes—
16#2541C411yescharacteristic
17#2461—24no—
18#2426—22no—
19#2361—22no—
20C4×#127C2212yes—
21C4×#124C2211yescharacteristic
22C42×Q8C2211yescharacteristic
23#1984—22no—
24#1939—22no—
25#1917—24no—
26#1839C2214yes—
27#1831C2214yes—
28#1767C2212yes—
29#1744C2212yes—
30#1272C2212yes—
31#1272C2212yes—
32#1270C2214yes—
33#1270C2214yes—
34#1270C2211yescharacteristic
35#1270C2211yescharacteristic
36#1270C2212yes—
37#559—44no—
38#554—44no—
39C2×C4⋊Q8C2×C411yescharacteristic
40C2×#127C2×C412yes—
41C2×#127C2×C411yescharacteristic
42C2×#125C2×C412yes—
43C2×#124—21no—
44C2×#124—22no—
45C2×#124C2×C411yescharacteristic
46C2×#124C2×C412yes—
47C2×#123—22no—
48C2×C22⋊Q8—21no—
49C2×C22⋊Q8—22no—
50C2×#121—22no—
51C2×C4×Q8C2×C411yescharacteristic
52C2×C4×Q8C2×C411yescharacteristic
53C2×C4×Q8D412yes—
54C2×C4×D4—21no—
55C2×C4×D4—24no—
56C2×C4×D4C2×C412yes—
57C2×C4×D4C2×C412yes—
58C2×C4×D4C2×C411yescharacteristic
59C22×C4⋊C4—21no—
60C2×C4⋊C8—22no—
61C2×C4⋊C8—24no—
62C2×#97—22no—
63C2×C4×C8—22no—
64#396—24no—
65#395—22no—
66#394—24no—
67#392—22no—
68#391C2×C412yes—
69#391C2311yescharacteristic
70#390—24no—
71#390—22no—
72#390C2×C412yes—
73#390C2×C412yes—
74#390C2311yescharacteristic
75#390C2312yes—
76#389—22no—
77#388—24no—
78#388—24no—
79#388—22no—
80#386—24no—
81#385—24no—
82#384C2×C412yes—
83#382C2×C411yescharacteristic
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×C411yescharacteristic
101C22×D4—21no—
102C22×D4—24no—
103C22×D4—21no—
104C22×D4C22×C412yes—
105C22×D4C22×C411yescharacteristic
106C23×C4—21no—
107C23×C4—22no—
108C23×C4—22no—
109C22×C8—22no—
110C22×C8—22no—
111C22×C8—24no—
112C4⋊Q8—42no—
113#127—24no—
114#127—42no—
115#127—42no—
116#125—44no—
117#124—22no—
118#124—41no—
119#124—44no—
120#124—44no—
121#124—82no—
122#124—82no—
123#123—48no—
124C22⋊Q8—48no—
125C22⋊Q8—44no—
126#121—48no—
127C4×Q8—24no—
128C4×Q8—44no—
129C4×Q8—42no—
130C4×D4—44no—
131C4×D4—42no—
132C4×D4—44no—
133C4×D4—44no—
134C4×D4—84no—
135C4×D4—84no—
136C2×C4⋊C4—22no—
137C2×C4⋊C4—21no—
138C2×C4⋊C4—21no—
139C2×C4⋊C4—22no—
140C2×C4⋊C4—24no—
141C2×C4⋊C4—44no—
142C2×C4⋊C4C22⋊C412yes—
143C2×C4⋊C4C22⋊C412yes—
144C2×C4⋊C4C22⋊C414yes—
145C2×C4⋊C4C22⋊C411yescharacteristic
146C2×C4⋊C4C22×C411yescharacteristic
147C2×C4⋊C4C22×C411yescharacteristic
148C2×C4⋊C4C22×C412yes—
149C2×C22⋊C4—22no—
150C2×C22⋊C4—21no—
151C2×C22⋊C4—22no—
152C2×C22⋊C4—22no—
153C2×C22⋊C4—24no—
154C2×C22⋊C4—22no—
155C2×C22⋊C4—22no—
156C2×C22⋊C4—22no—
157C2×C22⋊C4—24no—
158C2×C22⋊C4—24no—
159C2×C42—21no—
160C2×C42C22×C412yes—
161C2×C42C22×C411yescharacteristic
162C2×C42C22×C411yescharacteristic
163C2×C42C22×C412yes—
164C2×C42C2×D414yes—
165C2×C42C2×D412yes—
166C2×C42C2×D412yes—
167C2×C42C2×D414yes—
168C2×C42C2411yescharacteristic
169C4⋊C8—44no—
170C4⋊C8—84no—
171#97—82no—
172C4×C8—44no—
173#95—22no—
174#95—24no—
175#95—22no—
176#95C22⋊C414yes—
177#95C22⋊C412yes—
178C24—21no—
179C24—22no—
180C2×Q8—22no—
181C2×Q8—24no—
182C2×Q8—21no—
183C2×Q8—21no—
184C2×Q8—42no—
185C2×Q8—41no—
186C2×Q8—41no—
187C2×Q8—82no—
188C2×D4—22no—
189C2×D4—21no—
190C2×D4—22no—
191C2×D4—41no—
192C2×D4—41no—
193C2×D4—48no—
194C2×D4—42no—
195C2×D4—42no—
196C2×D4—48no—
197C2×D4—42no—
198C2×D4—42no—
199C2×D4—44no—
200C2×D4—84no—
201C2×D4—82no—
202C2×D4#13112yes—
203C22×C4—21no—
204C22×C4—21no—
205C22×C4—21no—
206C22×C4—22no—
207C22×C4—42no—
208C22×C4—44no—
209C22×C4—44no—
210C22×C4—44no—
211C22×C4—84no—
212C22×C4—84no—
213C22×C4C2×C22⋊C412yes—
214C22×C4C2×C22⋊C414yes—
215C22×C4C2×C22⋊C411yescharacteristic
216C22×C4C2×C22⋊C412yes—
217C22×C4C2×C22⋊C412yes—
218C22×C4C2×C22⋊C411yescharacteristic
219C22×C4C23×C411yescharacteristic
220C22×C4C22×D412yes—
221C2×C8—42no—
222C2×C8—42no—
223C2×C8—44no—
224C2×C8—42no—
225C2×C8—42no—
226C2×C8—44no—
227C2×C8—44no—
228C2×C8—42no—
229C2×C8—42no—
230C4⋊C4—24no—
231C4⋊C4—24no—
232C4⋊C4—44no—
233C4⋊C4—41no—
234C4⋊C4—41no—
235C4⋊C4—42no—
236C4⋊C4—42no—
237C4⋊C4—44no—
238C4⋊C4—44no—
239C4⋊C4—42no—
240C4⋊C4—84no—
241C4⋊C4#10014yes—
242C22⋊C4—44no—
243C22⋊C4—44no—
244C22⋊C4—42no—
245C22⋊C4—44no—
246C22⋊C4—48no—
247C22⋊C4—44no—
248C22⋊C4—42no—
249C22⋊C4—44no—
250C22⋊C4—42no—
251C22⋊C4—44no—
252C22⋊C4—44no—
253C22⋊C4—84no—
254C42—21no—
255C42—22no—
256C42—21no—
257C42—22no—
258C42—22no—
259C42—24no—
260C42—24no—
261C42—22no—
262C42—24no—
263C42—22no—
264C42—24no—
265C42—44no—
266C42—41no—
267C42—42no—
268C23—22no—
269C23—41no—
270C23—41no—
271C23—42no—
272C23—42no—
273C23—42no—
274C23—42no—
275C23—42no—
276C23—44no—
277C23—44no—
278C23C22×C22⋊C411yescharacteristic
279Q8—44no—
280Q8—81no—
281Q8—81no—
282Q8—162no—
283D4—42no—
284D4—42no—
285D4—82no—
286D4—82no—
287D4—816no—
288D4—84no—
289D4—162no—
290C2×C4—22no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—24no—
294C2×C4—22no—
295C2×C4—22no—
296C2×C4—24no—
297C2×C4—22no—
298C2×C4—21no—
299C2×C4—22no—
300C2×C4—24no—
301C2×C4—22no—
302C2×C4—22no—
303C2×C4—24no—
304C2×C4—24no—
305C2×C4—41no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—42no—
311C2×C4—44no—
312C2×C4—42no—
313C2×C4—42no—
314C2×C4—84no—
315C2×C4—164no—
316C2×C4—164no—
317C2×C4C2×#10014yes—
318C2×C4C2×#10012yes—
319C2×C4#42114yes—
320C2×C4C2×#13111yescharacteristic
321C2×C4#56811yescharacteristic
322C2×C4#57612yes—
323C2×C4#57712yes—
324C8—82no—
325C8—84no—
326C8—82no—
327C22—42no—
328C22—44no—
329C22—81no—
330C22—82no—
331C22—81no—
332C22—82no—
333C22—82no—
334C22—84no—
335C22—84no—
336C22—82no—
337C22#288011yescharacteristic
338C22#288411yescharacteristic
339C22C22×#10011yescharacteristic
340C22#289712yes—
341C22C2×#42112yes—
342C4—21no—
343C4—21no—
344C4—24no—
345C4—22no—
346C4—22no—
347C4—41no—
348C4—42no—
349C4—44no—
350C4—42no—
351C4—81no—
352C4—81no—
353C4—82no—
354C4—81no—
355C4#153614yes—
356C2—82no—
357C2—162no—
358C2#1742511yescharacteristic
359C2#1752212yes—
360C2#1760212yes—
361C2C2×#153611yescharacteristic
362C2#1777011yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radical#2541C4
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
3. #7716C2semidirect
4. #7679C2semidirect
5. #7651C2semidirect
6. #7635C2semidirect
7. #7603C2semidirect
8. #7567C2semidirect

Maximal subgroups

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

subgroupindex
#364962
#220222
#77162
#76792
#76512
#76352
#76032
#75672