Finite Group Databasebeta

← #165667

Subgroups — #165667

Order 512.

676 conjugacy classes of subgroups.

The subgroup lattice has 372 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36403C211yesmaximal, characteristic
2#22830C212yesmaximal, complemented
3#7726C212yesmaximal
4#7697C212yesmaximal, complemented
5#7692C212yesmaximal
6#7635C211yesmaximal, characteristic
7#7611C212yesmaximal, complemented
8#7601C211yesmaximal, characteristic
9#7581C212yesmaximal, complemented
10#2864C2211yescharacteristic
11#2860C2212yes—
12#2604—22no—
13#2596—22no—
14#2581C411yescharacteristic
15#2580C411yescharacteristic
16#2575—22no—
17#2509C412yes—
18#2448C412yes—
19#2373—22no—
20#2357C411yescharacteristic
21#2351C411yescharacteristic
22C4×#124C2211yescharacteristic
23#2295C2212yes—
24C42×Q8C2211yescharacteristic
25#1985—22no—
26#1917—22no—
27#1916—22no—
28#1839C2212yes—
29#1839C2212yes—
30#1831C2212yes—
31#1771—22no—
32#1767C2212yes—
33#1745C2212yes—
34#1744C2212yes—
35#1272C2211yescharacteristic
36#1272C2212yes—
37#1272C2211yescharacteristic
38#1271C2212yes—
39#1271C2212yes—
40#1271C2212yes—
41#1271C2212yes—
42#1270C2211yescharacteristic
43#1270C2212yes—
44#1270C2211yescharacteristic
45C2×#126—22no—
46C2×#124C2×C411yescharacteristic
47C2×C4×Q8C2×C412yes—
48C2×C4×Q8C2×C411yescharacteristic
49C2×C4×Q8C2×C411yescharacteristic
50C2×C4×Q8D412yes—
51C2×C4×D4C2×C411yescharacteristic
52C2×C4⋊C8—22no—
53C2×C4⋊C8—22no—
54C2×#97—22no—
55C2×#97—22no—
56C2×C4×C8—22no—
57#400—22no—
58#400—21no—
59#400—22no—
60#399—22no—
61#399—21no—
62#397—22no—
63#395—21no—
64#393—21no—
65#391C2×C412yes—
66#391C2311yescharacteristic
67#390C2×C412yes—
68#390C2×C412yes—
69#390C2311yescharacteristic
70#389—22no—
71#389—22no—
72#389C2×C412yes—
73#389C2×C412yes—
74#389C2×C412yes—
75#389C2×C412yes—
76#389C2312yes—
77#388—22no—
78#388—22no—
79#388—22no—
80#388—22no—
81#388—22no—
82#388—22no—
83#388—22no—
84#387—22no—
85#387—22no—
86#387—22no—
87#387—22no—
88#387—22no—
89#387—22no—
90#385—22no—
91#382—21no—
92#382C2×C411yescharacteristic
93#382C2×C411yescharacteristic
94#382C2×C411yescharacteristic
95#381—21no—
96C4×C4⋊C4—22no—
97C4×C4⋊C4—21no—
98C4×C4⋊C4C2×C412yes—
99C4×C4⋊C4C2×C411yescharacteristic
100C4×C4⋊C4C2×C412yes—
101C4×C4⋊C4D412yes—
102C4×C4⋊C4D412yes—
103C4×C4⋊C4D412yes—
104C4×C22⋊C4—22no—
105C4×C22⋊C4—21no—
106#376—22no—
107C43C2311yescharacteristic
108C43C2312yes—
109#336—22no—
110#336—22no—
111#336—22no—
112#336—22no—
113#336—22no—
114#336—22no—
115#336—22no—
116#336C2312yes—
117#336C2312yes—
118#336C2312yes—
119#336C2312yes—
120C22×Q8C22⋊C411yescharacteristic
121C22×Q8C22×C411yescharacteristic
122C22×D4C22×C411yescharacteristic
123C23×C4—21no—
124C22×C8—22no—
125C22×C8—22no—
126C22×C8—22no—
127C22×C8—22no—
128#126—44no—
129#126—44no—
130#124—22no—
131#124—41no—
132C4×Q8—24no—
133C4×Q8—42no—
134C4×Q8—42no—
135C4×Q8—42no—
136C4×Q8—42no—
137C4×Q8—42no—
138C4×D4—42no—
139C2×C4⋊C4—22no—
140C2×C4⋊C4—21no—
141C2×C4⋊C4—22no—
142C2×C4⋊C4—22no—
143C2×C4⋊C4—22no—
144C2×C4⋊C4—22no—
145C2×C4⋊C4—22no—
146C2×C4⋊C4C22⋊C412yes—
147C2×C4⋊C4C22⋊C411yescharacteristic
148C2×C4⋊C4C22⋊C412yes—
149C2×C4⋊C4C22⋊C412yes—
150C2×C4⋊C4C22⋊C412yes—
151C2×C4⋊C4C22×C412yes—
152C2×C4⋊C4C22×C411yescharacteristic
153C2×C4⋊C4C22×C411yescharacteristic
154C2×C4⋊C4C22×C412yes—
155C2×C22⋊C4—21no—
156C2×C22⋊C4—22no—
157C2×C22⋊C4—22no—
158C2×C22⋊C4—22no—
159C2×C22⋊C4—22no—
160C2×C22⋊C4—22no—
161C2×C22⋊C4—22no—
162C2×C42—22no—
163C2×C42—21no—
164C2×C42—22no—
165C2×C42C22×C411yescharacteristic
166C2×C42C22×C412yes—
167C2×C42C22×C411yescharacteristic
168C2×C42C22×C412yes—
169C2×C42C2×D412yes—
170C2×C42C2×D412yes—
171C2×C42C2×D412yes—
172C2×C42C2×D412yes—
173C2×C42C2×D412yes—
174C2×C42C2×D412yes—
175C2×C42C2411yescharacteristic
176C4⋊C8—44no—
177C4⋊C8—44no—
178#97—44no—
179#97—44no—
180C4×C8—44no—
181#95—22no—
182#95—22no—
183#95—21no—
184#95—22no—
185#95—21no—
186#95—22no—
187#95—22no—
188#95—22no—
189#95—21no—
190#95—21no—
191#95—22no—
192#95—22no—
193#95—22no—
194#95—22no—
195#95C22⋊C411yescharacteristic
196#95C22⋊C411yescharacteristic
197#95C22⋊C412yes—
198#95C22⋊C412yes—
199C24—21no—
200C2×Q8—21no—
201C2×Q8—41no—
202C2×Q8—42no—
203C2×Q8—41no—
204C2×Q8—41no—
205C2×Q8#13112yes—
206C2×D4—21no—
207C2×D4—21no—
208C2×D4—41no—
209C2×D4—41no—
210C22×C4—21no—
211C22×C4—21no—
212C22×C4—22no—
213C22×C4—22no—
214C22×C4—21no—
215C22×C4—42no—
216C22×C4—82no—
217C22×C4C2×C22⋊C412yes—
218C22×C4C2×C22⋊C412yes—
219C22×C4C2×C22⋊C412yes—
220C22×C4C2×C22⋊C411yescharacteristic
221C22×C4C2×C22⋊C412yes—
222C22×C4C2×C22⋊C412yes—
223C22×C4C2×C22⋊C411yescharacteristic
224C22×C4C23×C411yescharacteristic
225C22×C4C22×D412yes—
226C2×C8—24no—
227C2×C8—42no—
228C2×C8—42no—
229C2×C8—42no—
230C2×C8—42no—
231C2×C8—42no—
232C2×C8—42no—
233C2×C8—42no—
234C2×C8—42no—
235C2×C8—42no—
236C2×C8—42no—
237C2×C8—42no—
238C4⋊C4—24no—
239C4⋊C4—22no—
240C4⋊C4—22no—
241C4⋊C4—41no—
242C4⋊C4—42no—
243C4⋊C4—42no—
244C4⋊C4—42no—
245C4⋊C4—44no—
246C4⋊C4—42no—
247C4⋊C4—41no—
248C4⋊C4—41no—
249C4⋊C4—42no—
250C4⋊C4—44no—
251C4⋊C4—44no—
252C4⋊C4—42no—
253C4⋊C4—42no—
254C4⋊C4—82no—
255C4⋊C4—82no—
256C22⋊C4—44no—
257C22⋊C4—42no—
258C22⋊C4—44no—
259C22⋊C4—44no—
260C22⋊C4—42no—
261C22⋊C4—42no—
262C22⋊C4—82no—
263C22⋊C4—82no—
264C42—22no—
265C42—21no—
266C42—22no—
267C42—21no—
268C42—22no—
269C42—22no—
270C42—22no—
271C42—22no—
272C42—22no—
273C42—22no—
274C42—22no—
275C42—42no—
276C42—42no—
277C42—42no—
278C42—42no—
279C42—41no—
280C42—44no—
281C42—42no—
282C42—82no—
283C23—41no—
284C23—41no—
285C23—41no—
286C23—42no—
287C23—42no—
288C23C22×C22⋊C411yescharacteristic
289Q8—42no—
290Q8—42no—
291Q8—44no—
292D4—81no—
293D4—81no—
294C2×C4—21no—
295C2×C4—22no—
296C2×C4—21no—
297C2×C4—22no—
298C2×C4—22no—
299C2×C4—22no—
300C2×C4—22no—
301C2×C4—22no—
302C2×C4—22no—
303C2×C4—21no—
304C2×C4—22no—
305C2×C4—22no—
306C2×C4—22no—
307C2×C4—22no—
308C2×C4—22no—
309C2×C4—22no—
310C2×C4—22no—
311C2×C4—22no—
312C2×C4—22no—
313C2×C4—22no—
314C2×C4—22no—
315C2×C4—22no—
316C2×C4—22no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—42no—
320C2×C4—42no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—42no—
324C2×C4—42no—
325C2×C4—42no—
326C2×C4—42no—
327C2×C4—42no—
328C2×C4—42no—
329C2×C4—162no—
330C2×C4#41312yes—
331C2×C4C2×#13111yescharacteristic
332C2×C4#56811yescharacteristic
333C2×C4#57512yes—
334C2×C4#57512yes—
335C8—44no—
336C8—82no—
337C8—82no—
338C8—82no—
339C22—81no—
340C22—81no—
341C22—81no—
342C22—82no—
343C22—82no—
344C22#288011yescharacteristic
345C22#288411yescharacteristic
346C22C2×#41311yescharacteristic
347C22#289612yes—
348C22#290112yes—
349C4—21no—
350C4—21no—
351C4—22no—
352C4—22no—
353C4—22no—
354C4—42no—
355C4—42no—
356C4—41no—
357C4—42no—
358C4—42no—
359C4—42no—
360C4—42no—
361C4—81no—
362C4—81no—
363C4—82no—
364C4—82no—
365C2—161no—
366C2#1742311yescharacteristic
367C2#1751612yes—
368C2#1762412yes—
369C2#1768611yescharacteristic
370C2#1775711yescharacteristic
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. #22830C2semidirect
4. #7697C2semidirect
7. #7611C2semidirect
9. #7581C2semidirect

Maximal subgroups

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

subgroupindex
#364032
#228302
#77262
#76972
#76922
#76352
#76112
#76012
#75812