Finite Group Databasebeta

← #165081

Subgroups — #165081

Order 512.

1006 conjugacy classes of subgroups.

The subgroup lattice has 391 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36415C211yesmaximal, characteristic
2#22700C212yesmaximal, complemented
3#7710C214yesmaximal, complemented
4#7662C212yesmaximal, complemented
5#7651C211yesmaximal, characteristic, complemented
6#7631C212yesmaximal, complemented
7#7597C212yesmaximal, complemented
8#7569C211yesmaximal, characteristic, complemented
9C2×#555C411yescharacteristic
10C2×#547C411yescharacteristic
11C2×#436—22no—
12#2862C2211yescharacteristic
13#2850C2212yes—
14#2563C411yescharacteristic
15#2557—24no—
16#2547C411yescharacteristic
17#2535—22no—
18#2499C412yes—
19#2431C412yes—
20#2352—22no—
21C4×#127C2211yescharacteristic
22#2297C2212yes—
23C42×D4C2211yescharacteristic
24#1939—22no—
25#1851—24no—
26#1831C2212yes—
27#1797C2214yes—
28C4×C4⋊C8C2212yes—
29#1728C2214yes—
30#1272C2214yes—
31#1272C2214yes—
32#1270C2212yes—
33#1270C2212yes—
34#1270C2211yescharacteristic
35#1270C2211yescharacteristic
36#1270C2212yes—
37C22×#41C2×C412yes—
38C22×M16—24no—
39#555—44no—
40#547—44no—
41C2×C4⋊Q8C2×C411yescharacteristic
42C2×C4⋊Q8C2×C411yescharacteristic
43C2×#127C2×C411yescharacteristic
44C2×#124—22no—
45C2×#124C2×C412yes—
46C2×#123—22no—
47C2×C22⋊Q8C2×C412yes—
48C2×#121—21no—
49C2×#121—21no—
50C2×#121C2×C412yes—
51C2×C4×D4—22no—
52C2×C4×D4C2×C411yescharacteristic
53C2×C4×D4C2×C411yescharacteristic
54C2×C4×D4C2×C411yescharacteristic
55C2×C4×D4D412yes—
56C2×#117—21no—
57C2×#117C2×C412yes—
58C22×C4⋊C4—21no—
59C22×C42C2312yes—
60#436—48no—
61C2×C4⋊C8—22no—
62C2×C4⋊C8—22no—
63C2×C4⋊C8—24no—
64C2×C22⋊C8—24no—
65C2×C4×C8—22no—
66#397—24no—
67#397—22no—
68#393—22no—
69#392—24no—
70#391—22no—
71#391C2×C412yes—
72#391C2×C412yes—
73#391C2312yes—
74#390—22no—
75#390C2×C412yes—
76#390C2311yescharacteristic
77#390C2311yescharacteristic
78#388—24no—
79#387—24no—
80#386—24no—
81#386—24no—
82#386—24no—
83#384C2×C411yescharacteristic
84#382—22no—
85#382C2×C412yes—
86#380—22no—
87#380—22no—
88C4×C4⋊C4C2×C411yescharacteristic
89C4×C4⋊C4D412yes—
90C4×C22⋊C4—24no—
91C4×C22⋊C4C2×C412yes—
92C4×C22⋊C4D414yes—
93C43C2311yescharacteristic
94#336—22no—
95#336—24no—
96#336C2312yes—
97#336C2314yes—
98#336C2312yes—
99C2×#41—24no—
100C2×#41—42no—
101C2×#41—44no—
102C2×#41—82no—
103C22×Q8—21no—
104C22×Q8C22×C412yes—
105C22×D4—22no—
106C22×D4—22no—
107C22×D4—22no—
108C22×D4C22⋊C411yescharacteristic
109C22×D4C22×C411yescharacteristic
110C22×D4C22×C411yescharacteristic
111C23×C4—21no—
112C23×C4—21no—
113C23×C4—22no—
114C23×C4C22×C412yes—
115C23×C4C2×D414yes—
116C2×M16—48no—
117C2×M16—48no—
118C2×M16—84no—
119C22×C8—22no—
120C22×C8—24no—
121C22×C8—22no—
122C4⋊Q8—42no—
123C4⋊Q8—42no—
124#127—22no—
125#127—41no—
126#124—48no—
127#124—82no—
128#123—48no—
129C22⋊Q8—44no—
130#121—44no—
131#121—44no—
132#121—44no—
133C4×D4—24no—
134C4×D4—24no—
135C4×D4—24no—
136C4×D4—42no—
137C4×D4—42no—
138C4×D4—44no—
139C4×D4—44no—
140#117—44no—
141#117—44no—
142C2×C4⋊C4—21no—
143C2×C4⋊C4—22no—
144C2×C4⋊C4—21no—
145C2×C4⋊C4—22no—
146C2×C4⋊C4—21no—
147C2×C4⋊C4—44no—
148C2×C4⋊C4C22⋊C412yes—
149C2×C4⋊C4C22⋊C411yescharacteristic
150C2×C4⋊C4C22×C411yescharacteristic
151C2×C4⋊C4C22×C412yes—
152C2×C4⋊C4C22×C411yescharacteristic
153C2×C22⋊C4—22no—
154C2×C22⋊C4—22no—
155C2×C22⋊C4—22no—
156C2×C22⋊C4—21no—
157C2×C22⋊C4—21no—
158C2×C22⋊C4—22no—
159C2×C22⋊C4—22no—
160C2×C22⋊C4—24no—
161C2×C22⋊C4—24no—
162C2×C22⋊C4C22⋊C412yes—
163C2×C22⋊C4C22⋊C414yes—
164C2×C42—22no—
165C2×C42—82no—
166C2×C42C22×C411yescharacteristic
167C2×C42C22×C411yescharacteristic
168C2×C42C22×C412yes—
169C2×C42C2×D412yes—
170C2×C42C2×D414yes—
171C2×C42C2×D412yes—
172C2×C42C2411yescharacteristic
173C4⋊C8—44no—
174C4⋊C8—44no—
175C4⋊C8—84no—
176C22⋊C8—48no—
177C4×C8—44no—
178#95—24no—
179#95—24no—
180#95—24no—
181#95—24no—
182#95C22⋊C414yes—
183#95C22⋊C412yes—
184C24—21no—
185C24—21no—
186C24C2×C22⋊C412yes—
187#41—88no—
188#41—88no—
189C2×Q8—22no—
190C2×Q8—22no—
191C2×Q8—42no—
192C2×Q8—42no—
193C2×Q8—42no—
194C2×Q8—82no—
195C2×D4—21no—
196C2×D4—21no—
197C2×D4—21no—
198C2×D4—22no—
199C2×D4—24no—
200C2×D4—41no—
201C2×D4—41no—
202C2×D4—44no—
203C2×D4—41no—
204C2×D4—41no—
205C2×D4—44no—
206C2×D4—44no—
207C2×D4—44no—
208C2×D4—44no—
209C2×D4—82no—
210C2×D4—82no—
211C2×D4—84no—
212C2×D4#13112yes—
213C22×C4—22no—
214C22×C4—22no—
215C22×C4—24no—
216C22×C4—24no—
217C22×C4—24no—
218C22×C4—22no—
219C22×C4—22no—
220C22×C4—21no—
221C22×C4—21no—
222C22×C4—44no—
223C22×C4—42no—
224C22×C4—44no—
225C22×C4—44no—
226C22×C4—42no—
227C22×C4—44no—
228C22×C4—42no—
229C22×C4—44no—
230C22×C4—44no—
231C22×C4—82no—
232C22×C4C2×C22⋊C414yes—
233C22×C4C2×C22⋊C411yescharacteristic
234C22×C4C2×C22⋊C412yes—
235C22×C4C2×C22⋊C412yes—
236C22×C4C2×C22⋊C411yescharacteristic
237C22×C4C23×C411yescharacteristic
238C22×C4C22×D412yes—
239M16—816no—
240C2×C8—24no—
241C2×C8—42no—
242C2×C8—42no—
243C2×C8—42no—
244C2×C8—42no—
245C2×C8—44no—
246C2×C8—44no—
247C2×C8—42no—
248C2×C8—44no—
249C4⋊C4—24no—
250C4⋊C4—41no—
251C4⋊C4—42no—
252C4⋊C4—41no—
253C4⋊C4—42no—
254C4⋊C4—42no—
255C4⋊C4—44no—
256C4⋊C4—42no—
257C4⋊C4—44no—
258C4⋊C4#10014yes—
259C22⋊C4—24no—
260C22⋊C4—28no—
261C22⋊C4—42no—
262C22⋊C4—42no—
263C22⋊C4—44no—
264C22⋊C4—44no—
265C22⋊C4—42no—
266C22⋊C4—42no—
267C22⋊C4—42no—
268C22⋊C4—42no—
269C22⋊C4—44no—
270C22⋊C4—84no—
271C22⋊C4—84no—
272C42—21no—
273C42—21no—
274C42—22no—
275C42—42no—
276C42—44no—
277C42—41no—
278C42—42no—
279C42—44no—
280C42AGL(1,ℤ/8)14yes—
281C42#13714yes—
282C23—22no—
283C23—22no—
284C23—22no—
285C23—22no—
286C23—24no—
287C23—24no—
288C23—41no—
289C23—41no—
290C23—42no—
291C23—42no—
292C23—41no—
293C23—41no—
294C23—41no—
295C23—42no—
296C23—42no—
297C23C22×C22⋊C411yescharacteristic
298Q8—82no—
299Q8—82no—
300Q8—162no—
301D4—42no—
302D4—42no—
303D4—44no—
304D4—88no—
305D4—81no—
306D4—81no—
307D4—88no—
308D4—164no—
309C2×C4—22no—
310C2×C4—22no—
311C2×C4—22no—
312C2×C4—21no—
313C2×C4—22no—
314C2×C4—21no—
315C2×C4—22no—
316C2×C4—22no—
317C2×C4—24no—
318C2×C4—21no—
319C2×C4—24no—
320C2×C4—22no—
321C2×C4—24no—
322C2×C4—44no—
323C2×C4—44no—
324C2×C4—48no—
325C2×C4—48no—
326C2×C4—42no—
327C2×C4—42no—
328C2×C4—41no—
329C2×C4—42no—
330C2×C4—41no—
331C2×C4—42no—
332C2×C4—42no—
333C2×C4—42no—
334C2×C4—44no—
335C2×C4—88no—
336C2×C4—84no—
337C2×C4—84no—
338C2×C4C2×#10014yes—
339C2×C4C2×#10012yes—
340C2×C4#41714yes—
341C2×C4#41714yes—
342C2×C4C2×#13111yescharacteristic
343C2×C4#56811yescharacteristic
344C2×C4C2×AGL(1,ℤ/8)12yes—
345C2×C4C2×#13712yes—
346C8—44no—
347C8—82no—
348C8—84no—
349C22—42no—
350C22—42no—
351C22—42no—
352C22—42no—
353C22—44no—
354C22—44no—
355C22—44no—
356C22—81no—
357C22—81no—
358C22—82no—
359C22—81no—
360C22—81no—
361C22—81no—
362C22—82no—
363C22—82no—
364C22#288011yescharacteristic
365C22#288311yescharacteristic
366C22C22×#10011yescharacteristic
367C22C2×#41712yes—
368C22#290112yes—
369C4—21no—
370C4—21no—
371C4—24no—
372C4—24no—
373C4—22no—
374C4—22no—
375C4—41no—
376C4—44no—
377C4—82no—
378C4—82no—
379C4—81no—
380C4—81no—
381C4#156014yes—
382C2—82no—
383C2—82no—
384C2—161no—
385C2#1742111yescharacteristic
386C2#1751212yes—
387C2#1763012yes—
388C2#1768811yescharacteristic
389C2C2×#156011yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radical#36415C2
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C42×D4 ◁ G
LECSG ▷ C2×C42 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #22700C2semidirect
3. #7710C2semidirect
4. #7662C2semidirect
5. #7651C2semidirect
6. #7631C2semidirect
7. #7597C2semidirect
8. #7569C2semidirect

Maximal subgroups

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

subgroupindex
#364152
#227002
#77102
#76622
#76512
#76312
#75972
#75692