Finite Group Databasebeta

← #164341

Subgroups — #164341

Order 512.

987 conjugacy classes of subgroups.

The subgroup lattice has 367 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36496C211yesmaximal, characteristic
2#20573C212yesmaximal, complemented
3#7679C212yesmaximal, complemented
4#7676C214yesmaximal, complemented
5#7637C212yesmaximal, complemented
6#7635C211yesmaximal, characteristic, complemented
7#7569C212yesmaximal, complemented
8#7567C211yesmaximal, characteristic, complemented
9C2×#559C411yescharacteristic
10C2×#554C411yescharacteristic
11C2×#446—22no—
12#2861C2212yes—
13#2858C2211yescharacteristic
14#2567C411yescharacteristic
15#2566C412yes—
16#2544C412yes—
17#2541C411yescharacteristic
18#2461—24no—
19#2426—22no—
20#2361—22no—
21C4×#127C2212yes—
22C4×#124C2211yescharacteristic
23C42×Q8C2211yescharacteristic
24#1771—24no—
25C8×C4⋊C4—22no—
26#1767C2212yes—
27C4×C4⋊C8C2214yes—
28#1745C2214yes—
29#1744C2212yes—
30#1272C2212yes—
31#1272C2212yes—
32#1270C2211yescharacteristic
33#1270C2211yescharacteristic
34#1270C2212yes—
35#1270C2214yes—
36#1270C2214yes—
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—
60#446—216no—
61C2×C4⋊C8—24no—
62C2×C4⋊C8—22no—
63C2×#97—24no—
64C2×#97—22no—
65C2×C4×C8—22no—
66C2×C4×C8—24no—
67#396—24no—
68#395—22no—
69#394—24no—
70#392—22no—
71#391C2×C412yes—
72#391C2311yescharacteristic
73#390—24no—
74#390—22no—
75#390C2×C412yes—
76#390C2×C412yes—
77#390C2311yescharacteristic
78#390C2312yes—
79#389—22no—
80#388—24no—
81#388—24no—
82#388—22no—
83#386—24no—
84#385—24no—
85#384C2×C412yes—
86#382C2×C411yescharacteristic
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×C411yescharacteristic
102C22×D4—21no—
103C22×D4—24no—
104C22×D4—21no—
105C22×D4C22×C412yes—
106C22×D4C22×C411yescharacteristic
107C23×C4—21no—
108C23×C4—22no—
109C23×C4—22no—
110C22×C8—22no—
111C22×C8—22no—
112C22×C8—24no—
113C4⋊Q8—42no—
114#127—24no—
115#127—42no—
116#127—42no—
117#125—44no—
118#124—22no—
119#124—41no—
120#124—44no—
121#124—44no—
122#124—82no—
123#124—82no—
124#123—48no—
125C22⋊Q8—48no—
126C22⋊Q8—44no—
127#121—48no—
128C4×Q8—24no—
129C4×Q8—28no—
130C4×Q8—42no—
131C4×D4—44no—
132C4×D4—42no—
133C4×D4—44no—
134C4×D4—44no—
135C4×D4—84no—
136C4×D4—84no—
137C2×C4⋊C4—22no—
138C2×C4⋊C4—21no—
139C2×C4⋊C4—21no—
140C2×C4⋊C4—22no—
141C2×C4⋊C4—24no—
142C2×C4⋊C4—44no—
143C2×C4⋊C4C22⋊C412yes—
144C2×C4⋊C4C22⋊C414yes—
145C2×C4⋊C4C22⋊C411yescharacteristic
146C2×C4⋊C4C22⋊C412yes—
147C2×C4⋊C4C22×C411yescharacteristic
148C2×C4⋊C4C22×C411yescharacteristic
149C2×C4⋊C4C22×C412yes—
150C2×C22⋊C4—22no—
151C2×C22⋊C4—21no—
152C2×C22⋊C4—22no—
153C2×C22⋊C4—22no—
154C2×C22⋊C4—24no—
155C2×C22⋊C4—22no—
156C2×C22⋊C4—22no—
157C2×C22⋊C4—22no—
158C2×C22⋊C4—24no—
159C2×C22⋊C4—24no—
160C2×C42—21no—
161C2×C42C22×C412yes—
162C2×C42C22×C411yescharacteristic
163C2×C42C22×C411yescharacteristic
164C2×C42C22×C412yes—
165C2×C42C2×D414yes—
166C2×C42C2×D412yes—
167C2×C42C2×D414yes—
168C2×C42C2×D412yes—
169C2×C42C2411yescharacteristic
170C4⋊C8—28no—
171C4⋊C8—216no—
172#97—216no—
173#97—82no—
174C4×C8—28no—
175C4×C8—84no—
176#95—22no—
177#95—24no—
178#95—22no—
179#95C22⋊C414yes—
180#95C22⋊C412yes—
181C24—21no—
182C24—22no—
183C2×Q8—22no—
184C2×Q8—24no—
185C2×Q8—21no—
186C2×Q8—21no—
187C2×Q8—42no—
188C2×Q8—41no—
189C2×Q8—41no—
190C2×Q8—82no—
191C2×D4—22no—
192C2×D4—21no—
193C2×D4—22no—
194C2×D4—41no—
195C2×D4—41no—
196C2×D4—48no—
197C2×D4—42no—
198C2×D4—42no—
199C2×D4—42no—
200C2×D4—42no—
201C2×D4—48no—
202C2×D4—44no—
203C2×D4—84no—
204C2×D4—82no—
205C2×D4#13112yes—
206C22×C4—21no—
207C22×C4—21no—
208C22×C4—21no—
209C22×C4—22no—
210C22×C4—42no—
211C22×C4—44no—
212C22×C4—44no—
213C22×C4—44no—
214C22×C4—84no—
215C22×C4—84no—
216C22×C4C2×C22⋊C412yes—
217C22×C4C2×C22⋊C411yescharacteristic
218C22×C4C2×C22⋊C412yes—
219C22×C4C2×C22⋊C412yes—
220C22×C4C2×C22⋊C414yes—
221C22×C4C2×C22⋊C411yescharacteristic
222C22×C4C23×C411yescharacteristic
223C22×C4C22×D412yes—
224C2×C8—24no—
225C2×C8—24no—
226C2×C8—28no—
227C2×C8—42no—
228C2×C8—42no—
229C2×C8—44no—
230C2×C8—42no—
231C2×C8—44no—
232C2×C8—42no—
233C4⋊C4—24no—
234C4⋊C4—28no—
235C4⋊C4—41no—
236C4⋊C4—41no—
237C4⋊C4—42no—
238C4⋊C4—42no—
239C4⋊C4—44no—
240C4⋊C4—44no—
241C4⋊C4—42no—
242C4⋊C4—84no—
243C4⋊C4#10014yes—
244C4⋊C4#10418yes—
245C22⋊C4—44no—
246C22⋊C4—44no—
247C22⋊C4—42no—
248C22⋊C4—44no—
249C22⋊C4—44no—
250C22⋊C4—42no—
251C22⋊C4—44no—
252C22⋊C4—42no—
253C22⋊C4—44no—
254C22⋊C4—44no—
255C22⋊C4—48no—
256C22⋊C4—84no—
257C42—21no—
258C42—22no—
259C42—21no—
260C42—22no—
261C42—24no—
262C42—22no—
263C42—24no—
264C42—22no—
265C42—24no—
266C42—22no—
267C42—24no—
268C42—28no—
269C42—41no—
270C42—42no—
271C23—22no—
272C23—41no—
273C23—41no—
274C23—42no—
275C23—42no—
276C23—42no—
277C23—42no—
278C23—42no—
279C23—44no—
280C23—44no—
281C23C22×C22⋊C411yescharacteristic
282Q8—44no—
283Q8—81no—
284Q8—81no—
285Q8—162no—
286D4—42no—
287D4—42no—
288D4—816no—
289D4—82no—
290D4—82no—
291D4—84no—
292D4—162no—
293C2×C4—24no—
294C2×C4—22no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—24no—
298C2×C4—24no—
299C2×C4—22no—
300C2×C4—22no—
301C2×C4—22no—
302C2×C4—21no—
303C2×C4—22no—
304C2×C4—22no—
305C2×C4—22no—
306C2×C4—24no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—42no—
313C2×C4—44no—
314C2×C4—42no—
315C2×C4—42no—
316C2×C4—84no—
317C2×C4—164no—
318C2×C4—164no—
319C2×C4C2×#10014yes—
320C2×C4C2×#10012yes—
321C2×C4C2×#10418yes—
322C2×C4C2×#10414yes—
323C2×C4C2×#13111yescharacteristic
324C2×C4#56811yescharacteristic
325C2×C4#57612yes—
326C2×C4#57712yes—
327C8—44no—
328C8—48no—
329C8—44no—
330C22—42no—
331C22—44no—
332C22—81no—
333C22—82no—
334C22—81no—
335C22—82no—
336C22—82no—
337C22—84no—
338C22—84no—
339C22—82no—
340C22#288011yescharacteristic
341C22#288411yescharacteristic
342C22C22×#10011yescharacteristic
343C22#289712yes—
344C22C22×#10412yes—
345C4—21no—
346C4—21no—
347C4—24no—
348C4—28no—
349C4—22no—
350C4—22no—
351C4—41no—
352C4—42no—
353C4—42no—
354C4—81no—
355C4—81no—
356C4—82no—
357C4—81no—
358C4#152514yes—
359C2—82no—
360C2—162no—
361C2#1742511yescharacteristic
362C2#1751712yes—
363C2#1755012yes—
364C2C2×#152511yescharacteristic
365C2#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. #20573C2semidirect
3. #7679C2semidirect
4. #7676C2semidirect
5. #7637C2semidirect
6. #7635C2semidirect
7. #7569C2semidirect
8. #7567C2semidirect

Maximal subgroups

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

subgroupindex
#364962
#205732
#76792
#76762
#76372
#76352
#75692
#75672