Finite Group Databasebeta

← #165147

Subgroups — #165147

Order 512.

867 conjugacy classes of subgroups.

The subgroup lattice has 349 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36538C211yesmaximal, characteristic
2#22736C212yesmaximal, complemented
3#7696C214yesmaximal, complemented
4#7659C212yesmaximal
5#7653C211yesmaximal, characteristic, complemented
6#7635C212yesmaximal, complemented
7#7601C212yesmaximal, complemented
8#7571C211yesmaximal, characteristic, complemented
9C2×#559C411yescharacteristic
10C2×#557C411yescharacteristic
11C2×#446—22no—
12#2861C2212yes—
13#2858C2211yescharacteristic
14#2568C411yescharacteristic
15#2566C412yes—
16#2558—24no—
17#2548C411yescharacteristic
18#2545—22no—
19#2545C412yes—
20#2367—22no—
21C4×C4⋊Q8C2211yescharacteristic
22C4×#124C2212yes—
23C42×Q8C2211yescharacteristic
24#1940—24no—
25#1939—22no—
26#1831C2214yes—
27#1831C2212yes—
28#1767C2214yes—
29C4×C4⋊C8C2212yes—
30#1272C2212yes—
31#1272C2212yes—
32#1272C2211yescharacteristic
33#1272C2211yescharacteristic
34#1272C2212yes—
35#1270C2214yes—
36#1270C2214yes—
37#559—44no—
38#557—44no—
39C2×C4⋊Q8—21no—
40C2×C4⋊Q8C2×C411yescharacteristic
41C2×C4⋊Q8C2×C411yescharacteristic
42C2×#127C2×C411yescharacteristic
43C2×#125—22no—
44C2×#125C2×C412yes—
45C2×#124—21no—
46C2×#124—22no—
47C2×#124C2×C412yes—
48C2×#124C2×C412yes—
49C2×#123—22no—
50C2×C4×Q8—21no—
51C2×C4×Q8C2×C411yescharacteristic
52C2×C4×Q8C2×C411yescharacteristic
53C2×C4×Q8C2×C412yes—
54C2×C4×Q8C2×C411yescharacteristic
55C2×C4×Q8D412yes—
56C2×C4×D4—21no—
57C2×C4×D4C2×C412yes—
58#446—48no—
59C2×C4⋊C8—22no—
60C2×C4⋊C8—24no—
61C2×C4⋊C8—24no—
62C2×C4⋊C8—22no—
63C2×#97—24no—
64C2×C4×C8—22no—
65#400—22no—
66#396—24no—
67#396—24no—
68#394—22no—
69#391—22no—
70#391C2×C412yes—
71#391C2311yescharacteristic
72#391C2311yescharacteristic
73#390—22no—
74#390—22no—
75#390C2×C412yes—
76#390C2×C412yes—
77#390C2312yes—
78#389—22no—
79#388—24no—
80#388—24no—
81#388—24no—
82#388—24no—
83#385—24no—
84#384C2×C411yescharacteristic
85#382C2×C412yes—
86C4×C4⋊C4—22no—
87C4×C4⋊C4C2×C412yes—
88C4×C4⋊C4C2×C411yescharacteristic
89C4×C4⋊C4D412yes—
90C4×C4⋊C4D414yes—
91C4×C22⋊C4—24no—
92C43C2311yescharacteristic
93C43C2312yes—
94#336—24no—
95#336—22no—
96#336C2312yes—
97#336C2314yes—
98#336C2312yes—
99C22×Q8—21no—
100C22×Q8C22⋊C411yescharacteristic
101C22×Q8C22×C411yescharacteristic
102C22×Q8C22×C411yescharacteristic
103C22×D4—21no—
104C22×D4—21no—
105C22×D4C22×C412yes—
106C23×C4—22no—
107C22×C8—22no—
108C22×C8—22no—
109C22×C8—24no—
110C4⋊Q8—22no—
111C4⋊Q8—41no—
112C4⋊Q8—42no—
113C4⋊Q8—44no—
114#127—42no—
115#125—44no—
116#125—48no—
117#124—24no—
118#124—42no—
119#124—44no—
120#124—44no—
121#124—44no—
122#124—44no—
123#123—48no—
124C4×Q8—24no—
125C4×Q8—24no—
126C4×Q8—24no—
127C4×Q8—44no—
128C4×Q8—42no—
129C4×Q8—42no—
130C4×Q8—44no—
131C4×D4—44no—
132C4×D4—44no—
133C2×C4⋊C4—22no—
134C2×C4⋊C4—21no—
135C2×C4⋊C4—21no—
136C2×C4⋊C4—22no—
137C2×C4⋊C4—21no—
138C2×C4⋊C4—22no—
139C2×C4⋊C4—22no—
140C2×C4⋊C4—22no—
141C2×C4⋊C4C22⋊C412yes—
142C2×C4⋊C4C22⋊C412yes—
143C2×C4⋊C4C22⋊C414yes—
144C2×C4⋊C4C22⋊C411yescharacteristic
145C2×C4⋊C4C22×C411yescharacteristic
146C2×C4⋊C4C22×C411yescharacteristic
147C2×C4⋊C4C22×C412yes—
148C2×C22⋊C4—22no—
149C2×C22⋊C4—22no—
150C2×C22⋊C4—22no—
151C2×C22⋊C4—24no—
152C2×C22⋊C4—24no—
153C2×C22⋊C4—24no—
154C2×C22⋊C4—24no—
155C2×C42—21no—
156C2×C42—22no—
157C2×C42—21no—
158C2×C42C22×C412yes—
159C2×C42C22×C411yescharacteristic
160C2×C42C22×C412yes—
161C2×C42C22×C411yescharacteristic
162C2×C42C2×D414yes—
163C2×C42C2×D414yes—
164C2×C42C2×D412yes—
165C2×C42C2×D412yes—
166C2×C42C2411yescharacteristic
167C4⋊C8—48no—
168C4⋊C8—48no—
169C4⋊C8—44no—
170C4⋊C8—44no—
171#97—48no—
172C4×C8—44no—
173#95—24no—
174#95—24no—
175#95—22no—
176#95—22no—
177#95C22⋊C414yes—
178#95C22⋊C412yes—
179C24—22no—
180C2×Q8—21no—
181C2×Q8—22no—
182C2×Q8—24no—
183C2×Q8—21no—
184C2×Q8—21no—
185C2×Q8—41no—
186C2×Q8—41no—
187C2×Q8—41no—
188C2×Q8—41no—
189C2×Q8—42no—
190C2×Q8—44no—
191C2×Q8#13112yes—
192C2×D4—22no—
193C2×D4—22no—
194C2×D4—42no—
195C2×D4—42no—
196C2×D4—42no—
197C2×D4—42no—
198C2×D4—44no—
199C2×D4—82no—
200C22×C4—21no—
201C22×C4—21no—
202C22×C4—21no—
203C22×C4—22no—
204C22×C4—21no—
205C22×C4—44no—
206C22×C4—84no—
207C22×C4C2×C22⋊C414yes—
208C22×C4C2×C22⋊C411yescharacteristic
209C22×C4C2×C22⋊C412yes—
210C22×C4C2×C22⋊C412yes—
211C22×C4C2×C22⋊C412yes—
212C22×C4C2×C22⋊C411yescharacteristic
213C22×C4C23×C411yescharacteristic
214C22×C4C22×D412yes—
215C2×C8—24no—
216C2×C8—42no—
217C2×C8—42no—
218C2×C8—42no—
219C2×C8—44no—
220C2×C8—42no—
221C2×C8—44no—
222C2×C8—42no—
223C2×C8—44no—
224C4⋊C4—24no—
225C4⋊C4—24no—
226C4⋊C4—28no—
227C4⋊C4—44no—
228C4⋊C4—41no—
229C4⋊C4—41no—
230C4⋊C4—42no—
231C4⋊C4—42no—
232C4⋊C4—42no—
233C4⋊C4—44no—
234C4⋊C4—44no—
235C4⋊C4—42no—
236C4⋊C4—44no—
237C4⋊C4—82no—
238C4⋊C4#10014yes—
239C22⋊C4—48no—
240C22⋊C4—44no—
241C22⋊C4—44no—
242C22⋊C4—44no—
243C22⋊C4—44no—
244C22⋊C4—44no—
245C22⋊C4—48no—
246C22⋊C4—84no—
247C42—22no—
248C42—21no—
249C42—22no—
250C42—21no—
251C42—24no—
252C42—24no—
253C42—24no—
254C42—22no—
255C42—42no—
256C42—44no—
257C42—41no—
258C42—42no—
259C42—44no—
260C42—42no—
261C42AGL(1,ℤ/8)14yes—
262C42#13714yes—
263C23—42no—
264C23—42no—
265C23—42no—
266C23—44no—
267C23—44no—
268C23C22×C22⋊C411yescharacteristic
269Q8—42no—
270Q8—42no—
271Q8—44no—
272Q8—81no—
273Q8—81no—
274Q8—84no—
275D4—82no—
276D4—82no—
277D4—84no—
278D4—162no—
279C2×C4—22no—
280C2×C4—22no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—22no—
284C2×C4—22no—
285C2×C4—24no—
286C2×C4—24no—
287C2×C4—22no—
288C2×C4—22no—
289C2×C4—21no—
290C2×C4—24no—
291C2×C4—24no—
292C2×C4—24no—
293C2×C4—22no—
294C2×C4—41no—
295C2×C4—41no—
296C2×C4—41no—
297C2×C4—41no—
298C2×C4—41no—
299C2×C4—41no—
300C2×C4—41no—
301C2×C4—42no—
302C2×C4—44no—
303C2×C4—42no—
304C2×C4—42no—
305C2×C4—164no—
306C2×C4C2×#10014yes—
307C2×C4C2×#10012yes—
308C2×C4#41714yes—
309C2×C4#41714yes—
310C2×C4C2×#13111yescharacteristic
311C2×C4#56811yescharacteristic
312C2×C4C2×AGL(1,ℤ/8)12yes—
313C2×C4C2×#13712yes—
314C8—44no—
315C8—82no—
316C8—84no—
317C22—82no—
318C22—82no—
319C22—82no—
320C22—84no—
321C22—84no—
322C22#288011yescharacteristic
323C22#288411yescharacteristic
324C22C22×#10011yescharacteristic
325C22C2×#41712yes—
326C22#290112yes—
327C4—21no—
328C4—21no—
329C4—24no—
330C4—24no—
331C4—22no—
332C4—22no—
333C4—42no—
334C4—41no—
335C4—42no—
336C4—44no—
337C4—81no—
338C4—81no—
339C4—81no—
340C4—82no—
341C4#156014yes—
342C2—162no—
343C2#1742511yescharacteristic
344C2#1751512yes—
345C2#1762812yes—
346C2#1768511yescharacteristic
347C2C2×#156011yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC2×#127C2×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. #22736C2semidirect
3. #7696C2semidirect
5. #7653C2semidirect
6. #7635C2semidirect
7. #7601C2semidirect
8. #7571C2semidirect

Maximal subgroups

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

subgroupindex
#365382
#227362
#76962
#76592
#76532
#76352
#76012
#75712