Finite Group Databasebeta

← #165087

Subgroups — #165087

Order 512.

878 conjugacy classes of subgroups.

The subgroup lattice has 360 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36534C211yesmaximal, characteristic
2#22736C212yesmaximal, complemented
3#7696C214yesmaximal, complemented
4#7662C212yesmaximal, complemented
5#7651C211yesmaximal, characteristic
6#7635C212yesmaximal, complemented
7#7601C212yesmaximal, complemented
8#7569C211yesmaximal, characteristic
9C2×#556C411yescharacteristic
10C2×#554C411yescharacteristic
11C2×#446—22no—
12#2862C2211yescharacteristic
13#2861C2212yes—
14#2565C412yes—
15#2563C411yescharacteristic
16#2560—24no—
17#2547C411yescharacteristic
18#2544C412yes—
19#2536—22no—
20#2352—22no—
21C4×#127C2211yescharacteristic
22C4×#124C2212yes—
23C42×Q8C2211yescharacteristic
24#1940—24no—
25#1939—22no—
26#1831C2214yes—
27#1831C2212yes—
28#1767C2214yes—
29C4×C4⋊C8C2212yes—
30#1272C2214yes—
31#1272C2214yes—
32#1270C2212yes—
33#1270C2212yes—
34#1270C2211yescharacteristic
35#1270C2211yescharacteristic
36#1270C2212yes—
37#556—44no—
38#554—44no—
39C2×C4⋊Q8C2×C411yescharacteristic
40C2×C4⋊Q8C2×C411yescharacteristic
41C2×#127C2×C411yescharacteristic
42C2×#126—22no—
43C2×#125C2×C412yes—
44C2×#124C2×C412yes—
45C2×#124C2×C412yes—
46C2×C22⋊Q8—21no—
47C2×C22⋊Q8—21no—
48C2×C22⋊Q8—22no—
49C2×C4×Q8C2×C412yes—
50C2×C4×Q8C2×C411yescharacteristic
51C2×C4×Q8D412yes—
52C2×C4×D4—22no—
53C2×C4×D4C2×C411yescharacteristic
54C2×C4×D4C2×C411yescharacteristic
55C2×C4×D4C2×C412yes—
56C2×#117—21no—
57C22×C4⋊C4—21no—
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#398—24no—
66#397—22no—
67#394—24no—
68#393—22no—
69#391—22no—
70#391C2×C412yes—
71#391C2×C412yes—
72#391C2312yes—
73#390—22no—
74#390C2×C412yes—
75#390C2311yescharacteristic
76#390C2311yescharacteristic
77#388—24no—
78#388—24no—
79#388—24no—
80#387—24no—
81#386—24no—
82#384C2×C411yescharacteristic
83#382—22no—
84#382C2×C412yes—
85#380—22no—
86#380—22no—
87C4×C4⋊C4C2×C411yescharacteristic
88C4×C4⋊C4C2×C412yes—
89C4×C4⋊C4D414yes—
90C4×C4⋊C4D412yes—
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×C412yes—
102C22×D4—22no—
103C22×D4C22×C411yescharacteristic
104C22×D4C22×C411yescharacteristic
105C23×C4—21no—
106C23×C4—21no—
107C23×C4—22no—
108C22×C8—22no—
109C22×C8—22no—
110C22×C8—24no—
111C4⋊Q8—42no—
112C4⋊Q8—42no—
113#127—22no—
114#127—41no—
115#126—48no—
116#125—44no—
117#124—24no—
118#124—42no—
119#124—44no—
120C22⋊Q8—48no—
121C22⋊Q8—44no—
122C22⋊Q8—44no—
123C4×Q8—24no—
124C4×Q8—24no—
125C4×Q8—24no—
126C4×Q8—44no—
127C4×D4—42no—
128C4×D4—44no—
129C4×D4—42no—
130C4×D4—44no—
131C4×D4—44no—
132#117—44no—
133C2×C4⋊C4—22no—
134C2×C4⋊C4—21no—
135C2×C4⋊C4—21no—
136C2×C4⋊C4—22no—
137C2×C4⋊C4—22no—
138C2×C4⋊C4—22no—
139C2×C4⋊C4—21no—
140C2×C4⋊C4—44no—
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—21no—
151C2×C22⋊C4—21no—
152C2×C22⋊C4—22no—
153C2×C22⋊C4—24no—
154C2×C22⋊C4—22no—
155C2×C22⋊C4—22no—
156C2×C22⋊C4—24no—
157C2×C42—22no—
158C2×C42C22×C411yescharacteristic
159C2×C42C22×C412yes—
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—24no—
176#95—24no—
177#95C22⋊C414yes—
178#95C22⋊C412yes—
179C24—21no—
180C24—21no—
181C2×Q8—22no—
182C2×Q8—24no—
183C2×Q8—22no—
184C2×Q8—22no—
185C2×Q8—42no—
186C2×Q8—42no—
187C2×Q8—42no—
188C2×Q8—82no—
189C2×D4—21no—
190C2×D4—21no—
191C2×D4—21no—
192C2×D4—41no—
193C2×D4—41no—
194C2×D4—44no—
195C2×D4—41no—
196C2×D4—41no—
197C2×D4—44no—
198C2×D4—82no—
199C2×D4#13112yes—
200C22×C4—22no—
201C22×C4—22no—
202C22×C4—21no—
203C22×C4—21no—
204C22×C4—42no—
205C22×C4—42no—
206C22×C4—44no—
207C22×C4—44no—
208C22×C4—44no—
209C22×C4—84no—
210C22×C4C2×C22⋊C414yes—
211C22×C4C2×C22⋊C411yescharacteristic
212C22×C4C2×C22⋊C412yes—
213C22×C4C2×C22⋊C412yes—
214C22×C4C2×C22⋊C412yes—
215C22×C4C2×C22⋊C411yescharacteristic
216C22×C4C23×C411yescharacteristic
217C22×C4C22×D412yes—
218C2×C8—24no—
219C2×C8—42no—
220C2×C8—42no—
221C2×C8—42no—
222C2×C8—44no—
223C2×C8—42no—
224C2×C8—44no—
225C2×C8—42no—
226C2×C8—44no—
227C4⋊C4—24no—
228C4⋊C4—24no—
229C4⋊C4—28no—
230C4⋊C4—41no—
231C4⋊C4—41no—
232C4⋊C4—42no—
233C4⋊C4—44no—
234C4⋊C4—44no—
235C4⋊C4—42no—
236C4⋊C4—44no—
237C4⋊C4—42no—
238C4⋊C4—42no—
239C4⋊C4—44no—
240C4⋊C4#10014yes—
241C22⋊C4—48no—
242C22⋊C4—42no—
243C22⋊C4—44no—
244C22⋊C4—42no—
245C22⋊C4—44no—
246C22⋊C4—42no—
247C22⋊C4—42no—
248C22⋊C4—42no—
249C22⋊C4—42no—
250C22⋊C4—44no—
251C22⋊C4—84no—
252C42—22no—
253C42—21no—
254C42—22no—
255C42—21no—
256C42—24no—
257C42—24no—
258C42—22no—
259C42—24no—
260C42—44no—
261C42—42no—
262C42—41no—
263C42—44no—
264C42AGL(1,ℤ/8)14yes—
265C42#13714yes—
266C23—22no—
267C23—41no—
268C23—41no—
269C23—42no—
270C23—42no—
271C23—41no—
272C23—41no—
273C23—41no—
274C23—42no—
275C23—42no—
276C23C22×C22⋊C411yescharacteristic
277Q8—44no—
278Q8—82no—
279Q8—82no—
280Q8—162no—
281D4—42no—
282D4—42no—
283D4—88no—
284D4—81no—
285D4—81no—
286C2×C4—22no—
287C2×C4—22no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—22no—
291C2×C4—24no—
292C2×C4—22no—
293C2×C4—24no—
294C2×C4—21no—
295C2×C4—22no—
296C2×C4—24no—
297C2×C4—24no—
298C2×C4—24no—
299C2×C4—22no—
300C2×C4—42no—
301C2×C4—42no—
302C2×C4—41no—
303C2×C4—42no—
304C2×C4—41no—
305C2×C4—42no—
306C2×C4—42no—
307C2×C4—42no—
308C2×C4—44no—
309C2×C4—84no—
310C2×C4—84no—
311C2×C4—164no—
312C2×C4C2×#10014yes—
313C2×C4C2×#10012yes—
314C2×C4#41714yes—
315C2×C4#41714yes—
316C2×C4C2×#13111yescharacteristic
317C2×C4#56811yescharacteristic
318C2×C4C2×AGL(1,ℤ/8)12yes—
319C2×C4C2×#13712yes—
320C8—44no—
321C8—82no—
322C8—84no—
323C22—42no—
324C22—44no—
325C22—81no—
326C22—81no—
327C22—82no—
328C22—81no—
329C22—81no—
330C22—81no—
331C22—82no—
332C22—82no—
333C22#288011yescharacteristic
334C22#288411yescharacteristic
335C22C22×#10011yescharacteristic
336C22C2×#41712yes—
337C22#290112yes—
338C4—21no—
339C4—21no—
340C4—24no—
341C4—24no—
342C4—22no—
343C4—22no—
344C4—41no—
345C4—42no—
346C4—44no—
347C4—82no—
348C4—82no—
349C4—81no—
350C4—81no—
351C4#156014yes—
352C2—82no—
353C2—161no—
354C2#1742511yescharacteristic
355C2#1751212yes—
356C2#1762812yes—
357C2#1768511yescharacteristic
358C2C2×#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
4. #7662C2semidirect
6. #7635C2semidirect
7. #7601C2semidirect

Maximal subgroups

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

subgroupindex
#365342
#227362
#76962
#76622
#76512
#76352
#76012
#75692