Finite Group Databasebeta

← #165139

Subgroups — #165139

Order 512.

895 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#36276C211yesmaximal, characteristic
2#22839C212yesmaximal
3#7707C214yesmaximal, complemented
4#7662C212yesmaximal, complemented
5#7653C211yesmaximal, characteristic, complemented
6#7629C212yesmaximal, complemented
7#7595C212yesmaximal, complemented
8#7571C211yesmaximal, characteristic, complemented
9C2×#554C411yescharacteristic
10C2×D4×Q8C411yescharacteristic
11#2862C2211yescharacteristic
12#2857C2212yes—
13#2567C411yescharacteristic
14#2550—22no—
15#2549—24no—
16#2546C411yescharacteristic
17#2502C412yes—
18#2434C412yes—
19#2349—22no—
20C4×C4⋊Q8C2211yescharacteristic
21C2×#391C2212yes—
22C42×D4C2211yescharacteristic
23#1954—22no—
24#1924—24no—
25#1831C2212yes—
26#1817C2214yes—
27#1770—22no—
28C4×C4⋊C8C2212yes—
29#1749C2214yes—
30#1272C2212yes—
31#1272C2212yes—
32#1272C2214yes—
33#1272C2214yes—
34#1272C2211yescharacteristic
35#1272C2211yescharacteristic
36#1272C2212yes—
37C23×Q8C2×C412yes—
38#554—44no—
39D4×Q8—44no—
40C2×C4⋊Q8—21no—
41C2×C4⋊Q8C2×C411yescharacteristic
42C2×C4⋊Q8C2×C411yescharacteristic
43C2×C4⋊Q8C2×C412yes—
44C2×C4⋊Q8C2×C411yescharacteristic
45C2×#125—21no—
46C2×C22⋊Q8—22no—
47C2×C22⋊Q8—22no—
48C2×C22⋊Q8C2×C412yes—
49C2×C22⋊Q8C2×C412yes—
50C2×C4×Q8C2×C411yescharacteristic
51C2×C4×Q8C2×C411yescharacteristic
52C2×C4×D4—21no—
53C2×C4×D4—21no—
54C2×C4×D4C2×C411yescharacteristic
55C2×C4×D4D412yes—
56C22×C4⋊C4C2×C412yes—
57C22×C42C2312yes—
58C2×C4⋊C8—22no—
59C2×C4⋊C8—22no—
60C2×C22⋊C8—24no—
61C2×C22⋊C8—24no—
62C2×#97—24no—
63C2×C4×C8—22no—
64#398—24no—
65#398—22no—
66#395—22no—
67#395—24no—
68#391—22no—
69#391—24no—
70#391C2×C412yes—
71#391C2×C412yes—
72#391C2×C412yes—
73#391C2311yescharacteristic
74#391C2311yescharacteristic
75#391C2312yes—
76#389—22no—
77#389—22no—
78#389—22no—
79#389—24no—
80#385—24no—
81#385—24no—
82#384C2×C411yescharacteristic
83#384C2×C412yes—
84#382—22no—
85#380—24no—
86C4×C4⋊C4—22no—
87C4×C4⋊C4C2×C411yescharacteristic
88C4×C4⋊C4D412yes—
89C4×C22⋊C4C2×C412yes—
90C4×C22⋊C4D414yes—
91C2×#95—24no—
92C43C2311yescharacteristic
93#336—24no—
94#336—22no—
95#336C2312yes—
96#336C2314yes—
97#336C2312yes—
98C22×Q8—22no—
99C22×Q8—21no—
100C22×Q8—24no—
101C22×Q8—42no—
102C22×Q8—44no—
103C22×Q8—82no—
104C22×Q8C22×C411yescharacteristic
105C22×Q8C22×C412yes—
106C22×Q8C22×C411yescharacteristic
107C22×D4C22⋊C411yescharacteristic
108C23×C4—22no—
109C23×C4—22no—
110C23×C4C22×C412yes—
111C23×C4C2×D414yes—
112C22×C8—22no—
113C22×C8—22no—
114C22×C8—24no—
115C4⋊Q8—22no—
116C4⋊Q8—41no—
117C4⋊Q8—42no—
118C4⋊Q8—42no—
119C4⋊Q8—44no—
120C4⋊Q8—82no—
121#125—44no—
122C22⋊Q8—48no—
123C22⋊Q8—48no—
124C22⋊Q8—44no—
125C22⋊Q8—44no—
126C4×Q8—42no—
127C4×Q8—42no—
128C4×D4—24no—
129C4×D4—24no—
130C4×D4—24no—
131C4×D4—44no—
132C4×D4—44no—
133C2×C4⋊C4—22no—
134C2×C4⋊C4—21no—
135C2×C4⋊C4—21no—
136C2×C4⋊C4—21no—
137C2×C4⋊C4—21no—
138C2×C4⋊C4—21no—
139C2×C4⋊C4—22no—
140C2×C4⋊C4—22no—
141C2×C4⋊C4—22no—
142C2×C4⋊C4—24no—
143C2×C4⋊C4—22no—
144C2×C4⋊C4—24no—
145C2×C4⋊C4—44no—
146C2×C4⋊C4C22⋊C411yescharacteristic
147C2×C4⋊C4C22⋊C412yes—
148C2×C4⋊C4C22×C411yescharacteristic
149C2×C4⋊C4C22×C412yes—
150C2×C4⋊C4C22×C411yescharacteristic
151C2×C22⋊C4—22no—
152C2×C22⋊C4—22no—
153C2×C22⋊C4C22⋊C412yes—
154C2×C22⋊C4C22⋊C414yes—
155C2×C42—21no—
156C2×C42—21no—
157C2×C42—82no—
158C2×C42C22×C411yescharacteristic
159C2×C42C22×C411yescharacteristic
160C2×C42C22×C412yes—
161C2×C42C2×D412yes—
162C2×C42C2×D414yes—
163C2×C42C2×D412yes—
164C2×C42C2411yescharacteristic
165C4⋊C8—44no—
166C4⋊C8—44no—
167C22⋊C8—48no—
168C22⋊C8—84no—
169#97—84no—
170C4×C8—44no—
171#95—24no—
172#95—24no—
173#95—24no—
174#95—22no—
175#95—24no—
176#95—22no—
177#95C22⋊C414yes—
178#95C22⋊C412yes—
179C24C2×C22⋊C412yes—
180C2×Q8—21no—
181C2×Q8—21no—
182C2×Q8—22no—
183C2×Q8—21no—
184C2×Q8—22no—
185C2×Q8—44no—
186C2×Q8—44no—
187C2×Q8—42no—
188C2×Q8—41no—
189C2×Q8—41no—
190C2×Q8—41no—
191C2×Q8—42no—
192C2×Q8—41no—
193C2×Q8—42no—
194C2×Q8—82no—
195C2×Q8—82no—
196C2×Q8—88no—
197C2×Q8—88no—
198C2×Q8#13112yes—
199C2×D4—24no—
200C2×D4#10014yes—
201C22×C4—21no—
202C22×C4—24no—
203C22×C4—22no—
204C22×C4—22no—
205C22×C4—24no—
206C22×C4—24no—
207C22×C4—21no—
208C22×C4—22no—
209C22×C4—22no—
210C22×C4—21no—
211C22×C4—21no—
212C22×C4—42no—
213C22×C4—44no—
214C22×C4—44no—
215C22×C4—44no—
216C22×C4—44no—
217C22×C4—44no—
218C22×C4—82no—
219C22×C4—84no—
220C22×C4C2×C22⋊C414yes—
221C22×C4C2×C22⋊C411yescharacteristic
222C22×C4C2×C22⋊C412yes—
223C22×C4C2×C22⋊C412yes—
224C22×C4C2×C22⋊C411yescharacteristic
225C22×C4C23×C411yescharacteristic
226C22×C4C22×D412yes—
227C2×C8—24no—
228C2×C8—42no—
229C2×C8—42no—
230C2×C8—44no—
231C2×C8—44no—
232C2×C8—42no—
233C2×C8—42no—
234C2×C8—42no—
235C2×C8—44no—
236C4⋊C4—22no—
237C4⋊C4—24no—
238C4⋊C4—44no—
239C4⋊C4—41no—
240C4⋊C4—42no—
241C4⋊C4—41no—
242C4⋊C4—44no—
243C4⋊C4—42no—
244C4⋊C4—42no—
245C4⋊C4—44no—
246C4⋊C4—42no—
247C4⋊C4—44no—
248C4⋊C4—42no—
249C4⋊C4—42no—
250C4⋊C4—82no—
251C4⋊C4—84no—
252C4⋊C4—84no—
253C22⋊C4—24no—
254C22⋊C4—28no—
255C22⋊C4—44no—
256C22⋊C4—44no—
257C42—21no—
258C42—21no—
259C42—22no—
260C42—44no—
261C42—42no—
262C42—41no—
263C42—42no—
264C42—42no—
265C42—42no—
266C42#13714yes—
267C42#13714yes—
268C23—22no—
269C23—22no—
270C23—24no—
271C23—24no—
272C23C2×#10014yes—
273C23C22×C22⋊C411yescharacteristic
274Q8—42no—
275Q8—42no—
276Q8—81no—
277Q8—81no—
278Q8—82no—
279Q8—82no—
280Q8—88no—
281Q8—162no—
282D4—28no—
283C2×C4—22no—
284C2×C4—22no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—22no—
288C2×C4—22no—
289C2×C4—22no—
290C2×C4—22no—
291C2×C4—24no—
292C2×C4—22no—
293C2×C4—21no—
294C2×C4—22no—
295C2×C4—24no—
296C2×C4—22no—
297C2×C4—24no—
298C2×C4—44no—
299C2×C4—44no—
300C2×C4—41no—
301C2×C4—41no—
302C2×C4—41no—
303C2×C4—42no—
304C2×C4—41no—
305C2×C4—42no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—42no—
310C2×C4—42no—
311C2×C4—42no—
312C2×C4—42no—
313C2×C4—44no—
314C2×C4—84no—
315C2×C4—84no—
316C2×C4—88no—
317C2×C4—164no—
318C2×C4C2×#10012yes—
319C2×C4#41914yes—
320C2×C4#41914yes—
321C2×C4C2×#13111yescharacteristic
322C2×C4#56811yescharacteristic
323C2×C4C2×#13712yes—
324C2×C4C2×#13712yes—
325C8—44no—
326C8—82no—
327C8—84no—
328C22—24no—
329C22—24no—
330C22—24no—
331C22—44no—
332C22—44no—
333C22#288011yescharacteristic
334C22#288311yescharacteristic
335C22C22×#10011yescharacteristic
336C22C2×#41912yes—
337C22#290112yes—
338C4—21no—
339C4—21no—
340C4—24no—
341C4—22no—
342C4—22no—
343C4—42no—
344C4—42no—
345C4—41no—
346C4—44no—
347C4—82no—
348C4—81no—
349C4—81no—
350C4—81no—
351C4—82no—
352C4#155814yes—
353C2—44no—
354C2#1742111yescharacteristic
355C2#1751412yes—
356C2#1762712yes—
357C2#1768511yescharacteristic
358C2C2×#155811yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC22×D4C22⋊C4
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
3. #7707C2semidirect
4. #7662C2semidirect
5. #7653C2semidirect
6. #7629C2semidirect
7. #7595C2semidirect
8. #7571C2semidirect

Maximal subgroups

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

subgroupindex
#362762
#228392
#77072
#76622
#76532
#76292
#75952
#75712