Finite Group Databasebeta

← #223154

Subgroups — #223154

Order 512.

910 conjugacy classes of subgroups.

The subgroup lattice has 370 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36185C211yesmaximal, characteristic
2#21270C212yesmaximal, complemented
3#9890C212yesmaximal, complemented
4#9123C214yesmaximal, complemented
5#8721C212yesmaximal, complemented
6#8579C211yesmaximal, characteristic, complemented
7C4×#329C211yesmaximal, characteristic, complemented
8#8031C212yesmaximal, complemented
9C2×#556C411yescharacteristic
10C2×#555C411yescharacteristic
11C2×#432C2214yes—
12#2862C2211yescharacteristic
13#2850C2212yes—
14#2567C411yescharacteristic
15#2542C411yescharacteristic
16#2521—24no—
17#2500C412yes—
18#2499C412yes—
19#2427—22no—
20#2362—22no—
21C4×#124C2211yescharacteristic
22#2285C2212yes—
23C42×D4C2211yescharacteristic
24#1954—22no—
25#1939—22no—
26#1924—24no—
27#1767C2212yes—
28#1749C2214yes—
29C4×#97C2212yes—
30C2×#329C2212yes—
31C2×#329C2212yes—
32#1295C2211yescharacteristic
33#1295C2212yes—
34#1295C2214yes—
35#1295C2211yescharacteristic
36#1295C2214yes—
37#556—44no—
38#555—44no—
39C2×C4⋊Q8C2×C411yescharacteristic
40C2×C4⋊Q8C2×C411yescharacteristic
41C2×#126—22no—
42C2×#124—21no—
43C2×#124—22no—
44C2×#124C2311yescharacteristic
45C2×C22⋊Q8—22no—
46C2×C22⋊Q8C2×C412yes—
47C2×#121—21no—
48C2×#121C2×C412yes—
49C2×C4×Q8—21no—
50C2×C4×Q8C2×C411yescharacteristic
51C2×C4×D4C2×C411yescharacteristic
52C2×C4×D4C2×C411yescharacteristic
53C2×C4×D4D412yes—
54C2×#117—21no—
55C2×#117C2×C412yes—
56C2×#117C2×C412yes—
57C22×C42C2312yes—
58#432—28no—
59#432—44no—
60C2×C4⋊C8—24no—
61C2×C4⋊C8—22no—
62C2×C22⋊C8—24no—
63C2×C22⋊C8—24no—
64C2×#97—22no—
65C2×#97C2314yes—
66C2×#97C2312yes—
67C2×#97C2312yes—
68C2×C4×C8—22no—
69#399—22no—
70#397—24no—
71#395—22no—
72#394—24no—
73#391—22no—
74#391—22no—
75#391C2×C412yes—
76#391C2×C411yescharacteristic
77#390C2×C412yes—
78#390C2×C411yescharacteristic
79#389—22no—
80#388—24no—
81#388—22no—
82#388—24no—
83#387—24no—
84#386—24no—
85#384C2×C412yes—
86#384C2×C412yes—
87#382C2×C412yes—
88#382C2312yes—
89#382C2311yescharacteristic
90#380—24no—
91C4×C4⋊C4—24no—
92C4×C4⋊C4C2×C411yescharacteristic
93C4×C4⋊C4D412yes—
94C4×C22⋊C4—22no—
95C4×C22⋊C4C2×C412yes—
96C4×C22⋊C4D414yes—
97C43C2311yescharacteristic
98#336—22no—
99#336—24no—
100#329—24no—
101#329—42no—
102C22×Q8—21no—
103C22×Q8—21no—
104C22×Q8C22×C411yescharacteristic
105C22×D4—22no—
106C22×D4C22⋊C411yescharacteristic
107C22×D4C22×C411yescharacteristic
108C23×C4—21no—
109C23×C4C22×C412yes—
110C23×C4C2×D414yes—
111C22×C8—22no—
112C22×C8—22no—
113C22×C8—24no—
114C4⋊Q8—42no—
115C4⋊Q8—42no—
116#126—48no—
117#124—22no—
118#124—21no—
119#124—21no—
120#124—48no—
121#124—44no—
122C22⋊Q8—48no—
123C22⋊Q8—44no—
124#121—44no—
125#121—44no—
126C4×Q8—42no—
127C4×Q8—44no—
128C4×D4—24no—
129C4×D4—44no—
130C4×D4—42no—
131#117—44no—
132#117—44no—
133#117—44no—
134C2×C4⋊C4—21no—
135C2×C4⋊C4—22no—
136C2×C4⋊C4—21no—
137C2×C4⋊C4—24no—
138C2×C4⋊C4—21no—
139C2×C4⋊C4—24no—
140C2×C4⋊C4C22⋊C412yes—
141C2×C4⋊C4C22⋊C411yescharacteristic
142C2×C4⋊C4C22×C411yescharacteristic
143C2×C4⋊C4C22×C411yescharacteristic
144C2×C4⋊C4C22×C412yes—
145C2×C4⋊C4C22×C412yes—
146C2×C22⋊C4—22no—
147C2×C22⋊C4—22no—
148C2×C22⋊C4—22no—
149C2×C22⋊C4—22no—
150C2×C22⋊C4—21no—
151C2×C22⋊C4—24no—
152C2×C22⋊C4—22no—
153C2×C22⋊C4—22no—
154C2×C22⋊C4C22⋊C412yes—
155C2×C22⋊C4C22⋊C414yes—
156C2×C42—24no—
157C2×C42—22no—
158C2×C42—22no—
159C2×C42—22no—
160C2×C42—22no—
161C2×C42—21no—
162C2×C42C22×C411yescharacteristic
163C2×C42C22×C412yes—
164C2×C42C22×C411yescharacteristic
165C2×C42C2×D412yes—
166C2×C42C2×D414yes—
167C2×C42C2×D412yes—
168C2×C42C2411yescharacteristic
169C4⋊C8—48no—
170C4⋊C8—44no—
171C22⋊C8—48no—
172C22⋊C8—48no—
173#97—22no—
174#97—24no—
175#97—22no—
176#97—24no—
177#97—22no—
178#97—44no—
179#97#4114yes—
180C4×C8—44no—
181#95—22no—
182#95—24no—
183#95—24no—
184#95—22no—
185#95—24no—
186#95C22⋊C414yes—
187#95C22⋊C412yes—
188C24—21no—
189C24C2×C22⋊C412yes—
190C2×Q8—21no—
191C2×Q8—21no—
192C2×Q8—42no—
193C2×Q8—41no—
194C2×Q8—41no—
195C2×Q8—42no—
196C2×Q8—44no—
197C2×Q8—82no—
198C2×D4—21no—
199C2×D4—21no—
200C2×D4—22no—
201C2×D4—24no—
202C2×D4—41no—
203C2×D4—41no—
204C2×D4—44no—
205C2×D4—84no—
206C22×C4—22no—
207C22×C4—22no—
208C22×C4—24no—
209C22×C4—24no—
210C22×C4—24no—
211C22×C4—21no—
212C22×C4—22no—
213C22×C4—21no—
214C22×C4—22no—
215C22×C4—21no—
216C22×C4—42no—
217C22×C4—44no—
218C22×C4—44no—
219C22×C4—42no—
220C22×C4—44no—
221C22×C4C2×C22⋊C412yes—
222C22×C4C2×C22⋊C414yes—
223C22×C4C2×C22⋊C411yescharacteristic
224C22×C4C2×C22⋊C412yes—
225C22×C4C2×C22⋊C411yescharacteristic
226C22×C4C23×C411yescharacteristic
227C22×C4C22×D412yes—
228C2×C8—24no—
229C2×C8—24no—
230C2×C8—42no—
231C2×C8—42no—
232C2×C8—42no—
233C2×C8—42no—
234C2×C8—44no—
235C2×C8—44no—
236C2×C8—44no—
237C4⋊C4—24no—
238C4⋊C4—41no—
239C4⋊C4—41no—
240C4⋊C4—42no—
241C4⋊C4—42no—
242C4⋊C4—42no—
243C4⋊C4—48no—
244C4⋊C4—42no—
245C4⋊C4—42no—
246C4⋊C4—44no—
247C4⋊C4—84no—
248C4⋊C4#10014yes—
249C22⋊C4—24no—
250C22⋊C4—44no—
251C22⋊C4—42no—
252C22⋊C4—48no—
253C22⋊C4—44no—
254C22⋊C4—44no—
255C22⋊C4—44no—
256C22⋊C4—42no—
257C22⋊C4—42no—
258C22⋊C4—44no—
259C22⋊C4—82no—
260C42—22no—
261C42—22no—
262C42—24no—
263C42—24no—
264C42—22no—
265C42—24no—
266C42—42no—
267C42—41no—
268C42—44no—
269C42—44no—
270C42—42no—
271C42C2×#4112yes—
272C42H(5,𝔽2)11yescharacteristic
273C422−1+411yescharacteristic
274C23—22no—
275C23—22no—
276C23—22no—
277C23—24no—
278C23—24no—
279C23—41no—
280C23—41no—
281C23—41no—
282C23—42no—
283C23—42no—
284C23C22×C22⋊C411yescharacteristic
285Q8—81no—
286Q8—81no—
287Q8—84no—
288Q8—162no—
289D4—44no—
290D4—81no—
291D4—81no—
292D4—164no—
293C2×C4—22no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—22no—
297C2×C4—22no—
298C2×C4—22no—
299C2×C4—24no—
300C2×C4—21no—
301C2×C4—22no—
302C2×C4—22no—
303C2×C4—24no—
304C2×C4—22no—
305C2×C4—22no—
306C2×C4—24no—
307C2×C4—44no—
308C2×C4—44no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—42no—
314C2×C4—42no—
315C2×C4—41no—
316C2×C4—44no—
317C2×C4—42no—
318C2×C4—42no—
319C2×C4—44no—
320C2×C4—84no—
321C2×C4—84no—
322C2×C4—84no—
323C2×C4C2×#10014yes—
324C2×C4C2×#10012yes—
325C2×C4#42114yes—
326C2×C4C4×#4111yescharacteristic
327C2×C4#52011yescharacteristic
328C2×C4#54612yes—
329C2×C4#54712yes—
330C8—44no—
331C8—44no—
332C8—84no—
333C22—44no—
334C22—42no—
335C22—42no—
336C22—44no—
337C22—42no—
338C22—82no—
339C22—81no—
340C22—81no—
341C22—82no—
342C22—81no—
343C22#233911yescharacteristic
344C22C22×#10011yescharacteristic
345C22#288911yescharacteristic
346C22C2×#42112yes—
347C22#290112yes—
348C4—21no—
349C4—21no—
350C4—24no—
351C4—22no—
352C4—22no—
353C4—42no—
354C4—41no—
355C4—44no—
356C4—81no—
357C4—82no—
358C4—81no—
359C4—81no—
360C4—82no—
361C4#153614yes—
362C2—82no—
363C2—161no—
364C2C2×#153611yescharacteristic
365C2#1768312yes—
366C2#1768811yescharacteristic
367C2#2247911yescharacteristic
368C2#2250712yes—
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC2×#555C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C42×D4 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #21270C2semidirect
3. #9890C2semidirect
4. #9123C2semidirect
5. #8721C2semidirect
6. #8579C2semidirect
7. C4×#329C2semidirect
8. #8031C2semidirect

Maximal subgroups

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

subgroupindex
#361852
#212702
#98902
#91232
#87212
#85792
C4×#3292
#80312