Finite Group Databasebeta

← #223142

Subgroups — #223142

Order 512.

952 conjugacy classes of subgroups.

The subgroup lattice has 373 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36164C211yesmaximal, characteristic
2#21270C212yesmaximal, complemented
3#9890C212yesmaximal, complemented
4#9124C214yesmaximal, complemented
5#8718C212yesmaximal, complemented
6#8579C211yesmaximal, characteristic, complemented
7C4×#329C211yesmaximal, characteristic, complemented
8#8030C212yesmaximal, complemented
9C2×#554C411yescharacteristic
10C2×#547C411yescharacteristic
11C2×#432C2214yes—
12#2862C2211yescharacteristic
13#2850C2212yes—
14#2568C411yescharacteristic
15#2563C411yescharacteristic
16#2504C412yes—
17#2503C412yes—
18#2489—22no—
19#2488—24no—
20#2339—22no—
21C4×#124C2211yescharacteristic
22C2×#382C2212yes—
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—
37C22×#41—21no—
38#554—44no—
39#547—44no—
40C2×C4⋊Q8C2×C411yescharacteristic
41C2×C4⋊Q8C2×C411yescharacteristic
42C2×#125—21no—
43C2×#124C2311yescharacteristic
44C2×#123—22no—
45C2×C22⋊Q8—21no—
46C2×C22⋊Q8—22no—
47C2×C22⋊Q8C2×C412yes—
48C2×#121C2×C412yes—
49C2×C4×Q8C2×C411yescharacteristic
50C2×C4×D4—21no—
51C2×C4×D4C2×C411yescharacteristic
52C2×C4×D4C2×C411yescharacteristic
53C2×C4×D4D412yes—
54C22×C4⋊C4C2×C412yes—
55C22×C4⋊C4C2×C412yes—
56C22×C42C2312yes—
57#432—28no—
58#432—44no—
59C2×C4⋊C8—24no—
60C2×C4⋊C8—22no—
61C2×C22⋊C8—24no—
62C2×C22⋊C8—24no—
63C2×#97—22no—
64C2×#97C2314yes—
65C2×#97C2312yes—
66C2×#97C2312yes—
67C2×C4×C8—22no—
68#400—22no—
69#400—24no—
70#397—24no—
71#393—22no—
72#391—22no—
73#391—22no—
74#391—22no—
75#391C2×C412yes—
76#391C2×C411yescharacteristic
77#390C2×C412yes—
78#390C2×C411yescharacteristic
79#388—24no—
80#388—22no—
81#386—24no—
82#386—24no—
83#382—24no—
84#382C2×C412yes—
85#382C2×C412yes—
86#382C2×C412yes—
87#382C2312yes—
88#382C2311yescharacteristic
89#381—24no—
90#380—22no—
91C4×C4⋊C4C2×C411yescharacteristic
92C4×C4⋊C4D412yes—
93C4×C22⋊C4—22no—
94C4×C22⋊C4C2×C412yes—
95C4×C22⋊C4D414yes—
96C2×#95—24no—
97C43C2311yescharacteristic
98#336—22no—
99#336—24no—
100#329—24no—
101#329—42no—
102C2×#41—48no—
103C2×#41—44no—
104C22×Q8—21no—
105C22×Q8—21no—
106C22×Q8C22×C411yescharacteristic
107C22×D4—22no—
108C22×D4C22⋊C411yescharacteristic
109C22×D4C22×C411yescharacteristic
110C23×C4—22no—
111C23×C4—22no—
112C23×C4—21no—
113C23×C4C22×C412yes—
114C23×C4C2×D414yes—
115C22×C8—22no—
116C22×C8—22no—
117C22×C8—24no—
118C4⋊Q8—42no—
119C4⋊Q8—42no—
120#125—44no—
121#124—22no—
122#124—21no—
123#124—21no—
124#123—48no—
125C22⋊Q8—48no—
126C22⋊Q8—44no—
127C22⋊Q8—44no—
128#121—44no—
129C4×Q8—42no—
130C4×D4—24no—
131C4×D4—44no—
132C4×D4—42no—
133C4×D4—44no—
134C2×C4⋊C4—22no—
135C2×C4⋊C4—22no—
136C2×C4⋊C4—21no—
137C2×C4⋊C4—21no—
138C2×C4⋊C4—22no—
139C2×C4⋊C4—21no—
140C2×C4⋊C4—44no—
141C2×C4⋊C4—44no—
142C2×C4⋊C4C22⋊C411yescharacteristic
143C2×C4⋊C4C22⋊C412yes—
144C2×C4⋊C4C22×C411yescharacteristic
145C2×C4⋊C4C22×C411yescharacteristic
146C2×C4⋊C4C22×C412yes—
147C2×C4⋊C4C22×C412yes—
148C2×C22⋊C4—22no—
149C2×C22⋊C4—21no—
150C2×C22⋊C4—22no—
151C2×C22⋊C4—22no—
152C2×C22⋊C4—22no—
153C2×C22⋊C4—24no—
154C2×C22⋊C4C22⋊C412yes—
155C2×C22⋊C4C22⋊C414yes—
156C2×C42—24no—
157C2×C42—22no—
158C2×C42—22no—
159C2×C42—21no—
160C2×C42C22×C411yescharacteristic
161C2×C42C22×C412yes—
162C2×C42C22×C411yescharacteristic
163C2×C42C2×D412yes—
164C2×C42C2×D414yes—
165C2×C42C2×D412yes—
166C2×C42C2411yescharacteristic
167C4⋊C8—48no—
168C4⋊C8—44no—
169C22⋊C8—48no—
170C22⋊C8—48no—
171#97—22no—
172#97—24no—
173#97—22no—
174#97—24no—
175#97—22no—
176#97—44no—
177#97#4114yescomplemented
178C4×C8—44no—
179#95—24no—
180#95—24no—
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—
190#41—816no—
191C2×Q8—21no—
192C2×Q8—21no—
193C2×Q8—42no—
194C2×Q8—41no—
195C2×Q8—41no—
196C2×Q8—42no—
197C2×Q8—44no—
198C2×Q8—82no—
199C2×D4—21no—
200C2×D4—21no—
201C2×D4—24no—
202C2×D4—41no—
203C2×D4—41no—
204C2×D4—48no—
205C2×D4—44no—
206C2×D4#10014yes—
207C22×C4—24no—
208C22×C4—22no—
209C22×C4—22no—
210C22×C4—24no—
211C22×C4—24no—
212C22×C4—21no—
213C22×C4—22no—
214C22×C4—21no—
215C22×C4—21no—
216C22×C4—22no—
217C22×C4—42no—
218C22×C4—44no—
219C22×C4—44no—
220C22×C4—48no—
221C22×C4—44no—
222C22×C4—44no—
223C22×C4—42no—
224C22×C4—44no—
225C22×C4—84no—
226C22×C4C2×C22⋊C412yes—
227C22×C4C2×C22⋊C414yes—
228C22×C4C2×C22⋊C411yescharacteristic
229C22×C4C2×C22⋊C412yes—
230C22×C4C2×C22⋊C411yescharacteristic
231C22×C4C23×C411yescharacteristic
232C22×C4C22×D412yes—
233C2×C8—24no—
234C2×C8—24no—
235C2×C8—42no—
236C2×C8—42no—
237C2×C8—42no—
238C2×C8—42no—
239C2×C8—44no—
240C2×C8—44no—
241C2×C8—44no—
242C4⋊C4—22no—
243C4⋊C4—24no—
244C4⋊C4—41no—
245C4⋊C4—41no—
246C4⋊C4—42no—
247C4⋊C4—42no—
248C4⋊C4—44no—
249C4⋊C4—44no—
250C4⋊C4—42no—
251C4⋊C4—44no—
252C4⋊C4—42no—
253C4⋊C4—42no—
254C22⋊C4—24no—
255C22⋊C4—44no—
256C22⋊C4—42no—
257C22⋊C4—44no—
258C22⋊C4—44no—
259C22⋊C4—42no—
260C22⋊C4—42no—
261C22⋊C4—82no—
262C22⋊C4—84no—
263C42—22no—
264C42—22no—
265C42—24no—
266C42—24no—
267C42—22no—
268C42—24no—
269C42—42no—
270C42—41no—
271C42—42no—
272C42C2×#4112yes—
273C42H(5,𝔽2)11yescharacteristic
274C422−1+411yescharacteristic
275C23—22no—
276C23—22no—
277C23—24no—
278C23—24no—
279C23—41no—
280C23—41no—
281C23—41no—
282C23—42no—
283C23—42no—
284C23C2×#10014yes—
285C23C22×C22⋊C411yescharacteristic
286Q8—81no—
287Q8—81no—
288Q8—84no—
289Q8—162no—
290D4—28no—
291D4—81no—
292D4—81no—
293D4—88no—
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—22no—
304C2×C4—22no—
305C2×C4—24no—
306C2×C4—22no—
307C2×C4—22no—
308C2×C4—24no—
309C2×C4—44no—
310C2×C4—44no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—42no—
315C2×C4—41no—
316C2×C4—42no—
317C2×C4—41no—
318C2×C4—42no—
319C2×C4—44no—
320C2×C4—42no—
321C2×C4—44no—
322C2×C4—84no—
323C2×C4—84no—
324C2×C4—88no—
325C2×C4—84no—
326C2×C4—164no—
327C2×C4C2×#10012yes—
328C2×C4#42114yes—
329C2×C4C4×#4111yescharacteristic
330C2×C4#52011yescharacteristic
331C2×C4#54612yes—
332C2×C4#54712yes—
333C8—44no—
334C8—44no—
335C8—84no—
336C22—24no—
337C22—24no—
338C22—24no—
339C22—44no—
340C22—44no—
341C22—82no—
342C22—81no—
343C22—81no—
344C22—82no—
345C22—81no—
346C22#233911yescharacteristic
347C22C22×#10011yescharacteristic
348C22#288911yescharacteristic
349C22C2×#42112yes—
350C22#290112yes—
351C4—21no—
352C4—21no—
353C4—22no—
354C4—22no—
355C4—42no—
356C4—41no—
357C4—42no—
358C4—44no—
359C4—81no—
360C4—82no—
361C4—81no—
362C4—81no—
363C4—82no—
364C4#153314yes—
365C2—44no—
366C2—161no—
367C2C2×#153311yescharacteristic
368C2#1768212yes—
369C2#1768511yescharacteristic
370C2#2247911yescharacteristic
371C2#2250712yes—
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC2×#547C4
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. #9124C2semidirect
5. #8718C2semidirect
6. #8579C2semidirect
7. C4×#329C2semidirect
8. #8030C2semidirect
177. #97#41semidirect

Maximal subgroups

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

subgroupindex
#361642
#212702
#98902
#91242
#87182
#85792
C4×#3292
#80302