Finite Group Databasebeta

← #241176

Subgroups — #241176

Order 512.

1026 conjugacy classes of subgroups.

The subgroup lattice has 367 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35167C211yesmaximal, characteristic
2#21358C212yesmaximal
3#14150C211yesmaximal, characteristic
4#10569C214yesmaximal
5#10567C214yesmaximal
6#8026C212yesmaximal
7#7908C211yesmaximal, characteristic
8C2×#544—21no—
9C2×#431C2212yes—
10#2857—22no—
11#2659C2214yes—
12#2552—22no—
13#2551—22no—
14#2550—24no—
15#2316—21no—
16C2×#389—21no—
17#2285C2212yes—
18C2×#382C2211yescharacteristic
19C2×C4×C4⋊C4C2211yescharacteristic
20C2×C4×C4⋊C4C2211yescharacteristic
21#1947—22no—
22#1947—22no—
23#1852C2214yes—
24#1852C2214yes—
25C4×#107—24no—
26#1444C2212yes—
27#1442C2212yes—
28#1436—22no—
29C4⋊#95—21no—
30#1294C2218yes—
31#1280C2214yes—
32#544—44no—
33#544—44no—
34C2×C4⋊Q8—24no—
35C2×C4⋊Q8—22no—
36C2×#125—24no—
37C2×C22⋊Q8—24no—
38C2×#117—22no—
39C22×C4⋊C4—21no—
40C22×C4⋊C4—22no—
41C22×C4⋊C4D411yescharacteristic
42C22×C4⋊C4D411yescharacteristic
43C22×C4⋊C4C2311yescharacteristic
44C22×C4⋊C4C2311yescharacteristic
45C22×C42—21no—
46C22×C42D412yes—
47C22×C42C2311yescharacteristic
48#431—24no—
49#431Q818yescomplemented
50C2×#107—24no—
51C2×#107—24no—
52C2×C4⋊C8—22no—
53C2×C22⋊C8C2314yes—
54C2×C4×C8—22no—
55#398—28no—
56#395—28no—
57#391—24no—
58#389—24no—
59#389—24no—
60#389—24no—
61#389—24no—
62#389—44no—
63#384—22no—
64#384D412yes—
65#384D412yes—
66#382—22no—
67#382—22no—
68#382Q814yescomplemented
69#380—24no—
70#380—22no—
71#380C2314yes—
72C4×C4⋊C4—22no—
73C4×C4⋊C4—22no—
74C4×C4⋊C4—22no—
75C4×C4⋊C4—22no—
76C4×C4⋊C4—22no—
77C4×C4⋊C4D414yes—
78C2×#95—21no—
79C2×#95—22no—
80C2×#95—21no—
81C2×#95—22no—
82C2×#95—22no—
83C2×#95C2312yes—
84C2×#95C2312yes—
85#340—24no—
86#340—24no—
87C22×Q8—24no—
88C23×C4—21no—
89C23×C4—22no—
90C23×C4—21no—
91C23×C4C2×D412yes—
92C23×C4C2×D412yes—
93C23×C4C2×D411yescharacteristic
94C23×C4C2×D411yescharacteristic
95C23×C4C2411yescharacteristic
96C22×C8—24no—
97C4⋊Q8—416no—
98C4⋊Q8—84no—
99#125—416no—
100C22⋊Q8—48no—
101C22⋊Q8—48no—
102#117—44no—
103#117—44no—
104C2×C4⋊C4—22no—
105C2×C4⋊C4—22no—
106C2×C4⋊C4—22no—
107C2×C4⋊C4—22no—
108C2×C4⋊C4—22no—
109C2×C4⋊C4—22no—
110C2×C4⋊C4—24no—
111C2×C4⋊C4—28no—
112C2×C4⋊C4—22no—
113C2×C4⋊C4—22no—
114C2×C4⋊C4—22no—
115C2×C4⋊C4—22no—
116C2×C4⋊C4—24no—
117C2×C4⋊C4—24no—
118C2×C4⋊C4—24no—
119C2×C4⋊C4—42no—
120C2×C4⋊C4—42no—
121C2×C4⋊C4—42no—
122C2×C4⋊C4—42no—
123C2×C4⋊C4—44no—
124C2×C4⋊C4—44no—
125C2×C4⋊C4—82no—
126C2×C4⋊C4D814yes—
127C2×C4⋊C4Q1614yes—
128C2×C4⋊C4C2×D414yes—
129C2×C4⋊C4C2×D414yes—
130C2×C22⋊C4—24no—
131C2×C42—24no—
132C2×C42—22no—
133C2×C42—22no—
134C2×C42—42no—
135C2×C42—42no—
136C2×C42—42no—
137C2×C42—82no—
138C2×C42C2×D412yes—
139C2×C42C2×D412yes—
140C2×C42C2×Q814yes—
141C2×C42C2×Q812yes—
142C2×C42#4112yes—
143#107—48no—
144#107—48no—
145C4⋊C8—28no—
146C22⋊C8—24no—
147C22⋊C8C2×Q818yes—
148C4×C8—28no—
149#95—28no—
150#95—24no—
151#95—24no—
152#95—24no—
153#95—24no—
154#95—22no—
155#95—44no—
156#95—41no—
157#95—41no—
158#95—44no—
159#95—82no—
160#95C2×Q814yes—
161C24C22×D411yescharacteristic
162C2×Q8—48no—
163C2×Q8—84no—
164C2×Q8—84no—
165C22×C4—22no—
166C22×C4—22no—
167C22×C4—22no—
168C22×C4—22no—
169C22×C4—22no—
170C22×C4—22no—
171C22×C4—24no—
172C22×C4—24no—
173C22×C4—24no—
174C22×C4—41no—
175C22×C4—41no—
176C22×C4—41no—
177C22×C4—41no—
178C22×C4—42no—
179C22×C4—42no—
180C22×C4—42no—
181C22×C4—42no—
182C22×C4—42no—
183C22×C4—41no—
184C22×C4—42no—
185C22×C4—42no—
186C22×C4—41no—
187C22×C4—41no—
188C22×C4—41no—
189C22×C4—42no—
190C22×C4—41no—
191C22×C4—41no—
192C22×C4—41no—
193C22×C4—41no—
194C22×C4—41no—
195C22×C4—42no—
196C22×C4—41no—
197C22×C4—41no—
198C22×C4—41no—
199C22×C4—42no—
200C22×C4#12011yescharacteristic
201C22×C4#12011yescharacteristic
202C22×C4#12012yes—
203C22×C4C22⋊Q812yes—
204C22×C4C22⋊Q812yes—
205C22×C4C4⋊Q812yes—
206C22×C4C4⋊Q812yes—
207C22×C4C2×D814yes—
208C22×C4C2×D812yes—
209C22×C4C2×Q1614yes—
210C22×C4C2×Q1612yes—
211C22×C4AGL(1,ℤ/8)12yes—
212C22×C4#13712yes—
213C22×C4C22×D412yes—
214C22×C4C22×Q812yes—
215C22×C4C22×Q811yescharacteristic
216C22×C4C2×#4111yescharacteristic
217C2×C8—28no—
218C2×C8—44no—
219C2×C8—44no—
220C4⋊C4—28no—
221C4⋊C4—28no—
222C4⋊C4—24no—
223C4⋊C4—24no—
224C4⋊C4—44no—
225C4⋊C4—44no—
226C4⋊C4—44no—
227C4⋊C4—42no—
228C4⋊C4—42no—
229C4⋊C4—48no—
230C4⋊C4—416no—
231C4⋊C4—44no—
232C4⋊C4—44no—
233C4⋊C4—44no—
234C4⋊C4—48no—
235C4⋊C4—84no—
236C4⋊C4—84no—
237C4⋊C4—88no—
238C22⋊C4—44no—
239C22⋊C4—44no—
240C42—44no—
241C42—48no—
242C42—82no—
243C42—84no—
244C42C22⋊Q814yes—
245C42C22⋊Q814yes—
246C42C4⋊Q814yes—
247C42C4⋊Q814yes—
248C23—21no—
249C23—21no—
250C23—21no—
251C23—21no—
252C23C2×#12011yescharacteristic
253C23C2×C22⋊Q811yescharacteristic
254C23C2×C4⋊Q811yescharacteristic
255C23C22×D811yescharacteristic
256C23C22×Q1611yescharacteristic
257C23C2×AGL(1,ℤ/8)11yescharacteristic
258C23C2×#13711yescharacteristic
259Q8—168no—
260C2×C4—21no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—21no—
264C2×C4—22no—
265C2×C4—22no—
266C2×C4—24no—
267C2×C4—24no—
268C2×C4—42no—
269C2×C4—42no—
270C2×C4—42no—
271C2×C4—41no—
272C2×C4—41no—
273C2×C4—42no—
274C2×C4—42no—
275C2×C4—42no—
276C2×C4—42no—
277C2×C4—44no—
278C2×C4—44no—
279C2×C4—42no—
280C2×C4—44no—
281C2×C4—44no—
282C2×C4—42no—
283C2×C4—42no—
284C2×C4—42no—
285C2×C4—44no—
286C2×C4—44no—
287C2×C4—44no—
288C2×C4—44no—
289C2×C4—42no—
290C2×C4—42no—
291C2×C4—44no—
292C2×C4—44no—
293C2×C4—44no—
294C2×C4—82no—
295C2×C4—82no—
296C2×C4—82no—
297C2×C4—82no—
298C2×C4—82no—
299C2×C4—82no—
300C2×C4—81no—
301C2×C4—81no—
302C2×C4—81no—
303C2×C4—81no—
304C2×C4—81no—
305C2×C4—81no—
306C2×C4—81no—
307C2×C4—81no—
308C2×C4#39512yes—
309C2×C4#39512yes—
310C2×C4#39514yes—
311C2×C4#44714yes—
312C2×C4#45114yes—
313C2×C4#47414yes—
314C2×C4#47714yes—
315C2×C4#50014yes—
316C2×C4#50114yes—
317C2×C4C2×C22⋊Q812yes—
318C2×C4C2×C4⋊Q812yes—
319C8—48no—
320C22—21no—
321C22—21no—
322C22—21no—
323C22—21no—
324C22—21no—
325C22—21no—
326C22—21no—
327C22—21no—
328C22—21no—
329C22—21no—
330C22—21no—
331C22—21no—
332C22C2×#39511yescharacteristic
333C22C2×#44711yescharacteristic
334C22C2×#45111yescharacteristic
335C22#301911yescharacteristic
336C22#302011yescharacteristic
337C22#302511yescharacteristic
338C22#302511yescharacteristic
339C22C2×#47411yescharacteristic
340C22C2×#47711yescharacteristic
341C22C2×#50011yescharacteristic
342C22C2×#50111yescharacteristic
343C4—22no—
344C4—22no—
345C4—22no—
346C4—22no—
347C4—44no—
348C4—44no—
349C4—84no—
350C4—82no—
351C4—84no—
352C4—84no—
353C4—82no—
354C4#205718yes—
355C2—21no—
356C2—21no—
357C2—21no—
358C2—21no—
359C2#1821011yescharacteristic
360C2#1821311yescharacteristic
361C2#1821811yescharacteristic
362C2#1822111yescharacteristic
363C2C2×#205711yescharacteristic
364C2#2356111yescharacteristic
365C2#2356411yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutator, Frattini, cosocleC23×C4C24
prime radicalC24C22×D4
LCSG ▷ C23×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
49. #431Q8semidirect
68. #382Q8semidirect

Maximal subgroups

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

subgroupindex
#351672
#213582
#141502
#105692
#105672
#80262
#79082