Finite Group Databasebeta

← #237843

Subgroups — #237843

Order 512.

1108 conjugacy classes of subgroups.

The subgroup lattice has 358 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35167C211yesmaximal, characteristic
2#21833C212yesmaximal
3#14155C211yesmaximal, characteristic
4#10342C218yesmaximal
5#8014C212yesmaximal
6#7908C211yesmaximal, characteristic
7C2×#544—21no—
8C22×C4⋊Q8C2211yescharacteristic
9C2×#431C2212yes—
10#2857—22no—
11#2659C2214yes—
12#2552—22no—
13#2551—22no—
14#2550—24no—
15#2316—21no—
16C2×#391—21no—
17C2×#391—21no—
18#2285C2212yes—
19C2×C4×C4⋊C4C2211yescharacteristic
20C2×C4×C4⋊C4C2211yescharacteristic
21#1990—22no—
22#1990—22no—
23#1988—24no—
24#1823C2218yes—
25#1445C2214yes—
26C4⋊#95—22no—
27#1294C2218yes—
28#1280C2214yes—
29C23×Q8—21no—
30#544—44no—
31#544—44no—
32C2×C4⋊Q8—24no—
33C2×C4⋊Q8—22no—
34C2×C4⋊Q8—22no—
35C2×C4⋊Q8—22no—
36C2×C4⋊Q8—22no—
37C2×C4⋊Q8—24no—
38C2×C4⋊Q8Q814yescomplemented
39C2×#125—24no—
40C2×C22⋊Q8—24no—
41C2×#117—22no—
42C22×C4⋊C4—21no—
43C22×C4⋊C4—21no—
44C22×C4⋊C4D411yescharacteristic
45C22×C4⋊C4D411yescharacteristic
46C22×C4⋊C4C2312yes—
47C22×C42—21no—
48C22×C42D412yes—
49C22×C42C2311yescharacteristic
50#431—24no—
51#431—24no—
52C2×C4⋊C8—22no—
53C2×C22⋊C8C2314yes—
54C2×C4×C8—22no—
55#398—28no—
56#395—28no—
57#391—24no—
58#391—44no—
59#391—44no—
60#389—24no—
61#389—24no—
62#389—24no—
63#389—24no—
64#384—22no—
65#384D412yes—
66#384D412yes—
67#382—22no—
68#380—24no—
69#380—22no—
70#380C2314yes—
71C4×C4⋊C4—22no—
72C4×C4⋊C4—22no—
73C4×C4⋊C4—22no—
74C4×C4⋊C4—24no—
75C4×C4⋊C4D414yes—
76C2×#95—24no—
77C2×#95—24no—
78C2×#95C2312yes—
79C2×#95C2312yes—
80#340—28no—
81#340—28no—
82C22×Q8—24no—
83C22×Q8—24no—
84C22×Q8—42no—
85C22×Q8—42no—
86C22×Q8—42no—
87C22×Q8—42no—
88C22×Q8—42no—
89C22×Q8—42no—
90C23×C4—22no—
91C23×C4—22no—
92C23×C4C2×D411yescharacteristic
93C23×C4C2×D412yes—
94C23×C4C2×D412yes—
95C23×C4C2×D411yescharacteristic
96C23×C4C2411yescharacteristic
97C22×C8—24no—
98C4⋊Q8—28no—
99C4⋊Q8—28no—
100C4⋊Q8—416no—
101C4⋊Q8—44no—
102C4⋊Q8—44no—
103C4⋊Q8—84no—
104#125—416no—
105C22⋊Q8—48no—
106C22⋊Q8—48no—
107#117—44no—
108#117—44no—
109C2×C4⋊C4—22no—
110C2×C4⋊C4—22no—
111C2×C4⋊C4—22no—
112C2×C4⋊C4—22no—
113C2×C4⋊C4—24no—
114C2×C4⋊C4—24no—
115C2×C4⋊C4—24no—
116C2×C4⋊C4—24no—
117C2×C4⋊C4—28no—
118C2×C4⋊C4—22no—
119C2×C4⋊C4—22no—
120C2×C4⋊C4—22no—
121C2×C4⋊C4—22no—
122C2×C4⋊C4—24no—
123C2×C4⋊C4—22no—
124C2×C4⋊C4—24no—
125C2×C4⋊C4—24no—
126C2×C4⋊C4—42no—
127C2×C4⋊C4—41no—
128C2×C4⋊C4—41no—
129C2×C4⋊C4—42no—
130C2×C4⋊C4—42no—
131C2×C4⋊C4D814yes—
132C2×C4⋊C4SD1614yes—
133C2×C4⋊C4C2×D414yes—
134C2×C4⋊C4C2×D414yes—
135C2×C4⋊C4C2×Q814yes—
136C2×C22⋊C4—24no—
137C2×C42—24no—
138C2×C42—22no—
139C2×C42—22no—
140C2×C42—44no—
141C2×C42—42no—
142C2×C42—42no—
143C2×C42—42no—
144C2×C42C2×D412yes—
145C2×C42C2×D412yes—
146C2×C42C2×Q812yes—
147C2×C42#4114yes—
148C2×C42#4112yes—
149C4⋊C8—44no—
150C22⋊C8—24no—
151C22⋊C8—24no—
152C4×C8—44no—
153#95—28no—
154#95—24no—
155#95—24no—
156#95—24no—
157#95—24no—
158#95—48no—
159#95—48no—
160C24C22×D411yescharacteristic
161C2×Q8—48no—
162C2×Q8—48no—
163C2×Q8—48no—
164C2×Q8—48no—
165C2×Q8—84no—
166C2×Q8—84no—
167C2×Q8—84no—
168C2×Q8—84no—
169C2×Q8—84no—
170C2×Q8—84no—
171C22×C4—22no—
172C22×C4—22no—
173C22×C4—22no—
174C22×C4—22no—
175C22×C4—22no—
176C22×C4—22no—
177C22×C4—24no—
178C22×C4—24no—
179C22×C4—24no—
180C22×C4—24no—
181C22×C4—41no—
182C22×C4—41no—
183C22×C4—42no—
184C22×C4—42no—
185C22×C4—42no—
186C22×C4—42no—
187C22×C4—42no—
188C22×C4—42no—
189C22×C4—42no—
190C22×C4—42no—
191C22×C4—42no—
192C22×C4—42no—
193C22×C4—42no—
194C22×C4—42no—
195C22×C4—42no—
196C22×C4—42no—
197C22×C4—42no—
198C22×C4—42no—
199C22×C4#12011yescharacteristic
200C22×C4#12012yes—
201C22×C4#12011yescharacteristic
202C22×C4C22⋊Q812yes—
203C22×C4C22⋊Q812yes—
204C22×C4#12412yes—
205C22×C4#12412yes—
206C22×C4C2×D814yes—
207C22×C4C2×D812yes—
208C22×C4C2×SD1614yes—
209C22×C4C2×SD1612yes—
210C22×C4AGL(1,ℤ/8)12yes—
211C22×C4#13712yes—
212C22×C4C22×D412yes—
213C22×C4C22×Q811yescharacteristic
214C22×C4C2×#4112yes—
215C22×C4C2×#4111yescharacteristic
216C2×C8—44no—
217C2×C8—44no—
218C2×C8—44no—
219C4⋊C4—28no—
220C4⋊C4—28no—
221C4⋊C4—28no—
222C4⋊C4—44no—
223C4⋊C4—44no—
224C4⋊C4—48no—
225C4⋊C4—48no—
226C4⋊C4—416no—
227C4⋊C4—44no—
228C4⋊C4—44no—
229C4⋊C4—44no—
230C4⋊C4—44no—
231C4⋊C4—44no—
232C4⋊C4—44no—
233C4⋊C4—44no—
234C4⋊C4—84no—
235C4⋊C4—84no—
236C22⋊C4—44no—
237C22⋊C4—44no—
238C42—48no—
239C42—44no—
240C42—82no—
241C42—84no—
242C42C22⋊Q814yes—
243C42C22⋊Q814yes—
244C42#12414yes—
245C42#12414yes—
246C23—21no—
247C23—21no—
248C23—21no—
249C23—21no—
250C23C2×#12011yescharacteristic
251C23C2×C22⋊Q811yescharacteristic
252C23C2×#12411yescharacteristic
253C23C22×D811yescharacteristic
254C23C22×SD1611yescharacteristic
255C23C2×AGL(1,ℤ/8)11yescharacteristic
256C23C2×#13711yescharacteristic
257Q8—816no—
258Q8—168no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—22no—
262C2×C4—22no—
263C2×C4—24no—
264C2×C4—24no—
265C2×C4—24no—
266C2×C4—24no—
267C2×C4—41no—
268C2×C4—41no—
269C2×C4—42no—
270C2×C4—42no—
271C2×C4—42no—
272C2×C4—42no—
273C2×C4—42no—
274C2×C4—44no—
275C2×C4—44no—
276C2×C4—44no—
277C2×C4—44no—
278C2×C4—44no—
279C2×C4—44no—
280C2×C4—44no—
281C2×C4—44no—
282C2×C4—44no—
283C2×C4—44no—
284C2×C4—44no—
285C2×C4—44no—
286C2×C4—44no—
287C2×C4—44no—
288C2×C4—82no—
289C2×C4—82no—
290C2×C4—82no—
291C2×C4—82no—
292C2×C4—82no—
293C2×C4—82no—
294C2×C4—82no—
295C2×C4—82no—
296C2×C4—82no—
297C2×C4—82no—
298C2×C4#39312yes—
299C2×C4#39314yes—
300C2×C4#39312yes—
301C2×C4#44714yes—
302C2×C4#44814yes—
303C2×C4#45512yes—
304C2×C4#45612yes—
305C2×C4#47414yes—
306C2×C4#47614yes—
307C2×C4#48614yes—
308C2×C4#48914yes—
309C2×C4C2×C22⋊Q812yes—
310C2×C4C2×#12412yes—
311C8—84no—
312C22—21no—
313C22—21no—
314C22—21no—
315C22—21no—
316C22—21no—
317C22—21no—
318C22—21no—
319C22—21no—
320C22—21no—
321C22—21no—
322C22—21no—
323C22—21no—
324C22C2×#39311yescharacteristic
325C22C2×#44711yescharacteristic
326C22C2×#44811yescharacteristic
327C22C2×#45511yescharacteristic
328C22C2×#45611yescharacteristic
329C22#302811yescharacteristic
330C22#302811yescharacteristic
331C22C2×#47411yescharacteristic
332C22C2×#47611yescharacteristic
333C22C2×#48611yescharacteristic
334C22C2×#48911yescharacteristic
335C4—22no—
336C4—22no—
337C4—22no—
338C4—44no—
339C4—44no—
340C4—84no—
341C4—84no—
342C4—84no—
343C4—84no—
344C4#202318yes—
345C4#202814yes—
346C2—21no—
347C2—21no—
348C2—21no—
349C2—21no—
350C2#1839411yescharacteristic
351C2#1839611yescharacteristic
352C2#1840811yescharacteristic
353C2#1841011yescharacteristic
354C2C2×#202311yescharacteristic
355C2C2×#202811yescharacteristic
356C2#2347011yescharacteristic
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
38. C2×C4⋊Q8Q8semidirect

Maximal subgroups

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

subgroupindex
#351672
#218332
#141552
#103422
#80142
#79082