Finite Group Databasebeta

← #192089

Subgroups — #192089

Order 512.

1208 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#34182C211yesmaximal, characteristic
2#21365C212yesmaximal
3C2×#1467C211yesmaximal, characteristic
4#13850C211yesmaximal, characteristic
5#8857C214yesmaximal
6#8856C214yesmaximal
7#8022C212yesmaximal
8C2×#519—21no—
9C22×C4×Q8C2211yescharacteristic
10C2×#432C2212yes—
11#2351—24no—
12Q8×C4⋊C4—22no—
13Q8×C4⋊C4—22no—
14#2316—21no—
15#2316C2214yes—
16#2285—22no—
17#2285C2212yes—
18C2×C4×C4⋊C4C2211yescharacteristic
19C2×#376C2211yescharacteristic
20#1948—22no—
21#1947—22no—
22#1853C2214yes—
23#1852C2214yes—
24#1776—24no—
25#1467C418yescomplemented
26C2×#324C2214yes—
27#1446—21no—
28#1445—21no—
29#1441C2212yes—
30#1441C2212yes—
31C4×#95—22no—
32#1294C2218yes—
33C23×Q8D411yescharacteristic
34#519—48no—
35C2×C4×Q8—24no—
36C2×C4×Q8—24no—
37C2×C4×Q8—22no—
38C2×C4×Q8—22no—
39C2×C4×Q8—24no—
40C2×C4×Q8C2×C414yes—
41C2×C4×Q8Q814yes—
42C2×#117—22no—
43C2×#117—24no—
44C22×C4⋊C4D411yescharacteristic
45C22×C4⋊C4C2312yes—
46C22×C4⋊C4C2311yescharacteristic
47C22×C4⋊C4C2311yescharacteristic
48C22×C42—21no—
49C22×C42—21no—
50C22×C42D412yes—
51C22×C42C2311yescharacteristic
52#432—24no—
53#432C2×C418yes—
54C2×#107—24no—
55C2×#106—24no—
56C2×C4⋊C8—22no—
57C2×C22⋊C8C2314yes—
58C2×#97—22no—
59#389—24no—
60#389—24no—
61#389—28no—
62#389—28no—
63#384—22no—
64#384—22no—
65#384D412yes—
66#384D412yes—
67#382—24no—
68#382—22no—
69#380—24no—
70#380—22no—
71#380C2314yes—
72C4×C4⋊C4—24no—
73C4×C4⋊C4—24no—
74C4×C4⋊C4—22no—
75C4×C4⋊C4—22no—
76C4×C4⋊C4—22no—
77C4×C4⋊C4D414yes—
78#376—24no—
79C2×#95—22no—
80C2×#95—22no—
81C2×#95—22no—
82C2×#95—22no—
83C2×#95—21no—
84C2×#95—21no—
85C2×#95C2312yes—
86#340—24no—
87#340—24no—
88#324C2×C4116yes—
89C22×Q8—22no—
90C22×Q8—22no—
91C22×Q8—24no—
92C22×Q8—24no—
93C22×Q8C4⋊C414yes—
94C23×C4—21no—
95C23×C4—21no—
96C23×C4—22no—
97C23×C4C2×D411yescharacteristic
98C23×C4C2×D411yescharacteristic
99C23×C4C2×D412yes—
100C23×C4C2×D412yes—
101C23×C4C2411yescharacteristic
102C22×C8—24no—
103C4×Q8—28no—
104C4×Q8—28no—
105C4×Q8—28no—
106C4×Q8—28no—
107C4×Q8—416no—
108C4×Q8—416no—
109#117—48no—
110#117—48no—
111#117—48no—
112C2×C4⋊C4—22no—
113C2×C4⋊C4—22no—
114C2×C4⋊C4—24no—
115C2×C4⋊C4—24no—
116C2×C4⋊C4—22no—
117C2×C4⋊C4—22no—
118C2×C4⋊C4—24no—
119C2×C4⋊C4—24no—
120C2×C4⋊C4—22no—
121C2×C4⋊C4—22no—
122C2×C4⋊C4—22no—
123C2×C4⋊C4—22no—
124C2×C4⋊C4—24no—
125C2×C4⋊C4—24no—
126C2×C4⋊C4—42no—
127C2×C4⋊C4—42no—
128C2×C4⋊C4C4⋊C414yes—
129C2×C4⋊C4C22×C414yes—
130C2×C4⋊C4C2×D414yes—
131C2×C4⋊C4C2×D414yes—
132C2×C4⋊C4C2×Q814yes—
133C2×C22⋊C4—24no—
134C2×C42—22no—
135C2×C42—22no—
136C2×C42—24no—
137C2×C42—28no—
138C2×C42—24no—
139C2×C42—24no—
140C2×C42—42no—
141C2×C42—42no—
142C2×C42—44no—
143C2×C42—41no—
144C2×C42—41no—
145C2×C42C4⋊C418yes—
146C2×C42C22×C412yes—
147C2×C42C2×D412yes—
148C2×C42C2×D412yes—
149C2×C42C2×Q812yes—
150C2×C42#4114yes—
151#107—48no—
152#106—48no—
153C4⋊C8—28no—
154C22⋊C8—24no—
155C22⋊C8C22×C418yes—
156#97—28no—
157#95—28no—
158#95—24no—
159#95—24no—
160#95—24no—
161#95—44no—
162#95—44no—
163#95—42no—
164#95—42no—
165#95—44no—
166#95—44no—
167C24C22×D411yescharacteristic
168C2×Q8—28no—
169C2×Q8—216no—
170C2×Q8—28no—
171C2×Q8—28no—
172C2×Q8—216no—
173C22×C4—22no—
174C22×C4—22no—
175C22×C4—22no—
176C22×C4—22no—
177C22×C4—22no—
178C22×C4—22no—
179C22×C4—22no—
180C22×C4—22no—
181C22×C4—22no—
182C22×C4—22no—
183C22×C4—22no—
184C22×C4—22no—
185C22×C4—24no—
186C22×C4—24no—
187C22×C4—24no—
188C22×C4—41no—
189C22×C4—41no—
190C22×C4—42no—
191C22×C4—42no—
192C22×C4—42no—
193C22×C4—42no—
194C22×C4—42no—
195C22×C4—42no—
196C22×C4—42no—
197C22×C4—41no—
198C22×C4—41no—
199C22×C4—41no—
200C22×C4—41no—
201C22×C4—42no—
202C22×C4—42no—
203C22×C4—42no—
204C22×C4—42no—
205C22×C4—42no—
206C22×C4C2×C4⋊C412yes—
207C22×C4C2×C4⋊C412yes—
208C22×C4C2×C4⋊C414yes—
209C22×C4C2×C4⋊C414yes—
210C22×C4#12011yescharacteristic
211C22×C4#12012yes—
212C22×C4#12011yescharacteristic
213C22×C4#12312yes—
214C22×C4#12312yes—
215C22×C4AGL(1,ℤ/8)14yes—
216C22×C4#13714yes—
217C22×C4C23×C411yescharacteristic
218C22×C4C22×D412yes—
219C22×C4C22×Q811yescharacteristic
220C22×C4C2×#4112yes—
221C2×C8—28no—
222C2×C8—44no—
223C2×C8—44no—
224C4⋊C4—28no—
225C4⋊C4—28no—
226C4⋊C4—28no—
227C4⋊C4—24no—
228C4⋊C4—24no—
229C4⋊C4—44no—
230C4⋊C4—44no—
231C4⋊C4—42no—
232C4⋊C4—42no—
233C4⋊C4—44no—
234C4⋊C4—44no—
235C4⋊C4—48no—
236C4⋊C4—48no—
237C4⋊C4—44no—
238C4⋊C4—44no—
239C4⋊C4—84no—
240C4⋊C4—84no—
241C22⋊C4—48no—
242C42—28no—
243C42—28no—
244C42—48no—
245C42—48no—
246C42—416no—
247C42—82no—
248C42—82no—
249C42C4×D414yes—
250C42C4×D414yes—
251C42C22⋊Q814yes—
252C42C22⋊Q814yes—
253C23—21no—
254C23—21no—
255C23—21no—
256C23—21no—
257C23C22×C4⋊C411yescharacteristic
258C23C2×#12011yescharacteristic
259C23C2×#12311yescharacteristic
260C23C2×AGL(1,ℤ/8)12yes—
261C23C2×#13712yes—
262Q8—232no—
263C2×C4—24no—
264C2×C4—24no—
265C2×C4—24no—
266C2×C4—24no—
267C2×C4—24no—
268C2×C4—22no—
269C2×C4—24no—
270C2×C4—22no—
271C2×C4—24no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—22no—
275C2×C4—22no—
276C2×C4—24no—
277C2×C4—24no—
278C2×C4—24no—
279C2×C4—24no—
280C2×C4—41no—
281C2×C4—41no—
282C2×C4—42no—
283C2×C4—42no—
284C2×C4—42no—
285C2×C4—44no—
286C2×C4—44no—
287C2×C4—44no—
288C2×C4—44no—
289C2×C4—44no—
290C2×C4—44no—
291C2×C4—44no—
292C2×C4—44no—
293C2×C4—44no—
294C2×C4—44no—
295C2×C4—42no—
296C2×C4—42no—
297C2×C4—44no—
298C2×C4—44no—
299C2×C4—82no—
300C2×C4—82no—
301C2×C4—82no—
302C2×C4—82no—
303C2×C4—82no—
304C2×C4—82no—
305C2×C4—82no—
306C2×C4—81no—
307C2×C4—81no—
308C2×C4—81no—
309C2×C4—81no—
310C2×C4#38512yes—
311C2×C4#38514yes—
312C2×C4#38512yes—
313C2×C4#42818yes—
314C2×C4#45412yes—
315C2×C4#45612yes—
316C2×C4C2×C4×D412yes—
317C2×C4C2×C22⋊Q812yes—
318C8—48no—
319C22—22no—
320C22—22no—
321C22—22no—
322C22—22no—
323C22—21no—
324C22—21no—
325C22—21no—
326C22—21no—
327C22C2×#38511yescharacteristic
328C22C2×#42812yes—
329C22#300311yescharacteristic
330C22#300411yescharacteristic
331C22C2×#45411yescharacteristic
332C22C2×#45611yescharacteristic
333C22#302511yescharacteristic
334C22#302511yescharacteristic
335C22#309212yes—
336C4—24no—
337C4—28no—
338C4—22no—
339C4—22no—
340C4—22no—
341C4—44no—
342C4—84no—
343C4—84no—
344C4—82no—
345C4—84no—
346C4—82no—
347C4#185714yes—
348C2—22no—
349C2—22no—
350C2#1818611yescharacteristic
351C2#1818711yescharacteristic
352C2#1818811yescharacteristic
353C2#1818911yescharacteristic
354C2#2084411yescharacteristic
355C2C2×#185711yescharacteristic
356C2#2088811yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC24C22×D4
LCSG ▷ C22×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
25. #1467C4semidirect

Maximal subgroups

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

subgroupindex
#341822
#213652
C2×#14672
#138502
#88572
#88562
#80222