Finite Group Databasebeta

← #391828

Subgroups — #391828

Order 512.

1218 conjugacy classes of subgroups.

The subgroup lattice has 346 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2851C211yesmaximal, characteristic
2#20682C211yesmaximal, characteristic
3#20626C211yesmaximal, characteristic
4#11383C212yesmaximal, complemented
5#11355C212yesmaximal, complemented
6#11204C214yesmaximal, complemented
7#11198C214yesmaximal
8C22×#125C2211yescharacteristic
9C23×C42C2211yescharacteristic
10C2×#423C2212yes—
11C2×#407C2212yes—
12C2×#407C2212yes—
13#2851—22no—
14#2851—22no—
15#2851—22no—
16#2851—22no—
17C2×#389C2212yes—
18C2×#387C414yes—
19C2×#385C414yes—
20C2×#382C2211yescharacteristic
21C2×#380C2212yes—
22#1817—22no—
23#1817—22no—
24#1797—22no—
25#1797—22no—
26C2×#343C2218yes—
27C2×#343C2218yes—
28#1728C2212yes—
29#1728C2212yes—
30#1728C2212yes—
31C24×C4C2×C412yes—
32C24×C4C2311yescharacteristic
33C22×M16C2314yes—
34C22×M16C2314yes—
35C2×#125—21no—
36C2×#125—21no—
37C2×#125—22no—
38C2×#125—44no—
39C22×C4⋊C4C2×C414yes—
40C22×C4⋊C4C2×C414yes—
41C22×C4⋊C4C2312yes—
42C22×C4⋊C4C2312yes—
43C22×C22⋊C4C2×C414yes—
44C22×C42—44no—
45C22×C42C2×C412yes—
46C22×C42D412yes—
47C22×C42D412yes—
48C22×C42D412yes—
49C22×C42D412yes—
50C22×C42C2311yescharacteristic
51C22×C42C2311yescharacteristic
52#423—44no—
53C2×C4⋊C8—24no—
54#407—44no—
55#407—44no—
56C2×C22⋊C8—24no—
57C2×C22⋊C8—24no—
58#389—22no—
59#389—22no—
60#389—22no—
61#389—22no—
62#387—24no—
63#387—24no—
64#387—24no—
65#387—24no—
66#385—24no—
67#385—24no—
68#385—24no—
69#385—24no—
70#382—21no—
71#382—21no—
72#382—22no—
73#380—22no—
74#380—22no—
75#380—22no—
76#380—22no—
77C2×#95C2×C414yes—
78C2×#95C2×C414yes—
79C2×#95C2×C414yes—
80#343—48no—
81#343—48no—
82#336—22no—
83#336—22no—
84#336—24no—
85#336—22no—
86#336—22no—
87C25C22×C411yescharacteristic
88C23×C4—22no—
89C23×C4—24no—
90C23×C4—24no—
91C23×C4—22no—
92C23×C4—44no—
93C23×C4—44no—
94C23×C4—44no—
95C23×C4C22⋊C412yes—
96C23×C4C22⋊C412yes—
97C23×C4C22⋊C412yes—
98C23×C4C22⋊C412yes—
99C23×C4C22×C411yescharacteristic
100C23×C4C22×C412yes—
101C23×C4C22×C414yes—
102C23×C4C22×C414yes—
103C23×C4C22×C412yes—
104C23×C4C2×D412yes—
105C23×C4C2×D412yes—
106C23×C4C2411yescharacteristic
107C2×M16—24no—
108C2×M16—28no—
109C2×M16—28no—
110C2×M16—28no—
111C2×M16—28no—
112C2×M16—28no—
113C2×M16—24no—
114C22×C8—24no—
115C22×C8—24no—
116#125—44no—
117#125—44no—
118#125—48no—
119C2×C4⋊C4—24no—
120C2×C4⋊C4—24no—
121C2×C4⋊C4—24no—
122C2×C4⋊C4—24no—
123C2×C4⋊C4—22no—
124C2×C4⋊C4—22no—
125C2×C4⋊C4—22no—
126C2×C4⋊C4—22no—
127C2×C4⋊C4—28no—
128C2×C4⋊C4—24no—
129C2×C4⋊C4—24no—
130C2×C4⋊C4—48no—
131C2×C4⋊C4—48no—
132C2×C22⋊C4—24no—
133C2×C22⋊C4—24no—
134C2×C22⋊C4—48no—
135C2×C42—24no—
136C2×C42—24no—
137C2×C42—44no—
138C2×C42—44no—
139C2×C42—44no—
140C2×C42—44no—
141C2×C42—44no—
142C2×C42C22⋊C412yes—
143C2×C42C22⋊C412yes—
144C2×C42C22⋊C412yes—
145C2×C42C22⋊C412yes—
146C2×C42C2×D411yescharacteristic
147C2×C42C2×D411yescharacteristic
148C2×C42C2×D412yes—
149C2×C42C2×D411yescharacteristic
150C2×C42C2×D411yescharacteristic
151C2×C42C2×D412yes—
152C4⋊C8—48no—
153C22⋊C8—48no—
154C22⋊C8—48no—
155#95—24no—
156#95—24no—
157#95—24no—
158#95—24no—
159#95—24no—
160#95—24no—
161C24—22no—
162C24—24no—
163C24—22no—
164C24—22no—
165C24—42no—
166C24C2×C22⋊C412yes—
167C24C23×C411yescharacteristic
168C22×C4—24no—
169C22×C4—24no—
170C22×C4—22no—
171C22×C4—22no—
172C22×C4—22no—
173C22×C4—24no—
174C22×C4—24no—
175C22×C4—24no—
176C22×C4—24no—
177C22×C4—22no—
178C22×C4—24no—
179C22×C4—24no—
180C22×C4—22no—
181C22×C4—24no—
182C22×C4—24no—
183C22×C4—24no—
184C22×C4—24no—
185C22×C4—24no—
186C22×C4—24no—
187C22×C4—28no—
188C22×C4—28no—
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—42no—
200C22×C4—432no—
201C22×C4—48no—
202C22×C4—48no—
203C22×C4—44no—
204C22×C4—44no—
205C22×C4—44no—
206C22×C4—44no—
207C22×C4C2×C22⋊C412yes—
208C22×C4C2×C22⋊C412yes—
209C22×C4C2×C22⋊C412yes—
210C22×C4C2×C22⋊C412yes—
211C22×C4C2×C22⋊C412yes—
212C22×C4C22×D411yescharacteristic
213C22×C4C22×D411yescharacteristic
214C22×C4H(5,𝔽2)14yes—
215C22×C4H(5,𝔽2)12yes—
216C22×C42−1+414yes—
217C22×C42−1+412yes—
218M16—416no—
219M16—48no—
220M16—48no—
221C2×C8—44no—
222C2×C8—44no—
223C2×C8—44no—
224C2×C8—44no—
225C2×C8—44no—
226C2×C8—44no—
227C4⋊C4—48no—
228C4⋊C4—48no—
229C4⋊C4—48no—
230C4⋊C4—44no—
231C4⋊C4—44no—
232C4⋊C4—42no—
233C4⋊C4—42no—
234C4⋊C4—44no—
235C4⋊C4—48no—
236C22⋊C4—48no—
237C22⋊C4—48no—
238C42—24no—
239C42—22no—
240C42—22no—
241C42—22no—
242C42—22no—
243C42—24no—
244C42—42no—
245C42—42no—
246C42—42no—
247C42—42no—
248C23—22no—
249C23—22no—
250C23—28no—
251C23—22no—
252C23—22no—
253C23—22no—
254C23—22no—
255C23—22no—
256C23—22no—
257C23—48no—
258C23—48no—
259C23—44no—
260C23—44no—
261C23#41314yes—
262C23C22×C22⋊C411yescharacteristic
263C23#51812yes—
264C23#51912yes—
265C23#52012yes—
266C2×C4—22no—
267C2×C4—22no—
268C2×C4—22no—
269C2×C4—22no—
270C2×C4—22no—
271C2×C4—22no—
272C2×C4—22no—
273C2×C4—22no—
274C2×C4—22no—
275C2×C4—22no—
276C2×C4—22no—
277C2×C4—22no—
278C2×C4—22no—
279C2×C4—22no—
280C2×C4—48no—
281C2×C4—48no—
282C2×C4—48no—
283C2×C4—48no—
284C2×C4—48no—
285C2×C4—48no—
286C2×C4—44no—
287C2×C4—44no—
288C2×C4—44no—
289C2×C4—48no—
290C2×C4—44no—
291C2×C4—44no—
292C2×C4—44no—
293C2×C4—44no—
294C2×C4—44no—
295C2×C4—44no—
296C2×C4—88no—
297C2×C4—88no—
298C2×C4#41314yes—
299C2×C4#53412yes—
300C2×C4#53412yes—
301C2×C4#53511yescharacteristic
302C2×C4#53612yes—
303C2×C4#53611yescharacteristic
304C2×C4#53612yes—
305C2×C4#53712yes—
306C2×C4#56218yes—
307C2×C4#56318yes—
308C8—84no—
309C8—84no—
310C22—22no—
311C22—22no—
312C22—24no—
313C22—24no—
314C22—24no—
315C22—24no—
316C22—24no—
317C22—416no—
318C22—42no—
319C22—42no—
320C22—42no—
321C22—42no—
322C22#230811yescharacteristic
323C22#231011yescharacteristic
324C22#231111yescharacteristic
325C22#237814yes—
326C22C2×#41312yes—
327C22C2×#41312yes—
328C4—24no—
329C4—42no—
330C4—42no—
331C4—42no—
332C4—42no—
333C4—84no—
334C4—84no—
335C4#281414yes—
336C4#281612yes—
337C4#281712yes—
338C2—24no—
339C2—44no—
340C2#1466011yescharacteristic
341C2#2175412yes—
342C2#2183011yescharacteristic
343C2#2183111yescharacteristic
344C2#2188012yes—
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC24C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25C22×C4
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C42 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #11383C2semidirect
5. #11355C2semidirect
6. #11204C2semidirect

Maximal subgroups

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

subgroupindex
C2×#28512
#206822
#206262
#113832
#113552
#112042
#111982