Finite Group Databasebeta

← #129804

Subgroups — #129804

Order 512.

882 conjugacy classes of subgroups; only the 116 normal subgroups are shown.

normal subgroups only

This group has 882 conjugacy classes of subgroups — too many to draw. Only its 116 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2628C211yesmaximal, characteristic
2C2×#1354C214yesmaximal, complemented
3#10728C211yesmaximal, characteristic
4#8300C211yesmaximal, characteristic
5C22×C4⋊Q8C411yescharacteristic
6C22×#117C2211yescharacteristic
7C2×#407—21no—
8#2628—28no—
9C2×#400—21no—
10C2×#399—21no—
11C2×#395C2212yes—
12C2×#393C412yes—
13C2×#389—21no—
14C2×#382C411yescharacteristic
15C2×#380—21no—
16#1817—21no—
17C2×#343—21no—
18C2×#343C2212yes—
19C2×#324C2212yes—
20C2×#323—21no—
21#1354—28no—
22C23×Q8C2×C412yes—
23C24×C4D411yescharacteristic
24C22×M16—22no—
25C2×C4⋊Q8—24no—
26C2×C4⋊Q8—42no—
27C2×C4⋊Q8—42no—
28C2×#117—24no—
29C2×#117—42no—
30C2×#117—42no—
31C22×C4⋊C4—21no—
32C22×C4⋊C4—21no—
33C22×C4⋊C4—21no—
34C22×C4⋊C4—22no—
35C22×C4⋊C4D411yescharacteristic
36C22×C4⋊C4C2311yescharacteristic
37C22×C22⋊C4—22no—
38C22×C22⋊C4C2×C411yescharacteristic
39C22×C22⋊C4D411yescharacteristic
40C22×C42C2×C411yescharacteristic
41C22×C42D411yescharacteristic
42#407—24no—
43#407—42no—
44C2×C22⋊C8—22no—
45C2×C22⋊C8—42no—
46#400—28no—
47#399—28no—
48#395—24no—
49#395D418yes—
50#393—24no—
51#393—24no—
52#389—28no—
53#382—24no—
54#380—28no—
55C2×#95—22no—
56C2×#95—21no—
57C2×#95—22no—
58C2×#95C2×C412yes—
59#343—24no—
60#343—42no—
61#336—42no—
62#324—24no—
63#323—42no—
64C25C22⋊C411yescharacteristic
65C22×Q8—22no—
66C22×Q8—42no—
67C22×Q8—42no—
68C22×Q8—84no—
69C22×Q8C22⋊C414yes—
70C23×C4—22no—
71C23×C4—22no—
72C23×C4—22no—
73C23×C4—22no—
74C23×C4—22no—
75C23×C4—22no—
76C23×C4—41no—
77C23×C4—41no—
78C23×C4C22⋊C411yescharacteristic
79C23×C4C22⋊C411yescharacteristic
80C23×C4C22⋊C411yescharacteristic
81C23×C4SD1612yes—
82C23×C4Q1612yes—
83C23×C4C22×C411yescharacteristic
84C23×C4C2×D411yescharacteristic
85C23×C4C2×D411yescharacteristic
86C2×M16—24no—
87C2×M16—42no—
88C2×M16—42no—
89C2×M16—42no—
90C2×M16—42no—
91C2×M16—42no—
92C22×C8—42no—
93C22×C8—42no—
94C4⋊Q8—416no—
95#117—416no—
96C2×C4⋊C4—24no—
97C2×C4⋊C4—24no—
98C2×C4⋊C4—24no—
99C2×C4⋊C4—22no—
100C2×C4⋊C4—22no—
101C2×C4⋊C4—28no—
102C2×C4⋊C4—41no—
103C2×C4⋊C4—41no—
104C2×C4⋊C4—41no—
105C2×C4⋊C4—42no—
106C2×C4⋊C4—42no—
107C2×C4⋊C4—42no—
108C2×C4⋊C4—42no—
109C2×C4⋊C4—42no—
110C2×C4⋊C4—42no—
111C2×C4⋊C4—84no—
112C2×C4⋊C4C2×D414yes—
113C2×C22⋊C4—22no—
114C2×C22⋊C4—22no—
115C2×C22⋊C4—22no—
116C2×C22⋊C4—28no—
117C2×C22⋊C4—41no—
118C2×C22⋊C4—41no—
119C2×C22⋊C4—41no—
120C2×C22⋊C4—84no—
121C2×C42—22no—
122C2×C42—22no—
123C2×C42—22no—
124C2×C42—22no—
125C2×C42—22no—
126C2×C42—41no—
127C22⋊C8—48no—
128C22⋊C8—42no—
129C22⋊C8—42no—
130#95—24no—
131#95—28no—
132#95—22no—
133#95—28no—
134#95C22⋊C414yes—
135C24—21no—
136C24—21no—
137C24—21no—
138C24—21no—
139C24—21no—
140C24—21no—
141C24—42no—
142C24—42no—
143C24#10312yes—
144C24C2×C22⋊C411yescharacteristic
145C2×Q8—44no—
146C2×Q8—44no—
147C2×Q8—84no—
148C2×Q8—84no—
149C2×Q8—88no—
150C2×Q8—88no—
151C22×C4—21no—
152C22×C4—21no—
153C22×C4—21no—
154C22×C4—21no—
155C22×C4—21no—
156C22×C4—21no—
157C22×C4—21no—
158C22×C4—21no—
159C22×C4—21no—
160C22×C4—21no—
161C22×C4—21no—
162C22×C4—21no—
163C22×C4—21no—
164C22×C4—21no—
165C22×C4—24no—
166C22×C4—24no—
167C22×C4—24no—
168C22×C4—24no—
169C22×C4—21no—
170C22×C4—24no—
171C22×C4—24no—
172C22×C4—21no—
173C22×C4—21no—
174C22×C4—42no—
175C22×C4—42no—
176C22×C4—44no—
177C22×C4—44no—
178C22×C4—44no—
179C22×C4—44no—
180C22×C4—42no—
181C22×C4—42no—
182C22×C4—42no—
183C22×C4—42no—
184C22×C4—44no—
185C22×C4—44no—
186C22×C4—44no—
187C22×C4—44no—
188C22×C4—42no—
189C22×C4—42no—
190C22×C4—42no—
191C22×C4—42no—
192C22×C4#9912yes—
193C22×C4#9912yes—
194C22×C4#10312yes—
195C22×C4#10412yes—
196C22×C4C2×C22⋊C412yes—
197C22×C4#12012yes—
198C22×C4#12012yes—
199C22×C4C2×SD1611yescharacteristic
200C22×C4C2×Q1611yescharacteristic
201C22×C4#13712yes—
202M16—44no—
203M16—44no—
204M16—82no—
205M16—82no—
206C2×C8—44no—
207C2×C8—82no—
208C2×C8—82no—
209C2×C8—82no—
210C2×C8—84no—
211C4⋊C4—44no—
212C4⋊C4—44no—
213C4⋊C4—48no—
214C4⋊C4—48no—
215C4⋊C4—48no—
216C4⋊C4—88no—
217C22⋊C4—44no—
218C22⋊C4—44no—
219C22⋊C4—88no—
220C42—28no—
221C42—44no—
222C23—22no—
223C23—22no—
224C23—22no—
225C23—22no—
226C23—22no—
227C23—44no—
228C23—42no—
229C23—44no—
230C23—42no—
231C23—42no—
232C23—42no—
233C23—41no—
234C23—41no—
235C23—41no—
236C23—41no—
237C23—82no—
238C23—82no—
239C23—81no—
240C23—81no—
241C23#37911yescharacteristic
242C23C2×#9911yescharacteristic
243C23C2×#9911yescharacteristic
244C23C2×#10311yescharacteristic
245C23#41911yescharacteristic
246C23C2×#10411yescharacteristic
247C23#42111yescharacteristic
248Q8—88no—
249Q8—88no—
250C2×C4—21no—
251C2×C4—21no—
252C2×C4—22no—
253C2×C4—22no—
254C2×C4—22no—
255C2×C4—22no—
256C2×C4—21no—
257C2×C4—22no—
258C2×C4—21no—
259C2×C4—22no—
260C2×C4—22no—
261C2×C4—22no—
262C2×C4—22no—
263C2×C4—22no—
264C2×C4—22no—
265C2×C4—22no—
266C2×C4—22no—
267C2×C4—48no—
268C2×C4—48no—
269C2×C4—42no—
270C2×C4—42no—
271C2×C4—42no—
272C2×C4—42no—
273C2×C4—42no—
274C2×C4—42no—
275C2×C4—42no—
276C2×C4—42no—
277C2×C4—44no—
278C2×C4—44no—
279C2×C4—44no—
280C2×C4—44no—
281C2×C4—42no—
282C2×C4—42no—
283C2×C4—41no—
284C2×C4—44no—
285C2×C4—44no—
286C2×C4—81no—
287C2×C4—84no—
288C2×C4—84no—
289C2×C4—84no—
290C2×C4—84no—
291C2×C4—81no—
292C2×C4#32814yes—
293C2×C4#44812yes—
294C2×C4#45112yes—
295C8—84no—
296C8—162no—
297C22—22no—
298C22—22no—
299C22—21no—
300C22—21no—
301C22—21no—
302C22—21no—
303C22—22no—
304C22—22no—
305C22—44no—
306C22—42no—
307C22—44no—
308C22—82no—
309C22—82no—
310C22—82no—
311C22—82no—
312C22—81no—
313C22—81no—
314C22C2×#32811yescharacteristic
315C22#150811yescharacteristic
316C22#151111yescharacteristic
317C22#152011yescharacteristic
318C22#178111yescharacteristic
319C22#178711yescharacteristic
320C22#178911yescharacteristic
321C22#212212yes—
322C22#213512yes—
323C4—22no—
324C4—24no—
325C4—22no—
326C4—42no—
327C4—42no—
328C4—42no—
329C4—84no—
330C4—84no—
331C4—81no—
332C2—21no—
333C2—21no—
334C2—21no—
335C2—21no—
336C2—82no—
337C2#832911yescharacteristic
338C2#1172412yes—
339C2#1181912yes—
340C2#1269111yescharacteristic
341C2#1270411yescharacteristic
subgroupG/H
center, socleC23C2×#99
commutatorC23×C4C22×C4
Frattini, cosocleC22×C4⋊C4C23
prime radicalC25C22⋊C4
LCSG ▷ C23×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C4 ◁ C22×#117 ◁ G
LECSG ▷ C22×C4⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#1354C2semidirect

Maximal subgroups

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

subgroupindex
C2×#26282
C2×#13542
#107282
#83002