Finite Group Databasebeta

← #129797

Subgroups — #129797

Order 512.

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

normal subgroups only

This group has 916 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×#1351C214yesmaximal
3#10726C211yesmaximal, characteristic
4#8300C211yesmaximal, characteristic
5C22×C4⋊Q8C411yescharacteristic, complemented
6C22×#117C2211yescharacteristic
7#2628—28no—
8C2×#400—21no—
9C2×#399—21no—
10C2×#395C412yescomplemented
11C2×#393C2212yes—
12C2×#389—21no—
13C2×#382C411yescharacteristic, complemented
14C2×#381—21no—
15C2×#380—21no—
16#1817—21no—
17C2×#342—21no—
18C2×#342C2212yes—
19C2×#324—21no—
20C2×#323C2212yes—
21#1351—28no—
22C23×Q8C2×C412yes—
23C24×C4D411yescharacteristic
24C2×C4⋊Q8—24no—
25C2×C4⋊Q8—42no—
26C2×C4⋊Q8—42no—
27C2×#117—24no—
28C2×#117—42no—
29C2×#117—42no—
30C22×C4⋊C4—21no—
31C22×C4⋊C4—21no—
32C22×C4⋊C4—21no—
33C22×C4⋊C4—22no—
34C22×C4⋊C4C2×C411yescharacteristic
35C22×C4⋊C4D411yescharacteristic
36C22×C22⋊C4—22no—
37C22×C22⋊C4—22no—
38C22×C22⋊C4D411yescharacteristic
39C22×C22⋊C4C2311yescharacteristic
40C22×C42C2×C411yescharacteristic
41C22×C42D411yescharacteristic
42C2×C22⋊C8—22no—
43#400—28no—
44#399—28no—
45#395—24no—
46#395—24no—
47#393—24no—
48#393D418yes—
49#389—28no—
50#382—24no—
51#381—24no—
52#381—42no—
53#380—28no—
54C2×#95—22no—
55C2×#95—21no—
56C2×#95—22no—
57C2×#95—42no—
58C2×#95C2×C412yes—
59#342—24no—
60#342—42no—
61#336—42no—
62#324—42no—
63#323—24no—
64C25C2×D411yescharacteristic
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×C4—42no—
79C23×C4C22⋊C411yescharacteristic
80C23×C4C22⋊C411yescharacteristic
81C23×C4C22⋊C411yescharacteristic
82C23×C4C22⋊C411yescharacteristic
83C23×C4SD1612yes—
84C23×C4Q1612yes—
85C23×C4C22×C411yescharacteristic
86C23×C4C2×D411yescharacteristic
87C22×C8—42no—
88C4⋊Q8—416no—
89#117—416no—
90C2×C4⋊C4—24no—
91C2×C4⋊C4—24no—
92C2×C4⋊C4—24no—
93C2×C4⋊C4—22no—
94C2×C4⋊C4—22no—
95C2×C4⋊C4—22no—
96C2×C4⋊C4—28no—
97C2×C4⋊C4—41no—
98C2×C4⋊C4—41no—
99C2×C4⋊C4—41no—
100C2×C4⋊C4—42no—
101C2×C4⋊C4—42no—
102C2×C4⋊C4—42no—
103C2×C4⋊C4—42no—
104C2×C4⋊C4—42no—
105C2×C4⋊C4—42no—
106C2×C4⋊C4—84no—
107C2×C22⋊C4—22no—
108C2×C22⋊C4—22no—
109C2×C22⋊C4—24no—
110C2×C22⋊C4—28no—
111C2×C22⋊C4—41no—
112C2×C22⋊C4—41no—
113C2×C22⋊C4—41no—
114C2×C22⋊C4—42no—
115C2×C22⋊C4—42no—
116C2×C22⋊C4—42no—
117C2×C22⋊C4—42no—
118C2×C22⋊C4—42no—
119C2×C22⋊C4—84no—
120C2×C22⋊C4C2×D414yes—
121C2×C42—22no—
122C2×C42—22no—
123C2×C42—22no—
124C2×C42—22no—
125C2×C42—22no—
126C2×C42—41no—
127C22⋊C8—42no—
128C22⋊C8—42no—
129#95—28no—
130#95—24no—
131#95—22no—
132#95—28no—
133#95—48no—
134#95C22⋊C414yes—
135C24—21no—
136C24—21no—
137C24—21no—
138C24—21no—
139C24—42no—
140C24—42no—
141C24C2×C22⋊C411yescharacteristic
142C24#12012yes—
143C24C2×SD1611yescharacteristic
144C24C2×Q1611yescharacteristic
145C24#13712yes—
146C2×Q8—44no—
147C2×Q8—44no—
148C2×Q8—84no—
149C2×Q8—84no—
150C2×Q8—88no—
151C2×Q8—88no—
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—21no—
166C22×C4—21no—
167C22×C4—24no—
168C22×C4—24no—
169C22×C4—21no—
170C22×C4—24no—
171C22×C4—24no—
172C22×C4—24no—
173C22×C4—24no—
174C22×C4—21no—
175C22×C4—21no—
176C22×C4—21no—
177C22×C4—42no—
178C22×C4—42no—
179C22×C4—44no—
180C22×C4—44no—
181C22×C4—44no—
182C22×C4—44no—
183C22×C4—42no—
184C22×C4—42no—
185C22×C4—42no—
186C22×C4—42no—
187C22×C4—44no—
188C22×C4—44no—
189C22×C4—44no—
190C22×C4—44no—
191C22×C4—42no—
192C22×C4—42no—
193C22×C4—42no—
194C22×C4—42no—
195C22×C4—44no—
196C22×C4—82no—
197C22×C4—82no—
198C22×C4—84no—
199C22×C4—84no—
200C22×C4#9912yes—
201C22×C4#9912yes—
202C22×C4#10312yes—
203C22×C4#10312yes—
204C22×C4#10412yes—
205C22×C4C2×C22⋊C412yes—
206C22×C4#12012yes—
207C2×C8—82no—
208C2×C8—84no—
209C4⋊C4—44no—
210C4⋊C4—48no—
211C4⋊C4—44no—
212C4⋊C4—48no—
213C4⋊C4—48no—
214C4⋊C4—88no—
215C22⋊C4—44no—
216C22⋊C4—44no—
217C22⋊C4—44no—
218C22⋊C4—44no—
219C22⋊C4—88no—
220C22⋊C4—82no—
221C22⋊C4—82no—
222C42—28no—
223C42—44no—
224C23—22no—
225C23—22no—
226C23—21no—
227C23—21no—
228C23—22no—
229C23—22no—
230C23—22no—
231C23—21no—
232C23—21no—
233C23—22no—
234C23—42no—
235C23—42no—
236C23—42no—
237C23—42no—
238C23—44no—
239C23—42no—
240C23—42no—
241C23—82no—
242C23—81no—
243C23—81no—
244C23—82no—
245C23#37911yescharacteristic
246C23C2×#9911yescharacteristic
247C23C2×#9911yescharacteristic
248C23C2×#10311yescharacteristic
249C23#41911yescharacteristic
250C23C2×#10411yescharacteristic
251C23#42111yescharacteristic
252C23#44812yes—
253C23#45112yes—
254Q8—88no—
255Q8—88no—
256C2×C4—22no—
257C2×C4—22no—
258C2×C4—22no—
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—22no—
268C2×C4—48no—
269C2×C4—48no—
270C2×C4—42no—
271C2×C4—42no—
272C2×C4—42no—
273C2×C4—42no—
274C2×C4—42no—
275C2×C4—41no—
276C2×C4—41no—
277C2×C4—41no—
278C2×C4—41no—
279C2×C4—42no—
280C2×C4—42no—
281C2×C4—42no—
282C2×C4—44no—
283C2×C4—44no—
284C2×C4—42no—
285C2×C4—44no—
286C2×C4—44no—
287C2×C4—42no—
288C2×C4—41no—
289C2×C4—44no—
290C2×C4—44no—
291C2×C4—81no—
292C2×C4—84no—
293C2×C4—84no—
294C2×C4—84no—
295C2×C4—84no—
296C2×C4—81no—
297C2×C4—84no—
298C2×C4—88no—
299C2×C4—164no—
300C2×C4—162no—
301C2×C4—162no—
302C2×C4#32814yes—
303C8—162no—
304C22—22no—
305C22—22no—
306C22—21no—
307C22—21no—
308C22—21no—
309C22—21no—
310C22—22no—
311C22—22no—
312C22—22no—
313C22—22no—
314C22—42no—
315C22—42no—
316C22—44no—
317C22—42no—
318C22—42no—
319C22—42no—
320C22—42no—
321C22—82no—
322C22—82no—
323C22—81no—
324C22—81no—
325C22C2×#32811yescharacteristic
326C22#150811yescharacteristic
327C22#151111yescharacteristic
328C22#152011yescharacteristic
329C22#178111yescharacteristic
330C22#178711yescharacteristic
331C22#178911yescharacteristic
332C22#212212yes—
333C22#212912yes—
334C4—24no—
335C4—42no—
336C4—42no—
337C4—42no—
338C4—42no—
339C4—84no—
340C4—81no—
341C4—84no—
342C4—164no—
343C2—21no—
344C2—21no—
345C2—21no—
346C2—21no—
347C2—42no—
348C2—42no—
349C2#832911yescharacteristic
350C2#1171712yes—
351C2#1181212yes—
352C2#1269111yescharacteristic
353C2#1269811yescharacteristic
subgroupG/H
center, socleC23C2×#99
commutatorC23×C4C22×C4
Frattini, cosocleC22×C22⋊C4C23
prime radicalC25C2×D4
LCSG ▷ C23×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C4 ◁ C22×#117 ◁ G
LECSG ▷ C22×C22⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
5. C22×C4⋊Q8C4semidirect
10. C2×#395C4semidirect
13. C2×#382C4semidirect

Maximal subgroups

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

subgroupindex
C2×#26282
C2×#13512
#107262
#83002