Finite Group Databasebeta

← #129798

Subgroups — #129798

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×#1352C214yesmaximal
3#10726C211yesmaximal, characteristic
4#8300C211yesmaximal, characteristic
5C22×C4⋊Q8C411yescharacteristic, complemented
6C22×#117C2211yescharacteristic
7#2628—28no—
8C2×#400C412yescomplemented
9C2×#399C2212yes—
10C2×#395—21no—
11C2×#393—21no—
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#1352—28no—
22C23×Q8—21no—
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—22no—
33C22×C4⋊C4C2×C411yescharacteristic
34C22×C4⋊C4C2×C412yes—
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—24no—
44#400—24no—
45#399—24no—
46#399D418yes—
47#395—28no—
48#393—28no—
49#389—28no—
50#382—24no—
51#381—24no—
52#381—42no—
53#380—28no—
54C2×#95—21no—
55C2×#95—22no—
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—24no—
66C22×Q8—42no—
67C22×Q8—42no—
68C22×Q8—84no—
69C23×C4—22no—
70C23×C4—22no—
71C23×C4—22no—
72C23×C4—22no—
73C23×C4—22no—
74C23×C4—22no—
75C23×C4—41no—
76C23×C4—41no—
77C23×C4—42no—
78C23×C4C22⋊C411yescharacteristic
79C23×C4C22⋊C411yescharacteristic
80C23×C4C22⋊C411yescharacteristic
81C23×C4C22⋊C411yescharacteristic
82C23×C4SD1612yes—
83C23×C4Q1612yes—
84C23×C4C22×C411yescharacteristic
85C23×C4C2×D411yescharacteristic
86C22×C8—42no—
87C4⋊Q8—416no—
88#117—416no—
89C2×C4⋊C4—24no—
90C2×C4⋊C4—24no—
91C2×C4⋊C4—22no—
92C2×C4⋊C4—22no—
93C2×C4⋊C4—22no—
94C2×C4⋊C4—22no—
95C2×C4⋊C4—28no—
96C2×C4⋊C4—41no—
97C2×C4⋊C4—41no—
98C2×C4⋊C4—41no—
99C2×C4⋊C4—42no—
100C2×C4⋊C4—42no—
101C2×C4⋊C4—42no—
102C2×C4⋊C4—42no—
103C2×C4⋊C4—42no—
104C2×C4⋊C4—42no—
105C2×C4⋊C4—84no—
106C2×C4⋊C4C22⋊C414yes—
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—28no—
132#95—22no—
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—48no—
147C2×Q8—88no—
148C2×Q8—88no—
149C2×Q8—88no—
150C22×C4—21no—
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—21no—
168C22×C4—24no—
169C22×C4—24no—
170C22×C4—21no—
171C22×C4—24no—
172C22×C4—24no—
173C22×C4—21no—
174C22×C4—21no—
175C22×C4—42no—
176C22×C4—42no—
177C22×C4—44no—
178C22×C4—44no—
179C22×C4—44no—
180C22×C4—44no—
181C22×C4—42no—
182C22×C4—42no—
183C22×C4—42no—
184C22×C4—42no—
185C22×C4—44no—
186C22×C4—44no—
187C22×C4—44no—
188C22×C4—44no—
189C22×C4—42no—
190C22×C4—42no—
191C22×C4—42no—
192C22×C4—42no—
193C22×C4—44no—
194C22×C4—82no—
195C22×C4—82no—
196C22×C4—84no—
197C22×C4—84no—
198C22×C4#9912yes—
199C22×C4#9912yes—
200C22×C4#10312yes—
201C22×C4#10312yes—
202C22×C4#10412yes—
203C22×C4C2×C22⋊C412yes—
204C22×C4#12012yes—
205C2×C8—82no—
206C2×C8—84no—
207C4⋊C4—44no—
208C4⋊C4—44no—
209C4⋊C4—48no—
210C4⋊C4—44no—
211C4⋊C4—48no—
212C4⋊C4—44no—
213C4⋊C4—88no—
214C22⋊C4—44no—
215C22⋊C4—44no—
216C22⋊C4—44no—
217C22⋊C4—44no—
218C22⋊C4—88no—
219C22⋊C4—82no—
220C22⋊C4—82no—
221C42—28no—
222C42—44no—
223C23—22no—
224C23—22no—
225C23—21no—
226C23—21no—
227C23—22no—
228C23—22no—
229C23—22no—
230C23—21no—
231C23—21no—
232C23—22no—
233C23—42no—
234C23—42no—
235C23—42no—
236C23—42no—
237C23—44no—
238C23—42no—
239C23—42no—
240C23—82no—
241C23—81no—
242C23—81no—
243C23—82no—
244C23#37911yescharacteristic
245C23C2×#9911yescharacteristic
246C23C2×#9911yescharacteristic
247C23C2×#10311yescharacteristic
248C23#41911yescharacteristic
249C23C2×#10411yescharacteristic
250C23#42111yescharacteristic
251C23#44812yes—
252C23#45112yes—
253Q8—816no—
254C2×C4—22no—
255C2×C4—22no—
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—48no—
267C2×C4—48no—
268C2×C4—42no—
269C2×C4—42no—
270C2×C4—42no—
271C2×C4—42no—
272C2×C4—42no—
273C2×C4—41no—
274C2×C4—41no—
275C2×C4—41no—
276C2×C4—41no—
277C2×C4—42no—
278C2×C4—42no—
279C2×C4—42no—
280C2×C4—44no—
281C2×C4—44no—
282C2×C4—44no—
283C2×C4—44no—
284C2×C4—42no—
285C2×C4—42no—
286C2×C4—41no—
287C2×C4—44no—
288C2×C4—44no—
289C2×C4—81no—
290C2×C4—84no—
291C2×C4—84no—
292C2×C4—84no—
293C2×C4—84no—
294C2×C4—81no—
295C2×C4—84no—
296C2×C4—88no—
297C2×C4—164no—
298C2×C4—162no—
299C2×C4—162no—
300C2×C4#32814yes—
301C8—162no—
302C22—22no—
303C22—22no—
304C22—21no—
305C22—21no—
306C22—21no—
307C22—21no—
308C22—22no—
309C22—22no—
310C22—22no—
311C22—22no—
312C22—42no—
313C22—42no—
314C22—44no—
315C22—42no—
316C22—42no—
317C22—42no—
318C22—42no—
319C22—82no—
320C22—82no—
321C22—81no—
322C22—81no—
323C22C2×#32811yescharacteristic
324C22#150811yescharacteristic
325C22#151111yescharacteristic
326C22#152011yescharacteristic
327C22#178111yescharacteristic
328C22#178711yescharacteristic
329C22#178911yescharacteristic
330C22#212312yes—
331C22#212812yes—
332C4—24no—
333C4—42no—
334C4—42no—
335C4—42no—
336C4—42no—
337C4—84no—
338C4—81no—
339C4—84no—
340C4—164no—
341C2—21no—
342C2—21no—
343C2—21no—
344C2—21no—
345C2—42no—
346C2—42no—
347C2#832911yescharacteristic
348C2#1171812yes—
349C2#1181312yes—
350C2#1269211yescharacteristic
351C2#1269711yescharacteristic
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
8. C2×#400C4semidirect
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×#13522
#107262
#83002