Finite Group Databasebeta

← #97441

Subgroups — #97441

Order 512.

852 conjugacy classes of subgroups; only the 120 normal subgroups are shown.

normal subgroups only

This group has 852 conjugacy classes of subgroups — too many to draw. Only its 120 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×#2568C211yesmaximal, characteristic
2#10763C211yesmaximal, characteristic
3#8303C211yesmaximal, characteristic
4#7609C214yesmaximal
5C22×C4⋊Q8C411yescharacteristic
6C22×C4×Q8C2211yescharacteristic
7C22×#101—21no—
8#2568—28no—
9C2×#400—21no—
10C2×#400—21no—
11C2×#391—21no—
12C2×#391C412yes—
13C2×#391C2212yes—
14C2×#382—21no—
15C2×C4×C4⋊C4C411yescharacteristic
16#1817C2212yes—
17C2×#343—21no—
18#1728C2212yes—
19C2×#324—21no—
20C2×#324—21no—
21#1272—28no—
22C23×Q8C2×C412yes—
23C23×Q8D411yescharacteristic
24C22×M16—22no—
25C2×C4⋊Q8—24no—
26C2×C4⋊Q8—42no—
27C2×C4⋊Q8—42no—
28C2×C4×Q8—24no—
29C2×C4×Q8—42no—
30C2×C4×Q8—42no—
31C22×C4⋊C4—21no—
32C22×C4⋊C4—21no—
33C22×C4⋊C4C2×C412yes—
34C22×C4⋊C4D411yescharacteristic
35C22×C4⋊C4D411yescharacteristic
36C22×C4⋊C4D411yescharacteristic
37C22×C42—21no—
38C22×C42C2×C411yescharacteristic
39C22×C42C2×C411yescharacteristic
40C22×C42C2311yescharacteristic
41C2×#101—24no—
42C2×#101—42no—
43C2×#101—42no—
44C2×C22⋊C8—22no—
45#400—28no—
46#400—28no—
47#391—28no—
48#391—24no—
49#391—24no—
50#391—24no—
51#391D418yes—
52#382—28no—
53C4×C4⋊C4—24no—
54C2×#95—21no—
55C2×#95—22no—
56C2×#95—22no—
57C2×#95—22no—
58C2×#95—22no—
59#343—42no—
60#336—24no—
61#336—24no—
62#324—42no—
63#324—42no—
64C22×Q8—22no—
65C22×Q8—22no—
66C22×Q8—22no—
67C22×Q8—22no—
68C22×Q8—22no—
69C22×Q8—41no—
70C22×Q8—41no—
71C22×Q8—44no—
72C22×Q8—44no—
73C22×Q8—42no—
74C22×Q8—42no—
75C22×Q8C22⋊C414yes—
76C22×Q8SD1612yes—
77C22×Q8Q1612yes—
78C23×C4—22no—
79C23×C4—22no—
80C23×C4C22⋊C411yescharacteristic
81C23×C4C22⋊C411yescharacteristic
82C23×C4C22⋊C411yescharacteristic
83C23×C4C22⋊C411yescharacteristic
84C23×C4C22×C411yescharacteristic
85C23×C4C2×D411yescharacteristic
86C23×C4C2×D411yescharacteristic
87C2×M16—24no—
88C2×M16—42no—
89C2×M16—42no—
90C2×M16—42no—
91C2×M16—42no—
92C2×M16—42no—
93C22×C8—42no—
94C22×C8—42no—
95C4⋊Q8—416no—
96C4×Q8—416no—
97C2×C4⋊C4—24no—
98C2×C4⋊C4—24no—
99C2×C4⋊C4—22no—
100C2×C4⋊C4—22no—
101C2×C4⋊C4—22no—
102C2×C4⋊C4—22no—
103C2×C4⋊C4—22no—
104C2×C4⋊C4—22no—
105C2×C4⋊C4—22no—
106C2×C4⋊C4—41no—
107C2×C4⋊C4—41no—
108C2×C4⋊C4—41no—
109C2×C4⋊C4—42no—
110C2×C4⋊C4—42no—
111C2×C4⋊C4—42no—
112C2×C4⋊C4—42no—
113C2×C4⋊C4—44no—
114C2×C4⋊C4C22⋊C414yes—
115C2×C42—22no—
116C2×C42—22no—
117C2×C42—22no—
118C2×C42—24no—
119C2×C42—22no—
120C2×C42—41no—
121C2×C42—41no—
122C2×C42—41no—
123C2×C42—41no—
124C2×C42—42no—
125C2×C42—42no—
126C2×C42C2×D414yes—
127#101—48no—
128C22⋊C8—42no—
129C22⋊C8—42no—
130#95—24no—
131#95—28no—
132#95—28no—
133#95—28no—
134#95—28no—
135C24C2×C22⋊C411yescharacteristic
136C2×Q8—24no—
137C2×Q8—24no—
138C2×Q8—24no—
139C2×Q8—24no—
140C2×Q8—28no—
141C2×Q8—42no—
142C2×Q8—42no—
143C2×Q8—44no—
144C2×Q8—44no—
145C2×Q8—44no—
146C2×Q8—44no—
147C2×Q8—44no—
148C2×Q8—44no—
149C2×Q8—44no—
150C2×Q8—44no—
151C2×Q8—44no—
152C2×Q8—88no—
153C2×Q8—88no—
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—24no—
165C22×C4—21no—
166C22×C4—21no—
167C22×C4—21no—
168C22×C4—21no—
169C22×C4—21no—
170C22×C4—21no—
171C22×C4—21no—
172C22×C4—21no—
173C22×C4—21no—
174C22×C4—21no—
175C22×C4—24no—
176C22×C4—21no—
177C22×C4—24no—
178C22×C4—21no—
179C22×C4—21no—
180C22×C4—42no—
181C22×C4—42no—
182C22×C4—44no—
183C22×C4—44no—
184C22×C4—44no—
185C22×C4—44no—
186C22×C4—42no—
187C22×C4—42no—
188C22×C4—42no—
189C22×C4—42no—
190C22×C4—42no—
191C22×C4#9912yes—
192C22×C4#9912yes—
193C22×C4#10312yes—
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—44no—
214C4⋊C4—44no—
215C4⋊C4—48no—
216C4⋊C4—48no—
217C4⋊C4—44no—
218C42—28no—
219C42—44no—
220C42—48no—
221C42—44no—
222C23—22no—
223C23—22no—
224C23#37911yescharacteristic
225C23C2×#9911yescharacteristic
226C23C2×#9911yescharacteristic
227C23C2×#10311yescharacteristic
228C23#41911yescharacteristic
229C23C2×#10411yescharacteristic
230C23#42111yescharacteristic
231Q8—48no—
232Q8—48no—
233Q8—88no—
234Q8—88no—
235C2×C4—24no—
236C2×C4—24no—
237C2×C4—21no—
238C2×C4—21no—
239C2×C4—22no—
240C2×C4—22no—
241C2×C4—22no—
242C2×C4—22no—
243C2×C4—21no—
244C2×C4—22no—
245C2×C4—21no—
246C2×C4—22no—
247C2×C4—22no—
248C2×C4—22no—
249C2×C4—22no—
250C2×C4—22no—
251C2×C4—22no—
252C2×C4—22no—
253C2×C4—24no—
254C2×C4—22no—
255C2×C4—22no—
256C2×C4—22no—
257C2×C4—22no—
258C2×C4—22no—
259C2×C4—42no—
260C2×C4—42no—
261C2×C4—41no—
262C2×C4—41no—
263C2×C4—41no—
264C2×C4—41no—
265C2×C4—42no—
266C2×C4—42no—
267C2×C4—44no—
268C2×C4—41no—
269C2×C4—44no—
270C2×C4—44no—
271C2×C4—44no—
272C2×C4—44no—
273C2×C4—44no—
274C2×C4—81no—
275C2×C4—81no—
276C2×C4—84no—
277C2×C4—84no—
278C2×C4—84no—
279C2×C4—84no—
280C2×C4—81no—
281C2×C4—81no—
282C2×C4#32814yes—
283C2×C4#41314yes—
284C2×C4#44812yes—
285C2×C4#45112yes—
286C8—84no—
287C8—162no—
288C22—21no—
289C22—21no—
290C22—21no—
291C22—21no—
292C22—22no—
293C22—22no—
294C22—22no—
295C22—22no—
296C22C2×#32811yescharacteristic
297C22#150811yescharacteristic
298C22#151111yescharacteristic
299C22#152011yescharacteristic
300C22#178111yescharacteristic
301C22#178512yes—
302C22#178711yescharacteristic
303C22#178911yescharacteristic
304C22#179612yes—
305C4—24no—
306C4—22no—
307C4—24no—
308C4—22no—
309C4—42no—
310C4—42no—
311C4—42no—
312C4—81no—
313C4—84no—
314C4—84no—
315C2—21no—
316C2—21no—
317C2—21no—
318C2—21no—
319C2#832911yescharacteristic
320C2#836812yes—
321C2#839212yes—
322C2#852011yescharacteristic
323C2#855211yescharacteristic
subgroupG/H
center, socleC23#379
commutatorC23×C4C22×C4
Frattini, cosocleC22×C42C23
prime radicalC24C2×C22⋊C4
LCSG ▷ C23×C4 ▷ C23 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C23 ▷ C1

Maximal subgroups

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

subgroupindex
C2×#25682
#107632
#83032
#76092