Finite Group Databasebeta

← #140686

Subgroups — #140686

Order 512.

632 conjugacy classes of subgroups; only the 100 normal subgroups are shown.

normal subgroups only

This group has 632 conjugacy classes of subgroups — too many to draw. Only its 100 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
1#14155C211yesmaximal, characteristic
2C2×#1319C214yesmaximal
3#8560C211yesmaximal, characteristic
4#8153C211yesmaximal, characteristic
5C22×C4⋊Q8C411yescharacteristic
6C2×#391C412yes—
7C2×#391C2212yes—
8C2×C4×C4⋊C4C411yescharacteristic
9C2×C4×C4⋊C4C2211yescharacteristic
10#1797—21no—
11#1749—21no—
12C2×#324C2212yes—
13C2×#324C2212yes—
14#1445—21no—
15#1445—21no—
16C4⋊#95—21no—
17C4⋊#95—21no—
18#1319—28no—
19#1280—21no—
20#1280—21no—
21C23×Q8C2×C412yes—
22C2×C4⋊Q8—24no—
23C2×C4⋊Q8—41no—
24C2×C4⋊Q8—41no—
25C2×C4⋊Q8—42no—
26C22×C4⋊C4—21no—
27C22×C4⋊C4—21no—
28C22×C4⋊C4—21no—
29C22×C4⋊C4C2×C411yescharacteristic
30C22×C4⋊C4C2311yescharacteristic
31C22×C42C2×C411yescharacteristic
32C22×C42C2×C412yes—
33C22×C42D411yescharacteristic
34C22×C42D411yescharacteristic
35C2×C4⋊C8—41no—
36C2×C22⋊C8—22no—
37C2×C22⋊C8—22no—
38C2×#97—41no—
39#391—24no—
40#391—24no—
41#391—44no—
42C4×C4⋊C4—41no—
43C4×C4⋊C4—41no—
44C4×C4⋊C4—42no—
45C2×#95—22no—
46C2×#95—22no—
47C2×#95—21no—
48C2×#95—22no—
49C2×#95—22no—
50C2×#95D411yescharacteristic
51C2×#95D411yescharacteristic
52#336—41no—
53#336—41no—
54#324—22no—
55#324—24no—
56#324D414yes—
57C22×Q8—22no—
58C22×Q8—24no—
59C22×Q8—24no—
60C22×Q8—42no—
61C22×Q8—42no—
62C22×Q8—42no—
63C22×Q8—42no—
64C22×Q8M1614yes—
65C23×C4—22no—
66C23×C4—22no—
67C23×C4C22⋊C411yescharacteristic
68C23×C4C22⋊C411yescharacteristic
69C23×C4C22⋊C411yescharacteristic
70C23×C4C22⋊C411yescharacteristic
71C23×C4C22×C411yescharacteristic
72C23×C4C2×D411yescharacteristic
73C23×C4C2×D411yescharacteristic
74C22×C8—42no—
75C22×C8—42no—
76C4⋊Q8—48no—
77C4⋊Q8—48no—
78C2×C4⋊C4—24no—
79C2×C4⋊C4—24no—
80C2×C4⋊C4—24no—
81C2×C4⋊C4—22no—
82C2×C4⋊C4—22no—
83C2×C4⋊C4—41no—
84C2×C4⋊C4—41no—
85C2×C4⋊C4—41no—
86C2×C4⋊C4—42no—
87C2×C4⋊C4—42no—
88C2×C4⋊C4—42no—
89C2×C4⋊C4—42no—
90C2×C4⋊C4—41no—
91C2×C4⋊C4—41no—
92C2×C4⋊C4—42no—
93C2×C4⋊C4C2×D412yes—
94C2×C4⋊C4#4112yes—
95C2×C42—21no—
96C2×C42—21no—
97C2×C42—22no—
98C2×C42—22no—
99C2×C42—21no—
100C2×C42—21no—
101C2×C42—21no—
102C2×C42—21no—
103C2×C42—24no—
104C2×C42—42no—
105C2×C42—42no—
106C2×C42—42no—
107C2×C42—42no—
108C4⋊C8—82no—
109C22⋊C8—24no—
110C22⋊C8—42no—
111C22⋊C8—42no—
112C22⋊C8—42no—
113#97—82no—
114#95—41no—
115#95—41no—
116#95—44no—
117#95—42no—
118#95—44no—
119#95—44no—
120#95—42no—
121#95—42no—
122C24C2×C22⋊C411yescharacteristic
123C2×Q8—28no—
124C2×Q8—44no—
125C2×Q8—44no—
126C2×Q8—44no—
127C2×Q8—44no—
128C2×Q8—44no—
129C2×Q8—84no—
130C2×Q8—84no—
131C2×Q8—84no—
132C2×Q8—84no—
133C22×C4—21no—
134C22×C4—21no—
135C22×C4—21no—
136C22×C4—21no—
137C22×C4—21no—
138C22×C4—21no—
139C22×C4—22no—
140C22×C4—22no—
141C22×C4—21no—
142C22×C4—21no—
143C22×C4—21no—
144C22×C4—21no—
145C22×C4—21no—
146C22×C4—21no—
147C22×C4—21no—
148C22×C4—21no—
149C22×C4—24no—
150C22×C4—24no—
151C22×C4—21no—
152C22×C4—21no—
153C22×C4—21no—
154C22×C4—41no—
155C22×C4—41no—
156C22×C4—41no—
157C22×C4—41no—
158C22×C4—41no—
159C22×C4—41no—
160C22×C4—41no—
161C22×C4—41no—
162C22×C4—41no—
163C22×C4—41no—
164C22×C4—42no—
165C22×C4—42no—
166C22×C4—42no—
167C22×C4—42no—
168C22×C4—42no—
169C22×C4—42no—
170C22×C4—42no—
171C22×C4—42no—
172C22×C4—42no—
173C22×C4—42no—
174C22×C4—42no—
175C22×C4—42no—
176C22×C4#9911yescharacteristic
177C22×C4#9911yescharacteristic
178C22×C4#10012yes—
179C22×C4#10012yes—
180C22×C4C4×D412yes—
181C22×C4#12112yes—
182C22×C4#12411yescharacteristic
183C22×C4#12711yescharacteristic
184C22×C4C2×M1612yes—
185C22×C4#13112yes—
186C2×C8—44no—
187C2×C8—82no—
188C2×C8—82no—
189C2×C8—82no—
190C2×C8—82no—
191C2×C8—82no—
192C4⋊C4—42no—
193C4⋊C4—42no—
194C4⋊C4—44no—
195C4⋊C4—48no—
196C4⋊C4—44no—
197C4⋊C4—44no—
198C4⋊C4—48no—
199C42—24no—
200C42—24no—
201C42—42no—
202C42—42no—
203C42—81no—
204C42—81no—
205C42—84no—
206C23—21no—
207C23—21no—
208C23—21no—
209C23—21no—
210C23#39011yescharacteristic
211C23#40711yescharacteristic
212C23#40811yescharacteristic
213C23C2×#9911yescharacteristic
214C23#41011yescharacteristic
215C23C2×#10011yescharacteristic
216C23C2×#10011yescharacteristic
217Q8—88no—
218Q8—88no—
219C2×C4—24no—
220C2×C4—24no—
221C2×C4—21no—
222C2×C4—21no—
223C2×C4—22no—
224C2×C4—22no—
225C2×C4—21no—
226C2×C4—21no—
227C2×C4—22no—
228C2×C4—22no—
229C2×C4—22no—
230C2×C4—22no—
231C2×C4—22no—
232C2×C4—21no—
233C2×C4—21no—
234C2×C4—22no—
235C2×C4—22no—
236C2×C4—42no—
237C2×C4—41no—
238C2×C4—41no—
239C2×C4—41no—
240C2×C4—41no—
241C2×C4—41no—
242C2×C4—41no—
243C2×C4—41no—
244C2×C4—41no—
245C2×C4—42no—
246C2×C4—42no—
247C2×C4—42no—
248C2×C4—41no—
249C2×C4—41no—
250C2×C4—44no—
251C2×C4—44no—
252C2×C4—44no—
253C2×C4—42no—
254C2×C4—41no—
255C2×C4—41no—
256C2×C4—44no—
257C2×C4—44no—
258C2×C4—44no—
259C2×C4—41no—
260C2×C4—81no—
261C2×C4—81no—
262C2×C4—81no—
263C2×C4—82no—
264C2×C4—82no—
265C2×C4—82no—
266C2×C4—82no—
267C2×C4—82no—
268C2×C4—82no—
269C2×C4—82no—
270C2×C4—82no—
271C2×C4—81no—
272C2×C4#35412yes—
273C2×C4#35612yes—
274C2×C4#43514yes—
275C8—84no—
276C8—162no—
277C22—21no—
278C22—21no—
279C22—21no—
280C22—21no—
281C22—21no—
282C22—21no—
283C22—21no—
284C22—21no—
285C22—21no—
286C22—21no—
287C22—21no—
288C22—21no—
289C22#146211yescharacteristic
290C22#146611yescharacteristic
291C22#146911yescharacteristic
292C22#147111yescharacteristic
293C22#195511yescharacteristic
294C22#196411yescharacteristic
295C22#196511yescharacteristic
296C22C2×#35411yescharacteristic
297C22#212911yescharacteristic
298C22C2×#35611yescharacteristic
299C22#213511yescharacteristic
300C4—24no—
301C4—22no—
302C4—22no—
303C4—42no—
304C4—42no—
305C4—41no—
306C4—41no—
307C4—42no—
308C4—84no—
309C4—81no—
310C4—84no—
311C2—21no—
312C2—21no—
313C2—21no—
314C2—21no—
315C2#965311yescharacteristic
316C2#1150611yescharacteristic
317C2#1151011yescharacteristic
318C2#1156411yescharacteristic
319C2#1156811yescharacteristic
320C2#1322411yescharacteristic
321C2#1322711yescharacteristic
subgroupG/H
center, socleC23C2×#99
commutatorC23×C4C22×C4
Frattini, cosocleC22×C4⋊C4C23
prime radicalC24C2×C22⋊C4
LCSG ▷ C23×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C22×C4⋊C4 ▷ C23 ▷ C2 ▷ C1

Maximal subgroups

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

subgroupindex
#141552
C2×#13192
#85602
#81532