Finite Group Databasebeta

← #142056

Subgroups — #142056

Order 512.

616 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 616 conjugacy classes of subgroups — too many to draw. Only its 94 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#15128C211yesmaximal, characteristic
2#10802C211yesmaximal, characteristic
3#8153C211yesmaximal, characteristic
4#7251C214yesmaximal
5C22×C4⋊Q8C411yescharacteristic
6C2×#395—21no—
7C2×#389—21no—
8C2×C4×C4⋊C4C2211yescharacteristic
9#1749—21no—
10C2×#343—21no—
11C2×#343C2212yes—
12#1728—21no—
13C2×#324C2212yes—
14#1446—21no—
15#1444C412yes—
16#1441C2212yes—
17C4⋊#95—21no—
18C4⋊#95C411yescharacteristic
19#1280—21no—
20C23×Q8—21no—
21C22×M16—22no—
22C2×C4⋊Q8—41no—
23C2×C4⋊Q8—41no—
24C2×C4⋊Q8—44no—
25C22×C4⋊C4—21no—
26C22×C4⋊C4—21no—
27C22×C4⋊C4—22no—
28C22×C4⋊C4C2×C412yes—
29C22×C4⋊C4D411yescharacteristic
30C22×C4⋊C4C2311yescharacteristic
31C22×C42C2×C411yescharacteristic
32C22×C42D411yescharacteristic
33C22×C42D411yescharacteristic
34C2×C22⋊C8—22no—
35C2×#97—41no—
36#395—44no—
37#389—44no—
38C4×C4⋊C4—41no—
39C4×C4⋊C4—41no—
40C2×#95—21no—
41C2×#95—22no—
42C2×#95—22no—
43C2×#95—22no—
44C2×#95C2×C412yes—
45C2×#95C2×C411yescharacteristic
46C2×#95D411yescharacteristic
47#343—24no—
48#343—24no—
49#336—41no—
50#336—41no—
51#336—41no—
52#324—22no—
53#324Q814yescomplemented
54C22×Q8—42no—
55C22×Q8—44no—
56C22×Q8—42no—
57C22×Q8—42no—
58C22×Q8—82no—
59C23×C4—22no—
60C23×C4—22no—
61C23×C4C22⋊C411yescharacteristic
62C23×C4C22⋊C411yescharacteristic
63C23×C4C22⋊C411yescharacteristic
64C23×C4C22⋊C411yescharacteristic
65C23×C4C22×C411yescharacteristic
66C23×C4C2×D411yescharacteristic
67C23×C4C2×D411yescharacteristic
68C2×M16—24no—
69C2×M16—42no—
70C2×M16—42no—
71C2×M16—42no—
72C2×M16—42no—
73C2×M16—42no—
74C22×C8—42no—
75C22×C8—42no—
76C4⋊Q8—48no—
77C4⋊Q8—44no—
78C4⋊Q8—44no—
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—44no—
93C2×C4⋊C4—44no—
94C2×C4⋊C4—44no—
95C2×C4⋊C4—82no—
96C2×C4⋊C4C2×Q812yes—
97C2×C4⋊C4#4112yes—
98C2×C42—21no—
99C2×C42—21no—
100C2×C42—21no—
101C2×C42—21no—
102C2×C42—21no—
103C2×C42—22no—
104C2×C42—21no—
105C2×C42—41no—
106C2×C42—42no—
107C2×C42—42no—
108C22⋊C8—24no—
109C22⋊C8—42no—
110#97—82no—
111#95—41no—
112#95—41no—
113#95—42no—
114#95—44no—
115#95—42no—
116#95—44no—
117#95—82no—
118C24C2×C22⋊C411yescharacteristic
119C2×Q8—48no—
120C2×Q8—84no—
121C2×Q8—88no—
122C2×Q8—88no—
123C2×Q8—84no—
124C22×C4—21no—
125C22×C4—21no—
126C22×C4—21no—
127C22×C4—21no—
128C22×C4—21no—
129C22×C4—21no—
130C22×C4—22no—
131C22×C4—22no—
132C22×C4—21no—
133C22×C4—21no—
134C22×C4—21no—
135C22×C4—21no—
136C22×C4—21no—
137C22×C4—21no—
138C22×C4—24no—
139C22×C4—24no—
140C22×C4—21no—
141C22×C4—21no—
142C22×C4—21no—
143C22×C4—21no—
144C22×C4—21no—
145C22×C4—21no—
146C22×C4—41no—
147C22×C4—41no—
148C22×C4—41no—
149C22×C4—41no—
150C22×C4—41no—
151C22×C4—41no—
152C22×C4—41no—
153C22×C4—41no—
154C22×C4—41no—
155C22×C4—41no—
156C22×C4—42no—
157C22×C4—42no—
158C22×C4—42no—
159C22×C4—42no—
160C22×C4—42no—
161C22×C4—42no—
162C22×C4—42no—
163C22×C4—42no—
164C22×C4—42no—
165C22×C4—42no—
166C22×C4—42no—
167C22×C4—42no—
168C22×C4#9911yescharacteristic
169C22×C4#9911yescharacteristic
170C22×C4#10412yes—
171C22×C4C4×Q812yes—
172C22×C4C22⋊Q812yes—
173C22×C4#12411yescharacteristic
174C22×C4C4⋊Q811yescharacteristic
175C22×C4AGL(1,ℤ/8)12yes—
176C22×C4#13712yes—
177M16—44no—
178M16—44no—
179M16—82no—
180M16—82no—
181C2×C8—44no—
182C2×C8—44no—
183C2×C8—82no—
184C2×C8—82no—
185C2×C8—82no—
186C2×C8—82no—
187C4⋊C4—42no—
188C4⋊C4—42no—
189C4⋊C4—44no—
190C4⋊C4—48no—
191C4⋊C4—44no—
192C4⋊C4—44no—
193C4⋊C4—44no—
194C4⋊C4—44no—
195C4⋊C4—88no—
196C42—42no—
197C42—42no—
198C42—42no—
199C42—42no—
200C42—42no—
201C42—81no—
202C23—21no—
203C23—21no—
204C23—21no—
205C23—21no—
206C23#39111yescharacteristic
207C23C2×#9911yescharacteristic
208C23#41011yescharacteristic
209C23#41811yescharacteristic
210C23#41911yescharacteristic
211C23C2×#10411yescharacteristic
212C23#42111yescharacteristic
213Q8—816no—
214C2×C4—24no—
215C2×C4—24no—
216C2×C4—21no—
217C2×C4—21no—
218C2×C4—21no—
219C2×C4—21no—
220C2×C4—21no—
221C2×C4—21no—
222C2×C4—22no—
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—42no—
232C2×C4—42no—
233C2×C4—41no—
234C2×C4—41no—
235C2×C4—41no—
236C2×C4—41no—
237C2×C4—41no—
238C2×C4—41no—
239C2×C4—42no—
240C2×C4—42no—
241C2×C4—42no—
242C2×C4—44no—
243C2×C4—42no—
244C2×C4—44no—
245C2×C4—44no—
246C2×C4—44no—
247C2×C4—42no—
248C2×C4—44no—
249C2×C4—41no—
250C2×C4—44no—
251C2×C4—41no—
252C2×C4—41no—
253C2×C4—41no—
254C2×C4—41no—
255C2×C4—41no—
256C2×C4—81no—
257C2×C4—82no—
258C2×C4—82no—
259C2×C4—82no—
260C2×C4—82no—
261C2×C4—82no—
262C2×C4—82no—
263C2×C4—82no—
264C2×C4—82no—
265C2×C4—81no—
266C2×C4—81no—
267C2×C4AΣL(1,𝔽16)12yes—
268C2×C4#35612yes—
269C2×C4#48912yes—
270C2×C4#50112yes—
271C8—84no—
272C8—84no—
273C22—21no—
274C22—21no—
275C22—21no—
276C22—21no—
277C22—21no—
278C22—21no—
279C22—21no—
280C22—21no—
281C22—21no—
282C22—21no—
283C22—21no—
284C22—21no—
285C22#150611yescharacteristic
286C22#150711yescharacteristic
287C22#151111yescharacteristic
288C22#151411yescharacteristic
289C22#198611yescharacteristic
290C22#199311yescharacteristic
291C22#199611yescharacteristic
292C22C2×AΣL(1,𝔽16)11yescharacteristic
293C22#212311yescharacteristic
294C22C2×#35611yescharacteristic
295C22#213411yescharacteristic
296C4—22no—
297C4—22no—
298C4—22no—
299C4—22no—
300C4—42no—
301C4—42no—
302C4—42no—
303C4—84no—
304C4—81no—
305C4—84no—
306C4—81no—
307C2—21no—
308C2—21no—
309C2—21no—
310C2—21no—
311C2#988811yescharacteristic
312C2#1177911yescharacteristic
313C2#1178211yescharacteristic
314C2#1181511yescharacteristic
315C2#1181811yescharacteristic
316C2#1329611yescharacteristic
317C2#1330211yescharacteristic
subgroupG/H
center, socleC23C2×#99
commutatorC23×C4C22×C4
Frattini, cosocleC22×C4⋊C4C23
prime radicalC24C2×C22⋊C4
LCSG ▷ C23×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C22×C4⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
53. #324Q8semidirect

Maximal subgroups

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

subgroupindex
#151282
#108022
#81532
#72512