Finite Group Databasebeta

← #141982

Subgroups — #141982

Order 512.

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

normal subgroups only

This group has 612 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
2#8560C211yesmaximal, characteristic
3#8153C211yesmaximal, characteristic
4#7189C214yesmaximal
5C22×C4⋊Q8C411yescharacteristic
6C2×#391—21no—
7C2×#391—21no—
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⋊#95C412yes—
17C4⋊#95C2212yes—
18#1280—21no—
19#1280—21no—
20C23×Q8—21no—
21C2×C4⋊Q8—24no—
22C2×C4⋊Q8—41no—
23C2×C4⋊Q8—41no—
24C2×C4⋊Q8—42no—
25C22×C4⋊C4—21no—
26C22×C4⋊C4—21no—
27C22×C4⋊C4C2×C411yescharacteristic
28C22×C4⋊C4C2×C412yes—
29C22×C4⋊C4C2311yescharacteristic
30C22×C42C2×C411yescharacteristic
31C22×C42C2×C412yes—
32C22×C42D411yescharacteristic
33C22×C42D411yescharacteristic
34C2×C4⋊C8—41no—
35C2×C22⋊C8—22no—
36C2×C22⋊C8—22no—
37C2×#97—41no—
38#391—44no—
39#391—44no—
40C4×C4⋊C4—41no—
41C4×C4⋊C4—41no—
42C4×C4⋊C4—42no—
43C2×#95—22no—
44C2×#95—22no—
45C2×#95—21no—
46C2×#95—22no—
47C2×#95—22no—
48C2×#95D411yescharacteristic
49C2×#95D411yescharacteristic
50#336—41no—
51#336—41no—
52#324—22no—
53#324—24no—
54#324Q814yescomplemented
55C22×Q8—24no—
56C22×Q8—42no—
57C22×Q8—42no—
58C22×Q8—42no—
59C22×Q8—42no—
60C22×Q8—42no—
61C22×Q8—41no—
62C22×Q8—41no—
63C23×C4—22no—
64C23×C4—22no—
65C23×C4C22⋊C411yescharacteristic
66C23×C4C22⋊C411yescharacteristic
67C23×C4C22⋊C411yescharacteristic
68C23×C4C22⋊C411yescharacteristic
69C23×C4C22×C411yescharacteristic
70C23×C4C2×D411yescharacteristic
71C23×C4C2×D411yescharacteristic
72C22×C8—42no—
73C22×C8—42no—
74C4⋊Q8—48no—
75C4⋊Q8—48no—
76C2×C4⋊C4—24no—
77C2×C4⋊C4—24no—
78C2×C4⋊C4—22no—
79C2×C4⋊C4—22no—
80C2×C4⋊C4—22no—
81C2×C4⋊C4—41no—
82C2×C4⋊C4—41no—
83C2×C4⋊C4—41no—
84C2×C4⋊C4—42no—
85C2×C4⋊C4—42no—
86C2×C4⋊C4—42no—
87C2×C4⋊C4—42no—
88C2×C4⋊C4—42no—
89C2×C4⋊C4—42no—
90C2×C4⋊C4M1614yes—
91C2×C4⋊C4C2×Q812yes—
92C2×C4⋊C4#4112yes—
93C2×C42—21no—
94C2×C42—21no—
95C2×C42—22no—
96C2×C42—22no—
97C2×C42—21no—
98C2×C42—21no—
99C2×C42—21no—
100C2×C42—21no—
101C2×C42—24no—
102C2×C42—42no—
103C2×C42—42no—
104C2×C42—42no—
105C2×C42—42no—
106C4⋊C8—82no—
107C22⋊C8—24no—
108C22⋊C8—42no—
109C22⋊C8—42no—
110C22⋊C8—42no—
111#97—82no—
112#95—41no—
113#95—41no—
114#95—44no—
115#95—44no—
116#95—44no—
117#95—44no—
118#95—42no—
119C24C2×C22⋊C411yescharacteristic
120C2×Q8—44no—
121C2×Q8—44no—
122C2×Q8—48no—
123C2×Q8—48no—
124C2×Q8—84no—
125C2×Q8—84no—
126C2×Q8—82no—
127C2×Q8—82no—
128C2×Q8—82no—
129C2×Q8—82no—
130C22×C4—21no—
131C22×C4—21no—
132C22×C4—21no—
133C22×C4—21no—
134C22×C4—21no—
135C22×C4—21no—
136C22×C4—22no—
137C22×C4—22no—
138C22×C4—21no—
139C22×C4—21no—
140C22×C4—21no—
141C22×C4—21no—
142C22×C4—21no—
143C22×C4—21no—
144C22×C4—21no—
145C22×C4—21no—
146C22×C4—24no—
147C22×C4—24no—
148C22×C4—21no—
149C22×C4—21no—
150C22×C4—21no—
151C22×C4—41no—
152C22×C4—41no—
153C22×C4—41no—
154C22×C4—41no—
155C22×C4—41no—
156C22×C4—41no—
157C22×C4—41no—
158C22×C4—41no—
159C22×C4—41no—
160C22×C4—41no—
161C22×C4—42no—
162C22×C4—42no—
163C22×C4—42no—
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#9911yescharacteristic
174C22×C4#9911yescharacteristic
175C22×C4#10012yes—
176C22×C4#10112yes—
177C22×C4C4×Q812yes—
178C22×C4C22⋊Q812yes—
179C22×C4#12411yescharacteristic
180C22×C4C4⋊Q811yescharacteristic
181C22×C4C2×M1612yes—
182C22×C4#13112yes—
183C2×C8—44no—
184C2×C8—82no—
185C2×C8—82no—
186C2×C8—82no—
187C2×C8—82no—
188C2×C8—82no—
189C4⋊C4—42no—
190C4⋊C4—42no—
191C4⋊C4—44no—
192C4⋊C4—44no—
193C4⋊C4—48no—
194C4⋊C4—44no—
195C4⋊C4—48no—
196C42—24no—
197C42—24no—
198C42—42no—
199C42—42no—
200C42—81no—
201C42—81no—
202C42—84no—
203C23—21no—
204C23—21no—
205C23—21no—
206C23—21no—
207C23#39111yescharacteristic
208C23#40711yescharacteristic
209C23#40811yescharacteristic
210C23C2×#9911yescharacteristic
211C23#41011yescharacteristic
212C23C2×#10011yescharacteristic
213C23C2×#10111yescharacteristic
214Q8—88no—
215Q8—88no—
216C2×C4—24no—
217C2×C4—24no—
218C2×C4—21no—
219C2×C4—21no—
220C2×C4—22no—
221C2×C4—21no—
222C2×C4—21no—
223C2×C4—22no—
224C2×C4—22no—
225C2×C4—22no—
226C2×C4—22no—
227C2×C4—22no—
228C2×C4—22no—
229C2×C4—22no—
230C2×C4—22no—
231C2×C4—21no—
232C2×C4—21no—
233C2×C4—42no—
234C2×C4—41no—
235C2×C4—41no—
236C2×C4—41no—
237C2×C4—41no—
238C2×C4—41no—
239C2×C4—41no—
240C2×C4—41no—
241C2×C4—41no—
242C2×C4—42no—
243C2×C4—42no—
244C2×C4—42no—
245C2×C4—41no—
246C2×C4—41no—
247C2×C4—44no—
248C2×C4—44no—
249C2×C4—44no—
250C2×C4—42no—
251C2×C4—44no—
252C2×C4—44no—
253C2×C4—41no—
254C2×C4—41no—
255C2×C4—44no—
256C2×C4—41no—
257C2×C4—81no—
258C2×C4—81no—
259C2×C4—81no—
260C2×C4—82no—
261C2×C4—82no—
262C2×C4—82no—
263C2×C4—82no—
264C2×C4—82no—
265C2×C4—82no—
266C2×C4—82no—
267C2×C4—82no—
268C2×C4—81no—
269C2×C4#35312yes—
270C2×C4#35512yes—
271C2×C4#44614yes—
272C8—84no—
273C8—162no—
274C22—21no—
275C22—21no—
276C22—21no—
277C22—21no—
278C22—21no—
279C22—21no—
280C22—21no—
281C22—21no—
282C22—21no—
283C22—21no—
284C22—21no—
285C22—21no—
286C22#146211yescharacteristic
287C22#146711yescharacteristic
288C22#146911yescharacteristic
289C22#147211yescharacteristic
290C22#198511yescharacteristic
291C22#199611yescharacteristic
292C22#199711yescharacteristic
293C22C2×#35311yescharacteristic
294C22#212911yescharacteristic
295C22C2×#35511yescharacteristic
296C22#213511yescharacteristic
297C4—24no—
298C4—22no—
299C4—22no—
300C4—42no—
301C4—42no—
302C4—41no—
303C4—41no—
304C4—42no—
305C4—84no—
306C4—81no—
307C4—84no—
308C2—21no—
309C2—21no—
310C2—21no—
311C2—21no—
312C2#988211yescharacteristic
313C2#1150511yescharacteristic
314C2#1151011yescharacteristic
315C2#1157011yescharacteristic
316C2#1157311yescharacteristic
317C2#1329811yescharacteristic
318C2#1330111yescharacteristic
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

Complements

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

NH ≅ G/Ntype
54. #324Q8semidirect

Maximal subgroups

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

subgroupindex
#141552
#85602
#81532
#71892