Finite Group Databasebeta

← #98543

Subgroups — #98543

Order 512.

938 conjugacy classes of subgroups; only the 138 normal subgroups are shown.

normal subgroups only

This group has 938 conjugacy classes of subgroups — too many to draw. Only its 138 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×#2546C211yesmaximal, characteristic
2#8560C212yesmaximal
3#8342C212yesmaximal
4#7643C212yesmaximal
5C22×C4⋊Q8C411yescharacteristic
6C22×C4⋊Q8C411yescharacteristic
7C22×C4⋊Q8C2211yescharacteristic
8C22×C4×Q8C411yescharacteristic
9#2546—24no—
10#2546—24no—
11C2×#398—22no—
12C2×#391C411yescharacteristic
13C2×#391C2211yescharacteristic
14C2×#389—22no—
15C2×C4×C4⋊C4C2211yescharacteristic
16#1797C2214yes—
17C2×#333—24no—
18C2×#324—22no—
19#1280—22no—
20#1272—24no—
21C23×Q8C2×C411yescharacteristic
22C23×Q8C2×C411yescharacteristic
23C23×Q8C2×C411yescharacteristic
24C2×C4⋊Q8—22no—
25C2×C4⋊Q8—22no—
26C2×C4⋊Q8—22no—
27C2×C4⋊Q8—22no—
28C2×C4⋊Q8—22no—
29C2×C4⋊Q8—42no—
30C2×C4⋊Q8—42no—
31C2×C4⋊Q8—44no—
32C2×C4⋊Q8—44no—
33C2×C4⋊Q8D414yes—
34C2×C4×Q8—22no—
35C2×C4×Q8—22no—
36C2×C4×Q8—42no—
37C2×C4×Q8—42no—
38C22×C4⋊C4—21no—
39C22×C4⋊C4—21no—
40C22×C4⋊C4—22no—
41C22×C4⋊C4—21no—
42C22×C4⋊C4—22no—
43C22×C4⋊C4C2×C411yescharacteristic
44C22×C4⋊C4D412yes—
45C22×C42—21no—
46C22×C42C2×C411yescharacteristic
47C22×C42C2×C411yescharacteristic
48C22×C42C2311yescharacteristic
49C2×C4⋊C8—24no—
50C2×C22⋊C8—24no—
51#398—216no—
52#391—22no—
53#391—22no—
54#391—22no—
55#391D414yes—
56#389—216no—
57C4×C4⋊C4—22no—
58C4×C4⋊C4—22no—
59C2×#95—22no—
60C2×#95—22no—
61C2×#95D412yes—
62#336—24no—
63#333—48no—
64#324—44no—
65C22×Q8—24no—
66C22×Q8—21no—
67C22×Q8—21no—
68C22×Q8—21no—
69C22×Q8—42no—
70C22×Q8—42no—
71C22×Q8—42no—
72C22×Q8—42no—
73C22×Q8—42no—
74C22×Q8—41no—
75C22×Q8—41no—
76C22×Q8—41no—
77C22×Q8—41no—
78C22×Q8—41no—
79C22×Q8—41no—
80C22×Q8C22⋊C412yes—
81C22×Q8C22⋊C412yes—
82C22×Q8C22⋊C412yes—
83C23×C4—21no—
84C23×C4—21no—
85C23×C4—21no—
86C23×C4—21no—
87C23×C4C22⋊C411yescharacteristic
88C23×C4C22⋊C411yescharacteristic
89C23×C4C22⋊C412yes—
90C23×C4C22×C411yescharacteristic
91C23×C4C2×D412yes—
92C22×C8—44no—
93C4⋊Q8—28no—
94C4⋊Q8—44no—
95C4⋊Q8—48no—
96C4⋊Q8—48no—
97C4⋊Q8—48no—
98C4⋊Q8—48no—
99C4⋊Q8—44no—
100C4⋊Q8—44no—
101C4×Q8—48no—
102C4×Q8—48no—
103C2×C4⋊C4—24no—
104C2×C4⋊C4—24no—
105C2×C4⋊C4—24no—
106C2×C4⋊C4—24no—
107C2×C4⋊C4—28no—
108C2×C4⋊C4—24no—
109C2×C4⋊C4—21no—
110C2×C4⋊C4—28no—
111C2×C4⋊C4—41no—
112C2×C4⋊C4—41no—
113C2×C4⋊C4—42no—
114C2×C4⋊C4—48no—
115C2×C4⋊C4—42no—
116C2×C4⋊C4—48no—
117C2×C4⋊C4—42no—
118C2×C4⋊C4—42no—
119C2×C4⋊C4—42no—
120C2×C4⋊C4—42no—
121C2×C4⋊C4C22⋊C412yes—
122C2×C4⋊C4SD1614yes—
123C2×C4⋊C4Q1614yes—
124C2×C42—22no—
125C2×C42—22no—
126C2×C42—21no—
127C2×C42—21no—
128C2×C42—24no—
129C2×C42—21no—
130C2×C42—21no—
131C2×C42—42no—
132C2×C42—42no—
133C2×C42—42no—
134C2×C42—42no—
135C2×C42C2×D412yes—
136C2×C42C2×D412yes—
137C4⋊C8—48no—
138C22⋊C8—44no—
139C22⋊C8—44no—
140#95—28no—
141#95—28no—
142#95—24no—
143C24C2×C22⋊C411yescharacteristic
144C2×Q8—24no—
145C2×Q8—22no—
146C2×Q8—24no—
147C2×Q8—24no—
148C2×Q8—42no—
149C2×Q8—48no—
150C2×Q8—48no—
151C2×Q8—42no—
152C2×Q8—42no—
153C2×Q8—42no—
154C2×Q8—42no—
155C2×Q8—42no—
156C2×Q8—42no—
157C2×Q8—42no—
158C2×Q8—42no—
159C2×Q8—42no—
160C2×Q8—42no—
161C2×Q8—42no—
162C2×Q8—42no—
163C2×Q8—42no—
164C2×Q8—84no—
165C2×Q8—84no—
166C2×Q8—84no—
167C2×Q8—84no—
168C2×Q8#9914yes—
169C22×C4—21no—
170C22×C4—21no—
171C22×C4—21no—
172C22×C4—21no—
173C22×C4—22no—
174C22×C4—22no—
175C22×C4—22no—
176C22×C4—24no—
177C22×C4—22no—
178C22×C4—21no—
179C22×C4—22no—
180C22×C4—21no—
181C22×C4—22no—
182C22×C4—22no—
183C22×C4—22no—
184C22×C4—22no—
185C22×C4—42no—
186C22×C4—42no—
187C22×C4—42no—
188C22×C4—41no—
189C22×C4—41no—
190C22×C4—42no—
191C22×C4—42no—
192C22×C4—42no—
193C22×C4—42no—
194C22×C4—42no—
195C22×C4—44no—
196C22×C4—42no—
197C22×C4—42no—
198C22×C4—41no—
199C22×C4—41no—
200C22×C4—41no—
201C22×C4—41no—
202C22×C4—41no—
203C22×C4—41no—
204C22×C4#10012yes—
205C22×C4#10012yes—
206C22×C4#10314yes—
207C22×C4#10314yes—
208C22×C4C2×C22⋊C411yescharacteristic
209C22×C4C2×C22⋊C411yescharacteristic
210C22×C4#12014yes—
211C22×C4C2×SD1612yes—
212C22×C4C2×Q1612yes—
213C22×C4#13714yes—
214C2×C8—84no—
215C2×C8—84no—
216C2×C8—84no—
217C4⋊C4—28no—
218C4⋊C4—28no—
219C4⋊C4—416no—
220C4⋊C4—42no—
221C4⋊C4—42no—
222C4⋊C4—48no—
223C4⋊C4—48no—
224C4⋊C4—416no—
225C4⋊C4—48no—
226C42—24no—
227C42—24no—
228C42—24no—
229C42—42no—
230C42—48no—
231C42—42no—
232C42—42no—
233C23—22no—
234C23—21no—
235C23—21no—
236C23#37911yescharacteristic
237C23C2×#10011yescharacteristic
238C23C2×#10011yescharacteristic
239C23C2×#10312yes—
240C23#41912yes—
241Q8—44no—
242Q8—44no—
243Q8—48no—
244Q8—84no—
245Q8—84no—
246Q8—84no—
247Q8—84no—
248C2×C4—24no—
249C2×C4—24no—
250C2×C4—24no—
251C2×C4—24no—
252C2×C4—22no—
253C2×C4—24no—
254C2×C4—21no—
255C2×C4—22no—
256C2×C4—21no—
257C2×C4—21no—
258C2×C4—21no—
259C2×C4—24no—
260C2×C4—24no—
261C2×C4—22no—
262C2×C4—24no—
263C2×C4—42no—
264C2×C4—42no—
265C2×C4—44no—
266C2×C4—42no—
267C2×C4—44no—
268C2×C4—44no—
269C2×C4—44no—
270C2×C4—42no—
271C2×C4—42no—
272C2×C4—42no—
273C2×C4—42no—
274C2×C4—42no—
275C2×C4—42no—
276C2×C4—44no—
277C2×C4—44no—
278C2×C4—42no—
279C2×C4—42no—
280C2×C4—42no—
281C2×C4—42no—
282C2×C4—42no—
283C2×C4—42no—
284C2×C4—42no—
285C2×C4—82no—
286C2×C4—82no—
287C2×C4—82no—
288C2×C4—82no—
289C2×C4—82no—
290C2×C4—82no—
291C2×C4#33314yescomplemented
292C2×C4C2×#9912yes—
293C2×C4#41312yes—
294C2×C4#44814yes—
295C2×C4#45114yes—
296C8—164no—
297C22—21no—
298C22—21no—
299C22—21no—
300C22—21no—
301C22—22no—
302C22—22no—
303C22—22no—
304C22—22no—
305C22C2×#33311yescharacteristic
306C22#154812yes—
307C22#155811yescharacteristic
308C22#178111yescharacteristic
309C22#178311yescharacteristic
310C22#178511yescharacteristic
311C22#178712yes—
312C22#179612yes—
313C4—22no—
314C4—22no—
315C4—24no—
316C4—24no—
317C4—24no—
318C4—42no—
319C4—44no—
320C4—44no—
321C4—82no—
322C4—82no—
323C4—82no—
324C2—21no—
325C2—22no—
326C2—21no—
327C2#834211yescharacteristic
328C2#837912yes—
329C2#839212yes—
330C2#852411yescharacteristic
331C2#854911yescharacteristic
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

Complements

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

NH ≅ G/Ntype
291. C2×C4#333semidirect

Maximal subgroups

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

subgroupindex
C2×#25462
#85602
#83422
#76432