Finite Group Databasebeta

← #10804458

Subgroups — #10804458

Order 768.

1946 conjugacy classes of subgroups; only the 259 normal subgroups are shown.

normal subgroups only

This group has 1946 conjugacy classes of subgroups — too many to draw. Only its 259 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
1C22×#940C211yesmaximal, characteristic
2#78619C212yesmaximal, complemented
3C3×#1783C211yesmaximal, characteristic, complemented
4#72129C211yesmaximal, characteristic, complemented
5#70064C218yesmaximal, complemented
6#69988C212yesmaximal, complemented
7#21797—31nomaximal, Sylow
8C23×C3×D4C2211yescharacteristic
9C23×#67C414yes—
10C23×D12C2211yescharacteristic
11C2×#940—24no—
12C2×#940—24no—
13C2×#940C414yes—
14C2×#940C2214yes—
15C22×C4×Dic3C2211yescharacteristic
16C2×C3×#100C2214yescomplemented
17C3×#407C2212yes—
18C2×#833C2214yescomplemented
19#5207C2212yes—
20C2×#823C2218yescomplemented
21#5118C2214yescomplemented
22Dic3×M16C2214yescomplemented
23#4719—24no—
24#4718—24no—
25C22×#127—31no—
26#2920—32no—
27#1965—38no—
28#1783—32no—
29#1783—31no—
30#1783D311yescharacteristic, complemented
31C25×C3C2×C412yes—
32C24×D3C2×C412yes—
33C22×C3×D4—24no—
34C22×C3×D4—44no—
35C22×C3×D4C2×C412yes—
36C22×C3×D4C2312yes—
37C23×C3×C4C2311yescharacteristic
38C22×#67—24no—
39C22×#67—28no—
40C22×#67—24no—
41C22×#67—416no—
42C22×#67C2×C4116yes—
43C23×Dic3C2×C412yes—
44C22×D12—24no—
45C22×D12—44no—
46C22×D12C2×C412yes—
47C22×D12C2312yes—
48C2×C3×M16C2314yes—
49#940—24no—
50#940—28no—
51#940—24no—
52#940—24no—
53#940—28no—
54#940—24no—
55C2×C4×Dic3C2×C412yes—
56C2×C4×Dic3D414yes—
57C2×C4×Dic3D414yes—
58C2×C4×Dic3C2312yes—
59C2×#208C2314yes—
60C3×#100—44no—
61C3×C22⋊C8—24no—
62#833—44no—
63#830—24no—
64#823—48no—
65#815—24no—
66Dic3⋊C8—28no—
67C8×Dic3—24no—
68C23×D4—34no—
69C23×D4—31no—
70C23×D4C2×D311yescharacteristic
71C2×#127—34no—
72C2×#127—34no—
73C2×#127—64no—
74C2×#127—64no—
75C22×C42—31no—
76#432—64no—
77#431—64no—
78#423—34no—
79C2×#100—34no—
80C2×#100—38no—
81C2×#100C2×D314yescomplemented
82#407—32no—
83#407C2×D312yes—
84C4×M16—34no—
85C24×C3—24no—
86C24×C3—44no—
87C24×C3—48no—
88C24×C3C22×C411yescharacteristic
89C24×C3C22×C414yes—
90C23×D3—24no—
91C23×D3—44no—
92C23×D3—48no—
93C23×D3C22×C414yes—
94C2×C3×D4—24no—
95C2×C3×D4—24no—
96C2×C3×D4—24no—
97C2×C3×D4—24no—
98C2×C3×D4—44no—
99C2×C3×D4—416no—
100C22×C3×C4—24no—
101C22×C3×C4C22×C411yescharacteristic
102C22×C3×C4C2×D414yes—
103C22×C3×C4C2411yescharacteristic
104C2×#67—216no—
105C2×#67—216no—
106C2×#67—416no—
107C2×#67—432no—
108C2×#67—432no—
109C2×#67—416no—
110C22×Dic3—28no—
111C22×Dic3C22⋊C412yes—
112C22×Dic3C22⋊C414yes—
113C22×Dic3C22⋊C412yes—
114C22×Dic3C22×C414yes—
115C2×D12—24no—
116C2×D12—24no—
117C2×D12—28no—
118C2×D12—44no—
119C2×D12—416no—
120C3×M16—28no—
121C2×C3×C8—24no—
122C4×Dic3C22⋊C412yes—
123C4×Dic3C22⋊C414yes—
124C4×Dic3C22⋊C412yes—
125C4×Dic3C2×D412yes—
126C4×Dic3C2×D414yes—
127C4×Dic3C2×D412yes—
128#208—28no—
129C2×C3⋊C8—24no—
130C25—32no—
131C25C4×D312yes—
132C22×D4—24no—
133C22×D4—316no—
134C22×D4—32no—
135C22×D4—32no—
136C22×D4—44no—
137C22×D4—64no—
138C22×D4—68no—
139C22×D4—64no—
140C22×D4—64no—
141C22×D4—124no—
142C22×D4—1216no—
143C22×D4C4×D312yes—
144C22×D4C22×D312yes—
145C23×C4—32no—
146C23×C4C22×D311yescharacteristic
147C2×M16—34no—
148C2×M16C22×D314yes—
149#127—64no—
150#127—68no—
151#127—64no—
152#127—64no—
153#127—64no—
154#127—68no—
155C2×C42—32no—
156C2×C42—34no—
157C2×C42—34no—
158C2×C42—32no—
159C4⋊C8—68no—
160#100—44no—
161#100—128no—
162#100—124no—
163C22⋊C8—24no—
164C22⋊C8—64no—
165#97—64no—
166C4×C8—64no—
167C23×C3—22no—
168C23×C3—24no—
169C23×C3—24no—
170C23×C3—24no—
171C23×C3—24no—
172C23×C3—48no—
173C23×C3—48no—
174C23×C3—48no—
175C23×C3—48no—
176C23×C3—816no—
177C23×C3C2×C22⋊C412yes—
178C23×C3C23×C411yescharacteristic
179C22×D3—24no—
180C22×D3—28no—
181C22×D3—48no—
182C22×D3—48no—
183C22×D3—48no—
184C22×D3—48no—
185C22×D3—816no—
186C3×D4—48no—
187C3×D4—48no—
188C2×C3×C4—22no—
189C2×C3×C4—28no—
190C2×C3×C4C2×C22⋊C412yes—
191C2×C3×C4C22×D412yes—
192C2×C3×C4H(5,𝔽2)14yes—
193#67—432no—
194#67—432no—
195C2×Dic3—24no—
196C2×Dic3—28no—
197C2×Dic3—28no—
198C2×Dic3#10018yescomplemented
199C2×Dic3C2×C22⋊C414yes—
200C2×Dic3C2×C22⋊C414yes—
201D12—48no—
202D12—48no—
203C3×C8—44no—
204C3⋊C8—44no—
205C24—24no—
206C24—34no—
207C24—44no—
208C24—48no—
209C24—64no—
210C24—124no—
211C24—128no—
212C24C2×C4×D311yescharacteristic
213C24C2×C4×D314yes—
214C2×D4—24no—
215C2×D4—24no—
216C2×D4—24no—
217C2×D4—24no—
218C2×D4—44no—
219C2×D4—416no—
220C2×D4—616no—
221C2×D4—616no—
222C2×D4—64no—
223C2×D4—64no—
224C2×D4—68no—
225C2×D4—124no—
226C2×D4—1216no—
227C2×D4—1216no—
228C2×D4—1232no—
229C2×D4—1216no—
230C2×D4—1232no—
231C22×C4—24no—
232C22×C4—32no—
233C22×C4—34no—
234C22×C4—32no—
235C22×C4—34no—
236C22×C4—68no—
237C22×C4C2×C4×D311yescharacteristic
238C22×C4D3×D414yes—
239C22×C4C23×D311yescharacteristic
240M16—28no—
241M16—68no—
242C2×C8—24no—
243C2×C8—64no—
244C42—32no—
245C42—34no—
246C42—32no—
247C42—32no—
248C42—32no—
249C42—34no—
250C22×C3—28no—
251C22×C3—22no—
252C22×C3—22no—
253C22×C3—22no—
254C22×C3—44no—
255C22×C3—48no—
256C22×C3—84no—
257C22×C3—84no—
258C22×C3—816no—
259C22×C3C2×#10014yes—
260C22×C3C22×C22⋊C411yescharacteristic
261C22×C3#51812yes—
262C2×D3—44no—
263C2×D3—48no—
264C2×D3—816no—
265C2×D3—88no—
266C3×C4—22no—
267C3×C4#53514yes—
268Dic3—24no—
269Dic3C2×#10018yes—
270C23—22no—
271C23—24no—
272C23—24no—
273C23—24no—
274C23—24no—
275C23—48no—
276C23—48no—
277C23—48no—
278C23—48no—
279C23—64no—
280C23—68no—
281C23—816no—
282C23—128no—
283C23—128no—
284C23—128no—
285C23—128no—
286C23—2416no—
287C23D3×C22⋊C412yes—
288C23C22×C4×D311yescharacteristic
289D4—48no—
290D4—48no—
291D4—128no—
292D4—1232no—
293D4—1232no—
294D4—128no—
295C2×C4—22no—
296C2×C4—28no—
297C2×C4—34no—
298C2×C4—38no—
299C2×C4—34no—
300C2×C4—68no—
301C2×C4—68no—
302C2×C4—64no—
303C2×C4D3×C22⋊C412yes—
304C2×C4C2×D3×D412yes—
305C2×C4#100414yes—
306C8—44no—
307C8—124no—
308C2×C3—22no—
309C2×C3—24no—
310C2×C3—84no—
311C2×C3#230911yescharacteristic
312C2×C3C22×#10012yes—
313D3—84no—
314C22—28no—
315C22—22no—
316C22—22no—
317C22—22no—
318C22—44no—
319C22—48no—
320C22—84no—
321C22—84no—
322C22—816no—
323C22—124no—
324C22—128no—
325C22—248no—
326C22—2416no—
327C22D3×#10014yes—
328C22C2×D3×C22⋊C411yescharacteristic
329C22#548612yes—
330C4—22no—
331C4—38no—
332C4—64no—
333C4#559214yes—
334C3#2179711yescharacteristic, complemented, Sylow
335C2—22no—
336C2—24no—
337C2—84no—
338C2—244no—
339C2#7381911yescharacteristic
340C2C2×D3×#10012yes—
subgroupG/H
centerC22C2×D3×C22⋊C4
commutatorC23×C3C23×C4
FrattiniC22×C4C23×D3
FittingC3×#1783C2
socleC22×C3C22×C22⋊C4
prime radicalC22×#940C2
cosocleC22×C3×C4C24
hypercenter#1783D3
hypocenterC3#21797
LCSG ▷ C23×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1783

Complements

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

NH ≅ G/Ntype
2. #78619C2semidirect
3. C3×#1783C2semidirect
4. #72129C2semidirect
5. #70064C2semidirect
6. #69988C2semidirect
16. C2×C3×#100C22semidirect
18. C2×#833C22semidirect
20. C2×#823C22semidirect
21. #5118C22semidirect
22. Dic3×M16C22semidirect
30. #1783D3semidirect
81. C2×#100C2×D3semidirect
198. C2×Dic3#100semidirect
334. C3#21797semidirect

Maximal subgroups

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

subgroupindex
C22×#9402
#786192
C3×#17832
#721292
#700642
#699882
#217973