Finite Group Databasebeta

← #10804106

Subgroups — #10804106

Order 768.

1470 conjugacy classes of subgroups; only the 251 normal subgroups are shown.

normal subgroups only

This group has 1470 conjugacy classes of subgroups — too many to draw. Only its 251 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×#5828C211yesmaximal, characteristic
2C2×#4726C212yesmaximal, complemented
3C3×#1780C211yesmaximal, characteristic, complemented
4#72136C211yesmaximal, characteristic, complemented
5#69958C218yesmaximal, complemented
6#69911C212yesmaximal, complemented
7#21743—31nomaximal, Sylow
8C22×C3×#41C2211yescharacteristic
9C23×C4×D3C2211yescharacteristic
10#5828—28no—
11C2×#946C412yes—
12C2×#938C412yes—
13C2×#937C412yes—
14C2×#935C412yes—
15C2×C4×#67C2214yes—
16C2×#926C2211yescharacteristic
17C22×ΓL(1,𝔽25)C2214yes—
18C2×C3×C22⋊C8C2212yes—
19C2×C3×C22⋊C8C2214yescomplemented
20C2×#830C2212yes—
21C2×#830C2214yescomplemented
22C2×(C2×D3)⋊C8C2214yes—
23#4726—28no—
24#4550C2218yescomplemented
25C2×#525—31no—
26C2×#407—32no—
27#1870—38no—
28#1818—32no—
29#1780—31no—
30#1780D311yescharacteristic, complemented
31C24×D3C2×C411yescharacteristic
32C2×C3×#41—24no—
33C2×C3×#41—44no—
34C22×C3×Q8C2×C411yescharacteristic
35C22×C3×D4C2×C412yes—
36C22×C3×D4C2×C411yescharacteristic
37C23×C3×C4C2312yes—
38C23×C3×C4C2311yescharacteristic
39C22×#67C2×C414yes—
40C23×Dic3C2×C411yescharacteristic
41C22×C4×D3—44no—
42C22×C4×D3C2×C412yes—
43C22×C4×D3D418yes—
44C22×C4×D3C2312yes—
45C22×C3×C8C2314yes—
46C2×#217C2×C412yes—
47C2×#217C2×C414yes—
48#946—28no—
49#938—28no—
50#937—28no—
51#935—28no—
52C4×#67—28no—
53C4×#67—28no—
54C2×(C2×D3)⋊C4C2×C414yes—
55#926—24no—
56C2×C4⋊Dic3C2×C412yes—
57C2×Dic3⋊C4C2×C414yes—
58C2×C4×Dic3C2312yes—
59C22×C3⋊C8C2314yes—
60C2×ΓL(1,𝔽25)—28no—
61C2×ΓL(1,𝔽25)—48no—
62C3×C22⋊C8—24no—
63C3×C22⋊C8—44no—
64#830—24no—
65#830—44no—
66(C2×D3)⋊C8—28no—
67C22×#41C2×D311yescharacteristic
68C24×C4—31no—
69C22×M16—34no—
70#525—68no—
71C2×#123—32no—
72C2×C22⋊Q8—32no—
73C2×#121—32no—
74C2×#120—32no—
75C2×C4×D4—34no—
76C2×#117—31no—
77#407—68no—
78C2×C22⋊C8—32no—
79C2×C22⋊C8—34no—
80C2×C22⋊C8—34no—
81C2×C22⋊C8C2×D312yes—
82C2×C22⋊C8C2×D314yescomplemented
83#336—38no—
84C24×C3C22×C412yes—
85C24×C3C22×C411yescharacteristic
86C23×D3—42no—
87C23×D3—44no—
88C23×D3C22⋊C414yes—
89C23×D3C22×C412yes—
90C3×#41—416no—
91C2×C3×Q8—22no—
92C2×C3×Q8—42no—
93C2×C3×D4—24no—
94C2×C3×D4—22no—
95C2×C3×D4—42no—
96C2×C3×D4—42no—
97C2×C3×D4—42no—
98C22×C3×C4—24no—
99C22×C3×C4—24no—
100C22×C3×C4—24no—
101C22×C3×C4—24no—
102C22×C3×C4C22×C412yes—
103C22×C3×C4C22×C411yescharacteristic
104C22×C3×C4C2×D414yes—
105C22×C3×C4C2411yescharacteristic
106C2×#67—28no—
107C2×#67—48no—
108C22×Dic3—42no—
109C22×Dic3C22⋊C414yes—
110C22×Dic3C22×C414yes—
111C22×Dic3C22×C412yes—
112C2×C4×D3—28no—
113C2×C4×D3—416no—
114C2×C4×D3—44no—
115C2×C4×D3C22⋊C418yes—
116C2×C4×D3C2×D414yes—
117C2×C4×D3C2×D414yes—
118C2×C3×C8—24no—
119C2×C3×C8—28no—
120#217—24no—
121#217—28no—
122(C2×D3)⋊C4—28no—
123C4⋊Dic3—24no—
124Dic3⋊C4—28no—
125C4×Dic3—24no—
126C2×C3⋊C8—24no—
127C2×C3⋊C8—28no—
128ΓL(1,𝔽25)—416no—
129C25—31no—
130C2×#41—24no—
131C2×#41—44no—
132C22×Q8C4×D311yescharacteristic
133C22×D4—34no—
134C22×D4C4×D312yes—
135C22×D4C4×D311yescharacteristic
136C23×C4—31no—
137C23×C4—32no—
138C23×C4—32no—
139C23×C4—38no—
140C23×C4—124no—
141C23×C4C22×D312yes—
142C23×C4C22×D311yescharacteristic
143C2×M16—68no—
144C2×M16—128no—
145C22×C8—34no—
146C22×C8C22×D314yes—
147#123—68no—
148C22⋊Q8—68no—
149#121—68no—
150#120—68no—
151C4×D4—68no—
152C4×D4—68no—
153#117—64no—
154C2×C4⋊C4—32no—
155C2×C4⋊C4—34no—
156C2×C22⋊C4—32no—
157C2×C22⋊C4—34no—
158C2×C22⋊C4—34no—
159C2×C42—32no—
160C22⋊C8—24no—
161C22⋊C8—44no—
162C22⋊C8—68no—
163C22⋊C8—64no—
164C22⋊C8—124no—
165C23×C3—22no—
166C23×C3—24no—
167C23×C3—22no—
168C23×C3—22no—
169C23×C3—22no—
170C23×C3—28no—
171C23×C3C2×C22⋊C412yes—
172C23×C3C23×C411yescharacteristic
173C22×D3—24no—
174C22×D3—24no—
175C22×D3—24no—
176C22×D3—44no—
177C22×D3—48no—
178C22×D3—416no—
179C22×D3C2×C22⋊C414yes—
180C3×Q8—44no—
181C3×D4—44no—
182C3×D4—48no—
183C2×C3×C4—22no—
184C2×C3×C4—28no—
185C2×C3×C4—22no—
186C2×C3×C4—22no—
187C2×C3×C4—22no—
188C2×C3×C4—48no—
189C2×C3×C4C2×C22⋊C412yes—
190C2×C3×C4C22×D412yes—
191C2×C3×C4H(5,𝔽2)14yes—
192#67—416no—
193C2×Dic3—24no—
194C2×Dic3—24no—
195C2×Dic3—28no—
196C2×Dic3—44no—
197C2×Dic3C2×C22⋊C414yes—
198C4×D3—28no—
199C4×D3—28no—
200C4×D3—48no—
201C3×C8—44no—
202C3⋊C8—44no—
203C24—32no—
204C24—34no—
205C24—122no—
206C24—124no—
207C24C2×C4×D312yes—
208C24C2×C4×D311yescharacteristic
209#41—416no—
210C2×Q8—22no—
211C2×Q8—42no—
212C2×D4—24no—
213C2×D4—22no—
214C2×D4—42no—
215C2×D4—42no—
216C2×D4—42no—
217C2×D4—68no—
218C2×D4—128no—
219C22×C4—24no—
220C22×C4—24no—
221C22×C4—24no—
222C22×C4—24no—
223C22×C4—32no—
224C22×C4—34no—
225C22×C4—34no—
226C22×C4—34no—
227C22×C4—38no—
228C22×C4—34no—
229C22×C4—68no—
230C22×C4—1216no—
231C22×C4—124no—
232C22×C4—122no—
233C22×C4C2×C4×D312yes—
234C22×C4C2×C4×D311yescharacteristic
235C22×C4D3×D414yes—
236C22×C4C23×D311yescharacteristic
237M16—1216no—
238C2×C8—24no—
239C2×C8—28no—
240C2×C8—64no—
241C2×C8—68no—
242C4⋊C4—68no—
243C4⋊C4—64no—
244C22⋊C4—68no—
245C22⋊C4—68no—
246C22⋊C4—64no—
247C42—64no—
248C22×C3—22no—
249C22×C3—24no—
250C22×C3—44no—
251C22×C3—42no—
252C22×C3—44no—
253C22×C3—44no—
254C22×C3—44no—
255C22×C3C22×C22⋊C411yescharacteristic
256C22×C3#51812yes—
257C22×C3#56814yes—
258C2×D3—24no—
259C2×D3—24no—
260C2×D3—24no—
261C2×D3—416no—
262C2×D3—48no—
263C3×C4—22no—
264C3×C4—42no—
265C3×C4#53414yes—
266Dic3—24no—
267Dic3—44no—
268C23—22no—
269C23—24no—
270C23—22no—
271C23—22no—
272C23—22no—
273C23—28no—
274C23—34no—
275C23—64no—
276C23—64no—
277C23—64no—
278C23—124no—
279C23—128no—
280C23—1216no—
281C23D3×C22⋊C412yes—
282C23C22×C4×D311yescharacteristic
283Q8—44no—
284D4—44no—
285D4—48no—
286D4—1216no—
287C2×C4—22no—
288C2×C4—28no—
289C2×C4—22no—
290C2×C4—22no—
291C2×C4—22no—
292C2×C4—34no—
293C2×C4—48no—
294C2×C4—64no—
295C2×C4—68no—
296C2×C4—68no—
297C2×C4—64no—
298C2×C4—68no—
299C2×C4—128no—
300C2×C4—124no—
301C2×C4D3×C22⋊C412yes—
302C2×C4C2×D3×D412yes—
303C2×C4#100414yes—
304C8—44no—
305C8—124no—
306C2×C3—42no—
307C2×C3—82no—
308C2×C3#230811yescharacteristic
309C2×C3#287912yes—
310C2×C3#297814yes—
311D3—44no—
312C22—24no—
313C22—22no—
314C22—44no—
315C22—44no—
316C22—44no—
317C22—44no—
318C22—42no—
319C22—64no—
320C22—64no—
321C22—64no—
322C22—1216no—
323C22—128no—
324C22C2×D3×C22⋊C411yescharacteristic
325C22#548612yes—
326C22#575014yes—
327C4—22no—
328C4—42no—
329C4—64no—
330C4—124no—
331C4#558914yes—
332C3#2174311yescharacteristic, complemented, Sylow
333C2—42no—
334C2—82no—
335C2—124no—
336C2#7384911yescharacteristic
337C2#7866212yes—
338C2#7866614yes—
subgroupG/H
center, FrattiniC22×C4C23×D3
commutator, socleC23×C3C23×C4
FittingC3×#1780C2
prime radicalC2×#5828C2
cosocleC22×C3×C4C24
hypercenter#1780D3
hypocenterC3#21743
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C22×C4 ◁ #1780

Complements

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

NH ≅ G/Ntype
2. C2×#4726C2semidirect
3. C3×#1780C2semidirect
4. #72136C2semidirect
5. #69958C2semidirect
6. #69911C2semidirect
19. C2×C3×C22⋊C8C22semidirect
21. C2×#830C22semidirect
24. #4550C22semidirect
30. #1780D3semidirect
82. C2×C22⋊C8C2×D3semidirect
332. C3#21743semidirect

Maximal subgroups

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

subgroupindex
C2×#58282
C2×#47262
C3×#17802
#721362
#699582
#699112
#217433