Finite Group Databasebeta

← #10804486

Subgroups — #10804486

Order 768.

1194 conjugacy classes of subgroups; only the 235 normal subgroups are shown.

normal subgroups only

This group has 1194 conjugacy classes of subgroups — too many to draw. Only its 235 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×#5825C211yesmaximal, characteristic
2#78620C212yesmaximal, complemented
3C3×#1785C211yesmaximal, characteristic
4#72137C211yesmaximal, characteristic
5#70098C214yesmaximal, complemented
6#70065C214yesmaximal, complemented
7#69990C212yesmaximal, complemented
8#21802—31nomaximal, Sylow
9C22×C3×#41C2211yescharacteristic
10C23×Dic6C2211yescharacteristic
11#5825—28no—
12C2×#944C2212yes—
13C2×#936C2212yes—
14C2×#935C414yes—
15C2×#926C2211yescharacteristic
16C2×#924C414yes—
17C2×C3×#101C2212yes—
18C2×C3×#100C2212yes—
19C3×#407C2212yes—
20C2×#836C2212yes—
21C2×#833C2212yes—
22#5207C2212yes—
23C2×#824C2214yes—
24C2×#824C2214yes—
25#5118C2214yes—
26Dic3×M16C2214yes—
27#4720—28no—
28C2×#536—31no—
29#2921—32no—
30#1967—34no—
31#1966—34no—
32#1785—31no—
33#1785D311yescharacteristic
34#1784—32no—
35C2×C3×#41—24no—
36C2×C3×#41—42no—
37C2×C3×#41—42no—
38C22×C3×Q8C2311yescharacteristic
39C22×C3×D4C2×C412yes—
40C22×C3×D4C2311yescharacteristic
41C23×C3×C4C2×C412yes—
42C23×C3×C4C2311yescharacteristic
43C23×Dic3C2×C412yes—
44C22×Dic6—24no—
45C22×Dic6—42no—
46C22×Dic6—42no—
47C22×Dic6C2×C412yes—
48C22×Dic6C2312yes—
49C2×C3×M16C2314yes—
50C2×#217C2×C418yes—
51C2×#217C2×C412yes—
52#944—24no—
53#944—24no—
54#936—24no—
55#936—24no—
56#935—216no—
57#926D418yes—
58C2×C4⋊Dic3C2×C412yes—
59#924—216no—
60C2×Dic3⋊C4C2×C418yes—
61C2×C4×Dic3C2312yes—
62C2×#208C2314yes—
63C3×#101—42no—
64C3×#100—42no—
65C3×C22⋊C8—24no—
66#836—42no—
67#833—42no—
68#830—24no—
69#824—44no—
70#824—44no—
71#815—24no—
72Dic3⋊C8—28no—
73C8×Dic3—24no—
74C22×#41C2×D311yescharacteristic
75C23×Q8—31no—
76#536—68no—
77C2×C4⋊Q8—32no—
78C2×#124—32no—
79C2×#123—34no—
80C2×C22⋊Q8—34no—
81C2×#117—31no—
82#433—68no—
83#423—34no—
84C2×#101—32no—
85C2×#101—34no—
86C2×#101—34no—
87C2×#101C2×D312yes—
88C2×#100—32no—
89C2×#100C2×D312yes—
90#407—32no—
91#407C2×D312yes—
92C4×M16—34no—
93C24×C3C22×C412yes—
94C24×C3C22×C411yescharacteristic
95C3×#41—416no—
96C2×C3×Q8—22no—
97C2×C3×Q8—22no—
98C2×C3×Q8—22no—
99C2×C3×D4—24no—
100C2×C3×D4—22no—
101C2×C3×D4—22no—
102C2×C3×D4—22no—
103C2×C3×D4—42no—
104C2×C3×D4—42no—
105C22×C3×C4—22no—
106C22×C3×C4—22no—
107C22×C3×C4—24no—
108C22×C3×C4—42no—
109C22×C3×C4—42no—
110C22×C3×C4C22×C412yes—
111C22×C3×C4C22×C411yescharacteristic
112C22×C3×C4C2×D414yes—
113C22×C3×C4C2411yescharacteristic
114C22×Dic3—24no—
115C22×Dic3—42no—
116C22×Dic3—42no—
117C22×Dic3C22×C414yes—
118C22×Dic3C22×C414yes—
119C2×Dic6—24no—
120C2×Dic6—22no—
121C2×Dic6—22no—
122C2×Dic6—24no—
123C2×Dic6—24no—
124C2×Dic6—42no—
125C2×Dic6—42no—
126C2×Dic6—416no—
127C3×M16—24no—
128C3×M16—24no—
129C2×C3×C8—24no—
130#217—216no—
131#217C22⋊C418yes—
132C4⋊Dic3C22⋊C418yes—
133Dic3⋊C4—216no—
134C4×Dic3C2×D414yes—
135C4×Dic3C2×D414yes—
136#208—24no—
137#208—24no—
138C2×C3⋊C8—24no—
139C2×#41—24no—
140C2×#41—42no—
141C2×#41—42no—
142C22×Q8—32no—
143C22×Q8—32no—
144C22×Q8—64no—
145C22×Q8—122no—
146C22×Q8—122no—
147C22×Q8C22×D311yescharacteristic
148C22×D4C4×D312yes—
149C22×D4C22×D311yescharacteristic
150C23×C4—32no—
151C23×C4C4×D312yes—
152C23×C4C22×D311yescharacteristic
153C2×M16—34no—
154C2×M16C22×D314yes—
155C4⋊Q8—64no—
156C4⋊Q8—64no—
157#124—64no—
158#124—64no—
159#123—616no—
160C22⋊Q8—616no—
161#117—38no—
162C2×C4⋊C4—32no—
163C2×C4⋊C4—38no—
164C2×C22⋊C4—32no—
165C2×C22⋊C4—38no—
166C2×C42—32no—
167C4⋊C8—68no—
168#101—42no—
169#101—124no—
170#101—124no—
171#101—122no—
172#100—42no—
173#100—122no—
174C22⋊C8—24no—
175C22⋊C8—64no—
176#97—64no—
177C4×C8—64no—
178C23×C3—21no—
179C23×C3—21no—
180C23×C3—22no—
181C23×C3—22no—
182C23×C3—22no—
183C23×C3—42no—
184C23×C3—44no—
185C23×C3C2×C22⋊C412yes—
186C23×C3C23×C411yescharacteristic
187C3×Q8—44no—
188C3×D4—48no—
189C3×D4—44no—
190C2×C3×C4—21no—
191C2×C3×C4—21no—
192C2×C3×C4—28no—
193C2×C3×C4—22no—
194C2×C3×C4—22no—
195C2×C3×C4—22no—
196C2×C3×C4—48no—
197C2×C3×C4C2×C22⋊C412yes—
198C2×C3×C4C22×D412yes—
199C2×C3×C4H(5,𝔽2)12yes—
200C2×C3×C4H(5,𝔽2)12yes—
201C2×Dic3—24no—
202C2×Dic3—24no—
203C2×Dic3—24no—
204C2×Dic3—44no—
205C2×Dic3—48no—
206C2×Dic3C2×C22⋊C418yes—
207Dic6—48no—
208Dic6—44no—
209Dic6—44no—
210C3×C8—44no—
211C3⋊C8—44no—
212C24C2×C4×D312yes—
213C24C2×C4×D311yescharacteristic
214#41—416no—
215C2×Q8—22no—
216C2×Q8—22no—
217C2×Q8—22no—
218C2×Q8—64no—
219C2×Q8—62no—
220C2×Q8—62no—
221C2×Q8—64no—
222C2×Q8—64no—
223C2×Q8—122no—
224C2×Q8—122no—
225C2×Q8—1216no—
226C2×D4—24no—
227C2×D4—22no—
228C2×D4—22no—
229C2×D4—22no—
230C2×D4—42no—
231C2×D4—42no—
232C22×C4—22no—
233C22×C4—22no—
234C22×C4—24no—
235C22×C4—34no—
236C22×C4—34no—
237C22×C4—42no—
238C22×C4—42no—
239C22×C4—64no—
240C22×C4—122no—
241C22×C4—122no—
242C22×C4C2×C4×D312yes—
243C22×C4C2×C4×D311yescharacteristic
244C22×C4D3×D414yes—
245C22×C4C23×D311yescharacteristic
246M16—24no—
247M16—24no—
248M16—64no—
249M16—64no—
250C2×C8—24no—
251C2×C8—64no—
252C4⋊C4—38no—
253C4⋊C4—616no—
254C22⋊C4—38no—
255C22⋊C4—616no—
256C42—34no—
257C42—34no—
258C22×C3—22no—
259C22×C3—22no—
260C22×C3—24no—
261C22×C3—42no—
262C22×C3—42no—
263C22×C3—42no—
264C22×C3—44no—
265C22×C3—82no—
266C22×C3—82no—
267C22×C3C22×C22⋊C411yescharacteristic
268C22×C3#51811yescharacteristic
269C22×C3#51811yescharacteristic
270C3×C4—22no—
271C3×C4—42no—
272C3×C4#53514yes—
273Dic3—28no—
274Dic3—44no—
275C23—21no—
276C23—21no—
277C23—22no—
278C23—22no—
279C23—22no—
280C23—42no—
281C23—44no—
282C23D3×C22⋊C412yes—
283C23C22×C4×D311yescharacteristic
284Q8—44no—
285Q8—128no—
286Q8—124no—
287Q8—124no—
288D4—48no—
289D4—44no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—28no—
293C2×C4—22no—
294C2×C4—22no—
295C2×C4—22no—
296C2×C4—38no—
297C2×C4—48no—
298C2×C4—64no—
299C2×C4—64no—
300C2×C4—64no—
301C2×C4—124no—
302C2×C4—128no—
303C2×C4D3×C22⋊C412yes—
304C2×C4C2×D3×D412yes—
305C2×C4#100412yes—
306C2×C4#100412yes—
307C8—44no—
308C8—124no—
309C2×C3—21no—
310C2×C3—21no—
311C2×C3—42no—
312C2×C3—82no—
313C2×C3#230911yescharacteristic
314C2×C3#288912yes—
315C22—22no—
316C22—22no—
317C22—24no—
318C22—42no—
319C22—42no—
320C22—42no—
321C22—44no—
322C22—82no—
323C22—82no—
324C22C2×D3×C22⋊C411yescharacteristic
325C22#548611yescharacteristic
326C22#548611yescharacteristic
327C4—22no—
328C4—42no—
329C4—68no—
330C4—124no—
331C4#559214yes—
332C3#2180211yescharacteristic, complemented, Sylow
333C2—21no—
334C2—21no—
335C2—42no—
336C2—82no—
337C2#7381911yescharacteristic
338C2#7871512yes—
subgroupG/H
centerC22C2×D3×C22⋊C4
commutatorC23×C3C23×C4
FrattiniC22×C4C23×D3
FittingC3×#1785C2
socleC22×C3C22×C22⋊C4
prime radicalC22×C3×#41C22
cosocleC22×C3×C4C24
hypercenter#1785D3
hypocenterC3#21802
LCSG ▷ C23×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1785

Complements

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

NH ≅ G/Ntype
2. #78620C2semidirect
5. #70098C2semidirect
6. #70065C2semidirect
7. #69990C2semidirect
332. C3#21802semidirect

Maximal subgroups

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

subgroupindex
C2×#58252
#786202
C3×#17852
#721372
#700982
#700652
#699902
#218023