Finite Group Databasebeta

← #10682486

Subgroups — #10682486

Order 768.

620 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 620 conjugacy classes of subgroups — too many to draw. Only its 67 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
1C3×#1781C211yesmaximal, characteristic
2#72130C211yesmaximal, characteristic
3#72129C211yesmaximal, characteristic
4#7316—31nomaximal, Sylow
5C23×C3×D4C2211yescharacteristic
6C2×C3×#99—21no—
7C2×C3×#99—21no—
8C3×#379C412yescomplemented
9#5247C412yescomplemented
10C2×#834—21no—
11C2×#834—21no—
12C2×#833—21no—
13C2×#833—21no—
14#5207C412yescomplemented
15#1783—31no—
16#1781—31no—
17#1781D311yescharacteristic
18C25×C3C2×C411yescharacteristic
19C25×C3C2×C411yescharacteristic
20C22×C3×D4—42no—
21C22×C3×D4—42no—
22C22×C3×D4—42no—
23C22×C3×D4D411yescharacteristic
24C22×C3×D4D411yescharacteristic
25C22×C3×D4D411yescharacteristic
26C22×C3×D4Q811yescharacteristic
27C23×C3×C4C2×C411yescharacteristic
28C2×C3×C22⋊C4—22no—
29C2×C3×C22⋊C4—42no—
30C2×#217—22no—
31C2×#217—42no—
32C2×#208—22no—
33C3×#99—42no—
34C3×#99—42no—
35#834—42no—
36#834—42no—
37#833—42no—
38#833—42no—
39#830—42no—
40C23×D4C2×D311yescharacteristic
41C2×#100—61no—
42C2×#100—61no—
43C2×#99—21no—
44C2×#99—21no—
45C2×#99—61no—
46C2×#99—61no—
47#407—32no—
48#379—32no—
49#379Dic312yescomplemented
50C24×C3—22no—
51C24×C3—22no—
52C24×C3—41no—
53C24×C3—41no—
54C24×C3—41no—
55C24×C3—41no—
56C24×C3—82no—
57C24×C3—82no—
58C24×C3C4211yescharacteristic
59C24×C3C22⋊C411yescharacteristic
60C24×C3C22⋊C411yescharacteristic
61C24×C3C4⋊C411yescharacteristic
62C24×C3C4⋊C411yescharacteristic
63C2×C3×D4—22no—
64C2×C3×D4—22no—
65C2×C3×D4—22no—
66C2×C3×D4—22no—
67C2×C3×D4—22no—
68C2×C3×D4—22no—
69C2×C3×D4—41no—
70C2×C3×D4—41no—
71C2×C3×D4—41no—
72C2×C3×D4—41no—
73C2×C3×D4—41no—
74C2×C3×D4—41no—
75C2×C3×D4—416no—
76C22×C3×C4—22no—
77C22×C3×C4—41no—
78C22×C3×C4—41no—
79C22×C3×C4—42no—
80C22×C3×C4C22⋊C411yescharacteristic
81C22×C3×C4C4⋊C411yescharacteristic
82C22×Dic3—42no—
83C3×C22⋊C4—42no—
84C3×C22⋊C4—42no—
85C3×C22⋊C4—84no—
86#217—42no—
87#217—42no—
88#217—84no—
89#208—42no—
90#208—42no—
91C2×C3⋊C8—42no—
92C25C4×D311yescharacteristic
93C25C2×Dic311yescharacteristic
94C22×D4—42no—
95C22×D4—42no—
96C22×D4—42no—
97C22×D4Dic611yescharacteristic
98C22×D4D1211yescharacteristic
99C22×D4#6711yescharacteristic
100C22×D4#6711yescharacteristic
101C23×C4C4×D311yescharacteristic
102C2×M16—62no—
103C2×C22⋊C4—22no—
104C2×C22⋊C4—42no—
105C2×C22⋊C4—62no—
106C2×C22⋊C4—122no—
107#100—122no—
108#100—122no—
109#99—42no—
110#99—42no—
111#99—122no—
112#99—122no—
113C22⋊C8—122no—
114C23×C3—21no—
115C23×C3—21no—
116C23×C3—21no—
117C23×C3—21no—
118C23×C3—21no—
119C23×C3—21no—
120C23×C3—21no—
121C23×C3—21no—
122C23×C3—21no—
123C23×C3—21no—
124C23×C3—21no—
125C23×C3—21no—
126C23×C3—21no—
127C23×C3—42no—
128C23×C3—44no—
129C23×C3—44no—
130C23×C3—44no—
131C23×C3—44no—
132C23×C3—82no—
133C23×C3—82no—
134C23×C3—82no—
135C23×C3—82no—
136C23×C3—81no—
137C23×C3—81no—
138C23×C3—81no—
139C23×C3—81no—
140C23×C3—82no—
141C23×C3—82no—
142C23×C3—82no—
143C23×C3—82no—
144C23×C3—82no—
145C23×C3—82no—
146C23×C3#9511yescharacteristic
147C23×C3#9912yes—
148C23×C3#9912yes—
149C3×D4—44no—
150C3×D4—44no—
151C3×D4—44no—
152C3×D4—44no—
153C2×C3×C4—21no—
154C2×C3×C4—21no—
155C2×C3×C4—21no—
156C2×C3×C4—21no—
157C2×C3×C4—21no—
158C2×C3×C4—44no—
159C2×C3×C4—82no—
160C2×C3×C4—84no—
161C2×C3×C4#9912yes—
162C2×Dic3—82no—
163C2×Dic3—84no—
164C3⋊C8—82no—
165C24—22no—
166C24—22no—
167C24—41no—
168C24—41no—
169C24—41no—
170C24—41no—
171C24—82no—
172C24—82no—
173C24C4×Dic311yescharacteristic
174C24Dic3⋊C411yescharacteristic
175C24C4⋊Dic311yescharacteristic
176C24(C2×D3)⋊C411yescharacteristic
177C24#21711yescharacteristic
178C2×D4—22no—
179C2×D4—22no—
180C2×D4—22no—
181C2×D4—22no—
182C2×D4—22no—
183C2×D4—22no—
184C2×D4—41no—
185C2×D4—41no—
186C2×D4—41no—
187C2×D4—41no—
188C2×D4—41no—
189C2×D4—41no—
190C2×D4—416no—
191C22×C4—22no—
192C22×C4—41no—
193C22×C4—41no—
194C22×C4—42no—
195C22×C4—122no—
196C22×C4Dic3⋊C411yescharacteristic
197C22×C4(C2×D3)⋊C411yescharacteristic
198M16—122no—
199M16—122no—
200C2×C8—122no—
201C22⋊C4—42no—
202C22⋊C4—42no—
203C22⋊C4—84no—
204C22⋊C4—122no—
205C22⋊C4—122no—
206C22⋊C4—244no—
207C22×C3—22no—
208C22×C3—22no—
209C22×C3—22no—
210C22×C3—22no—
211C22×C3—22no—
212C22×C3—41no—
213C22×C3—42no—
214C22×C3—42no—
215C22×C3—41no—
216C22×C3—41no—
217C22×C3—41no—
218C22×C3—42no—
219C22×C3—42no—
220C22×C3—41no—
221C22×C3—41no—
222C22×C3—41no—
223C22×C3—81no—
224C22×C3—81no—
225C22×C3—81no—
226C22×C3—81no—
227C22×C3—81no—
228C22×C3—81no—
229C22×C3—81no—
230C22×C3—81no—
231C22×C3—88no—
232C22×C3—88no—
233C22×C3#34211yescharacteristic
234C22×C3#34211yescharacteristic
235C22×C3#34211yescharacteristic
236C3×C4—22no—
237C3×C4—41no—
238C3×C4—162no—
239Dic3—162no—
240C23—21no—
241C23—21no—
242C23—21no—
243C23—21no—
244C23—21no—
245C23—21no—
246C23—21no—
247C23—21no—
248C23—21no—
249C23—21no—
250C23—21no—
251C23—21no—
252C23—21no—
253C23—42no—
254C23—44no—
255C23—44no—
256C23—44no—
257C23—44no—
258C23—82no—
259C23—82no—
260C23—82no—
261C23—82no—
262C23—81no—
263C23—81no—
264C23—81no—
265C23—81no—
266C23—82no—
267C23—82no—
268C23—82no—
269C23—82no—
270C23—82no—
271C23—82no—
272C23#80612yes—
273C23C3⋊#9511yescharacteristic
274C23#83412yes—
275D4—44no—
276D4—44no—
277D4—44no—
278D4—44no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—44no—
285C2×C4—82no—
286C2×C4—84no—
287C2×C4—242no—
288C2×C4—244no—
289C2×C4#80612yes—
290C8—242no—
291C2×C3—21no—
292C2×C3—21no—
293C2×C3—21no—
294C2×C3—42no—
295C2×C3—42no—
296C2×C3—42no—
297C2×C3—81no—
298C2×C3—81no—
299C2×C3#130411yescharacteristic
300C22—22no—
301C22—22no—
302C22—22no—
303C22—22no—
304C22—22no—
305C22—41no—
306C22—42no—
307C22—42no—
308C22—41no—
309C22—41no—
310C22—41no—
311C22—42no—
312C22—42no—
313C22—41no—
314C22—41no—
315C22—41no—
316C22—81no—
317C22—81no—
318C22—81no—
319C22—81no—
320C22—81no—
321C22—81no—
322C22—81no—
323C22—81no—
324C22—88no—
325C22—88no—
326C22#452611yescharacteristic
327C22#452711yescharacteristic
328C22#452711yescharacteristic
329C4—22no—
330C4—41no—
331C4—162no—
332C4—482no—
333C3#731611yescharacteristic, complemented, Sylow
334C2—21no—
335C2—21no—
336C2—21no—
337C2—42no—
338C2—42no—
339C2—42no—
340C2—81no—
341C2—81no—
342C2C3⋊#130411yescharacteristic
subgroupG/H
centerC2C3⋊#1304
commutatorC24×C3C42
FrattiniC23×D4C2×D3
FittingC3×#1781C2
socleC2×C3#1304
prime radical, cosocleC23×C3×D4C22
hypercenter#1781D3
hypocenterC3#7316
LCSG ▷ C24×C3 ▷ C23×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2 ◁ C23 ◁ C23×D4 ◁ #1781

Complements

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

NH ≅ G/Ntype
8. C3×#379C4semidirect
9. #5247C4semidirect
14. #5207C4semidirect
49. #379Dic3semidirect
333. C3#7316semidirect

Maximal subgroups

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

subgroupindex
C3×#17812
#721302
#721292
#73163