Finite Group Databasebeta

← #10706145

Subgroups — #10706145

Order 768.

336 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 336 conjugacy classes of subgroups — too many to draw. Only its 81 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
1#80740C211yesmaximal, characteristic
2#69093C211yesmaximal, characteristic, complemented
3#68847C211yesmaximal, characteristic, complemented
4C3×#1348C211yesmaximal, characteristic, complemented
5#67776C211yesmaximal, characteristic, complemented
6#67526C211yesmaximal, characteristic
7#67485C211yesmaximal, characteristic
8#11860—31nomaximal, Sylow
9#5624—21no—
10C3×#467C2211yescharacteristic
11#5191C411yescharacteristic, complemented
12Dic3×Q16C411yescharacteristic, complemented
13#5115C2211yescharacteristic
14C2×C8×Dic3C2211yescharacteristic
15#5048—21no—
16#4975—21no—
17#4878C411yescharacteristic, complemented
18#4872C411yescharacteristic, complemented
19#4800—21no—
20#4788—21no—
21C3×C22⋊C16C2211yescharacteristic
22C3×#340C2211yescharacteristic
23#4551C2211yescharacteristic
24#4549C2211yescharacteristic
25#4529—21no—
26#4502—21no—
27Dic3⋊C16—21no—
28#3168—31no—
29#2163—31no—
30#1834—31no—
31#1348—31no—
32#1348D311yescharacteristic, complemented
33#1347—31no—
34#1347—31no—
35C2×C3×Q16C2×C411yescharacteristic
36C22×C3×C8C2311yescharacteristic
37C3×C22⋊Q8—21no—
38C2×C3×C4⋊C4—21no—
39#947—21no—
40Q8×Dic3—21no—
41C2×#215—21no—
42#930—21no—
43C2×C4⋊Dic3—21no—
44C2×Dic3⋊C4—41no—
45C2×Dic3⋊C4—41no—
46C2×C4×Dic3D411yescharacteristic
47C22×C3⋊C8D411yescharacteristic
48C2×D24C2×C411yescharacteristic
49#898—41no—
50#893—21no—
51#892—41no—
52#890—21no—
53#886—41no—
54#67⋊C4—41no—
55C2×C3×C16—21no—
56C3×#107C2×C411yescharacteristic
57C3×#103—21no—
58#835—21no—
59#821—21no—
60#818C2×C411yescharacteristic
61C8×Dic3C2×C411yescharacteristic
62C8×Dic3C2×C411yescharacteristic
63C8×Dic3D411yescharacteristic
64C8×Dic3D411yescharacteristic
65C2×C3⋊C16—21no—
66#809—21no—
67#808—21no—
68#529—61no—
69#499—31no—
70#495—31no—
71#488—61no—
72#467C2×D311yescharacteristic
73#466—31no—
74#465—61no—
75C4×Q16—31no—
76C4×SD16—61no—
77C4×D8—31no—
78C2×C4×C8—31no—
79#384—61no—
80C4⋊C16—61no—
81C22⋊C16—31no—
82C22⋊C16C2×D311yescharacteristic
83#346—61no—
84#340—31no—
85#340—61no—
86#340C2×D311yescharacteristic
87C2×C3×Q8—21no—
88C22×C3×C4—41no—
89C22×C3×C4C2×D411yescharacteristic
90C2×#67—41no—
91C22×Dic3—41no—
92C22×Dic3D811yescharacteristic
93C22×Dic3D811yescharacteristic
94C2×D12—21no—
95C2×C4×D3—41no—
96C3×Q16—41no—
97C2×C3×C8—41no—
98C2×C3×C8C22×C411yescharacteristic
99C2×C3×C8C2×D411yescharacteristic
100C3×C4⋊C4—21no—
101C3×C4⋊C4—41no—
102C3×C4⋊C4—41no—
103C3×C4⋊C4—41no—
104C3×C22⋊C4—41no—
105#215—42no—
106(C2×D3)⋊C4—41no—
107(C2×D3)⋊C4—41no—
108(C2×D3)⋊C4—41no—
109(C2×D3)⋊C4—41no—
110C4⋊Dic3—21no—
111C4⋊Dic3—41no—
112C4⋊Dic3—41no—
113C4⋊Dic3—41no—
114C4⋊Dic3—41no—
115Dic3⋊C4—41no—
116Dic3⋊C4—82no—
117Dic3⋊C4—82no—
118C4×Dic3—41no—
119C4×Dic3C22⋊C411yescharacteristic
120C4×Dic3C22⋊C411yescharacteristic
121C4×Dic3SD1611yescharacteristic
122C4×Dic3SD1611yescharacteristic
123C2×C3⋊C8—41no—
124C2×C3⋊C8C22⋊C411yescharacteristic
125C2×C3⋊C8C22⋊C411yescharacteristic
126D24—41no—
127C3×C16—41no—
128C3⋊C16—41no—
129C2×Q16C4×D311yescharacteristic
130C2×SD16—61no—
131C2×D8—31no—
132C22×C8—31no—
133C22×C8C22×D311yescharacteristic
134#126—121no—
135#125—61no—
136#124—61no—
137#123—121no—
138C22⋊Q8—21no—
139#121—61no—
140C4×Q8—61no—
141C4×D4—61no—
142C4×D4—121no—
143#117—121no—
144C2×C4⋊C4—21no—
145C2×C4⋊C4—61no—
146C2×C4⋊C4—121no—
147C2×C4⋊C4—121no—
148C2×C42—31no—
149C2×C16—21no—
150C2×C16—61no—
151#107—31no—
152#107C4×D311yescharacteristic
153#106—61no—
154#103—21no—
155#103—61no—
156#102—61no—
157#102—61no—
158C4×C8—31no—
159C4×C8—31no—
160C4×C8—31no—
161C4×C8—31no—
162C23×C3C2×D811yescharacteristic
163C22×D3—41no—
164C3×Q8—41no—
165C3×Q8—41no—
166C2×C3×C4—21no—
167C2×C3×C4—21no—
168C2×C3×C4—41no—
169C2×C3×C4—41no—
170C2×C3×C4—81no—
171C2×C3×C4—81no—
172C2×C3×C4—81no—
173C2×C3×C4C2×C22⋊C411yescharacteristic
174C2×C3×C4C2×SD1611yescharacteristic
175#67—81no—
176#67—81no—
177C2×Dic3—21no—
178C2×Dic3—21no—
179C2×Dic3—22no—
180C2×Dic3—41no—
181C2×Dic3—81no—
182C2×Dic3—81no—
183C2×Dic3—81no—
184C2×Dic3#10211yescharacteristic
185C2×Dic3#10211yescharacteristic
186C2×Dic3#10211yescharacteristic
187C2×Dic3#10211yescharacteristic
188D12—41no—
189D12—41no—
190C4×D3—82no—
191C3×C8—21no—
192C3×C8—21no—
193C3⋊C8—21no—
194C3⋊C8—21no—
195C2×Q8—21no—
196C2×D4—61no—
197C2×D4—121no—
198C22×C4—31no—
199C22×C4—31no—
200C22×C4—41no—
201C22×C4—121no—
202C22×C4—121no—
203C22×C4D3×D411yescharacteristic
204Q16—41no—
205SD16—122no—
206D8—121no—
207C2×C8—31no—
208C2×C8—31no—
209C2×C8—41no—
210C2×C8—121no—
211C2×C8C2×C4×D311yescharacteristic
212C2×C8D3×D411yescharacteristic
213C4⋊C4—21no—
214C4⋊C4—41no—
215C4⋊C4—41no—
216C4⋊C4—41no—
217C4⋊C4—61no—
218C4⋊C4—121no—
219C4⋊C4—121no—
220C4⋊C4—121no—
221C4⋊C4—121no—
222C4⋊C4—121no—
223C4⋊C4—242no—
224C4⋊C4—242no—
225C22⋊C4—41no—
226C22⋊C4—121no—
227C22⋊C4—121no—
228C22⋊C4—121no—
229C22⋊C4—121no—
230C42—31no—
231C42—31no—
232C42—31no—
233C42—31no—
234C42—121no—
235C16—41no—
236C16—121no—
237C22×C3—21no—
238C22×C3—21no—
239C22×C3C2×#10211yescharacteristic
240C22×C3#50911yescharacteristic
241C22×C3#51011yescharacteristic
242C2×D3—81no—
243C2×D3—81no—
244C2×D3—81no—
245C3×C4—21no—
246C3×C4—81no—
247C3×C4—81no—
248C3×C4—161no—
249C3×C4#41011yescharacteristic
250C3×C4#42111yescharacteristic
251Dic3—21no—
252Dic3—21no—
253Dic3—21no—
254Dic3—81no—
255Dic3—161no—
256Dic3#36012yes—
257C23—121no—
258C23D3×D811yescharacteristic
259Q8—41no—
260Q8—41no—
261D4—121no—
262D4—121no—
263D4—241no—
264D4—241no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—31no—
268C2×C4—31no—
269C2×C4—31no—
270C2×C4—31no—
271C2×C4—41no—
272C2×C4—41no—
273C2×C4—62no—
274C2×C4—61no—
275C2×C4—61no—
276C2×C4—81no—
277C2×C4—81no—
278C2×C4—81no—
279C2×C4—121no—
280C2×C4—242no—
281C2×C4—241no—
282C2×C4—241no—
283C2×C4—241no—
284C2×C4D3×C22⋊C411yescharacteristic
285C2×C4D3×SD1611yescharacteristic
286C8—21no—
287C8—21no—
288C8—61no—
289C8—61no—
290C2×C3—21no—
291C2×C3—21no—
292C2×C3#151711yescharacteristic
293C2×C3#213911yescharacteristic
294C2×C3C2×#36011yescharacteristic
295D3—161no—
296C22—21no—
297C22—21no—
298C22—241no—
299C22—241no—
300C22—241no—
301C22D3×#10211yescharacteristic
302C22#491411yescharacteristic
303C22#491811yescharacteristic
304C4—21no—
305C4—32no—
306C4—61no—
307C4—61no—
308C4—61no—
309C4—81no—
310C4—81no—
311C4—161no—
312C4—241no—
313C4—481no—
314C4#474011yescharacteristic
315C4#482111yescharacteristic
316C3#1186011yescharacteristic, complemented, Sylow
317C2—21no—
318C2—21no—
319C2—481no—
320C2#6828211yescharacteristic
321C2#6922111yescharacteristic
322C2D3×#36011yescharacteristic
subgroupG/H
centerC22D3×#102
commutatorC2×C3×C8C22×C4
FrattiniC22×C8C22×D3
FittingC3×#1348C2
socleC22×C3C2×#102
prime radical#5115C22
cosocleC22×C3×C8C23
hypercenter#1348D3
hypocenterC3#11860
LCSG ▷ C2×C3×C8 ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ #1348

Complements

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

NH ≅ G/Ntype
2. #69093C2semidirect
3. #68847C2semidirect
4. C3×#1348C2semidirect
5. #67776C2semidirect
11. #5191C4semidirect
12. Dic3×Q16C4semidirect
17. #4878C4semidirect
18. #4872C4semidirect
32. #1348D3semidirect
316. C3#11860semidirect

Maximal subgroups

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

subgroupindex
#807402
#690932
#688472
C3×#13482
#677762
#675262
#674852
#118603