Finite Group Databasebeta

← #10695077

Subgroups — #10695077

Order 768.

460 conjugacy classes of subgroups.

The subgroup lattice has 330 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3167C211yesmaximal, characteristic
2#71839C211yesmaximal, characteristic, complemented
3#71826C211yesmaximal, characteristic, complemented
4#71815C211yesmaximal, characteristic
5#71802C211yesmaximal, characteristic
6#69750C211yesmaximal, characteristic, complemented
7#69730C211yesmaximal, characteristic, complemented
8#9846—31nomaximal, Sylow
9C3×#533C2211yescharacteristic
10C3×#532C2211yescharacteristic
11C3×#498C411yescharacteristic, complemented
12C3×#492C411yescharacteristic, complemented
13C3×#487C411yescharacteristic, complemented
14C3×#486C411yescharacteristic, complemented
15C3×#465—22no—
16C3×C4×SD16—22no—
17C2×C3×C4×C8C2211yescharacteristic
18C2×#837—22no—
19C42×Dic3C2211yescharacteristic
20#4924C2211yescharacteristic
21#4924C2211yescharacteristic
22C2×C4×C3⋊C8C2211yescharacteristic
23#4550—22no—
24#4535—22no—
25D4⋊(C3⋊C8)—22no—
26#4532—22no—
27#4457—22no—
28#3167D311yescharacteristic
29#1994—31no—
30#1962—31no—
31#1744—31no—
32#1566—31no—
33#1493—31no—
34#1492—31no—
35C2×C3×#41—21no—
36C2×C3×SD16—22no—
37C22×C3×C8D412yes—
38C3×C4⋊Q8C2×C411yescharacteristic
39C3×C4⋊Q8C2×C411yescharacteristic
40C3×C4⋊Q8C2×C411yescharacteristic
41C3×#127C2×C411yescharacteristic
42C3×#125—21no—
43C3×#124—21no—
44C3×C22⋊Q8—21no—
45C3×C22⋊Q8—21no—
46C3×#121—21no—
47C3×#121—21no—
48C3×C4×Q8—21no—
49C3×C4×Q8—21no—
50C3×C4×D4—21no—
51C3×C4×D4—21no—
52C3×#117—21no—
53C2×C3×C42C2311yescharacteristic
54C2×C4×Dic3—21no—
55C2×C4×Dic3—21no—
56C2×C4×Dic3D412yes—
57C2×#208—21no—
58C2×#208—21no—
59C22×C3⋊C8—21no—
60C3×#106—22no—
61C3×#103—22no—
62C3×#102—22no—
63C3×C4×C8C2×C411yescharacteristic
64C3×C4×C8C2×C411yescharacteristic
65C3×C4×C8D412yes—
66#837—44no—
67C4⋊(C3⋊C8)—21no—
68C4⋊(C3⋊C8)—21no—
69C4⋊(C3⋊C8)—21no—
70C4⋊(C3⋊C8)—21no—
71C4×C3⋊C8—21no—
72C4×C3⋊C8D412yes—
73#533C2×D311yescharacteristic
74#532C2×D311yescharacteristic
75#498Dic311yescharacteristic, complemented
76#492Dic311yescharacteristic, complemented
77#487Dic311yescharacteristic, complemented
78#486Dic311yescharacteristic, complemented
79#465—22no—
80C4×SD16—22no—
81#423—31no—
82#423—31no—
83C2×#104—62no—
84C2×C4×C8—31no—
85C2×C4×C8C2×D311yescharacteristic
86C43—31no—
87#339—62no—
88#336—62no—
89#334—62no—
90Q8⋊C8—62no—
91D4⋊C8—62no—
92C3×#41—42no—
93C3×#41—42no—
94C2×C3×Q8—21no—
95C2×C3×Q8—21no—
96C2×C3×Q8—21no—
97C2×C3×D4—21no—
98C2×C3×D4—21no—
99C2×C3×D4—21no—
100C2×C3×D4—41no—
101C22×C3×C4—41no—
102C22×C3×C4C2×D411yescharacteristic
103C22×C3×C4C2×D411yescharacteristic
104C22×C3×C4#4111yescharacteristic
105C22×Dic3—22no—
106C3×SD16—44no—
107C2×C3×C8—42no—
108C2×C3×C8C22⋊C412yes—
109C2×C3×C8C22⋊C412yes—
110C3×C4⋊C4—21no—
111C3×C4⋊C4—21no—
112C3×C4⋊C4—21no—
113C3×C4⋊C4—21no—
114C3×C4⋊C4—21no—
115C3×C4⋊C4—21no—
116C3×C4⋊C4—41no—
117C3×C4⋊C4—41no—
118C3×C22⋊C4—41no—
119C3×C22⋊C4—41no—
120C3×C22⋊C4—41no—
121C3×C42—41no—
122C3×C42C22×C411yescharacteristic
123C3×C42C2×D411yescharacteristic
124C3×C42C2×D411yescharacteristic
125C3×C42#4111yescharacteristic
126C4×Dic3—22no—
127C4×Dic3—42no—
128C4×Dic3—42no—
129C4×Dic3SD1614yescomplemented
130#208—42no—
131#208—42no—
132C2×C3⋊C8—22no—
133C2×C3⋊C8—42no—
134C2×C3⋊C8—41no—
135C2×C3⋊C8—41no—
136C2×#41—21no—
137C2×SD16—22no—
138C2×M16—61no—
139C2×M16—61no—
140C22×C8—61no—
141C22×C8#6712yes—
142C4⋊Q8C2×Dic311yescharacteristic
143C4⋊Q8C2×Dic311yescharacteristic
144C4⋊Q8C2×Dic311yescharacteristic
145#127C2×Dic311yescharacteristic
146#125—21no—
147#124—21no—
148C22⋊Q8—21no—
149C22⋊Q8—21no—
150#121—21no—
151#121—21no—
152C4×Q8—21no—
153C4×Q8—21no—
154C4×D4—21no—
155C4×D4—21no—
156#117—21no—
157C2×C42—32no—
158C2×C42—61no—
159C2×C42—61no—
160C2×C42C22×D311yescharacteristic
161#106—22no—
162C4⋊C8—61no—
163C4⋊C8—61no—
164C4⋊C8—61no—
165C4⋊C8—61no—
166#104—124no—
167#103—22no—
168#102—22no—
169C4×C8—32no—
170C4×C8—61no—
171C4×C8C2×Dic311yescharacteristic
172C4×C8C2×Dic311yescharacteristic
173C4×C8#6712yes—
174C23×C3—41no—
175C23×C3#12111yescharacteristic
176C3×Q8—42no—
177C3×Q8—42no—
178C3×Q8—42no—
179C3×D4—42no—
180C3×D4—42no—
181C3×D4—42no—
182C3×D4—82no—
183C2×C3×C4—22no—
184C2×C3×C4—22no—
185C2×C3×C4—41no—
186C2×C3×C4—41no—
187C2×C3×C4—41no—
188C2×C3×C4—81no—
189C2×C3×C4—81no—
190C2×C3×C4C2×C22⋊C411yescharacteristic
191C2×C3×C4C4×D411yescharacteristic
192C2×C3×C4C4×D411yescharacteristic
193C2×C3×C4#12111yescharacteristic
194C2×C3×C4#12411yescharacteristic
195C2×C3×C4#12711yescharacteristic
196C2×C3×C4C2×SD1612yes—
197C2×C3×C4#13512yes—
198C2×Dic3—24no—
199C2×Dic3—42no—
200C2×Dic3—42no—
201C3×C8—22no—
202C3×C8—22no—
203C3⋊C8—42no—
204C3⋊C8—81no—
205C3⋊C8—81no—
206#41—42no—
207#41—42no—
208C2×Q8—21no—
209C2×Q8—21no—
210C2×Q8—21no—
211C2×D4—21no—
212C2×D4—21no—
213C2×D4—21no—
214C2×D4—41no—
215C22×C4—41no—
216C22×C4—62no—
217C22×C4D3×D411yescharacteristic
218C22×C4#23711yescharacteristic
219C22×C4C2×#6711yescharacteristic
220SD16—44no—
221M16—122no—
222M16—122no—
223C2×C8—42no—
224C2×C8—62no—
225C2×C8—122no—
226C2×C8—121no—
227C2×C8—121no—
228C2×C8#21712yes—
229C2×C8#21712yes—
230C4⋊C4—21no—
231C4⋊C4—21no—
232C4⋊C4—21no—
233C4⋊C4—21no—
234C4⋊C4—21no—
235C4⋊C4—21no—
236C4⋊C4—41no—
237C4⋊C4—41no—
238C22⋊C4—41no—
239C22⋊C4—41no—
240C22⋊C4—41no—
241C42—34no—
242C42—41no—
243C42—62no—
244C42—122no—
245C42—122no—
246C42D3×D411yescharacteristic
247C42#23711yescharacteristic
248C42C22×Dic311yescharacteristic
249C42C2×#6711yescharacteristic
250C22×C3—21no—
251C22×C3—21no—
252C22×C3—81no—
253C22×C3—81no—
254C22×C3—81no—
255C22×C3#39011yescharacteristic
256C22×C3#46512yes—
257C3×C4—21no—
258C3×C4—21no—
259C3×C4—81no—
260C3×C4—81no—
261C3×C4—81no—
262C3×C4#42112yes—
263C3×C4C4×SD1612yes—
264C3×C4#44412yes—
265C3×C4#49211yescharacteristic
266C3×C4#49511yescharacteristic
267Dic3—44no—
268C23—41no—
269C23#93811yescharacteristic
270Q8—42no—
271Q8—42no—
272Q8—42no—
273D4—42no—
274D4—42no—
275D4—42no—
276D4—82no—
277C2×C4—22no—
278C2×C4—22no—
279C2×C4—41no—
280C2×C4—41no—
281C2×C4—41no—
282C2×C4—64no—
283C2×C4—81no—
284C2×C4—81no—
285C2×C4—122no—
286C2×C4—122no—
287C2×C4D3×SD1612yes—
288C2×C4#91612yes—
289C2×C4D4×Dic311yescharacteristic
290C2×C4D4×Dic311yescharacteristic
291C2×C4#93611yescharacteristic
292C2×C4#93811yescharacteristic
293C2×C4#94011yescharacteristic
294C2×C4C2×#21711yescharacteristic
295C8—22no—
296C8—22no—
297C8—122no—
298C8—241no—
299C8—241no—
300C2×C3—22no—
301C2×C3—161no—
302C2×C3#196311yescharacteristic
303C2×C3#196811yescharacteristic
304C2×C3#197511yescharacteristic
305C22—21no—
306C22—21no—
307C22—81no—
308C22—81no—
309C22—81no—
310C22#517112yes—
311C22#522111yescharacteristic
312C4—21no—
313C4—21no—
314C4—81no—
315C4—81no—
316C4—81no—
317C4—124no—
318C4Dic3×SD1612yes—
319C4#516811yescharacteristic
320C4#517511yescharacteristic
321C4#519712yes—
322C4#523612yes—
323C3#984611yescharacteristic, complemented, Sylow
324C2—22no—
325C2—161no—
326C2#7181611yescharacteristic
327C2#7204911yescharacteristic
328C2#7208511yescharacteristic
subgroupG/H
centerC2×C4#940
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC3×#3167C2
socleC22×C3#390
prime radicalC3×#532C22
cosocleC2×C3×C42C23
hypercenter#3167D3
hypocenterC3#9846
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #3167

Complements

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

NH ≅ G/Ntype
2. #71839C2semidirect
3. #71826C2semidirect
6. #69750C2semidirect
7. #69730C2semidirect
11. C3×#498C4semidirect
12. C3×#492C4semidirect
13. C3×#487C4semidirect
14. C3×#486C4semidirect
75. #498Dic3semidirect
76. #492Dic3semidirect
77. #487Dic3semidirect
78. #486Dic3semidirect
129. C4×Dic3SD16semidirect
323. C3#9846semidirect

Maximal subgroups

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

subgroupindex
C3×#31672
#718392
#718262
#718152
#718022
#697502
#697302
#98463