Finite Group Databasebeta

← #10691190

Subgroups — #10691190

Order 768.

504 conjugacy classes of subgroups.

The subgroup lattice has 314 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2905C211yesmaximal, characteristic
2#82264C211yesmaximal, characteristic
3#71571C211yesmaximal, characteristic
4#70507C212yesmaximal, complemented
5#67713C212yesmaximal, complemented
6#9191—31nomaximal, Sylow
7C22×C3×M16—21no—
8C2×C3×#117C2211yescharacteristic
9C2×#869C2211yescharacteristic
10C3×#424—24no—
11C3×#423—21no—
12C3×#423C2212yes—
13#5212—21no—
14#4986—21no—
15#4979C412yes—
16#4949—21no—
17#4947C412yes—
18#4936C2211yescharacteristic
19#4928—21no—
20#4925C2212yes—
21#4528—21no—
22#4525—21no—
23#4481—24no—
24#2905D311yescharacteristic
25#2628—31no—
26#1548—32no—
27#1458—31no—
28#1286—32no—
29C23×C3×C4D411yescharacteristic
30C22×Dic6C2×C412yes—
31C2×C3×M16—22no—
32C2×C3×M16—22no—
33C2×C3×M16—42no—
34C2×C3×M16—42no—
35C22×C3×C8—22no—
36C3×#117—22no—
37C3×#117—22no—
38C2×C3×C4⋊C4C2×C411yescharacteristic
39C2×C3×C4⋊C4D411yescharacteristic
40C2×C3×C22⋊C4C2×C411yescharacteristic
41C2×C3×C22⋊C4D411yescharacteristic
42C2×C3×C42D411yescharacteristic
43C2×C3×C42C2311yescharacteristic
44C2×#217—22no—
45C2×C4⋊Dic3—21no—
46C2×C4⋊Dic3—21no—
47C2×C4⋊Dic3—21no—
48C2×Dic3⋊C4—22no—
49#869—22no—
50#869—22no—
51C3×C4⋊C8—24no—
52C3×C4⋊C8—22no—
53C3×C4⋊C8D414yes—
54C3⋊#95—22no—
55C3⋊#95—22no—
56C3⋊#95—21no—
57C3⋊#95C2×C412yes—
58#830—22no—
59C4⋊(C3⋊C8)—22no—
60C4×C3⋊C8—22no—
61C22×M16—21no—
62C2×C4⋊Q8—31no—
63C2×#117C2×D311yescharacteristic
64#431—32no—
65#424—24no—
66#423—21no—
67#423C2×D312yes—
68#400—61no—
69#399—61no—
70#395—32no—
71#393—32no—
72#389—61no—
73#382—31no—
74#380—61no—
75#333—64no—
76#324—61no—
77#323—61no—
78C24×C3C22⋊C411yescharacteristic
79C22×C3×C4—21no—
80C22×C3×C4—21no—
81C22×C3×C4—21no—
82C22×C3×C4—21no—
83C22×C3×C4C22⋊C411yescharacteristic
84C22×C3×C4C22⋊C411yescharacteristic
85C22×C3×C4C22⋊C411yescharacteristic
86C22×C3×C4SD1612yes—
87C22×C3×C4Q1612yes—
88C22×C3×C4C22×C411yescharacteristic
89C22×C3×C4C2×D411yescharacteristic
90C22×C3×C4C2×D411yescharacteristic
91C22×Dic3—22no—
92C22×Dic3—22no—
93C2×Dic6—24no—
94C2×Dic6—82no—
95C3×M16—44no—
96C3×M16—42no—
97C3×M16—42no—
98C2×C3×C8—24no—
99C2×C3×C8—44no—
100C3×C4⋊C4—22no—
101C3×C4⋊C4—22no—
102C3×C22⋊C4—22no—
103C3×C22⋊C4—22no—
104C3×C42—21no—
105C3×C42—21no—
106C3×C42C2×D412yes—
107C3×C42#4112yes—
108#217—82no—
109C4⋊Dic3—41no—
110C4⋊Dic3—42no—
111C4⋊Dic3—41no—
112C4⋊Dic3—41no—
113C4⋊Dic3—41no—
114Dic3⋊C4—82no—
115C2×C3⋊C8—42no—
116C22×Q8—32no—
117C23×C4#6711yescharacteristic
118C2×M16—22no—
119C2×M16—22no—
120C2×M16—42no—
121C2×M16—42no—
122C22×C8—22no—
123C4⋊Q8—62no—
124C4⋊Q8—62no—
125#117—22no—
126#117—22no—
127C2×C4⋊C4—61no—
128C2×C4⋊C4—61no—
129C2×C4⋊C4—61no—
130C2×C4⋊C4—62no—
131C2×C4⋊C4C4×D311yescharacteristic
132C2×C4⋊C4D1211yescharacteristic
133C2×C22⋊C4—62no—
134C2×C22⋊C4C4×D311yescharacteristic
135C2×C22⋊C4D1211yescharacteristic
136C2×C42#6711yescharacteristic
137C2×C42C22×D311yescharacteristic
138C4⋊C8—24no—
139C4⋊C8—22no—
140C4⋊C8—62no—
141C4⋊C8#6714yes—
142C22⋊C8—62no—
143C4×C8—62no—
144#95—32no—
145#95—62no—
146#95—62no—
147#95—61no—
148C23×C3—21no—
149C23×C3—21no—
150C23×C3—41no—
151C23×C3—41no—
152C23×C3#10312yes—
153C23×C3C2×C22⋊C411yescharacteristic
154C2×C3×C4—21no—
155C2×C3×C4—21no—
156C2×C3×C4—22no—
157C2×C3×C4—22no—
158C2×C3×C4—22no—
159C2×C3×C4—22no—
160C2×C3×C4—21no—
161C2×C3×C4—21no—
162C2×C3×C4—41no—
163C2×C3×C4—41no—
164C2×C3×C4—41no—
165C2×C3×C4—41no—
166C2×C3×C4#10312yes—
167C2×C3×C4C4×D412yes—
168C2×C3×C4#12011yescharacteristic
169C2×C3×C4#12112yes—
170C2×C3×C4#12311yescharacteristic
171C2×C3×C4C2×SD1611yescharacteristic
172C2×C3×C4C2×Q1611yescharacteristic
173C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
174C2×C3×C4#13711yescharacteristic
175C2×Dic3—42no—
176C2×Dic3—42no—
177C2×Dic3—42no—
178C2×Dic3—42no—
179C2×Dic3—42no—
180C2×Dic3—42no—
181Dic6—84no—
182C3×C8—44no—
183C3⋊C8—82no—
184C24(C2×D3)⋊C411yescharacteristic
185C2×Q8—64no—
186C2×Q8—242no—
187C22×C4—21no—
188C22×C4—21no—
189C22×C4—21no—
190C22×C4—21no—
191C22×C4—62no—
192C22×C4—62no—
193C22×C4(C2×D3)⋊C411yescharacteristic
194C22×C4(C2×D3)⋊C411yescharacteristic
195C22×C4(C2×D3)⋊C411yescharacteristic
196C22×C4#21412yes—
197C22×C4#21612yes—
198C22×C4C2×C4×D311yescharacteristic
199C22×C4C2×D1211yescharacteristic
200C22×C4C2×#6711yescharacteristic
201M16—44no—
202M16—42no—
203M16—42no—
204C2×C8—24no—
205C2×C8—44no—
206C2×C8—122no—
207C4⋊C4—22no—
208C4⋊C4—22no—
209C4⋊C4—121no—
210C4⋊C4—122no—
211C4⋊C4—121no—
212C4⋊C4—121no—
213C4⋊C4—121no—
214C4⋊C4—242no—
215C22⋊C4—22no—
216C22⋊C4—22no—
217C22⋊C4—242no—
218C42—21no—
219C42—21no—
220C42#23512yes—
221C42C2×#6712yes—
222C22×C3—21no—
223C22×C3—21no—
224C22×C3—22no—
225C22×C3—42no—
226C22×C3—41no—
227C22×C3—41no—
228C22×C3—81no—
229C22×C3#38611yescharacteristic
230C22×C3C2×#10311yescharacteristic
231C22×C3#41711yescharacteristic
232C3×C4—22no—
233C3×C4—22no—
234C3×C4—22no—
235C3×C4—41no—
236C3×C4—41no—
237C3×C4#45011yescharacteristic
238C3×C4#45111yescharacteristic
239C3×C4#48311yescharacteristic
240C3×C4#48411yescharacteristic
241Dic3—82no—
242Dic3—82no—
243C23—21no—
244C23—21no—
245C23—41no—
246C23—41no—
247C23#81012yes—
248C23C2×(C2×D3)⋊C411yescharacteristic
249Q8—244no—
250C2×C4—21no—
251C2×C4—21no—
252C2×C4—22no—
253C2×C4—22no—
254C2×C4—22no—
255C2×C4—22no—
256C2×C4—21no—
257C2×C4—21no—
258C2×C4—41no—
259C2×C4—41no—
260C2×C4—41no—
261C2×C4—41no—
262C2×C4—122no—
263C2×C4—122no—
264C2×C4—122no—
265C2×C4—122no—
266C2×C4—122no—
267C2×C4—122no—
268C2×C4#81012yes—
269C2×C4C4×#6712yes—
270C2×C4#92911yescharacteristic
271C2×C4#93012yes—
272C2×C4#93211yescharacteristic
273C2×C4C2×#21411yescharacteristic
274C2×C4#94211yescharacteristic
275C2×C4C2×#21611yescharacteristic
276C2×C4#95311yescharacteristic
277C8—44no—
278C8—242no—
279C2×C3—21no—
280C2×C3—21no—
281C2×C3—42no—
282C2×C3#187211yescharacteristic
283C2×C3#190911yescharacteristic
284C2×C3#191411yescharacteristic
285C22—21no—
286C22—21no—
287C22—22no—
288C22—42no—
289C22—41no—
290C22—41no—
291C22—81no—
292C22#496611yescharacteristic
293C22C2×#81011yescharacteristic
294C22#521311yescharacteristic
295C4—22no—
296C4—22no—
297C4—22no—
298C4—41no—
299C4—41no—
300C4—242no—
301C4—242no—
302C4#497011yescharacteristic
303C4#497111yescharacteristic
304C4#521811yescharacteristic
305C4#522811yescharacteristic
306C3#919111yescharacteristic, complemented, Sylow
307C2—21no—
308C2—21no—
309C2—42no—
310C2#7155911yescharacteristic
311C2#7202411yescharacteristic
312C2#7203011yescharacteristic
subgroupG/H
centerC22#5213
commutatorC22×C3×C4C22×C4
FrattiniC2×C42C22×D3
FittingC3×#2905C2
socleC22×C3#386
prime radicalC24×C3C22⋊C4
cosocleC2×C3×C42C23
hypercenter#2905D3
hypocenterC3#9191
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2905

Complements

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

NH ≅ G/Ntype
4. #70507C2semidirect
5. #67713C2semidirect
306. C3#9191semidirect

Maximal subgroups

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

subgroupindex
C3×#29052
#822642
#715712
#705072
#677132
#91913