Finite Group Databasebeta

← #10690227

Subgroups — #10690227

Order 768.

616 conjugacy classes of subgroups.

The subgroup lattice has 322 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#400C211yesmaximal, characteristic
2#82156C211yesmaximal, characteristic
3#71554C212yesmaximal
4C2×#4528C211yesmaximal, characteristic
5#67612C212yesmaximal
6#9023—31nomaximal, Sylow
7C22×C3×C4⋊C4C2211yescharacteristic
8C22×C3×C4⋊C4C2211yescharacteristic
9C2×#926—21no—
10C3×#400—22no—
11C3×#400C414yescomplemented
12C2×C3×#95—21no—
13C2×C3×#95—21no—
14C2×C3×#95C2211yescharacteristic
15#5212—21no—
16#5212C2212yes—
17C2×#830C2212yes—
18#4984—24no—
19#4529—24no—
20#4528—22no—
21C2×#400D311yescharacteristic
22#2316—31no—
23#1824—32no—
24C2×#324—31no—
25#1294—32no—
26C23×C3×C4—21no—
27C23×C3×C4—21no—
28C23×C3×C4D411yescharacteristic
29C23×C3×C4D411yescharacteristic
30C23×C3×C4C2311yescharacteristic
31C2×C3×C4⋊C4—22no—
32C2×C3×C4⋊C4—22no—
33C2×C3×C4⋊C4—22no—
34C2×C3×C4⋊C4—22no—
35C2×C3×C4⋊C4—21no—
36C2×C3×C4⋊C4C2×C412yes—
37C2×C3×C4⋊C4C2×C412yes—
38C2×C3×C4⋊C4D412yes—
39C2×#217—22no—
40#926—42no—
41#926—42no—
42C2×C4⋊Dic3—22no—
43C2×C4⋊Dic3—22no—
44C2×C4⋊Dic3D414yes—
45C2×C4×Dic3—22no—
46C22×C3⋊C8—22no—
47C3×#95—24no—
48C3×#95—24no—
49C3×#95—21no—
50C3×#95C2×C412yes—
51C3⋊#95—22no—
52C3⋊#95—22no—
53#830—22no—
54#830—22no—
55C2×#117—61no—
56C22×C4⋊C4C2×D311yescharacteristic
57C22×C4⋊C4C2×D311yescharacteristic
58C2×C22⋊C8—32no—
59#400—22no—
60#400Dic314yescomplemented
61#389—64no—
62#380—32no—
63#380—61no—
64C2×#95—21no—
65C2×#95—21no—
66C2×#95C2×D311yescharacteristic
67#340—64no—
68#324—62no—
69C24×C3C2×D411yescharacteristic
70C22×C3×C4—21no—
71C22×C3×C4—21no—
72C22×C3×C4—21no—
73C22×C3×C4—21no—
74C22×C3×C4—21no—
75C22×C3×C4—21no—
76C22×C3×C4—22no—
77C22×C3×C4—22no—
78C22×C3×C4—42no—
79C22×C3×C4—42no—
80C22×C3×C4—42no—
81C22×C3×C4—42no—
82C22×C3×C4—44no—
83C22×C3×C4—41no—
84C22×C3×C4—41no—
85C22×C3×C4—41no—
86C22×C3×C4—41no—
87C22×C3×C4C22⋊C412yes—
88C22×C3×C4C22⋊C412yes—
89C22×C3×C4SD1612yes—
90C22×C3×C4Q1612yes—
91C22×C3×C4C22×C411yescharacteristic
92C22×C3×C4C2×D412yes—
93C22×C3×C4C2×D411yescharacteristic
94C22×C3×C4#4112yes—
95C22×Dic3—24no—
96C3×C4⋊C4—24no—
97C3×C4⋊C4—24no—
98C3×C4⋊C4—24no—
99C3×C4⋊C4—42no—
100#217—42no—
101#217—42no—
102C4⋊Dic3—44no—
103C4⋊Dic3—44no—
104C4⋊Dic3—82no—
105C4×Dic3—44no—
106C2×C3⋊C8—44no—
107C2×C3⋊C8—42no—
108C23×C4—21no—
109C23×C4—21no—
110C23×C4#6711yescharacteristic
111C23×C4#6711yescharacteristic
112C23×C4C22×D311yescharacteristic
113C22×C8—62no—
114#117—122no—
115#117—122no—
116C2×C4⋊C4—22no—
117C2×C4⋊C4—22no—
118C2×C4⋊C4—22no—
119C2×C4⋊C4—22no—
120C2×C4⋊C4—21no—
121C2×C4⋊C4—34no—
122C2×C4⋊C4—62no—
123C2×C4⋊C4—62no—
124C2×C4⋊C4C2×Dic312yes—
125C2×C4⋊C4C2×Dic312yes—
126C2×C4⋊C4#6712yes—
127C2×C22⋊C4—62no—
128C2×C42—62no—
129C22⋊C8—62no—
130C22⋊C8—62no—
131#95—24no—
132#95—24no—
133#95—21no—
134#95—62no—
135#95—62no—
136#95C2×Dic312yes—
137C23×C3—22no—
138C23×C3—22no—
139C23×C3C2×C22⋊C411yescharacteristic
140C23×C3#12011yescharacteristic
141C23×C3#12311yescharacteristic
142C23×C3C2×SD1611yescharacteristic
143C23×C3C2×Q1611yescharacteristic
144C23×C3AGL(1,ℤ/8)11yescharacteristic
145C23×C3#13711yescharacteristic
146C2×C3×C4—22no—
147C2×C3×C4—22no—
148C2×C3×C4—22no—
149C2×C3×C4—22no—
150C2×C3×C4—24no—
151C2×C3×C4—22no—
152C2×C3×C4—22no—
153C2×C3×C4—22no—
154C2×C3×C4—41no—
155C2×C3×C4—41no—
156C2×C3×C4—42no—
157C2×C3×C4—42no—
158C2×C3×C4—42no—
159C2×C3×C4—42no—
160C2×C3×C4—42no—
161C2×C3×C4—44no—
162C2×C3×C4—42no—
163C2×C3×C4—42no—
164C2×C3×C4—81no—
165C2×C3×C4—82no—
166C2×C3×C4—82no—
167C2×C3×C4—82no—
168C2×C3×C4—82no—
169C2×C3×C4#10314yes—
170C2×C3×C4C4×D412yes—
171C2×C3×C4#12112yes—
172C2×Dic3—44no—
173C2×Dic3—44no—
174C2×Dic3—44no—
175C3⋊C8—82no—
176C24C2×#6711yescharacteristic
177C22×C4—21no—
178C22×C4—21no—
179C22×C4—21no—
180C22×C4—21no—
181C22×C4—21no—
182C22×C4—21no—
183C22×C4—22no—
184C22×C4—22no—
185C22×C4—42no—
186C22×C4—42no—
187C22×C4—42no—
188C22×C4—42no—
189C22×C4—44no—
190C22×C4—41no—
191C22×C4—41no—
192C22×C4—41no—
193C22×C4—41no—
194C22×C4—64no—
195C22×C4#21512yes—
196C22×C4#21612yes—
197C22×C4#21712yes—
198C22×C4#21712yes—
199C22×C4D3×D412yes—
200C22×C4#23712yes—
201C22×C4C22×Dic311yescharacteristic
202C22×C4C2×#6711yescharacteristic
203C2×C8—124no—
204C2×C8—122no—
205C4⋊C4—24no—
206C4⋊C4—24no—
207C4⋊C4—24no—
208C4⋊C4—42no—
209C4⋊C4—124no—
210C4⋊C4—124no—
211C4⋊C4—242no—
212C22⋊C4—122no—
213C22⋊C4—122no—
214C42—124no—
215C22×C3—22no—
216C22×C3—22no—
217C22×C3—22no—
218C22×C3—22no—
219C22×C3—21no—
220C22×C3—21no—
221C22×C3—21no—
222C22×C3—21no—
223C22×C3#38611yescharacteristic
224C22×C3C2×#10311yescharacteristic
225C22×C3#41711yescharacteristic
226C22×C3#45011yescharacteristic
227C22×C3#45111yescharacteristic
228C22×C3#45412yes—
229C22×C3#45812yes—
230C22×C3#48311yescharacteristic
231C22×C3#48411yescharacteristic
232C3×C4—24no—
233C3×C4—22no—
234C3×C4—22no—
235C3×C4—42no—
236C3×C4—82no—
237C3×C4—82no—
238Dic3—84no—
239C23—22no—
240C23—22no—
241C23#93511yescharacteristic
242C23#93711yescharacteristic
243C23C2×#21511yescharacteristic
244C23C2×#21611yescharacteristic
245C23#94911yescharacteristic
246C23#95111yescharacteristic
247C23C2×#21711yescharacteristic
248C2×C4—22no—
249C2×C4—22no—
250C2×C4—22no—
251C2×C4—22no—
252C2×C4—22no—
253C2×C4—22no—
254C2×C4—24no—
255C2×C4—22no—
256C2×C4—42no—
257C2×C4—41no—
258C2×C4—41no—
259C2×C4—42no—
260C2×C4—42no—
261C2×C4—42no—
262C2×C4—42no—
263C2×C4—44no—
264C2×C4—42no—
265C2×C4—42no—
266C2×C4—81no—
267C2×C4—82no—
268C2×C4—82no—
269C2×C4—82no—
270C2×C4—82no—
271C2×C4—124no—
272C2×C4—124no—
273C2×C4—124no—
274C2×C4#83514yes—
275C2×C4D4×Dic312yes—
276C2×C4#93912yes—
277C8—242no—
278C2×C3—21no—
279C2×C3—21no—
280C2×C3—21no—
281C2×C3—21no—
282C2×C3#161612yes—
283C2×C3#161712yes—
284C2×C3#187211yescharacteristic
285C2×C3#188811yescharacteristic
286C2×C3#189011yescharacteristic
287C22—21no—
288C22—21no—
289C22—21no—
290C22—21no—
291C22—22no—
292C22—22no—
293C22—22no—
294C22—22no—
295C22#504311yescharacteristic
296C22#504411yescharacteristic
297C22#504611yescharacteristic
298C22#505011yescharacteristic
299C22#505312yes—
300C22#506212yes—
301C22#522011yescharacteristic
302C22C2×#83511yescharacteristic
303C22#523211yescharacteristic
304C4—24no—
305C4—22no—
306C4—22no—
307C4—42no—
308C4—82no—
309C4—82no—
310C4—244no—
311C3#902311yescharacteristic, complemented, Sylow
312C2—21no—
313C2—21no—
314C2—21no—
315C2—21no—
316C2#6971812yes—
317C2#6971912yes—
318C2#7169211yescharacteristic
319C2#7170411yescharacteristic
320C2#7172411yescharacteristic
subgroupG/H
centerC23#937
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC2×C3×#400C2
socleC23×C3#120
prime radicalC24×C3C2×D4
cosocleC23×C3×C4C23
hypercenterC2×#400D3
hypocenterC3#9023
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#400

Complements

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

NH ≅ G/Ntype
11. C3×#400C4semidirect
60. #400Dic3semidirect
311. C3#9023semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#4002
#821562
#715542
C2×#45282
#676122
#90233