Finite Group Databasebeta

← #10692929

Subgroups — #10692929

Order 768.

610 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 610 conjugacy classes of subgroups — too many to draw. Only its 99 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#82177C211yesmaximal, characteristic
2#82155C211yesmaximal, characteristic, complemented
3C3×#1821C211yesmaximal, characteristic, complemented
4#69978C212yesmaximal
5#67682C212yesmaximal, complemented
6#9464—31nomaximal, Sylow
7C22×C3×C4⋊C4C2211yescharacteristic
8C2×#930C2211yescharacteristic
9C2×#929—21no—
10C2×#929—21no—
11C3×#407C2212yes—
12#5212—21no—
13#5211C2212yes—
14C4×#217—21no—
15C4×#217C2211yescharacteristic
16#4993—22no—
17#4988C412yescomplemented
18#4980—24no—
19#4975C412yescomplemented
20C3×#343—22no—
21#4470—24no—
22#2501—31no—
23#2313—31no—
24#1821D311yescharacteristic, complemented
25#1510—32no—
26#1284—32no—
27C23×C3×C4D411yescharacteristic
28C23×C3×C4D411yescharacteristic
29C23×C3×C4C2311yescharacteristic
30C22×#67—21no—
31C22×D12C2×C411yescharacteristic
32C2×C3×M16—22no—
33C2×C3×C4⋊C4—22no—
34C2×C3×C4⋊C4—22no—
35C2×C3×C4⋊C4C2×C412yes—
36C2×C3×C4⋊C4D412yes—
37C2×#217—21no—
38C2×#217—22no—
39#930—22no—
40#930—22no—
41#929—44no—
42#929—44no—
43C2×(C2×D3)⋊C4—22no—
44C2×(C2×D3)⋊C4—22no—
45C2×(C2×D3)⋊C4—21no—
46C2×C4⋊Dic3—22no—
47C2×C4⋊Dic3C2×C411yescharacteristic
48C2×Dic3⋊C4—22no—
49C2×Dic3⋊C4—22no—
50C2×C4×Dic3—22no—
51C2×C4×Dic3C2×C412yes—
52C3×C22⋊C8—22no—
53C3×C22⋊C8—22no—
54C3⋊#95—22no—
55C3⋊#95—22no—
56C3⋊#95—21no—
57C3⋊#95—22no—
58C3⋊#95D414yes—
59C2×#123—61no—
60C2×#123—61no—
61C2×#121—31no—
62C22×C4⋊C4C2×D311yescharacteristic
63#407C2×D312yes—
64#399—62no—
65#390—32no—
66#387—64no—
67#384—32no—
68#381—32no—
69#380—61no—
70C4×C22⋊C4—31no—
71C4×C22⋊C4—61no—
72#343—22no—
73#327—64no—
74C24×C3C2×D411yescharacteristic
75C23×D3—21no—
76C22×C3×C4—21no—
77C22×C3×C4—21no—
78C22×C3×C4—21no—
79C22×C3×C4—21no—
80C22×C3×C4—42no—
81C22×C3×C4—42no—
82C22×C3×C4—42no—
83C22×C3×C4C22⋊C412yes—
84C22×C3×C4C22⋊C412yes—
85C22×C3×C4C22×C411yescharacteristic
86C22×C3×C4C2×D412yes—
87C22×C3×C4C2×D411yescharacteristic
88C22×C3×C4#4112yes—
89C2×#67—42no—
90C2×#67—42no—
91C2×#67—41no—
92C2×#67—41no—
93C22×Dic3—22no—
94C22×Dic3—21no—
95C22×Dic3—24no—
96C2×D12—22no—
97C2×D12—42no—
98C3×M16—44no—
99C2×C3×C8—42no—
100C3×C4⋊C4—24no—
101C3×C4⋊C4—24no—
102#217—24no—
103#217—44no—
104(C2×D3)⋊C4—44no—
105(C2×D3)⋊C4—44no—
106(C2×D3)⋊C4—42no—
107C4⋊Dic3—22no—
108C4⋊Dic3—82no—
109Dic3⋊C4—44no—
110Dic3⋊C4—44no—
111C4×Dic3—42no—
112C4×Dic3—82no—
113C22×D4—31no—
114C22×D4—61no—
115C23×C4D1211yescharacteristic
116C23×C4D1211yescharacteristic
117C23×C4C22×D311yescharacteristic
118C2×M16—22no—
119#123—124no—
120#123—124no—
121#121—62no—
122#121—62no—
123C2×C4⋊C4—22no—
124C2×C4⋊C4—22no—
125C2×C4⋊C4—31no—
126C2×C4⋊C4—62no—
127C2×C4⋊C4—62no—
128C2×C4⋊C4—62no—
129C2×C4⋊C4C4×D312yes—
130C2×C4⋊C4#6712yes—
131C2×C22⋊C4—61no—
132C2×C22⋊C4—62no—
133C2×C22⋊C4—62no—
134C2×C22⋊C4—62no—
135C2×C22⋊C4—61no—
136C2×C42—32no—
137C2×C42—62no—
138C22⋊C8—22no—
139C22⋊C8—22no—
140#95—34no—
141#95—62no—
142#95—62no—
143#95—62no—
144#95—61no—
145C23×C3—22no—
146C23×C3—22no—
147C23×C3—41no—
148C23×C3C2×C22⋊C411yescharacteristic
149C23×C3#12011yescharacteristic
150C23×C3#12311yescharacteristic
151C22×D3—41no—
152C22×D3—41no—
153C22×D3—42no—
154C22×D3—41no—
155C22×D3—42no—
156C2×C3×C4—22no—
157C2×C3×C4—22no—
158C2×C3×C4—22no—
159C2×C3×C4—22no—
160C2×C3×C4—42no—
161C2×C3×C4—42no—
162C2×C3×C4—42no—
163C2×C3×C4—42no—
164C2×C3×C4—42no—
165C2×C3×C4—42no—
166C2×C3×C4—81no—
167C2×C3×C4—81no—
168C2×C3×C4C4×D412yes—
169C2×C3×C4#12112yes—
170#67—82no—
171#67—82no—
172C2×Dic3—24no—
173C2×Dic3—42no—
174C2×Dic3—42no—
175C2×Dic3—41no—
176C2×Dic3—41no—
177C2×Dic3—44no—
178C2×Dic3—44no—
179C2×Dic3—41no—
180C2×Dic3—44no—
181D12—82no—
182C3×C8—82no—
183C24—61no—
184C24C2×D1211yescharacteristic
185C2×D4—62no—
186C2×D4—122no—
187C2×D4—122no—
188C2×D4—122no—
189C2×D4—121no—
190C2×D4—121no—
191C22×C4—21no—
192C22×C4—21no—
193C22×C4—21no—
194C22×C4—21no—
195C22×C4—42no—
196C22×C4—42no—
197C22×C4—42no—
198C22×C4—62no—
199C22×C4—64no—
200C22×C4—61no—
201C22×C4(C2×D3)⋊C412yes—
202C22×C4(C2×D3)⋊C412yes—
203C22×C4C2×C4×D311yescharacteristic
204C22×C4D3×D412yes—
205C22×C4#23712yes—
206C22×C4C2×#6711yescharacteristic
207M16—44no—
208C2×C8—42no—
209C4⋊C4—24no—
210C4⋊C4—24no—
211C4⋊C4—62no—
212C4⋊C4—124no—
213C4⋊C4—124no—
214C4⋊C4—242no—
215C22⋊C4—64no—
216C22⋊C4—124no—
217C22⋊C4—124no—
218C22⋊C4—124no—
219C22⋊C4—122no—
220C42—122no—
221C42—242no—
222C22×C3—21no—
223C22×C3—21no—
224C22×C3—21no—
225C22×C3—21no—
226C22×C3—21no—
227C22×C3—21no—
228C22×C3—21no—
229C22×C3—44no—
230C22×C3#38611yescharacteristic
231C22×C3#41012yes—
232C22×C3#45312yes—
233C2×D3—82no—
234C2×D3—82no—
235C2×D3—81no—
236C2×D3—81no—
237C2×D3—81no—
238C3×C4—22no—
239C3×C4—22no—
240C3×C4—42no—
241C3×C4—42no—
242Dic3—44no—
243Dic3—84no—
244Dic3—81no—
245C23—22no—
246C23—22no—
247C23—41no—
248C23—121no—
249C23—121no—
250C23—122no—
251C23—121no—
252C23—122no—
253C23#88211yescharacteristic
254C23#88611yescharacteristic
255C23C2×(C2×D3)⋊C411yescharacteristic
256D4—242no—
257D4—242no—
258D4—242no—
259C2×C4—22no—
260C2×C4—22no—
261C2×C4—22no—
262C2×C4—22no—
263C2×C4—42no—
264C2×C4—42no—
265C2×C4—42no—
266C2×C4—42no—
267C2×C4—42no—
268C2×C4—42no—
269C2×C4—64no—
270C2×C4—81no—
271C2×C4—81no—
272C2×C4—122no—
273C2×C4—122no—
274C2×C4—124no—
275C2×C4—121no—
276C2×C4—121no—
277C2×C4—124no—
278C2×C4—121no—
279C2×C4—124no—
280C2×C4#67⋊C412yes—
281C2×C4#93912yes—
282C8—82no—
283C2×C3—22no—
284C2×C3—22no—
285C2×C3—41no—
286C2×C3#188711yescharacteristic
287C2×C3#189911yescharacteristic
288C2×C3#191211yescharacteristic
289D3—161no—
290C22—21no—
291C22—21no—
292C22—21no—
293C22—21no—
294C22—21no—
295C22—21no—
296C22—21no—
297C22—44no—
298C22—242no—
299C22—242no—
300C22—241no—
301C22—241no—
302C22—241no—
303C22#474712yes—
304C22#495811yescharacteristic
305C22#500612yes—
306C4—22no—
307C4—22no—
308C4—42no—
309C4—42no—
310C4—124no—
311C4—244no—
312C4—241no—
313C3#946411yescharacteristic, complemented, Sylow
314C2—22no—
315C2—22no—
316C2—41no—
317C2—481no—
318C2#6953511yescharacteristic
319C2#7007011yescharacteristic
320C2#7010511yescharacteristic
subgroupG/H
centerC22#4958
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC3×#1821C2
socleC22×C3#386
prime radicalC2×#930C22
cosocleC23×C3×C4C23
hypercenter#1821D3
hypocenterC3#9464
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1821

Complements

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

NH ≅ G/Ntype
2. #82155C2semidirect
3. C3×#1821C2semidirect
5. #67682C2semidirect
17. #4988C4semidirect
19. #4975C4semidirect
24. #1821D3semidirect
313. C3#9464semidirect

Maximal subgroups

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

subgroupindex
#821772
#821552
C3×#18212
#699782
#676822
#94643