Finite Group Databasebeta

← #10692932

Subgroups — #10692932

Order 768.

610 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 610 conjugacy classes of subgroups — too many to draw. Only its 107 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
1C3×#2501C211yesmaximal, characteristic
2#82155C211yesmaximal, characteristic, complemented
3#71674C212yesmaximal
4#71562C211yesmaximal, characteristic, complemented
5#67616C212yesmaximal, complemented
6#9464—31nomaximal, Sylow
7C2×C3×#123—21no—
8C2×C3×#123—21no—
9C2×C3×#121C2211yescharacteristic
10C22×C3×C4⋊C4C2211yescharacteristic
11C3×#399—22no—
12C3×#390C412yescomplemented
13C3×#384C412yescomplemented
14C3×C4×C22⋊C4C2211yescharacteristic
15#5212—21no—
16#5211C2212yes—
17C4×#217—21no—
18#5207C2212yes—
19#4980—24no—
20#4537—24no—
21#4530—22no—
22#2501D311yescharacteristic
23#2313—31no—
24#1821—31no—
25#1510—32no—
26#1284—32no—
27C22×C3×D4—21no—
28C22×C3×D4C2×C411yescharacteristic
29C23×C3×C4D411yescharacteristic
30C23×C3×C4D411yescharacteristic
31C23×C3×C4C2311yescharacteristic
32C3×#123—44no—
33C3×#123—44no—
34C3×#121—22no—
35C3×#121—22no—
36C2×C3×C4⋊C4—22no—
37C2×C3×C4⋊C4—22no—
38C2×C3×C4⋊C4—22no—
39C2×C3×C4⋊C4—22no—
40C2×C3×C4⋊C4C2×C411yescharacteristic
41C2×C3×C4⋊C4C2×C412yes—
42C2×C3×C4⋊C4D412yes—
43C2×C3×C22⋊C4—21no—
44C2×C3×C22⋊C4—22no—
45C2×C3×C22⋊C4—22no—
46C2×C3×C22⋊C4—21no—
47C2×C3×C42C2×C412yes—
48C2×#217—22no—
49C2×C4⋊Dic3—22no—
50C2×C4×Dic3—22no—
51C2×#208—22no—
52C3×#95—21no—
53C3⋊#95—22no—
54C3⋊#95—22no—
55C3⋊#95—22no—
56C3⋊#95D414yes—
57#830—22no—
58#830—22no—
59C2×#123—21no—
60C2×#123—21no—
61C2×#121C2×D311yescharacteristic
62C22×C4⋊C4C2×D311yescharacteristic
63#407—32no—
64#399—22no—
65#390Dic312yescomplemented
66#387—64no—
67#384Dic312yescomplemented
68#381—32no—
69#380—61no—
70C4×C22⋊C4—61no—
71C4×C22⋊C4C2×D311yescharacteristic
72#343—62no—
73#327—64no—
74C24×C3—21no—
75C24×C3C2×D411yescharacteristic
76C2×C3×D4—22no—
77C2×C3×D4—42no—
78C2×C3×D4—42no—
79C2×C3×D4—42no—
80C2×C3×D4—41no—
81C2×C3×D4—41no—
82C22×C3×C4—22no—
83C22×C3×C4—21no—
84C22×C3×C4—21no—
85C22×C3×C4—21no—
86C22×C3×C4—21no—
87C22×C3×C4—21no—
88C22×C3×C4—42no—
89C22×C3×C4—42no—
90C22×C3×C4—42no—
91C22×C3×C4C22⋊C412yes—
92C22×C3×C4C22⋊C412yes—
93C22×C3×C4C22×C411yescharacteristic
94C22×C3×C4C2×D412yes—
95C22×C3×C4C2×D411yescharacteristic
96C22×C3×C4#4112yes—
97C22×Dic3—24no—
98C3×C4⋊C4—24no—
99C3×C4⋊C4—24no—
100C3×C4⋊C4—22no—
101C3×C4⋊C4—44no—
102C3×C4⋊C4—44no—
103C3×C22⋊C4—24no—
104C3×C22⋊C4—44no—
105C3×C22⋊C4—44no—
106C3×C22⋊C4—42no—
107C3×C42—42no—
108#217—44no—
109C4⋊Dic3—82no—
110C4×Dic3—82no—
111#208—44no—
112C2×C3⋊C8—42no—
113C22×D4—21no—
114C22×D4C2×Dic311yescharacteristic
115C23×C4#6711yescharacteristic
116C23×C4#6711yescharacteristic
117C23×C4C22×D311yescharacteristic
118C2×M16—62no—
119#123—44no—
120#123—44no—
121#121—22no—
122#121—22no—
123C2×C4⋊C4—22no—
124C2×C4⋊C4—22no—
125C2×C4⋊C4—22no—
126C2×C4⋊C4—22no—
127C2×C4⋊C4—62no—
128C2×C4⋊C4C2×Dic311yescharacteristic
129C2×C4⋊C4C2×Dic312yes—
130C2×C4⋊C4#6712yes—
131C2×C22⋊C4—21no—
132C2×C22⋊C4—22no—
133C2×C22⋊C4—22no—
134C2×C22⋊C4—21no—
135C2×C22⋊C4—62no—
136C2×C42—62no—
137C2×C42C2×Dic312yes—
138C22⋊C8—62no—
139C22⋊C8—62no—
140#95—21no—
141#95—34no—
142#95—62no—
143#95—62no—
144#95—62no—
145C23×C3—22no—
146C23×C3—22no—
147C23×C3—41no—
148C23×C3—41no—
149C23×C3—41no—
150C23×C3—42no—
151C23×C3—41no—
152C23×C3—42no—
153C23×C3C2×C22⋊C411yescharacteristic
154C23×C3#12011yescharacteristic
155C23×C3#12311yescharacteristic
156C3×D4—82no—
157C3×D4—82no—
158C3×D4—82no—
159C2×C3×C4—22no—
160C2×C3×C4—22no—
161C2×C3×C4—24no—
162C2×C3×C4—22no—
163C2×C3×C4—22no—
164C2×C3×C4—42no—
165C2×C3×C4—42no—
166C2×C3×C4—42no—
167C2×C3×C4—42no—
168C2×C3×C4—42no—
169C2×C3×C4—42no—
170C2×C3×C4—42no—
171C2×C3×C4—42no—
172C2×C3×C4—41no—
173C2×C3×C4—41no—
174C2×C3×C4—41no—
175C2×C3×C4—81no—
176C2×C3×C4—81no—
177C2×C3×C4C4×D412yes—
178C2×C3×C4#12112yes—
179C2×Dic3—44no—
180C2×Dic3—44no—
181C2×Dic3—44no—
182C3⋊C8—82no—
183C24—21no—
184C24C2×#6711yescharacteristic
185C2×D4—22no—
186C2×D4—42no—
187C2×D4—42no—
188C2×D4—42no—
189C2×D4—41no—
190C2×D4—41no—
191C22×C4—21no—
192C22×C4—21no—
193C22×C4—22no—
194C22×C4—21no—
195C22×C4—21no—
196C22×C4—21no—
197C22×C4—42no—
198C22×C4—42no—
199C22×C4—42no—
200C22×C4—64no—
201C22×C4#21712yes—
202C22×C4#21712yes—
203C22×C4D3×D412yes—
204C22×C4#23712yes—
205C22×C4C22×Dic311yescharacteristic
206C22×C4C2×#6711yescharacteristic
207M16—124no—
208C2×C8—122no—
209C4⋊C4—24no—
210C4⋊C4—24no—
211C4⋊C4—22no—
212C4⋊C4—44no—
213C4⋊C4—44no—
214C4⋊C4—242no—
215C22⋊C4—24no—
216C22⋊C4—44no—
217C22⋊C4—44no—
218C22⋊C4—42no—
219C22⋊C4—124no—
220C42—42no—
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—82no—
231C22×C3—82no—
232C22×C3—81no—
233C22×C3—81no—
234C22×C3—81no—
235C22×C3#38611yescharacteristic
236C22×C3#41012yes—
237C22×C3#45312yes—
238C3×C4—22no—
239C3×C4—22no—
240C3×C4—44no—
241C3×C4—42no—
242C3×C4—42no—
243C3×C4—81no—
244Dic3—84no—
245C23—22no—
246C23—22no—
247C23—41no—
248C23—41no—
249C23—41no—
250C23—42no—
251C23—41no—
252C23—42no—
253C23#93511yescharacteristic
254C23#93711yescharacteristic
255C23C2×#21711yescharacteristic
256D4—82no—
257D4—82no—
258D4—82no—
259C2×C4—22no—
260C2×C4—22no—
261C2×C4—24no—
262C2×C4—22no—
263C2×C4—22no—
264C2×C4—42no—
265C2×C4—42no—
266C2×C4—42no—
267C2×C4—42no—
268C2×C4—42no—
269C2×C4—42no—
270C2×C4—42no—
271C2×C4—42no—
272C2×C4—41no—
273C2×C4—41no—
274C2×C4—41no—
275C2×C4—81no—
276C2×C4—81no—
277C2×C4—124no—
278C2×C4—124no—
279C2×C4—124no—
280C2×C4D4×Dic312yes—
281C2×C4#93912yes—
282C8—242no—
283C2×C3—22no—
284C2×C3—22no—
285C2×C3—41no—
286C2×C3—161no—
287C2×C3#188711yescharacteristic
288C2×C3#189911yescharacteristic
289C2×C3#191211yescharacteristic
290C22—21no—
291C22—21no—
292C22—21no—
293C22—21no—
294C22—21no—
295C22—21no—
296C22—21no—
297C22—44no—
298C22—82no—
299C22—82no—
300C22—81no—
301C22—81no—
302C22—81no—
303C22#507712yes—
304C22#522011yescharacteristic
305C22#524012yes—
306C4—22no—
307C4—22no—
308C4—44no—
309C4—42no—
310C4—42no—
311C4—81no—
312C4—244no—
313C3#946411yescharacteristic, complemented, Sylow
314C2—22no—
315C2—22no—
316C2—41no—
317C2—161no—
318C2#7170511yescharacteristic
319C2#7176311yescharacteristic
320C2#7186411yescharacteristic
subgroupG/H
centerC22#5220
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC3×#2501C2
socleC22×C3#386
prime radicalC2×C3×#121C22
cosocleC23×C3×C4C23
hypercenter#2501D3
hypocenterC3#9464
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2501

Complements

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

NH ≅ G/Ntype
2. #82155C2semidirect
4. #71562C2semidirect
5. #67616C2semidirect
12. C3×#390C4semidirect
13. C3×#384C4semidirect
65. #390Dic3semidirect
67. #384Dic3semidirect
313. C3#9464semidirect

Maximal subgroups

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

subgroupindex
C3×#25012
#821552
#716742
#715622
#676162
#94643