Finite Group Databasebeta

← #10691186

Subgroups — #10691186

Order 768.

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

normal subgroups only

This group has 436 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×#2634C211yesmaximal, characteristic
2#82436C211yesmaximal, characteristic
3#71735C212yesmaximal
4C4×#836C211yesmaximal, characteristic
5#67633C212yesmaximal
6#9190—31nomaximal, Sylow
7C2×C3×#125C2211yescharacteristic
8C2×C3×C4×Q8C2211yescharacteristic
9C3×#400—21no—
10C3×#400C411yescharacteristic
11C3×#400C411yescharacteristic
12C3×#398C411yescharacteristic
13C3×#398C411yescharacteristic
14C3×#395—21no—
15C3×#387—21no—
16C3×#387—21no—
17C3×#383C2211yescharacteristic
18C2×#836—21no—
19#5024—24no—
20#4926—21no—
21#4926C2212yes—
22#4925—21no—
23C4×#208C2212yes—
24#4543—24no—
25#4530—21no—
26#2983—31no—
27#2634D311yescharacteristic
28C4×#101—31no—
29#1528—32no—
30#1288—32no—
31C22×C3×Q8D411yescharacteristic
32C3×#125—22no—
33C3×#125—22no—
34C3×C4×Q8—22no—
35C3×C4×Q8—22no—
36C2×C3×C4⋊C4—21no—
37C2×C3×C4⋊C4—21no—
38C2×C3×C4⋊C4—21no—
39C2×C3×C4⋊C4—21no—
40C2×C3×C4⋊C4C2×C411yescharacteristic
41C2×C3×C4⋊C4C2×C411yescharacteristic
42C2×C3×C4⋊C4C2×C411yescharacteristic
43C2×C3×C4⋊C4C2×C411yescharacteristic
44C2×C3×C4⋊C4D411yescharacteristic
45C2×C3×C42D411yescharacteristic
46C2×C3×C42D411yescharacteristic
47C2×C3×C42C2311yescharacteristic
48C2×#208—22no—
49C3×#95—21no—
50C3×#95—21no—
51C3×#95—21no—
52C3×#95—21no—
53C3×#95—21no—
54C3×#95—21no—
55C3×#95—21no—
56C3×#95—21no—
57C3×#95C2×C411yescharacteristic
58C3×#95C2×C411yescharacteristic
59#836—24no—
60#830—22no—
61C4⋊(C3⋊C8)—22no—
62C4⋊(C3⋊C8)—22no—
63C4⋊(C3⋊C8)—22no—
64#803—22no—
65#803—22no—
66#803D414yes—
67C4×C3⋊C8—22no—
68C4×C3⋊C8—22no—
69C2×#125C2×D311yescharacteristic
70C2×C4×Q8C2×D311yescharacteristic
71#446—64no—
72#432—32no—
73#432—61no—
74#431—61no—
75C2×#101—61no—
76C4×M16—32no—
77#400—21no—
78#400Dic311yescharacteristic
79#400Dic311yescharacteristic
80#398Dic311yescharacteristic
81#398Dic311yescharacteristic
82#395—21no—
83#387—21no—
84#387—21no—
85#383C2×D311yescharacteristic
86#343—61no—
87#330—64no—
88C2×C3×Q8—22no—
89C2×C3×Q8—22no—
90C2×C3×Q8—41no—
91C22×C3×C4—21no—
92C22×C3×C4—21no—
93C22×C3×C4—21no—
94C22×C3×C4—21no—
95C22×C3×C4C22⋊C411yescharacteristic
96C22×C3×C4C22⋊C411yescharacteristic
97C22×C3×C4C22⋊C411yescharacteristic
98C22×C3×C4C22⋊C411yescharacteristic
99C22×C3×C4C22×C411yescharacteristic
100C22×C3×C4C2×D411yescharacteristic
101C22×C3×C4C2×D411yescharacteristic
102C3×C4⋊C4—22no—
103C3×C4⋊C4—22no—
104C3×C4⋊C4—22no—
105C3×C4⋊C4—22no—
106C3×C4⋊C4—22no—
107C3×C4⋊C4—42no—
108C3×C4⋊C4—42no—
109C3×C4⋊C4—41no—
110C3×C4⋊C4—42no—
111C3×C4⋊C4—41no—
112C3×C42—21no—
113C3×C42—21no—
114C3×C42—21no—
115C3×C42—21no—
116C3×C42C2×D412yes—
117C3×C42#4112yes—
118#208—24no—
119#208—42no—
120C2×C3⋊C8—24no—
121C2×C3⋊C8—42no—
122C22×Q8#6711yescharacteristic
123C2×M16—62no—
124#125—22no—
125#125—22no—
126C4×Q8—22no—
127C4×Q8—22no—
128C2×C4⋊C4—21no—
129C2×C4⋊C4—21no—
130C2×C4⋊C4—21no—
131C2×C4⋊C4—21no—
132C2×C4⋊C4C2×Dic311yescharacteristic
133C2×C4⋊C4C2×Dic311yescharacteristic
134C2×C4⋊C4C2×Dic311yescharacteristic
135C2×C4⋊C4C2×Dic311yescharacteristic
136C2×C4⋊C4#6711yescharacteristic
137C2×C42#6711yescharacteristic
138C2×C42#6711yescharacteristic
139C2×C42C22×D311yescharacteristic
140C4⋊C8—62no—
141C4⋊C8—62no—
142C4⋊C8—62no—
143#101—64no—
144C22⋊C8—62no—
145#97—34no—
146#97—62no—
147#97—62no—
148C4×C8—62no—
149C4×C8—62no—
150#95—21no—
151#95—21no—
152#95—21no—
153#95—21no—
154#95—21no—
155#95—21no—
156#95—21no—
157#95—21no—
158#95C2×Dic311yescharacteristic
159#95C2×Dic311yescharacteristic
160C23×C3C2×C22⋊C411yescharacteristic
161C3×Q8—44no—
162C2×C3×C4—21no—
163C2×C3×C4—21no—
164C2×C3×C4—21no—
165C2×C3×C4—21no—
166C2×C3×C4—21no—
167C2×C3×C4—21no—
168C2×C3×C4—21no—
169C2×C3×C4—21no—
170C2×C3×C4—21no—
171C2×C3×C4—41no—
172C2×C3×C4—41no—
173C2×C3×C4—41no—
174C2×C3×C4—41no—
175C2×C3×C4—41no—
176C2×C3×C4—41no—
177C2×C3×C4—41no—
178C2×C3×C4—41no—
179C2×C3×C4—42no—
180C2×C3×C4—41no—
181C2×C3×C4—41no—
182C2×C3×C4—41no—
183C2×C3×C4—41no—
184C2×C3×C4—41no—
185C2×C3×C4—41no—
186C2×C3×C4#10412yes—
187C2×C3×C4C4×D412yes—
188C2×C3×C4#12011yescharacteristic
189C2×C3×C4#12112yes—
190C2×C3×C4#12311yescharacteristic
191C3⋊C8—44no—
192C3⋊C8—44no—
193C2×Q8—22no—
194C2×Q8—22no—
195C2×Q8—41no—
196C22×C4—21no—
197C22×C4—21no—
198C22×C4—21no—
199C22×C4—21no—
200C22×C4#21711yescharacteristic
201C22×C4#21711yescharacteristic
202C22×C4#21711yescharacteristic
203C22×C4#21711yescharacteristic
204C22×C4C22×Dic311yescharacteristic
205C22×C4C2×#6711yescharacteristic
206C22×C4C2×#6711yescharacteristic
207M16—64no—
208M16—122no—
209C2×C8—64no—
210C2×C8—122no—
211C4⋊C4—22no—
212C4⋊C4—22no—
213C4⋊C4—22no—
214C4⋊C4—22no—
215C4⋊C4—22no—
216C4⋊C4—42no—
217C4⋊C4—42no—
218C4⋊C4—41no—
219C4⋊C4—41no—
220C4⋊C4—42no—
221C42—21no—
222C42—21no—
223C42—21no—
224C42—21no—
225C42D3×D412yes—
226C42#23712yes—
227C22×C3—21no—
228C22×C3—21no—
229C22×C3#38611yescharacteristic
230C22×C3C2×#10411yescharacteristic
231C22×C3#42111yescharacteristic
232C3×C4—22no—
233C3×C4—22no—
234C3×C4—21no—
235C3×C4—21no—
236C3×C4—22no—
237C3×C4—41no—
238C3×C4—41no—
239C3×C4—41no—
240C3×C4—81no—
241C3×C4—81no—
242C3×C4—81no—
243C3×C4—81no—
244C23C2×#21711yescharacteristic
245Q8—44no—
246C2×C4—21no—
247C2×C4—21no—
248C2×C4—21no—
249C2×C4—21no—
250C2×C4—21no—
251C2×C4—21no—
252C2×C4—21no—
253C2×C4—21no—
254C2×C4—21no—
255C2×C4—41no—
256C2×C4—41no—
257C2×C4—41no—
258C2×C4—41no—
259C2×C4—41no—
260C2×C4—41no—
261C2×C4—41no—
262C2×C4—41no—
263C2×C4—42no—
264C2×C4—41no—
265C2×C4—41no—
266C2×C4—41no—
267C2×C4—41no—
268C2×C4—41no—
269C2×C4—41no—
270C2×C4#83712yes—
271C2×C4D4×Dic312yes—
272C2×C4#93511yescharacteristic
273C2×C4#93711yescharacteristic
274C2×C4#93912yes—
275C8—124no—
276C8—124no—
277C2×C3—21no—
278C2×C3—21no—
279C2×C3#187111yescharacteristic
280C2×C3#191211yescharacteristic
281C2×C3#191511yescharacteristic
282C22—21no—
283C22—21no—
284C22#522011yescharacteristic
285C22C2×#83711yescharacteristic
286C22#523611yescharacteristic
287C4—22no—
288C4—22no—
289C4—21no—
290C4—21no—
291C4—22no—
292C4—41no—
293C4—41no—
294C4—41no—
295C4—81no—
296C4—81no—
297C4—81no—
298C4—81no—
299C3#919011yescharacteristic, complemented, Sylow
300C2—21no—
301C2—21no—
302C2#7171311yescharacteristic
303C2#7186411yescharacteristic
304C2#7186611yescharacteristic
subgroupG/H
centerC22#5220
commutatorC22×C3×C4C22×C4
FrattiniC2×C42C22×D3
FittingC3×#2634C2
socleC22×C3#386
prime radicalC23×C3C2×C22⋊C4
cosocleC2×C3×C42C23
hypercenter#2634D3
hypocenterC3#9190
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2634

Complements

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

NH ≅ G/Ntype
299. C3#9190semidirect

Maximal subgroups

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

subgroupindex
C3×#26342
#824362
#717352
C4×#8362
#676332
#91903