Finite Group Databasebeta

← #10693062

Subgroups — #10693062

Order 768.

522 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 522 conjugacy classes of subgroups — too many to draw. Only its 119 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×#2536C211yesmaximal, characteristic
2#82436C211yesmaximal, characteristic, complemented
3#71637C211yesmaximal, characteristic, complemented
4#71598C212yesmaximal
5#67631C212yesmaximal, complemented
6#9490—31nomaximal, Sylow
7C2×C3×C22⋊Q8—21no—
8C2×C3×C4×Q8C2211yescharacteristic
9C2×C3×C4×D4C2211yescharacteristic
10C3×#398C412yes—
11C3×#394C412yes—
12C3×#388—22no—
13C3×#382C2211yescharacteristic
14C3×#380—21no—
15C2×#836—21no—
16#5024—24no—
17#4926—22no—
18#4925C2212yes—
19#4924C2212yes—
20D4⋊(C3⋊C8)—24no—
21#4530—21no—
22#2983—31no—
23#2536D311yescharacteristic
24#1801—31no—
25#1486—32no—
26#1286—32no—
27C22×C3×Q8D411yescharacteristic
28C22×C3×D4C2×C411yescharacteristic
29C23×C3×C4—21no—
30C3×C22⋊Q8—44no—
31C3×C4×Q8—22no—
32C3×C4×Q8—22no—
33C3×C4×D4—22no—
34C3×C4×D4—22no—
35C2×C3×C4⋊C4—22no—
36C2×C3×C4⋊C4—21no—
37C2×C3×C4⋊C4C2×C411yescharacteristic
38C2×C3×C4⋊C4C2×C412yes—
39C2×C3×C4⋊C4D411yescharacteristic
40C2×C3×C22⋊C4—22no—
41C2×C3×C22⋊C4—22no—
42C2×C3×C22⋊C4—21no—
43C2×C3×C42D412yes—
44C2×C3×C42C2311yescharacteristic
45C2×#208—22no—
46C3×#95—22no—
47C3×#95—21no—
48C3×#95C2×C412yes—
49#836—42no—
50#830—22no—
51C4⋊(C3⋊C8)—24no—
52C4⋊(C3⋊C8)—22no—
53C4⋊(C3⋊C8)—22no—
54C4⋊(C3⋊C8)—22no—
55#803—24no—
56C4×C3⋊C8D414yes—
57C2×C22⋊Q8—21no—
58C2×C4×Q8C2×D311yescharacteristic
59C2×C4×D4C2×D311yescharacteristic
60#446—64no—
61#432—62no—
62#431—32no—
63#423—32no—
64C2×#101—61no—
65#398Dic312yes—
66#394Dic312yes—
67#388—22no—
68#382C2×D311yescharacteristic
69#380—21no—
70#343—61no—
71D4⋊C8—64no—
72C24×C3—21no—
73C2×C3×Q8—22no—
74C2×C3×Q8—21no—
75C2×C3×Q8—22no—
76C2×C3×Q8—21no—
77C2×C3×D4—22no—
78C2×C3×D4—42no—
79C22×C3×C4—22no—
80C22×C3×C4—21no—
81C22×C3×C4—42no—
82C22×C3×C4—42no—
83C22×C3×C4—41no—
84C22×C3×C4—41no—
85C22×C3×C4C22⋊C412yes—
86C22×C3×C4C22⋊C412yes—
87C22×C3×C4C22×C411yescharacteristic
88C22×C3×C4C2×D411yescharacteristic
89C22×C3×C4C2×D411yescharacteristic
90C3×C4⋊C4—24no—
91C3×C4⋊C4—22no—
92C3×C4⋊C4—22no—
93C3×C4⋊C4—44no—
94C3×C4⋊C4—42no—
95C3×C22⋊C4—44no—
96C3×C22⋊C4—44no—
97C3×C22⋊C4—41no—
98C3×C22⋊C4—41no—
99C3×C42—22no—
100C3×C42—22no—
101C3×C42C2×D412yes—
102C3×C42#4112yes—
103#208—42no—
104#208—42no—
105C2×C3⋊C8—24no—
106C2×C3⋊C8—42no—
107C22×Q8#6711yescharacteristic
108C22×D4C2×Dic311yescharacteristic
109C23×C4—21no—
110C2×M16—62no—
111C22⋊Q8—44no—
112C4×Q8—22no—
113C4×Q8—22no—
114C4×D4—22no—
115C4×D4—22no—
116C2×C4⋊C4—22no—
117C2×C4⋊C4—21no—
118C2×C4⋊C4C2×Dic311yescharacteristic
119C2×C4⋊C4C2×Dic312yes—
120C2×C4⋊C4#6711yescharacteristic
121C2×C22⋊C4—22no—
122C2×C22⋊C4—22no—
123C2×C22⋊C4—21no—
124C2×C42#6712yes—
125C2×C42C22×D311yescharacteristic
126C4⋊C8—64no—
127C4⋊C8—62no—
128C4⋊C8—62no—
129C4⋊C8—62no—
130#101—122no—
131C22⋊C8—62no—
132#97—64no—
133C4×C8—34no—
134#95—22no—
135#95—21no—
136#95C2×Dic312yes—
137C23×C3—22no—
138C23×C3—41no—
139C23×C3—41no—
140C23×C3—41no—
141C23×C3—41no—
142C23×C3—41no—
143C23×C3—41no—
144C23×C3C2×C22⋊C411yescharacteristic
145C3×Q8—42no—
146C3×Q8—42no—
147C3×D4—24no—
148C3×D4—42no—
149C2×C3×C4—22no—
150C2×C3×C4—22no—
151C2×C3×C4—22no—
152C2×C3×C4—22no—
153C2×C3×C4—41no—
154C2×C3×C4—41no—
155C2×C3×C4—42no—
156C2×C3×C4—42no—
157C2×C3×C4—42no—
158C2×C3×C4—41no—
159C2×C3×C4—41no—
160C2×C3×C4—42no—
161C2×C3×C4—41no—
162C2×C3×C4—82no—
163C2×C3×C4—82no—
164C2×C3×C4C4×D412yes—
165C2×C3×C4#12011yescharacteristic
166C2×C3×C4#12112yes—
167C2×C3×C4#12311yescharacteristic
168C2×C3×C4AGL(1,ℤ/8)12yes—
169C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
170C2×C3×C4#13711yescharacteristic
171C3⋊C8—44no—
172C3⋊C8—82no—
173C24—21no—
174C2×Q8—22no—
175C2×Q8—22no—
176C2×Q8—21no—
177C2×Q8—21no—
178C2×D4—22no—
179C2×D4—42no—
180C22×C4—22no—
181C22×C4—21no—
182C22×C4—42no—
183C22×C4—42no—
184C22×C4—41no—
185C22×C4—41no—
186C22×C4#21712yes—
187C22×C4#21712yes—
188C22×C4C22×Dic311yescharacteristic
189C22×C4C2×#6711yescharacteristic
190C22×C4C2×#6711yescharacteristic
191M16—122no—
192M16—122no—
193C2×C8—64no—
194C2×C8—122no—
195C4⋊C4—24no—
196C4⋊C4—22no—
197C4⋊C4—22no—
198C4⋊C4—44no—
199C4⋊C4—42no—
200C22⋊C4—44no—
201C22⋊C4—44no—
202C22⋊C4—41no—
203C22⋊C4—41no—
204C42—22no—
205C42—22no—
206C42D3×D412yes—
207C42#23712yes—
208C22×C3—21no—
209C22×C3—21no—
210C22×C3—42no—
211C22×C3—42no—
212C22×C3—42no—
213C22×C3—81no—
214C22×C3—81no—
215C22×C3—81no—
216C22×C3—81no—
217C22×C3#38611yescharacteristic
218C22×C3#41711yescharacteristic
219C22×C3#41811yescharacteristic
220C3×C4—22no—
221C3×C4—21no—
222C3×C4—21no—
223C3×C4—42no—
224C3×C4—42no—
225C3×C4—82no—
226C3×C4—81no—
227C3×C4#46812yes—
228C3×C4#46912yes—
229C23—22no—
230C23—41no—
231C23—41no—
232C23—41no—
233C23—41no—
234C23—41no—
235C23—41no—
236C23C2×#21711yescharacteristic
237Q8—42no—
238Q8—42no—
239D4—24no—
240D4—42no—
241C2×C4—22no—
242C2×C4—22no—
243C2×C4—22no—
244C2×C4—22no—
245C2×C4—41no—
246C2×C4—41no—
247C2×C4—42no—
248C2×C4—42no—
249C2×C4—42no—
250C2×C4—42no—
251C2×C4—41no—
252C2×C4—41no—
253C2×C4—41no—
254C2×C4—82no—
255C2×C4—82no—
256C2×C4#93212yes—
257C2×C4D4×Dic312yes—
258C2×C4#93511yescharacteristic
259C2×C4#93711yescharacteristic
260C2×C4#93912yes—
261C2×C4#94911yescharacteristic
262C2×C4#95111yescharacteristic
263C8—124no—
264C8—242no—
265C2×C3—21no—
266C2×C3—21no—
267C2×C3—82no—
268C2×C3#189111yescharacteristic
269C2×C3#189511yescharacteristic
270C2×C3#191211yescharacteristic
271C22—21no—
272C22—21no—
273C22—42no—
274C22—42no—
275C22—42no—
276C22—81no—
277C22—81no—
278C22—81no—
279C22—81no—
280C22#521511yescharacteristic
281C22#522011yescharacteristic
282C22#523211yescharacteristic
283C4—22no—
284C4—21no—
285C4—21no—
286C4—42no—
287C4—42no—
288C4—82no—
289C4—81no—
290C4#503912yes—
291C4#504212yes—
292C3#949011yescharacteristic, complemented, Sylow
293C2—21no—
294C2—21no—
295C2—82no—
296C2#7168211yescharacteristic
297C2#7171411yescharacteristic
298C2#7186411yescharacteristic
subgroupG/H
centerC22#5220
commutatorC22×C3×C4C22×C4
FrattiniC2×C42C22×D3
FittingC3×#2536C2
socleC22×C3#386
prime radicalC22×C3×D4C2×C4
cosocleC2×C3×C42C23
hypercenter#2536D3
hypocenterC3#9490
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2536

Complements

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

NH ≅ G/Ntype
2. #82436C2semidirect
3. #71637C2semidirect
5. #67631C2semidirect
292. C3#9490semidirect

Maximal subgroups

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

subgroupindex
C3×#25362
#824362
#716372
#715982
#676312
#94903