Finite Group Databasebeta

← #10693042

Subgroups — #10693042

Order 768.

542 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 542 conjugacy classes of subgroups — too many to draw. Only its 135 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#82435C211yesmaximal, characteristic, complemented
3#71637C211yesmaximal, characteristic, complemented
4#71598C212yesmaximal
5#67631C212yesmaximal, complemented
6#9486—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—
16Q8×C3⋊C8—24no—
17#4925—22no—
18#4925C2212yes—
19#4924C2212yes—
20D4⋊(C3⋊C8)—24no—
21#4530—21no—
22#2982—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—42no—
31C3×C22⋊Q8—42no—
32C3×C4×Q8—22no—
33C3×C4×Q8—22no—
34C3×C4×D4—22no—
35C3×C4×D4—22no—
36C2×C3×C4⋊C4—22no—
37C2×C3×C4⋊C4—21no—
38C2×C3×C4⋊C4C2×C411yescharacteristic
39C2×C3×C4⋊C4C2×C412yes—
40C2×C3×C4⋊C4D411yescharacteristic
41C2×C3×C22⋊C4—22no—
42C2×C3×C22⋊C4—22no—
43C2×C3×C22⋊C4—21no—
44C2×C3×C42D412yes—
45C2×C3×C42C2311yescharacteristic
46C2×#208—22no—
47C3×#95—22no—
48C3×#95—21no—
49C3×#95C2×C412yes—
50#836—42no—
51#830—22no—
52C4⋊(C3⋊C8)—24no—
53C4⋊(C3⋊C8)—22no—
54C4⋊(C3⋊C8)—22no—
55C4⋊(C3⋊C8)—22no—
56C4×C3⋊C8—24no—
57C4×C3⋊C8D414yes—
58C2×C22⋊Q8—21no—
59C2×C4×Q8C2×D311yescharacteristic
60C2×C4×D4C2×D311yescharacteristic
61C8×Q8—64no—
62#431—32no—
63#431—62no—
64#423—32no—
65C2×#101—61no—
66#398Dic312yes—
67#394Dic312yes—
68#388—22no—
69#382C2×D311yescharacteristic
70#380—21no—
71#343—61no—
72D4⋊C8—64no—
73C24×C3—21no—
74C2×C3×Q8—22no—
75C2×C3×Q8—22no—
76C2×C3×Q8D812yes—
77C2×C3×Q8SD1612yes—
78C2×C3×D4—22no—
79C2×C3×D4—42no—
80C22×C3×C4—22no—
81C22×C3×C4—21no—
82C22×C3×C4—42no—
83C22×C3×C4—41no—
84C22×C3×C4—41no—
85C22×C3×C4—42no—
86C22×C3×C4C22⋊C412yes—
87C22×C3×C4C22⋊C412yes—
88C22×C3×C4C22×C411yescharacteristic
89C22×C3×C4C2×D411yescharacteristic
90C22×C3×C4C2×D411yescharacteristic
91C3×C4⋊C4—24no—
92C3×C4⋊C4—22no—
93C3×C4⋊C4—22no—
94C3×C4⋊C4—44no—
95C3×C4⋊C4—41no—
96C3×C4⋊C4—41no—
97C3×C22⋊C4—44no—
98C3×C22⋊C4—44no—
99C3×C22⋊C4—41no—
100C3×C22⋊C4—41no—
101C3×C42—22no—
102C3×C42—22no—
103C3×C42C2×D412yes—
104C3×C42#4112yes—
105#208—42no—
106#208—42no—
107C2×C3⋊C8—24no—
108C2×C3⋊C8—42no—
109C22×Q8#6711yescharacteristic
110C22×D4C2×Dic311yescharacteristic
111C23×C4—21no—
112C2×M16—62no—
113C22⋊Q8—42no—
114C22⋊Q8—42no—
115C4×Q8—22no—
116C4×Q8—22no—
117C4×D4—22no—
118C4×D4—22no—
119C2×C4⋊C4—22no—
120C2×C4⋊C4—21no—
121C2×C4⋊C4C2×Dic311yescharacteristic
122C2×C4⋊C4C2×Dic312yes—
123C2×C4⋊C4#6711yescharacteristic
124C2×C22⋊C4—22no—
125C2×C22⋊C4—22no—
126C2×C22⋊C4—21no—
127C2×C42#6712yes—
128C2×C42C22×D311yescharacteristic
129C4⋊C8—64no—
130C4⋊C8—62no—
131C4⋊C8—62no—
132C4⋊C8—62no—
133#101—122no—
134C22⋊C8—62no—
135C4×C8—34no—
136C4×C8—64no—
137#95—22no—
138#95—21no—
139#95C2×Dic312yes—
140C23×C3—22no—
141C23×C3—41no—
142C23×C3—41no—
143C23×C3—41no—
144C23×C3—41no—
145C23×C3—41no—
146C23×C3—41no—
147C23×C3C2×C22⋊C411yescharacteristic
148C3×Q8—24no—
149C3×Q8—24no—
150C3×D4—24no—
151C3×D4—42no—
152C2×C3×C4—22no—
153C2×C3×C4—22no—
154C2×C3×C4—22no—
155C2×C3×C4—41no—
156C2×C3×C4—41no—
157C2×C3×C4—42no—
158C2×C3×C4—42no—
159C2×C3×C4—42no—
160C2×C3×C4—41no—
161C2×C3×C4—42no—
162C2×C3×C4—41no—
163C2×C3×C4—41no—
164C2×C3×C4—82no—
165C2×C3×C4—82no—
166C2×C3×C4#10214yes—
167C2×C3×C4C4×D412yes—
168C2×C3×C4#12011yescharacteristic
169C2×C3×C4#12112yes—
170C2×C3×C4#12311yescharacteristic
171C2×C3×C4C2×D811yescharacteristic
172C2×C3×C4C2×SD1611yescharacteristic
173C2×C3×C4#13512yes—
174C3⋊C8—44no—
175C3⋊C8—82no—
176C24—21no—
177C2×Q8—22no—
178C2×Q8—22no—
179C2×Q8#21312yes—
180C2×Q8#21412yes—
181C2×D4—22no—
182C2×D4—42no—
183C22×C4—22no—
184C22×C4—21no—
185C22×C4—41no—
186C22×C4—41no—
187C22×C4—42no—
188C22×C4—42no—
189C22×C4#21712yes—
190C22×C4#21712yes—
191C22×C4C22×Dic311yescharacteristic
192C22×C4C2×#6711yescharacteristic
193C22×C4C2×#6711yescharacteristic
194M16—122no—
195M16—122no—
196C2×C8—64no—
197C2×C8—122no—
198C4⋊C4—24no—
199C4⋊C4—22no—
200C4⋊C4—22no—
201C4⋊C4—44no—
202C4⋊C4—41no—
203C4⋊C4—41no—
204C22⋊C4—44no—
205C22⋊C4—44no—
206C22⋊C4—41no—
207C22⋊C4—41no—
208C42—22no—
209C42—22no—
210C42D3×D412yes—
211C42#23712yes—
212C22×C3—21no—
213C22×C3—21no—
214C22×C3—42no—
215C22×C3—42no—
216C22×C3—42no—
217C22×C3—81no—
218C22×C3—81no—
219C22×C3—81no—
220C22×C3—81no—
221C22×C3#38611yescharacteristic
222C22×C3C2×#10211yescharacteristic
223C22×C3#41611yescharacteristic
224C3×C4—22no—
225C3×C4—24no—
226C3×C4—21no—
227C3×C4—21no—
228C3×C4—42no—
229C3×C4—82no—
230C3×C4—81no—
231C3×C4#46512yes—
232C3×C4#46612yes—
233C23—22no—
234C23—41no—
235C23—41no—
236C23—41no—
237C23—41no—
238C23—41no—
239C23—41no—
240C23C2×#21711yescharacteristic
241Q8—24no—
242Q8—24no—
243D4—24no—
244D4—42no—
245C2×C4—22no—
246C2×C4—22no—
247C2×C4—22no—
248C2×C4—41no—
249C2×C4—41no—
250C2×C4—42no—
251C2×C4—42no—
252C2×C4—42no—
253C2×C4—42no—
254C2×C4—41no—
255C2×C4—41no—
256C2×C4—41no—
257C2×C4—82no—
258C2×C4—82no—
259C2×C4#83214yes—
260C2×C4C2×#21311yescharacteristic
261C2×C4C2×#21411yescharacteristic
262C2×C4D4×Dic312yes—
263C2×C4#93511yescharacteristic
264C2×C4#93711yescharacteristic
265C2×C4#93912yes—
266C2×C4#95012yes—
267C8—124no—
268C8—242no—
269C2×C3—21no—
270C2×C3—21no—
271C2×C3—82no—
272C2×C3#189111yescharacteristic
273C2×C3#189311yescharacteristic
274C2×C3#191211yescharacteristic
275C22—21no—
276C22—21no—
277C22—42no—
278C22—42no—
279C22—42no—
280C22—81no—
281C22—81no—
282C22—81no—
283C22—81no—
284C22C2×#83211yescharacteristic
285C22#522011yescharacteristic
286C22#523311yescharacteristic
287C4—24no—
288C4—22no—
289C4—21no—
290C4—21no—
291C4—42no—
292C4—82no—
293C4—81no—
294C4#503812yes—
295C4#504112yes—
296C3#948611yescharacteristic, complemented, Sylow
297C2—21no—
298C2—21no—
299C2—82no—
300C2#7168111yescharacteristic
301C2#7171411yescharacteristic
302C2#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#9486
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. #82435C2semidirect
3. #71637C2semidirect
5. #67631C2semidirect
296. C3#9486semidirect

Maximal subgroups

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

subgroupindex
C3×#25362
#824352
#716372
#715982
#676312
#94863