Finite Group Databasebeta

← #10692573

Subgroups — #10692573

Order 768.

730 conjugacy classes of subgroups; only the 139 normal subgroups are shown.

normal subgroups only

This group has 730 conjugacy classes of subgroups — too many to draw. Only its 139 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×#2541C211yesmaximal, characteristic
2#82301C211yesmaximal, characteristic, complemented
3#71759C212yesmaximal, complemented
4#71617C211yesmaximal, characteristic, complemented
5#67620C212yesmaximal, complemented
6#9397—31nomaximal, Sylow
7C2×C3×#127—21no—
8C2×C3×#127C2211yescharacteristic
9C2×C3×#121—22no—
10C2×C3×C4×D4C2211yescharacteristic
11C3×#392C414yes—
12C3×#390—21no—
13C3×C4×C4⋊C4C2211yescharacteristic
14C2×#833—21no—
15C2×#830—22no—
16D4×C3⋊C8—24no—
17#4925C2212yes—
18#4924C2212yes—
19#4544—24no—
20#4530—21no—
21#2972—31no—
22#2541D311yescharacteristic
23#1800—31no—
24#1545—32no—
25#1303—32no—
26C22×C3×D4—24no—
27C22×C3×D4—21no—
28C22×C3×D4—21no—
29C22×C3×D4C2×C412yes—
30C22×C3×D4D411yescharacteristic
31C23×C3×C4D412yes—
32C3×#127—22no—
33C3×#127—22no—
34C3×#127—42no—
35C3×#127—42no—
36C3×#121—48no—
37C3×C4×D4—22no—
38C3×C4×D4—22no—
39C2×C3×C4⋊C4—21no—
40C2×C3×C4⋊C4D411yescharacteristic
41C2×C3×C22⋊C4—24no—
42C2×C3×C22⋊C4C2×C412yes—
43C2×C3×C42—21no—
44C2×C3×C42C2311yescharacteristic
45C2×#208—22no—
46C22×C3⋊C8—24no—
47C3×#95C2×C412yes—
48#833—42no—
49#830—24no—
50#830—22no—
51C4⋊(C3⋊C8)—22no—
52C4⋊(C3⋊C8)—22no—
53C4⋊(C3⋊C8)D414yescomplemented
54C4×C3⋊C8—22no—
55C2×#127—21no—
56C2×#127C2×D311yescharacteristic
57C2×#121—22no—
58C2×C4×D4C2×D311yescharacteristic
59C8×D4—64no—
60#431—32no—
61#423—32no—
62C2×#100—61no—
63C2×C22⋊C8—62no—
64#392Dic314yes—
65#390—21no—
66C4×C4⋊C4C2×D311yescharacteristic
67#343—61no—
68#331—64no—
69C24×C3—22no—
70C24×C3C22⋊C412yes—
71C2×C3×D4—24no—
72C2×C3×D4—22no—
73C2×C3×D4—22no—
74C2×C3×D4—44no—
75C2×C3×D4—48no—
76C2×C3×D4—48no—
77C2×C3×D4—41no—
78C2×C3×D4—42no—
79C2×C3×D4—41no—
80C2×C3×D4—41no—
81C2×C3×D4—41no—
82C2×C3×D4—42no—
83C2×C3×D4—84no—
84C2×C3×D4—82no—
85C2×C3×D4D812yes—
86C2×C3×D4SD1612yes—
87C22×C3×C4—22no—
88C22×C3×C4—22no—
89C22×C3×C4—22no—
90C22×C3×C4—44no—
91C22×C3×C4C22⋊C412yes—
92C22×C3×C4C22×C411yescharacteristic
93C22×C3×C4C2×D411yescharacteristic
94C22×C3×C4C2×D411yescharacteristic
95C3×C4⋊C4—24no—
96C3×C4⋊C4—22no—
97C3×C22⋊C4—24no—
98C3×C22⋊C4—48no—
99C3×C42—41no—
100C3×C42—41no—
101C3×C42C2×D412yes—
102C3×C42#4112yes—
103#208—42no—
104#208—42no—
105C2×C3⋊C8—24no—
106C2×C3⋊C8—42no—
107C2×C3⋊C8—84no—
108C22×D4—24no—
109C22×D4—21no—
110C22×D4—21no—
111C22×D4C2×Dic312yes—
112C22×D4#6711yescharacteristic
113C23×C4#6712yes—
114C2×M16—62no—
115C22×C8—64no—
116#127—22no—
117#127—22no—
118#127—42no—
119#127—42no—
120#121—48no—
121C4×D4—22no—
122C4×D4—22no—
123C2×C4⋊C4—21no—
124C2×C4⋊C4#6711yescharacteristic
125C2×C22⋊C4—24no—
126C2×C22⋊C4C2×Dic312yes—
127C2×C42—21no—
128C2×C42C22×D311yescharacteristic
129C4⋊C8—34no—
130C4⋊C8—62no—
131C4⋊C8—62no—
132#100—122no—
133C22⋊C8—64no—
134C22⋊C8—62no—
135C4×C8—62no—
136#95C2×Dic312yes—
137C23×C3—22no—
138C23×C3—22no—
139C23×C3—42no—
140C23×C3—42no—
141C23×C3—44no—
142C23×C3—44no—
143C23×C3—42no—
144C23×C3—42no—
145C23×C3—42no—
146C23×C3#10214yes—
147C23×C3C2×C22⋊C411yescharacteristic
148C3×D4—24no—
149C3×D4—24no—
150C3×D4—84no—
151C3×D4—82no—
152C3×D4—816no—
153C3×D4—82no—
154C3×D4—82no—
155C3×D4—82no—
156C2×C3×C4—22no—
157C2×C3×C4—24no—
158C2×C3×C4—22no—
159C2×C3×C4—41no—
160C2×C3×C4—42no—
161C2×C3×C4—42no—
162C2×C3×C4—44no—
163C2×C3×C4—42no—
164C2×C3×C4—82no—
165C2×C3×C4C4×D412yes—
166C2×C3×C4#12011yescharacteristic
167C2×C3×C4#12112yes—
168C2×C3×C4#12311yescharacteristic
169C2×C3×C4C2×D811yescharacteristic
170C2×C3×C4C2×SD1611yescharacteristic
171C2×C3×C4#13512yes—
172C3⋊C8—44no—
173C3⋊C8—82no—
174C24—22no—
175C24#21712yes—
176C2×D4—24no—
177C2×D4—22no—
178C2×D4—22no—
179C2×D4—44no—
180C2×D4—48no—
181C2×D4—48no—
182C2×D4—41no—
183C2×D4—41no—
184C2×D4—41no—
185C2×D4—41no—
186C2×D4—42no—
187C2×D4—42no—
188C2×D4—84no—
189C2×D4—82no—
190C2×D4#21312yes—
191C2×D4#21412yes—
192C22×C4—22no—
193C22×C4—22no—
194C22×C4—22no—
195C22×C4—44no—
196C22×C4#21712yes—
197C22×C4C22×Dic311yescharacteristic
198C22×C4C2×#6711yescharacteristic
199C22×C4C2×#6711yescharacteristic
200M16—122no—
201M16—122no—
202C2×C8—64no—
203C2×C8—122no—
204C2×C8—244no—
205C4⋊C4—24no—
206C4⋊C4—22no—
207C22⋊C4—24no—
208C22⋊C4—48no—
209C42—41no—
210C42—41no—
211C42D3×D412yes—
212C42#23712yes—
213C22×C3—21no—
214C22×C3—21no—
215C22×C3—24no—
216C22×C3—44no—
217C22×C3—42no—
218C22×C3—42no—
219C22×C3—82no—
220C22×C3—84no—
221C22×C3—84no—
222C22×C3—82no—
223C22×C3—82no—
224C22×C3—82no—
225C22×C3#38611yescharacteristic
226C22×C3C2×#10211yescharacteristic
227C22×C3#41611yescharacteristic
228C3×C4—21no—
229C3×C4—22no—
230C3×C4—44no—
231C3×C4—42no—
232C3×C4#45312yes—
233C3×C4#46512yes—
234C3×C4#46612yes—
235C23—22no—
236C23—22no—
237C23—42no—
238C23—42no—
239C23—44no—
240C23—44no—
241C23—42no—
242C23—42no—
243C23—42no—
244C23#83214yes—
245C23C2×#21711yescharacteristic
246D4—24no—
247D4—24no—
248D4—84no—
249D4—82no—
250D4—816no—
251D4—82no—
252D4—82no—
253D4—82no—
254C2×C4—22no—
255C2×C4—24no—
256C2×C4—22no—
257C2×C4—41no—
258C2×C4—42no—
259C2×C4—42no—
260C2×C4—44no—
261C2×C4—42no—
262C2×C4—82no—
263C2×C4C2×#21311yescharacteristic
264C2×C4C2×#21411yescharacteristic
265C2×C4D4×Dic312yes—
266C2×C4#93511yescharacteristic
267C2×C4#93711yescharacteristic
268C2×C4#93912yes—
269C2×C4#95012yes—
270C8—124no—
271C8—242no—
272C2×C3—21no—
273C2×C3—21no—
274C2×C3—44no—
275C2×C3—162no—
276C2×C3#188711yescharacteristic
277C2×C3#189311yescharacteristic
278C2×C3#191411yescharacteristic
279C22—21no—
280C22—21no—
281C22—24no—
282C22—44no—
283C22—42no—
284C22—42no—
285C22—82no—
286C22—84no—
287C22—84no—
288C22—82no—
289C22—82no—
290C22—82no—
291C22C2×#83211yescharacteristic
292C22#522011yescharacteristic
293C22#523311yescharacteristic
294C4—21no—
295C4—22no—
296C4—44no—
297C4—42no—
298C4#503812yes—
299C4#504112yes—
300C4#507712yes—
301C3#939711yescharacteristic, complemented, Sylow
302C2—21no—
303C2—21no—
304C2—44no—
305C2—162no—
306C2#7168111yescharacteristic
307C2#7176311yescharacteristic
308C2#7185511yescharacteristic
subgroupG/H
centerC22#5220
commutatorC22×C3×C4C22×C4
FrattiniC2×C42C22×D3
Fitting, prime radicalC3×#2541C2
socleC22×C3#386
cosocleC2×C3×C42C23
hypercenter#2541D3
hypocenterC3#9397
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2541

Complements

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

NH ≅ G/Ntype
2. #82301C2semidirect
3. #71759C2semidirect
4. #71617C2semidirect
5. #67620C2semidirect
53. C4⋊(C3⋊C8)D4semidirect
301. C3#9397semidirect

Maximal subgroups

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

subgroupindex
C3×#25412
#823012
#717592
#716172
#676202
#93973