Finite Group Databasebeta

← #10692797

Subgroups — #10692797

Order 768.

556 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 556 conjugacy classes of subgroups — too many to draw. Only its 111 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×#3151C211yesmaximal, characteristic
2#83278C211yesmaximal, characteristic, complemented
3#71810C211yesmaximal, characteristic, complemented
4#71759C212yesmaximal, complemented
5#67669C212yesmaximal, complemented
6#9437—31nomaximal, Sylow
7C2×C3×AGL(1,ℤ/8)—22no—
8C3×#532C2211yescharacteristic
9C2×C3×#127C2211yescharacteristic
10C3×#496C414yes—
11C3×#493—21no—
12C3×#492—21no—
13C3×C4×M16C2211yescharacteristic
14#5234—22no—
15#5012—22no—
16D4×C3⋊C8—22no—
17#4925C2212yes—
18#4924C2212yes—
19#4546—21no—
20#4544—21no—
21#4544—22no—
22#4544—22no—
23#3151D311yescharacteristic
24#2979—31no—
25#1560—31no—
26#1545—32no—
27#1299—32no—
28C2×C3×#41D412yes—
29C22×C3×D4—21no—
30C22×C3×D4—22no—
31C3×AGL(1,ℤ/8)—44no—
32C3×AGL(1,ℤ/8)—44no—
33C2×C3×SD16—24no—
34C2×C3×D8—24no—
35C2×C3×M16—21no—
36C3×C4⋊Q8D411yescharacteristic
37C3×#127—22no—
38C3×#127—21no—
39C3×#127C2×C412yes—
40C3×#127D411yescharacteristic
41C3×#124C2×C412yes—
42C3×#121—22no—
43C3×C4×D4—22no—
44C2×C3×C42C2311yescharacteristic
45C2×#208—22no—
46C2×#208—22no—
47C22×C3⋊C8—22no—
48C3×#97C2×C412yes—
49C3×C4×C8—21no—
50#830—24no—
51#830—22no—
52C4⋊(C3⋊C8)—22no—
53C4⋊(C3⋊C8)—22no—
54C4⋊(C3⋊C8)D412yes—
55C4⋊(C3⋊C8)D412yes—
56C4×C3⋊C8—22no—
57C2×AGL(1,ℤ/8)—22no—
58#532C2×D311yescharacteristic
59C2×#127C2×D311yescharacteristic
60#496Dic314yes—
61#493—21no—
62#492—21no—
63#436—62no—
64C8×D4—62no—
65#431—32no—
66#423—32no—
67#408—62no—
68C4×M16C2×D311yescharacteristic
69#333—61no—
70#331—62no—
71#331—62no—
72#331—61no—
73C24×C3—41no—
74C3×#41—42no—
75C3×#41—42no—
76C2×C3×Q8C22⋊C412yes—
77C2×C3×D4—22no—
78C2×C3×D4—22no—
79C2×C3×D4—24no—
80C2×C3×D4—22no—
81C2×C3×D4—42no—
82C2×C3×D4—42no—
83C2×C3×D4—41no—
84C2×C3×D4—42no—
85C2×C3×D4—82no—
86C2×C3×D4—82no—
87C2×C3×D4C22⋊C412yes—
88C22×C3×C4—22no—
89C22×C3×C4C2×D411yescharacteristic
90C22×C3×C4#4112yes—
91C3×SD16—48no—
92C3×D8—48no—
93C3×M16—22no—
94C3×M16—22no—
95C2×C3×C8—22no—
96C3×C4⋊C4—22no—
97C3×C22⋊C4—24no—
98C3×C42C22×C411yescharacteristic
99C3×C42C2×D412yes—
100C3×C42C2×D411yescharacteristic
101#208—82no—
102#208—82no—
103C2×C3⋊C8—22no—
104C2×C3⋊C8—22no—
105C2×C3⋊C8—42no—
106C2×C3⋊C8—82no—
107C2×#41#6712yes—
108C22×D4—22no—
109C22×D4—21no—
110AGL(1,ℤ/8)—44no—
111AGL(1,ℤ/8)—44no—
112C2×SD16—24no—
113C2×D8—24no—
114C2×M16—21no—
115C2×M16—62no—
116C2×M16—62no—
117C22×C8—62no—
118C4⋊Q8#6711yescharacteristic
119#127—22no—
120#127—21no—
121#127C2×Dic312yes—
122#127#6711yescharacteristic
123#124C2×Dic312yes—
124#121—22no—
125C4×D4—22no—
126C2×C42C22×D311yescharacteristic
127C4⋊C8—32no—
128C4⋊C8—32no—
129C4⋊C8—62no—
130C4⋊C8—62no—
131C22⋊C8—64no—
132C22⋊C8—62no—
133#97C2×Dic312yes—
134C4×C8—21no—
135C4×C8—62no—
136C23×C3—22no—
137C23×C3—42no—
138C23×C3—82no—
139C23×C3—84no—
140C23×C3#12311yescharacteristic
141C3×Q8—42no—
142C3×D4—42no—
143C3×D4—44no—
144C3×D4—42no—
145C3×D4—42no—
146C3×D4—84no—
147C3×D4—82no—
148C3×D4—81no—
149C3×D4—81no—
150C3×D4—82no—
151C3×D4—82no—
152C2×C3×C4—21no—
153C2×C3×C4—21no—
154C2×C3×C4—22no—
155C2×C3×C4—22no—
156C2×C3×C4—22no—
157C2×C3×C4—42no—
158C2×C3×C4—42no—
159C2×C3×C4C2×C22⋊C411yescharacteristic
160C2×C3×C4C4×D412yes—
161C2×C3×C4#12011yescharacteristic
162C2×C3×C4#12112yes—
163C3×C8—24no—
164C3⋊C8—42no—
165C3⋊C8—42no—
166C3⋊C8—82no—
167C24—41no—
168#41—42no—
169#41—42no—
170C2×Q8#21712yes—
171C2×D4—22no—
172C2×D4—24no—
173C2×D4—22no—
174C2×D4—22no—
175C2×D4—42no—
176C2×D4—42no—
177C2×D4—41no—
178C2×D4—42no—
179C2×D4—82no—
180C2×D4—82no—
181C2×D4#21712yes—
182C22×C4—22no—
183C22×C4#23712yes—
184C22×C4C2×#6711yescharacteristic
185SD16—48no—
186D8—48no—
187M16—22no—
188M16—22no—
189M16—242no—
190M16—242no—
191C2×C8—22no—
192C2×C8—62no—
193C2×C8—62no—
194C2×C8—122no—
195C2×C8—242no—
196C4⋊C4—22no—
197C22⋊C4—24no—
198C42D3×D412yes—
199C42C22×Dic311yescharacteristic
200C42C2×#6711yescharacteristic
201C22×C3—21no—
202C22×C3—22no—
203C22×C3—42no—
204C22×C3—44no—
205C22×C3—82no—
206C22×C3—84no—
207C22×C3—164no—
208C22×C3#38611yescharacteristic
209C3×C4—21no—
210C3×C4—22no—
211C3×C4—42no—
212C3×C4#45311yescharacteristic
213C3×C4#45311yescharacteristic
214C3×C4#47112yes—
215C3×C4#47212yes—
216C23—22no—
217C23—42no—
218C23—82no—
219C23—84no—
220C23#93511yescharacteristic
221Q8—42no—
222D4—42no—
223D4—44no—
224D4—42no—
225D4—42no—
226D4—84no—
227D4—82no—
228D4—82no—
229D4—82no—
230D4—81no—
231D4—81no—
232C2×C4—22no—
233C2×C4—22no—
234C2×C4—21no—
235C2×C4—21no—
236C2×C4—22no—
237C2×C4—42no—
238C2×C4—42no—
239C2×C4D4×Dic312yes—
240C2×C4#93711yescharacteristic
241C2×C4#93912yes—
242C2×C4C2×#21711yescharacteristic
243C8—24no—
244C8—122no—
245C8—122no—
246C8—242no—
247C2×C3—41no—
248C2×C3—82no—
249C2×C3—162no—
250C2×C3#188711yescharacteristic
251C2×C3#190712yes—
252C22—21no—
253C22—22no—
254C22—42no—
255C22—44no—
256C22—82no—
257C22—84no—
258C22—164no—
259C22#522011yescharacteristic
260C4—21no—
261C4—22no—
262C4—42no—
263C4#507711yescharacteristic
264C4#507711yescharacteristic
265C4#519912yes—
266C4#520012yes—
267C3#943711yescharacteristic, complemented, Sylow
268C2—41no—
269C2—82no—
270C2—162no—
271C2#7176311yescharacteristic
272C2#7209112yes—
subgroupG/H
centerC22#5220
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
Fitting, prime radicalC3×#3151C2
socleC22×C3#386
cosocleC2×C3×C42C23
hypercenter#3151D3
hypocenterC3#9437
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3151

Complements

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

NH ≅ G/Ntype
2. #83278C2semidirect
3. #71810C2semidirect
4. #71759C2semidirect
5. #67669C2semidirect
267. C3#9437semidirect

Maximal subgroups

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

subgroupindex
C3×#31512
#832782
#718102
#717592
#676692
#94373