Finite Group Databasebeta

← #69527

Subgroups — #69527

Order 384.

316 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 316 conjugacy classes of subgroups — too many to draw. Only its 87 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
1C2×#924C211yesmaximal, characteristic
2C3×#380C211yesmaximal, characteristic
3C2×#830C211yesmaximal, characteristic
4C2×#810C211yesmaximal, characteristic, complemented
5C2×#810C211yesmaximal, characteristic, complemented
6C2×#807C211yesmaximal, characteristic, complemented
7#4529C211yesmaximal, characteristic, complemented
8#2036—31nomaximal, Sylow
9C23×C3×C4C2211yescharacteristic
10C22×Dic6C2211yescharacteristic
11C2×C3×C4⋊C4C2211yescharacteristic
12C2×C3×C4⋊C4C2211yescharacteristic
13C2×C3×C22⋊C4—21no—
14C2×#217—21no—
15C2×C4⋊Dic3C2211yescharacteristic
16#924—22no—
17#924—22no—
18C2×Dic3⋊C4—21no—
19C22×C3⋊C8C2211yescharacteristic
20C22×C3⋊C8C2211yescharacteristic
21C3×#95—21no—
22#830—22no—
23#810—22no—
24#810—22no—
25#807—22no—
26C2×C22⋊Q8—31no—
27C2×#107—31no—
28C2×#103—31no—
29C2×#103—31no—
30C2×C22⋊C8—31no—
31#380D311yescharacteristic
32#340—31no—
33C24×C3D411yescharacteristic
34C22×C3×C4—21no—
35C22×C3×C4—21no—
36C22×C3×C4—21no—
37C22×C3×C4—21no—
38C22×C3×C4—21no—
39C22×C3×C4—21no—
40C22×C3×C4—21no—
41C22×C3×C4—21no—
42C22×C3×C4D411yescharacteristic
43C22×C3×C4C2311yescharacteristic
44C22×Dic3—21no—
45C22×Dic3—21no—
46C2×Dic6—21no—
47C2×Dic6—21no—
48C2×Dic6—21no—
49C2×Dic6—21no—
50C2×Dic6—21no—
51C2×Dic6D412yes—
52C3×C4⋊C4—21no—
53C3×C4⋊C4—21no—
54C3×C4⋊C4—21no—
55C3×C4⋊C4—21no—
56C3×C22⋊C4—42no—
57#217—24no—
58C4⋊Dic3—21no—
59C4⋊Dic3D412yes—
60Dic3⋊C4—24no—
61C2×C3⋊C8—21no—
62C2×C3⋊C8—21no—
63C2×C3⋊C8—21no—
64C2×C3⋊C8—21no—
65C2×C3⋊C8—21no—
66C2×C3⋊C8D412yes—
67C22×Q8—31no—
68C23×C4C2×D311yescharacteristic
69C22×C8—31no—
70C22×C8—31no—
71C22⋊Q8—62no—
72C22⋊Q8—62no—
73C2×C4⋊C4—31no—
74C2×C4⋊C4—61no—
75C2×C4⋊C4C2×D311yescharacteristic
76C2×C4⋊C4C2×D311yescharacteristic
77C2×C22⋊C4—21no—
78C2×C22⋊C4—61no—
79#107—62no—
80#103—62no—
81#103—62no—
82C22⋊C8—62no—
83#95—21no—
84C23×C3—21no—
85C23×C3—21no—
86C23×C3—21no—
87C23×C3—21no—
88C23×C3—21no—
89C23×C3—21no—
90C23×C3Q1612yes—
91C23×C3C2×D411yescharacteristic
92C2×C3×C4—21no—
93C2×C3×C4—21no—
94C2×C3×C4—21no—
95C2×C3×C4—41no—
96C2×C3×C4—41no—
97C2×C3×C4—41no—
98C2×C3×C4—41no—
99C2×C3×C4—41no—
100C2×C3×C4—41no—
101C2×C3×C4—41no—
102C2×C3×C4—41no—
103C2×C3×C4—41no—
104C2×C3×C4—41no—
105C2×C3×C4C2×D411yescharacteristic
106C2×C3×C4C2×D411yescharacteristic
107C2×C3×C4C2×D411yescharacteristic
108C2×C3×C4#4111yescharacteristic
109C2×C3×C4#4111yescharacteristic
110C2×C3×C4#4111yescharacteristic
111C2×Dic3—22no—
112C2×Dic3—22no—
113C2×Dic3—41no—
114C2×Dic3—41no—
115C2×Dic3—41no—
116C2×Dic3—41no—
117Dic6—22no—
118Dic6—22no—
119Dic6—41no—
120Dic6—41no—
121C3⋊C8—22no—
122C3⋊C8—41no—
123C24#6711yescharacteristic
124C2×Q8—32no—
125C2×Q8—61no—
126C2×Q8—61no—
127C2×Q8—61no—
128C2×Q8—61no—
129C2×Q8—61no—
130C22×C4—21no—
131C22×C4—21no—
132C22×C4—21no—
133C22×C4—21no—
134C22×C4—21no—
135C22×C4—21no—
136C22×C4—21no—
137C22×C4—21no—
138C22×C4—61no—
139C22×C4—61no—
140C22×C4#6711yescharacteristic
141C22×C4C22×D311yescharacteristic
142C2×C8—32no—
143C2×C8—61no—
144C2×C8—61no—
145C2×C8—61no—
146C2×C8—61no—
147C2×C8—61no—
148C4⋊C4—21no—
149C4⋊C4—21no—
150C4⋊C4—21no—
151C4⋊C4—21no—
152C4⋊C4—32no—
153C4⋊C4—64no—
154C4⋊C4—61no—
155C22⋊C4—42no—
156C22⋊C4—64no—
157C22×C3—22no—
158C22×C3—22no—
159C22×C3—22no—
160C22×C3—41no—
161C22×C3—41no—
162C22×C3—41no—
163C22×C3—41no—
164C22×C3#12011yescharacteristic
165C22×C3#12111yescharacteristic
166C22×C3#12311yescharacteristic
167C22×C3C2×Q1611yescharacteristic
168C22×C3#13511yescharacteristic
169C22×C3AGL(1,ℤ/8)11yescharacteristic
170C22×C3#13711yescharacteristic
171C3×C4—41no—
172C3×C4—81no—
173C3×C4—81no—
174C3×C4#12111yescharacteristic
175C3×C4#12111yescharacteristic
176C3×C4#12311yescharacteristic
177C3×C4#12411yescharacteristic
178Dic3—42no—
179Dic3—42no—
180C23—21no—
181C23—21no—
182C23—21no—
183C23—21no—
184C23—21no—
185C23—21no—
186C23#21612yes—
187C23C2×#6711yescharacteristic
188Q8—62no—
189Q8—62no—
190Q8—121no—
191Q8—121no—
192C2×C4—21no—
193C2×C4—21no—
194C2×C4—21no—
195C2×C4—41no—
196C2×C4—41no—
197C2×C4—41no—
198C2×C4—41no—
199C2×C4—41no—
200C2×C4—41no—
201C2×C4—41no—
202C2×C4—41no—
203C2×C4—41no—
204C2×C4—41no—
205C2×C4—62no—
206C2×C4—62no—
207C2×C4—121no—
208C2×C4—121no—
209C2×C4—121no—
210C2×C4—121no—
211C2×C4#23511yescharacteristic
212C2×C4#23511yescharacteristic
213C2×C4D3×D411yescharacteristic
214C2×C4D3×D411yescharacteristic
215C2×C4D3×D411yescharacteristic
216C2×C4#23711yescharacteristic
217C8—62no—
218C8—121no—
219C2×C3—42no—
220C2×C3#39411yescharacteristic
221C2×C3#44911yescharacteristic
222C2×C3#45111yescharacteristic
223C2×C3#46711yescharacteristic
224C2×C3#46811yescharacteristic
225C2×C3#48411yescharacteristic
226C2×C3#48511yescharacteristic
227C22—22no—
228C22—22no—
229C22—22no—
230C22—41no—
231C22—41no—
232C22—41no—
233C22—41no—
234C22#92911yescharacteristic
235C22#93211yescharacteristic
236C22#93711yescharacteristic
237C22#93911yescharacteristic
238C22C2×#21611yescharacteristic
239C22#95011yescharacteristic
240C22#95111yescharacteristic
241C4—41no—
242C4—81no—
243C4—81no—
244C4—122no—
245C4—122no—
246C4#88311yescharacteristic
247C4#88411yescharacteristic
248C4#88411yescharacteristic
249C4#88511yescharacteristic
250C3#203611yescharacteristic, complemented, Sylow
251C2—42no—
252C2#495911yescharacteristic
253C2#497011yescharacteristic
254C2#499611yescharacteristic
255C2#503711yescharacteristic
256C2#504211yescharacteristic
257C2#505011yescharacteristic
258C2#505111yescharacteristic
subgroupG/H
centerC23C2×#67
commutator, cosocleC22×C3×C4C23
FrattiniC22×C4C22×D3
FittingC3×#380C2
socleC23×C3C2×D4
prime radicalC24×C3D4
hypercenter#380D3
hypocenterC3#2036
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ #380

Complements

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

NH ≅ G/Ntype
4. C2×#810C2semidirect
5. C2×#810C2semidirect
6. C2×#807C2semidirect
7. #4529C2semidirect
250. C3#2036semidirect

Maximal subgroups

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

subgroupindex
C2×#9242
C3×#3802
C2×#8302
C2×#8102
C2×#8072
#45292
#20363