Finite Group Databasebeta

← #69649

Subgroups — #69649

Order 384.

340 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 340 conjugacy classes of subgroups — too many to draw. Only its 99 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
1C22×#214C211yesmaximal, characteristic
2C3×#390C211yesmaximal, characteristic
3C2×#832C211yesmaximal, characteristic
4C2×#810C211yesmaximal, characteristic, complemented
5C2×#807C211yesmaximal, characteristic, complemented
6#4929C211yesmaximal, characteristic, complemented
7C2×#803C211yesmaximal, characteristic, complemented
8#1973—31nomaximal, Sylow
9C22×C3×D4C2211yescharacteristic
10C22×Dic6C2211yescharacteristic
11C2×C3×C4⋊C4C2211yescharacteristic
12C2×C3×C22⋊C4—22no—
13C2×C3×C42C2211yescharacteristic
14C2×#214—22no—
15C2×#214—22no—
16C2×#214C414yescomplemented
17C2×C4⋊Dic3C2211yescharacteristic
18C22×C3⋊C8C2211yescharacteristic
19C22×C3⋊C8C2211yescharacteristic
20#832—22no—
21C3⋊#95—22no—
22#810—22no—
23#807—22no—
24#803—22no—
25C22×SD16—31no—
26C2×#107—31no—
27C2×#103—31no—
28C2×#102—31no—
29C2×#97—31no—
30#391—31no—
31#390D311yescharacteristic
32C24×C3—21no—
33C2×C3×D4—21no—
34C2×C3×D4—21no—
35C2×C3×D4—21no—
36C2×C3×D4—21no—
37C2×C3×D4—21no—
38C2×C3×D4C2×C412yes—
39C22×C3×C4—21no—
40C22×C3×C4D412yes—
41C22×C3×C4C2311yescharacteristic
42C22×Dic3—21no—
43C22×Dic3—21no—
44C2×Dic6—21no—
45C2×Dic6—21no—
46C2×Dic6—21no—
47C2×Dic6—21no—
48C2×Dic6—21no—
49C2×Dic6C2×C412yes—
50C3×C4⋊C4—21no—
51C3×C4⋊C4—21no—
52C3×C22⋊C4—44no—
53C3×C42—21no—
54C3×C42—21no—
55#214—28no—
56C4⋊Dic3—21no—
57C4⋊Dic3—21no—
58C2×C3⋊C8—21no—
59C2×C3⋊C8—21no—
60C2×C3⋊C8C2×C412yes—
61C2×C3⋊C8D412yes—
62C2×C3⋊C8D412yes—
63C2×C3⋊C8D412yes—
64C22×Q8—31no—
65C22×D4C2×D311yescharacteristic
66C2×SD16—34no—
67C2×SD16—62no—
68C2×SD16—62no—
69C22×C8—31no—
70C22×C8—31no—
71C2×C4⋊C4—31no—
72C2×C4⋊C4C2×D311yescharacteristic
73C2×C22⋊C4—22no—
74C2×C42C2×D311yescharacteristic
75#107—62no—
76#103—62no—
77#102—62no—
78#97—62no—
79#95—62no—
80C23×C3—22no—
81C23×C3—41no—
82C23×C3—41no—
83C23×C3—42no—
84C23×C3—42no—
85C23×C3C2×D411yescharacteristic
86C3×D4—22no—
87C3×D4—22no—
88C3×D4—41no—
89C3×D4—41no—
90C2×C3×C4—22no—
91C2×C3×C4—22no—
92C2×C3×C4—22no—
93C2×C3×C4—41no—
94C2×C3×C4—42no—
95C2×C3×C4C22×C411yescharacteristic
96C2×C3×C4C2×D411yescharacteristic
97C2×C3×C4C2×D411yescharacteristic
98C2×C3×C4C2×D411yescharacteristic
99C2×C3×C4#4111yescharacteristic
100C2×C3×C4#4111yescharacteristic
101C2×Dic3—22no—
102C2×Dic3—41no—
103C2×Dic3—41no—
104C2×Dic3—41no—
105C2×Dic3—42no—
106Dic6—22no—
107Dic6—22no—
108Dic6—41no—
109Dic6—41no—
110C3⋊C8—22no—
111C3⋊C8C22⋊C414yescomplemented
112C24—21no—
113C2×Q8—32no—
114C2×Q8—61no—
115C2×Q8—61no—
116C2×Q8—61no—
117C2×Q8—61no—
118C2×Q8—61no—
119C2×D4—21no—
120C2×D4—21no—
121C2×D4—21no—
122C2×D4—21no—
123C2×D4—21no—
124C2×D4C4×D312yes—
125C22×C4—21no—
126C22×C4—61no—
127C22×C4—61no—
128C22×C4#6712yes—
129C22×C4C22×D311yescharacteristic
130SD16—68no—
131C2×C8—32no—
132C2×C8—32no—
133C2×C8—32no—
134C2×C8—32no—
135C2×C8—61no—
136C2×C8—61no—
137C4⋊C4—21no—
138C4⋊C4—21no—
139C4⋊C4—61no—
140C4⋊C4—61no—
141C22⋊C4—44no—
142C42—21no—
143C42—21no—
144C22×C3—42no—
145C22×C3—44no—
146C22×C3—82no—
147C22×C3—81no—
148C22×C3—81no—
149C22×C3C4×D411yescharacteristic
150C22×C3#12111yescharacteristic
151C22×C3#12711yescharacteristic
152C22×C3AGL(1,ℤ/8)12yes—
153C22×C3#13712yes—
154C3×C4—42no—
155C3×C4—81no—
156C3×C4C2×C22⋊C411yescharacteristic
157C3×C4C4×D411yescharacteristic
158C3×C4#12111yescharacteristic
159C3×C4#12411yescharacteristic
160Dic3—42no—
161Dic3—81no—
162C23—22no—
163C23—42no—
164C23—41no—
165C23—41no—
166C23—42no—
167C23C2×#6711yescharacteristic
168Q8—62no—
169Q8—62no—
170Q8—121no—
171Q8—121no—
172D4—22no—
173D4—22no—
174D4—41no—
175D4—41no—
176C2×C4—22no—
177C2×C4—22no—
178C2×C4—22no—
179C2×C4—42no—
180C2×C4—41no—
181C2×C4—62no—
182C2×C4—121no—
183C2×C4—121no—
184C2×C4—122no—
185C2×C4—121no—
186C2×C4C2×C4×D311yescharacteristic
187C2×C4#23511yescharacteristic
188C2×C4D3×D411yescharacteristic
189C2×C4D3×D411yescharacteristic
190C2×C4D3×D411yescharacteristic
191C2×C4#23711yescharacteristic
192C8—34no—
193C8—62no—
194C2×C3—82no—
195C2×C3#39011yescharacteristic
196C2×C3#44012yes—
197C2×C3#46812yes—
198C2×C3#49611yescharacteristic
199C2×C3#49711yescharacteristic
200C22—42no—
201C22—44no—
202C22—82no—
203C22—81no—
204C22—81no—
205C22C4×#6711yescharacteristic
206C22#93212yes—
207C22#93911yescharacteristic
208C22#94011yescharacteristic
209C22#95112yes—
210C4—42no—
211C4—81no—
212C4—122no—
213C4—241no—
214C4D3×C22⋊C411yescharacteristic
215C4#67⋊C411yescharacteristic
216C4#88411yescharacteristic
217C4#88511yescharacteristic
218C3#197311yescharacteristic, complemented, Sylow
219C2—82no—
220C2#495711yescharacteristic
221C2#501812yes—
222C2#504212yes—
223C2#506011yescharacteristic
224C2#507411yescharacteristic
subgroupG/H
centerC23C2×#67
commutatorC2×C3×C4C22×C4
FrattiniC22×C4C22×D3
FittingC3×#390C2
socleC23×C3C2×D4
prime radicalC22×C3×D4C22
cosocleC22×C3×C4C23
hypercenter#390D3
hypocenterC3#1973
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #390

Complements

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

NH ≅ G/Ntype
4. C2×#810C2semidirect
5. C2×#807C2semidirect
6. #4929C2semidirect
7. C2×#803C2semidirect
16. C2×#214C4semidirect
111. C3⋊C8C22⋊C4semidirect
218. C3#1973semidirect

Maximal subgroups

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

subgroupindex
C22×#2142
C3×#3902
C2×#8322
C2×#8102
C2×#8072
#49292
C2×#8032
#19733