Finite Group Databasebeta

← #70239

Subgroups — #70239

Order 384.

338 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 338 conjugacy classes of subgroups — too many to draw. Only its 107 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×Q8×Dic3C211yesmaximal, characteristic
2C22×#216C211yesmaximal, characteristic
3C2×C3×#103C211yesmaximal, characteristic
4C2×#816C211yesmaximal, characteristic
5#4973C211yesmaximal, characteristic
6#4550C211yesmaximal, characteristic
7#4529C211yesmaximal, characteristic
8#1878—31nomaximal, Sylow
9C22×C3×Q8C2211yescharacteristic
10C22×Dic6C2211yescharacteristic
11C22×C3×C8C2211yescharacteristic
12C2×C3×C4⋊C4C2211yescharacteristic
13Q8×Dic3—22no—
14Q8×Dic3—22no—
15C2×#216—22no—
16C2×#216—22no—
17C2×#216C414yescomplemented
18C2×C4⋊Dic3—21no—
19C2×C4⋊Dic3C2211yescharacteristic
20C2×C4×Dic3—21no—
21C2×C4×Dic3C2211yescharacteristic
22C22×C3⋊C8C2211yescharacteristic
23C3×#103—22no—
24C3⋊#95—21no—
25C3⋊#95—21no—
26#816—22no—
27C22×Q16—31no—
28C2×C4×Q8—31no—
29C2×#103—31no—
30C2×#103D311yescharacteristic
31#391—31no—
32#340—31no—
33#336—31no—
34C2×C3×Q8—21no—
35C2×C3×Q8—21no—
36C2×C3×Q8—21no—
37C2×C3×Q8C2×C412yes—
38C2×C3×Q8D412yes—
39C2×C3×Q8D412yes—
40C22×C3×C4—21no—
41C22×C3×C4—21no—
42C22×C3×C4C2311yescharacteristic
43C22×Dic3—21no—
44C22×Dic3—21no—
45C22×Dic3D411yescharacteristic
46C22×Dic3D411yescharacteristic
47C2×Dic6—21no—
48C2×Dic6—21no—
49C2×Dic6—21no—
50C2×Dic6—21no—
51C2×Dic6—21no—
52C2×Dic6C2×C412yes—
53C2×C3×C8—21no—
54C2×C3×C8—21no—
55C2×C3×C8—21no—
56C3×C4⋊C4—21no—
57C3×C4⋊C4—21no—
58#216—28no—
59C4⋊Dic3—24no—
60C4⋊Dic3—21no—
61C4⋊Dic3D412yes—
62C4×Dic3—24no—
63C4×Dic3—21no—
64C4×Dic3—21no—
65C2×C3⋊C8—21no—
66C2×C3⋊C8—21no—
67C2×C3⋊C8C2×C412yes—
68C22×Q8—31no—
69C22×Q8C2×D311yescharacteristic
70C2×Q16—34no—
71C2×Q16—62no—
72C2×Q16—62no—
73C22×C8—31no—
74C22×C8C2×D311yescharacteristic
75C4×Q8—62no—
76C4×Q8—62no—
77C2×C4⋊C4—31no—
78C2×C4⋊C4—61no—
79C2×C4⋊C4C2×D311yescharacteristic
80C2×C42—31no—
81C2×C42—61no—
82#103—22no—
83#103—62no—
84#95—61no—
85#95—61no—
86C23×C3C2×D411yescharacteristic
87C3×Q8—22no—
88C3×Q8—22no—
89C3×Q8—22no—
90C3×Q8C22⋊C414yes—
91C2×C3×C4—22no—
92C2×C3×C4—22no—
93C2×C3×C4—22no—
94C2×C3×C4—41no—
95C2×C3×C4—41no—
96C2×C3×C4—41no—
97C2×C3×C4C22×C411yescharacteristic
98C2×C3×C4C2×D411yescharacteristic
99C2×C3×C4C2×D411yescharacteristic
100C2×C3×C4C2×D411yescharacteristic
101C2×C3×C4#4111yescharacteristic
102C2×C3×C4#4111yescharacteristic
103C2×Dic3—21no—
104C2×Dic3—21no—
105C2×Dic3—22no—
106C2×Dic3—21no—
107C2×Dic3—22no—
108C2×Dic3—21no—
109C2×Dic3—21no—
110C2×Dic3—41no—
111C2×Dic3—41no—
112C2×Dic3—41no—
113C2×Dic3—41no—
114C2×Dic3Q1612yescomplemented
115Dic6—22no—
116Dic6—22no—
117Dic6—41no—
118Dic6—41no—
119C3×C8—41no—
120C3⋊C8—22no—
121C2×Q8—21no—
122C2×Q8—21no—
123C2×Q8—21no—
124C2×Q8—32no—
125C2×Q8—61no—
126C2×Q8—61no—
127C2×Q8—61no—
128C2×Q8—61no—
129C2×Q8—61no—
130C2×Q8C4×D312yes—
131C2×Q8D1212yes—
132C2×Q8#6712yes—
133C22×C4—21no—
134C22×C4—21no—
135C22×C4—31no—
136C22×C4—31no—
137C22×C4—61no—
138C22×C4—61no—
139C22×C4C22×D311yescharacteristic
140Q16—68no—
141C2×C8—21no—
142C2×C8—21no—
143C2×C8—21no—
144C2×C8—32no—
145C2×C8—61no—
146C2×C8—61no—
147C4⋊C4—21no—
148C4⋊C4—21no—
149C4⋊C4—32no—
150C4⋊C4—64no—
151C4⋊C4—61no—
152C42—64no—
153C42—61no—
154C42—61no—
155C22×C3C4×D411yescharacteristic
156C22×C3#12011yescharacteristic
157C22×C3#12111yescharacteristic
158C22×C3C2×Q1611yescharacteristic
159C22×C3#13511yescharacteristic
160C22×C3#13711yescharacteristic
161C22×C3#13711yescharacteristic
162C3×C4—24no—
163C3×C4—81no—
164C3×C4C2×C22⋊C411yescharacteristic
165C3×C4C4×D411yescharacteristic
166C3×C4#12111yescharacteristic
167C3×C4#12311yescharacteristic
168Dic3—22no—
169Dic3—42no—
170Dic3—41no—
171Dic3—42no—
172C23D3×D411yescharacteristic
173Q8—22no—
174Q8—22no—
175Q8—22no—
176Q8—62no—
177Q8—62no—
178Q8—121no—
179Q8—121no—
180Q8(C2×D3)⋊C414yes—
181C2×C4—22no—
182C2×C4—22no—
183C2×C4—22no—
184C2×C4—32no—
185C2×C4—41no—
186C2×C4—41no—
187C2×C4—41no—
188C2×C4—61no—
189C2×C4—61no—
190C2×C4—62no—
191C2×C4—62no—
192C2×C4—61no—
193C2×C4—61no—
194C2×C4—61no—
195C2×C4—121no—
196C2×C4—121no—
197C2×C4—121no—
198C2×C4—121no—
199C2×C4C2×C4×D311yescharacteristic
200C2×C4C2×D1211yescharacteristic
201C2×C4D3×D411yescharacteristic
202C2×C4#23711yescharacteristic
203C2×C4#23711yescharacteristic
204C2×C4C2×#6711yescharacteristic
205C8—41no—
206C8—62no—
207C2×C3#38611yescharacteristic
208C2×C3C4×Q1611yescharacteristic
209C2×C3#44111yescharacteristic
210C2×C3#45111yescharacteristic
211C2×C3#45211yescharacteristic
212C2×C3#46211yescharacteristic
213C2×C3#46411yescharacteristic
214C22#67⋊C411yescharacteristic
215C22#88211yescharacteristic
216C22#91511yescharacteristic
217C22#91611yescharacteristic
218C22D3×Q1611yescharacteristic
219C22#91811yescharacteristic
220C22#93911yescharacteristic
221C4—24no—
222C4—62no—
223C4—81no—
224C4—122no—
225C4—121no—
226C4—122no—
227C4#67⋊C411yescharacteristic
228C4#88611yescharacteristic
229C4C2×(C2×D3)⋊C411yescharacteristic
230C4#93911yescharacteristic
231C3#187811yescharacteristic, complemented, Sylow
232C2#478911yescharacteristic
233C2#479011yescharacteristic
234C2#480811yescharacteristic
235C2#480911yescharacteristic
236C2#495811yescharacteristic
237C2#516611yescharacteristic
238C2#518211yescharacteristic
subgroupG/H
centerC23D3×D4
commutatorC2×C3×C4C22×C4
FrattiniC22×C4C22×D3
FittingC2×C3×#103C2
socle, prime radicalC23×C3C2×D4
cosocleC22×C3×C4C23
hypercenterC2×#103D3
hypocenterC3#1878
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C22×C4 ◁ C2×#103

Complements

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

NH ≅ G/Ntype
17. C2×#216C4semidirect
114. C2×Dic3Q16semidirect
231. C3#1878semidirect

Maximal subgroups

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

subgroupindex
C2×Q8×Dic32
C22×#2162
C2×C3×#1032
C2×#8162
#49732
#45502
#45292
#18783