Finite Group Databasebeta

← #68910

Subgroups — #68910

Order 384.

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

normal subgroups only

This group has 290 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×#888C211yesmaximal, characteristic
2C2×#816C211yesmaximal, characteristic
3#4973C211yesmaximal, characteristic
4C2×#810C211yesmaximal, characteristic
5C3×#340C211yesmaximal, characteristic
6#4550C211yesmaximal, characteristic
7#4529C211yesmaximal, characteristic
8#2024—31nomaximal, Sylow
9C22×Dic6—21no—
10C22×Dic6C2211yescharacteristic
11C22×C3×C8C2211yescharacteristic
12C2×C3×C4⋊C4C2211yescharacteristic
13C2×C3×C4⋊C4C2211yescharacteristic
14C2×C4⋊Dic3C2211yescharacteristic
15C2×Dic3⋊C4—21no—
16C2×C4×Dic3C2211yescharacteristic
17C22×C3⋊C8C2211yescharacteristic
18#888—22no—
19#888—22no—
20C3⋊#95—21no—
21C3⋊#95—21no—
22#816—22no—
23#810—22no—
24C2×C4⋊Q8—31no—
25C2×#103—31no—
26C2×#103—31no—
27#391—31no—
28#340—31no—
29#340D311yescharacteristic
30#336—31no—
31C22×C3×C4—21no—
32C22×C3×C4—21no—
33C22×C3×C4C2311yescharacteristic
34C22×Dic3—21no—
35C22×Dic3—21no—
36C22×Dic3D411yescharacteristic
37C22×Dic3D411yescharacteristic
38C2×Dic6—21no—
39C2×Dic6—21no—
40C2×Dic6—21no—
41C2×Dic6—21no—
42C2×Dic6—24no—
43C2×Dic6—21no—
44C2×Dic6—42no—
45C2×Dic6—42no—
46C2×Dic6Q812yescomplemented
47C2×C3×C8—21no—
48C2×C3×C8—21no—
49C2×C3×C8—21no—
50C3×C4⋊C4—21no—
51C3×C4⋊C4—21no—
52C3×C4⋊C4—21no—
53C3×C4⋊C4D412yes—
54C4⋊Dic3—21no—
55C4⋊Dic3D412yes—
56Dic3⋊C4—24no—
57C4×Dic3—21no—
58C4×Dic3—21no—
59C2×C3⋊C8—21no—
60C2×C3⋊C8—21no—
61C2×C3⋊C8—21no—
62C22×Q8—31no—
63C22×Q8—61no—
64C22×C8—31no—
65C22×C8C2×D311yescharacteristic
66C4⋊Q8—62no—
67C4⋊Q8—62no—
68C2×C4⋊C4—31no—
69C2×C4⋊C4—61no—
70C2×C4⋊C4C2×D311yescharacteristic
71C2×C4⋊C4C2×D311yescharacteristic
72C2×C42—31no—
73#103—62no—
74#103—62no—
75#95—61no—
76#95—61no—
77C23×C3C2×D411yescharacteristic
78C2×C3×C4—22no—
79C2×C3×C4—41no—
80C2×C3×C4—41no—
81C2×C3×C4—41no—
82C2×C3×C4—41no—
83C2×C3×C4—41no—
84C2×C3×C4C2×D411yescharacteristic
85C2×C3×C4C2×D411yescharacteristic
86C2×C3×C4C2×Q811yescharacteristic
87C2×C3×C4#4111yescharacteristic
88C2×C3×C4#4111yescharacteristic
89C2×C3×C4#4111yescharacteristic
90C2×Dic3—21no—
91C2×Dic3—21no—
92C2×Dic3—21no—
93C2×Dic3—21no—
94C2×Dic3—22no—
95C2×Dic3—22no—
96C2×Dic3—41no—
97C2×Dic3—41no—
98C2×Dic3—41no—
99C2×Dic3—41no—
100C2×Dic3SD1612yes—
101C2×Dic3Q1612yes—
102Dic6—22no—
103Dic6—22no—
104Dic6—48no—
105Dic6—41no—
106Dic6—41no—
107C3×C8—41no—
108C3⋊C8—41no—
109C2×Q8—32no—
110C2×Q8—64no—
111C2×Q8—61no—
112C2×Q8—61no—
113C2×Q8—61no—
114C2×Q8—61no—
115C2×Q8—61no—
116C2×Q8—122no—
117C2×Q8—122no—
118C22×C4—21no—
119C22×C4—21no—
120C22×C4—31no—
121C22×C4—31no—
122C22×C4—61no—
123C22×C4—61no—
124C22×C4C22×D311yescharacteristic
125C2×C8—21no—
126C2×C8—21no—
127C2×C8—21no—
128C2×C8—61no—
129C2×C8—61no—
130C2×C8—61no—
131C4⋊C4—21no—
132C4⋊C4—21no—
133C4⋊C4—21no—
134C4⋊C4—32no—
135C4⋊C4—64no—
136C4⋊C4—61no—
137C4⋊C4D1212yes—
138C42—61no—
139C42—61no—
140C22×C3#12011yescharacteristic
141C22×C3C22⋊Q811yescharacteristic
142C22×C3#12411yescharacteristic
143C22×C3C2×SD1611yescharacteristic
144C22×C3C2×Q1611yescharacteristic
145C22×C3AGL(1,ℤ/8)11yescharacteristic
146C22×C3#13711yescharacteristic
147C3×C4—42no—
148C3×C4—81no—
149C3×C4C22⋊Q811yescharacteristic
150C3×C4C22⋊Q811yescharacteristic
151C3×C4#12411yescharacteristic
152C3×C4#12411yescharacteristic
153Dic3—22no—
154Dic3—22no—
155Dic3—42no—
156Dic3—42no—
157C23D3×D411yescharacteristic
158Q8—62no—
159Q8—62no—
160Q8—128no—
161Q8—121no—
162Q8—121no—
163C2×C4—22no—
164C2×C4—32no—
165C2×C4—32no—
166C2×C4—41no—
167C2×C4—41no—
168C2×C4—41no—
169C2×C4—41no—
170C2×C4—41no—
171C2×C4—61no—
172C2×C4—61no—
173C2×C4—61no—
174C2×C4—61no—
175C2×C4—62no—
176C2×C4—62no—
177C2×C4—121no—
178C2×C4—121no—
179C2×C4—121no—
180C2×C4—121no—
181C2×C4C2×D1211yescharacteristic
182C2×C4#23511yescharacteristic
183C2×C4#23511yescharacteristic
184C2×C4D3×D411yescharacteristic
185C2×C4#23711yescharacteristic
186C2×C4D3×Q811yescharacteristic
187C8—41no—
188C8—121no—
189C2×C3#39311yescharacteristic
190C2×C3#45011yescharacteristic
191C2×C3#45111yescharacteristic
192C2×C3#47511yescharacteristic
193C2×C3#47711yescharacteristic
194C2×C3#48711yescharacteristic
195C2×C3#48911yescharacteristic
196C22#88211yescharacteristic
197C22#88511yescharacteristic
198C22#89611yescharacteristic
199C22#91111yescharacteristic
200C22D3×SD1611yescharacteristic
201C22#91511yescharacteristic
202C22D3×Q1611yescharacteristic
203C4—42no—
204C4—62no—
205C4—62no—
206C4—81no—
207C4—122no—
208C4—122no—
209C4#87511yescharacteristic
210C4#88511yescharacteristic
211C4#89611yescharacteristic
212C4#89711yescharacteristic
213C3#202411yescharacteristic, complemented, Sylow
214C2#467311yescharacteristic
215C2#476511yescharacteristic
216C2#477611yescharacteristic
217C2#479211yescharacteristic
218C2#480911yescharacteristic
219C2#485411yescharacteristic
220C2#487511yescharacteristic
subgroupG/H
centerC23D3×D4
commutator, cosocleC22×C3×C4C23
FrattiniC22×C4C22×D3
FittingC3×#340C2
socle, prime radicalC23×C3C2×D4
hypercenter#340D3
hypocenterC3#2024
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C22×C4 ◁ #340

Complements

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

NH ≅ G/Ntype
46. C2×Dic6Q8semidirect
213. C3#2024semidirect

Maximal subgroups

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

subgroupindex
C2×#8882
C2×#8162
#49732
C2×#8102
C3×#3402
#45502
#45292
#20243