Finite Group Databasebeta

← #68909

Subgroups — #68909

Order 384.

374 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 374 conjugacy classes of subgroups — too many to draw. Only its 101 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×#204C211yesmaximal, characteristic
2C2×#887C211yesmaximal, characteristic, complemented
3#4988C211yesmaximal, characteristic
4C2×#810C211yesmaximal, characteristic, complemented
5C2×#809C211yesmaximal, characteristic
6C3×#340C211yesmaximal, characteristic, complemented
7#4550C211yesmaximal, characteristic, complemented
8#1880—31nomaximal, Sylow
9C22×D12C2211yescharacteristic
10C22×Dic6C2211yescharacteristic
11C22×C3×C8C2211yescharacteristic
12C2×C3×C4⋊C4C2211yescharacteristic
13C2×C3×C4⋊C4C2211yescharacteristic
14C2×(C2×D3)⋊C4—21no—
15C2×(C2×D3)⋊C4—21no—
16C2×Dic3⋊C4—21no—
17C2×C4×Dic3—21no—
18C2×C4×Dic3C2211yescharacteristic
19C22×C3⋊C8C2211yescharacteristic
20C2×#204—22no—
21C2×#204—22no—
22C2×#204C414yescomplemented
23#887—22no—
24#887—22no—
25#810—22no—
26#809—22no—
27C22×SD16—31no—
28C2×C4×Q8—31no—
29C2×#103—31no—
30C2×#102—31no—
31#390—31no—
32#340D311yescharacteristic, complemented
33#336—31no—
34C23×D3—21no—
35C22×C3×C4—21no—
36C22×C3×C4—21no—
37C22×C3×C4C2311yescharacteristic
38C22×Dic3—21no—
39C22×Dic3D411yescharacteristic
40C22×Dic3D411yescharacteristic
41C2×D12—21no—
42C2×D12—21no—
43C2×D12—21no—
44C2×D12—21no—
45C2×D12—21no—
46C2×D12C2×C412yes—
47C2×Dic6—21no—
48C2×Dic6—21no—
49C2×Dic6—21no—
50C2×Dic6C2×C412yes—
51C2×Dic6D412yes—
52C2×Dic6D412yes—
53C2×C3×C8—21no—
54C2×C3×C8—21no—
55C2×C3×C8C2×C412yes—
56C3×C4⋊C4—21no—
57C3×C4⋊C4—21no—
58C3×C4⋊C4—21no—
59C3×C4⋊C4D412yes—
60(C2×D3)⋊C4—42no—
61(C2×D3)⋊C4—42no—
62Dic3⋊C4—24no—
63C4×Dic3—24no—
64C4×Dic3—21no—
65C4×Dic3—21no—
66C2×C3⋊C8—21no—
67C2×C3⋊C8—21no—
68C2×C3⋊C8—21no—
69#204—28no—
70C22×Q8—31no—
71C22×D4—31no—
72C2×SD16—34no—
73C2×SD16—62no—
74C2×SD16—62no—
75C22×C8—31no—
76C22×C8C2×D311yescharacteristic
77C4×Q8—62no—
78C4×Q8—62no—
79C2×C4⋊C4—61no—
80C2×C4⋊C4C2×D311yescharacteristic
81C2×C4⋊C4C2×D311yescharacteristic
82C2×C22⋊C4—61no—
83C2×C22⋊C4—61no—
84C2×C42—31no—
85C2×C42—61no—
86#103—62no—
87#102—62no—
88C23×C3C2×D411yescharacteristic
89C22×D3—22no—
90C22×D3—41no—
91C22×D3—41no—
92C22×D3—41no—
93C22×D3—41no—
94C22×D3—41no—
95C22×D3—41no—
96C2×C3×C4—22no—
97C2×C3×C4—41no—
98C2×C3×C4—41no—
99C2×C3×C4—41no—
100C2×C3×C4—41no—
101C2×C3×C4—41no—
102C2×C3×C4C22×C411yescharacteristic
103C2×C3×C4C2×D411yescharacteristic
104C2×C3×C4C2×D411yescharacteristic
105C2×C3×C4C2×D411yescharacteristic
106C2×C3×C4#4111yescharacteristic
107C2×C3×C4#4111yescharacteristic
108C2×Dic3—22no—
109C2×Dic3—21no—
110C2×Dic3—22no—
111C2×Dic3—22no—
112C2×Dic3—21no—
113C2×Dic3—21no—
114C2×Dic3—21no—
115C2×Dic3—21no—
116C2×Dic3SD1612yescomplemented
117D12—22no—
118D12—22no—
119D12—41no—
120D12—41no—
121Dic6—22no—
122Dic6—22no—
123Dic6—22no—
124Dic6C22⋊C414yescomplemented
125C3×C8—22no—
126C3⋊C8—41no—
127C24—61no—
128C2×Q8—32no—
129C2×Q8—32no—
130C2×Q8—32no—
131C2×Q8—61no—
132C2×Q8—61no—
133C2×Q8—61no—
134C2×D4—32no—
135C2×D4—61no—
136C2×D4—61no—
137C2×D4—61no—
138C2×D4—61no—
139C2×D4—61no—
140C22×C4—21no—
141C22×C4—21no—
142C22×C4—31no—
143C22×C4—31no—
144C22×C4—61no—
145C22×C4C22×D311yescharacteristic
146SD16—68no—
147C2×C8—21no—
148C2×C8—21no—
149C2×C8—61no—
150C2×C8—61no—
151C2×C8—61no—
152C2×C8C4×D312yes—
153C4⋊C4—21no—
154C4⋊C4—21no—
155C4⋊C4—21no—
156C4⋊C4—64no—
157C4⋊C4D1212yes—
158C22⋊C4—122no—
159C22⋊C4—122no—
160C42—64no—
161C42—61no—
162C42—61no—
163C22×C3C4×D411yescharacteristic
164C22×C3#12011yescharacteristic
165C22×C3#12111yescharacteristic
166C22×C3C2×SD1611yescharacteristic
167C22×C3#13511yescharacteristic
168C22×C3AGL(1,ℤ/8)11yescharacteristic
169C22×C3#13711yescharacteristic
170C2×D3—42no—
171C2×D3—42no—
172C2×D3—42no—
173C2×D3—81no—
174C2×D3—81no—
175C2×D3—81no—
176C2×D3—81no—
177C3×C4—42no—
178C3×C4—81no—
179C3×C4C2×C22⋊C411yescharacteristic
180C3×C4C4×D411yescharacteristic
181C3×C4#12111yescharacteristic
182C3×C4#12311yescharacteristic
183Dic3—24no—
184Dic3—22no—
185Dic3—41no—
186C23—62no—
187C23—121no—
188C23—121no—
189C23—121no—
190C23—121no—
191C23—121no—
192C23—121no—
193C23D3×D411yescharacteristic
194Q8—34no—
195Q8—62no—
196Q8—62no—
197Q8—62no—
198D4—62no—
199D4—62no—
200D4—121no—
201D4—121no—
202C2×C4—22no—
203C2×C4—32no—
204C2×C4—41no—
205C2×C4—41no—
206C2×C4—41no—
207C2×C4—41no—
208C2×C4—41no—
209C2×C4—61no—
210C2×C4—62no—
211C2×C4—62no—
212C2×C4—62no—
213C2×C4—61no—
214C2×C4—61no—
215C2×C4—61no—
216C2×C4—61no—
217C2×C4C2×C4×D311yescharacteristic
218C2×C4C2×D1211yescharacteristic
219C2×C4#23511yescharacteristic
220C2×C4D3×D411yescharacteristic
221C2×C4D3×D411yescharacteristic
222C2×C4#23911yescharacteristic
223C8—22no—
224C8—121no—
225C2×C3#38611yescharacteristic
226C2×C3C4×SD1611yescharacteristic
227C2×C3#44011yescharacteristic
228C2×C3#44811yescharacteristic
229C2×C3#44911yescharacteristic
230C2×C3#46011yescharacteristic
231C2×C3#46411yescharacteristic
232D3—82no—
233C22—122no—
234C22—122no—
235C22—122no—
236C22—241no—
237C22—241no—
238C22—241no—
239C22—241no—
240C22#88211yescharacteristic
241C22#88411yescharacteristic
242C22#89311yescharacteristic
243C22#91111yescharacteristic
244C22D3×SD1611yescharacteristic
245C22#91611yescharacteristic
246C22#91811yescharacteristic
247C4—42no—
248C4—64no—
249C4—62no—
250C4—81no—
251C4—121no—
252C4C4×D1211yescharacteristic
253C4D3×C22⋊C411yescharacteristic
254C4#89411yescharacteristic
255C4#89511yescharacteristic
256C3#188011yescharacteristic, complemented, Sylow
257C2—242no—
258C2#466511yescharacteristic
259C2#476211yescharacteristic
260C2#478111yescharacteristic
261C2#479811yescharacteristic
262C2#480611yescharacteristic
263C2#485211yescharacteristic
264C2#488911yescharacteristic
subgroupG/H
centerC23D3×D4
commutatorC2×C3×C4C22×C4
FrattiniC22×C4C22×D3
FittingC3×#340C2
socleC23×C3C2×D4
prime radicalC22×D12C22
cosocleC22×C3×C4C23
hypercenter#340D3
hypocenterC3#1880
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C22×C4 ◁ #340

Complements

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

NH ≅ G/Ntype
2. C2×#887C2semidirect
4. C2×#810C2semidirect
6. C3×#340C2semidirect
7. #4550C2semidirect
22. C2×#204C4semidirect
32. #340D3semidirect
116. C2×Dic3SD16semidirect
124. Dic6C22⋊C4semidirect
256. C3#1880semidirect

Maximal subgroups

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

subgroupindex
C22×#2042
C2×#8872
#49882
C2×#8102
C2×#8092
C3×#3402
#45502
#18803