Finite Group Databasebeta

← #70213

Subgroups — #70213

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×#214C211yesmaximal, characteristic
2C2×#887C211yesmaximal, characteristic, complemented
3C2×C3×#102C211yesmaximal, characteristic
4#5221C211yesmaximal, characteristic
5C2×#816C211yesmaximal, characteristic, complemented
6#4550C211yesmaximal, characteristic, complemented
7#4529C211yesmaximal, characteristic, complemented
8#1880—31nomaximal, Sylow
9C22×C3×D4C2211yescharacteristic
10C22×Dic6C2211yescharacteristic
11C22×C3×C8C2211yescharacteristic
12C2×C3×C4⋊C4C2211yescharacteristic
13C2×#217—21no—
14C2×#217—21no—
15C2×#214—22no—
16C2×#214—22no—
17C2×#214C414yescomplemented
18C2×C4⋊Dic3C2211yescharacteristic
19C2×Dic3⋊C4—21no—
20C2×C4×Dic3—21no—
21C2×C4×Dic3C2211yescharacteristic
22C22×C3⋊C8C2211yescharacteristic
23#887—22no—
24#887—22no—
25C3×#102—22no—
26#816—22no—
27C22×SD16—31no—
28C2×C4×Q8—31no—
29C2×#103—31no—
30C2×#102D311yescharacteristic
31#390—31no—
32#340—31no—
33#336—31no—
34C24×C3—21no—
35C2×C3×D4—21no—
36C2×C3×D4—21no—
37C2×C3×D4—21no—
38C2×C3×D4—21no—
39C2×C3×D4—21no—
40C2×C3×D4C2×C412yes—
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×Dic6C2×C412yes—
51C2×Dic6D412yes—
52C2×Dic6D412yes—
53C2×C3×C8—21no—
54C2×C3×C8—21no—
55C2×C3×C8—21no—
56C3×C4⋊C4—21no—
57C3×C4⋊C4D412yes—
58#217—42no—
59#217—42no—
60#214—28no—
61C4⋊Dic3—21no—
62C4⋊Dic3—21no—
63Dic3⋊C4—24no—
64C4×Dic3—24no—
65C4×Dic3—21no—
66C4×Dic3—21no—
67C2×C3⋊C8—21no—
68C2×C3⋊C8—21no—
69C2×C3⋊C8C2×C412yes—
70C22×Q8—31no—
71C22×D4C2×D311yescharacteristic
72C2×SD16—34no—
73C2×SD16—62no—
74C2×SD16—62no—
75C22×C8—31no—
76C22×C8C2×D311yescharacteristic
77C4×Q8—62no—
78C4×Q8—62no—
79C2×C4⋊C4—31no—
80C2×C4⋊C4—61no—
81C2×C4⋊C4C2×D311yescharacteristic
82C2×C22⋊C4—61no—
83C2×C22⋊C4—61no—
84C2×C42—31no—
85C2×C42—61no—
86#103—62no—
87#102—22no—
88C23×C3—22no—
89C23×C3—41no—
90C23×C3—41no—
91C23×C3—41no—
92C23×C3—41no—
93C23×C3—41no—
94C23×C3—41no—
95C23×C3C2×D411yescharacteristic
96C3×D4—22no—
97C3×D4—22no—
98C3×D4—41no—
99C3×D4—41no—
100C2×C3×C4—22no—
101C2×C3×C4—41no—
102C2×C3×C4—41no—
103C2×C3×C4C22×C411yescharacteristic
104C2×C3×C4C2×D411yescharacteristic
105C2×C3×C4C2×D411yescharacteristic
106C2×C3×C4C2×D411yescharacteristic
107C2×C3×C4#4111yescharacteristic
108C2×C3×C4#4111yescharacteristic
109C2×Dic3—22no—
110C2×Dic3—21no—
111C2×Dic3—22no—
112C2×Dic3—22no—
113C2×Dic3—21no—
114C2×Dic3—21no—
115C2×Dic3—21no—
116C2×Dic3—21no—
117C2×Dic3—41no—
118C2×Dic3—41no—
119C2×Dic3—41no—
120C2×Dic3SD1612yescomplemented
121Dic6—22no—
122Dic6—22no—
123Dic6—22no—
124Dic6C22⋊C414yescomplemented
125C3×C8—41no—
126C3⋊C8—22no—
127C24—21no—
128C2×Q8—32no—
129C2×Q8—32no—
130C2×Q8—32no—
131C2×Q8—61no—
132C2×Q8—61no—
133C2×Q8—61no—
134C2×D4—21no—
135C2×D4—21no—
136C2×D4—21no—
137C2×D4—21no—
138C2×D4—21no—
139C2×D4C4×D312yes—
140C22×C4—21no—
141C22×C4—31no—
142C22×C4—31no—
143C22×C4—61no—
144C22×C4—61no—
145C22×C4C22×D311yescharacteristic
146SD16—68no—
147C2×C8—21no—
148C2×C8—21no—
149C2×C8—21no—
150C2×C8—32no—
151C2×C8—61no—
152C2×C8—61no—
153C4⋊C4—21no—
154C4⋊C4—64no—
155C4⋊C4—61no—
156C4⋊C4—61no—
157C4⋊C4#6712yes—
158C22⋊C4—122no—
159C22⋊C4—122no—
160C42—64no—
161C42—61no—
162C42—61no—
163C22×C3—42no—
164C22×C3—42no—
165C22×C3—42no—
166C22×C3—81no—
167C22×C3—81no—
168C22×C3—81no—
169C22×C3—81no—
170C22×C3C4×D411yescharacteristic
171C22×C3#12011yescharacteristic
172C22×C3#12111yescharacteristic
173C22×C3C2×SD1611yescharacteristic
174C22×C3#13511yescharacteristic
175C22×C3AGL(1,ℤ/8)11yescharacteristic
176C22×C3#13711yescharacteristic
177C3×C4—42no—
178C3×C4C2×C22⋊C411yescharacteristic
179C3×C4C4×D411yescharacteristic
180C3×C4#12111yescharacteristic
181C3×C4#12311yescharacteristic
182Dic3—24no—
183Dic3—22no—
184Dic3—41no—
185Dic3—81no—
186C23—22no—
187C23—41no—
188C23—41no—
189C23—41no—
190C23—41no—
191C23—41no—
192C23—41no—
193C23D3×D411yescharacteristic
194Q8—34no—
195Q8—62no—
196Q8—62no—
197Q8—62no—
198D4—22no—
199D4—22no—
200D4—41no—
201D4—41no—
202C2×C4—22no—
203C2×C4—32no—
204C2×C4—41no—
205C2×C4—41no—
206C2×C4—62no—
207C2×C4—62no—
208C2×C4—61no—
209C2×C4—61no—
210C2×C4—62no—
211C2×C4—61no—
212C2×C4—61no—
213C2×C4—61no—
214C2×C4—121no—
215C2×C4—121no—
216C2×C4—121no—
217C2×C4C2×C4×D311yescharacteristic
218C2×C4#23511yescharacteristic
219C2×C4D3×D411yescharacteristic
220C2×C4D3×D411yescharacteristic
221C2×C4#23711yescharacteristic
222C2×C4C2×#6711yescharacteristic
223C8—41no—
224C8—62no—
225C2×C3—82no—
226C2×C3#38611yescharacteristic
227C2×C3C4×SD1611yescharacteristic
228C2×C3#44011yescharacteristic
229C2×C3#44811yescharacteristic
230C2×C3#44911yescharacteristic
231C2×C3#46011yescharacteristic
232C2×C3#46411yescharacteristic
233C22—42no—
234C22—42no—
235C22—42no—
236C22—81no—
237C22—81no—
238C22—81no—
239C22—81no—
240C22#67⋊C411yescharacteristic
241C22#88411yescharacteristic
242C22#91111yescharacteristic
243C22#91211yescharacteristic
244C22D3×SD1611yescharacteristic
245C22#91511yescharacteristic
246C22#93711yescharacteristic
247C4—42no—
248C4—62no—
249C4—64no—
250C4—121no—
251C4—241no—
252C4D3×C22⋊C411yescharacteristic
253C4#88311yescharacteristic
254C4C4×#6711yescharacteristic
255C4#93811yescharacteristic
256C3#188011yescharacteristic, complemented, Sylow
257C2—82no—
258C2#475711yescharacteristic
259C2#475811yescharacteristic
260C2#476211yescharacteristic
261C2#476711yescharacteristic
262C2#495611yescharacteristic
263C2#515811yescharacteristic
264C2#517011yescharacteristic
subgroupG/H
centerC23D3×D4
commutatorC2×C3×C4C22×C4
FrattiniC22×C4C22×D3
FittingC2×C3×#102C2
socleC23×C3C2×D4
prime radicalC22×C3×D4C22
cosocleC22×C3×C4C23
hypercenterC2×#102D3
hypocenterC3#1880
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C22×C4 ◁ C2×#102

Complements

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

NH ≅ G/Ntype
2. C2×#887C2semidirect
5. C2×#816C2semidirect
6. #4550C2semidirect
7. #4529C2semidirect
17. C2×#214C4semidirect
120. 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×#2142
C2×#8872
C2×C3×#1022
#52212
C2×#8162
#45502
#45292
#18803