Finite Group Databasebeta

← #68104

Subgroups — #68104

Order 384.

256 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 256 conjugacy classes of subgroups — too many to draw. Only its 63 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×#885C211yesmaximal, characteristic
2#4719C211yesmaximal, characteristic, complemented
3#4718C211yesmaximal, characteristic, complemented
4C3×#323C211yesmaximal, characteristic, complemented
5#4525C211yesmaximal, characteristic, complemented
6#4468C211yesmaximal, characteristic, complemented
7#4468C211yesmaximal, characteristic, complemented
8#1469—31nomaximal, Sylow
9C22×#67C411yescharacteristic
10C22×Dic6C411yescharacteristic
11C2×C3×C22⋊C4C2211yescharacteristic
12C2×#217C411yescharacteristic
13C2×(C2×D3)⋊C4C411yescharacteristic
14C2×(C2×D3)⋊C4C2211yescharacteristic
15C2×C4×Dic3C2211yescharacteristic
16#885—21no—
17#885—21no—
18#885—21no—
19#885—21no—
20C3×C22⋊C8C2211yescharacteristic
21C3×C22⋊C8C2211yescharacteristic
22#830C2211yescharacteristic
23#830C2211yescharacteristic
24#815—21no—
25Dic3⋊C8—21no—
26Dic3⋊C8—21no—
27C8×Dic3—21no—
28C2×#124—31no—
29#432—31no—
30#431—31no—
31#323—31no—
32#323—31no—
33#323—31no—
34#323D311yescharacteristic, complemented
35C24×C3C2×C411yescharacteristic
36C23×D3C2×C411yescharacteristic
37C22×C3×C4C2×C411yescharacteristic
38C22×C3×C4C2311yescharacteristic
39C2×#67—21no—
40C2×#67—21no—
41C2×#67—42no—
42C22×Dic3C2×C411yescharacteristic
43C22×Dic3C2×C411yescharacteristic
44C22×Dic3C2×C411yescharacteristic
45C2×Dic6—21no—
46C2×Dic6—21no—
47C2×Dic6—42no—
48C2×C3×C8—21no—
49C2×C3×C8—21no—
50C3×C22⋊C4—21no—
51C3×C22⋊C4—21no—
52#217—21no—
53#217—21no—
54(C2×D3)⋊C4—21no—
55(C2×D3)⋊C4—21no—
56(C2×D3)⋊C4—21no—
57(C2×D3)⋊C4—21no—
58C4×Dic3D411yescharacteristic
59C4×Dic3D411yescharacteristic
60C4×Dic3D411yescharacteristic
61C4×Dic3D411yescharacteristic
62C2×C3⋊C8—21no—
63C2×C3⋊C8—21no—
64C22×Q8—31no—
65C22×D4—31no—
66#124—61no—
67#124—61no—
68#124—61no—
69#124—61no—
70C2×C22⋊C4—31no—
71C2×C22⋊C4—31no—
72C2×C22⋊C4—31no—
73C2×C22⋊C4C2×D311yescharacteristic
74C2×C42—31no—
75C4⋊C8—61no—
76C4⋊C8—61no—
77C22⋊C8—31no—
78C22⋊C8—31no—
79C22⋊C8C2×D311yescharacteristic
80C22⋊C8C2×D311yescharacteristic
81#97—61no—
82C4×C8—61no—
83C23×C3—21no—
84C23×C3—21no—
85C23×C3—21no—
86C23×C3—41no—
87C23×C3—41no—
88C23×C3C22×C411yescharacteristic
89C22×D3—21no—
90C22×D3—21no—
91C22×D3—21no—
92C22×D3—41no—
93C22×D3—41no—
94C2×C3×C4—21no—
95C2×C3×C4—41no—
96C2×C3×C4—41no—
97C2×C3×C4C2×D411yescharacteristic
98C2×C3×C4C2×D411yescharacteristic
99#67—42no—
100#67—42no—
101C2×Dic3—21no—
102C2×Dic3—21no—
103C2×Dic3—22no—
104C2×Dic3—21no—
105C2×Dic3—41no—
106C2×Dic3C22⋊C411yescharacteristic
107C2×Dic3C22⋊C411yescharacteristic
108C2×Dic3C22⋊C411yescharacteristic
109C2×Dic3C22⋊C411yescharacteristic
110Dic6—42no—
111Dic6—42no—
112C3×C8—41no—
113C3×C8—41no—
114C3⋊C8—41no—
115C3⋊C8—41no—
116C24—31no—
117C24C4×D311yescharacteristic
118C2×Q8—61no—
119C2×Q8—61no—
120C2×Q8—122no—
121C2×D4—61no—
122C2×D4—61no—
123C2×D4—122no—
124C22×C4—31no—
125C22×C4—31no—
126C22×C4—31no—
127C22×C4C4×D311yescharacteristic
128C22×C4C22×D311yescharacteristic
129C2×C8—21no—
130C2×C8—21no—
131C2×C8—61no—
132C2×C8—61no—
133C22⋊C4—21no—
134C22⋊C4—21no—
135C22⋊C4—61no—
136C22⋊C4—61no—
137C22⋊C4—61no—
138C22⋊C4—61no—
139C22⋊C4—61no—
140C22⋊C4—61no—
141C42—31no—
142C42—31no—
143C42—31no—
144C42—31no—
145C22×C3—21no—
146C22×C3—21no—
147C22×C3—41no—
148C22×C3—41no—
149C22×C3—41no—
150C22×C3—81no—
151C22×C3—81no—
152C22×C3C2×C22⋊C411yescharacteristic
153C22×C3#13111yescharacteristic
154C22×C3#13111yescharacteristic
155C2×D3—41no—
156C2×D3—41no—
157C2×D3—41no—
158C2×D3—81no—
159C2×D3—81no—
160C3×C4—21no—
161C3×C4—21no—
162C3×C4—41no—
163Dic3—21no—
164Dic3—21no—
165Dic3—21no—
166Dic3—41no—
167Dic3#10012yes—
168C23—21no—
169C23—21no—
170C23—21no—
171C23—41no—
172C23—41no—
173C23—61no—
174C23—61no—
175C23—61no—
176C23—121no—
177C23—121no—
178C23C2×C4×D311yescharacteristic
179Q8—122no—
180Q8—122no—
181D4—122no—
182D4—122no—
183C2×C4—21no—
184C2×C4—31no—
185C2×C4—31no—
186C2×C4—31no—
187C2×C4—31no—
188C2×C4—41no—
189C2×C4—41no—
190C2×C4—62no—
191C2×C4—61no—
192C2×C4—61no—
193C2×C4—61no—
194C2×C4—121no—
195C2×C4D3×D411yescharacteristic
196C2×C4D3×D411yescharacteristic
197C8—41no—
198C8—41no—
199C8—121no—
200C8—121no—
201C2×C3—21no—
202C2×C3—21no—
203C2×C3—81no—
204C2×C3#40811yescharacteristic
205C2×C3#41011yescharacteristic
206C2×C3C2×#10011yescharacteristic
207D3—81no—
208C22—21no—
209C22—21no—
210C22—41no—
211C22—41no—
212C22—41no—
213C22—81no—
214C22—81no—
215C22—121no—
216C22—121no—
217C22—121no—
218C22—241no—
219C22—241no—
220C22D3×C22⋊C411yescharacteristic
221C22#90111yescharacteristic
222C22#90711yescharacteristic
223C4—21no—
224C4—21no—
225C4—32no—
226C4—41no—
227C4—61no—
228C4—61no—
229C4—61no—
230C4—121no—
231C3#146911yescharacteristic, complemented, Sylow
232C2—21no—
233C2—21no—
234C2—81no—
235C2—241no—
236C2#472711yescharacteristic
237C2#474011yescharacteristic
238C2D3×#10011yescharacteristic
subgroupG/H
centerC22D3×C22⋊C4
commutatorC23×C3C22×C4
FrattiniC22×C4C22×D3
FittingC3×#323C2
socleC22×C3C2×C22⋊C4
prime radicalC22×#67C4
cosocleC22×C3×C4C23
hypercenter#323D3
hypocenterC3#1469
LCSG ▷ C23×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #323

Complements

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

NH ≅ G/Ntype
2. #4719C2semidirect
3. #4718C2semidirect
4. C3×#323C2semidirect
5. #4525C2semidirect
6. #4468C2semidirect
7. #4468C2semidirect
34. #323D3semidirect
231. C3#1469semidirect

Maximal subgroups

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

subgroupindex
C2×#8852
#47192
#47182
C3×#3232
#45252
#44682
#14693