Finite Group Databasebeta

← #68281

Subgroups — #68281

Order 384.

252 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 252 conjugacy classes of subgroups — too many to draw. Only its 61 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
1#5596C211yesmaximal, characteristic
2#4719C211yesmaximal, characteristic, complemented
3#4651C211yesmaximal, characteristic, complemented
4C3×#327C211yesmaximal, characteristic, complemented
5#4537C211yesmaximal, characteristic, complemented
6#4470C211yesmaximal, characteristic, complemented
7#4451C211yesmaximal, characteristic, complemented
8#1514—31nomaximal, Sylow
9C2×#237—21no—
10C3×#121C2211yescharacteristic
11#940C411yescharacteristic, complemented
12#939—21no—
13D4×Dic3C411yescharacteristic, complemented
14#930C2211yescharacteristic
15C2×C4×Dic3C2211yescharacteristic
16#893C411yescharacteristic, complemented
17#888C411yescharacteristic, complemented
18#885—21no—
19#67⋊C4—21no—
20C3×C22⋊C8C2211yescharacteristic
21C3×#95C2211yescharacteristic
22C3⋊#95—21no—
23C3⋊#95—21no—
24C3⋊#95C2211yescharacteristic
25#830C2211yescharacteristic
26#815—21no—
27Dic3⋊C8—21no—
28#532—31no—
29#432—31no—
30#383—31no—
31#327—31no—
32#327—31no—
33#327—31no—
34#327D311yescharacteristic, complemented
35C2×C3×D4—21no—
36C2×C3×D4C2×C411yescharacteristic
37C22×C3×C4—21no—
38C22×C3×C4C2311yescharacteristic
39C2×#67—21no—
40C2×#67—21no—
41C2×#67—21no—
42C22×Dic3—21no—
43C22×Dic3—21no—
44C22×Dic3D411yescharacteristic
45C22×Dic3D411yescharacteristic
46#237—42no—
47#237—42no—
48C2×D12C2×C411yescharacteristic
49C2×C4×D3—21no—
50C2×Dic6—21no—
51C2×C3×C8—21no—
52C3×C4⋊C4C2×C411yescharacteristic
53C3×C22⋊C4—21no—
54#217—21no—
55(C2×D3)⋊C4—21no—
56(C2×D3)⋊C4—21no—
57C4⋊Dic3C2×C411yescharacteristic
58Dic3⋊C4—21no—
59C4×Dic3C2×C411yescharacteristic
60C4×Dic3C2×C411yescharacteristic
61C4×Dic3D411yescharacteristic
62C4×Dic3D411yescharacteristic
63C2×C3⋊C8—21no—
64C2×#41—61no—
65C4⋊Q8—31no—
66#127—31no—
67#124—61no—
68#121—31no—
69#121—61no—
70#121C2×D311yescharacteristic
71C4×D4—31no—
72C4×D4—31no—
73C4×D4—61no—
74C2×C42—31no—
75C4⋊C8—61no—
76C22⋊C8—31no—
77C22⋊C8C2×D311yescharacteristic
78#97—61no—
79#95—31no—
80#95—61no—
81#95—61no—
82#95C2×D311yescharacteristic
83C23×C3—21no—
84C23×C3C2×D411yescharacteristic
85C22×D3—21no—
86C3×D4—41no—
87C3×D4—41no—
88C3×D4—41no—
89C2×C3×C4—21no—
90C2×C3×C4—41no—
91C2×C3×C4—41no—
92C2×C3×C4—41no—
93C2×C3×C4—41no—
94C2×C3×C4C22×C411yescharacteristic
95C2×C3×C4C2×D411yescharacteristic
96#67—42no—
97#67—42no—
98#67—41no—
99#67—41no—
100C2×Dic3—21no—
101C2×Dic3—21no—
102C2×Dic3—21no—
103C2×Dic3—41no—
104C2×Dic3—42no—
105C2×Dic3—41no—
106C2×Dic3—41no—
107C2×Dic3—41no—
108C2×Dic3C22⋊C411yescharacteristic
109C2×Dic3C22⋊C411yescharacteristic
110C2×Dic3C22⋊C411yescharacteristic
111C2×Dic3C22⋊C411yescharacteristic
112D12—41no—
113C4×D3—42no—
114Dic6—42no—
115C3×C8—41no—
116C3⋊C8—41no—
117#41—122no—
118#41—122no—
119C2×Q8—61no—
120C2×D4—21no—
121C2×D4—31no—
122C2×D4—61no—
123C2×D4—61no—
124C2×D4—61no—
125C2×D4C4×D311yescharacteristic
126C22×C4—21no—
127C22×C4—31no—
128C22×C4—31no—
129C22×C4—61no—
130C22×C4—61no—
131C22×C4—61no—
132C22×C4C22×D311yescharacteristic
133C2×C8—21no—
134C2×C8—61no—
135C4⋊C4—31no—
136C4⋊C4—61no—
137C4⋊C4C4×D311yescharacteristic
138C22⋊C4—21no—
139C22⋊C4—61no—
140C22⋊C4—61no—
141C22⋊C4—61no—
142C42—31no—
143C42—31no—
144C42—31no—
145C42—31no—
146C22×C3—21no—
147C22×C3—21no—
148C22×C3—41no—
149C22×C3—41no—
150C22×C3—41no—
151C22×C3C2×C22⋊C411yescharacteristic
152C22×C3AGL(1,ℤ/8)11yescharacteristic
153C22×C3AGL(1,ℤ/8)11yescharacteristic
154C2×D3—41no—
155C2×D3—41no—
156C2×D3—41no—
157C3×C4—21no—
158C3×C4—21no—
159C3×C4—41no—
160C3×C4—81no—
161Dic3—21no—
162Dic3—21no—
163Dic3—21no—
164Dic3—21no—
165Dic3—41no—
166Dic3—81no—
167C23—21no—
168C23—61no—
169C23D3×D411yescharacteristic
170Q8—122no—
171D4—41no—
172D4—41no—
173D4—41no—
174D4—121no—
175D4—122no—
176D4—122no—
177D4—121no—
178D4—121no—
179C2×C4—21no—
180C2×C4—31no—
181C2×C4—31no—
182C2×C4—31no—
183C2×C4—31no—
184C2×C4—41no—
185C2×C4—41no—
186C2×C4—41no—
187C2×C4—41no—
188C2×C4—61no—
189C2×C4—61no—
190C2×C4—61no—
191C2×C4—122no—
192C2×C4—121no—
193C2×C4—122no—
194C2×C4—121no—
195C2×C4—121no—
196C2×C4—121no—
197C2×C4C2×C4×D311yescharacteristic
198C2×C4D3×D411yescharacteristic
199C8—41no—
200C8—121no—
201C2×C3—21no—
202C2×C3—21no—
203C2×C3—81no—
204C2×C3#41011yescharacteristic
205C2×C3#41811yescharacteristic
206C2×C3#42111yescharacteristic
207D3—81no—
208C22—21no—
209C22—21no—
210C22—41no—
211C22—41no—
212C22—41no—
213C22—121no—
214C22—121no—
215C22—121no—
216C22D3×C22⋊C411yescharacteristic
217C22#91111yescharacteristic
218C22#91411yescharacteristic
219C4—21no—
220C4—21no—
221C4—41no—
222C4—61no—
223C4—61no—
224C4—61no—
225C4—61no—
226C4—81no—
227C4—121no—
228C4—241no—
229C3#151411yescharacteristic, complemented, Sylow
230C2—21no—
231C2—21no—
232C2—81no—
233C2—241no—
234C2#474011yescharacteristic
235C2#477011yescharacteristic
236C2#482111yescharacteristic
subgroupG/H
centerC22D3×C22⋊C4
commutatorC2×C3×C4C22×C4
FrattiniC22×C4C22×D3
FittingC3×#327C2
socleC22×C3C2×C22⋊C4
prime radical#5596C2
cosocleC22×C3×C4C23
hypercenter#327D3
hypocenterC3#1514
LCSG ▷ C2×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #327

Complements

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

NH ≅ G/Ntype
2. #4719C2semidirect
3. #4651C2semidirect
4. C3×#327C2semidirect
5. #4537C2semidirect
6. #4470C2semidirect
7. #4451C2semidirect
11. #940C4semidirect
13. D4×Dic3C4semidirect
16. #893C4semidirect
17. #888C4semidirect
34. #327D3semidirect
229. C3#1514semidirect

Maximal subgroups

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

subgroupindex
#55962
#47192
#46512
C3×#3272
#45372
#44702
#44512
#15143