Finite Group Databasebeta

← #68278

Subgroups — #68278

Order 384.

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

normal subgroups only

This group has 272 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
1#5596C211yesmaximal, characteristic
2#4719C211yesmaximal, characteristic, complemented
3#4648C211yesmaximal, characteristic, complemented
4C3×#327C211yesmaximal, characteristic, complemented
5#4537C211yesmaximal, characteristic, complemented
6#4470C211yesmaximal, characteristic, complemented
7#4451C211yesmaximal, characteristic, complemented
8#1505—31nomaximal, Sylow
9C2×#237—21no—
10C3×#121C2211yescharacteristic
11#940C411yescharacteristic, complemented
12#939—21no—
13D4×Dic3C411yescharacteristic, complemented
14#930C2211yescharacteristic
15C2×C4×Dic3—21no—
16C2×C4×Dic3C2211yescharacteristic
17#893C411yescharacteristic, complemented
18#888C411yescharacteristic, complemented
19#885—21no—
20#67⋊C4—21no—
21C3×C22⋊C8C2211yescharacteristic
22C3×#95C2211yescharacteristic
23C3⋊#95—21no—
24C3⋊#95C2211yescharacteristic
25#830C2211yescharacteristic
26#815—21no—
27Dic3⋊C8—21no—
28#532—31no—
29#432—31no—
30#376—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×Dic3—24no—
60C4×Dic3C2×C411yescharacteristic
61C4×Dic3C2×C411yescharacteristic
62C4×Dic3D411yescharacteristic
63C4×Dic3D411yescharacteristic
64C2×C3⋊C8—21no—
65C2×#41—61no—
66C4⋊Q8—31no—
67#127—31no—
68#124—61no—
69#121—31no—
70#121—61no—
71#121C2×D311yescharacteristic
72C4×D4—31no—
73C4×D4—31no—
74C4×D4—61no—
75C2×C42—31no—
76C2×C42—61no—
77C4⋊C8—61no—
78C22⋊C8—31no—
79C22⋊C8C2×D311yescharacteristic
80#97—61no—
81#95—31no—
82#95—61no—
83#95C2×D311yescharacteristic
84C23×C3—21no—
85C23×C3C2×D411yescharacteristic
86C22×D3—21no—
87C3×D4—41no—
88C3×D4—41no—
89C3×D4—41no—
90C2×C3×C4—22no—
91C2×C3×C4—21no—
92C2×C3×C4—41no—
93C2×C3×C4—41no—
94C2×C3×C4—41no—
95C2×C3×C4C22×C411yescharacteristic
96C2×C3×C4C2×D411yescharacteristic
97#67—42no—
98#67—41no—
99#67—41no—
100#67—42no—
101C2×Dic3—21no—
102C2×Dic3—21no—
103C2×Dic3—22no—
104C2×Dic3—22no—
105C2×Dic3—21no—
106C2×Dic3—42no—
107C2×Dic3—41no—
108C2×Dic3—41no—
109C2×Dic3C22⋊C411yescharacteristic
110C2×Dic3C22⋊C411yescharacteristic
111C2×Dic3C22⋊C411yescharacteristic
112C2×Dic3C22⋊C411yescharacteristic
113D12—41no—
114C4×D3—42no—
115Dic6—42no—
116C3×C8—41no—
117C3⋊C8—41no—
118#41—122no—
119#41—122no—
120C2×Q8—61no—
121C2×D4—21no—
122C2×D4—31no—
123C2×D4—61no—
124C2×D4—61no—
125C2×D4—61no—
126C2×D4C4×D311yescharacteristic
127C22×C4—21no—
128C22×C4—31no—
129C22×C4—31no—
130C22×C4—61no—
131C22×C4—61no—
132C22×C4—61no—
133C22×C4C22×D311yescharacteristic
134C2×C8—21no—
135C2×C8—61no—
136C4⋊C4—31no—
137C4⋊C4—61no—
138C4⋊C4C4×D311yescharacteristic
139C22⋊C4—21no—
140C22⋊C4—61no—
141C22⋊C4—61no—
142C22⋊C4—61no—
143C42—31no—
144C42—31no—
145C42—31no—
146C42—31no—
147C42—64no—
148C22×C3—21no—
149C22×C3—21no—
150C22×C3—41no—
151C22×C3—41no—
152C22×C3—41no—
153C22×C3C2×C22⋊C411yescharacteristic
154C22×C3AGL(1,ℤ/8)11yescharacteristic
155C22×C3AGL(1,ℤ/8)11yescharacteristic
156C2×D3—41no—
157C2×D3—41no—
158C2×D3—41no—
159C3×C4—21no—
160C3×C4—21no—
161C3×C4—42no—
162C3×C4—41no—
163Dic3—21no—
164Dic3—21no—
165Dic3—21no—
166Dic3—42no—
167Dic3—41no—
168Dic3#10412yes—
169C23—21no—
170C23—61no—
171C23D3×D411yescharacteristic
172Q8—122no—
173D4—41no—
174D4—41no—
175D4—41no—
176D4—121no—
177D4—122no—
178D4—122no—
179D4—121no—
180D4—121no—
181C2×C4—22no—
182C2×C4—21no—
183C2×C4—31no—
184C2×C4—31no—
185C2×C4—31no—
186C2×C4—31no—
187C2×C4—41no—
188C2×C4—41no—
189C2×C4—41no—
190C2×C4—62no—
191C2×C4—61no—
192C2×C4—61no—
193C2×C4—62no—
194C2×C4—61no—
195C2×C4—122no—
196C2×C4—122no—
197C2×C4—121no—
198C2×C4—121no—
199C2×C4C2×C4×D311yescharacteristic
200C2×C4D3×D411yescharacteristic
201C8—41no—
202C8—121no—
203C2×C3—21no—
204C2×C3—21no—
205C2×C3—81no—
206C2×C3#41011yescharacteristic
207C2×C3#41811yescharacteristic
208C2×C3C2×#10411yescharacteristic
209D3—81no—
210C22—21no—
211C22—21no—
212C22—41no—
213C22—41no—
214C22—41no—
215C22—121no—
216C22—121no—
217C22—121no—
218C22D3×C22⋊C411yescharacteristic
219C22#91111yescharacteristic
220C22#91411yescharacteristic
221C4—21no—
222C4—21no—
223C4—32no—
224C4—42no—
225C4—41no—
226C4—61no—
227C4—61no—
228C4—61no—
229C4—122no—
230C4—121no—
231C3#150511yescharacteristic, complemented, Sylow
232C2—21no—
233C2—21no—
234C2—81no—
235C2—241no—
236C2#474011yescharacteristic
237C2#477011yescharacteristic
238C2D3×#10411yescharacteristic
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#1505
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. #4648C2semidirect
4. C3×#327C2semidirect
5. #4537C2semidirect
6. #4470C2semidirect
7. #4451C2semidirect
11. #940C4semidirect
13. D4×Dic3C4semidirect
17. #893C4semidirect
18. #888C4semidirect
34. #327D3semidirect
231. C3#1505semidirect

Maximal subgroups

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

subgroupindex
#55962
#47192
#46482
C3×#3272
#45372
#44702
#44512
#15053