Finite Group Databasebeta

← #68277

Subgroups — #68277

Order 384.

284 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 284 conjugacy classes of subgroups — too many to draw. Only its 71 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#4718C211yesmaximal, characteristic, complemented
3#4649C211yesmaximal, characteristic, complemented
4C3×#327C211yesmaximal, characteristic, complemented
5#4537C211yesmaximal, characteristic, complemented
6#4470C211yesmaximal, characteristic, complemented
7#4451C211yesmaximal, characteristic, complemented
8#1502—31nomaximal, Sylow
9C2×#237—21no—
10C3×#121C2211yescharacteristic
11#940C411yescharacteristic, complemented
12#939—21no—
13D4×Dic3C411yescharacteristic, complemented
14#930C2211yescharacteristic
15C2×Dic3⋊C4—21no—
16C2×C4×Dic3—21no—
17C2×C4×Dic3C2211yescharacteristic
18#893C411yescharacteristic, complemented
19#888C411yescharacteristic, complemented
20#885—21no—
21#67⋊C4—21no—
22C3×C22⋊C8C2211yescharacteristic
23C3×#95C2211yescharacteristic
24C3⋊#95C2211yescharacteristic
25#830C2211yescharacteristic
26Dic3⋊C8—21no—
27C8×Dic3—21no—
28#532—31no—
29#431—31no—
30C4×C4⋊C4—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—24no—
59Dic3⋊C4—21no—
60C4×Dic3—42no—
61C4×Dic3C2×C411yescharacteristic
62C4×Dic3C2×C411yescharacteristic
63C4×Dic3D411yescharacteristic
64C4×Dic3D411yescharacteristic
65C2×C3⋊C8—21no—
66C2×#41—61no—
67C4⋊Q8—31no—
68#127—31no—
69#124—61no—
70#121—31no—
71#121—61no—
72#121C2×D311yescharacteristic
73C4×D4—31no—
74C4×D4—31no—
75C4×D4—61no—
76C2×C4⋊C4—61no—
77C2×C42—31no—
78C2×C42—61no—
79C4⋊C8—61no—
80C22⋊C8—31no—
81C22⋊C8C2×D311yescharacteristic
82C4×C8—61no—
83#95—31no—
84#95C2×D311yescharacteristic
85C23×C3—21no—
86C23×C3C2×D411yescharacteristic
87C22×D3—21no—
88C3×D4—41no—
89C3×D4—41no—
90C3×D4—41no—
91C2×C3×C4—22no—
92C2×C3×C4—21no—
93C2×C3×C4—41no—
94C2×C3×C4—41no—
95C2×C3×C4—41no—
96C2×C3×C4C22×C411yescharacteristic
97C2×C3×C4C2×D411yescharacteristic
98#67—41no—
99#67—41no—
100#67—42no—
101#67—42no—
102C2×Dic3—22no—
103C2×Dic3—22no—
104C2×Dic3—21no—
105C2×Dic3—41no—
106C2×Dic3—41no—
107C2×Dic3—42no—
108C2×Dic3C22⋊C411yescharacteristic
109C2×Dic3C22⋊C411yescharacteristic
110C2×Dic3C22⋊C411yescharacteristic
111C2×Dic3C22⋊C411yescharacteristic
112C2×Dic3D812yes—
113C2×Dic3SD1612yes—
114D12—41no—
115C4×D3—42no—
116Dic6—42no—
117C3×C8—41no—
118C3⋊C8—41no—
119#41—122no—
120#41—122no—
121C2×Q8—61no—
122C2×D4—21no—
123C2×D4—31no—
124C2×D4—61no—
125C2×D4—61no—
126C2×D4—61no—
127C2×D4C4×D311yescharacteristic
128C22×C4—21no—
129C22×C4—31no—
130C22×C4—31no—
131C22×C4—61no—
132C22×C4—61no—
133C22×C4—61no—
134C22×C4C22×D311yescharacteristic
135C2×C8—21no—
136C2×C8—61no—
137C4⋊C4—31no—
138C4⋊C4—61no—
139C4⋊C4—64no—
140C4⋊C4C4×D311yescharacteristic
141C22⋊C4—21no—
142C22⋊C4—61no—
143C22⋊C4—61no—
144C22⋊C4—61no—
145C42—31no—
146C42—31no—
147C42—31no—
148C42—31no—
149C42—122no—
150C22×C3—21no—
151C22×C3—21no—
152C22×C3—41no—
153C22×C3—41no—
154C22×C3—41no—
155C22×C3C2×C22⋊C411yescharacteristic
156C22×C3C2×D811yescharacteristic
157C22×C3C2×SD1611yescharacteristic
158C2×D3—41no—
159C2×D3—41no—
160C2×D3—41no—
161C3×C4—21no—
162C3×C4—21no—
163C3×C4—42no—
164C3×C4—41no—
165Dic3—21no—
166Dic3—42no—
167Dic3—41no—
168Dic3#10212yes—
169Dic3#10212yes—
170Dic3#10412yes—
171C23—21no—
172C23—61no—
173C23D3×D411yescharacteristic
174Q8—122no—
175D4—41no—
176D4—41no—
177D4—41no—
178D4—121no—
179D4—122no—
180D4—122no—
181D4—121no—
182D4—121no—
183C2×C4—22no—
184C2×C4—21no—
185C2×C4—31no—
186C2×C4—31no—
187C2×C4—31no—
188C2×C4—32no—
189C2×C4—32no—
190C2×C4—31no—
191C2×C4—41no—
192C2×C4—41no—
193C2×C4—41no—
194C2×C4—62no—
195C2×C4—62no—
196C2×C4—61no—
197C2×C4—122no—
198C2×C4—122no—
199C2×C4—121no—
200C2×C4—121no—
201C2×C4C2×C4×D311yescharacteristic
202C2×C4D3×D411yescharacteristic
203C8—41no—
204C8—121no—
205C2×C3—21no—
206C2×C3—21no—
207C2×C3—81no—
208C2×C3#41011yescharacteristic
209C2×C3C2×#10211yescharacteristic
210C2×C3C2×#10411yescharacteristic
211D3—81no—
212C22—21no—
213C22—21no—
214C22—41no—
215C22—41no—
216C22—41no—
217C22—121no—
218C22—121no—
219C22—121no—
220C22D3×C22⋊C411yescharacteristic
221C22D3×D811yescharacteristic
222C22D3×SD1611yescharacteristic
223C4—21no—
224C4—21no—
225C4—32no—
226C4—32no—
227C4—32no—
228C4—42no—
229C4—41no—
230C4—61no—
231C4—122no—
232C4—121no—
233C3#150211yescharacteristic, complemented, Sylow
234C2—21no—
235C2—21no—
236C2—81no—
237C2—241no—
238C2#474011yescharacteristic
239C2D3×#10211yescharacteristic
240C2D3×#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#1502
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. #4718C2semidirect
3. #4649C2semidirect
4. C3×#327C2semidirect
5. #4537C2semidirect
6. #4470C2semidirect
7. #4451C2semidirect
11. #940C4semidirect
13. D4×Dic3C4semidirect
18. #893C4semidirect
19. #888C4semidirect
34. #327D3semidirect
233. C3#1502semidirect

Maximal subgroups

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

subgroupindex
#55962
#47182
#46492
C3×#3272
#45372
#44702
#44512
#15023