Finite Group Databasebeta

← #68297

Subgroups — #68297

Order 384.

254 conjugacy classes of subgroups; only the 51 normal subgroups are shown.

normal subgroups only

This group has 254 conjugacy classes of subgroups — too many to draw. Only its 51 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#5597C211yesmaximal, characteristic
2#5033C211yesmaximal, characteristic, complemented
3#4736C211yesmaximal, characteristic, complemented
4#4661C211yesmaximal, characteristic, complemented
5C3×#327C211yesmaximal, characteristic, complemented
6#4469C211yesmaximal, characteristic, complemented
7#4451C211yesmaximal, characteristic, complemented
8#1610—31nomaximal, Sylow
9C2×#237—21no—
10C3×#121C2211yescharacteristic
11#939—21no—
12#939—21no—
13#930C2211yescharacteristic
14C2×C4⋊Dic3—21no—
15C2×C4⋊Dic3C2211yescharacteristic
16C2×Dic3⋊C4C2211yescharacteristic
17#896—21no—
18#884—21no—
19#878—21no—
20C3×C22⋊C8C2211yescharacteristic
21C3×#95C2211yescharacteristic
22#832—21no—
23C3⋊#95—21no—
24#830C2211yescharacteristic
25#821—21no—
26#818—21no—
27#807—21no—
28#537—31no—
29#480—31no—
30#480—31no—
31#398—31no—
32#327—31no—
33#327D311yescharacteristic, complemented
34#324—31no—
35C2×C3×D4—21no—
36C2×C3×D4D411yescharacteristic
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×D12D411yescharacteristic
49C2×C4×D3—21no—
50C2×Dic6—21no—
51C2×C3×C8—21no—
52C3×C4⋊C4D411yescharacteristic
53C3×C22⋊C4—21no—
54#217—21no—
55(C2×D3)⋊C4—21no—
56(C2×D3)⋊C4—21no—
57C4⋊Dic3—21no—
58C4⋊Dic3—21no—
59C4⋊Dic3—42no—
60C4⋊Dic3D411yescharacteristic
61Dic3⋊C4—21no—
62Dic3⋊C4—21no—
63Dic3⋊C4—21no—
64C2×C3⋊C8—21no—
65C2×#41—61no—
66C22⋊Q8—61no—
67C22⋊Q8—61no—
68#121—31no—
69#121—61no—
70#121—61no—
71#121—61no—
72#121C2×D311yescharacteristic
73C2×C4⋊C4—31no—
74C2×C4⋊C4—31no—
75C2×C4⋊C4—61no—
76#107—61no—
77#107—61no—
78#102—61no—
79#102—61no—
80C22⋊C8—31no—
81C22⋊C8C2×D311yescharacteristic
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×C4C2×D411yescharacteristic
96C2×C3×C4C2×D411yescharacteristic
97#67—42no—
98#67—41no—
99#67—41no—
100#67—81no—
101C2×Dic3—21no—
102C2×Dic3—21no—
103C2×Dic3—21no—
104C2×Dic3—21no—
105C2×Dic3—42no—
106C2×Dic3—41no—
107C2×Dic3—41no—
108C2×Dic3—41no—
109C2×Dic3—41no—
110C2×Dic3D812yes—
111D12—41no—
112C4×D3—42no—
113Dic6—42no—
114C3×C8—41no—
115C3⋊C8—41no—
116#41—122no—
117#41—122no—
118C2×Q8—61no—
119C2×D4—21no—
120C2×D4—31no—
121C2×D4—61no—
122C2×D4—61no—
123C2×D4—61no—
124C2×D4D1211yescharacteristic
125C22×C4—21no—
126C22×C4—31no—
127C22×C4—31no—
128C22×C4—61no—
129C22×C4—61no—
130C22×C4—61no—
131C22×C4C22×D311yescharacteristic
132C2×C8—21no—
133C2×C8—61no—
134C4⋊C4—31no—
135C4⋊C4—61no—
136C4⋊C4—61no—
137C4⋊C4—61no—
138C4⋊C4—61no—
139C4⋊C4—61no—
140C4⋊C4—122no—
141C4⋊C4D1211yescharacteristic
142C22⋊C4—21no—
143C22⋊C4—61no—
144C22⋊C4—61no—
145C22⋊C4—61no—
146C22×C3—21no—
147C22×C3—21no—
148C22×C3—41no—
149C22×C3—41no—
150C22×C3—41no—
151C22×C3#12011yescharacteristic
152C22×C3C2×D811yescharacteristic
153C22×C3AGL(1,ℤ/8)11yescharacteristic
154C2×D3—41no—
155C2×D3—41no—
156C2×D3—41no—
157C3×C4—21no—
158C3×C4—21no—
159C3×C4—42no—
160C3×C4—41no—
161Dic3—22no—
162Dic3—41no—
163Dic3—41no—
164Dic3—81no—
165C23—21no—
166C23—61no—
167C23D3×D411yescharacteristic
168Q8—122no—
169D4—41no—
170D4—41no—
171D4—41no—
172D4—121no—
173D4—122no—
174D4—121no—
175D4—121no—
176D4—241no—
177C2×C4—22no—
178C2×C4—21no—
179C2×C4—32no—
180C2×C4—41no—
181C2×C4—41no—
182C2×C4—41no—
183C2×C4—61no—
184C2×C4—61no—
185C2×C4—61no—
186C2×C4—61no—
187C2×C4—122no—
188C2×C4—122no—
189C2×C4—121no—
190C2×C4—121no—
191C2×C4—121no—
192C2×C4—121no—
193C2×C4C2×D1211yescharacteristic
194C2×C4D3×D411yescharacteristic
195C8—41no—
196C8—121no—
197C2×C3—21no—
198C2×C3—21no—
199C2×C3—81no—
200C2×C3#44711yescharacteristic
201C2×C3#45611yescharacteristic
202C2×C3#45811yescharacteristic
203D3—81no—
204C22—21no—
205C22—21no—
206C22—41no—
207C22—41no—
208C22—41no—
209C22—121no—
210C22—121no—
211C22—121no—
212C22#88211yescharacteristic
213C22D3×D811yescharacteristic
214C22#91411yescharacteristic
215C4—21no—
216C4—21no—
217C4—42no—
218C4—41no—
219C4—62no—
220C4—121no—
221C4—121no—
222C4—241no—
223C3#161011yescharacteristic, complemented, Sylow
224C2—21no—
225C2—21no—
226C2—81no—
227C2—241no—
228C2#474211yescharacteristic
229C2#477311yescharacteristic
230C2#482611yescharacteristic
subgroupG/H
centerC22#882
commutator, cosocleC22×C3×C4C23
FrattiniC22×C4C22×D3
FittingC3×#327C2
socleC22×C3#120
prime radical#5597C2
hypercenter#327D3
hypocenterC3#1610
LCSG ▷ C22×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. #5033C2semidirect
3. #4736C2semidirect
4. #4661C2semidirect
5. C3×#327C2semidirect
6. #4469C2semidirect
7. #4451C2semidirect
33. #327D3semidirect
223. C3#1610semidirect

Maximal subgroups

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

subgroupindex
#55972
#50332
#47362
#46612
C3×#3272
#44692
#44512
#16103