Finite Group Databasebeta

← #68299

Subgroups — #68299

Order 384.

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

normal subgroups only

This group has 246 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#5034C211yesmaximal, characteristic, complemented
3#4735C211yesmaximal, characteristic, complemented
4#4659C211yesmaximal, characteristic, complemented
5C3×#327C211yesmaximal, characteristic, complemented
6#4469C211yesmaximal, characteristic, complemented
7#4451C211yesmaximal, characteristic, complemented
8#1616—31nomaximal, Sylow
9C2×#237—21no—
10C3×#121C2211yescharacteristic
11#939—21no—
12#939—21no—
13#930C2211yescharacteristic
14C2×C4⋊Dic3C2211yescharacteristic
15C2×Dic3⋊C4C2211yescharacteristic
16#896—21no—
17#884—21no—
18#878—21no—
19C3×C22⋊C8C2211yescharacteristic
20C3×#95C2211yescharacteristic
21#832—21no—
22C3⋊#95—21no—
23C3⋊#95—21no—
24#830C2211yescharacteristic
25#821—21no—
26#817—21no—
27#808—21no—
28#537—31no—
29#481—31no—
30#481—31no—
31#400—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⋊Dic3D411yescharacteristic
60Dic3⋊C4—21no—
61Dic3⋊C4—21no—
62Dic3⋊C4—21no—
63C2×C3⋊C8—21no—
64C2×#41—61no—
65C22⋊Q8—61no—
66C22⋊Q8—61no—
67#121—31no—
68#121—61no—
69#121—61no—
70#121—61no—
71#121C2×D311yescharacteristic
72C2×C4⋊C4—31no—
73C2×C4⋊C4—31no—
74#106—61no—
75#106—61no—
76#102—61no—
77#102—61no—
78C22⋊C8—31no—
79C22⋊C8C2×D311yescharacteristic
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×C4C2×D411yescharacteristic
95C2×C3×C4C2×D411yescharacteristic
96#67—42no—
97#67—41no—
98#67—41no—
99#67—81no—
100C2×Dic3—21no—
101C2×Dic3—21no—
102C2×Dic3—21no—
103C2×Dic3—21no—
104C2×Dic3—42no—
105C2×Dic3—41no—
106C2×Dic3—42no—
107C2×Dic3—41no—
108C2×Dic3SD1612yes—
109D12—41no—
110C4×D3—42no—
111Dic6—42no—
112C3×C8—41no—
113C3⋊C8—41no—
114#41—122no—
115#41—122no—
116C2×Q8—61no—
117C2×D4—21no—
118C2×D4—31no—
119C2×D4—61no—
120C2×D4—61no—
121C2×D4—61no—
122C2×D4D1211yescharacteristic
123C22×C4—21no—
124C22×C4—31no—
125C22×C4—31no—
126C22×C4—61no—
127C22×C4—61no—
128C22×C4—61no—
129C22×C4C22×D311yescharacteristic
130C2×C8—21no—
131C2×C8—61no—
132C4⋊C4—31no—
133C4⋊C4—61no—
134C4⋊C4—61no—
135C4⋊C4—61no—
136C4⋊C4—61no—
137C4⋊C4—61no—
138C4⋊C4D1211yescharacteristic
139C22⋊C4—21no—
140C22⋊C4—61no—
141C22⋊C4—61no—
142C22⋊C4—61no—
143C22×C3—21no—
144C22×C3—21no—
145C22×C3—41no—
146C22×C3—41no—
147C22×C3—41no—
148C22×C3#12011yescharacteristic
149C22×C3C2×SD1611yescharacteristic
150C22×C3AGL(1,ℤ/8)11yescharacteristic
151C2×D3—41no—
152C2×D3—41no—
153C2×D3—41no—
154C3×C4—21no—
155C3×C4—21no—
156C3×C4—41no—
157C3×C4—81no—
158Dic3—22no—
159Dic3—41no—
160Dic3—41no—
161Dic3—81no—
162C23—21no—
163C23—61no—
164C23D3×D411yescharacteristic
165Q8—122no—
166D4—41no—
167D4—41no—
168D4—41no—
169D4—121no—
170D4—122no—
171D4—121no—
172D4—121no—
173D4—241no—
174C2×C4—21no—
175C2×C4—32no—
176C2×C4—41no—
177C2×C4—41no—
178C2×C4—41no—
179C2×C4—41no—
180C2×C4—61no—
181C2×C4—61no—
182C2×C4—61no—
183C2×C4—61no—
184C2×C4—122no—
185C2×C4—122no—
186C2×C4—121no—
187C2×C4—122no—
188C2×C4—121no—
189C2×C4C2×D1211yescharacteristic
190C2×C4D3×D411yescharacteristic
191C8—41no—
192C8—121no—
193C2×C3—21no—
194C2×C3—21no—
195C2×C3—81no—
196C2×C3#45011yescharacteristic
197C2×C3#45411yescharacteristic
198C2×C3#45811yescharacteristic
199D3—81no—
200C22—21no—
201C22—21no—
202C22—41no—
203C22—41no—
204C22—41no—
205C22—121no—
206C22—121no—
207C22—121no—
208C22#88211yescharacteristic
209C22#91111yescharacteristic
210C22D3×SD1611yescharacteristic
211C4—21no—
212C4—21no—
213C4—41no—
214C4—62no—
215C4—81no—
216C4—121no—
217C4—121no—
218C4—241no—
219C3#161611yescharacteristic, complemented, Sylow
220C2—21no—
221C2—21no—
222C2—81no—
223C2—241no—
224C2#474211yescharacteristic
225C2#477611yescharacteristic
226C2#482711yescharacteristic
subgroupG/H
centerC22#882
commutator, cosocleC22×C3×C4C23
FrattiniC22×C4C22×D3
FittingC3×#327C2
socleC22×C3#120
prime radical#5597C2
hypercenter#327D3
hypocenterC3#1616
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. #5034C2semidirect
3. #4735C2semidirect
4. #4659C2semidirect
5. C3×#327C2semidirect
6. #4469C2semidirect
7. #4451C2semidirect
33. #327D3semidirect
219. C3#1616semidirect

Maximal subgroups

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

subgroupindex
#55972
#50342
#47352
#46592
C3×#3272
#44692
#44512
#16163