Finite Group Databasebeta

← #70993

Subgroups — #70993

Order 384.

216 conjugacy classes of subgroups; only the 41 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 41 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#5735C211yesmaximal, characteristic, complemented
2C3×#456C211yesmaximal, characteristic, complemented
3#5246C211yesmaximal, characteristic, complemented
4#5245C211yesmaximal, characteristic
5#4548C211yesmaximal, characteristic, complemented
6#4478C211yesmaximal, characteristic, complemented
7#4477C211yesmaximal, characteristic, complemented
8#2203—31nomaximal, Sylow
9C3×2−1+4C2211yescharacteristic
10C3×#137—21no—
11C3×C4⋊Q8C2211yescharacteristic
12#951—21no—
13#949—21no—
14#947C2211yescharacteristic
15#946—21no—
16#944C2211yescharacteristic
17#898—21no—
18#896—21no—
19#894—21no—
20#889—21no—
21#876—21no—
22C3×#104—21no—
23C3×#101C2211yescharacteristic
24#837—21no—
25#837—21no—
26#836C2211yescharacteristic
27#836C2211yescharacteristic
28#563—31no—
29#456—31no—
30#456D311yescharacteristic, complemented
31#454—31no—
32#356—31no—
33#356—31no—
34#355—31no—
35C3×#41—21no—
36C3×#41—21no—
37C3×#41—21no—
38C3×#41—41no—
39C2×C3×Q8—41no—
40C2×C3×Q8D411yescharacteristic
41C2×C3×Q8C2311yescharacteristic
42C2×D12D411yescharacteristic
43C2×C4×D3—21no—
44C2×Dic6D411yescharacteristic
45C3×Q16—41no—
46C3×SD16—41no—
47C3×M16—21no—
48C3×C4⋊C4—21no—
49C3×C4⋊C4—21no—
50C3×C42D411yescharacteristic
51#216—41no—
52#215—41no—
53#214—41no—
54#213—41no—
55(C2×D3)⋊C4—21no—
56(C2×D3)⋊C4—21no—
57(C2×D3)⋊C4—21no—
58(C2×D3)⋊C4—21no—
59(C2×D3)⋊C4—21no—
60C4⋊Dic3—21no—
61Dic3⋊C4—21no—
62Dic3⋊C4—21no—
63Dic3⋊C4—21no—
64Dic3⋊C4—21no—
65Dic3⋊C4—21no—
66C4×Dic3D411yescharacteristic
67C4×Dic3D411yescharacteristic
68#208—21no—
69#208—21no—
702−1+4C2×D311yescharacteristic
71#137—21no—
72#137—61no—
73AGL(1,ℤ/8)—61no—
74C4⋊Q8—31no—
75C4⋊Q8C2×D311yescharacteristic
76#126—61no—
77#126—61no—
78#125—61no—
79#124—31no—
80#123—61no—
81C22⋊Q8—61no—
82C22⋊Q8—61no—
83#104—21no—
84#104—61no—
85#104—61no—
86#101—31no—
87#101—31no—
88#101C2×D311yescharacteristic
89C22×D3—21no—
90C3×Q8—21no—
91C3×Q8—21no—
92C3×Q8—21no—
93C3×Q8—41no—
94C3×Q8—41no—
95C3×D4—21no—
96C3×D4—21no—
97C3×D4—21no—
98C3×D4—41no—
99C2×C3×C4—21no—
100C2×C3×C4—21no—
101C2×C3×C4—41no—
102C2×C3×C4—41no—
103C2×C3×C4—41no—
104C2×C3×C4C2×D411yescharacteristic
105C2×C3×C4C2×D411yescharacteristic
106C2×C3×C4C2×D411yescharacteristic
107C2×Dic3—21no—
108C2×Dic3—21no—
109C2×Dic3—21no—
110D12—41no—
111C4×D3—81no—
112Dic6—41no—
113C3×C8—41no—
114C3⋊C8—41no—
115C3⋊C8—41no—
116#41—21no—
117#41—21no—
118#41—21no—
119#41—41no—
120C2×Q8—31no—
121C2×Q8—41no—
122C2×Q8#6711yescharacteristic
123C2×Q8C22×D311yescharacteristic
124C2×D4—31no—
125C22×C4—61no—
126Q16—41no—
127Q16—121no—
128SD16—41no—
129SD16—121no—
130SD16—121no—
131D8—121no—
132M16—21no—
133M16—61no—
134M16—61no—
135C4⋊C4—21no—
136C4⋊C4—21no—
137C4⋊C4—61no—
138C4⋊C4—61no—
139C4⋊C4—61no—
140C4⋊C4—61no—
141C4⋊C4—61no—
142C4⋊C4—61no—
143C22⋊C4—61no—
144C22⋊C4—61no—
145C22⋊C4—61no—
146C22⋊C4—61no—
147C22⋊C4—61no—
148C42—31no—
149C42—31no—
150C42#6711yescharacteristic
151C22×C3—41no—
152C22×C3#12011yescharacteristic
153C2×D3—42no—
154C2×D3—41no—
155C3×C4—21no—
156C3×C4—21no—
157C3×C4—21no—
158C3×C4—41no—
159C3×C4—41no—
160C3×C4—41no—
161Dic3—41no—
162Dic3—41no—
163Dic3—41no—
164C23—61no—
165Q8—21no—
166Q8—21no—
167Q8—21no—
168Q8—41no—
169Q8—41no—
170Q8—121no—
171D4—21no—
172D4—21no—
173D4—21no—
174D4—41no—
175D4—121no—
176C2×C4—21no—
177C2×C4—21no—
178C2×C4—41no—
179C2×C4—41no—
180C2×C4—41no—
181C2×C4—61no—
182C2×C4—61no—
183C2×C4—61no—
184C2×C4—241no—
185C2×C4D3×D411yescharacteristic
186C2×C4D3×D411yescharacteristic
187C2×C4C2×#6711yescharacteristic
188C8—41no—
189C8—121no—
190C8—121no—
191C2×C3—21no—
192C2×C3—81no—
193C2×C3B(4,𝔽2)11yescharacteristic
194D3—81no—
195C22—41no—
196C22—122no—
197C22—121no—
198C22#93711yescharacteristic
199C4—21no—
200C4—21no—
201C4—21no—
202C4—41no—
203C4—41no—
204C4—41no—
205C4—121no—
206C4—121no—
207C4—121no—
208C3#220311yescharacteristic, complemented, Sylow
209C2—21no—
210C2—81no—
211C2—241no—
212C2#503211yescharacteristic
subgroupG/H
centerC2#5032
commutator, cosocleC2×C3×Q8C23
FrattiniC2×Q8C22×D3
FittingC3×#456C2
socleC2×C3B(4,𝔽2)
prime radical#5245C2
hypercenter#456D3
hypocenterC3#2203
LCSG ▷ C2×C3×Q8 ▷ C22×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2 ◁ C22 ◁ C2×Q8 ◁ #456

Complements

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

NH ≅ G/Ntype
1. #5735C2semidirect
2. C3×#456C2semidirect
3. #5246C2semidirect
5. #4548C2semidirect
6. #4478C2semidirect
7. #4477C2semidirect
30. #456D3semidirect
208. C3#2203semidirect

Maximal subgroups

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

subgroupindex
#57352
C3×#4562
#52462
#52452
#45482
#44782
#44772
#22033