Finite Group Databasebeta

← #70988

Subgroups — #70988

Order 384.

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

normal subgroups only

This group has 236 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#5670C211yesmaximal, characteristic, complemented
2C3×#454C211yesmaximal, characteristic, complemented
3#5246C211yesmaximal, characteristic, complemented
4#5245C211yesmaximal, characteristic, complemented
5#4542C211yesmaximal, characteristic, complemented
6#4478C211yesmaximal, characteristic, complemented
7#4477C211yesmaximal, characteristic, complemented
8#2204—31nomaximal, Sylow
9C3×2−1+4C2211yescharacteristic
10C3×AGL(1,ℤ/8)—21no—
11C3×#124C2211yescharacteristic
12#951—21no—
13#949—21no—
14#947C2211yescharacteristic
15#944C2211yescharacteristic
16#939—21no—
17#885—21no—
18#884—21no—
19#67⋊C4—21no—
20#877—21no—
21#872—21no—
22C3×#104—21no—
23C3×#101C2211yescharacteristic
24#837—21no—
25#837—21no—
26#836C2211yescharacteristic
27#836C2211yescharacteristic
28#555—31no—
29#456—31no—
30#454—31no—
31#454D311yescharacteristic, complemented
32#356—31no—
33#355—31no—
34#355—31no—
35C3×#41—21no—
36C3×#41—21no—
37C3×#41—21no—
38C3×#41—41no—
39C2×C3×Q8—41no—
40C2×C3×Q8C2311yescharacteristic
41C2×C3×D4D411yescharacteristic
42C2×#67—21no—
43C2×#67—21no—
44C22×Dic3—21no—
45C2×D12D411yescharacteristic
46C2×C4×D3—21no—
47C2×Dic6D411yescharacteristic
48C3×SD16—41no—
49C3×D8—41no—
50C3×M16—21no—
51C3×C22⋊C4—21no—
52C3×C22⋊C4—21no—
53C3×C42D411yescharacteristic
54#217—21no—
55#216—41no—
56#215—41no—
57#214—41no—
58#213—41no—
59(C2×D3)⋊C4—21no—
60(C2×D3)⋊C4—21no—
61(C2×D3)⋊C4—21no—
62Dic3⋊C4—21no—
63Dic3⋊C4—21no—
64Dic3⋊C4—21no—
65C4×Dic3—21no—
66C4×Dic3D411yescharacteristic
67C4×Dic3D411yescharacteristic
68#208—21no—
69#208—21no—
702−1+4C2×D311yescharacteristic
71#137—61no—
72AGL(1,ℤ/8)—21no—
73AGL(1,ℤ/8)—61no—
74C4⋊Q8—31no—
75#124—31no—
76#124—61no—
77#124C2×D311yescharacteristic
78#121—61no—
79#121—61no—
80C4×D4—61no—
81#117—61no—
82#117—61no—
83#104—21no—
84#104—61no—
85#104—61no—
86#101—31no—
87#101—31no—
88#101C2×D311yescharacteristic
89C23×C3—21no—
90C22×D3—21no—
91C3×Q8—21no—
92C3×Q8—21no—
93C3×Q8—21no—
94C3×Q8—41no—
95C3×D4—21no—
96C3×D4—21no—
97C3×D4—21no—
98C3×D4—41no—
99C3×D4—41no—
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
107#67—42no—
108#67—81no—
109C2×Dic3—21no—
110C2×Dic3—21no—
111C2×Dic3—21no—
112C2×Dic3—42no—
113D12—41no—
114C4×D3—42no—
115Dic6—41no—
116C3×C8—41no—
117C3⋊C8—41no—
118C3⋊C8—41no—
119#41—21no—
120#41—21no—
121#41—21no—
122#41—41no—
123C2×Q8—31no—
124C2×Q8—41no—
125C2×Q8C22×D311yescharacteristic
126C2×D4—31no—
127C2×D4—61no—
128C2×D4—61no—
129C2×D4#6711yescharacteristic
130C22×C4—61no—
131C22×C4—61no—
132Q16—121no—
133SD16—41no—
134SD16—121no—
135SD16—121no—
136D8—41no—
137D8—121no—
138M16—21no—
139M16—61no—
140M16—61no—
141C4⋊C4—61no—
142C4⋊C4—61no—
143C4⋊C4—61no—
144C22⋊C4—21no—
145C22⋊C4—21no—
146C22⋊C4—61no—
147C22⋊C4—61no—
148C22⋊C4—61no—
149C22⋊C4—61no—
150C42—31no—
151C42—31no—
152C42—61no—
153C42#6711yescharacteristic
154C22×C3—41no—
155C22×C3—41no—
156C22×C3—41no—
157C22×C3—41no—
158C22×C3#12011yescharacteristic
159C2×D3—41no—
160C2×D3—41no—
161C2×D3—41no—
162C3×C4—21no—
163C3×C4—21no—
164C3×C4—21no—
165C3×C4—41no—
166C3×C4—41no—
167Dic3—22no—
168Dic3—41no—
169Dic3—41no—
170C23—21no—
171C23—61no—
172Q8—21no—
173Q8—21no—
174Q8—21no—
175Q8—41no—
176Q8—121no—
177D4—21no—
178D4—21no—
179D4—21no—
180D4—41no—
181D4—41no—
182D4—121no—
183D4—122no—
184D4—241no—
185C2×C4—21no—
186C2×C4—41no—
187C2×C4—41no—
188C2×C4—41no—
189C2×C4—61no—
190C2×C4—61no—
191C2×C4—61no—
192C2×C4—122no—
193C2×C4—122no—
194C2×C4D3×D411yescharacteristic
195C2×C4D3×D411yescharacteristic
196C2×C4C2×#6711yescharacteristic
197C8—41no—
198C8—121no—
199C8—121no—
200C2×C3—21no—
201C2×C3—81no—
202C2×C3—81no—
203C2×C3B(4,𝔽2)11yescharacteristic
204D3—81no—
205C22—41no—
206C22—41no—
207C22—41no—
208C22—41no—
209C22—121no—
210C22—121no—
211C22—121no—
212C22#93711yescharacteristic
213C4—21no—
214C4—21no—
215C4—21no—
216C4—41no—
217C4—41no—
218C4—62no—
219C4—121no—
220C4—121no—
221C3#220411yescharacteristic, complemented, Sylow
222C2—21no—
223C2—81no—
224C2—81no—
225C2—241no—
226C2#503211yescharacteristic
subgroupG/H
centerC2#5032
commutator, cosocleC2×C3×Q8C23
FrattiniC2×Q8C22×D3
FittingC3×#454C2
socleC2×C3B(4,𝔽2)
hypercenter#454D3
hypocenterC3#2204
LCSG ▷ C2×C3×Q8 ▷ C22×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2 ◁ C22 ◁ C2×Q8 ◁ #454

Complements

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

NH ≅ G/Ntype
1. #5670C2semidirect
2. C3×#454C2semidirect
3. #5246C2semidirect
4. #5245C2semidirect
5. #4542C2semidirect
6. #4478C2semidirect
7. #4477C2semidirect
31. #454D3semidirect
221. C3#2204semidirect

Maximal subgroups

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

subgroupindex
#56702
C3×#4542
#52462
#52452
#45422
#44782
#44772
#22043