Finite Group Databasebeta

← #85388

Subgroups — #85388

Order 384.

214 conjugacy classes of subgroups; only the 32 normal subgroups are shown.

normal subgroups only

This group has 214 conjugacy classes of subgroups — too many to draw. Only its 32 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#548⋊C3C211yesmaximal, characteristic, complemented
2#5427C211yesmaximal, characteristic, complemented
3#5424C211yesmaximal, characteristic, complemented
4#5421C211yesmaximal, characteristic, complemented
5#5420C211yesmaximal, characteristic, complemented
6#5394C211yesmaximal, characteristic, complemented
7#5386C211yesmaximal, characteristic, complemented
8#3321—31nomaximal, Sylow
9C2×#41⋊C3C2211yescharacteristic
10C22×SL(2,𝔽3)C2211yescharacteristic
11C2×GL(2,𝔽3)C2211yescharacteristic
12#885—41nomaximal
13C4×SL(2,𝔽3)C2211yescharacteristic
14#859C2211yescharacteristic
15#859C2211yescharacteristic
16#859C2211yescharacteristic
17#555—31no—
18#548D311yescharacteristic, complemented
19#489—31no—
20#485—31no—
21#482—31no—
22#475—31no—
23#469—31no—
24#465—31no—
25#463—31no—
26#449—31no—
27#448—31no—
28#440—31no—
29#435—31no—
30#419—31no—
31#416—31no—
32C2×#67—41no—
33C2×Dic6—41no—
34#41⋊C3—21no—
35C2×SL(2,𝔽3)—21no—
36C2×SL(2,𝔽3)C2311yescharacteristic
37GL(2,𝔽3)—21no—
38C3×C22⋊C4—41no—
39#217—41no—
40(C2×D3)⋊C4—41no—
41(C2×D3)⋊C4—41no—
42C4×Dic3—41no—
43C2×#41C2×D311yescharacteristic
44C22×Q8C2×D311yescharacteristic
45C2×SD16—31no—
46C2×SD16—31no—
47C2×D8—31no—
48C2×M16—31no—
49C22×C8—31no—
50C4⋊Q8—31no—
51#124—31no—
52#124—61no—
53#124—61no—
54C22⋊Q8—31no—
55#121—31no—
56#121—31no—
57#121—31no—
58#121—61no—
59C4×Q8C2×D311yescharacteristic
60C4×D4—31no—
61C4×D4—31no—
62#117—31no—
63#117—31no—
64#117—61no—
65#107—31no—
66#106—31no—
67#106—31no—
68C4⋊C8—31no—
69#103—31no—
70#103—31no—
71#103—31no—
72#103—31no—
73#103—31no—
74#103—31no—
75#102—31no—
76#102—31no—
77#102—31no—
78#102—31no—
79C22⋊C8—31no—
80C22⋊C8—31no—
81#97—31no—
82C23×C3—41no—
83C22×D3—41no—
84C2×C3×C4—41no—
85C2×C3×C4—41no—
86#67—82no—
87C2×Dic3—41no—
88C2×Dic3—41no—
89C2×Dic3—41no—
90Dic6—82no—
91SL(2,𝔽3)#4111yescharacteristic
92#41—21no—
93#41—61no—
94C2×Q8—21no—
95C2×Q8—31no—
96C2×Q8—61no—
97C2×Q8—61no—
98C2×Q8C22×D311yescharacteristic
99C2×D4—31no—
100C2×D4—31no—
101C2×D4—31no—
102C2×D4—61no—
103C2×D4—61no—
104C22×C4—31no—
105C22×C4—31no—
106C22×C4—61no—
107SD16—61no—
108SD16—61no—
109SD16—121no—
110D8—121no—
111M16—121no—
112C2×C8—31no—
113C2×C8—31no—
114C2×C8—31no—
115C2×C8—31no—
116C2×C8—121no—
117C4⋊C4—31no—
118C4⋊C4—31no—
119C4⋊C4—31no—
120C4⋊C4—31no—
121C4⋊C4—31no—
122C4⋊C4—61no—
123C22⋊C4—31no—
124C22⋊C4—61no—
125C22⋊C4—61no—
126C22⋊C4—61no—
127C22⋊C4—61no—
128C22⋊C4—61no—
129C22⋊C4—61no—
130C22⋊C4S411yescharacteristic
131C42—31no—
132C42—61no—
133C42—61no—
134C22×C3—41no—
135C22×C3—81no—
136C22×C3—81no—
137C22×C3—81no—
138C2×D3—81no—
139C2×D3—81no—
140C2×D3—81no—
141C3×C4—81no—
142C3×C4—81no—
143Dic3—42no—
144Dic3—81no—
145Dic3—81no—
146C23—31no—
147C23—61no—
148C23C2×S411yescharacteristic
149Q8—31no—
150Q8—61no—
151Q8—61no—
152Q8—241no—
153Q8#23711yescharacteristic
154D4—61no—
155D4—61no—
156D4—61no—
157D4—61no—
158D4—121no—
159D4—241no—
160D4—241no—
161C2×C4—31no—
162C2×C4—31no—
163C2×C4—31no—
164C2×C4—61no—
165C2×C4—61no—
166C2×C4—61no—
167C2×C4—61no—
168C2×C4—61no—
169C2×C4—61no—
170C2×C4—61no—
171C2×C4—122no—
172C2×C4C2×S411yescharacteristic
173C2×C4C2×S411yescharacteristic
174C8—61no—
175C8—61no—
176C8—61no—
177C8—61no—
178C2×C3—41no—
179C2×C3—41no—
180C2×C3—41no—
181C2×C3—161no—
182D3—161no—
183C22—21no—
184C22—21no—
185C22—21no—
186C22—61no—
187C22—61no—
188C22—61no—
189C22—121no—
190C22—121no—
191C22—121no—
192C22C22×S411yescharacteristic
193C4—21no—
194C4—21no—
195C4—31no—
196C4—31no—
197C4—61no—
198C4—61no—
199C4—121no—
200C4—121no—
201C4—121no—
202C3—41noSylow
203C2—41no—
204C2—121no—
205C2—241no—
206C2#591411yescharacteristic
207C2#592311yescharacteristic
208C2#592611yescharacteristic
subgroupG/H
center, Frattini, socleC22C22×S4
commutator, cosocleC2×SL(2,𝔽3)C23
Fitting#548D3
hypercenterC22⋊C4S4
hypocenterSL(2,𝔽3)#41
LCSG ▷ C2×SL(2,𝔽3) ▷ SL(2,𝔽3)
UCSC1 ◁ C22 ◁ C22⋊C4

Complements

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

NH ≅ G/Ntype
1. #548⋊C3C2semidirect
2. #5427C2semidirect
3. #5424C2semidirect
4. #5421C2semidirect
5. #5420C2semidirect
6. #5394C2semidirect
7. #5386C2semidirect
18. #548D3semidirect

Maximal subgroups

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

subgroupindex
#548⋊C32
#54272
#54242
#54212
#54202
#53942
#53862
#33213
#8854