Finite Group Databasebeta

← #85349

Subgroups — #85349

Order 384.

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

normal subgroups only

This group has 202 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
1C4×#41⋊C3C211yesmaximal, characteristic, complemented
2#5427C211yesmaximal, characteristic
3#5425C211yesmaximal, characteristic, complemented
4#5395C211yesmaximal, characteristic, complemented
5#5394C211yesmaximal, characteristic, complemented
6#5389C211yesmaximal, characteristic, complemented
7#5388C211yesmaximal, characteristic, complemented
8#3111—31nomaximal, Sylow
9C2×#41⋊C3C2211yescharacteristic
10C2×GL(2,𝔽3)C2211yescharacteristic
11C2×#226C2211yescharacteristic
12#876—41nomaximal
13C4×SL(2,𝔽3)C2211yescharacteristic
14C4×SL(2,𝔽3)C2211yescharacteristic
15#859C2211yescharacteristic
16#859C2211yescharacteristic
17#529—31no—
18C4×#41D311yescharacteristic, complemented
19#485—31no—
20#482—31no—
21#479—31no—
22#478—31no—
23#464—31no—
24#463—31no—
25#452—31no—
26#449—31no—
27#442—31no—
28#441—31no—
29#440—31no—
30#440—31no—
31#432—31no—
32#41⋊C3—21no—
33C2×SL(2,𝔽3)C2311yescharacteristic
34GL(2,𝔽3)—21no—
35#226—21no—
36C3×C42—41no—
37(C2×D3)⋊C4—41no—
38(C2×D3)⋊C4—41no—
39(C2×D3)⋊C4—41no—
40Dic3⋊C4—41no—
41Dic3⋊C4—41no—
42Dic3⋊C4—41no—
43C2×#41C2×D311yescharacteristic
44C2×Q16—31no—
45C2×SD16—31no—
46C2×SD16—31no—
47C2×D8—31no—
48#126—61no—
49#125—31no—
50#124—31no—
51#123—61no—
52C22⋊Q8—31no—
53#121—31no—
54C4×Q8—31no—
55C4×Q8C2×D311yescharacteristic
56C4×Q8C2×D311yescharacteristic
57C4×D4—31no—
58C4×D4—31no—
59C4×D4—31no—
60C4×D4—61no—
61#117—31no—
62#117—61no—
63C2×C42—31no—
64#107—31no—
65#107—31no—
66#106—31no—
67#106—31no—
68C4⋊C8—31no—
69C4⋊C8—31no—
70#103—31no—
71#103—31no—
72#103—31no—
73#103—31no—
74#102—31no—
75#102—31no—
76#102—31no—
77#102—31no—
78C22⋊C8—31no—
79C22⋊C8—31no—
80#97—31no—
81#97—31no—
82C22×D3—41no—
83C2×C3×C4—41no—
84C2×C3×C4—41no—
85C2×C3×C4—41no—
86C2×Dic3—41no—
87C2×Dic3—41no—
88C2×Dic3—41no—
89SL(2,𝔽3)#4111yescharacteristic
90#41—21no—
91#41—61no—
92C2×Q8—31no—
93C2×Q8C22×D311yescharacteristic
94C2×D4—31no—
95C2×D4—31no—
96C2×D4—61no—
97C22×C4—31no—
98C22×C4—31no—
99C22×C4—31no—
100C22×C4—61no—
101Q16—61no—
102Q16—61no—
103SD16—62no—
104SD16—61no—
105SD16—61no—
106D8—62no—
107C2×C8—31no—
108C2×C8—31no—
109C2×C8—31no—
110C2×C8—31no—
111C4⋊C4—31no—
112C4⋊C4—31no—
113C4⋊C4—31no—
114C4⋊C4—31no—
115C4⋊C4—61no—
116C4⋊C4—61no—
117C4⋊C4—61no—
118C4⋊C4—61no—
119C22⋊C4—61no—
120C22⋊C4—61no—
121C22⋊C4—61no—
122C22⋊C4—61no—
123C22⋊C4—61no—
124C22⋊C4—61no—
125C42—31no—
126C42—31no—
127C42—31no—
128C42—61no—
129C42S411yescharacteristic
130C22×C3—41no—
131C2×D3—81no—
132C2×D3—81no—
133C2×D3—81no—
134C3×C4—81no—
135C3×C4—81no—
136C3×C4—81no—
137Dic3—81no—
138Dic3—81no—
139Dic3—81no—
140C23—31no—
141C23—61no—
142Q8—31no—
143Q8—61no—
144Q8—61no—
145Q8#23511yescharacteristic
146D4—32no—
147D4—61no—
148D4—61no—
149D4—61no—
150D4—122no—
151C2×C4—31no—
152C2×C4—31no—
153C2×C4—31no—
154C2×C4—61no—
155C2×C4—61no—
156C2×C4—61no—
157C2×C4—61no—
158C2×C4—61no—
159C2×C4—121no—
160C2×C4—121no—
161C2×C4—122no—
162C2×C4C2×S411yescharacteristic
163C2×C4C2×S411yescharacteristic
164C2×C4C2×S411yescharacteristic
165C8—61no—
166C8—61no—
167C8—61no—
168C8—61no—
169C2×C3—41no—
170C2×C3—41no—
171C2×C3—41no—
172D3—161no—
173C22—61no—
174C22—61no—
175C22—61no—
176C22—121no—
177C22—121no—
178C22—121no—
179C22C22×S411yescharacteristic
180C4—21no—
181C4—21no—
182C4—21no—
183C4—31no—
184C4—31no—
185C4—61no—
186C4—61no—
187C4—121no—
188C4—121no—
189C4—121no—
190C3—41noSylow
191C2—121no—
192C2—241no—
193C2#591211yescharacteristic
194C2#592311yescharacteristic
195C2#592311yescharacteristic
subgroupG/H
center, Frattini, socleC22C22×S4
commutator, cosocleC2×SL(2,𝔽3)C23
FittingC4×#41D3
prime radical#5427C2
hypercenterC42S4
hypocenterSL(2,𝔽3)#41
LCSG ▷ C2×SL(2,𝔽3) ▷ SL(2,𝔽3)
UCSC1 ◁ C22 ◁ C42

Complements

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

NH ≅ G/Ntype
1. C4×#41⋊C3C2semidirect
3. #5425C2semidirect
4. #5395C2semidirect
5. #5394C2semidirect
6. #5389C2semidirect
7. #5388C2semidirect
18. C4×#41D3semidirect

Maximal subgroups

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

subgroupindex
C4×#41⋊C32
#54272
#54252
#53952
#53942
#53892
#53882
#31113
#8764