Finite Group Databasebeta

← #85354

Subgroups — #85354

Order 384.

273 conjugacy classes of subgroups; only the 46 normal subgroups are shown.

normal subgroups only

This group has 273 conjugacy classes of subgroups — too many to draw. Only its 46 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
1C2×#983C211yesmaximal, characteristic
2#519⋊C3C211yesmaximal, characteristic, complemented
3C2×#859C211yesmaximal, characteristic, complemented
4#5394C211yesmaximal, characteristic, complemented
5C4×GL(2,𝔽3)C211yesmaximal, characteristic, complemented
6#5388C211yesmaximal, characteristic, complemented
7C4×#226C211yesmaximal, characteristic, complemented
8#2947—31nomaximal, Sylow
9C22×SL(2,𝔽3)C2211yescharacteristic
10#983C414yescomplemented
11C2×GL(2,𝔽3)C2211yescharacteristic
12C2×#226C2211yescharacteristic
13#67⋊C4—41nomaximal
14C4×SL(2,𝔽3)C2211yescharacteristic
15C4×SL(2,𝔽3)C2211yescharacteristic
16#859C2211yescharacteristic
17#859C2211yescharacteristic
18C2×#137—31no—
19#520—31no—
20#519D311yescharacteristic, complemented
21#441—31no—
22#441—31no—
23#440—31no—
24#440—31no—
25C4×Q16—31no—
26C4×Q16—31no—
27C4×SD16—31no—
28C4×SD16—31no—
29#428—31no—
30#416—31no—
31C2×#103—31no—
32#405—31no—
33C2×#67—41no—
34C22×Dic3—41no—
35C2×C4×D3—41no—
36C2×SL(2,𝔽3)C2×C412yescomplemented
37C2×SL(2,𝔽3)C2311yescharacteristic
38GL(2,𝔽3)C2×C412yescomplemented
39#226C2×C412yescomplemented
40C3×C22⋊C4—41no—
41(C2×D3)⋊C4—41no—
42Dic3⋊C4—41no—
43C4×Dic3—41no—
44C2×#41—31no—
45C22×Q8C2×D311yescharacteristic
46#137—34no—
47#137—34no—
48C2×Q16—31no—
49C2×Q16—31no—
50C2×SD16—31no—
51C2×SD16—31no—
52C2×M16—31no—
53C22×C8—31no—
54C4×Q8—31no—
55C4×Q8—31no—
56C4×Q8—31no—
57C4×Q8—31no—
58C4×Q8C2×D311yescharacteristic
59C4×Q8C2×D311yescharacteristic
60C4×D4—31no—
61C4×D4—31no—
62C4×D4—61no—
63C4×D4—61no—
64#117—31no—
65#117—31no—
66#117—61no—
67C2×C4⋊C4—31no—
68C2×C4⋊C4—61no—
69#107—31no—
70#107—31no—
71#106—31no—
72#106—31no—
73#103—31no—
74#103—31no—
75#103—31no—
76#103—31no—
77#103—31no—
78#103—31no—
79#102—31no—
80#102—31no—
81#97—31no—
82#97—31no—
83C4×C8—31no—
84C4×C8—31no—
85C23×C3—41no—
86C22×D3—41no—
87C2×C3×C4—41no—
88C2×C3×C4—41no—
89#67—44no—
90C2×Dic3—41no—
91C2×Dic3—41no—
92C2×Dic3—41no—
93C2×Dic3—82no—
94C4×D3—82no—
95SL(2,𝔽3)C22×C411yescharacteristic
96#41—34no—
97#41—62no—
98C2×Q8—31no—
99C2×Q8—32no—
100C2×Q8—31no—
101C2×Q8C4×D312yescomplemented
102C2×Q8C22×D311yescharacteristic
103C2×D4—31no—
104C2×D4—61no—
105C22×C4—31no—
106C22×C4—61no—
107C22×C4—61no—
108C22×C4—61no—
109C22×C4—61no—
110Q16—34no—
111Q16—32no—
112Q16—32no—
113SD16—34no—
114SD16—32no—
115SD16—32no—
116M16—34no—
117C2×C8—31no—
118C2×C8—31no—
119C2×C8—31no—
120C2×C8—31no—
121C2×C8—62no—
122C4⋊C4—31no—
123C4⋊C4—31no—
124C4⋊C4—31no—
125C4⋊C4—31no—
126C4⋊C4—31no—
127C4⋊C4—31no—
128C4⋊C4—61no—
129C4⋊C4—61no—
130C4⋊C4—61no—
131C22⋊C4—31no—
132C22⋊C4—61no—
133C22⋊C4—61no—
134C22⋊C4S411yescharacteristic
135C42—31no—
136C42—31no—
137C42—61no—
138C42—61no—
139C42—61no—
140C22×C3—42no—
141C22×C3—41no—
142C22×C3—81no—
143C22×C3—81no—
144C2×D3—42no—
145C2×D3—81no—
146C2×D3—81no—
147C3×C4—81no—
148C3×C4—81no—
149Dic3—42no—
150Dic3—42no—
151Dic3—81no—
152C23—61no—
153C23C2×S411yescharacteristic
154Q8—32no—
155Q8—31no—
156Q8—32no—
157Q8—61no—
158Q8—61no—
159Q8C2×C4×D311yescharacteristic
160D4—32no—
161D4—61no—
162D4—64no—
163C2×C4—32no—
164C2×C4—32no—
165C2×C4—31no—
166C2×C4—31no—
167C2×C4—31no—
168C2×C4—31no—
169C2×C4—64no—
170C2×C4—61no—
171C2×C4—61no—
172C2×C4—61no—
173C2×C4—241no—
174C2×C4—241no—
175C2×C4—241no—
176C2×C4C2×S411yescharacteristic
177C2×C4C2×S411yescharacteristic
178C8—32no—
179C8—32no—
180C8—61no—
181C8—61no—
182C2×C3—41no—
183C2×C3—41no—
184C2×C3—41no—
185C2×C3—82no—
186D3—82no—
187C22—21no—
188C22—21no—
189C22—62no—
190C22—121no—
191C22—121no—
192C22C4×S412yes—
193C22C22×S411yescharacteristic
194C4—21no—
195C4—21no—
196C4—31no—
197C4—31no—
198C4—32no—
199C4—62no—
200C4—61no—
201C4—61no—
202C4—121no—
203C4—121no—
204C3—41noSylow
205C2—22no—
206C2—122no—
207C2C2×C4×S411yescharacteristic
208C2#592611yescharacteristic
209C2#592711yescharacteristic
subgroupG/H
center, Frattini, socleC22C22×S4
commutator, hypocenterSL(2,𝔽3)C22×C4
Fitting#519D3
prime radicalC2×#983C2
cosocleC2×SL(2,𝔽3)C23
hypercenterC22⋊C4S4
LCSG ▷ SL(2,𝔽3)
UCSC1 ◁ C22 ◁ C22⋊C4

Complements

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

NH ≅ G/Ntype
2. #519⋊C3C2semidirect
3. C2×#859C2semidirect
4. #5394C2semidirect
5. C4×GL(2,𝔽3)C2semidirect
6. #5388C2semidirect
7. C4×#226C2semidirect
10. #983C4semidirect
20. #519D3semidirect
36. C2×SL(2,𝔽3)C2×C4semidirect
38. GL(2,𝔽3)C2×C4semidirect
39. #226C2×C4semidirect
101. C2×Q8C4×D3semidirect

Maximal subgroups

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

subgroupindex
C2×#9832
#519⋊C32
C2×#8592
#53942
C4×GL(2,𝔽3)2
#53882
C4×#2262
#29473
#67⋊C44