Finite Group Databasebeta

← #85542

Subgroups — #85542

Order 384.

286 conjugacy classes of subgroups; only the 36 normal subgroups are shown.

normal subgroups only

This group has 286 conjugacy classes of subgroups — too many to draw. Only its 36 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#5945C211yesmaximal, characteristic
2C2×#984C211yesmaximal, characteristic, complemented
3#5430C212yesmaximal, complemented
4#5428C212yesmaximal, complemented
5#5423C211yesmaximal, characteristic, complemented
6#3022—31nomaximal, Sylow
72−1+4⋊C3—21no—
82−1+4⋊C3C2211yescharacteristic
9#994C2211yescharacteristic
10C2×#41⋊C3—21no—
11C2×#41⋊C3C2211yescharacteristic
12#984—21no—
13#984C2212yescomplemented
14C2×#226—21no—
15#951—41nomaximal
16GU(2,𝔽3)C2212yescomplemented
17#585D311yescharacteristic
18C2×#137—31no—
19C2×#137—31no—
20#536—31no—
21#456—32no—
22#456—32no—
23#455—32no—
24#455—32no—
25#421—31no—
26#421—31no—
27#413—31no—
28C3×#41—41no—
29C2×Dic6—41no—
30#41⋊C3—21no—
31#41⋊C3D411yescharacteristic
32#41⋊C3C2311yescharacteristic
33C2×SL(2,𝔽3)—21no—
34C2×SL(2,𝔽3)D411yescharacteristic
35#226—22no—
36#216—42no—
37#214—42no—
38#208—41no—
392−1+4—21no—
402−1+4—31no—
412−1+4C2×D311yescharacteristic
42H(5,𝔽2)—31no—
43H(5,𝔽2)—61no—
44H(5,𝔽2)C2×D311yescharacteristic
45C2×#41—21no—
46C2×#41—31no—
47C2×#41—31no—
48C2×#41—61no—
49C2×#41C2×D311yescharacteristic
50C22×Q8—31no—
51#137—32no—
52#137—32no—
53#137—32no—
54#137—32no—
55#137—61no—
56#137—61no—
57#137—61no—
58#137—61no—
59C2×Q16—61no—
60C2×Q16—61no—
61C2×SD16—61no—
62C2×SD16—61no—
63C2×M16—31no—
64C2×M16—31no—
65C4⋊Q8—32no—
66#124—32no—
67#123—62no—
68C22⋊Q8—62no—
69#117—31no—
70#104—32no—
71#104—32no—
72#104—32no—
73#104—32no—
74#101—32no—
75#100—32no—
76C3×Q8—41no—
77C3×D4—41no—
78C3×D4—81no—
79C2×C3×C4—41no—
80C2×C3×C4—81no—
81C2×Dic3—81no—
82Dic6—42no—
83Dic6—81no—
84SL(2,𝔽3)C2×D411yescharacteristic
85C3⋊C8—42no—
86#41—21no—
87#41—31no—
88#41—31no—
89#41—31no—
90#41—31no—
91#41—31no—
92#41—61no—
93#41—61no—
94#41—61no—
95#41—61no—
96#41—62no—
97#41—61no—
98#41—61no—
99#41—61no—
100#41—61no—
101#41#6711yescharacteristic
102#41S411yescharacteristic
103#41C22×D311yescharacteristic
104C2×Q8—21no—
105C2×Q8—31no—
106C2×Q8—31no—
107C2×Q8—32no—
108C2×Q8—61no—
109C2×Q8—61no—
110C2×Q8—61no—
111C2×Q8—61no—
112C2×Q8—61no—
113C2×Q8—61no—
114C2×Q8#6711yescharacteristic
115C2×D4—31no—
116C2×D4—31no—
117C2×D4—31no—
118C2×D4—32no—
119C2×D4—61no—
120C2×D4—61no—
121C2×D4—61no—
122C2×D4—62no—
123C22×C4—31no—
124C22×C4—32no—
125C22×C4—61no—
126C22×C4—61no—
127Q16—62no—
128Q16—62no—
129Q16—62no—
130Q16—62no—
131SD16—62no—
132SD16—62no—
133SD16—62no—
134SD16—62no—
135M16—32no—
136M16—32no—
137M16—61no—
138M16—61no—
139C2×C8—61no—
140C2×C8—61no—
141C4⋊C4—61no—
142C4⋊C4—64no—
143C22⋊C4—61no—
144C22⋊C4—64no—
145C42—32no—
146C22×C3—41no—
147C22×C3—81no—
148C3×C4—41no—
149C3×C4—41no—
150C3×C4—81no—
151Dic3—82no—
152C23—31no—
153C23—32no—
154C23—61no—
155Q8—31no—
156Q8—31no—
157Q8—61no—
158Q8—61no—
159Q8—62no—
160Q8—62no—
161Q8—121no—
162Q8—121no—
163Q8C2×#6711yescharacteristic
164Q8C2×S411yescharacteristic
165D4—21no—
166D4—31no—
167D4—31no—
168D4—31no—
169D4—61no—
170D4—61no—
171D4—61no—
172D4—61no—
173D4—61no—
174D4—62no—
175D4—61no—
176D4C2×S411yescharacteristic
177C2×C4—21no—
178C2×C4—31no—
179C2×C4—31no—
180C2×C4—31no—
181C2×C4—31no—
182C2×C4—31no—
183C2×C4—32no—
184C2×C4—61no—
185C2×C4—61no—
186C2×C4—61no—
187C2×C4—62no—
188C2×C4—61no—
189C2×C4—62no—
190C2×C4—62no—
191C2×C4—121no—
192C2×C4—121no—
193C2×C4C2×S411yescharacteristic
194C8—62no—
195C8—62no—
196C2×C3—41no—
197C2×C3—81no—
198C2×C3—161no—
199C22—21no—
200C22—31no—
201C22—31no—
202C22—61no—
203C22—121no—
204C22—122no—
205C22—241no—
206C22GL(2,ℤ/4)11yescharacteristic
207C4—21no—
208C4—31no—
209C4—31no—
210C4—61no—
211C4—122no—
212C4—122no—
213C4GL(2,ℤ/4)11yescharacteristic
214C4C22×S411yescharacteristic
215C3—41noSylow
216C2—21no—
217C2—41no—
218C2—61no—
219C2—61no—
220C2—121no—
221C2C2×GL(2,ℤ/4)11yescharacteristic
subgroupG/H
center, socleC2C2×GL(2,ℤ/4)
commutator, cosocle#41⋊C3C23
FrattiniC4C22×S4
Fitting#585D3
prime radical#5945C2
hypercenter#41S4
hypocenterSL(2,𝔽3)C2×D4
LCSG ▷ #41⋊C3 ▷ SL(2,𝔽3)
UCSC1 ◁ C2 ◁ C2×C4 ◁ #41

Complements

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

NH ≅ G/Ntype
2. C2×#984C2semidirect
3. #5430C2semidirect
4. #5428C2semidirect
5. #5423C2semidirect
13. #984C22semidirect
16. GU(2,𝔽3)C22semidirect

Maximal subgroups

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

subgroupindex
#59452
C2×#9842
#54302
#54282
#54232
#30223
#9514