Finite Group Databasebeta

← #3487⋊C3 #85606

Subgroups — #3487⋊C3 #85606

Order 384.

366 conjugacy classes of subgroups; only the 60 normal subgroups are shown.

normal subgroups only

This group has 366 conjugacy classes of subgroups — too many to draw. Only its 60 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
1C2×2−1+4⋊C3C211yesmaximal, characteristic, complemented
2C22×#41⋊C3C211yesmaximal, characteristic, complemented
3C4⋊(#41⋊C3)C212yesmaximal, complemented
4#548⋊C3C212yesmaximal, complemented
5#519⋊C3C211yesmaximal, characteristic, complemented
6#3487C311yesmaximal, characteristic, complemented, Sylow
72−1+4⋊C3—22no—
8C2×#41⋊C3—22no—
9C2×#41⋊C3C2212yes—
10C2×#41⋊C3C2211yescharacteristic, complemented
11C22×SL(2,𝔽3)C2211yescharacteristic
12C22×SL(2,𝔽3)C2211yescharacteristic, complemented
13C3×#123—41nomaximal
14C4×SL(2,𝔽3)C2212yescomplemented
15C2×2−1+4C2×C311yescharacteristic
16C22×#41C2×C311yescharacteristic, complemented
17#550—32no—
18#550C2×C312yescomplemented
19#548—32no—
20#548C2×C312yescomplemented
21#535—31no—
22#535—31no—
23#525—31no—
24#525—31no—
25#519C2×C311yescharacteristic, complemented
26C2×C3×D4—41no—
27C22×C3×C4—41no—
28#41⋊C3—22no—
29#41⋊C3—21no—
30C2×SL(2,𝔽3)—21no—
31C2×SL(2,𝔽3)D412yescomplemented
32C2×SL(2,𝔽3)C2311yescharacteristic
33C3×C4⋊C4—42no—
34C3×C22⋊C4—41no—
35C3×C22⋊C4—42no—
362−1+4—22no—
372−1+4—62no—
38C2×#41—22no—
39C2×#41—31no—
40C2×#41—31no—
41C2×#41—31no—
42C2×#41—32no—
43C2×#41—62no—
44C2×#41C22×C312yes—
45C2×#41C22×C311yescharacteristic
46C22×Q8—31no—
47C22×Q8C22×C311yescharacteristic
48C22×Q8C22×C311yescharacteristic
49C22×D4—31no—
50C23×C4—31no—
51#127—32no—
52#127—32no—
53#124—32no—
54#124—32no—
55#123—31no—
56#123A411yescharacteristic
57C22⋊Q8—32no—
58C22⋊Q8—32no—
59#121—32no—
60#121—34no—
61#121—32no—
62#121—32no—
63#121—32no—
64#120—32no—
65#120—31no—
66#120—31no—
67C4×Q8—32no—
68C4×Q8C22×C312yes—
69C4×D4—34no—
70C4×D4—32no—
71C4×D4—32no—
72#117—31no—
73#117—31no—
74C23×C3—41no—
75C23×C3—41no—
76C3×D4—82no—
77C2×C3×C4—42no—
78C2×C3×C4—42no—
79C2×C3×C4—41no—
80C2×C3×C4—82no—
81SL(2,𝔽3)C2×D411yescharacteristic
82C24—31no—
83#41—22no—
84#41—21no—
85#41—62no—
86#41—62no—
87#41—62no—
88#41—62no—
89#41—64no—
90#41—62no—
91#41—62no—
92#41—64no—
93#41—62no—
94#41—61no—
95C2×Q8—21no—
96C2×Q8—31no—
97C2×Q8—31no—
98C2×Q8—32no—
99C2×Q8—31no—
100C2×Q8—62no—
101C2×Q8—62no—
102C2×Q8—61no—
103C2×Q8C3×D412yes—
104C2×Q8C23×C311yescharacteristic
105C2×D4—32no—
106C2×D4—31no—
107C2×D4—32no—
108C2×D4—31no—
109C2×D4—31no—
110C2×D4—34no—
111C2×D4—32no—
112C2×D4—32no—
113C2×D4—62no—
114C2×D4—62no—
115C2×D4C2×A411yescharacteristic
116C22×C4—31no—
117C22×C4—32no—
118C22×C4—31no—
119C22×C4—31no—
120C22×C4—32no—
121C22×C4—32no—
122C22×C4—62no—
123C22×C4—62no—
124C22×C4C2×A411yescharacteristic
125C4⋊C4—32no—
126C4⋊C4—32no—
127C4⋊C4—32no—
128C4⋊C4C2×A412yes—
129C22⋊C4—34no—
130C22⋊C4—31no—
131C22⋊C4—32no—
132C22⋊C4—32no—
133C22⋊C4—32no—
134C22⋊C4C2×A411yescharacteristic
135C22⋊C4C2×A412yes—
136C42—32no—
137C42—32no—
138C22×C3—42no—
139C22×C3—41no—
140C22×C3—82no—
141C22×C3—81no—
142C22×C3—82no—
143C3×C4—82no—
144C3×C4—82no—
145C3×C4—81no—
146C23—32no—
147C23—31no—
148C23—62no—
149C23—122no—
150C23C22×A411yescharacteristic
151C23C22×A411yescharacteristic
152Q8—32no—
153Q8—31no—
154Q8—32no—
155Q8—62no—
156Q8—61no—
157Q8—61no—
158Q8C2×C3×D411yescharacteristic
159D4—22no—
160D4—62no—
161D4—62no—
162D4—62no—
163D4—61no—
164D4—61no—
165D4—64no—
166D4—62no—
167D4—122no—
168D4—122no—
169D4—124no—
170C2×C4—22no—
171C2×C4—32no—
172C2×C4—32no—
173C2×C4—31no—
174C2×C4—31no—
175C2×C4—32no—
176C2×C4—31no—
177C2×C4—62no—
178C2×C4—64no—
179C2×C4—62no—
180C2×C4—61no—
181C2×C4—61no—
182C2×C4—61no—
183C2×C4—61no—
184C2×C4—64no—
185C2×C4C22×A412yes—
186C2×C4C22×A411yescharacteristic
187C2×C4C22×A412yes—
188C2×C3—41no—
189C2×C3—42no—
190C2×C3—82no—
191C2×C3—161no—
192C22—22no—
193C22—21no—
194C22—22no—
195C22—61no—
196C22—62no—
197C22—62no—
198C22—62no—
199C22—62no—
200C22—124no—
201C22D4×A412yes—
202C22C23×A411yescharacteristic
203C4—22no—
204C4—21no—
205C4—22no—
206C4—32no—
207C4—31no—
208C4—31no—
209C4—61no—
210C4—62no—
211C3—41noSylow
212C2—22no—
213C2—41no—
214C2—121no—
215C2—122no—
216C2C2×D4×A411yescharacteristic
217C2#594512yes—
subgroupG/H
center, Frattini, socleC22C23×A4
commutator, cosocleC2×Q8C23×C3
Fitting#3487C3
hypercenter#123A4
hypocenterQ8C2×C3×D4
LCSG ▷ C2×Q8 ▷ Q8
UCSC1 ◁ C22 ◁ #123

Complements

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

NH ≅ G/Ntype
1. C2×2−1+4⋊C3C2semidirect
2. C22×#41⋊C3C2semidirect
3. C4⋊(#41⋊C3)C2semidirect
4. #548⋊C3C2semidirect
5. #519⋊C3C2semidirect
6. #3487C3semidirect
10. C2×#41⋊C3C22semidirect
12. C22×SL(2,𝔽3)C22semidirect
14. C4×SL(2,𝔽3)C22semidirect
16. C22×#41C2×C3semidirect
18. #550C2×C3semidirect
20. #548C2×C3semidirect
25. #519C2×C3semidirect
31. C2×SL(2,𝔽3)D4semidirect

Maximal subgroups

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

subgroupindex
C2×2−1+4⋊C32
C22×#41⋊C32
C4⋊(#41⋊C3)2
#548⋊C32
#519⋊C32
#34873
C3×#1234