Finite Group Databasebeta

← #3524⋊C3 #85621

Subgroups — #3524⋊C3 #85621

Order 384.

255 conjugacy classes of subgroups; only the 54 normal subgroups are shown.

normal subgroups only

This group has 255 conjugacy classes of subgroups — too many to draw. Only its 54 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#557⋊C3C211yesmaximal, characteristic, complemented
2C4⋊(#41⋊C3)C211yesmaximal, characteristic, complemented
3#548⋊C3C211yesmaximal, characteristic, complemented
4#548⋊C3C211yesmaximal, characteristic, complemented
5#520⋊C3C211yesmaximal, characteristic, complemented
6#519⋊C3C211yesmaximal, characteristic, complemented
7C4×#41⋊C3C211yesmaximal, characteristic, complemented
8#3524C311yesmaximal, characteristic, complemented, Sylow
9C2×#41⋊C3C2211yescharacteristic
10C2×#41⋊C3C2211yescharacteristic
11C22×SL(2,𝔽3)C2211yescharacteristic
12C3×#126—41nomaximal
13C4×SL(2,𝔽3)C2211yescharacteristic
14C4×SL(2,𝔽3)C2211yescharacteristic
15C4×SL(2,𝔽3)C2211yescharacteristic
16C4×SL(2,𝔽3)C2211yescharacteristic
17#559—31no—
18#557C2×C311yescharacteristic, complemented
19#555—31no—
20#553—31no—
21#552—31no—
22#550C2×C311yescharacteristic, complemented
23#548C2×C311yescharacteristic
24#548C2×C311yescharacteristic
25#542—31no—
26#540—31no—
27#529—31no—
28#529—31no—
29#520C2×C311yescharacteristic, complemented
30#519C2×C311yescharacteristic
31C4×#41C2×C311yescharacteristic, complemented
32#41⋊C3—21no—
33#41⋊C3—21no—
34C2×SL(2,𝔽3)—21no—
35C2×SL(2,𝔽3)C2311yescharacteristic
36C3×C4⋊C4—41no—
37C3×C4⋊C4—41no—
38C3×C4⋊C4—41no—
39C3×C22⋊C4—41no—
40C3×C22⋊C4—41no—
41C3×C22⋊C4—41no—
42C3×C42—41no—
43C2×#41C22×C311yescharacteristic
44C2×#41C22×C311yescharacteristic
45C22×Q8C22×C311yescharacteristic
46C4⋊Q8—31no—
47#127—31no—
48#126—32no—
49#126—31no—
50#126A411yescharacteristic
51#125—31no—
52#125—31no—
53#125—31no—
54#124—32no—
55#124—31no—
56#124—31no—
57#124—31no—
58#123—32no—
59#123—32no—
60#123—31no—
61#123—31no—
62C22⋊Q8—31no—
63C22⋊Q8—31no—
64C22⋊Q8—31no—
65#121—31no—
66#121—31no—
67#121—32no—
68#121—31no—
69#121—31no—
70#121—31no—
71C4×Q8—31no—
72C4×Q8—31no—
73C4×Q8—31no—
74C4×Q8C22×C311yescharacteristic
75C4×Q8C22×C311yescharacteristic
76C4×Q8C22×C311yescharacteristic
77C4×Q8C22×C311yescharacteristic
78C4×D4—32no—
79C4×D4—31no—
80C4×D4—31no—
81C4×D4—31no—
82C4×D4—31no—
83C4×D4—31no—
84C4×D4—31no—
85C4×D4—31no—
86#117—32no—
87#117—31no—
88#117—31no—
89#117—31no—
90#117—31no—
91C2×C4⋊C4—31no—
92C2×C42—31no—
93C23×C3—41no—
94C2×C3×C4—41no—
95C2×C3×C4—41no—
96C2×C3×C4—41no—
97C2×C3×C4—41no—
98C2×C3×C4—41no—
99C2×C3×C4—41no—
100SL(2,𝔽3)#4111yescharacteristic
101#41—21no—
102#41—21no—
103#41—61no—
104#41—61no—
105C2×Q8—21no—
106C2×Q8—31no—
107C2×Q8—31no—
108C2×Q8—61no—
109C2×Q8C23×C311yescharacteristic
110C2×D4—31no—
111C2×D4—31no—
112C2×D4—32no—
113C2×D4—31no—
114C2×D4—31no—
115C2×D4—31no—
116C22×C4—31no—
117C22×C4—32no—
118C22×C4—31no—
119C22×C4—31no—
120C22×C4—31no—
121C22×C4—31no—
122C4⋊C4—31no—
123C4⋊C4—31no—
124C4⋊C4—31no—
125C4⋊C4—31no—
126C4⋊C4—32no—
127C4⋊C4—31no—
128C4⋊C4—31no—
129C4⋊C4—31no—
130C4⋊C4—31no—
131C4⋊C4—31no—
132C4⋊C4—31no—
133C4⋊C4—31no—
134C4⋊C4C2×A411yescharacteristic
135C4⋊C4C2×A411yescharacteristic
136C4⋊C4C2×A411yescharacteristic
137C22⋊C4—32no—
138C22⋊C4—32no—
139C22⋊C4—32no—
140C22⋊C4—31no—
141C22⋊C4—31no—
142C22⋊C4—31no—
143C22⋊C4—31no—
144C22⋊C4—31no—
145C22⋊C4—31no—
146C22⋊C4—31no—
147C22⋊C4C2×A411yescharacteristic
148C22⋊C4C2×A411yescharacteristic
149C22⋊C4C2×A411yescharacteristic
150C42—32no—
151C42—31no—
152C42—31no—
153C42—31no—
154C42—31no—
155C42—31no—
156C42C2×A411yescharacteristic
157C22×C3—41no—
158C22×C3—81no—
159C22×C3—81no—
160C22×C3—81no—
161C3×C4—81no—
162C3×C4—81no—
163C3×C4—81no—
164C3×C4—81no—
165C3×C4—81no—
166C3×C4—81no—
167C23—31no—
168C23—31no—
169C23C22×A411yescharacteristic
170Q8—31no—
171Q8—61no—
172Q8—61no—
173Q8—121no—
174Q8C3×#4111yescharacteristic
175D4—61no—
176D4—61no—
177D4—61no—
178D4—61no—
179D4—121no—
180D4—121no—
181D4—121no—
182D4—122no—
183C2×C4—31no—
184C2×C4—31no—
185C2×C4—31no—
186C2×C4—31no—
187C2×C4—31no—
188C2×C4—31no—
189C2×C4—61no—
190C2×C4—61no—
191C2×C4—61no—
192C2×C4—61no—
193C2×C4—61no—
194C2×C4—61no—
195C2×C4—121no—
196C2×C4—121no—
197C2×C4—122no—
198C2×C4C22×A411yescharacteristic
199C2×C4C22×A411yescharacteristic
200C2×C4C22×A411yescharacteristic
201C2×C4C22×A411yescharacteristic
202C2×C4C22×A411yescharacteristic
203C2×C4C22×A411yescharacteristic
204C2×C3—41no—
205C2×C3—41no—
206C2×C3—41no—
207C2×C3—161no—
208C22—21no—
209C22—21no—
210C22—21no—
211C22—61no—
212C22—61no—
213C22—61no—
214C22—61no—
215C22—61no—
216C22—61no—
217C22C23×A411yescharacteristic
218C4—21no—
219C4—21no—
220C4—21no—
221C4—21no—
222C4—21no—
223C4—21no—
224C4—31no—
225C4—31no—
226C4—61no—
227C4—61no—
228C4—61no—
229C4—61no—
230C4—61no—
231C3—41noSylow
232C2—41no—
233C2—121no—
234C2—121no—
235C2A4×#4111yescharacteristic
236C2#594511yescharacteristic
237C2#594511yescharacteristic
subgroupG/H
center, Frattini, socleC22C23×A4
commutator, cosocleC2×Q8C23×C3
Fitting#3524C3
hypercenter#126A4
hypocenterQ8C3×#41
LCSG ▷ C2×Q8 ▷ Q8
UCSC1 ◁ C22 ◁ #126

Complements

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

NH ≅ G/Ntype
1. #557⋊C3C2semidirect
2. C4⋊(#41⋊C3)C2semidirect
3. #548⋊C3C2semidirect
4. #548⋊C3C2semidirect
5. #520⋊C3C2semidirect
6. #519⋊C3C2semidirect
7. C4×#41⋊C3C2semidirect
8. #3524C3semidirect
18. #557C2×C3semidirect
22. #550C2×C3semidirect
29. #520C2×C3semidirect
31. C4×#41C2×C3semidirect

Maximal subgroups

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

subgroupindex
#557⋊C32
C4⋊(#41⋊C3)2
#548⋊C32
#520⋊C32
#519⋊C32
C4×#41⋊C32
#35243
C3×#1264