Finite Group Databasebeta

← #3516⋊C3 #85616

Subgroups — #3516⋊C3 #85616

Order 384.

267 conjugacy classes of subgroups; only the 56 normal subgroups are shown.

normal subgroups only

This group has 267 conjugacy classes of subgroups — too many to draw. Only its 56 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
2D4×SL(2,𝔽3)C211yesmaximal, characteristic, complemented
3#548⋊C3C211yesmaximal, characteristic, complemented
4#548⋊C3C211yesmaximal, characteristic, complemented
5#520⋊C3C211yesmaximal, characteristic, complemented
6#519⋊C3C211yesmaximal, characteristic, complemented
7C2×C4×SL(2,𝔽3)C211yesmaximal, characteristic, complemented
8#3516C311yesmaximal, characteristic, complemented, Sylow
9C2×#41⋊C3C2211yescharacteristic
10C22×SL(2,𝔽3)C2211yescharacteristic
11C22×SL(2,𝔽3)C2211yescharacteristic
12C3×#123—41nomaximal
13C4×SL(2,𝔽3)C2211yescharacteristic
14C4×SL(2,𝔽3)C2211yescharacteristic
15C4×SL(2,𝔽3)C2211yescharacteristic
16C4×SL(2,𝔽3)C2211yescharacteristic
17#557C2×C311yescharacteristic, complemented
18#556—31no—
19#554—31no—
20#553—31no—
21#552—31no—
22D4×Q8C2×C311yescharacteristic
23#548C2×C311yescharacteristic, complemented
24#548C2×C311yescharacteristic, complemented
25#542—31no—
26#541—31no—
27#529—31no—
28#529—31no—
29#520C2×C311yescharacteristic, complemented
30#519C2×C311yescharacteristic, complemented
31C2×C4×Q8C2×C311yescharacteristic, complemented
32C2×C3×D4—41no—
33C22×C3×C4—41no—
34#41⋊C3—21no—
35C2×SL(2,𝔽3)—21no—
36C2×SL(2,𝔽3)—21no—
37C2×SL(2,𝔽3)C2311yescharacteristic
38C3×C4⋊C4—41no—
39C3×C4⋊C4—41no—
40C3×C22⋊C4—41no—
41C3×C22⋊C4—41no—
42C3×C22⋊C4—41no—
43C2×#41C22×C311yescharacteristic
44C22×Q8C22×C311yescharacteristic
45C22×Q8C22×C311yescharacteristic
46C4⋊Q8—31no—
47C4⋊Q8—31no—
48#126—32no—
49#126—32no—
50#126—31no—
51#126—31no—
52#125—31no—
53#125—31no—
54#125—32no—
55#125—31no—
56#124—31no—
57#124—31no—
58#124—31no—
59#123—32no—
60#123—31no—
61#123A411yescharacteristic
62C22⋊Q8—32no—
63C22⋊Q8—31no—
64C22⋊Q8—31no—
65C22⋊Q8—31no—
66C22⋊Q8—31no—
67C22⋊Q8—31no—
68#121—31no—
69#121—31no—
70#121—31no—
71C4×Q8—31no—
72C4×Q8—31no—
73C4×Q8—31no—
74C4×Q8—31no—
75C4×Q8—31no—
76C4×Q8C22×C311yescharacteristic
77C4×Q8C22×C311yescharacteristic
78C4×Q8C22×C311yescharacteristic
79C4×Q8C22×C311yescharacteristic
80C4×D4—32no—
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—
90C2×C4⋊C4—31no—
91C2×C4⋊C4—31no—
92C2×C42—31no—
93C23×C3—41no—
94C23×C3—41no—
95C3×D4—82no—
96C2×C3×C4—41no—
97C2×C3×C4—41no—
98C2×C3×C4—41no—
99C2×C3×C4—41no—
100C2×C3×C4—41no—
101C2×C3×C4—82no—
102SL(2,𝔽3)#4111yescharacteristic
103#41—21no—
104#41—61no—
105C2×Q8—21no—
106C2×Q8—21no—
107C2×Q8—31no—
108C2×Q8—31no—
109C2×Q8—31no—
110C2×Q8—61no—
111C2×Q8—61no—
112C2×Q8C23×C311yescharacteristic
113C2×D4—31no—
114C2×D4—31no—
115C2×D4—31no—
116C2×D4C2×A411yescharacteristic
117C22×C4—31no—
118C22×C4—31no—
119C22×C4—32no—
120C22×C4—31no—
121C22×C4—31no—
122C22×C4C2×A411yescharacteristic
123C4⋊C4—31no—
124C4⋊C4—31no—
125C4⋊C4—31no—
126C4⋊C4—31no—
127C4⋊C4—32no—
128C4⋊C4—32no—
129C4⋊C4—32no—
130C4⋊C4—31no—
131C4⋊C4—31no—
132C4⋊C4—31no—
133C4⋊C4—31no—
134C4⋊C4—31no—
135C4⋊C4—31no—
136C4⋊C4—31no—
137C4⋊C4—31no—
138C4⋊C4C2×A411yescharacteristic
139C4⋊C4C2×A411yescharacteristic
140C22⋊C4—32no—
141C22⋊C4—32no—
142C22⋊C4—31no—
143C22⋊C4—31no—
144C22⋊C4—31no—
145C22⋊C4—31no—
146C22⋊C4—31no—
147C22⋊C4—31no—
148C22⋊C4—31no—
149C22⋊C4C2×A411yescharacteristic
150C22⋊C4C2×A411yescharacteristic
151C22⋊C4C2×A411yescharacteristic
152C42—32no—
153C42—31no—
154C42—31no—
155C42—31no—
156C42—31no—
157C42—31no—
158C42—31no—
159C22×C3—42no—
160C22×C3—41no—
161C22×C3—81no—
162C22×C3—81no—
163C22×C3—81no—
164C22×C3—81no—
165C22×C3—81no—
166C3×C4—81no—
167C3×C4—81no—
168C3×C4—81no—
169C3×C4—81no—
170C3×C4—81no—
171C23—31no—
172C23C22×A411yescharacteristic
173C23C22×A411yescharacteristic
174Q8—31no—
175Q8—61no—
176Q8—61no—
177Q8—61no—
178Q8—61no—
179Q8—121no—
180Q8C3×#4111yescharacteristic
181D4—22no—
182D4—61no—
183D4—61no—
184D4—62no—
185D4—121no—
186C2×C4—22no—
187C2×C4—31no—
188C2×C4—31no—
189C2×C4—31no—
190C2×C4—31no—
191C2×C4—31no—
192C2×C4—31no—
193C2×C4—31no—
194C2×C4—61no—
195C2×C4—61no—
196C2×C4—61no—
197C2×C4—61no—
198C2×C4—61no—
199C2×C4—61no—
200C2×C4—121no—
201C2×C4—122no—
202C2×C4C22×A411yescharacteristic
203C2×C4C22×A411yescharacteristic
204C2×C4C22×A411yescharacteristic
205C2×C4C22×A411yescharacteristic
206C2×C4C22×A411yescharacteristic
207C2×C3—41no—
208C2×C3—41no—
209C2×C3—41no—
210C2×C3—82no—
211C2×C3—161no—
212C22—21no—
213C22—21no—
214C22—21no—
215C22—21no—
216C22—21no—
217C22—61no—
218C22—61no—
219C22—61no—
220C222−1+4⋊C312yes—
221C22C23×A411yescharacteristic
222C4—21no—
223C4—21no—
224C4—21no—
225C4—21no—
226C4—21no—
227C4—31no—
228C4—31no—
229C4—61no—
230C4—61no—
231C4—61no—
232C4—61no—
233C4—61no—
234C4—61no—
235C3—41noSylow
236C2—22no—
237C2—41no—
238C2—121no—
239C2A4×#4111yescharacteristic
240C2C2×2−1+4⋊C311yescharacteristic
241C2#594511yescharacteristic
subgroupG/H
center, Frattini, socleC22C23×A4
commutator, cosocleC2×Q8C23×C3
Fitting#3516C3
hypercenter#123A4
hypocenterQ8C3×#41
LCSG ▷ C2×Q8 ▷ Q8
UCSC1 ◁ C22 ◁ #123

Complements

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

NH ≅ G/Ntype
1. #557⋊C3C2semidirect
2. D4×SL(2,𝔽3)C2semidirect
3. #548⋊C3C2semidirect
4. #548⋊C3C2semidirect
5. #520⋊C3C2semidirect
6. #519⋊C3C2semidirect
7. C2×C4×SL(2,𝔽3)C2semidirect
8. #3516C3semidirect
17. #557C2×C3semidirect
23. #548C2×C3semidirect
24. #548C2×C3semidirect
29. #520C2×C3semidirect
30. #519C2×C3semidirect
31. C2×C4×Q8C2×C3semidirect

Maximal subgroups

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

subgroupindex
#557⋊C32
D4×SL(2,𝔽3)2
#548⋊C32
#520⋊C32
#519⋊C32
C2×C4×SL(2,𝔽3)2
#35163
C3×#1234