Finite Group Databasebeta

← C2×#2205⋊C3 #11709708

Subgroups — C2×#2205⋊C3 #11709708

Order 768.

286 conjugacy classes of subgroups; only the 23 normal subgroups are shown.

normal subgroups only

This group has 286 conjugacy classes of subgroups — too many to draw. Only its 23 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×Q8⋊SL(2,𝔽3)C211yesmaximal, characteristic, complemented
2#2205⋊C3C212yesmaximal, direct product factor
3C2×#2205C311yesmaximal, characteristic, complemented, Sylow
4C2×2−1+4⋊C3—41nomaximal
5Q8⋊SL(2,𝔽3)C2211yescharacteristic, complemented
6C2×Q82C2×C311yescharacteristic, complemented
7C2×#456—31no—
8#2205—32no—
9#2205C2×C312yescomplemented
10C2×#356—31no—
112−1+4⋊C3—42no—
122−1+4⋊C3—42no—
13C2×#41⋊C3—41no—
14C22×SL(2,𝔽3)—41no—
15C22×SL(2,𝔽3)—41no—
16C22×SL(2,𝔽3)—81no—
17C22×SL(2,𝔽3)—81no—
18C2×2−1+4A411yescharacteristic
19C2×#137—61no—
20Q82—31no—
21Q82—61no—
22Q82—61no—
23Q82C22×C311yescharacteristic, complemented
24C2×C4⋊Q8—31no—
25C2×C4⋊Q8—61no—
26C2×C4×Q8—61no—
27#456—32no—
28#456—32no—
29#456—61no—
30#456—61no—
31C2×#104—61no—
32C2×#101—31no—
33#356—32no—
34#356—32no—
35C2×C3×D4—161no—
36#41⋊C3—42no—
37C2×SL(2,𝔽3)—41no—
38C2×SL(2,𝔽3)—42no—
39C2×SL(2,𝔽3)—41no—
40C2×SL(2,𝔽3)—41no—
41C2×SL(2,𝔽3)—81no—
42C2×SL(2,𝔽3)—81no—
43C2×SL(2,𝔽3)—81no—
44C2×SL(2,𝔽3)—81no—
45C2×SL(2,𝔽3)—82no—
462−1+4—42no—
472−1+4—61no—
482−1+4C2×A412yes—
49C2×#41—41no—
50C2×#41—61no—
51C22×Q8—21no—
52C22×Q8—21no—
53C22×Q8—31no—
54C22×Q8—41no—
55C22×Q8C2×A411yescharacteristic
56#137—62no—
57#137—62no—
58#137—122no—
59C2×Q16—121no—
60C2×SD16—121no—
61C2×M16—61no—
62C4⋊Q8—31no—
63C4⋊Q8—31no—
64C4⋊Q8—31no—
65C4⋊Q8—31no—
66C4⋊Q8—61no—
67C4⋊Q8—62no—
68C4⋊Q8—61no—
69C4⋊Q8—62no—
70C4⋊Q8—62no—
71C4⋊Q8—61no—
72C4⋊Q8—61no—
73C4×Q8—62no—
74C4×Q8—61no—
75C4×Q8—61no—
76C4×Q8—61no—
77C4×Q8—62no—
78C4×Q8—61no—
79C2×C4⋊C4—62no—
80C2×C4⋊C4—61no—
81C2×C42—31no—
82C2×C42—61no—
83#104—62no—
84#104—62no—
85#101—32no—
86#101—61no—
87C23×C3—161no—
88C23×C3—161no—
89C3×D4—162no—
90C3×D4—162no—
91C2×C3×C4—161no—
92SL(2,𝔽3)—41no—
93SL(2,𝔽3)—81no—
94SL(2,𝔽3)—81no—
95#41—42no—
96#41—62no—
97#41—62no—
98#41—121no—
99#41—121no—
100#41—122no—
101C2×Q8—21no—
102C2×Q8—22no—
103C2×Q8—21no—
104C2×Q8—31no—
105C2×Q8—31no—
106C2×Q8—31no—
107C2×Q8—41no—
108C2×Q8—41no—
109C2×Q8—42no—
110C2×Q8—61no—
111C2×Q8—61no—
112C2×Q8—61no—
113C2×Q8—62no—
114C2×Q8—61no—
115C2×Q8—81no—
116C2×Q8—81no—
117C2×Q8—121no—
118C2×Q8—121no—
119C2×Q8—122no—
120C2×Q8C22×A411yescharacteristic
121C2×D4—41no—
122C2×D4—61no—
123C22×C4—31no—
124C22×C4—61no—
125C22×C4—61no—
126C22×C4—121no—
127Q16—122no—
128Q16—122no—
129SD16—122no—
130SD16—122no—
131M16—62no—
132M16—121no—
133C2×C8—121no—
134C4⋊C4—62no—
135C4⋊C4—62no—
136C4⋊C4—62no—
137C4⋊C4—62no—
138C4⋊C4—61no—
139C4⋊C4—61no—
140C4⋊C4—62no—
141C42—31no—
142C42—31no—
143C42—61no—
144C42—61no—
145C42—61no—
146C42—62no—
147C22×C3—162no—
148C22×C3—161no—
149C22×C3—161no—
150C22×C3—161no—
151C22×C3—322no—
152C22×C3—322no—
153C3×C4—162no—
154C23—41no—
155C23#120⋊C311yescharacteristic
156Q8—21no—
157Q8—41no—
158Q8—62no—
159Q8—61no—
160Q8—62no—
161Q8—62no—
162Q8—81no—
163Q8—121no—
164Q8—121no—
165Q8—121no—
166Q8—121no—
167Q8—241no—
168Q8—241no—
169D4—42no—
170D4—42no—
171D4—62no—
172D4—121no—
173C2×C4—31no—
174C2×C4—31no—
175C2×C4—41no—
176C2×C4—62no—
177C2×C4—62no—
178C2×C4—61no—
179C2×C4—61no—
180C2×C4—61no—
181C2×C4—61no—
182C2×C4—61no—
183C2×C4—61no—
184C2×C4—122no—
185C2×C4—122no—
186C2×C4—121no—
187C2×C4—121no—
188C8—122no—
189C2×C3—161no—
190C2×C3—162no—
191C2×C3—321no—
192C2×C3—321no—
193C2×C3—322no—
194C22—22no—
195C22—42no—
196C22—82no—
197C22B(4,𝔽2)⋊C311yescharacteristic
198C22B(4,𝔽2)⋊C311yescharacteristic
199C22C2×#120⋊C311yescharacteristic
200C4—42no—
201C4—61no—
202C4—62no—
203C4—61no—
204C4—61no—
205C4—61no—
206C4—121no—
207C4—121no—
208C3—161noSylow
209C2—21no—
210C2—21no—
211C2—82no—
212C2#2205⋊C312yesdirect product factor
213C2C2×B(4,𝔽2)⋊C311yescharacteristic
subgroupG/H
center, socleC22B(4,𝔽2)⋊C3
commutator, cosocle, hypocenterQ82C22×C3
FrattiniC2×Q8C22×A4
FittingC2×#2205C3
hypercenterC23#120⋊C3
LCSG ▷ Q82
UCSC1 ◁ C22 ◁ C23

Complements

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

NH ≅ G/Ntype
1. C2×Q8⋊SL(2,𝔽3)C2semidirect
2. #2205⋊C3C2direct (N × H)
3. C2×#2205C3semidirect
5. Q8⋊SL(2,𝔽3)C22semidirect
6. C2×Q82C2×C3semidirect
9. #2205C2×C3semidirect
23. Q82C22×C3semidirect
212. C2#2205⋊C3direct (N × H)

Maximal subgroups

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

subgroupindex
C2×Q8⋊SL(2,𝔽3)2
#2205⋊C32
C2×#22053
C2×2−1+4⋊C34