Finite Group Databasebeta

← D4×#6204 #13152194

Subgroups — D4×#6204 #13152194

Order 1728.

490 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 490 conjugacy classes of subgroups — too many to draw. Only its 99 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
1C22×#6204C212yesmaximal, complemented
2C4×#6204C211yesmaximal, characteristic, complemented
3C32×D4×Q8C311yesmaximal, characteristic
4D4×Q8⋊C9C313yesmaximal, complemented
5C2×D4×3−1+2—41nomaximal
6C2×#6204—22no—
7C2×#6204C2211yescharacteristic
8C22×C32×Q8C2×C312yes—
9C32×C4⋊Q8—31no—
10C32×C22⋊Q8—32no—
11C32×C4×Q8C2×C311yescharacteristic
12C32×C4×D4—31no—
13C22×Q8⋊C9C2×C316yescomplemented
14C4×Q8⋊C9C2×C313yescomplemented
15C23×3−1+2—42no—
16D4×3−1+2—44no—
17C2×C4×3−1+2—41no—
18#6204D411yescharacteristic, direct product factor
19C3×D4×Q8—31no—
20C3×D4×Q8C3211yescharacteristic
21C2×C32×Q8—22no—
22C2×C32×Q8—32no—
23C2×C32×Q8—62no—
24C2×C32×Q8C22×C311yescharacteristic
25C2×C32×D4A411yescharacteristic
26C22×C32×C4—32no—
27C32×C4⋊C4—32no—
28C32×C4⋊C4—31no—
29C32×C4⋊C4—31no—
30C32×C22⋊C4—32no—
31C32×C42—31no—
32C2×D4×C9—43no—
33C2×Q8⋊C9—26no—
34C2×Q8⋊C9C22×C313yes—
35C22×3−1+2—41no—
36C22×3−1+2—44no—
37C22×3−1+2—82no—
38C22×3−1+2—82no—
39C4×3−1+2—42no—
40C22×C3×Q8—32no—
41C22×C3×Q8C2×C3212yes—
42C3×C4⋊Q8—33no—
43C3×C4⋊Q8—31no—
44C3×C22⋊Q8—36no—
45C3×C22⋊Q8—32no—
46C3×C4×Q8—31no—
47C3×C4×Q8C2×C3211yescharacteristic
48C3×C4×D4—33no—
49C3×C4×D4—31no—
50C23×C32C2×A412yes—
51C32×Q8—31no—
52C32×Q8—62no—
53C32×Q8—62no—
54C32×Q8C3×D411yescharacteristic
55C32×D4SL(2,𝔽3)14yes—
56C2×C32×C4—31no—
57C2×C32×C4—31no—
58C2×C32×C4—32no—
59C2×C32×C4—62no—
60C2×C32×C4—62no—
61C2×C32×C4C2×A411yescharacteristic
62C23×C9—46no—
63D4×C9—412no—
64C2×C4×C9—43no—
65Q8⋊C9C3×D413yescomplemented
66D4×Q83−1+211yescharacteristic, complemented, Sylow
67C2×3−1+2—41no—
68C2×3−1+2—41no—
69C2×3−1+2—41no—
70C2×3−1+2—84no—
71C2×C3×Q8—22no—
72C2×C3×Q8—36no—
73C2×C3×Q8—32no—
74C2×C3×Q8—31no—
75C2×C3×Q8—62no—
76C2×C3×Q8—66no—
77C2×C3×Q8—62no—
78C2×C3×Q8C22×C3211yescharacteristic
79C2×C3×D4—31no—
80C2×C3×D4C3×A411yescharacteristic
81C22×C3×C4—36no—
82C22×C3×C4—32no—
83C3×C4⋊C4—31no—
84C3×C4⋊C4—33no—
85C3×C4⋊C4—33no—
86C3×C4⋊C4—36no—
87C3×C4⋊C4—32no—
88C3×C4⋊C4—31no—
89C3×C22⋊C4—36no—
90C3×C22⋊C4—32no—
91C3×C42—33no—
92C3×C42—31no—
93C22×C32—22no—
94C22×C32—22no—
95C22×C32C2×SL(2,𝔽3)14yes—
96C22×C32C22×A411yescharacteristic
97C32×C4—31no—
98C32×C4—31no—
99C32×C4—62no—
100C32×C4—61no—
101C32×C4C2×SL(2,𝔽3)12yes—
102C22×C9—43no—
103C22×C9—412no—
104C22×C9—86no—
105C22×C9—86no—
106C4×C9—46no—
107C22×Q8C2×3−1+212yescomplemented
108C4⋊Q8—31no—
109C22⋊Q8—32no—
110C4×Q8C2×3−1+211yescharacteristic, complemented
111C4×D4—31no—
1123−1+2—41noSylow
113C23×C3—32no—
114C23×C3C2×C3×A412yes—
115C3×Q8—31no—
116C3×Q8—33no—
117C3×Q8—31no—
118C3×Q8—62no—
119C3×Q8—62no—
120C3×Q8—66no—
121C3×Q8—66no—
122C3×Q8C32×D411yescharacteristic
123C3×D4—34no—
124C3×D4C3×SL(2,𝔽3)14yes—
125C2×C3×C4—33no—
126C2×C3×C4—31no—
127C2×C3×C4—33no—
128C2×C3×C4—36no—
129C2×C3×C4—31no—
130C2×C3×C4—32no—
131C2×C3×C4—31no—
132C2×C3×C4—62no—
133C2×C3×C4—62no—
134C2×C3×C4—66no—
135C2×C3×C4—66no—
136C2×C3×C4C2×C3×A411yescharacteristic
137C2×C32—24no—
138C2×C32D4×A411yescharacteristic
139C2×C32C22×SL(2,𝔽3)11yescharacteristic
140C2×C322−1+4⋊C311yescharacteristic
141C2×C9—43no—
142C2×C9—43no—
143C2×C9—43no—
144C2×C9—812no—
145C2×Q8—22no—
146C2×Q8—32no—
147C2×Q8—62no—
148C2×Q8C22×3−1+211yescharacteristic
149C2×D4#109411yescharacteristic
150C22×C4—32no—
151C4⋊C4—31no—
152C4⋊C4—32no—
153C4⋊C4—31no—
154C22⋊C4—32no—
155C42—31no—
156C22×C3—22no—
157C22×C3—22no—
158C22×C3—31no—
159C22×C3—34no—
160C22×C3—62no—
161C22×C3—62no—
162C22×C3C2×C3×SL(2,𝔽3)14yes—
163C22×C3C22×C3×A411yescharacteristic
164C3×C4—33no—
165C3×C4—33no—
166C3×C4—31no—
167C3×C4—31no—
168C3×C4—32no—
169C3×C4—66no—
170C3×C4—63no—
171C3×C4—62no—
172C3×C4—61no—
173C3×C4C2×C3×SL(2,𝔽3)12yes—
174C32D4×SL(2,𝔽3)11yescharacteristic
175C9—43no—
176C23C2×#109412yes—
177Q8—31no—
178Q8—62no—
179Q8—62no—
180Q8D4×3−1+211yescharacteristic, complemented
181D4#620414yesdirect product factor
182C2×C4—31no—
183C2×C4—31no—
184C2×C4—32no—
185C2×C4—62no—
186C2×C4—62no—
187C2×C4C2×#109411yescharacteristic
188C2×C3—24no—
189C2×C3—31no—
190C2×C3—31no—
191C2×C3—31no—
192C2×C3—64no—
193C2×C3C3×D4×A411yescharacteristic
194C2×C3C22×C3×SL(2,𝔽3)11yescharacteristic
195C2×C3C3×2−1+4⋊C311yescharacteristic
196C22—22no—
197C22—22no—
198C22C2×#620414yescomplemented
199C22C22×#109411yescharacteristic
200C4—31no—
201C4—31no—
202C4—62no—
203C4—61no—
204C4C2×#620412yescomplemented
205C3—31no—
206C3C3×D4×SL(2,𝔽3)11yescharacteristic
207C2—24no—
208C2D4×#109411yescharacteristic
209C2C22×#620411yescharacteristic
210C2#1172257911yescharacteristic
subgroupG/H
center, Frattini, socleC22×C3C22×C3×A4
commutator, cosocleC2×C3×Q8C22×C32
FittingC32×D4×Q8C3
prime radical, hypercenterC2×C32×D4A4
hypocenterQ8D4×3−1+2
LCSG ▷ C2×C3×Q8 ▷ Q8
UCSC1 ◁ C22×C3 ◁ C2×C32×D4

Complements

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

NH ≅ G/Ntype
1. C22×#6204C2semidirect
2. C4×#6204C2semidirect
4. D4×Q8⋊C9C3semidirect
13. C22×Q8⋊C9C2×C3semidirect
14. C4×Q8⋊C9C2×C3semidirect
18. #6204D4direct (N × H)
65. Q8⋊C9C3×D4semidirect
66. D4×Q83−1+2semidirect
107. C22×Q8C2×3−1+2semidirect
110. C4×Q8C2×3−1+2semidirect
180. Q8D4×3−1+2semidirect
181. D4#6204direct (N × H)
198. C22C2×#6204semidirect
204. C4C2×#6204semidirect

Maximal subgroups

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

subgroupindex
C22×#62042
C4×#62042
C32×D4×Q83
D4×Q8⋊C93
C2×D4×3−1+24