Finite Group Databasebeta

← C22×#88610 #13164487

Subgroups — C22×#88610 #13164487

Order 1728.

760 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 760 conjugacy classes of subgroups — too many to draw. Only its 94 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×Q8⋊H(3,𝔽3)C211yesmaximal, characteristic
2C2×#88610C216yesmaximal, direct product factor
3C22×C3×#226—33nomaximal
4C22×C3×#216—31nomaximal
5C22×#1084—41nomaximal
6C2×Q8⋊H(3,𝔽3)C2213yes—
7#88610C2214yesdirect product factor
8C22×C32×Q8D311yescharacteristic
9C22×C3×Dic6—31no—
10C22×C3×SL(2,𝔽3)D313yes—
11C2×C3×#226—318no—
12C2×C3×#216—36no—
13C2×C3×#216—36no—
14C22×C3×C3⋊C8—31no—
15C23×H(3,𝔽3)—41no—
16C2×#1084—46no—
17Q8⋊H(3,𝔽3)C2311yescharacteristic
18C22×#226—33no—
19C22×C3×Q16—91no—
20C22×#216—31no—
21C2×C32×Q8—33no—
22C2×C32×Q8C2×D313yes—
23C22×C32×C4—31no—
24C22×C3×Dic3—61no—
25C22×C3×Dic3—123no—
26C2×C3×Dic6—36no—
27C2×C3×Dic6—63no—
28C2×C3×SL(2,𝔽3)C2×D319yes—
29C3×#226—312no—
30C3×#216—312no—
31C3×#216—34no—
32C2×C3×C3⋊C8—36no—
33C22×H(3,𝔽3)—44no—
34C22×H(3,𝔽3)—43no—
35#1084—44no—
36C22×C3×Q8—31no—
37C22×C3×Q8—91no—
38C22×C3×Q8C3⋊D311yescharacteristic
39C22×Dic6—31no—
40C22×SL(2,𝔽3)—33no—
41C2×#226—318no—
42C2×C3×Q16—96no—
43C2×C3×Q16—96no—
44C22×C3×C8—91no—
45C2×#216—36no—
46C2×#216—36no—
47C22×C3⋊C8—31no—
48C23×C32—43no—
49C23×C32S411yescharacteristic
50C32×Q8—33no—
51C32×Q8—61no—
52C32×Q8C22×D311yescharacteristic
53C2×C32×C4—33no—
54C2×C32×C4—33no—
55C2×C3×Dic3—66no—
56C2×C3×Dic3—1218no—
57C3×Dic6—34no—
58C3×Dic6—66no—
59C3×SL(2,𝔽3)C22×D313yes—
60C3×C3⋊C8—34no—
61C22×Q16—91noSylow
62C2×H(3,𝔽3)—46no—
63C2×H(3,𝔽3)—41no—
64C2×C3×Q8—33no—
65C2×C3×Q8—33no—
66C2×C3×Q8—33no—
67C2×C3×Q8—63no—
68C2×C3×Q8—96no—
69C2×C3×Q8—183no—
70C2×C3×Q8C2×C3⋊D313yes—
71C22×C3×C4—31no—
72C22×C3×C4—31no—
73C22×C3×C4—61no—
74C22×C3×C4—181no—
75C22×Dic3—61no—
76C22×Dic3—123no—
77C2×Dic6—36no—
78C2×Dic6—63no—
79C2×SL(2,𝔽3)—39no—
80#226—312no—
81C3×Q16—912no—
82C3×Q16—94no—
83C2×C3×C8—96no—
84#216—312no—
85#216—34no—
86C2×C3⋊C8—36no—
87C22×C32—49no—
88C22×C32—412no—
89C22×C32#22614yescomplemented
90C22×C32C2×S413yes—
91C32×C4—31no—
92C32×C4—33no—
93C3×Dic3—64no—
94C3×Dic3—1212no—
95C22×Q8—91no—
96C22×Q8#27211yescharacteristic
97C2×Q16—96no—
98C2×Q16—96no—
99C22×C8—91no—
100H(3,𝔽3)—41noSylow
101C23×C3—31no—
102C23×C3—123no—
103C23×C3C3⋊S411yescharacteristic
104C3×Q8—33no—
105C3×Q8—33no—
106C3×Q8—31no—
107C3×Q8—63no—
108C3×Q8—61no—
109C3×Q8—63no—
110C3×Q8—94no—
111C3×Q8—186no—
112C3×Q8C22×C3⋊D311yescharacteristic
113C2×C3×C4—33no—
114C2×C3×C4—33no—
115C2×C3×C4—33no—
116C2×C3×C4—33no—
117C2×C3×C4—63no—
118C2×C3×C4—63no—
119C2×C3×C4—186no—
120C2×Dic3—66no—
121C2×Dic3—1218no—
122Dic6—34no—
123Dic6—66no—
124SL(2,𝔽3)—33no—
125C3×C8—94no—
126C3⋊C8—34no—
127C2×C32—418no—
128C2×C32—43no—
129C2×C32C2×#22616yescomplemented
130C2×C32C22×S411yescharacteristic
131C2×Q8—33no—
132C2×Q8—96no—
133C2×Q8—183no—
134C2×Q8C2×#27213yes—
135C22×C4—31no—
136C22×C4—181no—
137Q16—912no—
138Q16—94no—
139C2×C8—96no—
140C22×C3—34no—
141C22×C3—33no—
142C22×C3—1212no—
143C22×C3—129no—
144C22×C3C3⋊#22614yes—
145C22×C3C2×C3⋊S413yes—
146C3×C4—31no—
147C3×C4—33no—
148C3×C4—31no—
149C3×C4—33no—
150C3×C4—61no—
151C3×C4—63no—
152C3×C4—184no—
153Dic3—64no—
154Dic3—1212no—
155C32—43no—
156C32C22×#22611yescharacteristic, complemented
157C23#625811yescharacteristic
158Q8—33no—
159Q8—61no—
160Q8—94no—
161Q8—186no—
162Q8C22×#27211yescharacteristic
163C2×C4—33no—
164C2×C4—33no—
165C2×C4—186no—
166C8—94no—
167C2×C3—31no—
168C2×C3—36no—
169C2×C3—123no—
170C2×C3—1218no—
171C2×C3C2×C3⋊#22616yes—
172C2×C3C22×C3⋊S411yescharacteristic
173C22#8861014yesdirect product factor
174C22C2×#625813yes—
175C4—31no—
176C4—33no—
177C4—184no—
178C3—31no—
179C3—123no—
180C3C22×C3⋊#22611yescharacteristic
181C2C2×#8861016yesdirect product factor
182C2C22×#625811yescharacteristic
subgroupG/H
center, socle, hypercenterC23×C3C3⋊S4
commutator, cosocle, hypocenterQ8⋊H(3,𝔽3)C23
FrattiniC2×C3C22×C3⋊S4
FittingC22×C32×Q8D3
prime radicalC22×Q8⋊H(3,𝔽3)C2
LCSG ▷ Q8⋊H(3,𝔽3)
UCSC1 ◁ C23×C3

Complements

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

NH ≅ G/Ntype
2. C2×#88610C2direct (N × H)
7. #88610C22direct (N × H)
89. C22×C32#226semidirect
129. C2×C32C2×#226semidirect
156. C32C22×#226semidirect
173. C22#88610direct (N × H)
181. C2C2×#88610direct (N × H)

Maximal subgroups

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

subgroupindex
C22×Q8⋊H(3,𝔽3)2
C2×#886102
C22×C3×#2263
C22×C3×#2163
C22×#10844