Finite Group Databasebeta

← #13164556

Subgroups — #13164556

Order 1728.

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

normal subgroups only

This group has 330 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
1C32⋊#997C211yesmaximal, characteristic
2C3×#5933—33nomaximal
3C3×#5242—31nomaximal
4#88613—41nomaximal
5#88610—42nomaximal
6C32×H(5,𝔽2)D311yescharacteristic
7C3×#997D313yes—
8C3×#951—32no—
9C3×#936—31no—
10C3×#837—32no—
11C3×#833—31no—
12C2×C22⋊H(3,𝔽3)—41no—
13C2×C22⋊H(3,𝔽3)—81no—
14Q8⋊H(3,𝔽3)—42no—
15#5933—33no—
16C3×#455—91no—
17#5242—31no—
18C32×#41—32no—
19C2×C32×D4—31no—
20C2×C32×D4—61no—
21C2×C3×Dic6—31no—
22C3×SL(2,ℤ/4)—123no—
23C3×#226—126no—
24C3×#217—61no—
25C3×#217—61no—
26C3×#216—62no—
27C3×#214—62no—
28C3×C4×Dic3—31no—
29C3×#208—32no—
30C22⋊H(3,𝔽3)—41no—
31C22⋊H(3,𝔽3)—81no—
32#1084—161no—
33C3×H(5,𝔽2)—31no—
34C3×H(5,𝔽2)C3⋊D311yescharacteristic
35#997—33no—
36C3×#137—92no—
37C3×#124—91no—
38#951—32no—
39#936—31no—
40C3×#104—92no—
41C3×#100—91no—
42#837—32no—
43#833—31no—
44C23×C32—21no—
45C23×C32—31no—
46C23×C32S411yescharacteristic
47C2×C3×A4—43no—
48C2×C3×A4—83no—
49C32×Q8S412yes—
50C32×D4—32no—
51C32×D4—62no—
52C2×C32×C4—31no—
53C2×C32×C4—61no—
54C2×C3×Dic3—61no—
55C2×C3×Dic3—61no—
56C3×Dic6—62no—
57C3×SL(2,𝔽3)—46no—
58C3×C3⋊C8—62no—
59#455—91noSylow
60C2×H(3,𝔽3)—161no—
61C3×#41—32no—
62C3×#41—32no—
63C3×#41—62no—
64C2×C3×Q8—91no—
65C2×C3×D4—31no—
66C2×C3×D4—31no—
67C2×C3×D4—61no—
68C2×C3×D4—61no—
69C2×C3×D4—62no—
70C2×C3×D4—61no—
71C2×Dic6—31no—
72SL(2,ℤ/4)—123no—
73#226—126no—
74C3×Q16—182no—
75C3×SD16—182no—
76C3×M16—92no—
77C3×C22⋊C4—181no—
78C3×C22⋊C4—181no—
79C3×C42—91no—
80#217—61no—
81#217—61no—
82#216—62no—
83#214—62no—
84C4×Dic3—31no—
85#208—32no—
86C22×C32—31no—
87C22×C32—41no—
88C22×C32—61no—
89C22×C32—81no—
90C22×C32—121no—
91C3×A4—43no—
92C3×A4—83no—
93C32×C4—32no—
94C3×Dic3—121no—
95C3×Dic3—121no—
96C3×Dic3—483no—
97H(5,𝔽2)#27211yescharacteristic
98#137—92no—
99#124—91no—
100#104—92no—
101#100—91no—
102H(3,𝔽3)—161noSylow
103C23×C3—21no—
104C23×C3—31no—
105C23×C3—31no—
106C23×C3—31no—
107C23×C3—61no—
108C23×C3—61no—
109C23×C3C3⋊S411yescharacteristic
110C2×A4—123no—
111C2×A4—243no—
112C3×Q8—32no—
113C3×Q8—182no—
114C3×Q8C3⋊S412yes—
115C3×D4—32no—
116C3×D4—32no—
117C3×D4—62no—
118C3×D4—62no—
119C3×D4—62no—
120C3×D4—62no—
121C3×D4—62no—
122C2×C3×C4—31no—
123C2×C3×C4—31no—
124C2×C3×C4—61no—
125C2×C3×C4—61no—
126C2×C3×C4—62no—
127C2×C3×C4—61no—
128C2×C3×C4—181no—
129C2×C3×C4—181no—
130C2×Dic3—61no—
131C2×Dic3—61no—
132Dic6—62no—
133SL(2,𝔽3)—126no—
134C3×C8—182no—
135C3⋊C8—62no—
136C2×C32—61no—
137C2×C32—121no—
138C2×C32—163no—
139C2×C32#102011yescharacteristic
140#41—32no—
141C2×Q8—91no—
142C2×D4—31no—
143C2×D4—61no—
144Q16—182no—
145SD16—182no—
146M16—92no—
147C22⋊C4—181no—
148C22⋊C4—181no—
149C42—91no—
150C22×C3—31no—
151C22×C3—31no—
152C22×C3—41no—
153C22×C3—61no—
154C22×C3—62no—
155C22×C3—61no—
156C22×C3—61no—
157C22×C3—81no—
158C22×C3—121no—
159C22×C3—121no—
160C22×C3—121no—
161C22×C3—241no—
162C22×C3—241no—
163A4—123no—
164A4—243no—
165C3×C4—32no—
166C3×C4—32no—
167C3×C4—62no—
168C3×C4—361no—
169C3×C4—361no—
170Dic3—121no—
171Dic3—121no—
172Dic3—483no—
173C32—163no—
174C32#593311yescharacteristic, complemented
175C23—21no—
176C23—31no—
177C23#625811yescharacteristic
178Q8—182no—
179Q8#625812yes—
180D4—32no—
181D4—62no—
182C2×C4—31no—
183C2×C4—61no—
184C2×C4—181no—
185C2×C4—181no—
186C8—182no—
187C2×C3—31no—
188C2×C3—61no—
189C2×C3—61no—
190C2×C3—121no—
191C2×C3—121no—
192C2×C3—122no—
193C2×C3—121no—
194C2×C3—483no—
195C2×C3C3⋊#102011yescharacteristic
196C22—31no—
197C22—41no—
198C22—61no—
199C22—81no—
200C22—121no—
201C4—32no—
202C4—361no—
203C4—361no—
204C3—31no—
205C3—483no—
206C3C3⋊#593311yescharacteristic
207C2—61no—
208C2—121no—
209C2#1172405511yescharacteristic
subgroupG/H
center, Frattini, socle, hypercenterC2×C3C3⋊#1020
commutator, prime radical, cosocle, hypocenterC32⋊#997C2
FittingC32×H(5,𝔽2)D3
LCSG ▷ C32⋊#997
UCSC1 ◁ C2×C3

Complements

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

NH ≅ G/Ntype
174. C32#5933semidirect

Maximal subgroups

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

subgroupindex
C32⋊#9972
C3×#59333
C3×#52423
#886134
#886104