Finite Group Databasebeta

← #13147770

Subgroups — #13147770

Order 1728.

885 conjugacy classes of subgroups; only the 151 normal subgroups are shown.

normal subgroups only

This group has 885 conjugacy classes of subgroups — too many to draw. Only its 151 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
1C42×#272C211yesmaximal, characteristic
2#64079⋊C3C212yesmaximal, complemented
3#11721117C212yesmaximal, complemented
4#11721096C214yesmaximal
5#11721089C212yesmaximal
6#11721078C214yesmaximal, complemented
7#10593874—32nomaximal
8C2×C4×#272C2212yes—
9C2×C4×#272C2211yescharacteristic
10C2×#6196C2214yes—
11C42×H(3,𝔽3)C2211yescharacteristic
12C4×#1084C2212yes—
13C4×#1084C2211yescharacteristic
14C4⋊((C3⋊Dic3)⋊C3)C2214yes—
15#3750⋊C3C2218yes—
16#88451C2214yes—
17#88449C2214yes—
18#88447C2214yes—
19#64079D312yescomplemented
20C3×C42×D3—32no—
21C3×C42×D3—61no—
22C3×#870—32no—
23#63918—34no—
24#63851—38no—
25#63841—34no—
26#63837—38no—
27C22×#272C2311yescharacteristic
28C2×C4×H(3,𝔽3)C2312yes—
29C2×C4×H(3,𝔽3)C2311yescharacteristic
30C2×#1084C2311yescharacteristic
31C2×#1084C2312yes—
32C4×#272—24no—
33C4×#272Q814yescomplemented
34C2×(C3⋊Dic3)⋊C3C2318yes—
35#6196—28no—
36#5688—92no—
37#5470—91no—
38C2×C3×C4×D3—32no—
39C2×C3×C4×D3—34no—
40C2×C3×C4×D3—61no—
41C2×C3×C4×D3—62no—
42C2×#627—38no—
43C32×C42—21no—
44C32×C42C2×D312yes—
45C4⋊(C3⋊Dic3)C2×D314yes—
46#3750C2×D318yes—
47C3×C4⋊Dic3—34no—
48C3×Dic3⋊C4—38no—
49C3×C4×Dic3—34no—
50C3×C4×Dic3—32no—
51C3×C4×Dic3—62no—
52C3×C4×Dic3—61no—
53Dic3⋊Dic3—38no—
54(C2×D3)⋊Dic3—38no—
55Dic32—38no—
56C22×H(3,𝔽3)C2411yescharacteristic
57C2×#272—22no—
58C2×#272C2×Q812yes—
59C4×H(3,𝔽3)—22no—
60C4×H(3,𝔽3)C2×Q812yes—
61#1084C2×Q812yes—
62#1084#4114yes—
63(C3⋊Dic3)⋊C3—28no—
64C3×#125—92no—
65C2×C3×C42—91no—
66#926—92no—
67#924—94no—
68#896—98no—
69#892—94no—
70#888—94no—
71#887—98no—
72C42×D3—32no—
73C42×D3—61no—
74#870—32no—
75#870—92no—
76#869—92no—
77C4×Dic6—94no—
78C22×C3×D3—32no—
79C22×C3×D3—61no—
80C2×C32×C4—22no—
81C2×C32×C4—21no—
82C2×C32×C4C22×D314yes—
83C2×C32×C4C22×D312yes—
84C2×C3⋊Dic3C22×D318yes—
85C2×C3×Dic3—32no—
86C2×C3×Dic3—34no—
87C2×C3×Dic3—38no—
88C2×C3×Dic3—61no—
89C2×C3×Dic3—62no—
90C3×C4×D3—38no—
91C3×C4×D3—64no—
92C3×C4×D3—68no—
93C3×C4×D3—68no—
94#627—616no—
95#533—271noSylow
96C2×H(3,𝔽3)C22×Q811yescharacteristic
97C2×H(3,𝔽3)C2×#4112yes—
98#272—22no—
99C22×C3×C4—91no—
100C22×C3×C4—92no—
101C2×C4×D3—32no—
102C2×C4×D3—34no—
103C2×C4×D3—61no—
104C2×C4×D3—62no—
105C2×Dic6—94no—
106C2×Dic6—98no—
107C3×C4⋊C4—98no—
108C3×C4⋊C4—94no—
109C3×C42—32no—
110C3×C42—61no—
111C3×C42—92no—
112C3×C42—91no—
113C3×C42D3211yescharacteristic
114#217—94no—
115(C2×D3)⋊C4—98no—
116C4⋊Dic3—34no—
117C4⋊Dic3—94no—
118C4⋊Dic3—94no—
119Dic3⋊C4—38no—
120Dic3⋊C4—98no—
121Dic3⋊C4—98no—
122C4×Dic3—34no—
123C4×Dic3—32no—
124C4×Dic3—62no—
125C4×Dic3—61no—
126C4×Dic3—94no—
127C4×Dic3—98no—
128C22×C32—21no—
129C22×C32C23×D312yes—
130C2×C3×D3—34no—
131C2×C3×D3—62no—
132C2×C3×D3—64no—
133C2×C3×D3—64no—
134C32×C4—24no—
135C32×C4—24no—
136C32×C4—22no—
137C32×C4D3×Q814yes—
138C3⋊Dic3—28no—
139C3×Dic3—34no—
140C3×Dic3—38no—
141C3×Dic3—62no—
142C3×Dic3—64no—
143C3×Dic3—68no—
144C4⋊Q8—272no—
145#125—92no—
146C22⋊Q8—274no—
147C4×Q8—274no—
148#117—272no—
149C2×C42—91no—
150H(3,𝔽3)#53311yescharacteristic, complemented, Sylow
151C23×C3—91no—
152C22×D3—32no—
153C22×D3—61no—
154C2×C3×C4—32no—
155C2×C3×C4—34no—
156C2×C3×C4—62no—
157C2×C3×C4—61no—
158C2×C3×C4—91no—
159C2×C3×C4—92no—
160C2×C3×C4—94no—
161C2×C3×C4—98no—
162C2×C3×C4—184no—
163C2×C3×C4C2×D3212yes—
164C2×C3×C4C2×D3211yescharacteristic
165C2×Dic3—32no—
166C2×Dic3—34no—
167C2×Dic3—38no—
168C2×Dic3—61no—
169C2×Dic3—62no—
170C2×Dic3—98no—
171C4×D3—38no—
172C4×D3—64no—
173C4×D3—68no—
174C4×D3—68no—
175Dic6—1816no—
176Dic6—188no—
177C2×C32—21no—
178C2×C32—22no—
179C2×C32C2×D3×Q812yes—
180C2×C32D3×#4114yes—
181C3×D3—64no—
182C3×D3—64no—
183C2×Q8—274no—
184C22×C4—91no—
185C22×C4—92no—
186C4⋊C4—94no—
187C4⋊C4—98no—
188C4⋊C4—274no—
189C22⋊C4—274no—
190C42—92no—
191C42—91no—
192C42—274no—
193C42((C3⋊D3)⋊C3)⋊C211yescharacteristic
194C22×C3—32no—
195C22×C3—61no—
196C22×C3—92no—
197C22×C3—182no—
198C22×C3C22×D3211yescharacteristic
199C2×D3—34no—
200C2×D3—62no—
201C2×D3—64no—
202C2×D3—64no—
203C3×C4—22no—
204C3×C4—34no—
205C3×C4—64no—
206C3×C4—64no—
207C3×C4—62no—
208C3×C4—92no—
209C3×C4—94no—
210C3×C4—188no—
211C3×C4#379712yes—
212Dic3—38no—
213Dic3—34no—
214Dic3—62no—
215Dic3—64no—
216Dic3—68no—
217Dic3—188no—
218C32—21no—
219C32#568812yescomplemented
220C23—91no—
221Q8—548no—
222C2×C4—91no—
223C2×C4—92no—
224C2×C4—94no—
225C2×C4—98no—
226C2×C4—184no—
227C2×C4C2×((C3⋊D3)⋊C3)⋊C212yes—
228C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
229C2×C3—32no—
230C2×C3—34no—
231C2×C3—61no—
232C2×C3—62no—
233C2×C3—182no—
234C2×C3C2×#379711yescharacteristic
235C2×C3#6430612yes—
236D3—64no—
237D3—64no—
238C22—92no—
239C22—182no—
240C22C22×((C3⋊D3)⋊C3)⋊C211yescharacteristic
241C4—22no—
242C4—92no—
243C4—94no—
244C4—188no—
245C4#8865112yes—
246C3—32no—
247C3—61no—
248C3#1059385411yescharacteristic
249C2—182no—
250C2C2×#8865111yescharacteristic
251C2#1172249112yes—
subgroupG/H
centerC22C22×((C3⋊D3)⋊C3)⋊C2
commutator, cosocleC22×H(3,𝔽3)C24
Frattini, socleC22×C3C22×D32
FittingC42×H(3,𝔽3)C22
prime radicalC22×#272C23
hypercenterC42((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#533
LCSG ▷ C22×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C42

Complements

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

NH ≅ G/Ntype
2. #64079⋊C3C2semidirect
3. #11721117C2semidirect
6. #11721078C2semidirect
19. #64079D3semidirect
33. C4×#272Q8semidirect
150. H(3,𝔽3)#533semidirect
219. C32#5688semidirect

Maximal subgroups

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

subgroupindex
C42×#2722
#64079⋊C32
#117211172
#117210962
#117210892
#117210782
#105938743