Finite Group Databasebeta

← #13137976

Subgroups — #13137976

Order 1728.

438 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 438 conjugacy classes of subgroups — too many to draw. Only its 69 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
1#11721765C211yesmaximal, characteristic
2#11721152C211yesmaximal, characteristic
3#63635⋊C3C212yesmaximal, complemented
4#11720326C212yesmaximal, complemented
5#11720323C211yesmaximal, characteristic
6#10590439—32nomaximal
7C2×C4×#272C2211yescharacteristic
8C2×#6229C2211yescharacteristic
9C2×#6199—21no—
10C4⋊C4×H(3,𝔽3)C2211yescharacteristic
11#88542—21no—
12C4⋊#1084—21no—
13C4⋊((C3⋊Dic3)⋊C3)C2212yes—
14C2×(C3⋊(C3⋊C8))⋊C3C2212yes—
15#88449—21no—
16C3×#897—32no—
17C3×#897—61no—
18D3×C4⋊Dic3—32no—
19#63635D312yescomplemented
20C3×#808—32no—
21#63566—32no—
22#63565—32no—
23(C2×D3)⋊(C3⋊C8)—32no—
24C22×#272D411yescharacteristic
25C2×C4×H(3,𝔽3)—21no—
26C2×C4×H(3,𝔽3)C2311yescharacteristic
27C2×#1084—21no—
28C2×#1084D411yescharacteristic
29C4×#272—22no—
30#6229—22no—
31C2×(C3⋊Dic3)⋊C3—22no—
32#6199—42no—
33(C3⋊(C3⋊C8))⋊C3—22no—
34#5043—91no—
35#4862—92no—
36C2×C3×C4×D3—32no—
37C2×C3×C4×D3—61no—
38C2×C3×Dic6—32no—
39C2×C3×Dic6—61no—
40C2×D3×Dic3—62no—
41C32×C4⋊C4—21no—
42C32×C4⋊C4C2×D312yes—
43C4⋊(C3⋊Dic3)C2×D312yes—
44C2×C3⋊(C3⋊C8)C2×D312yes—
45C3×(C2×D3)⋊C4—62no—
46C3×(C2×D3)⋊C4—62no—
47C3×C4⋊Dic3—32no—
48C3×C4⋊Dic3—62no—
49C3×C4⋊Dic3—62no—
50C2×C3×C3⋊C8—32no—
51Dic3⋊Dic3—62no—
52C22×H(3,𝔽3)C2×D411yescharacteristic
53C2×#272—21no—
54C2×#272—21no—
55C2×#272SD1612yes—
56C4×H(3,𝔽3)—41no—
57C4×H(3,𝔽3)#4112yes—
58#1084—21no—
59#1084—41no—
60(C3⋊Dic3)⋊C3—42no—
61C3×C22⋊Q8—91no—
62C2×C4⋊Dic3—91no—
63#897—32no—
64#897—61no—
65D3×C4⋊C4—92no—
66C3×#106—92no—
67#835—92no—
68#830—91no—
69(C2×D3)⋊C8—92no—
70#816—92no—
71#810—92no—
72#808—32no—
73#808—92no—
74C22×C3×D3—32no—
75C22×C3×D3—61no—
76C2×C32×C4—22no—
77C2×C32×C4—22no—
78C2×C32×C4—21no—
79C2×C32×C4C22×D312yes—
80C2×C3⋊Dic3—22no—
81C2×C3×Dic3—32no—
82C2×C3×Dic3—61no—
83C2×C3×Dic3—62no—
84C2×C3×Dic3—62no—
85C2×C3×Dic3—62no—
86C3×C4×D3—62no—
87C3×C4×D3—62no—
88C3×C4×D3—122no—
89C3×Dic6—62no—
90C3×Dic6—62no—
91C3×Dic6—122no—
92D3×Dic3—124no—
93C3⋊(C3⋊C8)—22no—
94C3×C3⋊C8—62no—
95#483—271noSylow
96C2×H(3,𝔽3)#12311yescharacteristic
97C2×H(3,𝔽3)C2×SD1611yescharacteristic
98C2×H(3,𝔽3)#13711yescharacteristic
99#272—22no—
100C2×C3×Q8—91no—
101C22×C3×C4—91no—
102C22×Dic3—181no—
103C2×C4×D3—32no—
104C2×C4×D3—61no—
105C2×C4×D3—182no—
106C2×Dic6—32no—
107C2×Dic6—61no—
108C2×C3×C8—92no—
109C3×C4⋊C4—32no—
110C3×C4⋊C4—61no—
111C3×C4⋊C4—92no—
112C3×C4⋊C4—181no—
113C3×C4⋊C4D3211yescharacteristic
114C3×C22⋊C4—181no—
115(C2×D3)⋊C4—62no—
116(C2×D3)⋊C4—62no—
117C4⋊Dic3—32no—
118C4⋊Dic3—62no—
119C4⋊Dic3—62no—
120C4⋊Dic3—92no—
121C4⋊Dic3—181no—
122Dic3⋊C4—182no—
123C2×C3⋊C8—32no—
124C2×C3⋊C8—92no—
125C22×C32—21no—
126C22×C32D3×D412yes—
127C2×C3×D3—34no—
128C2×C3×D3—62no—
129C2×C3×D3—62no—
130C2×C3×D3—62no—
131C2×C3×D3—121no—
132C2×C3×D3—121no—
133C32×C4—22no—
134C32×C4—42no—
135C32×C4—42no—
136C32×C4#23512yes—
137C32×C4#23912yes—
138C3⋊Dic3—42no—
139C3×Dic3—62no—
140C3×Dic3—121no—
141C3×Dic3—122no—
142C3×Dic3—122no—
143C3×Dic3—122no—
144C22⋊Q8—91no—
145C2×C4⋊C4—271no—
146#106—92no—
147#103—272no—
148C22⋊C8—271no—
149H(3,𝔽3)#48311yescharacteristic, complemented, Sylow
150C23×C3—91no—
151C22×D3—32no—
152C22×D3—61no—
153C3×Q8—182no—
154C2×C3×C4—21no—
155C2×C3×C4—32no—
156C2×C3×C4—62no—
157C2×C3×C4—62no—
158C2×C3×C4—61no—
159C2×C3×C4—91no—
160C2×C3×C4—182no—
161C2×C3×C4—182no—
162C2×C3×C4—181no—
163C2×C3×C4C2×D3211yescharacteristic
164C2×Dic3—32no—
165C2×Dic3—61no—
166C2×Dic3—62no—
167C2×Dic3—62no—
168C2×Dic3—62no—
169C2×Dic3—182no—
170C2×Dic3—362no—
171C4×D3—62no—
172C4×D3—62no—
173C4×D3—122no—
174C4×D3—364no—
175Dic6—62no—
176Dic6—62no—
177Dic6—122no—
178C3×C8—182no—
179C3⋊C8—62no—
180C3⋊C8—182no—
181C2×C32—21no—
182C2×C32—21no—
183C2×C32—21no—
184C2×C32#89412yes—
185C2×C32D3×SD1612yes—
186C2×C32#91512yes—
187C3×D3—64no—
188C3×D3—122no—
189C2×Q8—91no—
190C22×C4—91no—
191C22×C4—541no—
192C2×C8—92no—
193C4⋊C4—92no—
194C4⋊C4—181no—
195C4⋊C4—541no—
196C4⋊C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
197C22⋊C4—181no—
198C22×C3—32no—
199C22×C3—61no—
200C22×C3—92no—
201C22×C3—181no—
202C22×C3—181no—
203C22×C3#381111yescharacteristic
204C2×D3—34no—
205C2×D3—62no—
206C2×D3—62no—
207C2×D3—62no—
208C2×D3—121no—
209C2×D3—121no—
210C3×C4—32no—
211C3×C4—32no—
212C3×C4—41no—
213C3×C4—62no—
214C3×C4—122no—
215C3×C4—122no—
216C3×C4—181no—
217C3×C4—362no—
218C3×C4—361no—
219C3×C4#379912yes—
220Dic3—62no—
221Dic3—121no—
222Dic3—122no—
223Dic3—122no—
224Dic3—122no—
225Dic3—362no—
226C32—21no—
227C32#486212yescomplemented
228C23—91no—
229Q8—182no—
230C2×C4—21no—
231C2×C4—91no—
232C2×C4—182no—
233C2×C4—181no—
234C2×C4—182no—
235C2×C4—1082no—
236C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
237C8—182no—
238C2×C3—32no—
239C2×C3—32no—
240C2×C3—32no—
241C2×C3—61no—
242C2×C3—61no—
243C2×C3—61no—
244C2×C3—182no—
245C2×C3#6389711yescharacteristic
246C2×C3#6393111yescharacteristic
247C2×C3#6394011yescharacteristic
248D3—64no—
249D3—122no—
250C22—92no—
251C22—181no—
252C22—181no—
253C22#8867711yescharacteristic
254C4—41no—
255C4—181no—
256C4—362no—
257C4—361no—
258C4#8865212yes—
259C3—32no—
260C3—61no—
261C3#1059044311yescharacteristic
262C2—182no—
263C2#1172115111yescharacteristic
264C2#1172118911yescharacteristic
265C2#1172122811yescharacteristic
subgroupG/H
centerC22#88677
commutator, cosocleC2×C4×H(3,𝔽3)C23
FrattiniC2×C3×C4C2×D32
FittingC4⋊C4×H(3,𝔽3)C22
socleC22×C3#3811
prime radicalC22×#272D4
hypercenterC4⋊C4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#483
LCSG ▷ C2×C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C2×C4 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
3. #63635⋊C3C2semidirect
4. #11720326C2semidirect
19. #63635D3semidirect
149. H(3,𝔽3)#483semidirect
227. C32#4862semidirect

Maximal subgroups

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

subgroupindex
#117217652
#117211522
#63635⋊C32
#117203262
#117203232
#105904393