Finite Group Databasebeta

← #13144486

Subgroups — #13144486

Order 1728.

660 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 660 conjugacy classes of subgroups — too many to draw. Only its 111 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
1H(3,𝔽3)×C22⋊Q8C211yesmaximal, characteristic, complemented
2C2×#88549C211yesmaximal, characteristic, complemented
3#11721830C211yesmaximal, characteristic
4C2×#88538C211yesmaximal, characteristic, complemented
5#11720583C211yesmaximal, characteristic, complemented
6#11720531C211yesmaximal, characteristic, complemented
7#11720529C211yesmaximal, characteristic, complemented
8C3×#5049—34nomaximal
9C2×Q8×H(3,𝔽3)C2211yescharacteristic
10C22×C4×H(3,𝔽3)C2211yescharacteristic
11C2×#6235—21no—
12C2×C4⋊#272C2211yescharacteristic
13C4⋊C4×H(3,𝔽3)—21no—
14C4⋊C4×H(3,𝔽3)C2211yescharacteristic
15C22⋊C4×H(3,𝔽3)—21no—
16#88549—22no—
17#88542—21no—
18C4⋊#1084C2211yescharacteristic
19#88538—22no—
20C2×#6180C2211yescharacteristic
21C2×#6180C2211yescharacteristic
22C32×C22⋊Q8D314yescomplemented
23C2×C3×#215—34no—
24C3×#930—34no—
25C2×C3×#208—34no—
26C3×#835—34no—
27C3×#809—34no—
28C3×#807—34no—
29C23×H(3,𝔽3)D411yescharacteristic
30C22×#272—21no—
31Q8×H(3,𝔽3)—21no—
32Q8×H(3,𝔽3)—21no—
33C2×C4×H(3,𝔽3)—21no—
34C2×C4×H(3,𝔽3)—21no—
35C2×C4×H(3,𝔽3)—21no—
36C2×C4×H(3,𝔽3)—21no—
37C2×C4×H(3,𝔽3)D411yescharacteristic
38C2×C4×H(3,𝔽3)C2311yescharacteristic
39#6235—42no—
40C2×#1084—21no—
41C4⋊#272—21no—
42C4⋊#272—21no—
43#6180—21no—
44#6180D412yescomplemented
45C3×#468—91no—
46#5049—34no—
47C2×C32×Q8C2×D314yes—
48C22×C32×C4C2×D314yes—
49C2×C3×#67—64no—
50C2×C3×D12—34no—
51C32×C4⋊C4—24no—
52C32×C4⋊C4C2×D314yes—
53C32×C22⋊C4—24no—
54C3×#215—68no—
55C3×(C2×D3)⋊C4—64no—
56C3×C4⋊Dic3—34no—
57C3×#208—68no—
58C2×C3×C3⋊C8—34no—
59C2×C3×C3⋊C8—34no—
60C22×H(3,𝔽3)—21no—
61C22×H(3,𝔽3)—21no—
62C22×H(3,𝔽3)—21no—
63C22×H(3,𝔽3)C2×D411yescharacteristic
64C2×#272—41no—
65C2×#272—41no—
66C2×#272—41no—
67C4×H(3,𝔽3)—21no—
68C4×H(3,𝔽3)—41no—
69C4×H(3,𝔽3)—41no—
70C4×H(3,𝔽3)C2×D411yescharacteristic
71C4×H(3,𝔽3)#4111yescharacteristic
72#1084—41no—
73C2×C3×SD16—91no—
74C2×C3×M16—91no—
75C3×C22⋊Q8—34no—
76C3×C22⋊Q8C3⋊D311yescharacteristic
77C3×#121—91no—
78C2×#215—34no—
79#930—34no—
80C2×#208—34no—
81C3×#107—91no—
82C3×#103—91no—
83C3×#102—91no—
84#835—34no—
85#809—34no—
86#807—34no—
87C23×C32#6714yes—
88C22×C3×D3—64no—
89C32×Q8—24no—
90C32×Q8—24no—
91C2×C32×C4—24no—
92C2×C32×C4—24no—
93C2×C32×C4—24no—
94C2×C32×C4—24no—
95C2×C32×C4#6714yes—
96C2×C32×C4C22×D314yes—
97C3×#67—128no—
98C2×C3×Dic3—64no—
99C3×D12—64no—
100C3×D12—64no—
101C3×C3⋊C8—38no—
102C3×C3⋊C8—64no—
103#468—91noSylow
104C2×H(3,𝔽3)—41no—
105C2×H(3,𝔽3)#12111yescharacteristic
106C2×H(3,𝔽3)AGL(1,ℤ/8)11yescharacteristic
107C2×H(3,𝔽3)#13711yescharacteristic
108#272—81no—
109C2×C3×Q8—34no—
110C2×C3×Q8C2×C3⋊D311yescharacteristic
111C2×C3×D4—91no—
112C2×C3×D4—181no—
113C22×C3×C4—34no—
114C22×C3×C4C2×C3⋊D311yescharacteristic
115C2×#67—64no—
116C2×D12—34no—
117C3×SD16—182no—
118C3×M16—182no—
119C2×C3×C8—91no—
120C2×C3×C8—91no—
121C3×C4⋊C4—21no—
122C3×C4⋊C4—34no—
123C3×C4⋊C4—64no—
124C3×C4⋊C4—91no—
125C3×C4⋊C4C2×C3⋊D311yescharacteristic
126C3×C22⋊C4—21no—
127C3×C22⋊C4—64no—
128C3×C22⋊C4—181no—
129#215—68no—
130(C2×D3)⋊C4—64no—
131C4⋊Dic3—34no—
132#208—68no—
133C2×C3⋊C8—34no—
134C2×C3⋊C8—34no—
135C22×C32—24no—
136C22×C32—24no—
137C22×C32—24no—
138C22×C32C2×#6714yes—
139C2×C3×D3—124no—
140C2×C3×D3—124no—
141C2×C3×D3—124no—
142C32×C4—24no—
143C32×C4—44no—
144C32×C4—44no—
145C32×C4D3×D414yes—
146C32×C4#23714yes—
147C3×Dic3—124no—
148C2×SD16—91no—
149C2×M16—91no—
150C22⋊Q8#27211yescharacteristic, complemented
151#121—91no—
152#107—91no—
153#103—91no—
154#102—91no—
155H(3,𝔽3)#46811yescharacteristic, complemented, Sylow
156C23×C3—34no—
157C23×C3—181no—
158C23×C3#63811yescharacteristic
159C22×D3—64no—
160C3×Q8—21no—
161C3×Q8—21no—
162C3×Q8—64no—
163C3×Q8—64no—
164C3×D4—181no—
165C3×D4—181no—
166C3×D4—362no—
167C2×C3×C4—21no—
168C2×C3×C4—21no—
169C2×C3×C4—21no—
170C2×C3×C4—21no—
171C2×C3×C4—34no—
172C2×C3×C4—34no—
173C2×C3×C4—64no—
174C2×C3×C4—64no—
175C2×C3×C4—64no—
176C2×C3×C4—64no—
177C2×C3×C4—181no—
178C2×C3×C4#63811yescharacteristic
179C2×C3×C4C22×C3⋊D311yescharacteristic
180#67—128no—
181C2×Dic3—64no—
182D12—64no—
183D12—64no—
184C3×C8—92no—
185C3×C8—181no—
186C3⋊C8—38no—
187C3⋊C8—64no—
188C2×C32—44no—
189C2×C32#93914yes—
190C2×C32#94214yes—
191C2×C32#94914yes—
192C3×D3—244no—
193C2×Q8C2×#27211yescharacteristic
194C2×D4—91no—
195C2×D4—181no—
196C22×C4C2×#27211yescharacteristic
197SD16—182no—
198M16—182no—
199C2×C8—91no—
200C2×C8—91no—
201C4⋊C4—21no—
202C4⋊C4—91no—
203C4⋊C4C2×#27211yescharacteristic
204C22⋊C4—21no—
205C22⋊C4—181no—
206C22×C3—21no—
207C22×C3—21no—
208C22×C3—21no—
209C22×C3—34no—
210C22×C3—64no—
211C22×C3—64no—
212C22×C3—64no—
213C22×C3—361no—
214C22×C3—361no—
215C22×C3—361no—
216C22×C3C2×#63811yescharacteristic
217C2×D3—124no—
218C2×D3—124no—
219C2×D3—124no—
220C3×C4—21no—
221C3×C4—34no—
222C3×C4—34no—
223C3×C4—41no—
224C3×C4—41no—
225C3×C4—64no—
226C3×C4—124no—
227C3×C4—124no—
228C3×C4—361no—
229C3×C4D4×C3⋊D311yescharacteristic
230C3×C4#383011yescharacteristic
231Dic3—124no—
232C32#504914yescomplemented
233C23—181no—
234C23#623511yescharacteristic
235Q8—21no—
236Q8—21no—
237D4—181no—
238D4—181no—
239D4—362no—
240C2×C4—21no—
241C2×C4—21no—
242C2×C4—21no—
243C2×C4—21no—
244C2×C4—181no—
245C2×C4#623511yescharacteristic
246C2×C4C22×#27211yescharacteristic
247C8—92no—
248C8—181no—
249C2×C3—34no—
250C2×C3—34no—
251C2×C3—34no—
252C2×C3—41no—
253C2×C3—124no—
254C2×C3—721no—
255C2×C3#6414811yescharacteristic
256C2×C3#6415111yescharacteristic
257C2×C3#6415811yescharacteristic
258D3—244no—
259C22—21no—
260C22—21no—
261C22—21no—
262C22—361no—
263C22—361no—
264C22—361no—
265C22C2×#623511yescharacteristic
266C4—21no—
267C4—41no—
268C4—41no—
269C4—361no—
270C4D4×#27211yescharacteristic
271C4#8874411yescharacteristic
272C3—34no—
273C3#1059274811yescharacteristic
274C2—41no—
275C2—721no—
276C2#1172184911yescharacteristic
277C2#1172185911yescharacteristic
278C2#1172187011yescharacteristic
subgroupG/H
center, socleC22×C3C2×#638
commutator, cosocleC2×C4×H(3,𝔽3)C23
FrattiniC2×C3×C4C22×C3⋊D3
FittingH(3,𝔽3)×C22⋊Q8C2
prime radical#11721830C2
hypercenterC3×C22⋊Q8C3⋊D3
hypocenterH(3,𝔽3)#468
LCSG ▷ C2×C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22×C3 ◁ C22×C3×C4 ◁ C3×C22⋊Q8

Complements

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

NH ≅ G/Ntype
1. H(3,𝔽3)×C22⋊Q8C2semidirect
2. C2×#88549C2semidirect
4. C2×#88538C2semidirect
5. #11720583C2semidirect
6. #11720531C2semidirect
7. #11720529C2semidirect
22. C32×C22⋊Q8D3semidirect
44. #6180D4semidirect
150. C22⋊Q8#272semidirect
155. H(3,𝔽3)#468semidirect
232. C32#5049semidirect

Maximal subgroups

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

subgroupindex
H(3,𝔽3)×C22⋊Q82
C2×#885492
#117218302
C2×#885382
#117205832
#117205312
#117205292
C3×#50493