Finite Group Databasebeta

← #13140519

Subgroups — #13140519

Order 1728.

472 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 472 conjugacy classes of subgroups — too many to draw. Only its 79 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
1D4×#1084C211yesmaximal, characteristic
2#11721117C211yesmaximal, characteristic, complemented
3#63659⋊C3C212yesmaximal
4#11720306C212yesmaximal, complemented
5#11720304C211yesmaximal, characteristic, complemented
6#10591410—32nomaximal
7C2×D4×H(3,𝔽3)C2211yescharacteristic
8C22×#1084—21no—
9C2×#6196—21no—
10#88552—21no—
11C4⋊#1084C2211yescharacteristic
12C4×#1084C2211yescharacteristic
13C4⋊((C3⋊Dic3)⋊C3)C2212yes—
14C2×(C3⋊(C3⋊C8))⋊C3C2212yes—
15#88449—21no—
16C3×D4×Dic3—32no—
17C3×D4×Dic3—61no—
18#63918—32no—
19#63659D312yes—
20C3×#832—32no—
21#63576—32no—
22#63575—32no—
23Dic3⋊(C3⋊C8)—32no—
24C23×H(3,𝔽3)—21no—
25D4×H(3,𝔽3)—21no—
26D4×H(3,𝔽3)Q812yescomplemented
27C2×C4×H(3,𝔽3)C2311yescharacteristic
28C2×#1084—21no—
29C2×#1084—42no—
30C2×#1084D411yescharacteristic
31C2×#1084D411yescharacteristic
32C2×(C3⋊Dic3)⋊C3—22no—
33#6196—42no—
34(C3⋊(C3⋊C8))⋊C3—22no—
35#5007—91no—
36#4759—92no—
37C2×C32×D4—21no—
38C2×C32×D4C2×D312yes—
39C22×C3×Dic3—62no—
40C22×C3×Dic3—62no—
41C2×#627—62no—
42C4⋊(C3⋊Dic3)C2×D312yes—
43C2×C3⋊(C3⋊C8)C2×D312yes—
44C3×#217—62no—
45C3×#217—62no—
46C3×C4⋊Dic3—32no—
47C3×C4⋊Dic3—32no—
48C3×C4⋊Dic3—61no—
49C3×C4×Dic3—32no—
50C3×C4×Dic3—61no—
51C2×C3×C3⋊C8—32no—
52Dic3⋊Dic3—62no—
53C22×H(3,𝔽3)—22no—
54C22×H(3,𝔽3)—41no—
55C22×H(3,𝔽3)—41no—
56C22×H(3,𝔽3)C2×D411yescharacteristic
57C4×H(3,𝔽3)C2×Q811yescharacteristic
58C4×H(3,𝔽3)#4111yescharacteristic
59#1084—21no—
60#1084—41no—
61#1084D812yes—
62(C3⋊Dic3)⋊C3—42no—
63C3×C4×D4—91no—
64D4×Dic3—32no—
65D4×Dic3—61no—
66#888—92no—
67#869—91no—
68C3×#102—92no—
69#832—32no—
70#832—92no—
71#818—92no—
72Dic3⋊C8—92no—
73#807—92no—
74#807—92no—
75C4⋊(C3⋊C8)—91no—
76C23×C32—22no—
77C23×C32—22no—
78C32×D4—22no—
79C32×D4—22no—
80C32×D4—22no—
81C32×D4Dic614yescomplemented
82C2×C32×C4—21no—
83C2×C32×C4C22×D312yes—
84C2×C3⋊Dic3—22no—
85C2×C3×Dic3—32no—
86C2×C3×Dic3—32no—
87C2×C3×Dic3—61no—
88C2×C3×Dic3—61no—
89C2×C3×Dic3—62no—
90C2×C3×Dic3—62no—
91C2×C3×Dic3—62no—
92C2×C3×Dic3—124no—
93C2×C3×Dic3—124no—
94#627—124no—
95C3⋊(C3⋊C8)—22no—
96C3×C3⋊C8—62no—
97#474—271noSylow
98C2×H(3,𝔽3)—42no—
99C2×H(3,𝔽3)C22⋊Q811yescharacteristic
100C2×H(3,𝔽3)C2×D811yescharacteristic
101C2×H(3,𝔽3)#13711yescharacteristic
102C2×C3×D4—32no—
103C2×C3×D4—61no—
104C2×C3×D4D3211yescharacteristic
105C22×C3×C4—181no—
106C22×Dic3—62no—
107C22×Dic3—62no—
108C2×Dic6—182no—
109C2×Dic6—181no—
110C2×C3×C8—92no—
111C3×C4⋊C4—92no—
112C3×C4⋊C4—91no—
113C3×C22⋊C4—181no—
114C3×C42—91no—
115#217—62no—
116#217—62no—
117C4⋊Dic3—32no—
118C4⋊Dic3—32no—
119C4⋊Dic3—61no—
120C4⋊Dic3—92no—
121C4⋊Dic3—181no—
122Dic3⋊C4—182no—
123C4×Dic3—32no—
124C4×Dic3—61no—
125C2×C3⋊C8—32no—
126C2×C3⋊C8—92no—
127C22×C32—24no—
128C22×C32—24no—
129C22×C32—21no—
130C22×C32—42no—
131C22×C32—42no—
132C22×C32—42no—
133C22×C32—42no—
134C22×C32D3×D412yes—
135C32×C4—21no—
136C32×C4—21no—
137C32×C4C2×Dic612yes—
138C32×C4#23712yes—
139C3⋊Dic3—42no—
140C3×Dic3—34no—
141C3×Dic3—62no—
142C3×Dic3—62no—
143C3×Dic3—121no—
144C3×Dic3—122no—
145C3×Dic3—122no—
146C3×Dic3—122no—
147C4⋊Q8—271no—
148C4×D4—91no—
149#107—272no—
150C4⋊C8—271no—
151#102—92no—
152H(3,𝔽3)#47411yescharacteristic, complemented, Sylow
153C23×C3—21no—
154C23×C3—62no—
155C23×C3—62no—
156C3×D4—21no—
157C3×D4—34no—
158C3×D4—62no—
159C3×D4—62no—
160C3×D4—62no—
161C3×D4#62712yes—
162C2×C3×C4—32no—
163C2×C3×C4—61no—
164C2×C3×C4—91no—
165C2×C3×C4—91no—
166C2×C3×C4—182no—
167C2×C3×C4—181no—
168C2×C3×C4—362no—
169C2×C3×C4C2×D3211yescharacteristic
170C2×Dic3—32no—
171C2×Dic3—32no—
172C2×Dic3—61no—
173C2×Dic3—61no—
174C2×Dic3—62no—
175C2×Dic3—62no—
176C2×Dic3—62no—
177C2×Dic3—124no—
178C2×Dic3—124no—
179C2×Dic3—182no—
180Dic6—364no—
181Dic6—362no—
182C3×C8—182no—
183C3⋊C8—62no—
184C3⋊C8—182no—
185C2×C32—21no—
186C2×C32—21no—
187C2×C32—21no—
188C2×C32—44no—
189C2×C32—44no—
190C2×C32#87812yes—
191C2×C32D3×D812yes—
192C2×C32#91512yes—
193C2×Q8—541no—
194C2×D4((C3⋊D3)⋊C3)⋊C211yescharacteristic
195C22×C4—181no—
196C2×C8—92no—
197C4⋊C4—92no—
198C4⋊C4—91no—
199C4⋊C4—541no—
200C22⋊C4—181no—
201C42—91no—
202C22×C3—22no—
203C22×C3—32no—
204C22×C3—41no—
205C22×C3—41no—
206C22×C3—64no—
207C22×C3—64no—
208C22×C3—61no—
209C22×C3—122no—
210C22×C3—122no—
211C22×C3—122no—
212C22×C3—122no—
213C22×C3#381111yescharacteristic
214C3×C4—32no—
215C3×C4—32no—
216C3×C4—61no—
217C3×C4—61no—
218C3×C4—92no—
219C3×C4—181no—
220C3×C4—362no—
221C3×C4—361no—
222C3×C4#380511yescharacteristic
223C3×C4C2×#62711yescharacteristic
224Dic3—34no—
225Dic3—62no—
226Dic3—62no—
227Dic3—121no—
228Dic3—122no—
229Dic3—122no—
230Dic3—122no—
231Dic3—362no—
232C32—21no—
233C32#475912yescomplemented
234C23—21no—
235Q8—1082no—
236D4—21no—
237D4#619612yescomplemented
238C2×C4—91no—
239C2×C4—91no—
240C2×C4—181no—
241C2×C4—182no—
242C2×C4—362no—
243C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
244C8—182no—
245C2×C3—32no—
246C2×C3—32no—
247C2×C3—32no—
248C2×C3—42no—
249C2×C3—61no—
250C2×C3—61no—
251C2×C3—61no—
252C2×C3—124no—
253C2×C3—124no—
254C2×C3#6392611yescharacteristic
255C2×C3#6393111yescharacteristic
256C2×C3#6398211yescharacteristic
257C22—22no—
258C22—41no—
259C22—41no—
260C22#8867711yescharacteristic
261C4—92no—
262C4—181no—
263C4—361no—
264C4—362no—
265C4C2×#619611yescharacteristic
266C4#8867211yescharacteristic
267C3—32no—
268C3—61no—
269C3#1059149411yescharacteristic
270C2—42no—
271C2#1172118911yescharacteristic
272C2#1172119011yescharacteristic
273C2#1172130911yescharacteristic
subgroupG/H
centerC22#88677
commutator, cosocleC2×C4×H(3,𝔽3)C23
FrattiniC2×C3×C4C2×D32
Fitting, prime radicalC2×D4×H(3,𝔽3)C22
socleC22×C3#3811
hypercenterC2×D4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#474
LCSG ▷ C2×C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
2. #11721117C2semidirect
4. #11720306C2semidirect
5. #11720304C2semidirect
26. D4×H(3,𝔽3)Q8semidirect
81. C32×D4Dic6semidirect
152. H(3,𝔽3)#474semidirect
233. C32#4759semidirect
237. D4#6196semidirect

Maximal subgroups

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

subgroupindex
D4×#10842
#117211172
#63659⋊C32
#117203062
#117203042
#105914103