Finite Group Databasebeta

← #13140521

Subgroups — #13140521

Order 1728.

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

normal subgroups only

This group has 514 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
2C2×#3754⋊C3C212yesmaximal
3#11721099C211yesmaximal, characteristic, complemented
4#11720309C212yesmaximal, complemented
5#11720304C211yesmaximal, characteristic, complemented
6#10591489—32nomaximal
7C2×D4×H(3,𝔽3)C2211yescharacteristic
8C22×#1084—21no—
9C2×#634⋊C3C2212yes—
10#88552—21no—
11C4⋊#1084C2211yescharacteristic
12C4×#1084C2211yescharacteristic
13#3754⋊C3—24no—
14C2×(C3⋊(C3⋊C8))⋊C3C2212yes—
15#88449—21no—
16#88449—21no—
17C2×#3754D312yes—
18C3×D4×Dic3—32no—
19C3×D4×Dic3—61no—
20C2×C3×#214—32no—
21#63866—32no—
22Dic3⋊(C3⋊C8)—32no—
23#63567—32no—
24#63565—32no—
25C23×H(3,𝔽3)—21no—
26D4×H(3,𝔽3)—21no—
27D4×H(3,𝔽3)D412yes—
28C2×C4×H(3,𝔽3)C2311yescharacteristic
29C2×#1084—21no—
30C2×#1084—42no—
31C2×#1084D411yescharacteristic
32C2×#1084D411yescharacteristic
33C2×(C3⋊Dic3)⋊C3—22no—
34#634⋊C3—22no—
35#634⋊C3—22no—
36(C3⋊(C3⋊C8))⋊C3—22no—
37#5162—92no—
38#5019—91no—
39C2×C32×D4—21no—
40C2×C32×D4C2×D312yes—
41C2×#634C2×D312yes—
42C22×C3×Dic3—62no—
43C22×C3×Dic3—62no—
44C2×C3×Dic6—32no—
45#3754—24no—
46C2×C3⋊(C3⋊C8)C2×D312yes—
47C3×#217—62no—
48C3×#217—62no—
49C3×#214—64no—
50C3×C4⋊Dic3—32no—
51C3×C4⋊Dic3—61no—
52C3×C4×Dic3—32no—
53C3×C4×Dic3—61no—
54C2×C3×C3⋊C8—32no—
55Dic3⋊Dic3—62no—
56Dic3⋊Dic3—62no—
57C22×H(3,𝔽3)—22no—
58C22×H(3,𝔽3)—41no—
59C22×H(3,𝔽3)—41no—
60C22×H(3,𝔽3)C2×D411yescharacteristic
61C4×H(3,𝔽3)C2×D411yescharacteristic
62C4×H(3,𝔽3)#4111yescharacteristic
63#1084—21no—
64#1084—41no—
65#1084SD1612yescomplemented
66(C3⋊Dic3)⋊C3—42no—
67C2×C3×SD16—92no—
68C3×C4×D4—91no—
69#944—92no—
70D4×Dic3—32no—
71D4×Dic3—61no—
72C2×#214—32no—
73C2×#214—92no—
74#869—91no—
75#835—92no—
76#816—92no—
77Dic3⋊C8—92no—
78#810—92no—
79C4⋊(C3⋊C8)—91no—
80C23×C32—22no—
81C23×C32—22no—
82C32×D4—22no—
83C32×D4—22no—
84C32×D4—22no—
85C32×D4#6714yes—
86C2×C32×C4—21no—
87C2×C32×C4C22×D312yes—
88C2×C3⋊Dic3—22no—
89#634—22no—
90#634—22no—
91C2×C3×Dic3—32no—
92C2×C3×Dic3—32no—
93C2×C3×Dic3—61no—
94C2×C3×Dic3—61no—
95C2×C3×Dic3—62no—
96C2×C3×Dic3—62no—
97C2×C3×Dic3—62no—
98C2×C3×Dic3—124no—
99C2×C3×Dic3—124no—
100C3×Dic6—62no—
101C3×Dic6—62no—
102C3⋊(C3⋊C8)—22no—
103C3×C3⋊C8—62no—
104#461—271noSylow
105C2×H(3,𝔽3)—42no—
106C2×H(3,𝔽3)#12111yescharacteristic
107C2×H(3,𝔽3)C2×SD1611yescharacteristic
108C2×H(3,𝔽3)#13711yescharacteristic
109C2×C3×Q8—92no—
110C2×C3×D4—32no—
111C2×C3×D4—61no—
112C2×C3×D4D3211yescharacteristic
113C22×C3×C4—181no—
114C22×Dic3—62no—
115C22×Dic3—62no—
116C2×Dic6—32no—
117C2×Dic6—92no—
118C3×SD16—184no—
119C2×C3×C8—92no—
120C3×C4⋊C4—91no—
121C3×C22⋊C4—181no—
122C3×C42—91no—
123#217—62no—
124#217—62no—
125#214—64no—
126#214—184no—
127C4⋊Dic3—32no—
128C4⋊Dic3—61no—
129C4⋊Dic3—181no—
130C4⋊Dic3—181no—
131Dic3⋊C4—182no—
132Dic3⋊C4—182no—
133C4×Dic3—32no—
134C4×Dic3—61no—
135C2×C3⋊C8—32no—
136C2×C3⋊C8—92no—
137C22×C32—24no—
138C22×C32—24no—
139C22×C32—21no—
140C22×C32—42no—
141C22×C32—42no—
142C22×C32—42no—
143C22×C32—42no—
144C22×C32D3×D412yes—
145C32×C4—21no—
146C32×C4—21no—
147C32×C4#23712yes—
148C32×C4C2×#6712yes—
149C3⋊Dic3—42no—
150C3×Dic3—34no—
151C3×Dic3—62no—
152C3×Dic3—62no—
153C3×Dic3—121no—
154C3×Dic3—122no—
155C3×Dic3—122no—
156C3×Dic3—122no—
157C2×SD16—92no—
158C4⋊Q8—271no—
159C4×D4—91no—
160C4⋊C8—271no—
161#103—272no—
162H(3,𝔽3)#46111yescharacteristic, complemented, Sylow
163C23×C3—21no—
164C23×C3—62no—
165C23×C3—62no—
166C3×Q8—182no—
167C3×Q8—182no—
168C3×D4—21no—
169C3×D4—34no—
170C3×D4—62no—
171C3×D4—62no—
172C3×D4—62no—
173C3×D4#62512yes—
174C2×C3×C4—32no—
175C2×C3×C4—61no—
176C2×C3×C4—91no—
177C2×C3×C4—91no—
178C2×C3×C4—182no—
179C2×C3×C4—181no—
180C2×C3×C4—362no—
181C2×C3×C4C2×D3211yescharacteristic
182C2×Dic3—32no—
183C2×Dic3—32no—
184C2×Dic3—61no—
185C2×Dic3—61no—
186C2×Dic3—62no—
187C2×Dic3—62no—
188C2×Dic3—62no—
189C2×Dic3—124no—
190C2×Dic3—124no—
191C2×Dic3—182no—
192Dic6—62no—
193Dic6—62no—
194Dic6—182no—
195Dic6—182no—
196C3×C8—182no—
197C3⋊C8—62no—
198C3⋊C8—182no—
199C2×C32—21no—
200C2×C32—21no—
201C2×C32—21no—
202C2×C32—44no—
203C2×C32—44no—
204C2×C32D3×SD1612yes—
205C2×C32#91512yes—
206C2×C32#93912yes—
207C2×Q8—92no—
208C2×D4((C3⋊D3)⋊C3)⋊C211yescharacteristic
209C22×C4—181no—
210SD16—184no—
211C2×C8—92no—
212C4⋊C4—91no—
213C4⋊C4—541no—
214C4⋊C4—541no—
215C22⋊C4—181no—
216C42—91no—
217C22×C3—22no—
218C22×C3—32no—
219C22×C3—41no—
220C22×C3—41no—
221C22×C3—64no—
222C22×C3—64no—
223C22×C3—61no—
224C22×C3—122no—
225C22×C3—122no—
226C22×C3—122no—
227C22×C3—122no—
228C22×C3#381111yescharacteristic
229C3×C4—32no—
230C3×C4—32no—
231C3×C4—61no—
232C3×C4—61no—
233C3×C4—92no—
234C3×C4—181no—
235C3×C4—362no—
236C3×C4—361no—
237C3×C4#380511yescharacteristic
238C3×C4C2×#62511yescharacteristic
239Dic3—34no—
240Dic3—62no—
241Dic3—62no—
242Dic3—121no—
243Dic3—122no—
244Dic3—122no—
245Dic3—122no—
246Dic3—362no—
247C32—21no—
248C32#516212yescomplemented
249C23—21no—
250Q8—182no—
251Q8—182no—
252D4—21no—
253D4#619812yes—
254C2×C4—91no—
255C2×C4—91no—
256C2×C4—181no—
257C2×C4—182no—
258C2×C4—362no—
259C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
260C8—182no—
261C2×C3—32no—
262C2×C3—32no—
263C2×C3—32no—
264C2×C3—42no—
265C2×C3—61no—
266C2×C3—61no—
267C2×C3—61no—
268C2×C3—124no—
269C2×C3—124no—
270C2×C3#6393011yescharacteristic
271C2×C3#6393111yescharacteristic
272C2×C3#6398011yescharacteristic
273C22—22no—
274C22—41no—
275C22—41no—
276C22#8867711yescharacteristic
277C4—92no—
278C4—181no—
279C4—362no—
280C4—361no—
281C4#8867211yescharacteristic
282C4C2×#619811yescharacteristic
283C3—32no—
284C3—61no—
285C3#1059149711yescharacteristic
286C2—42no—
287C2#1172118811yescharacteristic
288C2#1172118911yescharacteristic
289C2#1172131811yescharacteristic
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)#461
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
3. #11721099C2semidirect
4. #11720309C2semidirect
5. #11720304C2semidirect
65. #1084SD16semidirect
162. H(3,𝔽3)#461semidirect
248. C32#5162semidirect

Maximal subgroups

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

subgroupindex
D4×#10842
C2×#3754⋊C32
#117210992
#117203092
#117203042
#105914893