Finite Group Databasebeta

← #13143943

Subgroups — #13143943

Order 1728.

869 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 869 conjugacy classes of subgroups — too many to draw. Only its 125 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
1C2×#88744C211yesmaximal, characteristic
2C2×C8×#272C211yesmaximal, characteristic, complemented
3#11721760C211yesmaximal, characteristic, complemented
4H(3,𝔽3)×#102C211yesmaximal, characteristic, complemented
5#11720578C211yesmaximal, characteristic, complemented
6#11720549C211yesmaximal, characteristic, complemented
7#11720532C211yesmaximal, characteristic, complemented
8C3×#4772—34nomaximal
9C2×D4×H(3,𝔽3)C2211yescharacteristic
10C2×#6235—21no—
11C22×#1084—21no—
12#88744—22no—
13#88744C414yescomplemented
14C2×C4×#272C2211yescharacteristic
15C2×#6229C2211yescharacteristic
16C2×C8×H(3,𝔽3)C2211yescharacteristic
17C4⋊C4×H(3,𝔽3)C2211yescharacteristic
18#88542—21no—
19C4⋊#1084C2211yescharacteristic
20C4×#1084—21no—
21C2×#6180C2211yescharacteristic
22C8×#272—22no—
23C2×C3×#237—34no—
24C2×C3×D3×C8—34no—
25C3×#892—34no—
26C32×#102D314yescomplemented
27C3×#832—34no—
28C3×#816—34no—
29C3×#810—34no—
30C23×H(3,𝔽3)—21no—
31C22×#272D411yescharacteristic
32D4×H(3,𝔽3)—21no—
33D4×H(3,𝔽3)C2×C412yes—
34C2×C4×H(3,𝔽3)—21no—
35C2×C4×H(3,𝔽3)C2311yescharacteristic
36#6235—24no—
37C2×#1084—21no—
38C2×#1084—24no—
39C2×#1084—21no—
40C2×#1084D411yescharacteristic
41C4×#272C2×C412yes—
42C4×#272D412yes—
43#6229—21no—
44#6229C2×C412yes—
45C8×H(3,𝔽3)—21no—
46#6180—21no—
47C3×#416—91no—
48#4772—34no—
49C2×C32×D4C2×D314yes—
50C2×C3×#67—64no—
51C22×C3×Dic3—64no—
52C3×#237—316no—
53C3×#237—68no—
54C2×C3×C4×D3—34no—
55C2×C3×Dic6—34no—
56C2×C32×C8C2×D314yes—
57C32×C4⋊C4C2×D314yes—
58C3×(C2×D3)⋊C4—64no—
59C3×C4⋊Dic3—34no—
60C3×C4×Dic3—64no—
61C2×C3×C3⋊C8—34no—
62C3×D3×C8—68no—
63C22×H(3,𝔽3)—22no—
64C22×H(3,𝔽3)—41no—
65C22×H(3,𝔽3)—41no—
66C22×H(3,𝔽3)C2×D411yescharacteristic
67C2×#272—21no—
68C2×#272—21no—
69C2×#272C22⋊C412yes—
70C4×H(3,𝔽3)—41no—
71C4×H(3,𝔽3)C22×C411yescharacteristic
72C4×H(3,𝔽3)C2×D411yescharacteristic
73#1084—22no—
74#1084—41no—
75#1084C22⋊C412yes—
76C2×C3×#41—91no—
77C2×#237—34no—
78C22×C3×C8—91no—
79C3×#117—91no—
80C2×D3×C8—34no—
81#892—34no—
82C3×#103—91no—
83C3×#103—91no—
84C3×#102—34no—
85C3×#102—91no—
86C3×#102C3⋊D311yescharacteristic
87#832—34no—
88#816—34no—
89#810—34no—
90C23×C32—24no—
91C22×C3×D3—34no—
92C32×D4—24no—
93C32×D4C4×D318yes—
94C2×C32×C4—24no—
95C2×C32×C4C22×D314yes—
96C3×#67—616no—
97C2×C3×Dic3—34no—
98C2×C3×Dic3—616no—
99C2×C3×Dic3—64no—
100C2×C3×Dic3—64no—
101C3×C4×D3—38no—
102C3×C4×D3—38no—
103C3×Dic6—38no—
104C3×Dic6—64no—
105C32×C8—24no—
106C3×C3⋊C8—64no—
107#416—91noSylow
108C2×H(3,𝔽3)—42no—
109C2×H(3,𝔽3)C2×C22⋊C411yescharacteristic
110C2×H(3,𝔽3)#13511yescharacteristic
111C2×H(3,𝔽3)#13511yescharacteristic
112#272—22no—
113C3×#41—94no—
114C3×#41—182no—
115C2×C3×Q8—91no—
116C2×C3×D4—34no—
117C2×C3×D4—181no—
118C2×C3×D4C2×C3⋊D311yescharacteristic
119C22×C3×C4—91no—
120C22×C3×C4—181no—
121C2×#67—64no—
122C22×Dic3—64no—
123#237—316no—
124#237—68no—
125C2×C4×D3—34no—
126C2×Dic6—34no—
127C2×C3×C8—34no—
128C2×C3×C8—91no—
129C2×C3×C8—182no—
130C2×C3×C8C2×C3⋊D311yescharacteristic
131C3×C4⋊C4—34no—
132C3×C4⋊C4—91no—
133C3×C4⋊C4C2×C3⋊D311yescharacteristic
134C3×C22⋊C4—181no—
135C3×C42—181no—
136(C2×D3)⋊C4—64no—
137C4⋊Dic3—34no—
138C4×Dic3—64no—
139C2×C3⋊C8—34no—
140D3×C8—68no—
141C22×C32—28no—
142C22×C32—44no—
143C22×C32—44no—
144C22×C32D3×D414yes—
145C2×C3×D3—38no—
146C2×C3×D3—64no—
147C2×C3×D3—64no—
148C32×C4—44no—
149C32×C4C2×C4×D314yes—
150C32×C4D3×D414yes—
151C3×Dic3—38no—
152C3×Dic3—68no—
153C3×Dic3—124no—
154C2×#41—91no—
155C22×C8—91no—
156#117—91no—
157#103—91no—
158#103—91no—
159#102—91no—
160#102#27211yescharacteristic, complemented
161H(3,𝔽3)#41611yescharacteristic, complemented, Sylow
162C23×C3—21no—
163C23×C3—64no—
164C23×C3—91no—
165C22×D3—34no—
166C3×Q8—92no—
167C3×Q8—181no—
168C3×D4—21no—
169C3×D4—38no—
170C3×D4—64no—
171C3×D4—184no—
172C3×D4C4×C3⋊D312yes—
173C2×C3×C4—21no—
174C2×C3×C4—34no—
175C2×C3×C4—64no—
176C2×C3×C4—91no—
177C2×C3×C4—92no—
178C2×C3×C4—92no—
179C2×C3×C4—184no—
180C2×C3×C4—181no—
181C2×C3×C4—181no—
182C2×C3×C4C22×C3⋊D311yescharacteristic
183#67—616no—
184C2×Dic3—34no—
185C2×Dic3—616no—
186C2×Dic3—64no—
187C2×Dic3—64no—
188C4×D3—38no—
189C4×D3—38no—
190Dic6—38no—
191Dic6—64no—
192C3×C8—21no—
193C3×C8—64no—
194C3×C8—181no—
195C3⋊C8—64no—
196C2×C32—48no—
197C2×C32D3×C22⋊C414yes—
198C2×C32#91214yes—
199C2×C32#91614yes—
200C3×D3—68no—
201#41—94no—
202#41—182no—
203C2×Q8—91no—
204C2×D4—181no—
205C2×D4C2×#27211yescharacteristic
206C22×C4—91no—
207C22×C4—181no—
208C2×C8—91no—
209C2×C8—182no—
210C2×C8C2×#27211yescharacteristic
211C4⋊C4—91no—
212C4⋊C4C2×#27211yescharacteristic
213C22⋊C4—181no—
214C42—181no—
215C22×C3—22no—
216C22×C3—34no—
217C22×C3—41no—
218C22×C3—41no—
219C22×C3—68no—
220C22×C3—92no—
221C22×C3—124no—
222C22×C3—124no—
223C22×C3—181no—
224C22×C3—181no—
225C22×C3D4×C3⋊D311yescharacteristic
226C2×D3—38no—
227C2×D3—64no—
228C2×D3—64no—
229C3×C4—34no—
230C3×C4—34no—
231C3×C4—41no—
232C3×C4—92no—
233C3×C4—124no—
234C3×C4—182no—
235C3×C4—361no—
236C3×C4C2×C4×C3⋊D311yescharacteristic
237C3×C4D4×C3⋊D311yescharacteristic
238Dic3—38no—
239Dic3—68no—
240Dic3—124no—
241C32#477214yescomplemented
242C23—21no—
243C23—91no—
244Q8—92no—
245Q8—181no—
246D4—21no—
247D4—184no—
248D4C4×#27212yes—
249C2×C4—21no—
250C2×C4—91no—
251C2×C4—92no—
252C2×C4—92no—
253C2×C4—184no—
254C2×C4—181no—
255C2×C4—181no—
256C2×C4C22×#27211yescharacteristic
257C8—21no—
258C8—181no—
259C2×C3—34no—
260C2×C3—34no—
261C2×C3—34no—
262C2×C3—42no—
263C2×C3—128no—
264C2×C3—182no—
265C2×C3C22⋊C4×C3⋊D311yescharacteristic
266C2×C3#6412111yescharacteristic
267C2×C3#6412511yescharacteristic
268D3—68no—
269C22—22no—
270C22—41no—
271C22—41no—
272C22—92no—
273C22—181no—
274C22—181no—
275C22D4×#27211yescharacteristic
276C4—41no—
277C4—92no—
278C4—182no—
279C4—361no—
280C4C2×C4×#27211yescharacteristic
281C4D4×#27211yescharacteristic
282C3—34no—
283C3#1059247111yescharacteristic
284C2—42no—
285C2—182no—
286C2C22⋊C4×#27211yescharacteristic
287C2#1172179411yescharacteristic
288C2#1172180211yescharacteristic
subgroupG/H
center, socleC22×C3D4×C3⋊D3
commutatorC4×H(3,𝔽3)C22×C4
FrattiniC2×C3×C4C22×C3⋊D3
FittingH(3,𝔽3)×#102C2
prime radicalC2×#88744C2
cosocleC2×C4×H(3,𝔽3)C23
hypercenterC3×#102C3⋊D3
hypocenterH(3,𝔽3)#416
LCSG ▷ C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22×C3 ◁ C2×C3×C4 ◁ C3×#102

Complements

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

NH ≅ G/Ntype
2. C2×C8×#272C2semidirect
3. #11721760C2semidirect
4. H(3,𝔽3)×#102C2semidirect
5. #11720578C2semidirect
6. #11720549C2semidirect
7. #11720532C2semidirect
13. #88744C4semidirect
26. C32×#102D3semidirect
160. #102#272semidirect
161. H(3,𝔽3)#416semidirect
241. C32#4772semidirect

Maximal subgroups

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

subgroupindex
C2×#887442
C2×C8×#2722
#117217602
H(3,𝔽3)×#1022
#117205782
#117205492
#117205322
C3×#47723