Finite Group Databasebeta

← #13144007

Subgroups — #13144007

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×#88748C211yesmaximal, characteristic
2C2×C8×#272C211yesmaximal, characteristic, complemented
3#11721760C211yesmaximal, characteristic, complemented
4H(3,𝔽3)×#103C211yesmaximal, characteristic, complemented
5#11720583C211yesmaximal, characteristic, complemented
6#11720554C211yesmaximal, characteristic, complemented
7#11720531C211yesmaximal, characteristic, complemented
8C3×#4804—34nomaximal
9C2×Q8×H(3,𝔽3)C2211yescharacteristic
10#88748—22no—
11#88748C414yescomplemented
12C2×C4⋊#272—21no—
13C2×C4⋊#272C2211yescharacteristic
14C2×C4×#272—21no—
15C2×C4×#272C2211yescharacteristic
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×#239—34no—
24C2×C3×D3×C8—34no—
25C3×#892—34no—
26C32×#103D314yescomplemented
27C3×#835—34no—
28C3×#821—34no—
29C3×#809—34no—
30C22×#272—21no—
31C22×#272D411yescharacteristic
32Q8×H(3,𝔽3)—21no—
33Q8×H(3,𝔽3)C2×C412yes—
34C2×C4×H(3,𝔽3)—21no—
35C2×C4×H(3,𝔽3)—21no—
36C2×C4×H(3,𝔽3)C2311yescharacteristic
37C2×#1084—21no—
38C2×#1084D411yescharacteristic
39C4⋊#272—24no—
40C4⋊#272—21no—
41C4⋊#272C2×C412yes—
42C4×#272—24no—
43C4×#272C2×C412yes—
44C4×#272D412yes—
45C8×H(3,𝔽3)—21no—
46#6180—21no—
47C3×#416—91no—
48#4804—34no—
49C2×C32×Q8C2×D314yes—
50C3×#239—316no—
51C3×#239—68no—
52C2×C3×D12—34no—
53C2×C3×D12—64no—
54C2×C3×C4×D3—34no—
55C2×C3×C4×D3—64no—
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)C2×D411yescharacteristic
64C2×#272—21no—
65C2×#272—21no—
66C2×#272—22no—
67C2×#272—41no—
68C2×#272—41no—
69C2×#272C22⋊C412yes—
70C4×H(3,𝔽3)—22no—
71C4×H(3,𝔽3)—41no—
72C4×H(3,𝔽3)C22×C411yescharacteristic
73C4×H(3,𝔽3)C2×D411yescharacteristic
74#1084—41no—
75#1084C22⋊C412yes—
76C2×C3×#41—91no—
77C2×#239—34no—
78C22×C3×C8—91no—
79C3×#117—91no—
80C2×D3×C8—34no—
81#892—34no—
82C3×#103—34no—
83C3×#103—91no—
84C3×#103C3⋊D311yescharacteristic
85C3×#102—91no—
86C3×#102—91no—
87#835—34no—
88#821—34no—
89#809—34no—
90C22×C3×D3—34no—
91C22×C3×D3—64no—
92C32×Q8—24no—
93C32×Q8C4×D318yes—
94C2×C32×C4—24no—
95C2×C32×C4—24no—
96C2×C32×C4C22×D314yes—
97C2×C3×Dic3—34no—
98C2×C3×Dic3—64no—
99C3×D12—38no—
100C3×D12—616no—
101C3×D12—64no—
102C3×C4×D3—38no—
103C3×C4×D3—38no—
104C3×C4×D3—616no—
105C32×C8—24no—
106C3×C3⋊C8—64no—
107#416—91noSylow
108C2×H(3,𝔽3)C2×C22⋊C411yescharacteristic
109C2×H(3,𝔽3)#13511yescharacteristic
110C2×H(3,𝔽3)#13511yescharacteristic
111#272—22no—
112#272—42no—
113C3×#41—94no—
114C3×#41—182no—
115C2×C3×Q8—34no—
116C2×C3×Q8C2×C3⋊D311yescharacteristic
117C2×C3×D4—91no—
118C2×C3×D4—181no—
119C22×C3×C4—91no—
120C22×C3×C4—181no—
121#239—316no—
122#239—68no—
123C2×D12—34no—
124C2×D12—64no—
125C2×C4×D3—34no—
126C2×C4×D3—64no—
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×C32D3×D414yes—
142C2×C3×D3—38no—
143C2×C3×D3—64no—
144C2×C3×D3—64no—
145C2×C3×D3—68no—
146C2×C3×D3—124no—
147C2×C3×D3—124no—
148C32×C4—28no—
149C32×C4—44no—
150C32×C4C2×C4×D314yes—
151C32×C4D3×D414yes—
152C3×Dic3—38no—
153C3×Dic3—124no—
154C2×#41—91no—
155C22×C8—91no—
156#117—91no—
157#103—91no—
158#103#27211yescharacteristic, complemented
159#102—91no—
160#102—91no—
161H(3,𝔽3)#41611yescharacteristic, complemented, Sylow
162C23×C3—91no—
163C23×C3—181no—
164C22×D3—34no—
165C22×D3—64no—
166C3×Q8—21no—
167C3×Q8—38no—
168C3×Q8—64no—
169C3×Q8C4×C3⋊D312yes—
170C3×D4—92no—
171C3×D4—184no—
172C3×D4—181no—
173C2×C3×C4—21no—
174C2×C3×C4—21no—
175C2×C3×C4—34no—
176C2×C3×C4—64no—
177C2×C3×C4—64no—
178C2×C3×C4—91no—
179C2×C3×C4—92no—
180C2×C3×C4—92no—
181C2×C3×C4—184no—
182C2×C3×C4—181no—
183C2×C3×C4C22×C3⋊D311yescharacteristic
184C2×Dic3—34no—
185C2×Dic3—64no—
186D12—38no—
187D12—64no—
188D12—616no—
189C4×D3—38no—
190C4×D3—38no—
191C4×D3—616no—
192C3×C8—21no—
193C3×C8—64no—
194C3×C8—181no—
195C3⋊C8—64no—
196C2×C32D3×C22⋊C414yes—
197C2×C32#91614yes—
198C2×C32#91914yes—
199C3×D3—68no—
200C3×D3—128no—
201#41—94no—
202#41—182no—
203C2×Q8C2×#27211yescharacteristic
204C2×D4—91no—
205C2×D4—181no—
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—34no—
216C22×C3—92no—
217C22×C3—181no—
218C22×C3—181no—
219C22×C3—182no—
220C22×C3—361no—
221C22×C3—361no—
222C22×C3D4×C3⋊D311yescharacteristic
223C2×D3—38no—
224C2×D3—64no—
225C2×D3—64no—
226C2×D3—68no—
227C2×D3—124no—
228C2×D3—124no—
229C3×C4—22no—
230C3×C4—34no—
231C3×C4—34no—
232C3×C4—41no—
233C3×C4—68no—
234C3×C4—92no—
235C3×C4—124no—
236C3×C4—361no—
237C3×C4C2×C4×C3⋊D311yescharacteristic
238C3×C4D4×C3⋊D311yescharacteristic
239Dic3—38no—
240Dic3—124no—
241C32#480414yescomplemented
242C23—91no—
243C23—181no—
244Q8—21no—
245Q8C4×#27212yes—
246D4—92no—
247D4—184no—
248D4—181no—
249C2×C4—21no—
250C2×C4—21no—
251C2×C4—91no—
252C2×C4—92no—
253C2×C4—92no—
254C2×C4—184no—
255C2×C4—181no—
256C2×C4C22×#27211yescharacteristic
257C8—21no—
258C8—181no—
259C2×C3—34no—
260C2×C3—34no—
261C2×C3—34no—
262C2×C3—182no—
263C2×C3—362no—
264C2×C3C22⋊C4×C3⋊D311yescharacteristic
265C2×C3#6412511yescharacteristic
266C2×C3#6412811yescharacteristic
267D3—68no—
268D3—128no—
269C22—92no—
270C22—181no—
271C22—181no—
272C22—182no—
273C22—361no—
274C22—361no—
275C22D4×#27211yescharacteristic
276C4—22no—
277C4—41no—
278C4—92no—
279C4—361no—
280C4C2×C4×#27211yescharacteristic
281C4D4×#27211yescharacteristic
282C3—34no—
283C3#1059250311yescharacteristic
284C2—182no—
285C2—362no—
286C2C22⋊C4×#27211yescharacteristic
287C2#1172180211yescharacteristic
288C2#1172180811yescharacteristic
subgroupG/H
center, socleC22×C3D4×C3⋊D3
commutatorC4×H(3,𝔽3)C22×C4
FrattiniC2×C3×C4C22×C3⋊D3
FittingH(3,𝔽3)×#103C2
prime radicalC2×#88748C2
cosocleC2×C4×H(3,𝔽3)C23
hypercenterC3×#103C3⋊D3
hypocenterH(3,𝔽3)#416
LCSG ▷ C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22×C3 ◁ C2×C3×C4 ◁ C3×#103

Complements

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

NH ≅ G/Ntype
2. C2×C8×#272C2semidirect
3. #11721760C2semidirect
4. H(3,𝔽3)×#103C2semidirect
5. #11720583C2semidirect
6. #11720554C2semidirect
7. #11720531C2semidirect
11. #88748C4semidirect
26. C32×#103D3semidirect
158. #103#272semidirect
161. H(3,𝔽3)#416semidirect
241. C32#4804semidirect

Maximal subgroups

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

subgroupindex
C2×#887482
C2×C8×#2722
#117217602
H(3,𝔽3)×#1032
#117205832
#117205542
#117205312
C3×#48043