Finite Group Databasebeta

← #13143952

Subgroups — #13143952

Order 1728.

891 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 891 conjugacy classes of subgroups — too many to draw. Only its 113 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, complemented
2C2×#88545C211yesmaximal, characteristic, complemented
3C2×#88528C211yesmaximal, characteristic, complemented
4#11721763C211yesmaximal, characteristic, complemented
5H(3,𝔽3)×#102C211yesmaximal, characteristic, complemented
6#11720553C211yesmaximal, characteristic, complemented
7#11720532C211yesmaximal, characteristic, complemented
8C3×#4781—34nomaximal
9C2×D4×H(3,𝔽3)C2211yescharacteristic
10C2×#6235—21no—
11C22×#1084—21no—
12#88744—22no—
13#88744—22no—
14C2×C4⋊#272—21no—
15C2×C4⋊#272C2211yescharacteristic
16C2×C4×#272C2211yescharacteristic
17C2×#6229C2211yescharacteristic
18C2×C8×H(3,𝔽3)C2211yescharacteristic
19C4⋊C4×H(3,𝔽3)C2211yescharacteristic
20#88545—22no—
21#88542—21no—
22C2×#6180C2211yescharacteristic
23#88528—22no—
24C2×C3×#237—34no—
25C2×C3×#213—34no—
26C2×C3×#204—34no—
27C3×#895—34no—
28C32×#102D314yescomplemented
29C3×(C2×D3)⋊C8—34no—
30C3×#810—34no—
31C23×H(3,𝔽3)—21no—
32C22×#272—21no—
33C22×#272D411yescharacteristic
34D4×H(3,𝔽3)—21no—
35D4×H(3,𝔽3)D412yescomplemented
36C2×C4×H(3,𝔽3)—21no—
37C2×C4×H(3,𝔽3)C2311yescharacteristic
38#6235—24no—
39C2×#1084—24no—
40C2×#1084—21no—
41C2×#1084D411yescharacteristic
42C4⋊#272—21no—
43C4⋊#272—21no—
44C4⋊#272—42no—
45C4×#272—21no—
46C4×#272—21no—
47#6229—21no—
48#6229D412yescomplemented
49C8×H(3,𝔽3)—21no—
50#6180—21no—
51C3×#449—91no—
52#4781—34no—
53C2×C32×D4C2×D314yes—
54C2×C3×#67—64no—
55C22×C3×Dic3—64no—
56C3×#237—68no—
57C3×#237—68no—
58C2×C3×D12—34no—
59C2×C3×D12—64no—
60C2×C3×C4×D3—34no—
61C2×C3×Dic6—34no—
62C2×C32×C8C2×D314yes—
63C32×C4⋊C4C2×D314yes—
64C3×#213—68no—
65C3×(C2×D3)⋊C4—64no—
66C2×C3×C3⋊C8—34no—
67C3×#204—68no—
68C22×H(3,𝔽3)—22no—
69C22×H(3,𝔽3)—41no—
70C22×H(3,𝔽3)—41no—
71C22×H(3,𝔽3)C2×D411yescharacteristic
72C2×#272—21no—
73C2×#272—21no—
74C2×#272—21no—
75C2×#272—41no—
76C2×#272—41no—
77C2×#272—41no—
78C4×H(3,𝔽3)—41no—
79C4×H(3,𝔽3)C2×D411yescharacteristic
80C4×H(3,𝔽3)C2×D411yescharacteristic
81#1084—22no—
82#1084—21no—
83C2×C3×#41—91no—
84C2×#237—34no—
85C2×C3×SD16—91no—
86C2×C3×D8—91no—
87C3×#121—91no—
88C2×#213—34no—
89C2×#204—34no—
90#895—34no—
91C3×#103—91no—
92C3×#102—34no—
93C3×#102C3⋊D311yescharacteristic
94C3×C22⋊C8—91no—
95(C2×D3)⋊C8—34no—
96#810—34no—
97C23×C32—24no—
98C22×C3×D3—34no—
99C22×C3×D3—64no—
100C32×D4—24no—
101C32×D4D1218yescomplemented
102C2×C32×C4—24no—
103C2×C32×C4C22×D314yes—
104C3×#67—616no—
105C2×C3×Dic3—34no—
106C2×C3×Dic3—616no—
107C2×C3×Dic3—64no—
108C3×D12—64no—
109C3×D12—64no—
110C3×D12—128no—
111C3×C4×D3—64no—
112C3×C4×D3—64no—
113C3×Dic6—38no—
114C3×Dic6—64no—
115C32×C8—24no—
116C3×C3⋊C8—64no—
117#449—91noSylow
118C2×H(3,𝔽3)—42no—
119C2×H(3,𝔽3)#12011yescharacteristic
120C2×H(3,𝔽3)#13511yescharacteristic
121C2×H(3,𝔽3)AGL(1,ℤ/8)11yescharacteristic
122#272—41no—
123#272—81no—
124C3×#41—182no—
125C3×#41—182no—
126C2×C3×Q8—91no—
127C2×C3×D4—34no—
128C2×C3×D4—91no—
129C2×C3×D4—181no—
130C2×C3×D4—181no—
131C2×C3×D4C2×C3⋊D311yescharacteristic
132C22×C3×C4—91no—
133C22×C3×C4—181no—
134C2×#67—64no—
135C22×Dic3—64no—
136#237—68no—
137#237—68no—
138C2×D12—34no—
139C2×D12—64no—
140C2×C4×D3—34no—
141C2×Dic6—34no—
142C3×SD16—182no—
143C3×D8—182no—
144C2×C3×C8—34no—
145C2×C3×C8—91no—
146C2×C3×C8C2×C3⋊D311yescharacteristic
147C3×C4⋊C4—34no—
148C3×C4⋊C4C2×C3⋊D311yescharacteristic
149C3×C22⋊C4—181no—
150#213—68no—
151(C2×D3)⋊C4—64no—
152C2×C3⋊C8—34no—
153#204—68no—
154C22×C32—28no—
155C22×C32—44no—
156C22×C32—44no—
157C22×C32D3×D414yes—
158C2×C3×D3—64no—
159C2×C3×D3—64no—
160C2×C3×D3—64no—
161C2×C3×D3—124no—
162C2×C3×D3—124no—
163C2×C3×D3—124no—
164C32×C4—44no—
165C32×C4C2×D1214yes—
166C32×C4D3×D414yes—
167C3×Dic3—68no—
168C3×Dic3—64no—
169C2×#41—91no—
170C2×SD16—91no—
171C2×D8—91no—
172#121—91no—
173#103—91no—
174#102#27211yescharacteristic, complemented
175C22⋊C8—91no—
176H(3,𝔽3)#44911yescharacteristic, complemented, Sylow
177C23×C3—21no—
178C23×C3—64no—
179C23×C3—91no—
180C23×C3—181no—
181C22×D3—34no—
182C22×D3—64no—
183C3×Q8—92no—
184C3×Q8—181no—
185C3×D4—21no—
186C3×D4—38no—
187C3×D4—64no—
188C3×D4—184no—
189C3×D4—181no—
190C3×D4—181no—
191C3×D4—362no—
192C3×D4C4⋊(C3⋊D3)12yes—
193C2×C3×C4—21no—
194C2×C3×C4—34no—
195C2×C3×C4—64no—
196C2×C3×C4—91no—
197C2×C3×C4—184no—
198C2×C3×C4—181no—
199C2×C3×C4—181no—
200C2×C3×C4—181no—
201C2×C3×C4C22×C3⋊D311yescharacteristic
202#67—616no—
203C2×Dic3—34no—
204C2×Dic3—64no—
205C2×Dic3—616no—
206D12—64no—
207D12—64no—
208D12—128no—
209C4×D3—64no—
210C4×D3—64no—
211Dic6—38no—
212Dic6—64no—
213C3×C8—21no—
214C3×C8—64no—
215C3×C8—181no—
216C3⋊C8—64no—
217C2×C32—48no—
218C2×C32#88214yes—
219C2×C32#91114yes—
220C2×C32#91614yes—
221C3×D3—124no—
222C3×D3—244no—
223#41—182no—
224#41—182no—
225C2×Q8—91no—
226C2×D4—91no—
227C2×D4—181no—
228C2×D4—181no—
229C2×D4C2×#27211yescharacteristic
230C22×C4—91no—
231C22×C4—181no—
232SD16—182no—
233D8—182no—
234C2×C8—91no—
235C2×C8C2×#27211yescharacteristic
236C4⋊C4C2×#27211yescharacteristic
237C22⋊C4—181no—
238C22×C3—22no—
239C22×C3—34no—
240C22×C3—41no—
241C22×C3—41no—
242C22×C3—68no—
243C22×C3—124no—
244C22×C3—124no—
245C22×C3—181no—
246C22×C3—181no—
247C22×C3—181no—
248C22×C3—361no—
249C22×C3—361no—
250C22×C3—361no—
251C22×C3D4×C3⋊D311yescharacteristic
252C2×D3—64no—
253C2×D3—64no—
254C2×D3—64no—
255C2×D3—124no—
256C2×D3—124no—
257C2×D3—124no—
258C3×C4—34no—
259C3×C4—34no—
260C3×C4—41no—
261C3×C4—124no—
262C3×C4—181no—
263C3×C4—182no—
264C3×C4C2×C4⋊(C3⋊D3)11yescharacteristic
265C3×C4D4×C3⋊D311yescharacteristic
266Dic3—68no—
267Dic3—64no—
268C32#478114yescomplemented
269C23—21no—
270C23—91no—
271C23—181no—
272Q8—92no—
273Q8—181no—
274D4—21no—
275D4—184no—
276D4—181no—
277D4—181no—
278D4—362no—
279D4C4⋊#27212yescomplemented
280C2×C4—21no—
281C2×C4—91no—
282C2×C4—184no—
283C2×C4—181no—
284C2×C4—181no—
285C2×C4—181no—
286C2×C4C22×#27211yescharacteristic
287C8—21no—
288C8—181no—
289C2×C3—34no—
290C2×C3—34no—
291C2×C3—34no—
292C2×C3—42no—
293C2×C3—128no—
294C2×C3—361no—
295C2×C3—721no—
296C2×C3#6409111yescharacteristic
297C2×C3#6412011yescharacteristic
298C2×C3#6412511yescharacteristic
299D3—124no—
300D3—244no—
301C22—22no—
302C22—41no—
303C22—41no—
304C22—181no—
305C22—181no—
306C22—181no—
307C22—361no—
308C22—361no—
309C22—361no—
310C22D4×#27211yescharacteristic
311C4—41no—
312C4—181no—
313C4—182no—
314C4C2×C4⋊#27211yescharacteristic
315C4D4×#27211yescharacteristic
316C3—34no—
317C3#1059248011yescharacteristic
318C2—42no—
319C2—361no—
320C2—721no—
321C2#1172173811yescharacteristic
322C2#1172179311yescharacteristic
323C2#1172180211yescharacteristic
subgroupG/H
center, socleC22×C3D4×C3⋊D3
commutator, cosocleC2×C4×H(3,𝔽3)C23
FrattiniC2×C3×C4C22×C3⋊D3
FittingH(3,𝔽3)×#102C2
hypercenterC3×#102C3⋊D3
hypocenterH(3,𝔽3)#449
LCSG ▷ C2×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
1. C2×#88744C2semidirect
2. C2×#88545C2semidirect
3. C2×#88528C2semidirect
4. #11721763C2semidirect
5. H(3,𝔽3)×#102C2semidirect
6. #11720553C2semidirect
7. #11720532C2semidirect
28. C32×#102D3semidirect
35. D4×H(3,𝔽3)D4semidirect
48. #6229D4semidirect
101. C32×D4D12semidirect
174. #102#272semidirect
176. H(3,𝔽3)#449semidirect
268. C32#4781semidirect
279. D4C4⋊#272semidirect

Maximal subgroups

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

subgroupindex
C2×#887442
C2×#885452
C2×#885282
#117217632
H(3,𝔽3)×#1022
#117205532
#117205322
C3×#47813