Finite Group Databasebeta

← #13144468

Subgroups — #13144468

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×#88740C211yesmaximal, characteristic, complemented
2H(3,𝔽3)×#121C211yesmaximal, characteristic, complemented
3C2×#88546C211yesmaximal, characteristic, complemented
4C2×#88545C211yesmaximal, characteristic, complemented
5#11720573C211yesmaximal, characteristic, complemented
6#11720532C211yesmaximal, characteristic, complemented
7#11720531C211yesmaximal, characteristic, complemented
8C3×#5037—34nomaximal
9C2×D4×H(3,𝔽3)—21no—
10C2×D4×H(3,𝔽3)C2211yescharacteristic
11C22×C4×H(3,𝔽3)C2211yescharacteristic
12C2×#6235—21no—
13#88740—22no—
14#88740—22no—
15C2×C4⋊#272C2211yescharacteristic
16C2×C4×#272—21no—
17C2×#6229C2211yescharacteristic
18C4⋊C4×H(3,𝔽3)C2211yescharacteristic
19C22⋊C4×H(3,𝔽3)—21no—
20#88546—22no—
21#88545—22no—
22C2×#6180C2211yescharacteristic
23C2×#6180C2211yescharacteristic
24C2×C3×#235—34no—
25C32×#121D314yescomplemented
26C2×C3×#214—34no—
27C2×C3×#213—34no—
28C3×#830—34no—
29C3×#810—34no—
30C3×#809—34no—
31C23×H(3,𝔽3)—21no—
32C23×H(3,𝔽3)D411yescharacteristic
33C22×#272—21no—
34D4×H(3,𝔽3)—21no—
35D4×H(3,𝔽3)—21no—
36D4×H(3,𝔽3)—42no—
37C2×C4×H(3,𝔽3)—21no—
38C2×C4×H(3,𝔽3)—21no—
39C2×C4×H(3,𝔽3)—21no—
40C2×C4×H(3,𝔽3)D411yescharacteristic
41C2×C4×H(3,𝔽3)C2311yescharacteristic
42#6235—24no—
43C2×#1084—21no—
44C4⋊#272—21no—
45C4⋊#272D412yescomplemented
46C4×#272—24no—
47#6229—21no—
48#6229D412yescomplemented
49#6180—21no—
50#6180—21no—
51C3×#449—91no—
52#5037—34no—
53C2×C32×D4—24no—
54C2×C32×D4C2×D314yes—
55C22×C32×C4C2×D314yes—
56C2×C3×#67—64no—
57C3×#235—68no—
58C3×#235—68no—
59C2×C3×D12—34no—
60C2×C3×C4×D3—64no—
61C2×C3×Dic6—34no—
62C32×C4⋊C4C2×D314yes—
63C32×C22⋊C4—24no—
64C3×#214—68no—
65C3×#213—68no—
66C2×C3×C3⋊C8—34no—
67C2×C3×C3⋊C8—34no—
68C22×H(3,𝔽3)—21no—
69C22×H(3,𝔽3)—21no—
70C22×H(3,𝔽3)—21no—
71C22×H(3,𝔽3)—41no—
72C22×H(3,𝔽3)—41no—
73C22×H(3,𝔽3)—41no—
74C22×H(3,𝔽3)C2×D411yescharacteristic
75C2×#272—22no—
76C2×#272—41no—
77C2×#272—41no—
78C4×H(3,𝔽3)—21no—
79C4×H(3,𝔽3)—41no—
80C4×H(3,𝔽3)C2×D411yescharacteristic
81C4×H(3,𝔽3)C2×D411yescharacteristic
82#1084—22no—
83C2×C3×#41—91no—
84C2×#235—34no—
85C2×C3×SD16—91no—
86C2×C3×D8—91no—
87C3×#121—34no—
88C3×#121C3⋊D311yescharacteristic
89C2×#214—34no—
90C2×#213—34no—
91C3×#103—91no—
92C3×#102—91no—
93C3×C22⋊C8—91no—
94#830—34no—
95#810—34no—
96#809—34no—
97C23×C32—24no—
98C23×C32#6714yes—
99C22×C3×D3—64no—
100C32×D4—24no—
101C32×D4—24no—
102C32×D4—48no—
103C2×C32×C4—24no—
104C2×C32×C4—24no—
105C2×C32×C4—24no—
106C2×C32×C4#6714yes—
107C2×C32×C4C22×D314yes—
108C3×#67—616no—
109C2×C3×Dic3—64no—
110C3×D12—38no—
111C3×D12—64no—
112C3×C4×D3—616no—
113C3×Dic6—38no—
114C3×Dic6—64no—
115C3×C3⋊C8—64no—
116C3×C3⋊C8—64no—
117#449—91noSylow
118C2×H(3,𝔽3)—41no—
119C2×H(3,𝔽3)—81no—
120C2×H(3,𝔽3)#12011yescharacteristic
121C2×H(3,𝔽3)#13511yescharacteristic
122C2×H(3,𝔽3)AGL(1,ℤ/8)11yescharacteristic
123#272—42no—
124C3×#41—182no—
125C3×#41—182no—
126C2×C3×Q8—91no—
127C2×C3×D4—21no—
128C2×C3×D4—34no—
129C2×C3×D4—64no—
130C2×C3×D4—91no—
131C2×C3×D4—181no—
132C2×C3×D4C2×C3⋊D311yescharacteristic
133C22×C3×C4—34no—
134C22×C3×C4—181no—
135C22×C3×C4C2×C3⋊D311yescharacteristic
136C2×#67—64no—
137#235—68no—
138#235—68no—
139C2×D12—34no—
140C2×C4×D3—64no—
141C2×Dic6—34no—
142C3×SD16—182no—
143C3×D8—182no—
144C2×C3×C8—91no—
145C2×C3×C8—91no—
146C3×C4⋊C4—34no—
147C3×C4⋊C4C2×C3⋊D311yescharacteristic
148C3×C22⋊C4—21no—
149C3×C22⋊C4—64no—
150#214—68no—
151#213—68no—
152C2×C3⋊C8—34no—
153C2×C3⋊C8—34no—
154C22×C32—24no—
155C22×C32—24no—
156C22×C32—24no—
157C22×C32—44no—
158C22×C32—44no—
159C22×C32—44no—
160C22×C32C2×#6714yes—
161C2×C3×D3—68no—
162C2×C3×D3—124no—
163C2×C3×D3—124no—
164C32×C4—24no—
165C32×C4—44no—
166C32×C4D3×D414yes—
167C32×C4D3×D414yes—
168C3×Dic3—68no—
169C2×#41—91no—
170C2×SD16—91no—
171C2×D8—91no—
172#121#27211yescharacteristic, complemented
173#103—91no—
174#102—91no—
175C22⋊C8—91no—
176H(3,𝔽3)#44911yescharacteristic, complemented, Sylow
177C23×C3—21no—
178C23×C3—34no—
179C23×C3—64no—
180C23×C3—181no—
181C23×C3#63811yescharacteristic
182C22×D3—64no—
183C3×Q8—92no—
184C3×Q8—181no—
185C3×D4—21no—
186C3×D4—21no—
187C3×D4—42no—
188C3×D4—64no—
189C3×D4—64no—
190C3×D4—92no—
191C3×D4—128no—
192C3×D4—184no—
193C3×D4—181no—
194C2×C3×C4—21no—
195C2×C3×C4—21no—
196C2×C3×C4—21no—
197C2×C3×C4—34no—
198C2×C3×C4—34no—
199C2×C3×C4—64no—
200C2×C3×C4—64no—
201C2×C3×C4—64no—
202C2×C3×C4—184no—
203C2×C3×C4—181no—
204C2×C3×C4#63811yescharacteristic
205C2×C3×C4C22×C3⋊D311yescharacteristic
206#67—616no—
207C2×Dic3—64no—
208D12—38no—
209D12—64no—
210C4×D3—616no—
211Dic6—38no—
212Dic6—64no—
213C3×C8—181no—
214C3×C8—181no—
215C3⋊C8—64no—
216C3⋊C8—64no—
217C2×C32—44no—
218C2×C32—84no—
219C2×C32#93214yes—
220C2×C32#93714yes—
221C2×C32#95014yes—
222C3×D3—128no—
223#41—182no—
224#41—182no—
225C2×Q8—91no—
226C2×D4—21no—
227C2×D4—91no—
228C2×D4—181no—
229C2×D4C2×#27211yescharacteristic
230C22×C4—181no—
231C22×C4C2×#27211yescharacteristic
232SD16—182no—
233D8—182no—
234C2×C8—91no—
235C2×C8—91no—
236C4⋊C4C2×#27211yescharacteristic
237C22⋊C4—21no—
238C22×C3—21no—
239C22×C3—21no—
240C22×C3—21no—
241C22×C3—34no—
242C22×C3—41no—
243C22×C3—41no—
244C22×C3—41no—
245C22×C3—64no—
246C22×C3—64no—
247C22×C3—64no—
248C22×C3—124no—
249C22×C3—124no—
250C22×C3—124no—
251C22×C3—182no—
252C22×C3—361no—
253C22×C3—361no—
254C22×C3C2×#63811yescharacteristic
255C2×D3—68no—
256C2×D3—124no—
257C2×D3—124no—
258C3×C4—21no—
259C3×C4—34no—
260C3×C4—34no—
261C3×C4—41no—
262C3×C4—64no—
263C3×C4—124no—
264C3×C4—182no—
265C3×C4D4×C3⋊D311yescharacteristic
266C3×C4D4×C3⋊D311yescharacteristic
267Dic3—68no—
268C32#503714yescomplemented
269C23—21no—
270C23—181no—
271C23#623511yescharacteristic
272Q8—92no—
273Q8—181no—
274D4—21no—
275D4—21no—
276D4—42no—
277D4—92no—
278D4—184no—
279D4—181no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—184no—
284C2×C4—181no—
285C2×C4#623511yescharacteristic
286C2×C4C22×#27211yescharacteristic
287C8—181no—
288C8—181no—
289C2×C3—34no—
290C2×C3—34no—
291C2×C3—34no—
292C2×C3—41no—
293C2×C3—81no—
294C2×C3—124no—
295C2×C3—244no—
296C2×C3—362no—
297C2×C3#6414111yescharacteristic
298C2×C3#6414611yescharacteristic
299C2×C3#6415911yescharacteristic
300D3—128no—
301C22—21no—
302C22—21no—
303C22—21no—
304C22—41no—
305C22—41no—
306C22—41no—
307C22—182no—
308C22—361no—
309C22—361no—
310C22C2×#623511yescharacteristic
311C4—21no—
312C4—41no—
313C4—182no—
314C4D4×#27211yescharacteristic
315C4D4×#27211yescharacteristic
316C3—34no—
317C3#1059273611yescharacteristic
318C2—41no—
319C2—81no—
320C2—362no—
321C2#1172184211yescharacteristic
322C2#1172184711yescharacteristic
323C2#1172187111yescharacteristic
subgroupG/H
center, socleC22×C3C2×#638
commutator, cosocleC2×C4×H(3,𝔽3)C23
FrattiniC2×C3×C4C22×C3⋊D3
FittingH(3,𝔽3)×#121C2
hypercenterC3×#121C3⋊D3
hypocenterH(3,𝔽3)#449
LCSG ▷ C2×C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22×C3 ◁ C22×C3×C4 ◁ C3×#121

Complements

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

NH ≅ G/Ntype
1. C2×#88740C2semidirect
2. H(3,𝔽3)×#121C2semidirect
3. C2×#88546C2semidirect
4. C2×#88545C2semidirect
5. #11720573C2semidirect
6. #11720532C2semidirect
7. #11720531C2semidirect
25. C32×#121D3semidirect
45. C4⋊#272D4semidirect
48. #6229D4semidirect
172. #121#272semidirect
176. H(3,𝔽3)#449semidirect
268. C32#5037semidirect

Maximal subgroups

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

subgroupindex
C2×#887402
H(3,𝔽3)×#1212
C2×#885462
C2×#885452
#117205732
#117205322
#117205312
C3×#50373