Finite Group Databasebeta

← #13144865

Subgroups — #13144865

Order 1728.

891 conjugacy classes of subgroups; only the 137 normal subgroups are shown.

normal subgroups only

This group has 891 conjugacy classes of subgroups — too many to draw. Only its 137 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×#41×H(3,𝔽3)C211yesmaximal, characteristic, complemented
2C2×#88549C211yesmaximal, characteristic, complemented
3C2×#88545C211yesmaximal, characteristic, complemented
4#11721830C211yesmaximal, characteristic, complemented
5#11720583C211yesmaximal, characteristic, complemented
6#11720578C211yesmaximal, characteristic, complemented
7#11720573C211yesmaximal, characteristic, complemented
8C3×#5238—34nomaximal
9#41×H(3,𝔽3)—22no—
10#41×H(3,𝔽3)—22no—
11C2×Q8×H(3,𝔽3)C2211yescharacteristic
12C2×D4×H(3,𝔽3)—21no—
13C2×D4×H(3,𝔽3)C2211yescharacteristic
14C22×C4×H(3,𝔽3)—21no—
15C22×C4×H(3,𝔽3)C2211yescharacteristic
16C2×#6235—21no—
17C2×C4⋊#272C2211yescharacteristic
18#88549—22no—
19#88545—22no—
20#88542—21no—
21C4⋊#1084C2211yescharacteristic
22C2×#6180C2211yescharacteristic
23C2×#6180C2211yescharacteristic
24C2×C32×#41D314yescomplemented
25C2×C3×#215—34no—
26C2×C3×#213—34no—
27C3×#930—34no—
28C3×#835—34no—
29C3×#832—34no—
30C3×#830—34no—
31C23×H(3,𝔽3)—21no—
32C23×H(3,𝔽3)D411yescharacteristic
33C22×#272—21no—
34Q8×H(3,𝔽3)—21no—
35Q8×H(3,𝔽3)D412yescomplemented
36D4×H(3,𝔽3)—24no—
37D4×H(3,𝔽3)—21no—
38D4×H(3,𝔽3)D412yescomplemented
39C2×C4×H(3,𝔽3)—24no—
40C2×C4×H(3,𝔽3)—21no—
41C2×C4×H(3,𝔽3)—21no—
42C2×C4×H(3,𝔽3)—21no—
43C2×C4×H(3,𝔽3)D411yescharacteristic
44C2×C4×H(3,𝔽3)C2311yescharacteristic
45#6235—42no—
46C2×#1084—21no—
47C4⋊#272—21no—
48C4⋊#272—21no—
49#6180—21no—
50#6180—21no—
51C3×#449—91no—
52#5238—34no—
53C32×#41—28no—
54C32×#41—28no—
55C2×C32×Q8C2×D314yes—
56C2×C32×D4—24no—
57C2×C32×D4C2×D314yes—
58C22×C32×C4—24no—
59C22×C32×C4C2×D314yes—
60C2×C3×#67—64no—
61C2×C3×D12—34no—
62C3×#215—68no—
63C3×#213—68no—
64C3×(C2×D3)⋊C4—64no—
65C3×C4⋊Dic3—34no—
66C2×C3×C3⋊C8—34no—
67C2×C3×C3⋊C8—34no—
68C22×H(3,𝔽3)—21no—
69C22×H(3,𝔽3)—22no—
70C22×H(3,𝔽3)—21no—
71C22×H(3,𝔽3)—21no—
72C22×H(3,𝔽3)—41no—
73C22×H(3,𝔽3)—41no—
74C22×H(3,𝔽3)C2×D411yescharacteristic
75C2×#272—41no—
76C2×#272—41no—
77C2×#272—41no—
78C4×H(3,𝔽3)—21no—
79C4×H(3,𝔽3)—22no—
80C4×H(3,𝔽3)C2×D411yescharacteristic
81C4×H(3,𝔽3)C2×D411yescharacteristic
82#1084—41no—
83C2×C3×#41—34no—
84C2×C3×#41C3⋊D311yescharacteristic
85C2×C3×SD16—91no—
86C2×C3×D8—91no—
87C3×#121—91no—
88C2×#215—34no—
89C2×#213—34no—
90#930—34no—
91C3×#103—91no—
92C3×#102—91no—
93C3×C22⋊C8—91no—
94#835—34no—
95#832—34no—
96#830—34no—
97C23×C32—24no—
98C23×C32#6714yes—
99C22×C3×D3—64no—
100C32×Q8—24no—
101C32×Q8#6718yescomplemented
102C32×D4—216no—
103C32×D4—24no—
104C32×D4#6718yescomplemented
105C2×C32×C4—216no—
106C2×C32×C4—24no—
107C2×C32×C4—24no—
108C2×C32×C4—24no—
109C2×C32×C4#6714yes—
110C2×C32×C4C22×D314yes—
111C3×#67—128no—
112C2×C3×Dic3—64no—
113C3×D12—64no—
114C3×D12—64no—
115C3×C3⋊C8—64no—
116C3×C3⋊C8—64no—
117#449—91noSylow
118C2×H(3,𝔽3)—41no—
119C2×H(3,𝔽3)—42no—
120C2×H(3,𝔽3)#12011yescharacteristic
121C2×H(3,𝔽3)#13511yescharacteristic
122C2×H(3,𝔽3)AGL(1,ℤ/8)11yescharacteristic
123#272—81no—
124C3×#41—22no—
125C3×#41—22no—
126C3×#41—68no—
127C3×#41—68no—
128C2×C3×Q8—34no—
129C2×C3×Q8C2×C3⋊D311yescharacteristic
130C2×C3×D4—21no—
131C2×C3×D4—34no—
132C2×C3×D4—64no—
133C2×C3×D4—91no—
134C2×C3×D4—181no—
135C2×C3×D4C2×C3⋊D311yescharacteristic
136C22×C3×C4—21no—
137C22×C3×C4—34no—
138C22×C3×C4—64no—
139C22×C3×C4C2×C3⋊D311yescharacteristic
140C2×#67—64no—
141C2×D12—34no—
142C3×SD16—182no—
143C3×D8—182no—
144C2×C3×C8—91no—
145C2×C3×C8—91no—
146C3×C4⋊C4—91no—
147C3×C22⋊C4—181no—
148#215—68no—
149#213—68no—
150(C2×D3)⋊C4—64no—
151C4⋊Dic3—34no—
152C2×C3⋊C8—34no—
153C2×C3⋊C8—34no—
154C22×C32—24no—
155C22×C32—24no—
156C22×C32—24no—
157C22×C32—28no—
158C22×C32—44no—
159C22×C32—44no—
160C22×C32C2×#6714yes—
161C2×C3×D3—124no—
162C2×C3×D3—124no—
163C2×C3×D3—124no—
164C32×C4—24no—
165C32×C4—28no—
166C32×C4C2×#6714yes—
167C32×C4C2×#6714yes—
168C3×Dic3—124no—
169C2×#41#27211yescharacteristic, complemented
170C2×SD16—91no—
171C2×D8—91no—
172#121—91no—
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—21no—
184C3×Q8—38no—
185C3×Q8—64no—
186C3×Q8#63812yes—
187C3×D4—24no—
188C3×D4—21no—
189C3×D4—38no—
190C3×D4—64no—
191C3×D4—616no—
192C3×D4—181no—
193C3×D4—181no—
194C3×D4—362no—
195C3×D4#63812yes—
196C2×C3×C4—24no—
197C2×C3×C4—21no—
198C2×C3×C4—21no—
199C2×C3×C4—21no—
200C2×C3×C4—34no—
201C2×C3×C4—34no—
202C2×C3×C4—64no—
203C2×C3×C4—64no—
204C2×C3×C4—64no—
205C2×C3×C4—616no—
206C2×C3×C4—181no—
207C2×C3×C4#63811yescharacteristic
208C2×C3×C4C22×C3⋊D311yescharacteristic
209#67—128no—
210C2×Dic3—64no—
211D12—64no—
212D12—64no—
213C3×C8—181no—
214C3×C8—181no—
215C3⋊C8—64no—
216C3⋊C8—64no—
217C2×C32—44no—
218C2×C32—48no—
219C2×C32#94914yes—
220C2×C32#95014yes—
221C2×C32#95314yes—
222C3×D3—244no—
223#41—22no—
224#41—22no—
225C2×Q8C2×#27211yescharacteristic
226C2×D4—21no—
227C2×D4—91no—
228C2×D4—181no—
229C2×D4C2×#27211yescharacteristic
230C22×C4—21no—
231C22×C4C2×#27211yescharacteristic
232SD16—182no—
233D8—182no—
234C2×C8—91no—
235C2×C8—91no—
236C4⋊C4—91no—
237C22⋊C4—181no—
238C22×C3—21no—
239C22×C3—21no—
240C22×C3—21no—
241C22×C3—22no—
242C22×C3—34no—
243C22×C3—41no—
244C22×C3—41no—
245C22×C3—64no—
246C22×C3—68no—
247C22×C3—64no—
248C22×C3—64no—
249C22×C3—124no—
250C22×C3—124no—
251C22×C3—361no—
252C22×C3—361no—
253C22×C3—361no—
254C22×C3C2×#63811yescharacteristic
255C2×D3—124no—
256C2×D3—124no—
257C2×D3—124no—
258C3×C4—21no—
259C3×C4—22no—
260C3×C4—34no—
261C3×C4—34no—
262C3×C4—64no—
263C3×C4—68no—
264C3×C4—361no—
265C3×C4C2×#63811yescharacteristic
266C3×C4C2×#63811yescharacteristic
267Dic3—124no—
268C32#523814yescomplemented
269C23—21no—
270C23—181no—
271C23#623511yescharacteristic
272Q8—21no—
273Q8#623512yescomplemented
274D4—24no—
275D4—21no—
276D4—181no—
277D4—181no—
278D4—362no—
279D4#623512yescomplemented
280C2×C4—24no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—21no—
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—42no—
294C2×C3—124no—
295C2×C3—128no—
296C2×C3—721no—
297C2×C3#6415811yescharacteristic
298C2×C3#6415911yescharacteristic
299C2×C3#6416211yescharacteristic
300D3—244no—
301C22—21no—
302C22—21no—
303C22—21no—
304C22—22no—
305C22—41no—
306C22—41no—
307C22—361no—
308C22—361no—
309C22—361no—
310C22C2×#623511yescharacteristic
311C4—21no—
312C4—22no—
313C4—361no—
314C4C2×#623511yescharacteristic
315C4C2×#623511yescharacteristic
316C3—34no—
317C3#1059293711yescharacteristic
318C2—41no—
319C2—42no—
320C2—721no—
321C2#1172187011yescharacteristic
322C2#1172187111yescharacteristic
323C2#1172187611yescharacteristic
subgroupG/H
center, socleC22×C3C2×#638
commutator, cosocleC2×C4×H(3,𝔽3)C23
FrattiniC2×C3×C4C22×C3⋊D3
FittingC2×#41×H(3,𝔽3)C2
hypercenterC2×C3×#41C3⋊D3
hypocenterH(3,𝔽3)#449
LCSG ▷ C2×C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22×C3 ◁ C22×C3×C4 ◁ C2×C3×#41

Complements

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

NH ≅ G/Ntype
1. C2×#41×H(3,𝔽3)C2semidirect
2. C2×#88549C2semidirect
3. C2×#88545C2semidirect
4. #11721830C2semidirect
5. #11720583C2semidirect
6. #11720578C2semidirect
7. #11720573C2semidirect
24. C2×C32×#41D3semidirect
35. Q8×H(3,𝔽3)D4semidirect
38. D4×H(3,𝔽3)D4semidirect
101. C32×Q8#67semidirect
104. C32×D4#67semidirect
169. C2×#41#272semidirect
176. H(3,𝔽3)#449semidirect
268. C32#5238semidirect
273. Q8#6235semidirect
279. D4#6235semidirect

Maximal subgroups

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

subgroupindex
C2×#41×H(3,𝔽3)2
C2×#885492
C2×#885452
#117218302
#117205832
#117205782
#117205732
C3×#52383