Finite Group Databasebeta

← #13137365

Subgroups — #13137365

Order 1728.

426 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 426 conjugacy classes of subgroups — too many to draw. Only its 75 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
1#11721721C211yesmaximal, characteristic
2#64078⋊C3C211yesmaximal, characteristic, complemented
3#63630⋊C3C211yesmaximal, characteristic, complemented
4#11720326C212yesmaximal, complemented
5#11720324C212yesmaximal
6#10590280—31nomaximal
7#10590276—31nomaximal
8C2×C4⋊#272C2211yescharacteristic
9C2×#6229C2211yescharacteristic
10C2×#634⋊C3—21no—
11C42×H(3,𝔽3)C2211yescharacteristic
12#88542—21no—
13#88542—21no—
14C4⋊((C3⋊Dic3)⋊C3)—21no—
15C4⋊((C3⋊Dic3)⋊C3)C2212yes—
16C2×(C3⋊(C3⋊C8))⋊C3C2212yes—
17#64078D311yescharacteristic, complemented
18C3×#875—31no—
19C3×#875—31no—
20C3×#875—61no—
21C3×#869—31no—
22#63630D311yescharacteristic, complemented
23C3×#803—31no—
24#63566—32no—
25#63565—32no—
26#63563—32no—
27#63562—32no—
28C22×#272—21no—
29C2×C4×H(3,𝔽3)D411yescharacteristic
30C2×C4×H(3,𝔽3)D411yescharacteristic
31C2×C4×H(3,𝔽3)C2311yescharacteristic
32C2×#1084—21no—
33C4⋊#272—22no—
34#6229—22no—
35C2×(C3⋊Dic3)⋊C3—22no—
36#634⋊C3—42no—
37(C3⋊(C3⋊C8))⋊C3—22no—
38#5089—91no—
39#5055—91no—
40#4714—91no—
41C2×#634—21no—
42C2×C3×D12—31no—
43C2×C3×D12—31no—
44C2×C3×D12—61no—
45C2×C3×Dic6—31no—
46C2×C3×Dic6—31no—
47C2×C3×Dic6—61no—
48C2×C3×Dic6—61no—
49C32×C42—21no—
50C32×C42C2×D311yescharacteristic
51C32×C42C2×D311yescharacteristic
52C4⋊(C3⋊Dic3)—21no—
53C4⋊(C3⋊Dic3)C2×D312yes—
54C2×C3⋊(C3⋊C8)C2×D312yes—
55C3×(C2×D3)⋊C4—62no—
56C3×(C2×D3)⋊C4—62no—
57C3×(C2×D3)⋊C4—61no—
58C3×(C2×D3)⋊C4—61no—
59C3×(C2×D3)⋊C4—61no—
60C3×(C2×D3)⋊C4—61no—
61C3×C4⋊Dic3—32no—
62C3×C4⋊Dic3—61no—
63C2×C3×C3⋊C8—32no—
64C22×H(3,𝔽3)C2×D411yescharacteristic
65C2×#272—41no—
66C2×#272—41no—
67C2×#272—41no—
68C4×H(3,𝔽3)—21no—
69C4×H(3,𝔽3)—21no—
70C4×H(3,𝔽3)#4112yes—
71#1084—41no—
72(C3⋊Dic3)⋊C3—42no—
73C3×C4⋊Q8—91no—
74C3×#124—91no—
75#875—31no—
76#875—31no—
77#875—61no—
78#869—31no—
79#869—91no—
80C3×#97—91no—
81#835—92no—
82#832—92no—
83#821—92no—
84#816—92no—
85#810—92no—
86#809—92no—
87#803—31no—
88#803—91no—
89C22×C3×D3—62no—
90C22×C3×D3—61no—
91C22×C3×D3—61no—
92C2×C32×C4—21no—
93C2×C32×C4—21no—
94C2×C32×C4—21no—
95C2×C32×C4D1211yescharacteristic
96C2×C32×C4D1211yescharacteristic
97C2×C32×C4#6711yescharacteristic
98C2×C32×C4#6711yescharacteristic
99C2×C32×C4C22×D311yescharacteristic
100C2×C32×C4C22×D311yescharacteristic
101C2×C3⋊Dic3—22no—
102#634—42no—
103C2×C3×Dic3—62no—
104C2×C3×Dic3—62no—
105C2×C3×Dic3—61no—
106C2×C3×Dic3—61no—
107C3×D12—62no—
108C3×D12—62no—
109C3×D12—122no—
110C3×Dic6—62no—
111C3×Dic6—62no—
112C3×Dic6—122no—
113C3×Dic6—122no—
114C3⋊(C3⋊C8)—22no—
115C3×C3⋊C8—62no—
116#489—271noSylow
117C2×H(3,𝔽3)#12411yescharacteristic
118C2×H(3,𝔽3)AGL(1,ℤ/8)11yescharacteristic
119C2×H(3,𝔽3)#13711yescharacteristic
120#272—81no—
121C2×C3×Q8—91no—
122C2×C3×Q8—181no—
123C2×C3×D4—91no—
124C2×D12—31no—
125C2×D12—31no—
126C2×D12—61no—
127C2×Dic6—31no—
128C2×Dic6—31no—
129C2×Dic6—61no—
130C2×Dic6—61no—
131C2×Dic6—181no—
132C2×C3×C8—92no—
133C3×C4⋊C4—92no—
134C3×C4⋊C4—181no—
135C3×C22⋊C4—181no—
136C3×C22⋊C4—181no—
137C3×C42—31no—
138C3×C42—31no—
139C3×C42—61no—
140C3×C42D3211yescharacteristic
141(C2×D3)⋊C4—61no—
142(C2×D3)⋊C4—61no—
143(C2×D3)⋊C4—61no—
144(C2×D3)⋊C4—61no—
145(C2×D3)⋊C4—62no—
146(C2×D3)⋊C4—62no—
147C4⋊Dic3—32no—
148C4⋊Dic3—61no—
149C4⋊Dic3—92no—
150C4⋊Dic3—181no—
151C2×C3⋊C8—32no—
152C2×C3⋊C8—92no—
153C22×C32—21no—
154C22×C32C2×D1211yescharacteristic
155C22×C32C2×#6711yescharacteristic
156C2×C3×D3—122no—
157C2×C3×D3—122no—
158C2×C3×D3—122no—
159C2×C3×D3—121no—
160C2×C3×D3—121no—
161C2×C3×D3—121no—
162C2×C3×D3—121no—
163C2×C3×D3—121no—
164C2×C3×D3—121no—
165C32×C4—21no—
166C32×C4—21no—
167C32×C4—21no—
168C32×C4—21no—
169C32×C4—22no—
170C32×C4—41no—
171C32×C4—41no—
172C32×C4#23512yes—
173C32×C4#23912yes—
174C3⋊Dic3—42no—
175C3×Dic3—122no—
176C3×Dic3—121no—
177C3×Dic3—122no—
178C3×Dic3—121no—
179C4⋊Q8—91no—
180#124—91no—
181#103—272no—
182#102—272no—
183#97—91no—
184H(3,𝔽3)#48911yescharacteristic, complemented, Sylow
185C23×C3—181no—
186C22×D3—61no—
187C22×D3—61no—
188C22×D3—62no—
189C3×Q8—182no—
190C3×Q8—362no—
191C3×D4—182no—
192C2×C3×C4—31no—
193C2×C3×C4—31no—
194C2×C3×C4—31no—
195C2×C3×C4—31no—
196C2×C3×C4—31no—
197C2×C3×C4—31no—
198C2×C3×C4—61no—
199C2×C3×C4—61no—
200C2×C3×C4—61no—
201C2×C3×C4—182no—
202C2×C3×C4—181no—
203C2×C3×C4#62611yescharacteristic
204C2×C3×C4#62611yescharacteristic
205C2×C3×C4C2×D3211yescharacteristic
206C2×Dic3—61no—
207C2×Dic3—61no—
208C2×Dic3—62no—
209C2×Dic3—62no—
210C2×Dic3—182no—
211D12—62no—
212D12—62no—
213D12—122no—
214Dic6—62no—
215Dic6—62no—
216Dic6—122no—
217Dic6—122no—
218Dic6—362no—
219C3×C8—182no—
220C3⋊C8—62no—
221C3⋊C8—182no—
222C2×C32—21no—
223C2×C32—21no—
224C2×C32—21no—
225C2×C32#87511yescharacteristic
226C2×C32#90811yescharacteristic
227C2×C32#90911yescharacteristic
228C2×C32#93211yescharacteristic
229C2×C32#94211yescharacteristic
230C2×C32#94711yescharacteristic
231C3×D3—242no—
232C3×D3—241no—
233C3×D3—241no—
234C2×Q8—91no—
235C2×Q8—181no—
236C2×D4—91no—
237C2×C8—92no—
238C4⋊C4—92no—
239C4⋊C4—181no—
240C22⋊C4—181no—
241C22⋊C4—181no—
242C42((C3⋊D3)⋊C3)⋊C211yescharacteristic
243C22×C3—31no—
244C22×C3—31no—
245C22×C3—61no—
246C22×C3—361no—
247C22×C3—361no—
248C22×C3—361no—
249C22×C3C2×#62611yescharacteristic
250C2×D3—121no—
251C2×D3—121no—
252C2×D3—121no—
253C2×D3—121no—
254C2×D3—121no—
255C2×D3—121no—
256C2×D3—122no—
257C2×D3—122no—
258C2×D3—122no—
259C3×C4—21no—
260C3×C4—21no—
261C3×C4—32no—
262C3×C4—32no—
263C3×C4—61no—
264C3×C4—61no—
265C3×C4—61no—
266C3×C4—61no—
267C3×C4—62no—
268C3×C4—121no—
269C3×C4—121no—
270C3×C4—362no—
271C3×C4—361no—
272C3×C4#379912yes—
273Dic3—121no—
274Dic3—121no—
275Dic3—122no—
276Dic3—122no—
277Dic3—362no—
278C32—21no—
279C32#471411yescharacteristic, complemented
280C32#508911yescharacteristic, complemented
281C23—181no—
282Q8—182no—
283Q8—362no—
284D4—182no—
285C2×C4—181no—
286C2×C4—182no—
287C2×C4#620011yescharacteristic
288C2×C4#620011yescharacteristic
289C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
290C8—182no—
291C2×C3—31no—
292C2×C3—31no—
293C2×C3—31no—
294C2×C3—31no—
295C2×C3—31no—
296C2×C3—31no—
297C2×C3—61no—
298C2×C3—61no—
299C2×C3—61no—
300C2×C3—721no—
301C2×C3#6383011yescharacteristic
302C2×C3#6383311yescharacteristic
303C2×C3#6386411yescharacteristic
304D3—241no—
305D3—241no—
306D3—242no—
307C22—361no—
308C22—361no—
309C22—361no—
310C22C2×#620011yescharacteristic
311C4—21no—
312C4—21no—
313C4—362no—
314C4—361no—
315C4#8865212yes—
316C3—31no—
317C3—31no—
318C3—61no—
319C3#1059029411yescharacteristic
320C2—721no—
321C2#1172106911yescharacteristic
322C2#1172107111yescharacteristic
323C2#1172111411yescharacteristic
subgroupG/H
centerC22C2×#6200
commutator, cosocleC2×C4×H(3,𝔽3)C23
FrattiniC2×C3×C4C2×D32
FittingC42×H(3,𝔽3)C22
socleC22×C3C2×#626
prime radicalC2×C4⋊#272C22
hypercenterC42((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#489
LCSG ▷ C2×C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C42

Complements

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

NH ≅ G/Ntype
2. #64078⋊C3C2semidirect
3. #63630⋊C3C2semidirect
4. #11720326C2semidirect
17. #64078D3semidirect
22. #63630D3semidirect
184. H(3,𝔽3)#489semidirect
279. C32#4714semidirect
280. C32#5089semidirect

Maximal subgroups

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

subgroupindex
#117217212
#64078⋊C32
#63630⋊C32
#117203262
#117203242
#105902803
#105902763