Finite Group Databasebeta

← #13159562

Subgroups — #13159562

Order 1728.

836 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#3743C211yesmaximal, characteristic
2#11723437C211yesmaximal, characteristic
3C2×C3⋊D3×C3⋊C8C211yesmaximal, characteristic
4#11722184C211yesmaximal, characteristic, complemented
5C32×Dic3⋊C8C211yesmaximal, characteristic, complemented
6#11722050C211yesmaximal, characteristic, complemented
7Dic3⋊(C3⋊(C3⋊C8))C211yesmaximal, characteristic, complemented
8#10592538—31nomaximal
9#10590695—34nomaximal
10C2×C3×C4×C3⋊D3C2211yescharacteristic
11C2×C33×C8C2211yescharacteristic
12C4×C3⋊(C3⋊Dic3)C2211yescharacteristic
13C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
14C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
15C3×#3743—22no—
16C32×C4×Dic3C2211yescharacteristic
17C2×C32×C3⋊C8C2211yescharacteristic
18#88807C411yescharacteristic
19#88807C411yescharacteristic
20#88803C411yescharacteristic
21#88803C411yescharacteristic
22C3⋊D3×C3⋊C8—22no—
23C2×#3743D311yescharacteristic
24C2×C8×C3⋊D3—31no—
25#64081—31no—
26C2×C3×ΓL(1,𝔽25)—34no—
27#63855—34no—
28C2×D3×C3⋊C8—34no—
29C32×C4⋊C8—31no—
30#63641—31no—
31#63641—31no—
32#63641—34no—
33#63641—64no—
34C4⋊(C3⋊(C3⋊C8))—31no—
35C3×Dic3⋊C8D314yescomplemented
36#63572—34no—
37Dic3⋊(C3⋊C8)—34no—
38C22×C3×C3⋊D3C2×C411yescharacteristic
39C2×C33×C4C2311yescharacteristic
40C2×C3⋊(C3⋊Dic3)C2×C411yescharacteristic
41C2×C3⋊(C3⋊Dic3)C2×C411yescharacteristic
42C2×C3×C3⋊Dic3C2×C411yescharacteristic
43C3×C4×C3⋊D3D412yes—
44C3×C4×C3⋊D3Q812yes—
45C2×C32×Dic3C2×C411yescharacteristic
46C2×C32×Dic3C2×C411yescharacteristic
47C33×C8—21no—
48C3⋊(C3⋊(C3⋊C8))—21no—
49C3×C3⋊(C3⋊C8)—21no—
50C32×C3⋊C8—21no—
51#5119—91no—
52#4839—94no—
53C2×C4×C3⋊D3C2×D311yescharacteristic
54C2×C3×C4×D3—34no—
55C2×C32×C8—24no—
56C2×C32×C8—31no—
57C2×C32×C8C2×D314yes—
58C2×C32×C8C2×D311yescharacteristic
59C32×C42—31no—
60#3752—31no—
61#3752—31no—
62#3750—31no—
63#3750—31no—
64C4×C3⋊Dic3—31no—
65C4×C3⋊Dic3—34no—
66C4×C3⋊Dic3—64no—
67C2×C3⋊(C3⋊C8)—31no—
68C2×C3⋊(C3⋊C8)—34no—
69C2×C3⋊(C3⋊C8)—64no—
70C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
71#3743—22no—
72C8×C3⋊D3—62no—
73C3×C4×Dic3C2×D314yes—
74C2×C3×C3⋊C8—34no—
75C2×C3×C3⋊C8C2×D314yes—
76C3×ΓL(1,𝔽25)—68no—
77Dic3⋊Dic3—34no—
78Dic3⋊Dic3—34no—
79(C2×D3)⋊Dic3—34no—
80(C2×D3)⋊Dic3—34no—
81D3×C3⋊C8—68no—
82C22×C33C22×C411yescharacteristic
83C2×C3×C3⋊D3—21no—
84C2×C3×C3⋊D3—21no—
85C2×C3×C3⋊D3C4⋊C412yes—
86C33×C4C2×D411yescharacteristic
87C33×C4C2×Q811yescharacteristic
88C3⋊(C3⋊Dic3)—21no—
89C3⋊(C3⋊Dic3)—21no—
90C3×C3⋊Dic3C4⋊C412yes—
91C32×Dic3—21no—
92C32×Dic3—21no—
93C2×C3×M16—91no—
94#926—91no—
95C22×C3⋊C8—91no—
96C2×ΓL(1,𝔽25)—34no—
97C2×D3×C8—94no—
98#872—94no—
99C3×C4⋊C8—34no—
100Dic3⋊C8—94no—
101Dic3⋊C8—94no—
102Dic3⋊C8—91no—
103Dic3⋊C8—91no—
104Dic3⋊C8—91no—
105Dic3⋊C8—184no—
106Dic3⋊C8C3⋊D311yescharacteristic, complemented
107C4⋊(C3⋊C8)—94no—
108C22×C3⋊D3C4×D311yescharacteristic
109C22×C3×D3—34no—
110C2×C32×C4—24no—
111C2×C32×C4—31no—
112C2×C32×C4—31no—
113C2×C32×C4C22×D314yes—
114C2×C32×C4C22×D311yescharacteristic
115C2×C3⋊Dic3—31no—
116C2×C3⋊Dic3—31no—
117C2×C3⋊Dic3—34no—
118C2×C3⋊Dic3—34no—
119C2×C3⋊Dic3—64no—
120C2×C3⋊Dic3—64no—
121C2×C3⋊Dic3C4×D311yescharacteristic
122C4×C3⋊D3Dic612yes—
123C4×C3⋊D3#6712yes—
124C2×C3×Dic3—34no—
125C2×C3×Dic3C4×D314yes—
126C2×C3×Dic3C4×D314yes—
127C3×C4×D3—38no—
128C3×C4×D3—38no—
129C32×C8—21no—
130C32×C8—24no—
131C32×C8—44no—
132C32×C8—61no—
133C3⋊(C3⋊C8)—21no—
134C3⋊(C3⋊C8)—61no—
135C3⋊(C3⋊C8)—64no—
136C3⋊(C3⋊C8)—124no—
137C3×C3⋊C8—24no—
138C3×C3⋊C8—64no—
139#424—271noSylow
140C2×C33C2×C4⋊C411yescharacteristic
141C2×C33#13111yescharacteristic
142C2×C33#13111yescharacteristic
143C3×C3⋊D3—22no—
144C22×C3×C4—91no—
145C2×C4×D3—34no—
146C3×M16—182no—
147C2×C3×C8—24no—
148C2×C3×C8—34no—
149C2×C3×C8—91no—
150C2×C3×C8D3214yes—
151C2×C3×C8C2×C3⋊D311yescharacteristic
152C3×C42—34no—
153#217—91no—
154#217—91no—
155(C2×D3)⋊C4—94no—
156(C2×D3)⋊C4—94no—
157C4⋊Dic3—91no—
158C4⋊Dic3—91no—
159Dic3⋊C4—94no—
160Dic3⋊C4—94no—
161C4×Dic3—94no—
162C4×Dic3—91no—
163C4×Dic3—184no—
164C4×Dic3C2×C3⋊D311yescharacteristic
165C2×C3⋊C8—34no—
166C2×C3⋊C8—94no—
167C2×C3⋊C8—91no—
168C2×C3⋊C8—184no—
169C2×C3⋊C8—182no—
170C2×C3⋊C8C2×C3⋊D311yescharacteristic
171ΓL(1,𝔽25)—68no—
172D3×C8—188no—
173C22×C32—24no—
174C22×C32C2×C4×D314yes—
175C22×C32C2×C4×D311yescharacteristic
176C2×C3⋊D3—21no—
177C2×C3⋊D3—21no—
178C2×C3⋊D3Dic3⋊C412yes—
179C2×C3×D3—38no—
180C2×C3×D3—64no—
181C2×C3×D3—64no—
182C32×C4—24no—
183C32×C4—24no—
184C32×C4—61no—
185C32×C4—61no—
186C32×C4C2×Dic611yescharacteristic
187C32×C4D3×D414yes—
188C32×C4D3×Q814yes—
189C32×C4C2×#6711yescharacteristic
190C3⋊Dic3—68no—
191C3⋊Dic3—64no—
192C3⋊Dic3—61no—
193C3⋊Dic3—61no—
194C3⋊Dic3—64no—
195C3⋊Dic3—124no—
196C3⋊Dic3Dic3⋊C412yes—
197C3×Dic3—24no—
198C3×Dic3—24no—
199C3×Dic3—38no—
200C2×M16—91no—
201C22×C8—271no—
202#117—271no—
203C4⋊C8—31no—
204C4⋊C8—271no—
205C4⋊C8—271no—
206C4⋊C8—271no—
207C33#42411yescharacteristic, complemented, Sylow
208C23×C3—91no—
209C22×D3—34no—
210C2×C3×C4—24no—
211C2×C3×C4—34no—
212C2×C3×C4—34no—
213C2×C3×C4—91no—
214C2×C3×C4—92no—
215C2×C3×C4—92no—
216C2×C3×C4C2×D3214yes—
217C2×C3×C4C22×C3⋊D311yescharacteristic
218C2×Dic3—34no—
219C2×Dic3—94no—
220C2×Dic3—94no—
221C2×Dic3—91no—
222C2×Dic3—91no—
223C2×Dic3—184no—
224C2×Dic3—184no—
225C2×Dic3C4×C3⋊D311yescharacteristic
226C2×Dic3C4×C3⋊D311yescharacteristic
227C4×D3—38no—
228C4×D3—38no—
229C3×C8—24no—
230C3×C8—21no—
231C3×C8—44no—
232C3×C8—64no—
233C3×C8—181no—
234C3⋊C8—21no—
235C3⋊C8—64no—
236C3⋊C8—184no—
237C3⋊C8—181no—
238C3⋊C8—364no—
239C2×C32—24no—
240C2×C32—24no—
241C2×C32—24no—
242C2×C32D3×C4⋊C414yes—
243C2×C32#90114yes—
244C2×C32#90714yes—
245C2×C32#90711yescharacteristic
246C2×C32#90711yescharacteristic
247C2×C32C2×Dic3⋊C411yescharacteristic
248C3⋊D3—22no—
249C3×D3—68no—
250C22×C4—91no—
251M16—182no—
252C2×C8—31no—
253C2×C8—91no—
254C2×C8—271no—
255C2×C8—542no—
256C2×C8D3×C3⋊D311yescharacteristic
257C4⋊C4—271no—
258C4⋊C4—271no—
259C22⋊C4—271no—
260C22⋊C4—271no—
261C42—31no—
262C42—271no—
263C22×C3—24no—
264C22×C3—92no—
265C22×C3—181no—
266C22×C3—181no—
267C22×C3C4×D3214yes—
268C22×C3C2×C4×C3⋊D311yescharacteristic
269C2×D3—38no—
270C2×D3—64no—
271C2×D3—64no—
272C3×C4—24no—
273C3×C4—24no—
274C3×C4—64no—
275C3×C4—64no—
276C3×C4—92no—
277C3×C4D3×Dic614yes—
278C3×C4D3×#6714yes—
279C3×C4D4×C3⋊D311yescharacteristic
280C3×C4Q8×C3⋊D311yescharacteristic
281Dic3—21no—
282Dic3—21no—
283Dic3—38no—
284Dic3—188no—
285Dic3—184no—
286Dic3—184no—
287Dic3—181no—
288Dic3—181no—
289Dic3—364no—
290C32—24no—
291C32#483914yescomplemented
292C32#511911yescharacteristic, complemented
293C23—91no—
294C2×C4—31no—
295C2×C4—31no—
296C2×C4—91no—
297C2×C4—92no—
298C2×C4—92no—
299C2×C4—271no—
300C2×C4—271no—
301C2×C4C2×D3×C3⋊D311yescharacteristic
302C8—21no—
303C8—61no—
304C8—181no—
305C8—541no—
306C2×C3—24no—
307C2×C3—24no—
308C2×C3—24no—
309C2×C3—182no—
310C2×C3#6380414yes—
311C2×C3#6380514yes—
312C2×C3D3×Dic3⋊C414yes—
313C2×C3C4⋊C4×C3⋊D311yescharacteristic
314C2×C3#6411011yescharacteristic
315C2×C3#6411611yescharacteristic
316D3—68no—
317C22—92no—
318C22—181no—
319C22—181no—
320C22C4×D3×C3⋊D311yescharacteristic
321C4—61no—
322C4—61no—
323C4—92no—
324C4—541no—
325C4—541no—
326C4C3⋊D3×Dic611yescharacteristic
327C4C3⋊D3×#6711yescharacteristic
328C3—24no—
329C3#1059069514yescomplemented
330C3#1059253811yescharacteristic, complemented
331C2—182no—
332C2#1172333211yescharacteristic
333C2#1172333411yescharacteristic
334C2C3⋊D3×Dic3⋊C411yescharacteristic
subgroupG/H
center, FrattiniC2×C4C2×D3×C3⋊D3
commutator, socleC22×C33C22×C4
FittingC2×C33×C8C22
prime radicalC22×C3×C3⋊D3C2×C4
cosocleC2×C33×C4C23
hypercenterC2×C8D3×C3⋊D3
hypocenterC33#424
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C2×C4 ◁ C2×C8

Complements

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

NH ≅ G/Ntype
4. #11722184C2semidirect
5. C32×Dic3⋊C8C2semidirect
6. #11722050C2semidirect
7. Dic3⋊(C3⋊(C3⋊C8))C2semidirect
35. C3×Dic3⋊C8D3semidirect
106. Dic3⋊C8C3⋊D3semidirect
207. C33#424semidirect
291. C32#4839semidirect
292. C32#5119semidirect
329. C3#10590695semidirect
330. C3#10592538semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#37432
#117234372
C2×C3⋊D3×C3⋊C82
#117221842
C32×Dic3⋊C82
#117220502
Dic3⋊(C3⋊(C3⋊C8))2
#105925383
#105906953