Finite Group Databasebeta

← #13158920

Subgroups — #13158920

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
1C3×#64081C211yesmaximal, characteristic
2C2×#88787C211yesmaximal, characteristic
3C2×C3⋊D3×C3⋊C8C211yesmaximal, characteristic
4C4⋊(C3⋊(C3⋊(C3⋊C8)))C211yesmaximal, characteristic, complemented
5C32×C4⋊(C3⋊C8)C211yesmaximal, characteristic, complemented
6#11722051C211yesmaximal, characteristic, complemented
7#11722051C211yesmaximal, characteristic, complemented
8#10592538—31nomaximal
9#10590324—34nomaximal
10C2×C3×C4×C3⋊D3C2211yescharacteristic
11C33×C42C2211yescharacteristic
12C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
13C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
14C3×#3752C411yescharacteristic
15C3×#3752C411yescharacteristic
16C3×#3750C411yescharacteristic
17C3×#3750C411yescharacteristic
18C3×C4×C3⋊Dic3C2211yescharacteristic
19C2×C32×C3⋊C8C2211yescharacteristic
20C2×C32×C3⋊C8C2211yescharacteristic
21#88787—22no—
22C3⋊D3×C3⋊C8—22no—
23C2×#3743—31no—
24C2×C8×C3⋊D3—31no—
25#64081D311yescharacteristic
26C3×#872—34no—
27C2×#3711—34no—
28C2×D3×C3⋊C8—34no—
29C32×C4⋊C8—31no—
30#63641—31no—
31#63641—31no—
32C4⋊(C3⋊(C3⋊C8))—31no—
33C4⋊(C3⋊(C3⋊C8))—34no—
34C4⋊(C3⋊(C3⋊C8))—64no—
35C3×C4⋊(C3⋊C8)D314yescomplemented
36Dic3⋊(C3⋊C8)—34no—
37Dic3⋊(C3⋊C8)—34no—
38C22×C3×C3⋊D3C2×C411yescharacteristic
39C2×C33×C4C2×C411yescharacteristic
40C2×C33×C4C2×C411yescharacteristic
41C2×C33×C4C2311yescharacteristic
42C2×C3×C3⋊Dic3C2×C411yescharacteristic
43C2×C3×C3⋊Dic3C2×C411yescharacteristic
44C2×C3×C3⋊Dic3C2×C411yescharacteristic
45C3×C4×C3⋊D3D412yes—
46C3×C4×C3⋊D3Q812yes—
47C3⋊(C3⋊(C3⋊C8))—21no—
48C3⋊(C3⋊(C3⋊C8))—21no—
49C32×C3⋊C8—21no—
50C32×C3⋊C8—21no—
51#4999—91no—
52#4839—94no—
53C2×C4×C3⋊D3C2×D311yescharacteristic
54C2×C3×C4×D3—34no—
55C2×C32×C8—31no—
56C2×C32×C8—31no—
57C32×C42—24no—
58C32×C42C2×D314yes—
59C32×C42C2×D311yescharacteristic
60#3752Dic311yescharacteristic
61#3752Dic311yescharacteristic
62#3750Dic311yescharacteristic
63#3750Dic311yescharacteristic
64C4×C3⋊Dic3C2×D311yescharacteristic
65C2×C3⋊(C3⋊C8)—31no—
66C2×C3⋊(C3⋊C8)—31no—
67C2×C3⋊(C3⋊C8)—34no—
68C2×C3⋊(C3⋊C8)—34no—
69C2×C3⋊(C3⋊C8)—64no—
70C2×C3⋊(C3⋊C8)—64no—
71#3743—62no—
72C8×C3⋊D3—62no—
73C3×(C2×D3)⋊C4—34no—
74C3×(C2×D3)⋊C4—34no—
75C3×Dic3⋊C4—34no—
76C3×Dic3⋊C4—34no—
77C3×C4×Dic3—34no—
78C2×C3×C3⋊C8C2×D314yes—
79C2×C3×C3⋊C8C2×D314yes—
80#3711—68no—
81D3×C3⋊C8—68no—
82C22×C33C22×C411yescharacteristic
83C2×C3×C3⋊D3—21no—
84C2×C3×C3⋊D3—21no—
85C2×C3×C3⋊D3C4⋊C412yes—
86C33×C4—21no—
87C33×C4—21no—
88C33×C4C2×D411yescharacteristic
89C33×C4C2×Q811yescharacteristic
90C3×C3⋊Dic3—21no—
91C3×C3⋊Dic3—21no—
92C3×C3⋊Dic3C4⋊C412yes—
93C3×#117—91no—
94C2×#208—91no—
95C22×C3⋊C8—91no—
96C2×ΓL(1,𝔽25)—94no—
97C2×D3×C8—94no—
98#872—34no—
99C3×C4⋊C8—34no—
100Dic3⋊C8—94no—
101Dic3⋊C8—94no—
102C4⋊(C3⋊C8)—94no—
103C4⋊(C3⋊C8)—91no—
104C4⋊(C3⋊C8)—91no—
105C4⋊(C3⋊C8)—91no—
106C4⋊(C3⋊C8)—184no—
107C4⋊(C3⋊C8)C3⋊D311yescharacteristic, complemented
108C22×C3⋊D3C2×Dic311yescharacteristic
109C22×C3×D3—34no—
110C2×C32×C4—24no—
111C2×C32×C4—24no—
112C2×C32×C4—24no—
113C2×C32×C4C4×D314yes—
114C2×C32×C4C4×D314yes—
115C2×C32×C4C2×Dic311yescharacteristic
116C2×C32×C4C2×Dic311yescharacteristic
117C2×C32×C4C22×D314yes—
118C2×C32×C4C22×D311yescharacteristic
119C2×C3⋊Dic3C2×Dic311yescharacteristic
120C2×C3⋊Dic3C2×Dic311yescharacteristic
121C2×C3⋊Dic3C2×Dic311yescharacteristic
122C4×C3⋊D3Dic612yes—
123C4×C3⋊D3D1212yes—
124C2×C3×Dic3—34no—
125C2×C3×Dic3—34no—
126C2×C3×Dic3—34no—
127C3×C4×D3—38no—
128C3×C4×D3—38no—
129C32×C8—61no—
130C32×C8—61no—
131C3⋊(C3⋊C8)—61no—
132C3⋊(C3⋊C8)—61no—
133C3⋊(C3⋊C8)—64no—
134C3⋊(C3⋊C8)—64no—
135C3⋊(C3⋊C8)—124no—
136C3⋊(C3⋊C8)—124no—
137C3×C3⋊C8—24no—
138C3×C3⋊C8—24no—
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—
146C2×C3×C8—34no—
147C2×C3×C8—34no—
148C3×C4⋊C4—91no—
149C3×C4⋊C4—91no—
150C3×C22⋊C4—91no—
151C3×C22⋊C4—91no—
152C3×C42—24no—
153C3×C42—91no—
154C3×C42D3214yes—
155C3×C42C2×C3⋊D311yescharacteristic
156(C2×D3)⋊C4—34no—
157(C2×D3)⋊C4—34no—
158Dic3⋊C4—34no—
159Dic3⋊C4—34no—
160C4×Dic3—34no—
161#208—182no—
162C2×C3⋊C8—94no—
163C2×C3⋊C8—94no—
164C2×C3⋊C8—91no—
165C2×C3⋊C8—91no—
166C2×C3⋊C8—184no—
167C2×C3⋊C8—184no—
168C2×C3⋊C8—182no—
169C2×C3⋊C8C2×C3⋊D311yescharacteristic
170C2×C3⋊C8C2×C3⋊D311yescharacteristic
171ΓL(1,𝔽25)—188no—
172D3×C8—188no—
173C22×C32—24no—
174C22×C32C2×C4×D314yes—
175C22×C32C22×Dic311yescharacteristic
176C2×C3⋊D3—21no—
177C2×C3⋊D3—21no—
178C2×C3⋊D3C4⋊Dic312yes—
179C2×C3×D3—38no—
180C2×C3×D3—64no—
181C2×C3×D3—64no—
182C32×C4—28no—
183C32×C4—24no—
184C32×C4—21no—
185C32×C4—21no—
186C32×C4—24no—
187C32×C4—24no—
188C32×C4—24no—
189C32×C4—44no—
190C32×C4C2×Dic611yescharacteristic
191C32×C4C2×D1211yescharacteristic
192C32×C4D3×D414yes—
193C32×C4D3×Q814yes—
194C3⋊Dic3—21no—
195C3⋊Dic3—21no—
196C3⋊Dic3C4⋊Dic312yes—
197C3×Dic3—38no—
198C3×Dic3—64no—
199C3×Dic3—64no—
200C2×M16—271no—
201C22×C8—271no—
202#117—91no—
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—24no—
212C2×C3×C4—24no—
213C2×C3×C4—91no—
214C2×C3×C4—92no—
215C2×C3×C4—92no—
216C2×C3×C4—91no—
217C2×C3×C4—91no—
218C2×C3×C4D3×Dic314yes—
219C2×C3×C4D3×Dic314yes—
220C2×C3×C4C4×C3⋊D311yescharacteristic
221C2×C3×C4C4×C3⋊D311yescharacteristic
222C2×C3×C4C2×D3214yes—
223C2×C3×C4C22×C3⋊D311yescharacteristic
224C2×Dic3—34no—
225C2×Dic3—34no—
226C2×Dic3—34no—
227C4×D3—38no—
228C4×D3—38no—
229C3×C8—64no—
230C3×C8—64no—
231C3⋊C8—21no—
232C3⋊C8—21no—
233C3⋊C8—184no—
234C3⋊C8—184no—
235C3⋊C8—181no—
236C3⋊C8—181no—
237C3⋊C8—364no—
238C3⋊C8—364no—
239C2×C32—24no—
240C2×C32—24no—
241C2×C32—24no—
242C2×C32D3×C4⋊C414yes—
243C2×C32#90114yes—
244C2×C32#90714yes—
245C2×C32C2×C4⋊Dic311yescharacteristic
246C2×C32#94811yescharacteristic
247C2×C32#94811yescharacteristic
248C3⋊D3—22no—
249C3×D3—68no—
250C22×C4—91no—
251M16—542no—
252C2×C8—31no—
253C2×C8—31no—
254C2×C8—271no—
255C2×C8—271no—
256C2×C8—542no—
257C4⋊C4—91no—
258C4⋊C4—91no—
259C22⋊C4—91no—
260C22⋊C4—91no—
261C42—91no—
262C42D3×C3⋊D311yescharacteristic
263C22×C3—24no—
264C22×C3—92no—
265C22×C3—181no—
266C22×C3—181no—
267C22×C3C2×D3×Dic314yes—
268C22×C3C2×C4×C3⋊D311yescharacteristic
269C2×D3—38no—
270C2×D3—64no—
271C2×D3—64no—
272C3×C4—28no—
273C3×C4—24no—
274C3×C4—24no—
275C3×C4—21no—
276C3×C4—21no—
277C3×C4—24no—
278C3×C4—24no—
279C3×C4—44no—
280C3×C4—92no—
281C3×C4—181no—
282C3×C4—181no—
283C3×C4D3×Dic614yes—
284C3×C4D3×D1214yes—
285C3×C4D4×C3⋊D311yescharacteristic
286C3×C4Q8×C3⋊D311yescharacteristic
287Dic3—38no—
288Dic3—64no—
289Dic3—64no—
290C32—24no—
291C32#483914yescomplemented
292C32#499911yescharacteristic, complemented
293C23—91no—
294C2×C4—91no—
295C2×C4—92no—
296C2×C4—92no—
297C2×C4—91no—
298C2×C4—91no—
299C2×C4Dic3×C3⋊D311yescharacteristic
300C2×C4Dic3×C3⋊D311yescharacteristic
301C2×C4C2×D3×C3⋊D311yescharacteristic
302C8—61no—
303C8—61no—
304C8—541no—
305C8—541no—
306C2×C3—24no—
307C2×C3—24no—
308C2×C3—24no—
309C2×C3—182no—
310C2×C3#6381414yes—
311C2×C3#6381514yes—
312C2×C3D3×C4⋊Dic314yes—
313C2×C3C4⋊C4×C3⋊D311yescharacteristic
314C2×C3#6411011yescharacteristic
315C2×C3#6411611yescharacteristic
316D3—68no—
317C22—92no—
318C22—181no—
319C22—181no—
320C22C2×Dic3×C3⋊D311yescharacteristic
321C4—21no—
322C4—21no—
323C4—92no—
324C4—181no—
325C4—181no—
326C4C3⋊D3×Dic611yescharacteristic
327C4C3⋊D3×D1211yescharacteristic
328C3—24no—
329C3#1059032414yescomplemented
330C3#1059253811yescharacteristic, complemented
331C2—182no—
332C2#1172335211yescharacteristic
333C2#1172335411yescharacteristic
334C2C3⋊D3×C4⋊Dic311yescharacteristic
subgroupG/H
center, FrattiniC2×C4C2×D3×C3⋊D3
commutator, socleC22×C33C22×C4
FittingC33×C42C22
prime radicalC22×C3×C3⋊D3C2×C4
cosocleC2×C33×C4C23
hypercenterC42D3×C3⋊D3
hypocenterC33#424
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C2×C4 ◁ C42

Complements

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

NH ≅ G/Ntype
4. C4⋊(C3⋊(C3⋊(C3⋊C8)))C2semidirect
5. C32×C4⋊(C3⋊C8)C2semidirect
6. #11722051C2semidirect
7. #11722051C2semidirect
35. C3×C4⋊(C3⋊C8)D3semidirect
107. C4⋊(C3⋊C8)C3⋊D3semidirect
207. C33#424semidirect
291. C32#4839semidirect
292. C32#4999semidirect
329. C3#10590324semidirect
330. C3#10592538semidirect

Maximal subgroups

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

subgroupindex
C3×#640812
C2×#887872
C2×C3⋊D3×C3⋊C82
C4⋊(C3⋊(C3⋊(C3⋊C8)))2
C32×C4⋊(C3⋊C8)2
#117220512
#105925383
#105903243