Finite Group Databasebeta

← #13158934

Subgroups — #13158934

Order 1728.

904 conjugacy classes of subgroups.

The subgroup lattice has 372 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C32×C4×D12C211yesmaximal, characteristic
2C2×#88786C211yesmaximal, characteristic, complemented
3C2×D3×C3⋊(C3⋊C8)C211yesmaximal, characteristic, complemented
4#11722173C211yesmaximal, characteristic, complemented
5C3×C4⋊(C3⋊(C3⋊C8))C211yesmaximal, characteristic, complemented
6(C2×D3)⋊(C3⋊(C3⋊C8))C211yesmaximal, characteristic, complemented
7(C2×D3)⋊(C3⋊(C3⋊C8))C211yesmaximal, characteristic, complemented
8#10592710—31nomaximal
9#10590330—34nomaximal
10C2×C32×D12C411yescharacteristic
11C2×C32×C4×D3C2211yescharacteristic
12C2×C32×C4×D3C2211yescharacteristic
13C33×C42C2211yescharacteristic
14C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
15C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
16C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
17C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
18C32×(C2×D3)⋊C4C411yescharacteristic
19C32×(C2×D3)⋊C4C411yescharacteristic
20C32×C4⋊Dic3C411yescharacteristic
21#88786—22no—
22D3×C3⋊(C3⋊C8)—22no—
23C32×C4×D4—31no—
24C2×#3748—31no—
25C22×C3⋊(C3⋊C8)—31no—
26C3×C4×D12D314yes—
27C2×#3711—34no—
28C2×D3×C3⋊C8—34no—
29#63657—31no—
30#63657—31no—
31C4⋊(C3⋊(C3⋊C8))D311yescharacteristic, complemented
32#63630—31no—
33#63630—34no—
34#63630—64no—
35C3×C4⋊(C3⋊C8)—34no—
36(C2×D3)⋊(C3⋊C8)—34no—
37(C2×D3)⋊(C3⋊C8)—34no—
38C22×C32×D3C2×C411yescharacteristic
39C22×C32×D3C2×C411yescharacteristic
40C2×C33×C4C2×C411yescharacteristic
41C2×C33×C4C2×C411yescharacteristic
42C2×C33×C4C2311yescharacteristic
43C2×C32×Dic3C2×C411yescharacteristic
44C2×C32×Dic3C2×C411yescharacteristic
45C32×D12—22no—
46C32×C4×D3—21no—
47C32×C4×D3—21no—
48C32×C4×D3—21no—
49C32×C4×D3—21no—
50C3⋊(C3⋊(C3⋊C8))—21no—
51C3⋊(C3⋊(C3⋊C8))D412yescomplemented
52C3×C3⋊(C3⋊C8)—21no—
53C3×C3⋊(C3⋊C8)—21no—
54#5011—94no—
55#4836—91no—
56C2×C32×D4—31no—
57C22×C32×C4—31no—
58C22×C32×C4—31no—
59C2×C3×D12Dic314yes—
60C2×C3×C4×D3C2×D314yes—
61C2×C3×C4×D3C2×D314yes—
62C32×C4⋊C4—31no—
63C32×C22⋊C4—31no—
64C32×C22⋊C4—31no—
65C32×C42—24no—
66C32×C42C2×D311yescharacteristic
67C32×C42C2×D314yes—
68#3748—62no—
69C2×C3⋊(C3⋊C8)—31no—
70C2×C3⋊(C3⋊C8)—34no—
71C2×C3⋊(C3⋊C8)—31no—
72C2×C3⋊(C3⋊C8)—34no—
73C2×C3⋊(C3⋊C8)—62no—
74C2×C3⋊(C3⋊C8)—64no—
75C2×C3⋊(C3⋊C8)—64no—
76C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
77C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
78C3×(C2×D3)⋊C4Dic314yes—
79C3×(C2×D3)⋊C4Dic314yes—
80C3×C4⋊Dic3Dic314yes—
81C2×C3×C3⋊C8—34no—
82C2×C3×C3⋊C8—34no—
83#3711—68no—
84D3×C3⋊C8—68no—
85C22×C33C22×C411yescharacteristic
86C2×C32×D3—21no—
87C2×C32×D3—21no—
88C2×C32×D3—21no—
89C2×C32×D3—21no—
90C2×C32×D3—21no—
91C2×C32×D3M1612yescomplemented
92C33×C4—21no—
93C33×C4—21no—
94C33×C4C2×D411yescharacteristic
95C33×C4#4111yescharacteristic
96C32×Dic3—21no—
97C32×Dic3—21no—
98C3×C4×D4—34no—
99C2×#208—94no—
100C22×C3⋊C8—94no—
101C2×ΓL(1,𝔽25)—91no—
102C2×D3×C8—91no—
103C4×D12C3⋊D311yescharacteristic
104C3×C4⋊C8—91no—
105#830—94no—
106#830—94no—
107(C2×D3)⋊C8—91no—
108(C2×D3)⋊C8—91no—
109C4⋊(C3⋊C8)—34no—
110#803—94no—
111#803—91no—
112#803—184no—
113C23×C32—31no—
114C23×C32—31no—
115C22×C3×D3C2×Dic314yes—
116C22×C3×D3C2×Dic314yes—
117C32×D4—62no—
118C2×C32×C4—24no—
119C2×C32×C4—24no—
120C2×C32×C4—24no—
121C2×C32×C4—31no—
122C2×C32×C4—31no—
123C2×C32×C4—61no—
124C2×C32×C4—61no—
125C2×C32×C4—61no—
126C2×C32×C4—61no—
127C2×C32×C4C4×D311yescharacteristic
128C2×C32×C4C4×D311yescharacteristic
129C2×C32×C4C2×Dic314yes—
130C2×C32×C4C2×Dic314yes—
131C2×C32×C4C22×D314yes—
132C2×C32×C4C22×D311yescharacteristic
133C2×C3×Dic3C2×Dic314yes—
134C2×C3×Dic3C2×Dic314yes—
135C3×D12—28no—
136C3×C4×D3—24no—
137C3×C4×D3—24no—
138C3×C4×D3—24no—
139C3×C4×D3—24no—
140C3⋊(C3⋊C8)—21no—
141C3⋊(C3⋊C8)—21no—
142C3⋊(C3⋊C8)—32no—
143C3⋊(C3⋊C8)—38no—
144C3⋊(C3⋊C8)—68no—
145C3⋊(C3⋊C8)—68no—
146C3⋊(C3⋊C8)—64no—
147C3⋊(C3⋊C8)—61no—
148C3×C3⋊C8—64no—
149C3×C3⋊C8—64no—
150#435—271noSylow
151C2×C33C4×D411yescharacteristic
152C2×C33C2×M1611yescharacteristic
153C2×C33#13111yescharacteristic
154C32×D3—22no—
155C32×D3—41no—
156C2×C3×D4—34no—
157C22×C3×C4—34no—
158C22×C3×C4—34no—
159C2×D12C3⋊Dic311yescharacteristic
160C2×C4×D3C2×C3⋊D311yescharacteristic
161C2×C4×D3C2×C3⋊D311yescharacteristic
162C2×C3×C8—91no—
163C2×C3×C8—91no—
164C3×C4⋊C4—34no—
165C3×C22⋊C4—34no—
166C3×C22⋊C4—34no—
167C3×C42—24no—
168C3×C42D3214yes—
169C3×C42C2×C3⋊D311yescharacteristic
170(C2×D3)⋊C4C3⋊Dic311yescharacteristic
171(C2×D3)⋊C4C3⋊Dic311yescharacteristic
172C4⋊Dic3C3⋊Dic311yescharacteristic
173#208—188no—
174C2×C3⋊C8—34no—
175C2×C3⋊C8—34no—
176C2×C3⋊C8—94no—
177C2×C3⋊C8—91no—
178C2×C3⋊C8—94no—
179C2×C3⋊C8—91no—
180C2×C3⋊C8—184no—
181C2×C3⋊C8—184no—
182C2×C3⋊C8—188no—
183ΓL(1,𝔽25)—182no—
184D3×C8—182no—
185C22×C32—24no—
186C22×C32—32no—
187C22×C32—61no—
188C22×C32—61no—
189C22×C32—61no—
190C22×C32—61no—
191C22×C32—61no—
192C22×C32C2×C4×D311yescharacteristic
193C22×C32C22×Dic314yes—
194C2×C3×D3—24no—
195C2×C3×D3—24no—
196C2×C3×D3—24no—
197C2×C3×D3—24no—
198C2×C3×D3—24no—
199C2×C3×D3#20818yescomplemented
200C32×C4—21no—
201C32×C4—21no—
202C32×C4—24no—
203C32×C4—24no—
204C32×C4—24no—
205C32×C4—24no—
206C32×C4—44no—
207C32×C4—44no—
208C32×C4—61no—
209C32×C4—61no—
210C32×C4D3×D414yes—
211C32×C4D3×D411yescharacteristic
212C32×C4#23714yes—
213C32×C4#23911yescharacteristic
214C3×Dic3—24no—
215C3×Dic3—24no—
216C2×M16—271no—
217C22×C8—271no—
218C4×D4—31no—
219C4⋊C8—91no—
220C22⋊C8—271no—
221C22⋊C8—271no—
222#97—271no—
223C33#43511yescharacteristic, complemented, Sylow
224C23×C3—34no—
225C23×C3—34no—
226C22×D3C2×C3⋊Dic311yescharacteristic
227C22×D3C2×C3⋊Dic311yescharacteristic
228C3×D4—68no—
229C2×C3×C4—24no—
230C2×C3×C4—24no—
231C2×C3×C4—24no—
232C2×C3×C4—34no—
233C2×C3×C4—34no—
234C2×C3×C4—64no—
235C2×C3×C4—64no—
236C2×C3×C4—64no—
237C2×C3×C4—64no—
238C2×C3×C4D3×Dic314yes—
239C2×C3×C4D3×Dic314yes—
240C2×C3×C4C2×C3⋊Dic311yescharacteristic
241C2×C3×C4C2×C3⋊Dic311yescharacteristic
242C2×C3×C4C2×D3214yes—
243C2×C3×C4C22×C3⋊D311yescharacteristic
244C2×Dic3C2×C3⋊Dic311yescharacteristic
245C2×Dic3C2×C3⋊Dic311yescharacteristic
246D12—22no—
247C4×D3—21no—
248C4×D3—21no—
249C4×D3—21no—
250C4×D3—21no—
251C3×C8—181no—
252C3×C8—181no—
253C3⋊C8—64no—
254C3⋊C8—64no—
255C3⋊C8—98no—
256C3⋊C8—92no—
257C3⋊C8—188no—
258C3⋊C8—188no—
259C3⋊C8—184no—
260C3⋊C8—181no—
261C2×C32—24no—
262C2×C32—24no—
263C2×C32—24no—
264C2×C32—62no—
265C2×C32—121no—
266C2×C32#89311yescharacteristic
267C2×C32#90111yescharacteristic
268C2×C32D3×M1611yescharacteristic
269C2×C32C2×#20814yes—
270C2×C32D4×Dic314yes—
271C2×C32#94814yes—
272C3×D3—28no—
273C3×D3—44no—
274C2×D4—31no—
275C22×C4—31no—
276C22×C4—31no—
277M16—542no—
278C2×C8—91no—
279C2×C8—91no—
280C2×C8—271no—
281C2×C8—271no—
282C2×C8—542no—
283C4⋊C4—31no—
284C22⋊C4—31no—
285C22⋊C4—31no—
286C42D3×C3⋊D311yescharacteristic
287C22×C3—24no—
288C22×C3—38no—
289C22×C3—64no—
290C22×C3—64no—
291C22×C3—64no—
292C22×C3—64no—
293C22×C3—64no—
294C22×C3C2×D3×Dic314yes—
295C22×C3C22×C3⋊Dic311yescharacteristic
296C2×D3—21no—
297C2×D3—21no—
298C2×D3—21no—
299C2×D3—21no—
300C2×D3—21no—
301C2×D3#374812yescomplemented
302C3×C4—24no—
303C3×C4—24no—
304C3×C4—21no—
305C3×C4—21no—
306C3×C4—24no—
307C3×C4—24no—
308C3×C4—44no—
309C3×C4—44no—
310C3×C4—64no—
311C3×C4—64no—
312C3×C4#379614yes—
313C3×C4#380214yes—
314C3×C4D4×C3⋊D311yescharacteristic
315C3×C4#383011yescharacteristic
316Dic3—21no—
317Dic3—21no—
318C32—24no—
319C32#483611yescharacteristic, complemented
320C32#501114yescomplemented
321C23—31no—
322C23—31no—
323D4—62no—
324C2×C4—31no—
325C2×C4—31no—
326C2×C4—61no—
327C2×C4—61no—
328C2×C4—61no—
329C2×C4—61no—
330C2×C4D3×C3⋊Dic311yescharacteristic
331C2×C4D3×C3⋊Dic311yescharacteristic
332C2×C4C2×D3×C3⋊D311yescharacteristic
333C8—181no—
334C8—181no—
335C8—272no—
336C8—541no—
337C2×C3—24no—
338C2×C3—24no—
339C2×C3—24no—
340C2×C3—68no—
341C2×C3—124no—
342C2×C3D3×#20814yes—
343C2×C3#6381414yes—
344C2×C3#6389814yes—
345C2×C3C2×#374811yescharacteristic
346C2×C3D4×C3⋊Dic311yescharacteristic
347C2×C3#6415711yescharacteristic
348D3—22no—
349D3—41no—
350C22—32no—
351C22—61no—
352C22—61no—
353C22—61no—
354C22—61no—
355C22—61no—
356C22C2×D3×C3⋊Dic311yescharacteristic
357C4—21no—
358C4—21no—
359C4—61no—
360C4—61no—
361C4#8901511yescharacteristic
362C4#8902611yescharacteristic
363C3—24no—
364C3#1059033014yescomplemented
365C3#1059271011yescharacteristic, complemented
366C2—62no—
367C2—121no—
368C2D3×#374811yescharacteristic
369C2#1172334911yescharacteristic
370C2#1172349911yescharacteristic
subgroupG/H
center, FrattiniC2×C4C2×D3×C3⋊D3
commutator, socleC22×C33C22×C4
FittingC33×C42C22
prime radicalC2×C32×D12C4
cosocleC2×C33×C4C23
hypercenterC42D3×C3⋊D3
hypocenterC33#435
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C2×C4 ◁ C42

Complements

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

NH ≅ G/Ntype
2. C2×#88786C2semidirect
3. C2×D3×C3⋊(C3⋊C8)C2semidirect
4. #11722173C2semidirect
5. C3×C4⋊(C3⋊(C3⋊C8))C2semidirect
6. (C2×D3)⋊(C3⋊(C3⋊C8))C2semidirect
7. (C2×D3)⋊(C3⋊(C3⋊C8))C2semidirect
31. C4⋊(C3⋊(C3⋊C8))D3semidirect
51. C3⋊(C3⋊(C3⋊C8))D4semidirect
91. C2×C32×D3M16semidirect
199. C2×C3×D3#208semidirect
223. C33#435semidirect
301. C2×D3#3748semidirect
319. C32#4836semidirect
320. C32#5011semidirect
364. C3#10590330semidirect
365. C3#10592710semidirect

Maximal subgroups

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

subgroupindex
C32×C4×D122
C2×#887862
C2×D3×C3⋊(C3⋊C8)2
#117221732
C3×C4⋊(C3⋊(C3⋊C8))2
(C2×D3)⋊(C3⋊(C3⋊C8))2
#105927103
#105903303