Finite Group Databasebeta

← #13160469

Subgroups — #13160469

Order 1728.

1596 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
1C2×C4×C3⋊(C3⋊Dic3)C211yesmaximal, characteristic
2C2×C3×C4⋊(C3⋊Dic3)C211yesmaximal, characteristic
3C2×C32×C4⋊Dic3C211yesmaximal, characteristic
4C2×Dic3×C3⋊Dic3C212yesmaximal
5#11722061C212yesmaximal
6C4⋊C4×C3⋊Dic3—31nomaximal
7#10591274—34nomaximal
8C22×C33×C4C2211yescharacteristic
9C22×C3⋊(C3⋊Dic3)C2211yescharacteristic
10C22×C3⋊(C3⋊Dic3)C2211yescharacteristic
11C22×C3×C3⋊Dic3C2212yes—
12C22×C32×Dic3C2212yes—
13C4×C3⋊(C3⋊Dic3)C414yescomplemented
14C3×C4⋊(C3⋊Dic3)C414yescomplemented
15C32×C4⋊Dic3C414yescomplemented
16Dic3×C3⋊Dic3—24no—
17C2×C32×C4⋊C4—31no—
18C2×C4⋊(C3⋊Dic3)D311yescharacteristic
19C2×C4×C3⋊Dic3—31no—
20C2×C4×C3⋊Dic3—32no—
21C2×C4×C3⋊Dic3—34no—
22C2×C4×C3⋊Dic3—64no—
23C2×C3×C4⋊Dic3—34no—
24C2×C3×C4⋊Dic3D314yes—
25C2×Dic32—38no—
26#63658—32no—
27#63579—38no—
28C23×C33C2311yescharacteristic
29C2×C33×C4C2×C412yes—
30C2×C33×C4C2×C412yes—
31C2×C33×C4C2×C412yes—
32C2×C3⋊(C3⋊Dic3)—21no—
33C2×C3⋊(C3⋊Dic3)—21no—
34C2×C3⋊(C3⋊Dic3)C2×C412yes—
35C2×C3⋊(C3⋊Dic3)C2×C412yes—
36C2×C3⋊(C3⋊Dic3)D412yes—
37C2×C3⋊(C3⋊Dic3)Q812yes—
38C2×C3×C3⋊Dic3—22no—
39C2×C3×C3⋊Dic3—22no—
40C2×C3×C3⋊Dic3C2×C414yes—
41C2×C32×Dic3—22no—
42C2×C32×Dic3—22no—
43C2×C32×Dic3C2×C414yes—
44Dic3×C4⋊C4—94no—
45Dic3×C4⋊C4—91no—
46C22×C32×C4—24no—
47C22×C32×C4—32no—
48C22×C32×C4C2×D314yes—
49C22×C32×C4C2×D311yescharacteristic
50C22×C3⋊Dic3—31no—
51C22×C3⋊Dic3—34no—
52C22×C3⋊Dic3—31no—
53C22×C3⋊Dic3—34no—
54C22×C3⋊Dic3—64no—
55C22×C3⋊Dic3—64no—
56C22×C3⋊Dic3C2×D312yes—
57C22×C3×Dic3—38no—
58C22×C3×Dic3C2×D318yes—
59C32×C4⋊C4—34no—
60C4⋊(C3⋊Dic3)Dic314yescomplemented
61C4×C3⋊Dic3—34no—
62C4×C3⋊Dic3—316no—
63C4×C3⋊Dic3—64no—
64C4×C3⋊Dic3—616no—
65C3×C4⋊Dic3—316no—
66C3×C4⋊Dic3Dic3116yescomplemented
67Dic32—616no—
68C22×C33C22×C411yescharacteristic
69C22×C33C22×C411yescharacteristic
70C22×C33C22×C411yescharacteristic
71C22×C33C2×D411yescharacteristic
72C22×C33C2×Q811yescharacteristic
73C22×C33#4111yescharacteristic
74C22×C33#4111yescharacteristic
75C33×C4C4214yescomplemented
76C3⋊(C3⋊Dic3)—22no—
77C3⋊(C3⋊Dic3)C4⋊C414yescomplemented
78C3×C3⋊Dic3—24no—
79C32×Dic3—24no—
80C2×C3×C4⋊C4—34no—
81C2×C3×C4⋊C4—91no—
82C2×C4⋊Dic3—34no—
83C2×C4⋊Dic3C3⋊D311yescharacteristic
84C2×C4×Dic3—94no—
85C2×C4×Dic3—91no—
86C2×C4×Dic3—98no—
87C2×C4×Dic3—92no—
88C2×C4×Dic3—184no—
89C3⋊#95—98no—
90C3⋊#95—92no—
91C23×C32—24no—
92C23×C32C22×D314yes—
93C23×C32C22×D311yescharacteristic
94C2×C32×C4—28no—
95C2×C32×C4—28no—
96C2×C32×C4—28no—
97C2×C32×C4—34no—
98C2×C32×C4—62no—
99C2×C32×C4—62no—
100C2×C32×C4C4×D318yes—
101C2×C32×C4C4×D318yes—
102C2×C32×C4C4×D312yes—
103C2×C32×C4C4×D312yes—
104C2×C32×C4C2×Dic318yes—
105C2×C32×C4C2×Dic312yes—
106C2×C3⋊Dic3—22no—
107C2×C3⋊Dic3—22no—
108C2×C3⋊Dic3—32no—
109C2×C3⋊Dic3—32no—
110C2×C3⋊Dic3—32no—
111C2×C3⋊Dic3—32no—
112C2×C3⋊Dic3—38no—
113C2×C3⋊Dic3—38no—
114C2×C3⋊Dic3—38no—
115C2×C3⋊Dic3—38no—
116C2×C3⋊Dic3—68no—
117C2×C3⋊Dic3—68no—
118C2×C3⋊Dic3—68no—
119C2×C3⋊Dic3—68no—
120C2×C3⋊Dic3—68no—
121C2×C3⋊Dic3—68no—
122C2×C3⋊Dic3—64no—
123C2×C3⋊Dic3—61no—
124C2×C3⋊Dic3—64no—
125C2×C3⋊Dic3—61no—
126C2×C3⋊Dic3C2×Dic314yes—
127C2×C3×Dic3—28no—
128C2×C3×Dic3—28no—
129C2×C3×Dic3—316no—
130C2×C3×Dic3—68no—
131C2×C3×Dic3—68no—
132C2×C3×Dic3C2×Dic3116yes—
133C4×C4⋊C4—271noSylow
134C2×C33C2×C4211yescharacteristic
135C2×C33C2×C4⋊C411yescharacteristic
136C2×C33#11711yescharacteristic
137C2×C33C4×D411yescharacteristic
138C2×C33C4×D411yescharacteristic
139C2×C33C4×Q811yescharacteristic
140C2×C33C4×Q811yescharacteristic
141C22×C3×C4—24no—
142C22×C3×C4—38no—
143C22×C3×C4—92no—
144C22×C3×C4D3214yes—
145C22×C3×C4C2×C3⋊D311yescharacteristic
146C22×Dic3—38no—
147C22×Dic3—94no—
148C22×Dic3—91no—
149C22×Dic3—94no—
150C22×Dic3—91no—
151C22×Dic3—184no—
152C22×Dic3—184no—
153C22×Dic3C2×C3⋊D312yes—
154C3×C4⋊C4—316no—
155C3×C4⋊C4—94no—
156C4⋊Dic3—316no—
157C4⋊Dic3C3⋊Dic314yescomplemented
158C4×Dic3—916no—
159C4×Dic3—94no—
160C4×Dic3—1816no—
161C4×Dic3—184no—
162C4×Dic3—1816no—
163C22×C32—24no—
164C22×C32—24no—
165C22×C32—24no—
166C22×C32—24no—
167C22×C32—24no—
168C22×C32—24no—
169C22×C32—24no—
170C22×C32C2×C4×D314yes—
171C22×C32C2×C4×D314yes—
172C22×C32C2×C4×D311yescharacteristic
173C22×C32C2×C4×D311yescharacteristic
174C22×C32D3×D414yes—
175C22×C32D3×D411yescharacteristic
176C22×C32#23714yes—
177C22×C32#23711yescharacteristic
178C22×C32D3×Q814yes—
179C22×C32D3×Q811yescharacteristic
180C22×C32#23914yes—
181C22×C32#23911yescharacteristic
182C22×C32C22×Dic314yes—
183C22×C32C22×Dic311yescharacteristic
184C32×C4—216no—
185C32×C4—64no—
186C32×C4C4×Dic3116yescomplemented
187C32×C4C4×Dic314yescomplemented
188C3⋊Dic3—24no—
189C3⋊Dic3—34no—
190C3⋊Dic3—316no—
191C3⋊Dic3—616no—
192C3⋊Dic3—616no—
193C3⋊Dic3—68no—
194C3⋊Dic3—62no—
195C3×Dic3—216no—
196C3×Dic3—616no—
197C2×C4⋊C4—31no—
198C2×C4⋊C4—91no—
199C2×C42—271no—
200C2×C42—272no—
201#95—272no—
202C33C4×C4⋊C411yescharacteristic, complemented, Sylow
203C23×C3—24no—
204C23×C3C2×D3214yes—
205C23×C3C22×C3⋊D311yescharacteristic
206C2×C3×C4—28no—
207C2×C3×C4—28no—
208C2×C3×C4—28no—
209C2×C3×C4—316no—
210C2×C3×C4—68no—
211C2×C3×C4—68no—
212C2×C3×C4—94no—
213C2×C3×C4—182no—
214C2×C3×C4—182no—
215C2×C3×C4D3×Dic318yes—
216C2×C3×C4D3×Dic318yes—
217C2×C3×C4#62418yes—
218C2×C3×C4C4×C3⋊D312yes—
219C2×C3×C4C4×C3⋊D312yes—
220C2×C3×C4C2×C3⋊Dic312yes—
221C2×Dic3—22no—
222C2×Dic3—22no—
223C2×Dic3—316no—
224C2×Dic3—68no—
225C2×Dic3—68no—
226C2×Dic3—98no—
227C2×Dic3—98no—
228C2×Dic3—98no—
229C2×Dic3—92no—
230C2×Dic3—92no—
231C2×Dic3—92no—
232C2×Dic3—98no—
233C2×Dic3—92no—
234C2×Dic3—188no—
235C2×Dic3—188no—
236C2×Dic3—188no—
237C2×Dic3—188no—
238C2×Dic3—188no—
239C2×Dic3—188no—
240C2×Dic3—184no—
241C2×Dic3—184no—
242C2×Dic3—181no—
243C2×Dic3—181no—
244C2×Dic3C2×C3⋊Dic314yes—
245C2×C32—24no—
246C2×C32—24no—
247C2×C32—24no—
248C2×C32—24no—
249C2×C32—24no—
250C2×C32—24no—
251C2×C32—24no—
252C2×C32#88714yes—
253C2×C32#88711yescharacteristic
254C2×C32D3×C4⋊C414yes—
255C2×C32D3×C4⋊C411yescharacteristic
256C2×C32#89214yes—
257C2×C32#89211yescharacteristic
258C2×C32#89314yes—
259C2×C32#89311yescharacteristic
260C2×C32C2×C4×Dic314yes—
261C2×C32C2×C4×Dic311yescharacteristic
262C2×C32D4×Dic314yes—
263C2×C32D4×Dic311yescharacteristic
264C2×C32Q8×Dic314yes—
265C2×C32Q8×Dic311yescharacteristic
266C22×C4—32no—
267C22×C4—92no—
268C22×C4—271no—
269C22×C4—271no—
270C22×C4D3×C3⋊D311yescharacteristic
271C4⋊C4—34no—
272C4⋊C4—94no—
273C42—274no—
274C42—544no—
275C22×C3—24no—
276C22×C3—24no—
277C22×C3—24no—
278C22×C3—24no—
279C22×C3—24no—
280C22×C3—24no—
281C22×C3—24no—
282C22×C3#379614yes—
283C22×C3#379614yes—
284C22×C3#379714yes—
285C22×C3#380214yes—
286C22×C3C2×D3×Dic314yes—
287C22×C3C2×D3×Dic314yes—
288C22×C3C2×#62414yes—
289C22×C3C2×C4×C3⋊D311yescharacteristic
290C22×C3C2×C4×C3⋊D311yescharacteristic
291C22×C3D4×C3⋊D311yescharacteristic
292C22×C3#383011yescharacteristic
293C22×C3Q8×C3⋊D311yescharacteristic
294C22×C3#383211yescharacteristic
295C22×C3C22×C3⋊Dic311yescharacteristic
296C3×C4—216no—
297C3×C4—616no—
298C3×C4—184no—
299C3×C4Dic32116yescomplemented
300C3×C4C4×C3⋊Dic314yescomplemented
301Dic3—24no—
302Dic3—616no—
303Dic3—94no—
304Dic3—916no—
305Dic3—1816no—
306Dic3—1816no—
307Dic3—182no—
308Dic3—188no—
309C32—24no—
310C32Dic3×C4⋊C414yescomplemented
311C32Dic3×C4⋊C411yescharacteristic, complemented
312C23C2×D3×C3⋊D311yescharacteristic
313C2×C4—34no—
314C2×C4—62no—
315C2×C4—62no—
316C2×C4—94no—
317C2×C4—182no—
318C2×C4—182no—
319C2×C4—272no—
320C2×C4—272no—
321C2×C4—272no—
322C2×C4—272no—
323C2×C4—541no—
324C2×C4—541no—
325C2×C4D3×C3⋊Dic312yes—
326C2×C4Dic3×C3⋊D312yes—
327C2×C4#628912yes—
328C2×C3—24no—
329C2×C3—24no—
330C2×C3—24no—
331C2×C3—24no—
332C2×C3—24no—
333C2×C3—24no—
334C2×C3—24no—
335C2×C3#6384314yes—
336C2×C3#6384314yes—
337C2×C3#6384914yes—
338C2×C3#6389814yes—
339C2×C3#6389814yes—
340C2×C3#6390014yes—
341C2×C3C2×Dic3214yes—
342C2×C3#6409611yescharacteristic
343C2×C3C4⋊C4×C3⋊D311yescharacteristic
344C2×C3#6410111yescharacteristic
345C2×C3#6410211yescharacteristic
346C2×C3C2×C4×C3⋊Dic311yescharacteristic
347C2×C3D4×C3⋊Dic311yescharacteristic
348C2×C3Q8×C3⋊Dic311yescharacteristic
349C22#8901511yescharacteristic
350C22#8901711yescharacteristic
351C22#8901911yescharacteristic
352C22#8902611yescharacteristic
353C22C2×D3×C3⋊Dic311yescharacteristic
354C22C2×Dic3×C3⋊D311yescharacteristic
355C22C2×#628911yescharacteristic
356C4—64no—
357C4—184no—
358C4—274no—
359C4—542no—
360C4Dic3×C3⋊Dic314yescomplemented
361C3—24no—
362C3#1059127414yescomplemented
363C3C4⋊C4×C3⋊Dic311yescharacteristic, complemented
364C2#1172339711yescharacteristic
365C2#1172341011yescharacteristic
366C2#1172341311yescharacteristic
367C2#1172349911yescharacteristic
368C2#1172350111yescharacteristic
369C2#1172350311yescharacteristic
370C2C2×Dic3×C3⋊Dic311yescharacteristic
subgroupG/H
center, FrattiniC23C2×D3×C3⋊D3
commutatorC2×C33C2×C42
FittingC22×C33×C4C22
socle, prime radical, cosocleC23×C33C23
hypercenterC22×C4D3×C3⋊D3
hypocenterC33C4×C4⋊C4
LCSG ▷ C2×C33 ▷ C33
UCSC1 ◁ C23 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
13. C4×C3⋊(C3⋊Dic3)C4semidirect
14. C3×C4⋊(C3⋊Dic3)C4semidirect
15. C32×C4⋊Dic3C4semidirect
60. C4⋊(C3⋊Dic3)Dic3semidirect
66. C3×C4⋊Dic3Dic3semidirect
75. C33×C4C42semidirect
77. C3⋊(C3⋊Dic3)C4⋊C4semidirect
157. C4⋊Dic3C3⋊Dic3semidirect
186. C32×C4C4×Dic3semidirect
187. C32×C4C4×Dic3semidirect
202. C33C4×C4⋊C4semidirect
299. C3×C4Dic32semidirect
300. C3×C4C4×C3⋊Dic3semidirect
310. C32Dic3×C4⋊C4semidirect
311. C32Dic3×C4⋊C4semidirect
360. C4Dic3×C3⋊Dic3semidirect
362. C3#10591274semidirect
363. C3C4⋊C4×C3⋊Dic3semidirect

Maximal subgroups

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

subgroupindex
C2×C4×C3⋊(C3⋊Dic3)2
C2×C3×C4⋊(C3⋊Dic3)2
C2×C32×C4⋊Dic32
C2×Dic3×C3⋊Dic32
#117220612
C4⋊C4×C3⋊Dic33
#105912743