Finite Group Databasebeta

← C4×#88814 #13161842

Subgroups — C4×#88814 #13161842

Order 1728.

1538 conjugacy classes of subgroups.

The subgroup lattice has 426 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×C4×C3⋊D3C211yesmaximal, characteristic
2C2×C3×C4×C3⋊Dic3C212yesmaximal, complemented
3C2×#88814C212yesmaximal
4#11722082C212yesmaximal, complemented
5C4×#3722—31nomaximal
6C4×(C2×D3)⋊Dic3—32nomaximal
7C23×C3×C3⋊D3C2211yescharacteristic
8C22×C33×C4C2211yescharacteristic
9C22×C3×C3⋊Dic3C2211yescharacteristic
10C22×C3×C3⋊Dic3C2214yes—
11C2×C3×C4×C3⋊D3—24no—
12C2×C3×C4×C3⋊D3C414yescomplemented
13C3×C4×C3⋊Dic3—24no—
14#88814C418yesdirect product factor
15C22×C4×C3⋊D3D311yescharacteristic
16C22×C3×C4×D3—32no—
17C22×C3×C4×D3—61no—
18C2×C4×C3⋊Dic3D312yescomplemented
19C2×C3×C4×Dic3—32no—
20C2×C3×C4×Dic3—32no—
21C2×C3×C4×Dic3—62no—
22C2×#3722—32no—
23C2×(C2×D3)⋊Dic3—34no—
24#63579—34no—
25#63579—32no—
26C23×C33C2311yescharacteristic
27C22×C3×C3⋊D3—22no—
28C22×C3×C3⋊D3—22no—
29C22×C3×C3⋊D3C2×C414yes—
30C22×C3×C3⋊D3C2×C412yes—
31C2×C33×C4C2×C412yes—
32C2×C33×C4D414yes—
33C2×C3×C3⋊Dic3—24no—
34C2×C3×C3⋊Dic3—22no—
35C2×C3×C3⋊Dic3—24no—
36C2×C3×C3⋊Dic3C2×C412yes—
37C2×C3×C3⋊Dic3C2×C418yescomplemented
38C3×C4×C3⋊D3—28no—
39C4×#217—91no—
40C4×(C2×D3)⋊C4—92no—
41C23×C3⋊D3C2×D311yescharacteristic
42C23×C3×D3—32no—
43C23×C3×D3—61no—
44C22×C32×C4—22no—
45C22×C32×C4—21no—
46C22×C32×C4—41no—
47C22×C32×C4C2×D312yes—
48C22×C32×C4C2×D311yescharacteristic
49C22×C3⋊Dic3C2×D314yes—
50C22×C3⋊Dic3C2×D311yescharacteristic
51C2×C4×C3⋊D3—24no—
52C2×C4×C3⋊D3Dic314yescomplemented
53C22×C3×Dic3—34no—
54C22×C3×Dic3—34no—
55C22×C3×Dic3—32no—
56C22×C3×Dic3—64no—
57C22×C3×Dic3—61no—
58C2×C3×C4×D3—38no—
59C2×C3×C4×D3—64no—
60C2×C3×C4×D3—68no—
61C2×C3×C4×D3—64no—
62C2×C3×C4×D3—64no—
63C4×C3⋊Dic3—24no—
64C3×C4×Dic3—68no—
65C3×C4×Dic3—64no—
66C3×C4×Dic3—64no—
67#3722—38no—
68(C2×D3)⋊Dic3—316no—
69C22×C33C22×C411yescharacteristic
70C22×C33C22×C412yes—
71C22×C33C2×D412yes—
72C22×C33#4112yes—
73C2×C3×C3⋊D3—22no—
74C2×C3×C3⋊D3—22no—
75C2×C3×C3⋊D3—28no—
76C2×C3×C3⋊D3C4214yescomplemented
77C33×C4C22⋊C414yescomplemented
78C3×C3⋊Dic3—22no—
79C3×C3⋊Dic3—28no—
80C23×C3×C4—91no—
81C22×C4×D3—32no—
82C22×C4×D3—61no—
83C2×C3×C42—92no—
84C2×#217—92no—
85C2×(C2×D3)⋊C4—94no—
86C2×C4×Dic3—32no—
87C2×C4×Dic3—32no—
88C2×C4×Dic3—62no—
89C3⋊#95—94no—
90C3⋊#95—92no—
91C23×C32—21no—
92C23×C32—22no—
93C23×C32—41no—
94C23×C32C22×D312yes—
95C23×C32C22×D311yescharacteristic
96C22×C3⋊D3—22no—
97C22×C3⋊D3—22no—
98C22×C3⋊D3C4×D314yes—
99C22×C3⋊D3C2×Dic312yes—
100C22×C3×D3—38no—
101C22×C3×D3—34no—
102C22×C3×D3—64no—
103C22×C3×D3—62no—
104C22×C3×D3—64no—
105C22×C3×D3—64no—
106C22×C3×D3—62no—
107C22×C3×D3—64no—
108C22×C3×D3—62no—
109C2×C32×C4—24no—
110C2×C32×C4—22no—
111C2×C32×C4—24no—
112C2×C32×C4—24no—
113C2×C32×C4—24no—
114C2×C32×C4—42no—
115C2×C32×C4—44no—
116C2×C32×C4C4×D314yes—
117C2×C32×C4D1214yes—
118C2×C32×C4C2×Dic312yes—
119C2×C32×C4#6714yes—
120C2×C32×C4#6714yes—
121C2×C3⋊Dic3—24no—
122C2×C3⋊Dic3—24no—
123C2×C3⋊Dic3—22no—
124C2×C3⋊Dic3C4×D318yescomplemented
125C2×C3⋊Dic3C2×Dic312yes—
126C4×C3⋊D3—28no—
127C2×C3×Dic3—38no—
128C2×C3×Dic3—38no—
129C2×C3×Dic3—34no—
130C2×C3×Dic3—62no—
131C2×C3×Dic3—68no—
132C2×C3×Dic3—68no—
133C2×C3×Dic3—64no—
134C2×C3×Dic3—68no—
135C2×C3×Dic3—64no—
136C2×C3×Dic3—62no—
137C2×C3×Dic3—64no—
138C2×C3×Dic3—64no—
139C2×C3×Dic3—64no—
140C2×C3×Dic3—62no—
141C3×C4×D3—616no—
142C3×C4×D3—616no—
143C4×C22⋊C4—271noSylow
144C2×C33C2×C4211yescharacteristic
145C2×C33C2×C22⋊C411yescharacteristic
146C2×C33#11711yescharacteristic
147C2×C33C4×D414yes—
148C3×C3⋊D3—24no—
149C24×C3—91no—
150C23×D3—32no—
151C23×D3—61no—
152C22×C3×C4—22no—
153C22×C3×C4—21no—
154C22×C3×C4—41no—
155C22×C3×C4—91no—
156C22×C3×C4—94no—
157C22×C3×C4—94no—
158C22×C3×C4—184no—
159C22×C3×C4D3212yes—
160C22×C3×C4D3211yescharacteristic
161C22×Dic3—32no—
162C22×Dic3—34no—
163C22×Dic3—34no—
164C22×Dic3—64no—
165C22×Dic3—61no—
166C2×C4×D3—38no—
167C2×C4×D3—64no—
168C2×C4×D3—68no—
169C2×C4×D3—64no—
170C2×C4×D3—64no—
171C3×C42—184no—
172#217—98no—
173(C2×D3)⋊C4—916no—
174C4×Dic3—68no—
175C4×Dic3—64no—
176C4×Dic3—64no—
177C22×C32—24no—
178C22×C32—22no—
179C22×C32—24no—
180C22×C32—22no—
181C22×C32—22no—
182C22×C32—21no—
183C22×C32—22no—
184C22×C32—22no—
185C22×C32—22no—
186C22×C32—41no—
187C22×C32—42no—
188C22×C32—42no—
189C22×C32—42no—
190C22×C32C2×C4×D314yes—
191C22×C32C2×C4×D312yes—
192C22×C32C2×C4×D312yes—
193C22×C32C2×D1212yes—
194C22×C32#23514yes—
195C22×C32#23512yes—
196C22×C32C22×Dic311yescharacteristic
197C22×C32C2×#6712yes—
198C22×C32C2×#6712yes—
199C2×C3⋊D3—22no—
200C2×C3⋊D3—28no—
201C2×C3⋊D3—22no—
202C2×C3⋊D3C4×Dic314yescomplemented
203C2×C3×D3—38no—
204C2×C3×D3—64no—
205C2×C3×D3—64no—
206C2×C3×D3—64no—
207C2×C3×D3—616no—
208C2×C3×D3—64no—
209C2×C3×D3—68no—
210C2×C3×D3—64no—
211C2×C3×D3—68no—
212C32×C4—28no—
213C32×C4—24no—
214C32×C4—44no—
215C32×C4(C2×D3)⋊C418yescomplemented
216C32×C4#21714yescomplemented
217C3⋊Dic3—28no—
218C3⋊Dic3—22no—
219C3×Dic3—64no—
220C3×Dic3—616no—
221C3×Dic3—68no—
222C3×Dic3—64no—
223C3×Dic3—68no—
224C23×C4—91no—
225C2×C22⋊C4—272no—
226C2×C42—92no—
227#95—272no—
228C33C4×C22⋊C411yescharacteristic, complemented, Sylow
229C23×C3—22no—
230C23×C3—21no—
231C23×C3—41no—
232C23×C3—94no—
233C23×C3—92no—
234C23×C3—182no—
235C23×C3—182no—
236C23×C3C2×D3212yes—
237C23×C3C2×D3211yescharacteristic
238C22×D3—38no—
239C22×D3—34no—
240C22×D3—64no—
241C22×D3—62no—
242C22×D3—64no—
243C22×D3—64no—
244C22×D3—64no—
245C22×D3—62no—
246C22×D3—62no—
247C2×C3×C4—24no—
248C2×C3×C4—22no—
249C2×C3×C4—24no—
250C2×C3×C4—24no—
251C2×C3×C4—24no—
252C2×C3×C4—42no—
253C2×C3×C4—44no—
254C2×C3×C4—98no—
255C2×C3×C4—92no—
256C2×C3×C4—184no—
257C2×C3×C4—184no—
258C2×C3×C4—182no—
259C2×C3×C4—188no—
260C2×C3×C4D3×Dic314yes—
261C2×C3×C4#62412yes—
262C2×C3×C4#62514yes—
263C2×C3×C4#62614yes—
264C2×C3×C4#62614yes—
265C2×Dic3—38no—
266C2×Dic3—38no—
267C2×Dic3—34no—
268C2×Dic3—68no—
269C2×Dic3—62no—
270C2×Dic3—68no—
271C2×Dic3—64no—
272C2×Dic3—68no—
273C2×Dic3—64no—
274C2×Dic3—64no—
275C2×Dic3—64no—
276C2×Dic3—62no—
277C2×Dic3—64no—
278C2×Dic3—62no—
279C4×D3—616no—
280C4×D3—616no—
281C2×C32—22no—
282C2×C32—21no—
283C2×C32—22no—
284C2×C32—21no—
285C2×C32—24no—
286C2×C32—24no—
287C2×C32—24no—
288C2×C32—22no—
289C2×C32—21no—
290C2×C32—41no—
291C2×C32—41no—
292C2×C32—41no—
293C2×C32—44no—
294C2×C32C42×D312yes—
295C2×C32#87212yes—
296C2×C32C4×D1214yes—
297C2×C32C2×C4×Dic311yescharacteristic
298C2×C32#92611yescharacteristic
299C2×C32C2×(C2×D3)⋊C412yes—
300C2×C32C4×#6714yes—
301C2×C32C4×#6714yes—
302C2×C32C2×#21711yescharacteristic
303C3⋊D3—24no—
304C3×D3—68no—
305C3×D3—68no—
306C24—91no—
307C22×C4—91no—
308C22×C4—94no—
309C22×C4—94no—
310C22×C4—184no—
311C22×C4#111311yescharacteristic
312C22⋊C4—278no—
313C42—184no—
314C22×C3—24no—
315C22×C3—22no—
316C22×C3—24no—
317C22×C3—22no—
318C22×C3—21no—
319C22×C3—22no—
320C22×C3—22no—
321C22×C3—22no—
322C22×C3—22no—
323C22×C3—42no—
324C22×C3—42no—
325C22×C3—41no—
326C22×C3—42no—
327C22×C3—94no—
328C22×C3—182no—
329C22×C3—188no—
330C22×C3—182no—
331C22×C3#379814yes—
332C22×C3#379812yes—
333C22×C3C4×D3214yes—
334C22×C3C4×D3212yes—
335C22×C3C2×D3×Dic312yes—
336C22×C3C2×#62411yescharacteristic
337C22×C3C2×#62512yes—
338C22×C3C2×#62612yes—
339C22×C3C2×#62612yes—
340C2×D3—38no—
341C2×D3—64no—
342C2×D3—64no—
343C2×D3—64no—
344C2×D3—616no—
345C2×D3—64no—
346C2×D3—68no—
347C2×D3—68no—
348C2×D3—64no—
349C3×C4—24no—
350C3×C4—28no—
351C3×C4—44no—
352C3×C4—188no—
353C3×C4—182no—
354C3×C4(C2×D3)⋊Dic318yescomplemented
355C3×C4#372214yescomplemented
356Dic3—64no—
357Dic3—616no—
358Dic3—68no—
359Dic3—68no—
360Dic3—64no—
361C32—22no—
362C32—21no—
363C32—41no—
364C32C4×(C2×D3)⋊C412yescomplemented
365C32C4×#21711yescharacteristic, complemented
366C23—94no—
367C23—92no—
368C23—182no—
369C23—182no—
370C23C2×#111311yescharacteristic
371C2×C4—98no—
372C2×C4—92no—
373C2×C4—184no—
374C2×C4—184no—
375C2×C4—182no—
376C2×C4—188no—
377C2×C4#629412yes—
378C2×C4#629514yes—
379C2×C3—21no—
380C2×C3—22no—
381C2×C3—21no—
382C2×C3—22no—
383C2×C3—24no—
384C2×C3—24no—
385C2×C3—24no—
386C2×C3—21no—
387C2×C3—22no—
388C2×C3—41no—
389C2×C3—41no—
390C2×C3—44no—
391C2×C3—41no—
392C2×C3—184no—
393C2×C3#6385512yes—
394C2×C3#6387411yescharacteristic
395C2×C3C4×D3×Dic312yes—
396C2×C3C4×#62411yescharacteristic
397C2×C3C4×#62514yes—
398C2×C3C4×#62614yes—
399C2×C3C4×#62614yes—
400C2×C3C2×(C2×D3)⋊Dic312yes—
401C2×C3C2×#372211yescharacteristic
402D3—68no—
403D3—68no—
404C22—94no—
405C22—182no—
406C22—188no—
407C22—182no—
408C22#8904212yes—
409C22C4×#111312yes—
410C22C2×#629411yescharacteristic
411C22C2×#629512yes—
412C4—188no—
413C4—182no—
414C4#8881414yesdirect product factor
415C3—21no—
416C3—22no—
417C3—41no—
418C3C4×(C2×D3)⋊Dic312yescomplemented
419C3C4×#372211yescharacteristic, complemented
420C2—184no—
421C2#1172367311yescharacteristic
422C2C4×#629411yescharacteristic
423C2C4×#629514yes—
424C2C2×#8881411yescharacteristic
subgroupG/H
center, hypercenterC22×C4#1113
commutatorC2×C33C2×C42
FrattiniC23C2×#1113
FittingC22×C33×C4C22
socle, cosocleC23×C33C23
prime radicalC23×C3×C3⋊D3C22
hypocenterC33C4×C22⋊C4
LCSG ▷ C2×C33 ▷ C33
UCSC1 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
2. C2×C3×C4×C3⋊Dic3C2semidirect
4. #11722082C2semidirect
12. C2×C3×C4×C3⋊D3C4semidirect
14. #88814C4direct (N × H)
18. C2×C4×C3⋊Dic3D3semidirect
37. C2×C3×C3⋊Dic3C2×C4semidirect
52. C2×C4×C3⋊D3Dic3semidirect
76. C2×C3×C3⋊D3C42semidirect
77. C33×C4C22⋊C4semidirect
124. C2×C3⋊Dic3C4×D3semidirect
202. C2×C3⋊D3C4×Dic3semidirect
215. C32×C4(C2×D3)⋊C4semidirect
216. C32×C4#217semidirect
228. C33C4×C22⋊C4semidirect
354. C3×C4(C2×D3)⋊Dic3semidirect
355. C3×C4#3722semidirect
364. C32C4×(C2×D3)⋊C4semidirect
365. C32C4×#217semidirect
414. C4#88814direct (N × H)
418. C3C4×(C2×D3)⋊Dic3semidirect
419. C3C4×#3722semidirect

Maximal subgroups

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

subgroupindex
C22×C3×C4×C3⋊D32
C2×C3×C4×C3⋊Dic32
C2×#888142
#117220822
C4×#37223
C4×(C2×D3)⋊Dic33