Finite Group Databasebeta

← C2×#11722073 #13161805

Subgroups — C2×#11722073 #13161805

Order 1728.

1170 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⋊D3)C211yesmaximal, characteristic
2C2×C3×#3748C212yesmaximal, complemented
3#11722073C214yesmaximal, direct product factor
4C2×#63559—31nomaximal
5C2×#63558—32nomaximal
6C23×C3×C3⋊D3C412yes—
7C22×C33×C4C2211yescharacteristic
8C2×C3×C4⋊(C3⋊D3)—22no—
9C2×C3×C4⋊(C3⋊D3)—22no—
10C2×C3×C4⋊(C3⋊D3)C412yes—
11C2×C3×C4⋊(C3⋊D3)C2212yes—
12C3×#3748—22no—
13C3×#3748C2214yescomplemented
14C2×C3×C3⋊(C3⋊C8)—22no—
15C22×C4⋊(C3⋊D3)D311yescharacteristic
16C22×C3×D12—32no—
17C22×C3×D12—61no—
18C2×#3748D312yescomplemented
19C2×C3×#208—32no—
20C2×C3×#208—32no—
21C2×C3×#208—62no—
22#63559—34no—
23#63558—38no—
24C23×C33C2×C411yescharacteristic
25C22×C3×C3⋊D3—22no—
26C22×C3×C3⋊D3—22no—
27C22×C3×C3⋊D3—44no—
28C22×C3×C3⋊D3C2×C414yes—
29C2×C33×C4C2×C411yescharacteristic
30C2×C33×C4D412yes—
31C2×C33×C4D412yes—
32C2×C33×C4C2311yescharacteristic
33C3×C4⋊(C3⋊D3)—24no—
34C3×C4⋊(C3⋊D3)—24no—
35C3×C3⋊(C3⋊C8)—24no—
36C2×#833—91no—
37C2×#823—92no—
38C23×C3⋊D3Dic312yes—
39C23×C3×D3—34no—
40C23×C3×D3—62no—
41C22×C32×C4—22no—
42C22×C32×C4—21no—
43C22×C32×C4—41no—
44C22×C32×C4C2×D312yes—
45C22×C32×C4C2×D311yescharacteristic
46C2×C4⋊(C3⋊D3)—22no—
47C2×C4⋊(C3⋊D3)—22no—
48C2×C4⋊(C3⋊D3)Dic312yes—
49C2×C4⋊(C3⋊D3)C2×D312yes—
50C2×C3×D12—34no—
51C2×C3×D12—34no—
52C2×C3×D12—62no—
53C2×C3×D12—62no—
54C2×C3×D12—64no—
55C2×C3×D12—64no—
56C2×C3×D12—62no—
57C2×C3×D12—62no—
58C2×C3×D12—62no—
59C2×C3×D12—62no—
60#3748—22no—
61#3748C2×D314yescomplemented
62C2×C3⋊(C3⋊C8)—22no—
63C3×#208—34no—
64C3×#208—34no—
65C3×#208—64no—
66C3×#208—64no—
67C3×#208—62no—
68C3×#208—62no—
69C2×C3×C3⋊C8—64no—
70C2×C3×C3⋊C8—62no—
71C2×C3×C3⋊C8—62no—
72C22×C33—22no—
73C22×C33C22⋊C411yescharacteristic
74C22×C33C22⋊C411yescharacteristic
75C22×C33C22×C411yescharacteristic
76C2×C3×C3⋊D3—24no—
77C2×C3×C3⋊D3—44no—
78C2×C3×C3⋊D3—44no—
79C2×C3×C3⋊D3—44no—
80C33×C4C22⋊C412yes—
81C33×C4C2×D412yes—
82C22×C3×D4—91no—
83C22×D12—32no—
84C22×D12—61no—
85C2×C3×M16—92no—
86C2×#208—32no—
87C2×#208—32no—
88C2×#208—62no—
89#833—94no—
90#823—98no—
91C23×C32—22no—
92C23×C32—21no—
93C23×C32—41no—
94C23×C32C4×D312yes—
95C23×C32C2×Dic311yescharacteristic
96C22×C3⋊D3—22no—
97C22×C3⋊D3—22no—
98C22×C3⋊D3—44no—
99C22×C3⋊D3C2×Dic314yes—
100C22×C3×D3—38no—
101C22×C3×D3—64no—
102C22×C3×D3—64no—
103C22×C3×D3—64no—
104C22×C3×D3—64no—
105C22×C3×D3—64no—
106C22×C3×D3—128no—
107C22×C3×D3—128no—
108C2×C32×C4—22no—
109C2×C32×C4—21no—
110C2×C32×C4—22no—
111C2×C32×C4—22no—
112C2×C32×C4—22no—
113C2×C32×C4—22no—
114C2×C32×C4—22no—
115C2×C32×C4—22no—
116C2×C32×C4—22no—
117C2×C32×C4—21no—
118C2×C32×C4—41no—
119C2×C32×C4—42no—
120C2×C32×C4—42no—
121C2×C32×C4—41no—
122C2×C32×C4C4×D312yes—
123C2×C32×C4D1212yes—
124C2×C32×C4D1212yes—
125C2×C32×C4C2×Dic311yescharacteristic
126C2×C32×C4#6712yes—
127C2×C32×C4#6712yes—
128C2×C32×C4#6712yes—
129C2×C32×C4#6712yes—
130C2×C32×C4C22×D312yes—
131C2×C32×C4C22×D311yescharacteristic
132C4⋊(C3⋊D3)—24no—
133C4⋊(C3⋊D3)—24no—
134C3×D12—68no—
135C3×D12—68no—
136C3×D12—64no—
137C3×D12—64no—
138C3×D12—64no—
139C3×D12—64no—
140C3⋊(C3⋊C8)—24no—
141C3×C3⋊C8—68no—
142C3×C3⋊C8—64no—
143C3×C3⋊C8—64no—
144C2×#100—271noSylow
145C2×C33—21no—
146C2×C33—21no—
147C2×C33#10012yescomplemented
148C2×C33C2×C22⋊C411yescharacteristic
149C3×C3⋊D3—44no—
150C24×C3—92no—
151C23×D3—34no—
152C23×D3—62no—
153C2×C3×D4—92no—
154C2×C3×D4—92no—
155C2×C3×D4—182no—
156C2×C3×D4—182no—
157C22×C3×C4—22no—
158C22×C3×C4—21no—
159C22×C3×C4—41no—
160C22×C3×C4D3212yes—
161C22×C3×C4D3211yescharacteristic
162C2×D12—34no—
163C2×D12—34no—
164C2×D12—62no—
165C2×D12—62no—
166C2×D12—64no—
167C2×D12—64no—
168C2×D12—62no—
169C2×D12—62no—
170C2×D12—62no—
171C2×D12—62no—
172C3×M16—94no—
173C3×M16—182no—
174C2×C3×C8—182no—
175#208—34no—
176#208—34no—
177#208—64no—
178#208—64no—
179#208—62no—
180#208—62no—
181C2×C3⋊C8—64no—
182C2×C3⋊C8—62no—
183C2×C3⋊C8—62no—
184C22×C32—22no—
185C22×C32—24no—
186C22×C32—24no—
187C22×C32—22no—
188C22×C32—22no—
189C22×C32—21no—
190C22×C32—21no—
191C22×C32—22no—
192C22×C32—21no—
193C22×C32—41no—
194C22×C32—41no—
195C22×C32—41no—
196C22×C32—44no—
197C22×C32—44no—
198C22×C32(C2×D3)⋊C412yes—
199C22×C32(C2×D3)⋊C412yes—
200C22×C32#21711yescharacteristic
201C22×C32#21711yescharacteristic
202C22×C32C2×C4×D312yes—
203C22×C32C22×Dic311yescharacteristic
204C2×C3⋊D3—24no—
205C2×C3⋊D3—44no—
206C2×C3⋊D3—44no—
207C2×C3⋊D3—44no—
208C2×C3×D3—68no—
209C2×C3×D3—68no—
210C2×C3×D3—128no—
211C2×C3×D3—128no—
212C2×C3×D3—128no—
213C2×C3×D3—128no—
214C2×C3×D3—128no—
215C2×C3×D3—128no—
216C32×C4—22no—
217C32×C4—22no—
218C32×C4—22no—
219C32×C4—22no—
220C32×C4—22no—
221C32×C4—22no—
222C32×C4—42no—
223C32×C4—42no—
224C32×C4(C2×D3)⋊C412yes—
225C32×C4(C2×D3)⋊C412yes—
226C32×C4#21712yes—
227C32×C4C2×D1212yes—
228C32×C4C2×#6712yes—
229C32×C4C2×#6712yes—
230C22×D4—91no—
231C2×M16—92no—
232#100—274no—
233C33C2×#10011yescharacteristic, complemented, Sylow
234C23×C3—21no—
235C23×C3—22no—
236C23×C3—41no—
237C23×C3—94no—
238C23×C3—182no—
239C23×C3—182no—
240C23×C3—364no—
241C23×C3D3×Dic312yes—
242C23×C3#62411yescharacteristic
243C22×D3—38no—
244C22×D3—64no—
245C22×D3—64no—
246C22×D3—64no—
247C22×D3—64no—
248C22×D3—64no—
249C22×D3—128no—
250C22×D3—128no—
251C3×D4—184no—
252C3×D4—184no—
253C2×C3×C4—22no—
254C2×C3×C4—21no—
255C2×C3×C4—22no—
256C2×C3×C4—22no—
257C2×C3×C4—22no—
258C2×C3×C4—22no—
259C2×C3×C4—22no—
260C2×C3×C4—22no—
261C2×C3×C4—22no—
262C2×C3×C4—21no—
263C2×C3×C4—41no—
264C2×C3×C4—42no—
265C2×C3×C4—42no—
266C2×C3×C4—41no—
267C2×C3×C4D3×Dic312yes—
268C2×C3×C4#62411yescharacteristic
269C2×C3×C4#62512yes—
270C2×C3×C4#62512yes—
271C2×C3×C4#62612yes—
272C2×C3×C4#62612yes—
273C2×C3×C4#62612yes—
274C2×C3×C4#62612yes—
275C2×C3×C4C2×D3212yes—
276C2×C3×C4C2×D3211yescharacteristic
277D12—68no—
278D12—68no—
279D12—64no—
280D12—64no—
281D12—64no—
282D12—64no—
283C3×C8—184no—
284C3⋊C8—68no—
285C3⋊C8—64no—
286C3⋊C8—64no—
287C2×C32—21no—
288C2×C32—21no—
289C2×C32—22no—
290C2×C32—22no—
291C2×C32—22no—
292C2×C32—22no—
293C2×C32—24no—
294C2×C32—22no—
295C2×C32—22no—
296C2×C32—21no—
297C2×C32—42no—
298C2×C32—41no—
299C2×C32—42no—
300C2×C32—42no—
301C2×C32—42no—
302C2×C32—42no—
303C2×C32#82314yescomplemented
304C2×C32#83312yescomplemented
305C2×C32C2×(C2×D3)⋊C412yes—
306C2×C32C2×#21711yescharacteristic
307C3⋊D3—44no—
308C3×D3—128no—
309C3×D3—128no—
310C24—92no—
311C2×D4—92no—
312C2×D4—92no—
313C2×D4—182no—
314C2×D4—182no—
315C22×C4#111311yescharacteristic
316M16—94no—
317M16—182no—
318C2×C8—182no—
319C22×C3—24no—
320C22×C3—24no—
321C22×C3—22no—
322C22×C3—21no—
323C22×C3—22no—
324C22×C3—21no—
325C22×C3—22no—
326C22×C3—21no—
327C22×C3—22no—
328C22×C3—44no—
329C22×C3—44no—
330C22×C3—41no—
331C22×C3—41no—
332C22×C3—41no—
333C22×C3—184no—
334C22×C3—364no—
335C22×C3—364no—
336C22×C3—364no—
337C22×C3(C2×D3)⋊Dic312yes—
338C22×C3(C2×D3)⋊Dic312yes—
339C22×C3#372211yescharacteristic
340C22×C3#372211yescharacteristic
341C22×C3C2×D3×Dic312yes—
342C22×C3C2×#62411yescharacteristic
343C2×D3—68no—
344C2×D3—68no—
345C2×D3—128no—
346C2×D3—128no—
347C2×D3—128no—
348C2×D3—128no—
349C2×D3—128no—
350C2×D3—128no—
351C3×C4—22no—
352C3×C4—22no—
353C3×C4—22no—
354C3×C4—22no—
355C3×C4—22no—
356C3×C4—22no—
357C3×C4—42no—
358C3×C4—42no—
359C3×C4(C2×D3)⋊Dic312yes—
360C3×C4(C2×D3)⋊Dic312yes—
361C3×C4#372212yes—
362C3×C4C2×#62512yes—
363C3×C4C2×#62612yes—
364C3×C4C2×#62612yes—
365C32—22no—
366C32—21no—
367C32—41no—
368C32C2×#82312yescomplemented
369C32C2×#83311yescharacteristic, complemented
370C23—94no—
371C23—182no—
372C23—182no—
373C23—364no—
374C23#629411yescharacteristic
375D4—184no—
376D4—184no—
377C2×C4#629411yescharacteristic
378C2×C4#629512yes—
379C2×C4#629512yes—
380C2×C4C2×#111311yescharacteristic
381C8—184no—
382C2×C3—22no—
383C2×C3—22no—
384C2×C3—22no—
385C2×C3—22no—
386C2×C3—21no—
387C2×C3—21no—
388C2×C3—22no—
389C2×C3—24no—
390C2×C3—21no—
391C2×C3—22no—
392C2×C3—42no—
393C2×C3—42no—
394C2×C3—42no—
395C2×C3—42no—
396C2×C3—42no—
397C2×C3—41no—
398C2×C3—364no—
399C2×C3#6355814yescomplemented
400C2×C3#6355912yescomplemented
401C2×C3C2×(C2×D3)⋊Dic312yes—
402C2×C3C2×#372211yescharacteristic
403D3—128no—
404D3—128no—
405C22—22no—
406C22—184no—
407C22—364no—
408C22—364no—
409C22—364no—
410C22#8881411yescharacteristic
411C22#8881411yescharacteristic
412C22C2×#629411yescharacteristic
413C4#8881412yes—
414C4C2×#629512yes—
415C3—21no—
416C3—22no—
417C3—41no—
418C3C2×#6355812yescomplemented
419C3C2×#6355911yescharacteristic, complemented
420C2—21no—
421C2—21no—
422C2—364no—
423C2#1172207312yesdirect product factor
424C2C2×#8881411yescharacteristic
subgroupG/H
centerC22#88814
commutatorC22×C33C22×C4
FrattiniC2×C4C2×#1113
FittingC22×C33×C4C22
socleC22×C33C22⋊C4
prime radicalC22×C3×C4⋊(C3⋊D3)C2
cosocleC2×C33×C4C23
hypercenterC22×C4#1113
hypocenterC33C2×#100
LCSG ▷ C22×C33 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
2. C2×C3×#3748C2semidirect
3. #11722073C2direct (N × H)
13. C3×#3748C22semidirect
18. C2×#3748D3semidirect
61. #3748C2×D3semidirect
147. C2×C33#100semidirect
233. C33C2×#100semidirect
303. C2×C32#823semidirect
304. C2×C32#833semidirect
368. C32C2×#823semidirect
369. C32C2×#833semidirect
399. C2×C3#63558semidirect
400. C2×C3#63559semidirect
418. C3C2×#63558semidirect
419. C3C2×#63559semidirect
423. C2#11722073direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×C4⋊(C3⋊D3)2
C2×C3×#37482
#117220732
C2×#635593
C2×#635583