Finite Group Databasebeta

← C4×#89021 #13175154

Subgroups — C4×#89021 #13175154

Order 1728.

2140 conjugacy classes of subgroups.

The subgroup lattice has 512 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#89021C211yesmaximal, characteristic
2C4×#6308C211yesmaximal, characteristic, complemented
3C3×C4×C4⋊(C3⋊D3)C211yesmaximal, characteristic, complemented
4C32×C42×D3C211yesmaximal, characteristic, complemented
5#11723506C212yesmaximal, complemented
6C4×#6290C212yesmaximal, complemented
7Dic3×C4⋊(C3⋊D3)C211yesmaximal, characteristic, complemented
8#11723464C212yesmaximal, complemented
9C4×Dic3×C3⋊D3C212yesmaximal, complemented
10#11723424C211yesmaximal, characteristic, complemented
11#11723399C211yesmaximal, characteristic, complemented
12C4×#3828—31nomaximal
13C4×#3795—34nomaximal
14C2×#6308C2211yescharacteristic
15C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
16C2×C3×C4×C3⋊D3C2212yes—
17C2×C32×C4×D3C2211yescharacteristic
18C2×C32×C4×D3C2211yescharacteristic
19C2×C32×C4×D3C2211yescharacteristic
20C2×#6290C2212yes—
21C2×Dic3×C3⋊D3C2212yes—
22C2×Dic3×C3⋊D3C2212yes—
23#89021C418yesdirect product factor
24C33×C42C2211yescharacteristic
25C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
26#88857C2212yes—
27C4×C3⋊(C3⋊Dic3)C2212yes—
28C3×#3752C2212yes—
29C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
30C32×C4×Dic3C2211yescharacteristic
31C32×C4×Dic3C2211yescharacteristic
32C32×C4×Dic3C2211yescharacteristic
33#88806C2212yes—
34#88803C2212yes—
35(C2×D3)⋊(C3⋊Dic3)C2212yes—
36Dic3×C3⋊Dic3C2212yes—
37Dic3×C3⋊Dic3C2212yes—
38C2×#3828—31no—
39C2×#3795—34no—
40C2×C32×C42—31no—
41C4×#638—32no—
42C4×#638—32no—
43#64135—31no—
44C4×C4⋊(C3⋊D3)—31no—
45C4×C4⋊(C3⋊D3)D311yescharacteristic, complemented
46#64081—32no—
47C42×C3⋊D3—32no—
48C4×#634—31no—
49C4×#634—31no—
50C4×#634—34no—
51C4×#634—64no—
52C3×C4×D12—34no—
53C3×C42×D3D314yescomplemented
54C4×#625—38no—
55Dic3×D12—34no—
56#63879—38no—
57#63877—38no—
58C4×D3×Dic3—38no—
59#63844—34no—
60#63841—34no—
61C22×C3×C3⋊D3C2312yes—
62C22×C32×D3C2311yescharacteristic
63C2×C33×C4C2311yescharacteristic
64C2×C33×C4C2311yescharacteristic
65C2×C33×C4C2311yescharacteristic
66C2×C3⋊(C3⋊Dic3)C2312yes—
67C2×C3⋊(C3⋊Dic3)C2312yes—
68#6308C2×C414yescomplemented
69C2×C3×C3⋊Dic3C2312yes—
70C3×C4⋊(C3⋊D3)C2×C414yescomplemented
71C3×C4×C3⋊D3—24no—
72C2×C32×Dic3C2311yescharacteristic
73C2×C32×Dic3C2311yescharacteristic
74C2×C32×Dic3C2311yescharacteristic
75C32×C4×D3—22no—
76C32×C4×D3—22no—
77C32×C4×D3C2×C414yescomplemented
78#6290C2×C418yescomplemented
79Dic3×C3⋊D3—24no—
80Dic3×C3⋊D3C2×C418yescomplemented
81C4×#237—91no—
82C4×#235—94no—
83C22×C32×C4—31no—
84C22×C32×C4—31no—
85C22×C32×C4—31no—
86C2×#638—32no—
87#3828—38no—
88C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
89C2×C4×C3⋊D3—32no—
90C2×C4×C3⋊D3—32no—
91C2×C4×C3⋊D3C2×D312yes—
92C2×#634—31no—
93C2×#634—34no—
94C2×#634—64no—
95C2×C3×D12—34no—
96C2×C3×C4×D3—38no—
97C2×C3×C4×D3C2×D314yes—
98C2×C3×C4×D3C2×D314yes—
99C2×C3×C4×D3C2×D314yes—
100C2×#625—38no—
101C2×D3×Dic3—38no—
102C2×D3×Dic3—38no—
103#3795—332no—
104C32×C42—24no—
105C32×C42—31no—
106C32×C42—31no—
107C32×C42—31no—
108C32×C42C2×D314yes—
109C32×C42C2×D311yescharacteristic
110#3757—32no—
111#3752—32no—
112#3752C2×D312yes—
113C4⋊(C3⋊Dic3)—31no—
114C4⋊(C3⋊Dic3)—34no—
115C4⋊(C3⋊Dic3)—64no—
116C4⋊(C3⋊Dic3)C2×D311yescharacteristic
117#3750—32no—
118#3750—38no—
119#3750—32no—
120#3750—68no—
121C4×C3⋊Dic3—32no—
122C4×C3⋊Dic3—38no—
123C4×C3⋊Dic3—32no—
124C4×C3⋊Dic3—32no—
125C4×C3⋊Dic3—68no—
126C3×(C2×D3)⋊C4—38no—
127C3×C4⋊Dic3—34no—
128C3×C4×Dic3C2×D314yes—
129C3×C4×Dic3C2×D314yes—
130C3×C4×Dic3C2×D314yes—
131#3724—38no—
132(C2×D3)⋊Dic3—38no—
133(C2×D3)⋊Dic3—38no—
134Dic32—38no—
135Dic32—38no—
136C22×C33C2411yescharacteristic
137C2×C3×C3⋊D3—22no—
138C2×C3×C3⋊D3—22no—
139C2×C3×C3⋊D3C22×C414yes—
140C2×C32×D3—21no—
141C2×C32×D3—21no—
142C2×C32×D3C22×C412yes—
143C33×C4—21no—
144C33×C4C22×C412yes—
145C33×C4#4112yescomplemented
146C3⋊(C3⋊Dic3)—22no—
147C3⋊(C3⋊Dic3)C22×C414yes—
148C3×C3⋊Dic3—22no—
149C32×Dic3—21no—
150C32×Dic3C22×C412yes—
151C32×Dic3#4112yescomplemented
152C2×#237—91no—
153C2×#235—94no—
154C3×C4×D4—91no—
155C2×C3×C42—34no—
156D4×Dic3—91no—
157C4×#67—98no—
158C4×#67—98no—
159C4×#67—92no—
160#926—94no—
161C2×C4×Dic3—92no—
162#892—91no—
163#887—91no—
164#67⋊C4—92no—
165#877—92no—
166C4×D12—34no—
167C4×D12—94no—
168#872—98no—
169C42×D3—98no—
170C42×D3C3⋊D311yescharacteristic, complemented
171C4×Dic6—94no—
172C4×Dic6—94no—
173C4×Dic6—91no—
174C4×Dic6—184no—
175C23×C32—31no—
176C22×C3⋊D3C22×D312yes—
177C22×C3×D3—38no—
178C22×C3×D3C22×D314yes—
179C2×C32×C4—24no—
180C2×C32×C4—24no—
181C2×C32×C4—24no—
182C2×C32×C4—31no—
183C2×C32×C4—31no—
184C2×C32×C4—34no—
185C2×C32×C4—31no—
186C2×C32×C4—62no—
187C2×C32×C4—62no—
188C2×C32×C4C22×D314yes—
189C2×C32×C4C22×D314yes—
190C2×C32×C4C22×D314yes—
191C2×C32×C4C22×D311yescharacteristic
192C2×C32×C4C22×D311yescharacteristic
193C2×C32×C4C22×D311yescharacteristic
194#638—38no—
195C2×C3⋊Dic3—32no—
196C2×C3⋊Dic3—32no—
197C2×C3⋊Dic3—38no—
198C2×C3⋊Dic3—38no—
199C2×C3⋊Dic3—68no—
200C2×C3⋊Dic3—68no—
201C2×C3⋊Dic3C22×D312yes—
202C4⋊(C3⋊D3)C4×D314yescomplemented
203C4×C3⋊D3—24no—
204C4×C3⋊D3—38no—
205C4×C3⋊D3—64no—
206#634—34no—
207#634—316no—
208#634—616no—
209C2×C3×Dic3—38no—
210C2×C3×Dic3C22×D314yes—
211C2×C3×Dic3C22×D314yes—
212C2×C3×Dic3C22×D314yes—
213C3×D12—316no—
214C3×C4×D3—28no—
215C3×C4×D3—28no—
216C3×C4×D3—616no—
217C3×C4×D3C4×D3116yescomplemented
218#625—332no—
219D3×Dic3—332no—
220D3×Dic3—616no—
221C4×#41—271noSylow
222C2×C33C23×C411yescharacteristic
223C2×C33C2×#4111yescharacteristic
224C2×C33C2×#4111yescharacteristic
225C3×C3⋊D3—24no—
226C32×D3—22no—
227C2×C3×D4—91no—
228C22×C3×C4—34no—
229C22×C3×C4—34no—
230C22×C3×C4—34no—
231C22×C3×C4—92no—
232C2×#67—98no—
233C2×#67—92no—
234C22×Dic3—92no—
235C22×Dic3—92no—
236#237—98no—
237#235—932no—
238C2×D12—34no—
239C2×C4×D3—38no—
240C2×C4×D3—98no—
241C2×C4×D3—98no—
242C2×C4×D3C2×C3⋊D311yescharacteristic
243C2×C4×D3C2×C3⋊D311yescharacteristic
244C2×C4×D3C2×C3⋊D311yescharacteristic
245C2×Dic6—94no—
246C2×Dic6—91no—
247C2×Dic6—184no—
248C3×C4⋊C4—91no—
249C3×C22⋊C4—92no—
250C3×C42—24no—
251C3×C42—34no—
252C3×C42—34no—
253C3×C42—34no—
254C3×C42D3214yes—
255C3×C42C2×C3⋊D311yescharacteristic
256#217—98no—
257#217—92no—
258(C2×D3)⋊C4—38no—
259(C2×D3)⋊C4—98no—
260(C2×D3)⋊C4—92no—
261C4⋊Dic3—34no—
262C4⋊Dic3—94no—
263C4⋊Dic3—91no—
264C4⋊Dic3—184no—
265Dic3⋊C4—98no—
266Dic3⋊C4—98no—
267Dic3⋊C4—92no—
268Dic3⋊C4—92no—
269Dic3⋊C4—188no—
270C4×Dic3—98no—
271C4×Dic3—98no—
272C4×Dic3—98no—
273C4×Dic3—92no—
274C4×Dic3—92no—
275C4×Dic3—92no—
276C4×Dic3—188no—
277C4×Dic3C2×C3⋊D311yescharacteristic
278C4×Dic3C2×C3⋊D311yescharacteristic
279C4×Dic3C2×C3⋊D311yescharacteristic
280C22×C32—24no—
281C22×C32—32no—
282C22×C32—61no—
283C22×C32—61no—
284C22×C32C23×D314yes—
285C22×C32C23×D311yescharacteristic
286C2×C3⋊D3—22no—
287C2×C3⋊D3—22no—
288C2×C3⋊D3C2×C4×D314yes—
289C2×C3×D3—24no—
290C2×C3×D3—24no—
291C2×C3×D3—316no—
292C2×C3×D3—68no—
293C2×C3×D3—68no—
294C2×C3×D3C2×C4×D318yes—
295C32×C4—21no—
296C32×C4—28no—
297C32×C4—28no—
298C32×C4—24no—
299C32×C4—32no—
300C32×C4—32no—
301C32×C4—44no—
302C32×C4—61no—
303C32×C4C2×C4×D318yes—
304C32×C4C2×C4×D312yes—
305C32×C4#23518yescomplemented
306C32×C4#23712yescomplemented
307C3⋊Dic3—22no—
308C3⋊Dic3—34no—
309C3⋊Dic3—316no—
310C3⋊Dic3—616no—
311C3⋊Dic3—62no—
312C3⋊Dic3—68no—
313C3⋊Dic3—128no—
314C3×Dic3—24no—
315C3×Dic3—68no—
316C3×Dic3C2×C4×D318yes—
317C3×Dic3#23518yescomplemented
318C2×#41—271no—
319C4×Q8—271no—
320C4×Q8—271no—
321C4×D4—91no—
322C4×D4—271no—
323C4×D4—272no—
324C4×D4—272no—
325#117—272no—
326#117—271no—
327C2×C42—31no—
328C2×C42—272no—
329C33C4×#4111yescharacteristic, complemented, Sylow
330C23×C3—34no—
331C23×C3—92no—
332C22×D3—38no—
333C22×D3C22×C3⋊D311yescharacteristic
334C3×D4—94no—
335C2×C3×C4—24no—
336C2×C3×C4—24no—
337C2×C3×C4—24no—
338C2×C3×C4—34no—
339C2×C3×C4—34no—
340C2×C3×C4—316no—
341C2×C3×C4—34no—
342C2×C3×C4—68no—
343C2×C3×C4—68no—
344C2×C3×C4—92no—
345C2×C3×C4—184no—
346C2×C3×C4C2×D3214yes—
347C2×C3×C4C2×D3214yes—
348C2×C3×C4C2×D3214yes—
349C2×C3×C4C22×C3⋊D311yescharacteristic
350C2×C3×C4C22×C3⋊D311yescharacteristic
351C2×C3×C4C22×C3⋊D311yescharacteristic
352#67—932no—
353#67—98no—
354C2×Dic3—38no—
355C2×Dic3—98no—
356C2×Dic3—92no—
357C2×Dic3—98no—
358C2×Dic3—98no—
359C2×Dic3—92no—
360C2×Dic3—188no—
361C2×Dic3—188no—
362C2×Dic3—184no—
363C2×Dic3C22×C3⋊D311yescharacteristic
364C2×Dic3C22×C3⋊D311yescharacteristic
365C2×Dic3C22×C3⋊D311yescharacteristic
366D12—316no—
367C4×D3—22no—
368C4×D3—22no—
369C4×D3—616no—
370C4×D3—932no—
371C4×D3—1816no—
372C4×D3C4×C3⋊D314yescomplemented
373Dic6—916no—
374Dic6—94no—
375Dic6—1816no—
376C2×C32—24no—
377C2×C32—24no—
378C2×C32—24no—
379C2×C32—62no—
380C2×C32C22×C4×D314yes—
381C2×C32C22×C4×D311yescharacteristic
382C2×C32C2×#23514yes—
383C2×C32C2×#23514yes—
384C2×C32C2×#23711yescharacteristic
385C2×C32D3×#4111yescharacteristic
386C3⋊D3—24no—
387C3×D3—28no—
388C3×D3—616no—
389#41—278no—
390C2×Q8—271no—
391C2×D4—91no—
392C2×D4—272no—
393C22×C4—31no—
394C22×C4—31no—
395C22×C4—31no—
396C22×C4—92no—
397C22×C4—272no—
398C22×C4—272no—
399C4⋊C4—91no—
400C4⋊C4—272no—
401C4⋊C4—272no—
402C4⋊C4—271no—
403C22⋊C4—92no—
404C22⋊C4—272no—
405C22⋊C4—272no—
406C42—31no—
407C42—31no—
408C42—31no—
409C42—272no—
410C42—272no—
411C42—272no—
412C42D3×C3⋊D311yescharacteristic
413C22×C3—24no—
414C22×C3—38no—
415C22×C3—64no—
416C22×C3—64no—
417C22×C3—94no—
418C22×C3—182no—
419C22×C3—182no—
420C22×C3C22×D3214yes—
421C22×C3C23×C3⋊D311yescharacteristic
422C2×D3—21no—
423C2×D3—21no—
424C2×D3—316no—
425C2×D3—68no—
426C2×D3—68no—
427C2×D3C2×C4×C3⋊D312yes—
428C3×C4—24no—
429C3×C4—21no—
430C3×C4—28no—
431C3×C4—28no—
432C3×C4—38no—
433C3×C4—38no—
434C3×C4—44no—
435C3×C4—64no—
436C3×C4—182no—
437C3×C4#379518yescomplemented
438C3×C4C4×D3218yes—
439C3×C4C2×C4×C3⋊D312yes—
440C3×C4#382812yescomplemented
441Dic3—21no—
442Dic3—68no—
443Dic3—916no—
444Dic3—94no—
445Dic3—1816no—
446Dic3—188no—
447Dic3—182no—
448Dic3—368no—
449Dic3C2×C4×C3⋊D312yes—
450Dic3#382812yescomplemented
451C32—24no—
452C32C4×#23514yescomplemented
453C32C4×#23711yescharacteristic, complemented
454C23—31no—
455C23—92no—
456Q8—274no—
457D4—94no—
458D4—278no—
459C2×C4—31no—
460C2×C4—31no—
461C2×C4—34no—
462C2×C4—31no—
463C2×C4—62no—
464C2×C4—62no—
465C2×C4—92no—
466C2×C4—184no—
467C2×C4—278no—
468C2×C4—272no—
469C2×C4—272no—
470C2×C4—544no—
471C2×C4C2×D3×C3⋊D311yescharacteristic
472C2×C4C2×D3×C3⋊D311yescharacteristic
473C2×C4C2×D3×C3⋊D311yescharacteristic
474C2×C3—24no—
475C2×C3—24no—
476C2×C3—24no—
477C2×C3—68no—
478C2×C3—184no—
479C2×C3C2×#379514yes—
480C2×C3C2×C4×D3214yes—
481C2×C3D3×#23514yes—
482C2×C3C22×C4×C3⋊D311yescharacteristic
483C2×C3C2×#382811yescharacteristic
484C2×C3C2×#382811yescharacteristic
485D3—22no—
486D3—616no—
487C22—32no—
488C22—61no—
489C22—61no—
490C22—94no—
491C22—182no—
492C22—182no—
493C22C22×D3×C3⋊D311yescharacteristic
494C4—21no—
495C4—32no—
496C4—32no—
497C4—61no—
498C4—182no—
499C4—274no—
500C4—542no—
501C4#8902112yesdirect product factor
502C4C4×D3×C3⋊D312yes—
503C3—24no—
504C3C4×#379514yescomplemented
505C3C4×#382811yescharacteristic, complemented
506C2—62no—
507C2—184no—
508C2C2×#8902111yescharacteristic
509C2C2×C4×D3×C3⋊D311yescharacteristic
510C2D3×#382811yescharacteristic
subgroupG/H
centerC2×C4C2×D3×C3⋊D3
commutatorC2×C33C23×C4
FrattiniC22C22×D3×C3⋊D3
FittingC33×C42C22
socle, cosocleC22×C33C24
prime radicalC2×#89021C2
hypercenterC42D3×C3⋊D3
hypocenterC33C4×#41
LCSG ▷ C2×C33 ▷ C33
UCSC1 ◁ C2×C4 ◁ C42

Complements

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

NH ≅ G/Ntype
2. C4×#6308C2semidirect
3. C3×C4×C4⋊(C3⋊D3)C2semidirect
4. C32×C42×D3C2semidirect
5. #11723506C2semidirect
6. C4×#6290C2semidirect
7. Dic3×C4⋊(C3⋊D3)C2semidirect
8. #11723464C2semidirect
9. C4×Dic3×C3⋊D3C2semidirect
10. #11723424C2semidirect
11. #11723399C2semidirect
23. #89021C4direct (N × H)
45. C4×C4⋊(C3⋊D3)D3semidirect
53. C3×C42×D3D3semidirect
68. #6308C2×C4semidirect
70. C3×C4⋊(C3⋊D3)C2×C4semidirect
77. C32×C4×D3C2×C4semidirect
78. #6290C2×C4semidirect
80. Dic3×C3⋊D3C2×C4semidirect
145. C33×C4#41semidirect
151. C32×Dic3#41semidirect
170. C42×D3C3⋊D3semidirect
202. C4⋊(C3⋊D3)C4×D3semidirect
217. C3×C4×D3C4×D3semidirect
305. C32×C4#235semidirect
306. C32×C4#237semidirect
317. C3×Dic3#235semidirect
329. C33C4×#41semidirect
372. C4×D3C4×C3⋊D3semidirect
437. C3×C4#3795semidirect
440. C3×C4#3828semidirect
450. Dic3#3828semidirect
452. C32C4×#235semidirect
453. C32C4×#237semidirect
501. C4#89021direct (N × H)
504. C3C4×#3795semidirect
505. C3C4×#3828semidirect

Maximal subgroups

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

subgroupindex
C2×#890212
C4×#63082
C3×C4×C4⋊(C3⋊D3)2
C32×C42×D32
#117235062
C4×#62902
Dic3×C4⋊(C3⋊D3)2
#117234642
C4×Dic3×C3⋊D32
#117234242
#117233992
C4×#38283
C4×#37953