Finite Group Databasebeta

← C2×#11724562 #13181591

Subgroups — C2×#11724562 #13181591

Order 1728.

5420 conjugacy classes of subgroups.

The subgroup lattice has 516 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#89076C211yesmaximal, characteristic, complemented
2C22×C3×#634C211yesmaximal, characteristic, complemented
3C2×C32×#235C211yesmaximal, characteristic, complemented
4C2×#89028C214yesmaximal, complemented
5C2×#89022C212yesmaximal, complemented
6C2×#89019C212yesmaximal, complemented
7#11724562C2116yesmaximal, direct product factor
8C2×#89015C212yesmaximal, complemented
9C2×D3×#634C212yesmaximal, complemented
10C2×#64367—31nomaximal
11C2×#64297—34nomaximal
12C22×C33×C4C2211yescharacteristic
13C2×#6312C2212yes—
14#89076C2218yescomplemented
15C2×C4⋊(C3⋊(C3⋊D3))C2211yescharacteristic
16C2×C4×C3⋊(C3⋊D3)C2212yes—
17C2×#6308C2211yescharacteristic
18C22×C3×C3⋊Dic3C2212yes—
19C2×C3×#634C2212yes—
20C2×C3×#634C2212yes—
21C2×C3×#634C2218yescomplemented
22C2×C32×#67C2212yes—
23C32×#235C2218yescomplemented
24C2×C32×D12C2211yescharacteristic
25C2×C32×C4×D3C2212yes—
26C2×C32×Dic6C2211yescharacteristic
27C2×#6293C2214yes—
28C2×#6291C2214yes—
29C2×#6289C2214yes—
30#89028C22132yescomplemented
31C2×D3×C3⋊Dic3C2214yes—
32#89022C22116yescomplemented
33#89019C22116yescomplemented
34#89015C22116yescomplemented
35D3×#634C22116yescomplemented
36C2×C32×#41—31no—
37#64367—316no—
38C2×Q8×C3⋊D3—32no—
39C2×#3830—34no—
40C2×#3830—32no—
41C2×#3828—31no—
42C2×#3828—32no—
43C2×#3828—34no—
44C2×#3828—64no—
45C22×#634—32no—
46C22×#634D311yescharacteristic, complemented
47C2×C3×#235D314yescomplemented
48C22×C3×Dic6—34no—
49C2×#3804—316no—
50C2×#3799—38no—
51C2×#3797—38no—
52#64297—364no—
53C2×#3796—38no—
54C2×D3×Dic6—38no—
55C23×C33C2311yescharacteristic
56C22×C3⋊(C3⋊D3)C2312yes—
57C22×C32×D3C2312yes—
58C2×C33×C4C2311yescharacteristic
59C2×C33×C4C2311yescharacteristic
60C2×C33×C4C2314yes—
61#6312C2318yes—
62C2×C3⋊(C3⋊Dic3)C2312yes—
63C4⋊(C3⋊(C3⋊D3))C2314yes—
64C4×C3⋊(C3⋊D3)C2318yes—
65#6308C2314yes—
66C2×C3×C3⋊Dic3C2314yes—
67C2×C3×C3⋊Dic3C2318yes—
68C3×#634C2318yes—
69C3×#634C2318yes—
70C32×#67C2318yes—
71C2×C32×Dic3C2312yes—
72C32×D12C2314yes—
73C32×C4×D3C2318yes—
74C32×Dic6C2314yes—
75#6293C23116yes—
76#6291C23116yes—
77#6289C23116yes—
78D3×C3⋊Dic3C23116yes—
79C2×#1010—94no—
80C2×#1007—91no—
81C32×#41—38no—
82C2×C32×Q8—31no—
83C2×C32×D4—32no—
84C2×C32×D4—31no—
85C22×C32×C4—24no—
86C22×C32×C4—32no—
87C22×C32×C4C2×D314yes—
88C22×C32×C4C2×D311yescharacteristic
89C2×#638—32no—
90C2×#638—38no—
91C2×#638—34no—
92C2×#638—68no—
93C22×C3⋊Dic3—34no—
94C22×C3⋊Dic3C2×D312yes—
95Q8×C3⋊D3—316no—
96#3830—332no—
97#3830—316no—
98#3828—38no—
99#3828—332no—
100#3828—316no—
101#3828—632no—
102C2×C4⋊(C3⋊D3)—31no—
103C2×C4⋊(C3⋊D3)—34no—
104C2×C4⋊(C3⋊D3)—64no—
105C2×C4×C3⋊D3—32no—
106C2×C4×C3⋊D3—38no—
107C2×C4×C3⋊D3—34no—
108C2×C4×C3⋊D3—68no—
109C2×#634—31no—
110C2×#634—34no—
111C2×#634—34no—
112C2×#634—316no—
113C2×#634—64no—
114C2×#634C2×D312yes—
115C2×#634C2×D312yes—
116C2×#634C2×D318yescomplemented
117C2×C3×#67C2×D318yes—
118C22×C3×Dic3—38no—
119C3×#235C2×D3132yescomplemented
120C2×C3×D12C2×D314yes—
121C2×C3×C4×D3C2×D318yes—
122C2×C3×Dic6—38no—
123C2×C3×Dic6—38no—
124C2×C3×Dic6—332no—
125C2×C3×Dic6C2×D314yes—
126C2×#627—316no—
127C2×#626—316no—
128C2×#624—316no—
129#3804—3128no—
130C2×D3×Dic3—316no—
131#3799—364no—
132#3797—364no—
133#3796—364no—
134D3×Dic6—364no—
135C22×C33—22no—
136C22×C33C2412yes—
137C22×C33C2411yescharacteristic
138C2×C3⋊(C3⋊D3)—24no—
139C2×C3⋊(C3⋊D3)C2414yes—
140C2×C32×D3—24no—
141C2×C32×D3C2414yes—
142C33×C4C2412yes—
143C33×C4C2412yes—
144C3⋊(C3⋊Dic3)C2414yes—
145C3×C3⋊Dic3C2418yes—
146C32×Dic3C2414yes—
147C2×C3×#41—34no—
148C22×C3×Q8—91no—
149#1010—964no—
150#1007—916no—
151C2×#239—92no—
152C2×#239—92no—
153C2×D3×Q8—98no—
154C2×D3×Q8—92no—
155C2×D3×Q8—92no—
156C2×#237—98no—
157C2×#237—916no—
158C2×#235—98no—
159C2×#235—94no—
160C2×#235—91no—
161C2×#235—94no—
162C2×#235—184no—
163C2×#235C3⋊D311yescharacteristic, complemented
164C22×Dic6—34no—
165C22×Dic6—98no—
166C23×C32—24no—
167C23×C32—32no—
168C23×C32C22×D314yes—
169C23×C32C22×D311yescharacteristic
170C22×C3⋊D3—32no—
171C22×C3⋊D3—38no—
172C22×C3⋊D3—68no—
173C22×C3×D3C22×D318yes—
174C32×Q8—34no—
175C32×D4—38no—
176C32×D4—34no—
177C2×C32×C4—24no—
178C2×C32×C4—24no—
179C2×C32×C4—216no—
180C2×C32×C4—32no—
181C2×C32×C4—38no—
182C2×C32×C4C22×D314yes—
183C2×C32×C4C22×D314yes—
184C2×C32×C4C22×D3116yes—
185C2×C32×C4C22×D311yescharacteristic
186C2×C32×C4C22×D311yescharacteristic
187C2×C32×C4C22×D314yes—
188#638—38no—
189#638—316no—
190#638—332no—
191#638—632no—
192C2×C3⋊Dic3—32no—
193C2×C3⋊Dic3—316no—
194C2×C3⋊Dic3—38no—
195C2×C3⋊Dic3—68no—
196C2×C3⋊Dic3C22×D314yes—
197C2×C3⋊Dic3C22×D318yes—
198C4⋊(C3⋊D3)—34no—
199C4⋊(C3⋊D3)—316no—
200C4⋊(C3⋊D3)—616no—
201C4×C3⋊D3—38no—
202C4×C3⋊D3—316no—
203C4×C3⋊D3—332no—
204C4×C3⋊D3—632no—
205#634—34no—
206#634—316no—
207#634—316no—
208#634—616no—
209#634C22×D318yes—
210#634C22×D318yes—
211C3×#67C22×D3132yes—
212C2×C3×Dic3—316no—
213C2×C3×Dic3—332no—
214C2×C3×Dic3C22×D318yes—
215C3×D12C22×D3116yes—
216C3×C4×D3C22×D3132yes—
217C3×Dic6—332no—
218C3×Dic6—332no—
219C3×Dic6C22×D3116yes—
220#627—364no—
221#626—364no—
222#624—364no—
223D3×Dic3—364no—
224C2×2−1+4—271noSylow
225C2×C33—22no—
226C2×C332−1+412yescomplemented
227C2×C33C2511yescharacteristic
228C3⋊(C3⋊D3)—24no—
229C32×D3—24no—
230C3×#41—332no—
231C2×C3×Q8—34no—
232C2×C3×Q8—92no—
233C2×C3×Q8—92no—
234C2×C3×Q8—98no—
235C2×C3×D4—38no—
236C2×C3×D4—34no—
237C22×C3×C4—24no—
238C22×C3×C4—38no—
239C22×C3×C4—92no—
240C22×C3×C4D3214yes—
241C22×C3×C4C2×C3⋊D311yescharacteristic
242C2×#67—916no—
243C2×#67—98no—
244C2×#67—92no—
245C2×#67—188no—
246C2×#67C2×C3⋊D312yes—
247C22×Dic3—38no—
248C22×Dic3—916no—
249#239—916no—
250#239—916no—
251D3×Q8—964no—
252D3×Q8—916no—
253D3×Q8—916no—
254#237—964no—
255#237—9128no—
256#235—964no—
257#235—932no—
258#235—98no—
259#235—932no—
260#235—1832no—
261#235C2×C3⋊D318yescomplemented
262C2×D12—94no—
263C2×D12—94no—
264C2×D12—91no—
265C2×D12—184no—
266C2×D12C2×C3⋊D311yescharacteristic
267C2×C4×D3—916no—
268C2×C4×D3—94no—
269C2×C4×D3—98no—
270C2×C4×D3—92no—
271C2×C4×D3—94no—
272C2×C4×D3—188no—
273C2×C4×D3C2×C3⋊D312yes—
274C2×Dic6—38no—
275C2×Dic6—38no—
276C2×Dic6—332no—
277C2×Dic6—964no—
278C2×Dic6—916no—
279C2×Dic6—94no—
280C2×Dic6—91no—
281C2×Dic6—94no—
282C2×Dic6—184no—
283C2×Dic6C2×C3⋊D311yescharacteristic
284C22×C32—22no—
285C22×C32—24no—
286C22×C32—28no—
287C22×C32—28no—
288C22×C32—34no—
289C22×C32—48no—
290C22×C32—64no—
291C22×C32C23×D318yes—
292C22×C32C23×D314yes—
293C22×C32C23×D312yes—
294C22×C32C23×D311yescharacteristic
295C2×C3⋊D3—34no—
296C2×C3⋊D3—316no—
297C2×C3⋊D3—616no—
298C2×C3⋊D3—64no—
299C2×C3⋊D3—616no—
300C2×C3⋊D3—1216no—
301C2×C3×D3—216no—
302C2×C3×D3C23×D3116yes—
303C32×C4—28no—
304C32×C4—28no—
305C32×C4—34no—
306C32×C4C23×D318yes—
307C32×C4C23×D318yes—
308C32×C4C23×D312yes—
309C32×C4C23×D312yes—
310C3⋊Dic3—34no—
311C3⋊Dic3—316no—
312C3⋊Dic3—616no—
313C3⋊Dic3C23×D318yes—
314C3×Dic3—332no—
315C3×Dic3C23×D3116yes—
3162−1+4—2716no—
317C2×#41—31no—
318C2×#41—272no—
319C2×#41—274no—
320C2×#41—272no—
321C2×#41—271no—
322C22×Q8—91no—
323C22×Q8—272no—
324C22×Q8—272no—
325C33C2×2−1+411yescharacteristic, complemented, Sylow
326C23×C3—24no—
327C23×C3—38no—
328C23×C3C2×D3214yes—
329C23×C3C22×C3⋊D311yescharacteristic
330C22×D3—98no—
331C22×D3—92no—
332C22×D3—188no—
333C22×D3C22×C3⋊D312yes—
334C3×Q8—316no—
335C3×Q8—98no—
336C3×Q8—98no—
337C3×D4—332no—
338C3×D4—316no—
339C2×C3×C4—24no—
340C2×C3×C4—24no—
341C2×C3×C4—216no—
342C2×C3×C4—38no—
343C2×C3×C4—332no—
344C2×C3×C4—94no—
345C2×C3×C4—98no—
346C2×C3×C4C2×D3214yes—
347C2×C3×C4C2×D3214yes—
348C2×C3×C4C2×D32116yes—
349C2×C3×C4C22×C3⋊D311yescharacteristic
350C2×C3×C4C22×C3⋊D311yescharacteristic
351C2×C3×C4C22×C3⋊D314yes—
352#67—964no—
353#67—932no—
354#67—98no—
355#67—1832no—
356#67C22×C3⋊D318yes—
357C2×Dic3—316no—
358C2×Dic3—332no—
359C2×Dic3—964no—
360C2×Dic3—98no—
361C2×Dic3—92no—
362C2×Dic3—188no—
363C2×Dic3C22×C3⋊D312yes—
364D12—916no—
365D12—916no—
366D12—94no—
367D12—1816no—
368D12C22×C3⋊D314yes—
369C4×D3—964no—
370C4×D3—916no—
371C4×D3—932no—
372C4×D3—98no—
373C4×D3—916no—
374C4×D3—1832no—
375C4×D3C22×C3⋊D318yes—
376Dic6—332no—
377Dic6—332no—
378Dic6—964no—
379Dic6—916no—
380Dic6—94no—
381Dic6—916no—
382Dic6—1816no—
383Dic6C22×C3⋊D314yes—
384C2×C32—22no—
385C2×C32—28no—
386C2×C32—24no—
387C2×C32—28no—
388C2×C32—48no—
389C2×C32—64no—
390C2×C32#100712yescomplemented
391C2×C32#101018yescomplemented
392C2×C32C24×D314yes—
393C2×C32C24×D311yescharacteristic
394C3⋊D3—64no—
395C3⋊D3—616no—
396C3⋊D3—1216no—
397C3×D3—216no—
398#41—38no—
399#41—2716no—
400#41—2732no—
401#41—2716no—
402#41—278no—
403C2×Q8—31no—
404C2×Q8—92no—
405C2×Q8—92no—
406C2×Q8—98no—
407C2×Q8—274no—
408C2×Q8—2716no—
409C2×Q8—2716no—
410C2×Q8—271no—
411C2×D4—32no—
412C2×D4—31no—
413C2×D4—274no—
414C2×D4—272no—
415C2×D4—271no—
416C22×C4—32no—
417C22×C4—92no—
418C22×C4—274no—
419C22×C4—274no—
420C22×C4—272no—
421C22×C4D3×C3⋊D311yescharacteristic
422C22×C3—28no—
423C22×C3—22no—
424C22×C3—28no—
425C22×C3—24no—
426C22×C3—316no—
427C22×C3—48no—
428C22×C3—616no—
429C22×C3C22×D3218yes—
430C22×C3C22×D3214yes—
431C22×C3C23×C3⋊D312yes—
432C22×C3C23×C3⋊D311yescharacteristic
433C2×D3—24no—
434C2×D3—916no—
435C2×D3—94no—
436C2×D3—1816no—
437C2×D3—1816no—
438C2×D3—184no—
439C2×D3—3616no—
440C2×D3C23×C3⋊D314yes—
441C3×C4—28no—
442C3×C4—28no—
443C3×C4—316no—
444C3×C4—98no—
445C3×C4C22×D3218yes—
446C3×C4C22×D3218yes—
447C3×C4C23×C3⋊D312yes—
448C3×C4C23×C3⋊D312yes—
449Dic3—332no—
450Dic3—916no—
451Dic3—94no—
452Dic3—1816no—
453Dic3C23×C3⋊D314yes—
454C32—24no—
455C32C2×#100711yescharacteristic, complemented
456C32C2×#101014yescomplemented
457C23—32no—
458C23—272no—
459C23C2×D3×C3⋊D311yescharacteristic
460Q8—34no—
461Q8—98no—
462Q8—98no—
463Q8—2716no—
464Q8—274no—
465D4—38no—
466D4—34no—
467D4—2716no—
468D4—278no—
469D4—274no—
470C2×C4—32no—
471C2×C4—38no—
472C2×C4—98no—
473C2×C4—94no—
474C2×C4—2716no—
475C2×C4—2716no—
476C2×C4—272no—
477C2×C4—278no—
478C2×C4C2×D3×C3⋊D314yes—
479C2×C4C2×D3×C3⋊D311yescharacteristic
480C2×C4C2×D3×C3⋊D311yescharacteristic
481C2×C3—28no—
482C2×C3—22no—
483C2×C3—28no—
484C2×C3—24no—
485C2×C3—48no—
486C2×C3—616no—
487C2×C3#6429718yescomplemented
488C2×C3#6436712yescomplemented
489C2×C3C23×D3214yes—
490C2×C3C24×C3⋊D311yescharacteristic
491D3—24no—
492D3—1816no—
493D3—184no—
494D3—3616no—
495C22—22no—
496C22—34no—
497C22—64no—
498C22—274no—
499C22—544no—
500C22C22×D3×C3⋊D312yes—
501C22C22×D3×C3⋊D311yescharacteristic
502C4—34no—
503C4—98no—
504C4—274no—
505C4C22×D3×C3⋊D312yes—
506C4C22×D3×C3⋊D312yes—
507C3—24no—
508C3C2×#6429714yescomplemented
509C3C2×#6436711yescharacteristic, complemented
510C2—22no—
511C2—64no—
512C2—544no—
513C2#1172456212yesdirect product factor
514C2C23×D3×C3⋊D311yescharacteristic
subgroupG/H
centerC22C22×D3×C3⋊D3
commutator, cosocleC2×C33C25
FrattiniC2C23×D3×C3⋊D3
FittingC22×C33×C4C22
socleC22×C33C24
hypercenterC22×C4D3×C3⋊D3
hypocenterC33C2×2−1+4
LCSG ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
1. C2×#89076C2semidirect
2. C22×C3×#634C2semidirect
3. C2×C32×#235C2semidirect
4. C2×#89028C2semidirect
5. C2×#89022C2semidirect
6. C2×#89019C2semidirect
7. #11724562C2direct (N × H)
8. C2×#89015C2semidirect
9. C2×D3×#634C2semidirect
14. #89076C22semidirect
21. C2×C3×#634C22semidirect
23. C32×#235C22semidirect
30. #89028C22semidirect
32. #89022C22semidirect
33. #89019C22semidirect
34. #89015C22semidirect
35. D3×#634C22semidirect
46. C22×#634D3semidirect
47. C2×C3×#235D3semidirect
116. C2×#634C2×D3semidirect
119. C3×#235C2×D3semidirect
163. C2×#235C3⋊D3semidirect
226. C2×C332−1+4semidirect
261. #235C2×C3⋊D3semidirect
325. C33C2×2−1+4semidirect
390. C2×C32#1007semidirect
391. C2×C32#1010semidirect
455. C32C2×#1007semidirect
456. C32C2×#1010semidirect
487. C2×C3#64297semidirect
488. C2×C3#64367semidirect
508. C3C2×#64297semidirect
509. C3C2×#64367semidirect
513. C2#11724562direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#890762
C22×C3×#6342
C2×C32×#2352
C2×#890282
C2×#890222
C2×#890192
#117245622
C2×#890152
C2×D3×#6342
C2×#643673
C2×#642973