Finite Group Databasebeta

← #13178743

Subgroups — #13178743

Order 1728.

1588 conjugacy classes of subgroups.

The subgroup lattice has 562 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#3832C211yesmaximal, characteristic, complemented
2C2×C3×Q8×C3⋊D3C211yesmaximal, characteristic, complemented
3C3×Q8×C3⋊Dic3C211yesmaximal, characteristic, complemented
4#11723696C213yesmaximal, complemented
5C4×#6295C213yesmaximal, complemented
6#11723677C213yesmaximal, complemented
7#11723674C213yesmaximal, complemented
8#10595414—31nomaximal
9#10595409—31nomaximal
10#10595342—31nomaximal
11C2×C33×Q8C2211yescharacteristic
12C3×#3832—24no—
13C3×Q8×C3⋊D3—24no—
14C2×C3×C4⋊(C3⋊D3)C2213yes—
15C2×C3×C4×C3⋊D3C2213yes—
16C2×C3×C4×C3⋊D3C2213yes—
17C2×C3×#634C2213yes—
18C2×#6295C2213yes—
19C3×C4⋊(C3⋊Dic3)C2213yes—
20C3×C4×C3⋊Dic3C2213yes—
21#88815C2213yes—
22#88814C2216yes—
23#88814C2213yes—
24#88814C2211yescharacteristic
25C2×#3832D311yescharacteristic, complemented
26C2×Q8×C3⋊D3D311yescharacteristic, complemented
27C2×C3×#239—31no—
28C2×C3×#239—31no—
29C2×C3×#239—61no—
30C2×C3×D3×Q8—31no—
31C2×C3×D3×Q8—31no—
32C2×C3×D3×Q8—61no—
33Q8×C3⋊Dic3D311yescharacteristic, complemented
34C3×Q8×Dic3—31no—
35C3×Q8×Dic3—31no—
36C3×Q8×Dic3—61no—
37#63916—33no—
38#63909—33no—
39#63908—33no—
40C4×#626—33no—
41C4×#626—33no—
42C4×#625—33no—
43#63871—33no—
44#63864—33no—
45#63863—33no—
46#63860—33no—
47#63857—33no—
48#63856—33no—
49C22×C3×C3⋊D3C2313yes—
50C22×C3×C3⋊D3C2311yescharacteristic
51C33×Q8D414yescomplemented
52C2×C33×C4C2313yes—
53C2×C3×C3⋊Dic3C2311yescharacteristic
54C2×C3×C3⋊Dic3C2313yes—
55C2×C3×C3⋊Dic3C2313yes—
56C2×C3×C3⋊Dic3C2311yescharacteristic
57C3×C4⋊(C3⋊D3)—26no—
58C3×C4×C3⋊D3—26no—
59C3×C4×C3⋊D3—26no—
60C3×#634—26no—
61#6295—26no—
62#5831—91no—
63#5817—91no—
64#5576—91no—
65C2×C32×Q8—21no—
66C2×C32×Q8—21no—
67C2×C32×Q8—21no—
68C2×C32×Q8—41no—
69C2×C32×Q8C2×D311yescharacteristic
70C2×C32×Q8C2×D311yescharacteristic
71C2×C32×Q8C2×D311yescharacteristic
72#3832—24no—
73Q8×C3⋊D3—24no—
74C2×C4⋊(C3⋊D3)C2×D313yes—
75C2×C4×C3⋊D3C2×D313yes—
76C2×C4×C3⋊D3C2×D313yes—
77C2×#634C2×D313yes—
78C3×#239—68no—
79C3×#239—64no—
80C3×#239—64no—
81C3×D3×Q8—68no—
82C3×D3×Q8—64no—
83C3×D3×Q8—64no—
84C2×C3×D12—33no—
85C2×C3×D12—33no—
86C2×C3×D12—63no—
87C2×C3×C4×D3—33no—
88C2×C3×C4×D3—33no—
89C2×C3×C4×D3—33no—
90C2×C3×C4×D3—33no—
91C2×C3×C4×D3—63no—
92C2×C3×C4×D3—63no—
93C2×C3×Dic6—33no—
94C2×C3×Dic6—33no—
95C2×C3×Dic6—63no—
96C2×#626—33no—
97C2×#626—33no—
98C2×#625—33no—
99C4⋊(C3⋊Dic3)C2×D313yes—
100C4×C3⋊Dic3C2×D313yes—
101C3×C4⋊Dic3—33no—
102C3×C4⋊Dic3—33no—
103C3×C4⋊Dic3—63no—
104C3×C4×Dic3—33no—
105C3×C4×Dic3—33no—
106C3×C4×Dic3—63no—
107#3724—33no—
108Dic3⋊Dic3—33no—
109Dic3⋊Dic3—33no—
110#3722—33no—
111#3722—36no—
112#3722—31no—
113(C2×D3)⋊Dic3—36no—
114(C2×D3)⋊Dic3—36no—
115(C2×D3)⋊Dic3—33no—
116(C2×D3)⋊Dic3—33no—
117(C2×D3)⋊Dic3—31no—
118(C2×D3)⋊Dic3—31no—
119C22×C33C2411yescharacteristic
120C2×C3×C3⋊D3—21no—
121C2×C3×C3⋊D3—21no—
122C2×C3×C3⋊D3—23no—
123C2×C3×C3⋊D3—23no—
124C2×C3×C3⋊D3—23no—
125C2×C3×C3⋊D3#4112yescomplemented
126C33×C4C2×D416yes—
127C3×C3⋊Dic3—23no—
128C3×C3⋊Dic3—21no—
129C3×C3⋊Dic3—23no—
130C3×C3⋊Dic3—21no—
131C2×C3×#41—91no—
132C22×C3×Q8—91no—
133C2×#239—31no—
134C2×#239—31no—
135C2×#239—61no—
136C2×D3×Q8—31no—
137C2×D3×Q8—31no—
138C2×D3×Q8—61no—
139C3×C4×Q8—91no—
140#947—93no—
141#946—93no—
142Q8×Dic3—31no—
143Q8×Dic3—31no—
144Q8×Dic3—61no—
145#938—93no—
146#936—93no—
147#930—93no—
148C4×#67—93no—
149C4×#67—93no—
150#924—93no—
151#897—93no—
152#895—93no—
153#875—93no—
154C4×D12—93no—
155C22×C3⋊D3C22×D313yes—
156C22×C3⋊D3C22×D311yescharacteristic
157C22×C3×D3—31no—
158C22×C3×D3—31no—
159C22×C3×D3—33no—
160C22×C3×D3—33no—
161C22×C3×D3—61no—
162C22×C3×D3—63no—
163C32×Q8—24no—
164C32×Q8—24no—
165C32×Q8—24no—
166C32×Q8—44no—
167C32×Q8D1214yescomplemented
168C32×Q8#6714yescomplemented
169C32×Q8#6714yescomplemented
170C2×C32×C4—23no—
171C2×C32×C4—23no—
172C2×C32×C4—23no—
173C2×C32×C4—43no—
174C2×C32×C4C22×D313yes—
175C2×C32×C4C22×D313yes—
176C2×C32×C4C22×D313yes—
177C2×C3⋊Dic3C22×D311yescharacteristic
178C2×C3⋊Dic3C22×D313yes—
179C2×C3⋊Dic3C22×D313yes—
180C2×C3⋊Dic3C22×D311yescharacteristic
181C4⋊(C3⋊D3)—26no—
182C4×C3⋊D3—26no—
183C4×C3⋊D3—26no—
184#634—26no—
185C2×C3×Dic3—31no—
186C2×C3×Dic3—33no—
187C2×C3×Dic3—33no—
188C2×C3×Dic3—31no—
189C2×C3×Dic3—31no—
190C2×C3×Dic3—33no—
191C2×C3×Dic3—33no—
192C2×C3×Dic3—31no—
193C2×C3×Dic3—61no—
194C2×C3×Dic3—63no—
195C2×C3×Dic3—61no—
196C2×C3×Dic3—63no—
197C3×D12—612no—
198C3×D12—66no—
199C3×D12—66no—
200C3×C4×D3—612no—
201C3×C4×D3—612no—
202C3×C4×D3—66no—
203C3×C4×D3—66no—
204C3×C4×D3—66no—
205C3×C4×D3—66no—
206C3×Dic6—612no—
207C3×Dic6—66no—
208C3×Dic6—66no—
209#626—66no—
210#626—66no—
211#625—66no—
212#548—271noSylow
213C2×C33C22×D411yescharacteristic
214C2×C33C2×#4111yescharacteristic
215C2×C332−1+411yescharacteristic
216C3×C3⋊D3—22no—
217C3×C3⋊D3—43no—
218C3×#41—184no—
219C2×C3×Q8—21no—
220C2×C3×Q8—21no—
221C2×C3×Q8—21no—
222C2×C3×Q8—41no—
223C2×C3×Q8—93no—
224C2×C3×Q8—184no—
225C2×C3×Q8D3211yescharacteristic
226C2×C3×Q8D3211yescharacteristic
227C2×C3×Q8D3211yescharacteristic
228C2×C3×D4—93no—
229C22×C3×C4—93no—
230C22×C3×C4—93no—
231C2×#67—93no—
232C2×#67—93no—
233#239—68no—
234#239—64no—
235#239—64no—
236D3×Q8—68no—
237D3×Q8—64no—
238D3×Q8—64no—
239C2×D12—33no—
240C2×D12—33no—
241C2×D12—63no—
242C2×D12—93no—
243C2×C4×D3—33no—
244C2×C4×D3—33no—
245C2×C4×D3—33no—
246C2×C4×D3—33no—
247C2×C4×D3—63no—
248C2×C4×D3—63no—
249C2×Dic6—33no—
250C2×Dic6—33no—
251C2×Dic6—63no—
252C3×C4⋊C4—93no—
253C3×C42—93no—
254#217—96no—
255#217—91no—
256#217—93no—
257(C2×D3)⋊C4—91no—
258(C2×D3)⋊C4—93no—
259(C2×D3)⋊C4—96no—
260(C2×D3)⋊C4—91no—
261(C2×D3)⋊C4—93no—
262(C2×D3)⋊C4—96no—
263C4⋊Dic3—33no—
264C4⋊Dic3—33no—
265C4⋊Dic3—63no—
266C4⋊Dic3—93no—
267Dic3⋊C4—93no—
268Dic3⋊C4—93no—
269C4×Dic3—33no—
270C4×Dic3—33no—
271C4×Dic3—63no—
272C22×C32—21no—
273C22×C32—21no—
274C22×C32—21no—
275C22×C32—41no—
276C22×C32C23×D311yescharacteristic
277C22×C32C23×D311yescharacteristic
278C22×C32C23×D311yescharacteristic
279C2×C3⋊D3—21no—
280C2×C3⋊D3—21no—
281C2×C3⋊D3—23no—
282C2×C3⋊D3—23no—
283C2×C3⋊D3—23no—
284C2×C3⋊D3#23912yescomplemented
285C2×C3×D3—32no—
286C2×C3×D3—32no—
287C2×C3×D3—62no—
288C2×C3×D3—62no—
289C2×C3×D3—62no—
290C2×C3×D3—61no—
291C2×C3×D3—61no—
292C2×C3×D3—61no—
293C2×C3×D3—61no—
294C2×C3×D3—66no—
295C2×C3×D3—63no—
296C2×C3×D3—63no—
297C2×C3×D3—63no—
298C2×C3×D3—63no—
299C2×C3×D3—63no—
300C2×C3×D3—63no—
301C2×C3×D3—123no—
302C2×C3×D3—123no—
303C32×C4—26no—
304C32×C4—26no—
305C32×C4—26no—
306C32×C4—46no—
307C32×C4C2×D1216yes—
308C32×C4C2×#6716yes—
309C32×C4C2×#6716yes—
310C3⋊Dic3—21no—
311C3⋊Dic3—23no—
312C3⋊Dic3—21no—
313C3⋊Dic3—23no—
314C3×Dic3—66no—
315C3×Dic3—62no—
316C3×Dic3—62no—
317C3×Dic3—63no—
318C3×Dic3—61no—
319C3×Dic3—63no—
320C3×Dic3—61no—
321C3×Dic3—63no—
322C3×Dic3—61no—
323C3×Dic3—63no—
324C3×Dic3—61no—
325C3×Dic3—123no—
326C2×#41—91no—
327C22×Q8—91no—
328#124—273no—
329C22⋊Q8—273no—
330#121—273no—
331C4×Q8—91no—
332C4×D4—273no—
333C33#54811yescharacteristic, complemented, Sylow
334C23×C3—91no—
335C23×C3—93no—
336C22×D3—31no—
337C22×D3—31no—
338C22×D3—33no—
339C22×D3—33no—
340C22×D3—61no—
341C22×D3—63no—
342C3×Q8—24no—
343C3×Q8—24no—
344C3×Q8—24no—
345C3×Q8—44no—
346C3×Q8—186no—
347C3×Q8#62514yescomplemented
348C3×Q8#62614yescomplemented
349C3×Q8#62614yescomplemented
350C3×D4—186no—
351C2×C3×C4—23no—
352C2×C3×C4—23no—
353C2×C3×C4—23no—
354C2×C3×C4—43no—
355C2×C3×C4—91no—
356C2×C3×C4—93no—
357C2×C3×C4—93no—
358C2×C3×C4—91no—
359C2×C3×C4—186no—
360C2×C3×C4—186no—
361C2×C3×C4C2×D3213yes—
362C2×C3×C4C2×D3213yes—
363C2×C3×C4C2×D3213yes—
364#67—186no—
365#67—186no—
366C2×Dic3—31no—
367C2×Dic3—31no—
368C2×Dic3—33no—
369C2×Dic3—33no—
370C2×Dic3—31no—
371C2×Dic3—33no—
372C2×Dic3—31no—
373C2×Dic3—33no—
374C2×Dic3—61no—
375C2×Dic3—63no—
376C2×Dic3—61no—
377C2×Dic3—63no—
378D12—612no—
379D12—66no—
380D12—66no—
381D12—186no—
382C4×D3—612no—
383C4×D3—612no—
384C4×D3—66no—
385C4×D3—66no—
386C4×D3—66no—
387C4×D3—66no—
388Dic6—612no—
389Dic6—66no—
390Dic6—66no—
391C2×C32—21no—
392C2×C32—21no—
393C2×C32—21no—
394C2×C32—21no—
395C2×C32—21no—
396C2×C32—21no—
397C2×C32—21no—
398C2×C32—21no—
399C2×C32—21no—
400C2×C32—41no—
401C2×C32—41no—
402C2×C32—41no—
403C2×C32C22×D1211yescharacteristic
404C2×C32C2×#23911yescharacteristic
405C2×C32#100711yescharacteristic
406C2×C32#100711yescharacteristic
407C2×C32D3×#4111yescharacteristic
408C2×C32D3×#4111yescharacteristic
409C2×C32#101011yescharacteristic
410C2×C32C22×#6711yescharacteristic
411C2×C32C22×#6711yescharacteristic
412C3⋊D3—22no—
413C3⋊D3—43no—
414C3×D3—64no—
415C3×D3—62no—
416C3×D3—62no—
417C3×D3—126no—
418C3×D3—123no—
419C3×D3—123no—
420#41—184no—
421C2×Q8—93no—
422C2×Q8—184no—
423C2×Q8#111311yescharacteristic
424C2×D4—93no—
425C2×D4—273no—
426C22×C4—93no—
427C22×C4—93no—
428C4⋊C4—93no—
429C4⋊C4—273no—
430C22⋊C4—271no—
431C22⋊C4—273no—
432C22⋊C4—276no—
433C42—93no—
434C22×C3—21no—
435C22×C3—21no—
436C22×C3—21no—
437C22×C3—41no—
438C22×C3—92no—
439C22×C3—181no—
440C22×C3—181no—
441C22×C3—183no—
442C22×C3—183no—
443C22×C3—183no—
444C22×C3C22×D3211yescharacteristic
445C22×C3C22×D3211yescharacteristic
446C22×C3C22×D3211yescharacteristic
447C2×D3—32no—
448C2×D3—32no—
449C2×D3—62no—
450C2×D3—62no—
451C2×D3—62no—
452C2×D3—61no—
453C2×D3—61no—
454C2×D3—61no—
455C2×D3—61no—
456C2×D3—66no—
457C2×D3—63no—
458C2×D3—63no—
459C2×D3—63no—
460C2×D3—63no—
461C2×D3—63no—
462C2×D3—63no—
463C2×D3—123no—
464C2×D3—123no—
465C3×C4—26no—
466C3×C4—26no—
467C3×C4—26no—
468C3×C4—46no—
469C3×C4—181no—
470C3×C4—183no—
471C3×C4—181no—
472C3×C4—183no—
473C3×C4C2×#62516yes—
474C3×C4C2×#62616yes—
475C3×C4C2×#62616yes—
476Dic3—66no—
477Dic3—62no—
478Dic3—62no—
479Dic3—61no—
480Dic3—61no—
481Dic3—63no—
482Dic3—63no—
483Dic3—63no—
484Dic3—61no—
485Dic3—63no—
486Dic3—61no—
487Dic3—123no—
488C32—21no—
489C32—21no—
490C32—21no—
491C32—41no—
492C32#557611yescharacteristic, complemented
493C32#581711yescharacteristic, complemented
494C32#583111yescharacteristic, complemented
495C23—91no—
496C23—93no—
497Q8—186no—
498Q8#629514yescomplemented
499D4—186no—
500D4—546no—
501C2×C4—91no—
502C2×C4—93no—
503C2×C4—93no—
504C2×C4—91no—
505C2×C4—186no—
506C2×C4—186no—
507C2×C4C2×#111313yes—
508C2×C3—21no—
509C2×C3—21no—
510C2×C3—21no—
511C2×C3—21no—
512C2×C3—21no—
513C2×C3—21no—
514C2×C3—21no—
515C2×C3—21no—
516C2×C3—21no—
517C2×C3—41no—
518C2×C3—41no—
519C2×C3—41no—
520C2×C3—182no—
521C2×C3—363no—
522C2×C3#6431511yescharacteristic
523C2×C3#6431511yescharacteristic
524C2×C3#6431611yescharacteristic
525C2×C3D3×#23911yescharacteristic
526C2×C3D3×#23911yescharacteristic
527C2×C3#6431911yescharacteristic
528C2×C3C22×#62511yescharacteristic
529C2×C3C22×#62611yescharacteristic
530C2×C3C22×#62611yescharacteristic
531D3—64no—
532D3—62no—
533D3—62no—
534D3—126no—
535D3—123no—
536D3—123no—
537C22—92no—
538C22—181no—
539C22—181no—
540C22—183no—
541C22—183no—
542C22—183no—
543C22C22×#111311yescharacteristic
544C4—181no—
545C4—183no—
546C4—181no—
547C4—183no—
548C4C2×#629516yes—
549C3—21no—
550C3—21no—
551C3—21no—
552C3—41no—
553C3#1059534211yescharacteristic, complemented
554C3#1059540911yescharacteristic, complemented
555C3#1059541411yescharacteristic, complemented
556C2—182no—
557C2—363no—
558C2#1172462411yescharacteristic
559C2#1172462611yescharacteristic
560C2C22×#629511yescharacteristic
subgroupG/H
center, FrattiniC22C22×#1113
commutator, socle, cosocleC22×C33C24
FittingC2×C33×Q8C22
hypercenterC2×Q8#1113
hypocenterC33#548
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×Q8

Complements

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

NH ≅ G/Ntype
1. C2×C3×#3832C2semidirect
2. C2×C3×Q8×C3⋊D3C2semidirect
3. C3×Q8×C3⋊Dic3C2semidirect
4. #11723696C2semidirect
5. C4×#6295C2semidirect
6. #11723677C2semidirect
7. #11723674C2semidirect
25. C2×#3832D3semidirect
26. C2×Q8×C3⋊D3D3semidirect
33. Q8×C3⋊Dic3D3semidirect
51. C33×Q8D4semidirect
125. C2×C3×C3⋊D3#41semidirect
167. C32×Q8D12semidirect
168. C32×Q8#67semidirect
169. C32×Q8#67semidirect
284. C2×C3⋊D3#239semidirect
333. C33#548semidirect
347. C3×Q8#625semidirect
348. C3×Q8#626semidirect
349. C3×Q8#626semidirect
492. C32#5576semidirect
493. C32#5817semidirect
494. C32#5831semidirect
498. Q8#6295semidirect
553. C3#10595342semidirect
554. C3#10595409semidirect
555. C3#10595414semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#38322
C2×C3×Q8×C3⋊D32
C3×Q8×C3⋊Dic32
#117236962
C4×#62952
#117236772
#117236742
#105954143
#105954093
#105953423