Finite Group Databasebeta

← #13175192

Subgroups — #13175192

Order 1728.

2020 conjugacy classes of subgroups.

The subgroup lattice has 548 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#89021C211yesmaximal, characteristic, complemented
2C2×D3×#634C211yesmaximal, characteristic, complemented
3C4×#6308C211yesmaximal, characteristic, complemented
4C3×#64084C211yesmaximal, characteristic, complemented
5C32×C4×D12C211yesmaximal, characteristic, complemented
6#11723530C212yesmaximal, complemented
7#11723515C211yesmaximal, characteristic, complemented
8#11723508C212yesmaximal, complemented
9#11723502C212yesmaximal, complemented
10#11723474C212yesmaximal, complemented
11#11723425C211yesmaximal, characteristic, complemented
12#10596023—31nomaximal
13#10593905—34nomaximal
14C2×#6308C2211yescharacteristic
15C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
16C2×C3×#634C2211yescharacteristic
17C2×C32×D12C2211yescharacteristic
18C2×C32×C4×D3C2211yescharacteristic
19C2×C32×C4×D3C2211yescharacteristic
20C2×#6293C2212yes—
21C2×#6290C2212yes—
22C2×#6290C2212yes—
23C2×Dic3×C3⋊D3C2212yes—
24C2×D3×C3⋊Dic3C2212yes—
25#89021—24no—
26D3×#634—24no—
27C33×C42C2211yescharacteristic
28C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
29#88857C2212yes—
30C4×C3⋊(C3⋊Dic3)C2212yes—
31C3×#3752C2212yes—
32C3×#3752C2212yes—
33C32×(C2×D3)⋊C4C2211yescharacteristic
34C32×(C2×D3)⋊C4C2211yescharacteristic
35C32×C4⋊Dic3C2211yescharacteristic
36#88806C2212yes—
37#88803C2212yes—
38(C2×D3)⋊(C3⋊Dic3)C2212yes—
39C2×#3828—31no—
40C22×#634—31no—
41C2×#3795—34no—
42C2×D3×Dic6—34no—
43C32×C4×D4—31no—
44#64148—32no—
45#64139—31no—
46#64133—31no—
47#64105—32no—
48#64094—32no—
49#64090—32no—
50#64084D311yescharacteristic, complemented
51C4×#634—31no—
52C4×#634—34no—
53C4×#634—64no—
54C3×#875—34no—
55C3×C4×D12D314yescomplemented
56#63915—38no—
57#63908—34no—
58#63902—38no—
59#63899—38no—
60#63885—38no—
61#63856—34no—
62C22×C3×C3⋊D3C2312yes—
63C22×C32×D3C2311yescharacteristic
64C22×C32×D3C2311yescharacteristic
65C2×C33×C4C2311yescharacteristic
66C2×C33×C4C2311yescharacteristic
67C2×C33×C4C2311yescharacteristic
68C2×C3⋊(C3⋊Dic3)C2312yes—
69C2×C3⋊(C3⋊Dic3)C2312yes—
70#6308D414yescomplemented
71C2×C3×C3⋊Dic3C2312yes—
72C3×C4⋊(C3⋊D3)—22no—
73C3×#634—22no—
74C2×C32×Dic3C2311yescharacteristic
75C2×C32×Dic3C2311yescharacteristic
76C32×D12—22no—
77C32×C4×D3—22no—
78C32×C4×D3—22no—
79#6293—24no—
80#6290—24no—
81#6290—24no—
82Dic3×C3⋊D3—24no—
83D3×C3⋊Dic3—24no—
84#5677—91no—
85#5552—94no—
86C2×C32×D4—31no—
87C22×C32×C4—31no—
88C22×C32×C4—31no—
89C2×#638—32no—
90C2×#638—32no—
91C22×C3⋊Dic3—32no—
92#3828—64no—
93C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
94C2×C4×C3⋊D3—32no—
95C2×#634—31no—
96C2×#634—34no—
97C2×#634—32no—
98C2×#634—64no—
99C2×#634—64no—
100C2×#634C2×D311yescharacteristic
101C2×C3×D12—34no—
102C2×C3×D12C2×D314yes—
103C2×C3×C4×D3C2×D314yes—
104C2×C3×C4×D3C2×D314yes—
105C2×C3×Dic6—34no—
106C2×#627—38no—
107C2×#625—38no—
108C2×#625—38no—
109C2×D3×Dic3—38no—
110C2×D3×Dic3—38no—
111#3795—616no—
112D3×Dic6—616no—
113C32×C4⋊C4—31no—
114C32×C22⋊C4—31no—
115C32×C22⋊C4—31no—
116C32×C42—24no—
117C32×C42C2×D314yes—
118C32×C42C2×D311yescharacteristic
119#3757—32no—
120#3752—32no—
121#3752C2×D312yes—
122#3752C2×D312yes—
123C4⋊(C3⋊Dic3)—31no—
124C4⋊(C3⋊Dic3)—34no—
125C4⋊(C3⋊Dic3)—64no—
126#3750—32no—
127#3750—38no—
128#3750—32no—
129#3750—68no—
130C4×C3⋊Dic3—32no—
131C4×C3⋊Dic3—38no—
132C4×C3⋊Dic3—68no—
133C3×(C2×D3)⋊C4—38no—
134C3×(C2×D3)⋊C4—38no—
135C3×(C2×D3)⋊C4C2×D314yes—
136C3×(C2×D3)⋊C4C2×D314yes—
137C3×C4⋊Dic3C2×D314yes—
138#3724—38no—
139(C2×D3)⋊Dic3—38no—
140(C2×D3)⋊Dic3—38no—
141C22×C33C2411yescharacteristic
142C2×C3×C3⋊D3—22no—
143C2×C3×C3⋊D3—22no—
144C2×C3×C3⋊D3—22no—
145C2×C32×D3—21no—
146C2×C32×D3—21no—
147C2×C32×D3—21no—
148C2×C32×D3—21no—
149C2×C32×D3—21no—
150C2×C32×D3#4112yescomplemented
151C33×C4—21no—
152C33×C4—21no—
153C33×C4C2×D412yes—
154C3⋊(C3⋊Dic3)—22no—
155C3⋊(C3⋊Dic3)C2×D414yes—
156C3×C3⋊Dic3—22no—
157C32×Dic3—21no—
158C32×Dic3—21no—
159C2×D3×Q8—91no—
160C2×#237—91no—
161C2×#235—94no—
162C22×Dic6—94no—
163C3×#124—91no—
164C3×C4×D4—34no—
165#946—91no—
166#939—98no—
167#938—91no—
168#930—94no—
169#924—94no—
170#896—98no—
171#885—98no—
172#885—92no—
173#884—92no—
174#67⋊C4—98no—
175#67⋊C4—92no—
176#878—92no—
177#875—34no—
178C4×D12C3⋊D311yescharacteristic, complemented
179C4×Dic6—94no—
180C4×Dic6—91no—
181C4×Dic6—184no—
182C23×C32—31no—
183C23×C32—31no—
184C22×C3⋊D3C22×D312yes—
185C22×C3×D3—38no—
186C22×C3×D3C22×D314yes—
187C22×C3×D3C22×D314yes—
188C32×D4—62no—
189C2×C32×C4—24no—
190C2×C32×C4—24no—
191C2×C32×C4—24no—
192C2×C32×C4—31no—
193C2×C32×C4—31no—
194C2×C32×C4—62no—
195C2×C32×C4—62no—
196C2×C32×C4C22×D314yes—
197C2×C32×C4C22×D314yes—
198C2×C32×C4C22×D314yes—
199C2×C32×C4C22×D311yescharacteristic
200C2×C32×C4C22×D311yescharacteristic
201C2×C32×C4C22×D311yescharacteristic
202#638—64no—
203#638—64no—
204C2×C3⋊Dic3—32no—
205C2×C3⋊Dic3—32no—
206C2×C3⋊Dic3—38no—
207C2×C3⋊Dic3—38no—
208C2×C3⋊Dic3—68no—
209C2×C3⋊Dic3—68no—
210C2×C3⋊Dic3—64no—
211C2×C3⋊Dic3C22×D312yes—
212C4⋊(C3⋊D3)—22no—
213C4×C3⋊D3—64no—
214#634—22no—
215#634—34no—
216#634—316no—
217#634—616no—
218#634—64no—
219C2×C3×Dic3—38no—
220C2×C3×Dic3C22×D314yes—
221C2×C3×Dic3C22×D314yes—
222C3×D12—28no—
223C3×D12—68no—
224C3×C4×D3—28no—
225C3×C4×D3—28no—
226C3×Dic6—68no—
227#627—616no—
228#625—616no—
229#625—616no—
230D3×Dic3—616no—
231D3×Dic3—616no—
232#548—271noSylow
233C2×C33C22×D411yescharacteristic
234C2×C33C2×#4111yescharacteristic
235C2×C332−1+411yescharacteristic
236C3×C3⋊D3—42no—
237C32×D3—22no—
238C32×D3—41no—
239C2×C3×Q8—91no—
240C2×C3×D4—34no—
241C2×C3×D4—91no—
242C22×C3×C4—34no—
243C22×C3×C4—34no—
244C2×#67—98no—
245C2×#67—98no—
246C2×#67—92no—
247C2×#67—92no—
248C22×Dic3—98no—
249C22×Dic3—92no—
250D3×Q8—184no—
251#237—184no—
252#235—1816no—
253C2×D12—34no—
254C2×D12C2×C3⋊D311yescharacteristic
255C2×C4×D3—98no—
256C2×C4×D3—92no—
257C2×C4×D3C2×C3⋊D311yescharacteristic
258C2×C4×D3C2×C3⋊D311yescharacteristic
259C2×Dic6—34no—
260C2×Dic6—94no—
261C2×Dic6—91no—
262C2×Dic6—98no—
263C2×Dic6—92no—
264C2×Dic6—184no—
265C2×Dic6—1816no—
266C3×C4⋊C4—34no—
267C3×C22⋊C4—34no—
268C3×C22⋊C4—34no—
269C3×C22⋊C4—92no—
270C3×C22⋊C4—92no—
271C3×C42—24no—
272C3×C42D3214yes—
273C3×C42C2×C3⋊D311yescharacteristic
274#217—98no—
275#217—92no—
276(C2×D3)⋊C4—38no—
277(C2×D3)⋊C4—38no—
278(C2×D3)⋊C4—98no—
279(C2×D3)⋊C4—92no—
280(C2×D3)⋊C4C2×C3⋊D311yescharacteristic
281(C2×D3)⋊C4C2×C3⋊D311yescharacteristic
282C4⋊Dic3—94no—
283C4⋊Dic3—91no—
284C4⋊Dic3—184no—
285C4⋊Dic3C2×C3⋊D311yescharacteristic
286Dic3⋊C4—98no—
287Dic3⋊C4—98no—
288Dic3⋊C4—92no—
289Dic3⋊C4—92no—
290Dic3⋊C4—188no—
291C4×Dic3—98no—
292C4×Dic3—92no—
293C4×Dic3—188no—
294C22×C32—24no—
295C22×C32—32no—
296C22×C32—61no—
297C22×C32—61no—
298C22×C32—61no—
299C22×C32—61no—
300C22×C32—61no—
301C22×C32C23×D314yes—
302C22×C32C23×D311yescharacteristic
303C2×C3⋊D3—22no—
304C2×C3⋊D3—22no—
305C2×C3⋊D3—22no—
306C2×C3×D3—24no—
307C2×C3×D3—24no—
308C2×C3×D3—24no—
309C2×C3×D3—24no—
310C2×C3×D3—24no—
311C2×C3×D3—68no—
312C2×C3×D3—68no—
313C2×C3×D3—68no—
314C2×C3×D3#23518yescomplemented
315C32×C4—28no—
316C32×C4—21no—
317C32×C4—24no—
318C32×C4—21no—
319C32×C4—28no—
320C32×C4—24no—
321C32×C4—44no—
322C32×C4—61no—
323C32×C4—61no—
324C32×C4D3×D418yes—
325C32×C4D3×D412yes—
326C3⋊Dic3—22no—
327C3⋊Dic3—34no—
328C3⋊Dic3—316no—
329C3⋊Dic3—616no—
330C3⋊Dic3—62no—
331C3⋊Dic3—68no—
332C3⋊Dic3—128no—
333C3×Dic3—24no—
334C3×Dic3—24no—
335C3×Dic3—68no—
336C2×#41—271no—
337C22×Q8—271no—
338#124—91no—
339#124—272no—
340C22⋊Q8—272no—
341C22⋊Q8—271no—
342#121—272no—
343#121—271no—
344C4×Q8—271no—
345C4×D4—31no—
346C4×D4—272no—
347C33#54811yescharacteristic, complemented, Sylow
348C23×C3—34no—
349C23×C3—34no—
350C23×C3—92no—
351C22×D3—38no—
352C22×D3C22×C3⋊D311yescharacteristic
353C22×D3C22×C3⋊D311yescharacteristic
354C3×Q8—182no—
355C3×D4—68no—
356C3×D4—182no—
357C2×C3×C4—24no—
358C2×C3×C4—24no—
359C2×C3×C4—24no—
360C2×C3×C4—34no—
361C2×C3×C4—34no—
362C2×C3×C4—68no—
363C2×C3×C4—68no—
364C2×C3×C4—92no—
365C2×C3×C4C2×D3214yes—
366C2×C3×C4C2×D3214yes—
367C2×C3×C4C2×D3214yes—
368C2×C3×C4C22×C3⋊D311yescharacteristic
369C2×C3×C4C22×C3⋊D311yescharacteristic
370C2×C3×C4C22×C3⋊D311yescharacteristic
371#67—184no—
372#67—1816no—
373#67—1816no—
374#67—184no—
375C2×Dic3—38no—
376C2×Dic3—98no—
377C2×Dic3—92no—
378C2×Dic3—98no—
379C2×Dic3—92no—
380C2×Dic3—188no—
381C2×Dic3—188no—
382C2×Dic3—184no—
383C2×Dic3—1816no—
384C2×Dic3C22×C3⋊D311yescharacteristic
385C2×Dic3C22×C3⋊D311yescharacteristic
386D12—22no—
387D12—68no—
388C4×D3—22no—
389C4×D3—22no—
390C4×D3—1816no—
391C4×D3—184no—
392Dic6—68no—
393Dic6—916no—
394Dic6—94no—
395Dic6—1816no—
396Dic6—184no—
397Dic6—1816no—
398C2×C32—24no—
399C2×C32—24no—
400C2×C32—24no—
401C2×C32—62no—
402C2×C32—121no—
403C2×C32C2×#23514yes—
404C2×C32C2×D3×D414yes—
405C2×C32C2×D3×D411yescharacteristic
406C2×C32D3×#4111yescharacteristic
407C2×C32#101014yes—
408C2×C32#101011yescharacteristic
409C3⋊D3—42no—
410C3×D3—28no—
411C3×D3—44no—
412C3×D3—128no—
413#41—544no—
414C2×Q8—91no—
415C2×Q8—272no—
416C2×Q8—271no—
417C2×Q8—544no—
418C2×D4—31no—
419C2×D4—91no—
420C2×D4—272no—
421C2×D4—272no—
422C22×C4—31no—
423C22×C4—31no—
424C22×C4—272no—
425C22×C4—272no—
426C4⋊C4—31no—
427C4⋊C4—272no—
428C4⋊C4—272no—
429C4⋊C4—271no—
430C22⋊C4—31no—
431C22⋊C4—31no—
432C22⋊C4—92no—
433C22⋊C4—92no—
434C22⋊C4—272no—
435C22⋊C4—272no—
436C42—272no—
437C42D3×C3⋊D311yescharacteristic
438C22×C3—24no—
439C22×C3—38no—
440C22×C3—64no—
441C22×C3—64no—
442C22×C3—64no—
443C22×C3—64no—
444C22×C3—64no—
445C22×C3—182no—
446C22×C3—182no—
447C22×C3—182no—
448C22×C3C22×D3214yes—
449C22×C3C23×C3⋊D311yescharacteristic
450C2×D3—21no—
451C2×D3—21no—
452C2×D3—21no—
453C2×D3—21no—
454C2×D3—21no—
455C2×D3—68no—
456C2×D3—68no—
457C2×D3—68no—
458C2×D3#382812yescomplemented
459C3×C4—28no—
460C3×C4—24no—
461C3×C4—24no—
462C3×C4—21no—
463C3×C4—21no—
464C3×C4—28no—
465C3×C4—44no—
466C3×C4—64no—
467C3×C4—64no—
468C3×C4—182no—
469C3×C4#380218yes—
470C3×C4D4×C3⋊D312yes—
471Dic3—21no—
472Dic3—21no—
473Dic3—68no—
474Dic3—916no—
475Dic3—94no—
476Dic3—1816no—
477Dic3—188no—
478Dic3—182no—
479Dic3—368no—
480C32—24no—
481C32#555214yescomplemented
482C32#567711yescharacteristic, complemented
483C23—31no—
484C23—31no—
485C23—92no—
486Q8—182no—
487Q8—274no—
488Q8—544no—
489D4—62no—
490D4—182no—
491D4—544no—
492D4—544no—
493C2×C4—31no—
494C2×C4—31no—
495C2×C4—62no—
496C2×C4—62no—
497C2×C4—92no—
498C2×C4—272no—
499C2×C4—272no—
500C2×C4—544no—
501C2×C4—544no—
502C2×C4C2×D3×C3⋊D311yescharacteristic
503C2×C4C2×D3×C3⋊D311yescharacteristic
504C2×C4C2×D3×C3⋊D311yescharacteristic
505C2×C3—24no—
506C2×C3—24no—
507C2×C3—24no—
508C2×C3—68no—
509C2×C3—124no—
510C2×C3—362no—
511C2×C3#6429814yes—
512C2×C3C2×#380214yes—
513C2×C3D3×#23514yes—
514C2×C3C2×#382811yescharacteristic
515C2×C3C2×D4×C3⋊D311yescharacteristic
516C2×C3#6436711yescharacteristic
517D3—22no—
518D3—41no—
519D3—128no—
520C22—32no—
521C22—61no—
522C22—61no—
523C22—61no—
524C22—61no—
525C22—61no—
526C22—182no—
527C22—182no—
528C22—182no—
529C22C22×D3×C3⋊D311yescharacteristic
530C4—21no—
531C4—21no—
532C4—61no—
533C4—61no—
534C4—182no—
535C4—274no—
536C4—542no—
537C4#8902612yes—
538C3—24no—
539C3#1059390514yescomplemented
540C3#1059602311yescharacteristic, complemented
541C2—62no—
542C2—121no—
543C2—362no—
544C2#1172456311yescharacteristic
545C2C2×#8902611yescharacteristic
546C2D3×#382811yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×C3⋊D3
commutator, socle, cosocleC22×C33C24
FittingC33×C42C22
hypercenterC42D3×C3⋊D3
hypocenterC33#548
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22 ◁ C42

Complements

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

NH ≅ G/Ntype
1. C2×#89021C2semidirect
2. C2×D3×#634C2semidirect
3. C4×#6308C2semidirect
4. C3×#64084C2semidirect
5. C32×C4×D12C2semidirect
6. #11723530C2semidirect
7. #11723515C2semidirect
8. #11723508C2semidirect
9. #11723502C2semidirect
10. #11723474C2semidirect
11. #11723425C2semidirect
50. #64084D3semidirect
55. C3×C4×D12D3semidirect
70. #6308D4semidirect
150. C2×C32×D3#41semidirect
178. C4×D12C3⋊D3semidirect
314. C2×C3×D3#235semidirect
347. C33#548semidirect
458. C2×D3#3828semidirect
481. C32#5552semidirect
482. C32#5677semidirect
539. C3#10593905semidirect
540. C3#10596023semidirect

Maximal subgroups

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

subgroupindex
C2×#890212
C2×D3×#6342
C4×#63082
C3×#640842
C32×C4×D122
#117235302
#117235152
#117235082
#117235022
#117234742
#117234252
#105960233
#105939053