Finite Group Databasebeta

← #13178720

Subgroups — #13178720

Order 1728.

1072 conjugacy classes of subgroups.

The subgroup lattice has 576 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#64155C211yesmaximal, characteristic
2C3×#64153C212yesmaximal, complemented
3#11723688C212yesmaximal
4#11723683C212yesmaximal
5#11723682C211yesmaximal, characteristic
6#11723679C212yesmaximal, complemented
7#11723671C211yesmaximal, characteristic
8#11723669C212yesmaximal, complemented
9#11723666C212yesmaximal, complemented
10#10595375—31nomaximal
11#10595369—32nomaximal
12C2×C33×Q8C2211yescharacteristic
13C2×C3×C4×C3⋊D3C2211yescharacteristic
14C2×C3×#634C2212yes—
15C2×#6296C2211yescharacteristic
16C2×#6294C2211yescharacteristic
17C2×#6294C2211yescharacteristic
18C3×#3752C2212yes—
19C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
20C3×#3750C2212yes—
21C3×#3750C2212yes—
22C3×#3750C2212yes—
23C3×#3750C2212yes—
24C3×#3750C2212yes—
25C3×C4×C3⋊Dic3C2212yes—
26#88815C2212yes—
27#88815C2212yes—
28#88815C2212yes—
29#88815C2212yes—
30#88815C2211yescharacteristic
31#88814C2212yes—
32#88813C2212yes—
33#64155D311yescharacteristic
34#64153D312yescomplemented
35C3×#946—32no—
36C3×#946—61no—
37C3×#944—32no—
38C3×#944—32no—
39C3×#944—62no—
40#63896—32no—
41#63888—32no—
42#63887—32no—
43#63883—31no—
44#63878—32no—
45#63877—32no—
46D3×Dic3⋊C4—32no—
47#63870—32no—
48#63867—32no—
49#63867—32no—
50#63865—31no—
51#63851—32no—
52#63847—32no—
53#63846—32no—
54#63845—32no—
55#63840—32no—
56#63839—32no—
57#63839—32no—
58#63836—32no—
59C22×C3×C3⋊D3C2311yescharacteristic
60C33×Q8—41no—
61C2×C33×C4C2312yes—
62C2×C33×C4C2311yescharacteristic
63C2×C3×C3⋊Dic3C2312yes—
64C2×C3×C3⋊Dic3C2312yes—
65C2×C3×C3⋊Dic3C2312yes—
66C2×C3×C3⋊Dic3C2311yescharacteristic
67C2×C3×C3⋊Dic3C2312yes—
68C2×C3×C3⋊Dic3C2312yes—
69C3×C4×C3⋊D3—41no—
70C3×#634—42no—
71#6296—41no—
72#6294—41no—
73#6294Q814yescomplemented
74#5729—92no—
75#5623—91no—
76C2×C32×Q8—22no—
77C2×C32×Q8—21no—
78C2×C32×Q8—41no—
79C2×C32×Q8C2×D311yescharacteristic
80C2×C32×Q8C2×D312yes—
81C2×C4×C3⋊D3C2×D311yescharacteristic
82C2×#634C2×D312yes—
83C2×C3×C4×D3—32no—
84C2×C3×C4×D3—61no—
85C2×C3×Dic6—32no—
86C2×C3×Dic6—32no—
87C2×C3×Dic6—62no—
88C2×#627—31no—
89C2×#627—32no—
90C2×#624—31no—
91C2×#624—31no—
92C2×D3×Dic3—32no—
93C2×D3×Dic3—32no—
94#3752C2×D312yes—
95C4⋊(C3⋊Dic3)C2×D311yescharacteristic
96#3750C2×D312yes—
97#3750C2×D312yes—
98#3750C2×D312yes—
99#3750C2×D312yes—
100#3750C2×D312yes—
101C4×C3⋊Dic3C2×D312yes—
102C3×(C2×D3)⋊C4—32no—
103C3×(C2×D3)⋊C4—32no—
104C3×(C2×D3)⋊C4—62no—
105C3×C4⋊Dic3—32no—
106C3×C4⋊Dic3—61no—
107C3×Dic3⋊C4—32no—
108C3×Dic3⋊C4—32no—
109C3×Dic3⋊C4—32no—
110C3×Dic3⋊C4—32no—
111C3×Dic3⋊C4—32no—
112C3×Dic3⋊C4—32no—
113C3×Dic3⋊C4—32no—
114C3×Dic3⋊C4—32no—
115C3×Dic3⋊C4—32no—
116C3×Dic3⋊C4—32no—
117C3×Dic3⋊C4—62no—
118C3×Dic3⋊C4—62no—
119C3×Dic3⋊C4—62no—
120C3×Dic3⋊C4—62no—
121C3×Dic3⋊C4—62no—
122C3×C4×Dic3—32no—
123C3×C4×Dic3—32no—
124C3×C4×Dic3—62no—
125#3724—32no—
126#3724—32no—
127#3724—32no—
128#3724—32no—
129#3724—31no—
130Dic3⋊Dic3—32no—
131Dic3⋊Dic3—32no—
132Dic3⋊Dic3—32no—
133Dic3⋊Dic3—32no—
134Dic3⋊Dic3—32no—
135Dic3⋊Dic3—32no—
136Dic3⋊Dic3—32no—
137Dic3⋊Dic3—32no—
138Dic3⋊Dic3—32no—
139#3722—32no—
140(C2×D3)⋊Dic3—32no—
141(C2×D3)⋊Dic3—32no—
142Dic32—32no—
143Dic32—32no—
144Dic32—32no—
145C22×C33C2411yescharacteristic
146C2×C3×C3⋊D3—21no—
147C2×C3×C3⋊D3—21no—
148C2×C3×C3⋊D3C2×Q812yes—
149C33×C4—22no—
150C33×C4—21no—
151C3×C3⋊Dic3—22no—
152C3×C3⋊Dic3—22no—
153C3×C3⋊Dic3—21no—
154C3×C3⋊Dic3—22no—
155C3×C3⋊Dic3—22no—
156C3×C3⋊Dic3C2×Q814yes—
157C3×C4⋊Q8—92no—
158C3×C22⋊Q8—91no—
159#946—32no—
160#946—61no—
161#946—92no—
162#944—32no—
163#944—32no—
164#944—62no—
165#944—92no—
166#924—91no—
167C2×Dic3⋊C4—91no—
168#896—92no—
169#896—92no—
170#892—92no—
171D3×C4⋊C4—92no—
172#890—92no—
173#889—92no—
174#889—92no—
175#889—92no—
176#889—92no—
177#888—92no—
178#888—92no—
179#878—92no—
180#877—92no—
181#872—92no—
182#870—92no—
183C22×C3⋊D3C22×D311yescharacteristic
184C22×C3×D3—32no—
185C22×C3×D3—61no—
186C32×Q8—42no—
187C32×Q8—41no—
188C32×Q8—44no—
189C32×Q8—42no—
190C32×Q8—44no—
191C2×C32×C4—22no—
192C2×C32×C4—21no—
193C2×C32×C4—22no—
194C2×C32×C4—22no—
195C2×C32×C4—22no—
196C2×C32×C4—42no—
197C2×C32×C4—41no—
198C2×C32×C4C22×D312yes—
199C2×C32×C4C22×D312yes—
200C2×C32×C4C22×D311yescharacteristic
201C2×C32×C4C22×D312yes—
202C2×C32×C4C22×D312yes—
203C2×C3⋊Dic3C22×D312yes—
204C2×C3⋊Dic3C22×D312yes—
205C2×C3⋊Dic3C22×D312yes—
206C2×C3⋊Dic3C22×D312yes—
207C2×C3⋊Dic3C22×D311yescharacteristic
208C2×C3⋊Dic3C22×D312yes—
209C4×C3⋊D3—41no—
210#634—42no—
211C2×C3×Dic3—32no—
212C2×C3×Dic3—32no—
213C2×C3×Dic3—32no—
214C2×C3×Dic3—32no—
215C2×C3×Dic3—32no—
216C2×C3×Dic3—32no—
217C2×C3×Dic3—32no—
218C2×C3×Dic3—32no—
219C2×C3×Dic3—32no—
220C2×C3×Dic3—32no—
221C2×C3×Dic3—32no—
222C2×C3×Dic3—62no—
223C2×C3×Dic3—62no—
224C2×C3×Dic3—62no—
225C2×C3×Dic3—62no—
226C2×C3×Dic3—61no—
227C2×C3×Dic3—62no—
228C3×C4×D3—122no—
229C3×C4×D3—122no—
230C3×Dic6—122no—
231C3×Dic6—122no—
232C3×Dic6—124no—
233#627—121no—
234#627—122no—
235#624—34no—
236#624—121no—
237D3×Dic3—38no—
238D3×Dic3—122no—
239#544—271noSylow
240C2×C33C22×Q811yescharacteristic
241C2×C33H(5,𝔽2)11yescharacteristic
242C2×C332−1+411yescharacteristic
243C3×C3⋊D3—22no—
244C2×C3×Q8—22no—
245C2×C3×Q8—21no—
246C2×C3×Q8—41no—
247C2×C3×Q8—92no—
248C2×C3×Q8D3211yescharacteristic
249C2×C3×Q8D3212yes—
250C22×C3×C4—91no—
251C22×Dic3—91no—
252C22×Dic3—91no—
253C2×C4×D3—32no—
254C2×C4×D3—61no—
255C2×C4×D3—92no—
256C2×C4×D3—92no—
257C2×Dic6—32no—
258C2×Dic6—32no—
259C2×Dic6—62no—
260C2×Dic6—91no—
261C2×Dic6—92no—
262C3×C4⋊C4—92no—
263C3×C4⋊C4—92no—
264C3×C4⋊C4—91no—
265C3×C4⋊C4—92no—
266C3×C4⋊C4—92no—
267C3×C4⋊C4—92no—
268C3×C22⋊C4—92no—
269C3×C42—92no—
270#217—92no—
271(C2×D3)⋊C4—32no—
272(C2×D3)⋊C4—32no—
273(C2×D3)⋊C4—62no—
274(C2×D3)⋊C4—92no—
275(C2×D3)⋊C4—92no—
276C4⋊Dic3—32no—
277C4⋊Dic3—61no—
278C4⋊Dic3—92no—
279C4⋊Dic3—92no—
280C4⋊Dic3—92no—
281C4⋊Dic3—91no—
282C4⋊Dic3—92no—
283Dic3⋊C4—32no—
284Dic3⋊C4—32no—
285Dic3⋊C4—32no—
286Dic3⋊C4—32no—
287Dic3⋊C4—32no—
288Dic3⋊C4—32no—
289Dic3⋊C4—32no—
290Dic3⋊C4—32no—
291Dic3⋊C4—32no—
292Dic3⋊C4—32no—
293Dic3⋊C4—62no—
294Dic3⋊C4—62no—
295Dic3⋊C4—62no—
296Dic3⋊C4—62no—
297Dic3⋊C4—62no—
298Dic3⋊C4—92no—
299Dic3⋊C4—92no—
300Dic3⋊C4—92no—
301Dic3⋊C4—92no—
302Dic3⋊C4—92no—
303Dic3⋊C4—92no—
304Dic3⋊C4—92no—
305Dic3⋊C4—92no—
306Dic3⋊C4—92no—
307C4×Dic3—32no—
308C4×Dic3—32no—
309C4×Dic3—62no—
310C4×Dic3—92no—
311C4×Dic3—92no—
312C4×Dic3—92no—
313C22×C32—22no—
314C22×C32—21no—
315C22×C32—41no—
316C22×C32C23×D312yes—
317C22×C32C23×D311yescharacteristic
318C2×C3⋊D3—21no—
319C2×C3⋊D3—21no—
320C2×C3⋊D3D3×Q812yes—
321C2×C3×D3—34no—
322C2×C3×D3—62no—
323C2×C3×D3—62no—
324C2×C3×D3—62no—
325C2×C3×D3—121no—
326C2×C3×D3—121no—
327C32×C4—22no—
328C32×C4—21no—
329C32×C4—24no—
330C32×C4—22no—
331C32×C4—22no—
332C32×C4—22no—
333C32×C4—22no—
334C32×C4—42no—
335C32×C4—42no—
336C32×C4—42no—
337C32×C4—44no—
338C32×C4—42no—
339C3⋊Dic3—22no—
340C3⋊Dic3—22no—
341C3⋊Dic3—22no—
342C3⋊Dic3—21no—
343C3⋊Dic3—22no—
344C3⋊Dic3D3×Q814yes—
345C3×Dic3—34no—
346C3×Dic3—34no—
347C3×Dic3—64no—
348C3×Dic3—64no—
349C3×Dic3—62no—
350C3×Dic3—62no—
351C3×Dic3—62no—
352C3×Dic3—62no—
353C3×Dic3—62no—
354C3×Dic3—62no—
355C3×Dic3—62no—
356C3×Dic3—62no—
357C3×Dic3—62no—
358C3×Dic3—122no—
359C3×Dic3—122no—
360C3×Dic3—121no—
361C3×Dic3—122no—
362C4⋊Q8—92no—
363C4⋊Q8—272no—
364#125—272no—
365#125—272no—
366C22⋊Q8—91no—
367C22⋊Q8—272no—
368C22⋊Q8—271no—
369#117—272no—
370C2×C4⋊C4—271no—
371C33#54411yescharacteristic, complemented, Sylow
372C23×C3—91no—
373C22×D3—32no—
374C22×D3—61no—
375C3×Q8—41no—
376C3×Q8—42no—
377C3×Q8—44no—
378C3×Q8—42no—
379C3×Q8—44no—
380C3×Q8—362no—
381C2×C3×C4—22no—
382C2×C3×C4—21no—
383C2×C3×C4—22no—
384C2×C3×C4—22no—
385C2×C3×C4—22no—
386C2×C3×C4—42no—
387C2×C3×C4—41no—
388C2×C3×C4—92no—
389C2×C3×C4—92no—
390C2×C3×C4—92no—
391C2×C3×C4—92no—
392C2×C3×C4—91no—
393C2×C3×C4—92no—
394C2×C3×C4—361no—
395C2×C3×C4C2×D3212yes—
396C2×C3×C4C2×D3212yes—
397C2×C3×C4C2×D3211yescharacteristic
398C2×C3×C4C2×D3212yes—
399C2×C3×C4C2×D3212yes—
400C2×Dic3—32no—
401C2×Dic3—32no—
402C2×Dic3—32no—
403C2×Dic3—32no—
404C2×Dic3—32no—
405C2×Dic3—32no—
406C2×Dic3—32no—
407C2×Dic3—32no—
408C2×Dic3—32no—
409C2×Dic3—32no—
410C2×Dic3—32no—
411C2×Dic3—62no—
412C2×Dic3—62no—
413C2×Dic3—62no—
414C2×Dic3—62no—
415C2×Dic3—61no—
416C2×Dic3—62no—
417C2×Dic3—94no—
418C2×Dic3—361no—
419C4×D3—98no—
420C4×D3—122no—
421C4×D3—122no—
422C4×D3—362no—
423Dic6—122no—
424Dic6—122no—
425Dic6—124no—
426Dic6—361no—
427Dic6—362no—
428C2×C32—22no—
429C2×C32—21no—
430C2×C32—22no—
431C2×C32—21no—
432C2×C32—22no—
433C2×C32—21no—
434C2×C32—41no—
435C2×C32—41no—
436C2×C32—41no—
437C2×C32#100412yes—
438C2×C32#100411yescharacteristic
439C2×C32C2×D3×Q812yes—
440C2×C32C2×D3×Q811yescharacteristic
441C2×C32#100712yes—
442C2×C32#101011yescharacteristic
443C3⋊D3—22no—
444C3×D3—64no—
445C3×D3—122no—
446C2×Q8—92no—
447C2×Q8—271no—
448C2×Q8#111311yescharacteristic
449C22×C4—91no—
450C22×C4—271no—
451C22×C4—271no—
452C4⋊C4—92no—
453C4⋊C4—92no—
454C4⋊C4—91no—
455C4⋊C4—92no—
456C4⋊C4—92no—
457C4⋊C4—92no—
458C4⋊C4—272no—
459C4⋊C4—272no—
460C4⋊C4—271no—
461C4⋊C4—272no—
462C4⋊C4—272no—
463C22⋊C4—92no—
464C22⋊C4—272no—
465C42—92no—
466C42—272no—
467C22×C3—22no—
468C22×C3—21no—
469C22×C3—41no—
470C22×C3—92no—
471C22×C3—181no—
472C22×C3—181no—
473C22×C3C22×D3212yes—
474C22×C3C22×D3211yescharacteristic
475C2×D3—34no—
476C2×D3—62no—
477C2×D3—62no—
478C2×D3—62no—
479C2×D3—121no—
480C2×D3—121no—
481C3×C4—24no—
482C3×C4—22no—
483C3×C4—22no—
484C3×C4—22no—
485C3×C4—22no—
486C3×C4—21no—
487C3×C4—22no—
488C3×C4—42no—
489C3×C4—42no—
490C3×C4—42no—
491C3×C4—42no—
492C3×C4—44no—
493C3×C4—94no—
494C3×C4—182no—
495C3×C4—182no—
496C3×C4—181no—
497C3×C4—182no—
498C3×C4—182no—
499Dic3—34no—
500Dic3—34no—
501Dic3—64no—
502Dic3—64no—
503Dic3—62no—
504Dic3—62no—
505Dic3—62no—
506Dic3—62no—
507Dic3—62no—
508Dic3—62no—
509Dic3—62no—
510Dic3—62no—
511Dic3—62no—
512Dic3—122no—
513Dic3—122no—
514Dic3—121no—
515Dic3—122no—
516C32—22no—
517C32—21no—
518C32—41no—
519C32#562311yescharacteristic, complemented
520C32#572912yescomplemented
521C23—91no—
522Q8—41no—
523Q8—362no—
524Q8—1081no—
525C2×C4—92no—
526C2×C4—92no—
527C2×C4—92no—
528C2×C4—92no—
529C2×C4—91no—
530C2×C4—92no—
531C2×C4—274no—
532C2×C4—361no—
533C2×C4—1081no—
534C2×C4C2×#111312yes—
535C2×C4C2×#111311yescharacteristic
536C2×C3—22no—
537C2×C3—21no—
538C2×C3—22no—
539C2×C3—21no—
540C2×C3—22no—
541C2×C3—21no—
542C2×C3—41no—
543C2×C3—41no—
544C2×C3—41no—
545C2×C3—182no—
546C2×C3#6431512yes—
547C2×C3#6431611yescharacteristic
548C2×C3D32×Q812yes—
549C2×C3D32×Q811yescharacteristic
550C2×C3#6433012yes—
551C2×C3#6433011yescharacteristic
552D3—64no—
553D3—122no—
554C22—92no—
555C22—181no—
556C22—181no—
557C22C22×#111311yescharacteristic
558C4—21no—
559C4—22no—
560C4—94no—
561C4—182no—
562C4—182no—
563C4—181no—
564C4—182no—
565C4—182no—
566C3—21no—
567C3—22no—
568C3—41no—
569C3#1059536912yescomplemented
570C3#1059537511yescharacteristic, complemented
571C2—182no—
572C2#1172462411yescharacteristic
573C2Q8×#111311yescharacteristic
574C2#1172463311yescharacteristic
subgroupG/H
center, FrattiniC22C22×#1113
commutator, socle, cosocleC22×C33C24
FittingC2×C33×Q8C22
prime radicalC22×C3×C3⋊D3C23
hypercenterC2×Q8#1113
hypocenterC33#544
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×Q8

Complements

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

NH ≅ G/Ntype
2. C3×#64153C2semidirect
6. #11723679C2semidirect
8. #11723669C2semidirect
9. #11723666C2semidirect
34. #64153D3semidirect
73. #6294Q8semidirect
371. C33#544semidirect
519. C32#5623semidirect
520. C32#5729semidirect
569. C3#10595369semidirect
570. C3#10595375semidirect

Maximal subgroups

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

subgroupindex
C3×#641552
C3×#641532
#117236882
#117236832
#117236822
#117236792
#117236712
#117236692
#117236662
#105953753
#105953693