Finite Group Databasebeta

← #13169396

Subgroups — #13169396

Order 1728.

982 conjugacy classes of subgroups.

The subgroup lattice has 596 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11723671C211yesmaximal, characteristic
2#11723404C211yesmaximal, characteristic
3#11723400C212yesmaximal, complemented
4C3×#63865C211yesmaximal, characteristic
5C3×#63847C212yesmaximal, complemented
6#11723035C211yesmaximal, characteristic, complemented
7#11723027C212yesmaximal
8#11723023C212yesmaximal
9#11723018C211yesmaximal, characteristic, complemented
10#11722945C211yesmaximal, characteristic, complemented
11Dic3×#627C211yesmaximal, characteristic, complemented
12#10594599—31nomaximal
13#10594336—32nomaximal
14C2×C3×C4×C3⋊D3C2211yescharacteristic
15C2×#6296C2211yescharacteristic
16C2×C3×#627C2211yescharacteristic
17C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
18C3×#3750C2212yes—
19C32×C4⋊Dic3C2211yescharacteristic
20C32×Dic3⋊C4C2212yes—
21#88815C2211yescharacteristic
22#88814C2212yes—
23#88806C2212yes—
24Dic3⋊(C3⋊Dic3)C2212yes—
25Dic3⋊(C3⋊Dic3)C2212yes—
26#88803C2211yescharacteristic
27#88803C2211yescharacteristic
28Dic3×C3⋊Dic3C2211yescharacteristic
29Dic3×C3⋊Dic3C2212yes—
30Dic3×C3⋊Dic3C2211yescharacteristic
31C3×#3724C2212yes—
32C3×#3724C2211yescharacteristic
33C3×#3724C2212yes—
34C3×Dic3⋊Dic3C2212yes—
35C3×#3722C2212yes—
36C3×Dic32C2212yes—
37#64101—31no—
38#64099—32no—
39C3×#896—32no—
40C3×#889—32no—
41C3×#889—32no—
42#63919—31no—
43C4×#627—31no—
44#63870—31no—
45#63868—32no—
46#63865—31no—
47#63865D311yescharacteristic
48#63862—32no—
49#63861—32no—
50#63861—32no—
51#63859—32no—
52#63851—32no—
53#63848—32no—
54#63847D312yescomplemented
55#63845—32no—
56#63845—32no—
57#63845—32no—
58#63845—62no—
59Dic3×Dic6—32no—
60#63841—32no—
61#63841—61no—
62#63839—32no—
63#63838—32no—
64#63838—31no—
65C22×C3×C3⋊D3C2311yescharacteristic
66C2×C33×C4C2311yescharacteristic
67C2×C3⋊(C3⋊Dic3)C2311yescharacteristic
68C2×C3⋊(C3⋊Dic3)C2311yescharacteristic
69C2×C3×C3⋊Dic3C2312yes—
70C2×C3×C3⋊Dic3C2312yes—
71C2×C3×C3⋊Dic3C2311yescharacteristic
72C3×C4×C3⋊D3—22no—
73C2×C32×Dic3C2312yes—
74C2×C32×Dic3C2311yescharacteristic
75C2×C32×Dic3C2312yes—
76C2×C32×Dic3C2311yescharacteristic
77#6296—22no—
78C3×#627—22no—
79#5696—92no—
80#5625—91no—
81C2×C4×C3⋊D3C2×D311yescharacteristic
82C2×C3×C4×D3—32no—
83C2×C3×C4×D3—61no—
84C2×C3×Dic6—32no—
85C2×#627—32no—
86C2×#627—31no—
87C2×#627C2×D311yescharacteristic
88C32×C4⋊C4—31no—
89C32×C4⋊C4—32no—
90#3752—31no—
91#3752—31no—
92C4⋊(C3⋊Dic3)—32no—
93C4⋊(C3⋊Dic3)—31no—
94C4⋊(C3⋊Dic3)—32no—
95C4⋊(C3⋊Dic3)—32no—
96C4⋊(C3⋊Dic3)—61no—
97C4⋊(C3⋊Dic3)—62no—
98C4⋊(C3⋊Dic3)—121no—
99#3750—32no—
100#3750C2×D312yes—
101C4×C3⋊Dic3—32no—
102C4×C3⋊Dic3—31no—
103C4×C3⋊Dic3—31no—
104C3×(C2×D3)⋊C4—32no—
105C3×(C2×D3)⋊C4—32no—
106C3×C4⋊Dic3—21no—
107C3×C4⋊Dic3—32no—
108C3×C4⋊Dic3C2×D312yes—
109C3×Dic3⋊C4—22no—
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—32no—
118C3×Dic3⋊C4—62no—
119C3×Dic3⋊C4C2×D312yes—
120C3×Dic3⋊C4C2×D312yes—
121C3×C4×Dic3—32no—
122C3×C4×Dic3—32no—
123#3724—31no—
124#3724—32no—
125#3724—32no—
126#3724—62no—
127#3724C2×D312yes—
128#3724C2×D311yescharacteristic
129#3724C2×D312yes—
130Dic3⋊Dic3—32no—
131Dic3⋊Dic3—32no—
132Dic3⋊Dic3—32no—
133Dic3⋊Dic3—32no—
134Dic3⋊Dic3—32no—
135Dic3⋊Dic3—62no—
136Dic3⋊Dic3—62no—
137Dic3⋊Dic3C2×D312yes—
138#3722—32no—
139#3722C2×D312yes—
140(C2×D3)⋊Dic3—32no—
141(C2×D3)⋊Dic3—32no—
142(C2×D3)⋊Dic3—32no—
143(C2×D3)⋊Dic3—32no—
144(C2×D3)⋊Dic3—61no—
145(C2×D3)⋊Dic3—61no—
146Dic32—32no—
147Dic32—32no—
148Dic32—32no—
149Dic32—32no—
150Dic32—61no—
151Dic32—61no—
152Dic32—62no—
153Dic32C2×D312yes—
154C22×C33C2411yescharacteristic
155C2×C3×C3⋊D3—21no—
156C2×C3×C3⋊D3—21no—
157C2×C3×C3⋊D3—21no—
158C33×C4—21no—
159C3⋊(C3⋊Dic3)—21no—
160C3⋊(C3⋊Dic3)—21no—
161C3×C3⋊Dic3—22no—
162C3×C3⋊Dic3—22no—
163C3×C3⋊Dic3#4112yes—
164C32×Dic3—22no—
165C32×Dic3—22no—
166C32×Dic3—21no—
167C32×Dic3—21no—
168C3×#125—92no—
169C3×C22⋊Q8—91no—
170Q8×Dic3—91no—
171#944—91no—
172#926—91no—
173#924—91no—
174#898—92no—
175#898—92no—
176#898—92no—
177#896—32no—
178#896—92no—
179#892—92no—
180#892—181no—
181#890—91no—
182#890—92no—
183#890—92no—
184#890—182no—
185#889—32no—
186#889—32no—
187#889—92no—
188#889—92no—
189#888—92no—
190#887—92no—
191#887—91no—
192#879—92no—
193#879—92no—
194#876—92no—
195C4×Dic6—92no—
196C22×C3⋊D3C22×D311yescharacteristic
197C22×C3×D3—32no—
198C22×C3×D3—61no—
199C2×C32×C4—21no—
200C2×C32×C4—22no—
201C2×C32×C4—31no—
202C2×C32×C4—32no—
203C2×C32×C4—32no—
204C2×C32×C4—31no—
205C2×C32×C4—41no—
206C2×C32×C4C22×D311yescharacteristic
207C2×C32×C4C22×D312yes—
208C2×C3⋊Dic3—32no—
209C2×C3⋊Dic3—31no—
210C2×C3⋊Dic3—32no—
211C2×C3⋊Dic3—31no—
212C2×C3⋊Dic3—61no—
213C2×C3⋊Dic3—61no—
214C2×C3⋊Dic3—62no—
215C2×C3⋊Dic3—62no—
216C2×C3⋊Dic3—121no—
217C2×C3⋊Dic3—121no—
218C2×C3⋊Dic3C22×D312yes—
219C2×C3⋊Dic3C22×D312yes—
220C2×C3⋊Dic3C22×D311yescharacteristic
221C4×C3⋊D3—22no—
222C2×C3×Dic3—22no—
223C2×C3×Dic3—21no—
224C2×C3×Dic3—21no—
225C2×C3×Dic3—22no—
226C2×C3×Dic3—32no—
227C2×C3×Dic3—32no—
228C2×C3×Dic3—32no—
229C2×C3×Dic3—32no—
230C2×C3×Dic3—32no—
231C2×C3×Dic3—61no—
232C2×C3×Dic3—62no—
233C2×C3×Dic3—62no—
234C2×C3×Dic3C22×D312yes—
235C2×C3×Dic3C22×D312yes—
236C2×C3×Dic3C22×D312yes—
237C2×C3×Dic3C22×D312yes—
238C2×C3×Dic3C22×D312yes—
239C2×C3×Dic3C22×D312yes—
240C3×C4×D3—64no—
241C3×C4×D3—64no—
242C3×Dic6—64no—
243#627—22no—
244#627—64no—
245#627—62no—
246#541—271noSylow
247C2×C33C2×#4111yescharacteristic
248C2×C332−1+411yescharacteristic
249C2×C332−1+411yescharacteristic
250C3×C3⋊D3—41no—
251C2×C3×Q8—91no—
252C22×C3×C4—91no—
253C2×C4×D3—32no—
254C2×C4×D3—61no—
255C2×Dic6—32no—
256C2×Dic6—92no—
257C2×Dic6—91no—
258C3×C4⋊C4—32no—
259C3×C4⋊C4—32no—
260C3×C4⋊C4—32no—
261C3×C4⋊C4—62no—
262C3×C4⋊C4—61no—
263C3×C4⋊C4—91no—
264C3×C4⋊C4—92no—
265C3×C4⋊C4—92no—
266C3×C4⋊C4—92no—
267C3×C4⋊C4—92no—
268C3×C22⋊C4—92no—
269C3×C42—92no—
270#217—92no—
271#217—91no—
272#217—91no—
273(C2×D3)⋊C4—32no—
274(C2×D3)⋊C4—32no—
275(C2×D3)⋊C4—92no—
276(C2×D3)⋊C4—92no—
277(C2×D3)⋊C4—92no—
278(C2×D3)⋊C4—92no—
279(C2×D3)⋊C4—181no—
280(C2×D3)⋊C4—181no—
281C4⋊Dic3—32no—
282C4⋊Dic3—91no—
283C4⋊Dic3—92no—
284C4⋊Dic3—92no—
285C4⋊Dic3—92no—
286C4⋊Dic3—92no—
287C4⋊Dic3—92no—
288C4⋊Dic3—91no—
289C4⋊Dic3—182no—
290C4⋊Dic3—182no—
291C4⋊Dic3—182no—
292C4⋊Dic3—181no—
293C4⋊Dic3—361no—
294C4⋊Dic3D3211yescharacteristic
295Dic3⋊C4—32no—
296Dic3⋊C4—32no—
297Dic3⋊C4—32no—
298Dic3⋊C4—32no—
299Dic3⋊C4—32no—
300Dic3⋊C4—32no—
301Dic3⋊C4—32no—
302Dic3⋊C4—32no—
303Dic3⋊C4—62no—
304Dic3⋊C4—92no—
305Dic3⋊C4—92no—
306Dic3⋊C4—92no—
307Dic3⋊C4—92no—
308Dic3⋊C4—92no—
309Dic3⋊C4—92no—
310Dic3⋊C4—182no—
311Dic3⋊C4D3212yes—
312C4×Dic3—32no—
313C4×Dic3—32no—
314C4×Dic3—92no—
315C4×Dic3—92no—
316C4×Dic3—92no—
317C4×Dic3—92no—
318C4×Dic3—92no—
319C4×Dic3—91no—
320C4×Dic3—91no—
321C4×Dic3—182no—
322C4×Dic3—181no—
323C4×Dic3—181no—
324C22×C32—22no—
325C22×C32—21no—
326C22×C32—41no—
327C22×C32C23×D311yescharacteristic
328C22×C32C23×D312yes—
329C2×C3⋊D3—21no—
330C2×C3⋊D3—21no—
331C2×C3⋊D3—21no—
332C2×C3×D3—62no—
333C2×C3×D3—62no—
334C2×C3×D3—62no—
335C2×C3×D3—62no—
336C2×C3×D3—121no—
337C2×C3×D3—121no—
338C32×C4—21no—
339C32×C4—22no—
340C32×C4—22no—
341C32×C4—42no—
342C32×C4—42no—
343C32×C4—62no—
344C32×C4—62no—
345C32×C4—61no—
346C32×C4—61no—
347C3⋊Dic3—22no—
348C3⋊Dic3—22no—
349C3⋊Dic3—62no—
350C3⋊Dic3—62no—
351C3⋊Dic3—61no—
352C3⋊Dic3—61no—
353C3⋊Dic3—122no—
354C3⋊Dic3—122no—
355C3⋊Dic3—121no—
356C3⋊Dic3—121no—
357C3⋊Dic3—241no—
358C3⋊Dic3—241no—
359C3⋊Dic3#23712yes—
360C3×Dic3—24no—
361C3×Dic3—22no—
362C3×Dic3—22no—
363C3×Dic3—22no—
364C3×Dic3—22no—
365C3×Dic3—22no—
366C3×Dic3—22no—
367C3×Dic3—34no—
368C3×Dic3—41no—
369C3×Dic3—41no—
370C3×Dic3—42no—
371C3×Dic3—62no—
372C3×Dic3—62no—
373C3×Dic3—62no—
374C3×Dic3—62no—
375C3×Dic3—62no—
376C3×Dic3—122no—
377C3×Dic3—122no—
378C4⋊Q8—271no—
379#126—272no—
380#126—272no—
381#125—92no—
382#125—272no—
383#125—271no—
384C22⋊Q8—91no—
385C22⋊Q8—271no—
386C4×Q8—271no—
387C4×Q8—271no—
388#117—271no—
389C33#54111yescharacteristic, complemented, Sylow
390C23×C3—91no—
391C22×D3—32no—
392C22×D3—61no—
393C3×Q8—182no—
394C2×C3×C4—21no—
395C2×C3×C4—22no—
396C2×C3×C4—32no—
397C2×C3×C4—32no—
398C2×C3×C4—32no—
399C2×C3×C4—32no—
400C2×C3×C4—32no—
401C2×C3×C4—32no—
402C2×C3×C4—41no—
403C2×C3×C4—62no—
404C2×C3×C4—61no—
405C2×C3×C4—61no—
406C2×C3×C4—62no—
407C2×C3×C4—91no—
408C2×C3×C4—92no—
409C2×C3×C4—92no—
410C2×C3×C4—182no—
411C2×C3×C4C2×D3211yescharacteristic
412C2×C3×C4C2×D3212yes—
413C2×Dic3—32no—
414C2×Dic3—32no—
415C2×Dic3—32no—
416C2×Dic3—32no—
417C2×Dic3—32no—
418C2×Dic3—61no—
419C2×Dic3—62no—
420C2×Dic3—62no—
421C2×Dic3—92no—
422C2×Dic3—91no—
423C2×Dic3—92no—
424C2×Dic3—91no—
425C2×Dic3—182no—
426C2×Dic3—181no—
427C2×Dic3—181no—
428C2×Dic3—182no—
429C2×Dic3—361no—
430C2×Dic3—361no—
431C2×Dic3C2×D3211yescharacteristic
432C2×Dic3C2×D3212yes—
433C2×Dic3C2×D3211yescharacteristic
434C2×Dic3C2×D3212yes—
435C4×D3—64no—
436C4×D3—64no—
437Dic6—64no—
438Dic6—184no—
439Dic6—182no—
440C2×C32—22no—
441C2×C32—21no—
442C2×C32—21no—
443C2×C32—22no—
444C2×C32—22no—
445C2×C32—21no—
446C2×C32—41no—
447C2×C32—41no—
448C2×C32—41no—
449C2×C32C2×#23711yescharacteristic
450C2×C32#100712yes—
451C2×C32#100711yescharacteristic
452C2×C32D3×#4112yes—
453C2×C32#101011yescharacteristic
454C2×C32#101012yes—
455C3⋊D3—41no—
456C3×D3—122no—
457C3×D3—122no—
458C2×Q8—91no—
459C2×Q8—271no—
460C22×C4—91no—
461C4⋊C4—31no—
462C4⋊C4—32no—
463C4⋊C4—91no—
464C4⋊C4—92no—
465C4⋊C4—92no—
466C4⋊C4—92no—
467C4⋊C4—92no—
468C4⋊C4—272no—
469C4⋊C4—272no—
470C4⋊C4—272no—
471C4⋊C4—271no—
472C4⋊C4—271no—
473C22⋊C4—92no—
474C22⋊C4—271no—
475C22⋊C4—271no—
476C22⋊C4—272no—
477C42—92no—
478C42—271no—
479C42—271no—
480C42—272no—
481C22×C3—22no—
482C22×C3—21no—
483C22×C3—41no—
484C22×C3—181no—
485C22×C3—181no—
486C22×C3—181no—
487C22×C3C22×D3212yes—
488C22×C3C22×D3211yescharacteristic
489C2×D3—62no—
490C2×D3—62no—
491C2×D3—62no—
492C2×D3—62no—
493C2×D3—121no—
494C2×D3—121no—
495C3×C4—22no—
496C3×C4—21no—
497C3×C4—22no—
498C3×C4—42no—
499C3×C4—42no—
500C3×C4—64no—
501C3×C4—62no—
502C3×C4—62no—
503C3×C4—62no—
504C3×C4—62no—
505C3×C4—62no—
506C3×C4—62no—
507C3×C4—92no—
508C3×C4—121no—
509C3×C4—121no—
510C3×C4—122no—
511C3×C4—182no—
512C3×C4—182no—
513Dic3—22no—
514Dic3—21no—
515Dic3—21no—
516Dic3—22no—
517Dic3—34no—
518Dic3—62no—
519Dic3—62no—
520Dic3—62no—
521Dic3—62no—
522Dic3—62no—
523Dic3—122no—
524Dic3—122no—
525Dic3—182no—
526Dic3—182no—
527Dic3—181no—
528Dic3—181no—
529Dic3—362no—
530Dic3—362no—
531Dic3—361no—
532Dic3—361no—
533Dic3—721no—
534Dic3—721no—
535C32—22no—
536C32—21no—
537C32—41no—
538C32#562511yescharacteristic, complemented
539C32#569612yescomplemented
540C23—91no—
541Q8—182no—
542Q8—542no—
543C2×C4—31no—
544C2×C4—32no—
545C2×C4—32no—
546C2×C4—31no—
547C2×C4—91no—
548C2×C4—92no—
549C2×C4—92no—
550C2×C4—182no—
551C2×C4—271no—
552C2×C4—271no—
553C2×C4D3311yescharacteristic
554C2×C3—22no—
555C2×C3—21no—
556C2×C3—22no—
557C2×C3—21no—
558C2×C3—21no—
559C2×C3—22no—
560C2×C3—41no—
561C2×C3—41no—
562C2×C3—41no—
563C2×C3—361no—
564C2×C3#6429812yes—
565C2×C3#6430611yescharacteristic
566C2×C3#6430911yescharacteristic
567C2×C3D3×#23712yes—
568C2×C3#6431612yes—
569C2×C3#6431611yescharacteristic
570D3—122no—
571D3—122no—
572C22—181no—
573C22—181no—
574C22—181no—
575C22C2×D3311yescharacteristic
576C4—21no—
577C4—62no—
578C4—61no—
579C4—61no—
580C4—62no—
581C4—92no—
582C4—182no—
583C4—182no—
584C4—541no—
585C4—541no—
586C3—21no—
587C3—22no—
588C3—41no—
589C3#1059433612yescomplemented
590C3#1059459911yescharacteristic, complemented
591C2—361no—
592C2#1172439211yescharacteristic
593C2#1172439911yescharacteristic
594C2#1172440711yescharacteristic
subgroupG/H
center, FrattiniC22C2×D33
commutator, socle, cosocleC22×C33C24
FittingC2×C33×C4C23
prime radicalC22×C3×C3⋊D3C23
hypercenterC2×C4D33
hypocenterC33#541
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
3. #11723400C2semidirect
5. C3×#63847C2semidirect
6. #11723035C2semidirect
9. #11723018C2semidirect
10. #11722945C2semidirect
11. Dic3×#627C2semidirect
54. #63847D3semidirect
389. C33#541semidirect
538. C32#5625semidirect
539. C32#5696semidirect
589. C3#10594336semidirect
590. C3#10594599semidirect

Maximal subgroups

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

subgroupindex
#117236712
#117234042
#117234002
C3×#638652
C3×#638472
#117230352
#117230272
#117230232
#117230182
#117229452
Dic3×#6272
#105945993
#105943363