Finite Group Databasebeta

← #13150745

Subgroups — #13150745

Order 1728.

832 conjugacy classes of subgroups.

The subgroup lattice has 580 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11721826C211yesmaximal, characteristic, complemented
2#64139⋊C3C211yesmaximal, characteristic
3#64133⋊C3C211yesmaximal, characteristic, complemented
4#11721311C212yesmaximal, complemented
5#11721304C212yesmaximal, complemented
6#11721136C211yesmaximal, characteristic, complemented
7#11721133C212yesmaximal, complemented
8#11721117C211yesmaximal, characteristic, complemented
9#11721113C211yesmaximal, characteristic, complemented
10#11721111C212yesmaximal, complemented
11#11721084C211yesmaximal, characteristic, complemented
12#10594983—31nomaximal
13#10594814—31nomaximal
14C22×C4×H(3,𝔽3)C2211yescharacteristic
15C2×#638⋊C3C2212yes—
16C2×C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic
17C2×#634⋊C3C2211yescharacteristic
18C2×((C3⋊D3)⋊C3)⋊C4C2212yes—
19C2×#6196C2212yes—
20#88552C2211yescharacteristic
21#88552C2211yescharacteristic
22C4⋊#1084C2211yescharacteristic
23C4⋊#1084C2211yescharacteristic
24C4×#1084C2211yescharacteristic
25C4×#1084C2211yescharacteristic
26#3757⋊C3C2212yes—
27#3752⋊C3C2212yes—
28C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
29C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
30#3750⋊C3C2212yes—
31#88450C2212yes—
32#88450C2212yes—
33#88450C2212yes—
34#88449C2212yes—
35#88448C2212yes—
36#88447C2212yes—
37#64139D311yescharacteristic
38#64133D311yescharacteristic, complemented
39C3×#930—31no—
40C3×#926—31no—
41C3×#926—31no—
42C3×#926—61no—
43C3×#924—31no—
44#63975—32no—
45#63969—32no—
46#63959—32no—
47#63953—32no—
48#63918—31no—
49#63918—31no—
50#63899—32no—
51#63895—32no—
52#63894—31no—
53Dic3×D12—31no—
54#63864—31no—
55#63860—31no—
56#63858—32no—
57#63848—32no—
58#63844—31no—
59Dic3×Dic6—31no—
60C23×H(3,𝔽3)C2311yescharacteristic
61C22×(C3⋊D3)⋊C3C2312yes—
62C2×C4×H(3,𝔽3)—22no—
63C2×C4×H(3,𝔽3)C2311yescharacteristic
64C2×C4×H(3,𝔽3)C2311yescharacteristic
65C2×#1084C2311yescharacteristic
66C2×#1084C2311yescharacteristic
67C2×#1084C2311yescharacteristic
68C2×#1084C2311yescharacteristic
69#638⋊C3—42no—
70C2×(C3⋊Dic3)⋊C3C2312yes—
71C2×(C3⋊Dic3)⋊C3C2312yes—
72C2×(C3⋊Dic3)⋊C3C2312yes—
73C4⋊((C3⋊D3)⋊C3)—22no—
74#634⋊C3—22no—
75((C3⋊D3)⋊C3)⋊C4—42no—
76#6196—42no—
77#5643—91no—
78#5603—91no—
79#5534—91no—
80C22×C32×C4—21no—
81C22×C32×C4C2×D311yescharacteristic
82C22×C32×C4C2×D311yescharacteristic
83C2×#638C2×D312yes—
84C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
85C2×#634C2×D311yescharacteristic
86C2×C3×#67—32no—
87C2×C3×D12—31no—
88C2×C3×Dic6—31no—
89C2×#627—32no—
90C2×#627—32no—
91C2×#624—32no—
92C2×D3×Dic3—32no—
93#3757C2×D312yes—
94#3752C2×D312yes—
95C4⋊(C3⋊Dic3)C2×D311yescharacteristic
96C4⋊(C3⋊Dic3)C2×D311yescharacteristic
97#3750C2×D312yes—
98C3×#217—32no—
99C3×#217—31no—
100C3×#217—31no—
101C3×#217—31no—
102C3×#217—31no—
103C3×#217—61no—
104C3×#217—61no—
105C3×(C2×D3)⋊C4—32no—
106C3×C4⋊Dic3—31no—
107C3×C4⋊Dic3—31no—
108C3×C4⋊Dic3—31no—
109C3×C4⋊Dic3—31no—
110C3×C4⋊Dic3—31no—
111C3×C4⋊Dic3—31no—
112C3×C4⋊Dic3—61no—
113C3×C4⋊Dic3—61no—
114C3×Dic3⋊C4—32no—
115C3×C4×Dic3—31no—
116C3×C4×Dic3—31no—
117C3×C4×Dic3—31no—
118C3×C4×Dic3—31no—
119C3×C4×Dic3—61no—
120C3×C4×Dic3—61no—
121#3724—32no—
122Dic3⋊Dic3—32no—
123Dic3⋊Dic3—32no—
124Dic3⋊Dic3—32no—
125#3722—32no—
126#3722—32no—
127#3722—32no—
128(C2×D3)⋊Dic3—32no—
129(C2×D3)⋊Dic3—32no—
130(C2×D3)⋊Dic3—32no—
131Dic32—32no—
132Dic32—32no—
133C22×H(3,𝔽3)—21no—
134C22×H(3,𝔽3)—21no—
135C22×H(3,𝔽3)—21no—
136C22×H(3,𝔽3)C2411yescharacteristic
137C2×(C3⋊D3)⋊C3—22no—
138C2×(C3⋊D3)⋊C3—22no—
139C2×(C3⋊D3)⋊C3—22no—
140C4×H(3,𝔽3)—21no—
141C4×H(3,𝔽3)#4112yes—
142#1084—21no—
143#1084—21no—
144#1084—21no—
145#1084—21no—
146(C3⋊Dic3)⋊C3—22no—
147(C3⋊Dic3)⋊C3—22no—
148(C3⋊Dic3)⋊C3—22no—
149C3×C22⋊Q8—91no—
150C3×#121—91no—
151C3×#117—91no—
152#947—91no—
153Q8×Dic3—91no—
154#936—91no—
155#936—92no—
156#935—92no—
157D4×Dic3—91no—
158#930—31no—
159#930—91no—
160#926—31no—
161#926—31no—
162#926—61no—
163#924—31no—
164#924—91no—
165#898—92no—
166#898—92no—
167#897—92no—
168#896—92no—
169#893—91no—
170#888—91no—
171#888—91no—
172#887—91no—
173#885—92no—
174#885—92no—
175#883—92no—
176#883—92no—
177#879—92no—
178#878—92no—
179#875—91no—
180C4×D12—91no—
181#869—91no—
182C4×Dic6—91no—
183C23×C32—21no—
184C23×C32C22×D311yescharacteristic
185C23×C32C22×D311yescharacteristic
186C22×C3⋊D3C22×D312yes—
187C22×C3×D3—32no—
188C2×C32×C4—24no—
189C2×C32×C4—22no—
190C2×C32×C4—22no—
191C2×C32×C4—21no—
192C2×C32×C4—21no—
193C2×C32×C4C22×D311yescharacteristic
194C2×C32×C4C22×D311yescharacteristic
195C2×C32×C4C22×D311yescharacteristic
196C2×C32×C4C22×D311yescharacteristic
197#638—42no—
198C2×C3⋊Dic3C22×D312yes—
199C2×C3⋊Dic3C22×D312yes—
200C2×C3⋊Dic3C22×D312yes—
201C4⋊(C3⋊D3)—22no—
202#634—22no—
203C3×#67—122no—
204C2×C3×Dic3—32no—
205C2×C3×Dic3—32no—
206C2×C3×Dic3—32no—
207C2×C3×Dic3—31no—
208C2×C3×Dic3—31no—
209C2×C3×Dic3—31no—
210C2×C3×Dic3—31no—
211C2×C3×Dic3—31no—
212C2×C3×Dic3—31no—
213C2×C3×Dic3—31no—
214C2×C3×Dic3—31no—
215C2×C3×Dic3—61no—
216C2×C3×Dic3—61no—
217C2×C3×Dic3—61no—
218C2×C3×Dic3—61no—
219C3×D12—62no—
220C3×Dic6—62no—
221#627—122no—
222#627—122no—
223#624—122no—
224D3×Dic3—122no—
225#542—271noSylow
226C2×H(3,𝔽3)—41no—
227C2×H(3,𝔽3)C2×#4111yescharacteristic
228C2×H(3,𝔽3)H(5,𝔽2)11yescharacteristic
229C2×H(3,𝔽3)2−1+411yescharacteristic
230(C3⋊D3)⋊C3—42no—
231C2×C3×Q8—91no—
232C2×C3×D4—91no—
233C2×C3×D4—92no—
234C22×C3×C4—31no—
235C22×C3×C4—31no—
236C22×C3×C4—61no—
237C22×C3×C4D3211yescharacteristic
238C2×#67—32no—
239C2×#67—92no—
240C22×Dic3—92no—
241C2×D12—31no—
242C2×D12—91no—
243C2×C4×D3—92no—
244C2×C4×D3—92no—
245C2×Dic6—31no—
246C2×Dic6—91no—
247C2×Dic6—92no—
248C2×Dic6—92no—
249C2×Dic6—92no—
250C3×C4⋊C4—91no—
251C3×C4⋊C4—91no—
252C3×C4⋊C4—92no—
253C3×C4⋊C4—91no—
254C3×C4⋊C4—91no—
255C3×C22⋊C4—92no—
256C3×C22⋊C4—92no—
257C3×C22⋊C4—91no—
258C3×C22⋊C4—91no—
259C3×C42—91no—
260C3×C42—91no—
261#217—31no—
262#217—31no—
263#217—31no—
264#217—31no—
265#217—32no—
266#217—61no—
267#217—61no—
268#217—92no—
269#217—92no—
270#217—92no—
271#217—92no—
272(C2×D3)⋊C4—32no—
273(C2×D3)⋊C4—92no—
274(C2×D3)⋊C4—92no—
275(C2×D3)⋊C4—92no—
276(C2×D3)⋊C4—92no—
277(C2×D3)⋊C4—92no—
278(C2×D3)⋊C4—92no—
279(C2×D3)⋊C4—92no—
280C4⋊Dic3—31no—
281C4⋊Dic3—31no—
282C4⋊Dic3—31no—
283C4⋊Dic3—31no—
284C4⋊Dic3—31no—
285C4⋊Dic3—31no—
286C4⋊Dic3—61no—
287C4⋊Dic3—61no—
288C4⋊Dic3—91no—
289C4⋊Dic3—91no—
290C4⋊Dic3—92no—
291C4⋊Dic3—92no—
292Dic3⋊C4—32no—
293Dic3⋊C4—92no—
294Dic3⋊C4—92no—
295Dic3⋊C4—92no—
296Dic3⋊C4—92no—
297Dic3⋊C4—92no—
298C4×Dic3—31no—
299C4×Dic3—31no—
300C4×Dic3—31no—
301C4×Dic3—31no—
302C4×Dic3—61no—
303C4×Dic3—61no—
304C4×Dic3—92no—
305C4×Dic3—92no—
306C4×Dic3—92no—
307C22×C32—22no—
308C22×C32—21no—
309C22×C32—21no—
310C22×C32—21no—
311C22×C32—21no—
312C22×C32—21no—
313C22×C32—21no—
314C22×C32—21no—
315C22×C32—41no—
316C22×C32—41no—
317C22×C32C23×D311yescharacteristic
318C22×C32C23×D311yescharacteristic
319C2×C3⋊D3—22no—
320C2×C3⋊D3—22no—
321C2×C3⋊D3—22no—
322C2×C3×D3—62no—
323C2×C3×D3—62no—
324C2×C3×D3—62no—
325C32×C4—22no—
326C32×C4—21no—
327C32×C4—21no—
328C32×C4—22no—
329C32×C4#23712yes—
330C32×C4#23712yes—
331C3⋊Dic3—22no—
332C3⋊Dic3—22no—
333C3⋊Dic3—22no—
334C3×Dic3—62no—
335C3×Dic3—62no—
336C3×Dic3—61no—
337C3×Dic3—61no—
338C3×Dic3—61no—
339C3×Dic3—61no—
340C3×Dic3—62no—
341C3×Dic3—61no—
342C3×Dic3—61no—
343C3×Dic3—61no—
344C3×Dic3—61no—
345C3×Dic3—121no—
346C3×Dic3—121no—
347C3×Dic3—121no—
348C3×Dic3—121no—
349C4⋊Q8—271no—
350#126—272no—
351#124—271no—
352#124—272no—
353#123—272no—
354C22⋊Q8—91no—
355C22⋊Q8—272no—
356#121—91no—
357C4×Q8—271no—
358C4×D4—271no—
359#117—91no—
360H(3,𝔽3)#54211yescharacteristic, complemented, Sylow
361C23×C3—31no—
362C23×C3—31no—
363C23×C3—61no—
364C23×C3—92no—
365C23×C3C2×D3211yescharacteristic
366C22×D3—32no—
367C22×D3—92no—
368C3×Q8—182no—
369C3×D4—182no—
370C3×D4—362no—
371C2×C3×C4—22no—
372C2×C3×C4—31no—
373C2×C3×C4—31no—
374C2×C3×C4—31no—
375C2×C3×C4—31no—
376C2×C3×C4—62no—
377C2×C3×C4—62no—
378C2×C3×C4—64no—
379C2×C3×C4—61no—
380C2×C3×C4—61no—
381C2×C3×C4—92no—
382C2×C3×C4—92no—
383C2×C3×C4—92no—
384C2×C3×C4—91no—
385C2×C3×C4—91no—
386C2×C3×C4—91no—
387C2×C3×C4—91no—
388C2×C3×C4C2×D3211yescharacteristic
389C2×C3×C4C2×D3211yescharacteristic
390#67—122no—
391#67—362no—
392C2×Dic3—31no—
393C2×Dic3—31no—
394C2×Dic3—31no—
395C2×Dic3—31no—
396C2×Dic3—31no—
397C2×Dic3—31no—
398C2×Dic3—31no—
399C2×Dic3—31no—
400C2×Dic3—32no—
401C2×Dic3—32no—
402C2×Dic3—32no—
403C2×Dic3—61no—
404C2×Dic3—61no—
405C2×Dic3—61no—
406C2×Dic3—61no—
407C2×Dic3—92no—
408C2×Dic3—92no—
409C2×Dic3—92no—
410C2×Dic3—362no—
411D12—62no—
412D12—182no—
413C4×D3—362no—
414C4×D3—362no—
415Dic6—62no—
416Dic6—182no—
417Dic6—362no—
418Dic6—362no—
419Dic6—362no—
420C2×C32—21no—
421C2×C32—21no—
422C2×C32—21no—
423C2×C32—42no—
424C2×C32—41no—
425C2×C32—41no—
426C2×C32C2×#23711yescharacteristic
427C2×C32C2×#23711yescharacteristic
428C2×C32#100411yescharacteristic
429C2×C32#100711yescharacteristic
430C2×C32#100911yescharacteristic
431C2×C32#101011yescharacteristic
432C3⋊D3—42no—
433C3×D3—122no—
434C2×Q8—91no—
435C2×Q8—272no—
436C2×D4—92no—
437C2×D4—91no—
438C22×C4—272no—
439C22×C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
440C4⋊C4—91no—
441C4⋊C4—91no—
442C4⋊C4—91no—
443C4⋊C4—91no—
444C4⋊C4—92no—
445C4⋊C4—272no—
446C4⋊C4—272no—
447C22⋊C4—91no—
448C22⋊C4—91no—
449C22⋊C4—92no—
450C22⋊C4—92no—
451C22⋊C4—272no—
452C22⋊C4—272no—
453C22⋊C4—272no—
454C42—91no—
455C42—91no—
456C42—272no—
457C22×C3—21no—
458C22×C3—21no—
459C22×C3—21no—
460C22×C3—31no—
461C22×C3—31no—
462C22×C3—61no—
463C22×C3—61no—
464C22×C3—61no—
465C22×C3—61no—
466C22×C3—61no—
467C22×C3—61no—
468C22×C3—62no—
469C22×C3—61no—
470C22×C3—121no—
471C22×C3—121no—
472C22×C3—182no—
473C22×C3—182no—
474C22×C3—182no—
475C22×C3C22×D3211yescharacteristic
476C2×D3—62no—
477C2×D3—62no—
478C2×D3—62no—
479C2×D3—182no—
480C2×D3—182no—
481C2×D3—182no—
482C3×C4—21no—
483C3×C4—32no—
484C3×C4—32no—
485C3×C4—61no—
486C3×C4—61no—
487C3×C4—62no—
488C3×C4—62no—
489C3×C4—182no—
490C3×C4—182no—
491C3×C4—182no—
492C3×C4—181no—
493C3×C4—181no—
494C3×C4—181no—
495C3×C4—181no—
496C3×C4#380512yes—
497Dic3—61no—
498Dic3—61no—
499Dic3—61no—
500Dic3—61no—
501Dic3—61no—
502Dic3—61no—
503Dic3—61no—
504Dic3—61no—
505Dic3—62no—
506Dic3—62no—
507Dic3—62no—
508Dic3—121no—
509Dic3—121no—
510Dic3—121no—
511Dic3—121no—
512Dic3—182no—
513Dic3—182no—
514Dic3—182no—
515C32—21no—
516C32#560311yescharacteristic, complemented
517C32#564311yescharacteristic, complemented
518C23—92no—
519C23C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
520Q8—182no—
521Q8—1082no—
522D4—182no—
523D4—362no—
524C2×C4—22no—
525C2×C4—91no—
526C2×C4—91no—
527C2×C4—91no—
528C2×C4—91no—
529C2×C4—92no—
530C2×C4—92no—
531C2×C4—92no—
532C2×C4—1082no—
533C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
534C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
535C2×C3—31no—
536C2×C3—31no—
537C2×C3—31no—
538C2×C3—31no—
539C2×C3—31no—
540C2×C3—31no—
541C2×C3—41no—
542C2×C3—61no—
543C2×C3—61no—
544C2×C3—61no—
545C2×C3—121no—
546C2×C3—121no—
547C2×C3—122no—
548C2×C3—362no—
549C2×C3#6429711yescharacteristic
550C2×C3#6430711yescharacteristic
551C2×C3C2×#380511yescharacteristic
552D3—122no—
553D3—362no—
554C22—21no—
555C22—21no—
556C22—21no—
557C22—182no—
558C22—182no—
559C22—182no—
560C22C22×((C3⋊D3)⋊C3)⋊C211yescharacteristic
561C4—21no—
562C4—182no—
563C4—181no—
564C4—181no—
565C4—181no—
566C4—182no—
567C4—182no—
568C4—181no—
569C4#8867212yes—
570C3—31no—
571C3—31no—
572C3—61no—
573C3#1059498411yescharacteristic
574C2—41no—
575C2—362no—
576C2#1172248211yescharacteristic
577C2#1172249211yescharacteristic
578C2C2×#8867211yescharacteristic
subgroupG/H
centerC22C22×((C3⋊D3)⋊C3)⋊C2
commutator, cosocleC22×H(3,𝔽3)C24
Frattini, socleC22×C3C22×D32
FittingC22×C4×H(3,𝔽3)C22
prime radical#64139⋊C3C2
hypercenterC22×C4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#542
LCSG ▷ C22×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
1. #11721826C2semidirect
3. #64133⋊C3C2semidirect
4. #11721311C2semidirect
5. #11721304C2semidirect
6. #11721136C2semidirect
7. #11721133C2semidirect
8. #11721117C2semidirect
9. #11721113C2semidirect
10. #11721111C2semidirect
11. #11721084C2semidirect
38. #64133D3semidirect
360. H(3,𝔽3)#542semidirect
516. C32#5603semidirect
517. C32#5643semidirect

Maximal subgroups

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

subgroupindex
#117218262
#64139⋊C32
#64133⋊C32
#117213112
#117213042
#117211362
#117211332
#117211172
#117211132
#117211112
#117210842
#105949833
#105948143