Finite Group Databasebeta

← #13150663

Subgroups — #13150663

Order 1728.

818 conjugacy classes of subgroups.

The subgroup lattice has 561 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11721826C211yesmaximal, characteristic, complemented
2C4×#638⋊C3C211yesmaximal, characteristic
3#64133⋊C3C211yesmaximal, characteristic, complemented
4#11721304C212yesmaximal
5#11721297C212yesmaximal, complemented
6#11721116C211yesmaximal, characteristic, complemented
7#11721112C212yesmaximal, complemented
8#11721106C211yesmaximal, characteristic, complemented
9#11721082C211yesmaximal, characteristic, complemented
10#11721081C211yesmaximal, characteristic, complemented
11#11721078C212yesmaximal, complemented
12#10594840—31nomaximal
13#10594809—31nomaximal
14C22×C4×H(3,𝔽3)C2211yescharacteristic
15C2×#638⋊C3C2211yescharacteristic
16C2×C4×(C3⋊D3)⋊C3C2211yescharacteristic
17C2×#634⋊C3C2211yescharacteristic
18C2×#6196C2212yes—
19#88552C2212yes—
20C4⋊#1084C2211yescharacteristic
21C4⋊#1084C2211yescharacteristic
22C4×#1084C2212yes—
23#3757⋊C3C2211yescharacteristic
24#3757⋊C3C2212yes—
25#3752⋊C3C2211yescharacteristic
26C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
27#3750⋊C3C2211yescharacteristic
28#3750⋊C3C2212yes—
29C4×(C3⋊Dic3)⋊C3C2211yescharacteristic
30#88450C2212yes—
31#88450C2212yes—
32#88449C2212yes—
33#88448C2212yes—
34#88448C2212yes—
35#88447C2212yes—
36#88447C2212yes—
37C4×#638D311yescharacteristic
38#64133D311yescharacteristic, complemented
39C3×C4×#67—31no—
40C3×#926—31no—
41C3×#926—31no—
42C3×#926—61no—
43C3×#924—31no—
44#63959—32no—
45#63956—32no—
46#63956—32no—
47#63953—32no—
48#63919—31no—
49#63918—31no—
50#63871—31no—
51#63868—32no—
52#63859—32no—
53#63857—31no—
54#63849—31no—
55#63844—31no—
56#63843—31no—
57#63841—31no—
58#63837—32no—
59#63837—32no—
60C23×H(3,𝔽3)C2311yescharacteristic
61C22×(C3⋊D3)⋊C3C2311yescharacteristic
62C2×C4×H(3,𝔽3)—22no—
63C2×C4×H(3,𝔽3)C2311yescharacteristic
64C2×C4×H(3,𝔽3)C2311yescharacteristic
65C2×#1084C2312yes—
66C2×#1084C2312yes—
67#638⋊C3—22no—
68C2×(C3⋊Dic3)⋊C3C2312yes—
69C2×(C3⋊Dic3)⋊C3C2312yes—
70C2×(C3⋊Dic3)⋊C3C2311yescharacteristic
71C2×(C3⋊Dic3)⋊C3C2311yescharacteristic
72C2×(C3⋊Dic3)⋊C3C2311yescharacteristic
73C4×(C3⋊D3)⋊C3—22no—
74#634⋊C3—22no—
75#6196—24no—
76#5641—91no—
77#5542—91no—
78#5533—91no—
79C22×C32×C4—21no—
80C22×C32×C4C2×D311yescharacteristic
81C22×C32×C4C2×D311yescharacteristic
82C2×#638C2×D311yescharacteristic
83C2×C4×C3⋊D3C2×D311yescharacteristic
84C2×#634C2×D311yescharacteristic
85C2×C3×#67—31no—
86C2×C3×C4×D3—31no—
87C2×C3×Dic6—31no—
88C2×#627—32no—
89C2×#627—32no—
90#3757C2×D312yes—
91#3757C2×D311yescharacteristic
92#3752C2×D311yescharacteristic
93C4⋊(C3⋊Dic3)C2×D311yescharacteristic
94#3750C2×D311yescharacteristic
95#3750C2×D312yes—
96C4×C3⋊Dic3C2×D311yescharacteristic
97C3×#217—32no—
98C3×#217—32no—
99C3×#217—32no—
100C3×#217—31no—
101C3×#217—62no—
102C3×(C2×D3)⋊C4—31no—
103C3×C4⋊Dic3—31no—
104C3×C4⋊Dic3—31no—
105C3×C4⋊Dic3—31no—
106C3×C4⋊Dic3—31no—
107C3×C4⋊Dic3—31no—
108C3×C4⋊Dic3—61no—
109C3×C4⋊Dic3—61no—
110C3×Dic3⋊C4—32no—
111C3×Dic3⋊C4—31no—
112C3×C4×Dic3—32no—
113C3×C4×Dic3—32no—
114C3×C4×Dic3—31no—
115C3×C4×Dic3—62no—
116#3724—32no—
117#3724—32no—
118Dic3⋊Dic3—32no—
119Dic3⋊Dic3—32no—
120Dic3⋊Dic3—32no—
121Dic3⋊Dic3—32no—
122#3722—32no—
123#3722—32no—
124(C2×D3)⋊Dic3—32no—
125(C2×D3)⋊Dic3—32no—
126Dic32—32no—
127Dic32—32no—
128Dic32—32no—
129Dic32—32no—
130C22×H(3,𝔽3)—21no—
131C22×H(3,𝔽3)—21no—
132C22×H(3,𝔽3)—21no—
133C22×H(3,𝔽3)C2411yescharacteristic
134C2×(C3⋊D3)⋊C3—21no—
135C2×(C3⋊D3)⋊C3—21no—
136C2×(C3⋊D3)⋊C3—21no—
137C4×H(3,𝔽3)—21no—
138C4×H(3,𝔽3)#4112yes—
139#1084—22no—
140#1084—22no—
141(C3⋊Dic3)⋊C3—22no—
142(C3⋊Dic3)⋊C3—22no—
143(C3⋊Dic3)⋊C3—21no—
144(C3⋊Dic3)⋊C3—21no—
145(C3⋊Dic3)⋊C3#4112yes—
146C3×C22⋊Q8—91no—
147C3×C4×D4—91no—
148C3×#117—91no—
149#946—91no—
150Q8×Dic3—91no—
151#936—92no—
152C4×#67—31no—
153C4×#67—91no—
154#926—31no—
155#926—31no—
156#926—61no—
157#926—91no—
158#924—31no—
159#924—91no—
160#924—91no—
161#898—92no—
162#897—91no—
163#892—91no—
164#892—91no—
165#888—91no—
166#888—91no—
167#887—91no—
168#887—92no—
169#887—92no—
170#885—92no—
171#885—92no—
172#879—92no—
173#879—92no—
174#879—92no—
175#879—92no—
176#876—92no—
177#869—91no—
178C4×Dic6—92no—
179C4×Dic6—91no—
180C23×C32—21no—
181C23×C32C22×D311yescharacteristic
182C23×C32C22×D311yescharacteristic
183C22×C3⋊D3C22×D311yescharacteristic
184C22×C3×D3—31no—
185C2×C32×C4—24no—
186C2×C32×C4—22no—
187C2×C32×C4—22no—
188C2×C32×C4—21no—
189C2×C32×C4—21no—
190C2×C32×C4C22×D311yescharacteristic
191C2×C32×C4C22×D311yescharacteristic
192C2×C32×C4C22×D311yescharacteristic
193C2×C32×C4C22×D311yescharacteristic
194#638—22no—
195C2×C3⋊Dic3C22×D311yescharacteristic
196C2×C3⋊Dic3C22×D312yes—
197C2×C3⋊Dic3C22×D312yes—
198C2×C3⋊Dic3C22×D311yescharacteristic
199C2×C3⋊Dic3C22×D311yescharacteristic
200C4×C3⋊D3—22no—
201#634—22no—
202C3×#67—62no—
203C2×C3×Dic3—31no—
204C2×C3×Dic3—31no—
205C2×C3×Dic3—32no—
206C2×C3×Dic3—32no—
207C2×C3×Dic3—31no—
208C2×C3×Dic3—32no—
209C2×C3×Dic3—32no—
210C2×C3×Dic3—32no—
211C2×C3×Dic3—32no—
212C2×C3×Dic3—62no—
213C2×C3×Dic3—62no—
214C3×C4×D3—62no—
215C3×Dic6—62no—
216#627—64no—
217#627—64no—
218#556—271noSylow
219C2×H(3,𝔽3)—41no—
220C2×H(3,𝔽3)C2×#4111yescharacteristic
221C2×H(3,𝔽3)C2×#4111yescharacteristic
222C2×H(3,𝔽3)2−1+411yescharacteristic
223(C3⋊D3)⋊C3—41no—
224C2×C3×Q8—91no—
225C2×C3×D4—91no—
226C22×C3×C4—31no—
227C22×C3×C4—31no—
228C22×C3×C4—61no—
229C22×C3×C4—91no—
230C22×C3×C4D3211yescharacteristic
231C2×#67—31no—
232C2×#67—91no—
233C2×C4×D3—31no—
234C2×C4×D3—91no—
235C2×Dic6—31no—
236C2×Dic6—91no—
237C2×Dic6—92no—
238C2×Dic6—92no—
239C2×Dic6—92no—
240C3×C4⋊C4—91no—
241C3×C4⋊C4—92no—
242C3×C4⋊C4—91no—
243C3×C4⋊C4—91no—
244C3×C4⋊C4—91no—
245C3×C22⋊C4—91no—
246C3×C22⋊C4—91no—
247C3×C22⋊C4—92no—
248C3×C22⋊C4—92no—
249C3×C42—91no—
250C3×C42—92no—
251#217—32no—
252#217—32no—
253#217—31no—
254#217—32no—
255#217—62no—
256#217—91no—
257#217—92no—
258#217—92no—
259#217—92no—
260(C2×D3)⋊C4—31no—
261(C2×D3)⋊C4—91no—
262(C2×D3)⋊C4—92no—
263(C2×D3)⋊C4—92no—
264(C2×D3)⋊C4—92no—
265(C2×D3)⋊C4—92no—
266C4⋊Dic3—31no—
267C4⋊Dic3—31no—
268C4⋊Dic3—31no—
269C4⋊Dic3—31no—
270C4⋊Dic3—31no—
271C4⋊Dic3—61no—
272C4⋊Dic3—61no—
273C4⋊Dic3—91no—
274C4⋊Dic3—92no—
275C4⋊Dic3—92no—
276C4⋊Dic3—92no—
277Dic3⋊C4—31no—
278Dic3⋊C4—32no—
279Dic3⋊C4—92no—
280Dic3⋊C4—91no—
281Dic3⋊C4—92no—
282Dic3⋊C4—92no—
283Dic3⋊C4—92no—
284Dic3⋊C4—92no—
285Dic3⋊C4—92no—
286Dic3⋊C4—92no—
287C4×Dic3—32no—
288C4×Dic3—32no—
289C4×Dic3—31no—
290C4×Dic3—62no—
291C4×Dic3—91no—
292C4×Dic3—92no—
293C4×Dic3—92no—
294C4×Dic3—92no—
295C4×Dic3—92no—
296C4×Dic3—92no—
297C4×Dic3—92no—
298C22×C32—22no—
299C22×C32—21no—
300C22×C32—21no—
301C22×C32—21no—
302C22×C32—21no—
303C22×C32—21no—
304C22×C32—21no—
305C22×C32—21no—
306C22×C32—41no—
307C22×C32—41no—
308C22×C32C23×D311yescharacteristic
309C22×C32C23×D311yescharacteristic
310C2×C3⋊D3—21no—
311C2×C3⋊D3—21no—
312C2×C3⋊D3—21no—
313C2×C3×D3—61no—
314C2×C3×D3—61no—
315C2×C3×D3—61no—
316C32×C4—22no—
317C32×C4—21no—
318C32×C4—21no—
319C32×C4—22no—
320C32×C4#23712yes—
321C32×C4#23712yes—
322C3⋊Dic3—22no—
323C3⋊Dic3—21no—
324C3⋊Dic3—22no—
325C3⋊Dic3—21no—
326C3⋊Dic3#23512yes—
327C3×Dic3—32no—
328C3×Dic3—62no—
329C3×Dic3—62no—
330C3×Dic3—62no—
331C3×Dic3—61no—
332C3×Dic3—61no—
333C3×Dic3—62no—
334C3×Dic3—62no—
335C3×Dic3—62no—
336C3×Dic3—122no—
337C3×Dic3—122no—
338C4⋊Q8—271no—
339#126—272no—
340#126—272no—
341#124—272no—
342C22⋊Q8—91no—
343C22⋊Q8—271no—
344C4×Q8—272no—
345C4×Q8—271no—
346C4×D4—91no—
347#117—91no—
348#117—271no—
349H(3,𝔽3)#55611yescharacteristic, complemented, Sylow
350C23×C3—31no—
351C23×C3—31no—
352C23×C3—61no—
353C23×C3—91no—
354C23×C3C2×D3211yescharacteristic
355C22×D3—31no—
356C22×D3—91no—
357C3×Q8—182no—
358C3×D4—182no—
359C2×C3×C4—22no—
360C2×C3×C4—31no—
361C2×C3×C4—31no—
362C2×C3×C4—31no—
363C2×C3×C4—31no—
364C2×C3×C4—62no—
365C2×C3×C4—62no—
366C2×C3×C4—64no—
367C2×C3×C4—61no—
368C2×C3×C4—61no—
369C2×C3×C4—91no—
370C2×C3×C4—91no—
371C2×C3×C4—92no—
372C2×C3×C4—92no—
373C2×C3×C4—91no—
374C2×C3×C4—92no—
375C2×C3×C4—92no—
376C2×C3×C4—182no—
377C2×C3×C4C2×D3211yescharacteristic
378C2×C3×C4C2×D3211yescharacteristic
379#67—62no—
380#67—182no—
381C2×Dic3—31no—
382C2×Dic3—32no—
383C2×Dic3—32no—
384C2×Dic3—32no—
385C2×Dic3—32no—
386C2×Dic3—31no—
387C2×Dic3—31no—
388C2×Dic3—32no—
389C2×Dic3—32no—
390C2×Dic3—62no—
391C2×Dic3—62no—
392C2×Dic3—91no—
393C2×Dic3—91no—
394C2×Dic3—92no—
395C2×Dic3—91no—
396C2×Dic3—92no—
397C4×D3—62no—
398C4×D3—182no—
399Dic6—62no—
400Dic6—182no—
401Dic6—184no—
402Dic6—184no—
403Dic6—184no—
404C2×C32—21no—
405C2×C32—21no—
406C2×C32—21no—
407C2×C32—42no—
408C2×C32—41no—
409C2×C32—41no—
410C2×C32C2×#23511yescharacteristic
411C2×C32C2×#23711yescharacteristic
412C2×C32C2×#23711yescharacteristic
413C2×C32#100711yescharacteristic
414C2×C32D3×#4111yescharacteristic
415C2×C32#101011yescharacteristic
416C3⋊D3—41no—
417C3×D3—121no—
418C2×Q8—91no—
419C2×Q8—272no—
420C2×D4—91no—
421C22×C4—91no—
422C22×C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
423C4⋊C4—91no—
424C4⋊C4—91no—
425C4⋊C4—91no—
426C4⋊C4—91no—
427C4⋊C4—92no—
428C4⋊C4—272no—
429C4⋊C4—272no—
430C4⋊C4—272no—
431C22⋊C4—92no—
432C22⋊C4—91no—
433C22⋊C4—92no—
434C22⋊C4—91no—
435C22⋊C4—272no—
436C22⋊C4—272no—
437C42—92no—
438C42—91no—
439C42—272no—
440C42—272no—
441C22×C3—21no—
442C22×C3—21no—
443C22×C3—21no—
444C22×C3—31no—
445C22×C3—31no—
446C22×C3—61no—
447C22×C3—61no—
448C22×C3—61no—
449C22×C3—61no—
450C22×C3—61no—
451C22×C3—61no—
452C22×C3—62no—
453C22×C3—61no—
454C22×C3—121no—
455C22×C3—121no—
456C22×C3—181no—
457C22×C3—181no—
458C22×C3—181no—
459C22×C3C22×D3211yescharacteristic
460C2×D3—61no—
461C2×D3—61no—
462C2×D3—61no—
463C2×D3—181no—
464C2×D3—181no—
465C2×D3—181no—
466C3×C4—21no—
467C3×C4—32no—
468C3×C4—32no—
469C3×C4—61no—
470C3×C4—61no—
471C3×C4—62no—
472C3×C4—62no—
473C3×C4—92no—
474C3×C4—182no—
475C3×C4—181no—
476C3×C4—182no—
477C3×C4—181no—
478C3×C4—182no—
479C3×C4—182no—
480C3×C4#380512yes—
481Dic3—32no—
482Dic3—62no—
483Dic3—62no—
484Dic3—62no—
485Dic3—62no—
486Dic3—61no—
487Dic3—62no—
488Dic3—62no—
489Dic3—61no—
490Dic3—92no—
491Dic3—122no—
492Dic3—122no—
493Dic3—182no—
494Dic3—182no—
495Dic3—181no—
496Dic3—181no—
497C32—21no—
498C32#554211yescharacteristic, complemented
499C32#564111yescharacteristic, complemented
500C23—91no—
501C23C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
502Q8—182no—
503Q8—544no—
504D4—182no—
505C2×C4—22no—
506C2×C4—91no—
507C2×C4—92no—
508C2×C4—91no—
509C2×C4—92no—
510C2×C4—91no—
511C2×C4—92no—
512C2×C4—92no—
513C2×C4—182no—
514C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
515C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
516C2×C3—31no—
517C2×C3—31no—
518C2×C3—31no—
519C2×C3—31no—
520C2×C3—31no—
521C2×C3—31no—
522C2×C3—41no—
523C2×C3—61no—
524C2×C3—61no—
525C2×C3—61no—
526C2×C3—121no—
527C2×C3—121no—
528C2×C3—122no—
529C2×C3—361no—
530C2×C3#6429711yescharacteristic
531C2×C3D3×#23511yescharacteristic
532C2×C3C2×#380511yescharacteristic
533D3—121no—
534D3—361no—
535C22—21no—
536C22—21no—
537C22—21no—
538C22—181no—
539C22—181no—
540C22—181no—
541C22C22×((C3⋊D3)⋊C3)⋊C211yescharacteristic
542C4—21no—
543C4—92no—
544C4—181no—
545C4—182no—
546C4—182no—
547C4—181no—
548C4—182no—
549C4—182no—
550C4#8867212yes—
551C3—31no—
552C3—31no—
553C3—61no—
554C3#1059484111yescharacteristic
555C2—41no—
556C2—361no—
557C2#1172248211yescharacteristic
558C2#1172249011yescharacteristic
559C2C2×#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 radicalC2×#638⋊C3C22
hypercenterC22×C4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#556
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
5. #11721297C2semidirect
6. #11721116C2semidirect
7. #11721112C2semidirect
8. #11721106C2semidirect
9. #11721082C2semidirect
10. #11721081C2semidirect
11. #11721078C2semidirect
38. #64133D3semidirect
349. H(3,𝔽3)#556semidirect
498. C32#5542semidirect
499. C32#5641semidirect

Maximal subgroups

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

subgroupindex
#117218262
C4×#638⋊C32
#64133⋊C32
#117213042
#117212972
#117211162
#117211122
#117211062
#117210822
#117210812
#117210782
#105948403
#105948093