Finite Group Databasebeta

← #13161956

Subgroups — #13161956

Order 1728.

990 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 990 conjugacy classes of subgroups — too many to draw. Only its 103 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#64361C211yesmaximal, characteristic
2C3×#64144C211yesmaximal, characteristic, complemented
3#11723720C212yesmaximal, complemented
4C3×#63661C211yesmaximal, characteristic, complemented
5#11722083C212yesmaximal, complemented
6#10591535—31nomaximal
7#10591534—31nomaximal
8#10591533—31nomaximal
9C2×C33×D4C2211yescharacteristic
10C2×C3×#638—21no—
11C2×C3×#638C2212yes—
12C22×C3×C3⋊Dic3C2211yescharacteristic, complemented
13C3×#3830—22no—
14C3×D4×C3⋊D3—22no—
15C3×#3828—21no—
16C3×#3757—21no—
17C3×#3757C2211yescharacteristic, complemented
18C3×#3757C2212yescomplemented
19C3×#3750—21no—
20#88815—22no—
21#88814—22no—
22#64361D311yescharacteristic
23C3×#1004—31no—
24C3×#1004—31no—
25C3×#1004—61no—
26#64144D311yescharacteristic, complemented
27C3×#935—31no—
28C3×#935—31no—
29C3×#935—61no—
30#63966—32no—
31#63962—32no—
32#63961—32no—
33#63661D311yescharacteristic, complemented
34C3×#834—31no—
35C3×#834—31no—
36C3×#834—61no—
37#63581—32no—
38#63580—32no—
39#63580—32no—
40C23×C33D411yescharacteristic
41C23×C33C2311yescharacteristic
42C22×C3×C3⋊D3—21no—
43C22×C3×C3⋊D3D412yes—
44C33×D4—22no—
45C2×C33×C4D411yescharacteristic
46C3×#638—22no—
47C3×#638—22no—
48C3×#638—22no—
49C2×C3×C3⋊Dic3—21no—
50C2×C3×C3⋊Dic3—21no—
51C2×C3×C3⋊Dic3—21no—
52C2×C3×C3⋊Dic3—22no—
53C2×C3×C3⋊Dic3—22no—
54C2×C3×C3⋊Dic3D412yes—
55C3×C4⋊(C3⋊D3)—21no—
56C3×C4×C3⋊D3—22no—
57C3×#634—21no—
58#5243—91no—
59#5053—91no—
60#4742—91no—
61C2×C32×D4—21no—
62C2×C32×D4—21no—
63C2×C32×D4—21no—
64C2×C32×D4—41no—
65C2×C32×D4C2×D311yescharacteristic
66C2×C32×D4C2×D311yescharacteristic
67C2×C32×D4C2×D311yescharacteristic
68C2×#638—21no—
69C2×#638C2×D312yes—
70C22×C3⋊Dic3C2×D311yescharacteristic, complemented
71#3830—22no—
72D4×C3⋊D3—22no—
73#3828—21no—
74C2×C3×#67—32no—
75C2×C3×#67—32no—
76C2×C3×#67—62no—
77C2×C3×#67—62no—
78C2×C3×#67—61no—
79C2×C3×#67—61no—
80C22×C3×Dic3—31no—
81C22×C3×Dic3—31no—
82C22×C3×Dic3—61no—
83C3×#237—62no—
84C3×#237—62no—
85C3×#237—64no—
86C3×D3×D4—62no—
87C3×D3×D4—62no—
88C3×D3×D4—64no—
89C3×#235—62no—
90C3×#235—61no—
91C3×#235—61no—
92#3757—21no—
93#3757C2×D311yescharacteristic, complemented
94#3757C2×D312yescomplemented
95#3750—21no—
96C3×#217—31no—
97C3×#217—31no—
98C3×#217—32no—
99C3×#217—32no—
100C3×#217—61no—
101C3×#217—62no—
102C3×#217—61no—
103C3×#217—61no—
104C3×#217—62no—
105C3×Dic3⋊C4—62no—
106C3×Dic3⋊C4—61no—
107C3×Dic3⋊C4—61no—
108#3724—62no—
109Dic3⋊Dic3—62no—
110Dic3⋊Dic3—62no—
111#3722—62no—
112(C2×D3)⋊Dic3—62no—
113(C2×D3)⋊Dic3—62no—
114C22×C33—21no—
115C22×C33—41no—
116C22×C33—41no—
117C22×C33C2×D411yescharacteristic
118C22×C33C2×D412yes—
119C2×C3×C3⋊D3—22no—
120C2×C3×C3⋊D3—42no—
121C2×C3×C3⋊D3—81no—
122C33×C4—21no—
123C3×C3⋊Dic3—22no—
124C3×C3⋊Dic3—41no—
125C3×C3⋊Dic3—42no—
126C3×C3⋊Dic3—41no—
127C3×H(5,𝔽2)—91no—
128#1004—31no—
129#1004—31no—
130#1004—61no—
131C3×#123—91no—
132#935—31no—
133#935—31no—
134#935—61no—
135#935—92no—
136#929—92no—
137#886—92no—
138C3×#99—91no—
139#834—31no—
140#834—31no—
141#834—61no—
142#834—92no—
143#806—92no—
144#806—92no—
145C23×C32—21no—
146C23×C32—21no—
147C23×C32—21no—
148C23×C32—21no—
149C23×C32—21no—
150C23×C32—21no—
151C23×C32—41no—
152C23×C32—41no—
153C23×C32D1211yescharacteristic
154C23×C32#6711yescharacteristic
155C23×C32#6711yescharacteristic
156C23×C32C22×D311yescharacteristic
157C23×C32C22×D311yescharacteristic
158C23×C32C22×D311yescharacteristic
159C22×C3⋊D3—21no—
160C22×C3⋊D3#6712yes—
161C22×C3×D3—32no—
162C22×C3×D3—32no—
163C22×C3×D3—62no—
164C22×C3×D3—62no—
165C22×C3×D3—61no—
166C22×C3×D3—61no—
167C32×D4—22no—
168C32×D4—24no—
169C32×D4—22no—
170C32×D4—22no—
171C32×D4—42no—
172C32×D4—42no—
173C32×D4—44no—
174C2×C32×C4—21no—
175C2×C32×C4—21no—
176C2×C32×C4—21no—
177C2×C32×C4—41no—
178C2×C32×C4D1211yescharacteristic
179C2×C32×C4#6711yescharacteristic
180C2×C32×C4#6711yescharacteristic
181#638—22no—
182#638—22no—
183#638—22no—
184C2×C3⋊Dic3—21no—
185C2×C3⋊Dic3—21no—
186C2×C3⋊Dic3—21no—
187C2×C3⋊Dic3—22no—
188C2×C3⋊Dic3—22no—
189C2×C3⋊Dic3#6712yes—
190C4⋊(C3⋊D3)—21no—
191C4×C3⋊D3—22no—
192#634—21no—
193C3×#67—64no—
194C3×#67—64no—
195C3×#67—62no—
196C3×#67—62no—
197C3×#67—64no—
198C3×#67—62no—
199C3×#67—62no—
200C3×#67—62no—
201C3×#67—62no—
202C2×C3×Dic3—32no—
203C2×C3×Dic3—32no—
204C2×C3×Dic3—62no—
205C2×C3×Dic3—62no—
206C2×C3×Dic3—61no—
207C2×C3×Dic3—61no—
208C2×C3×Dic3—62no—
209C2×C3×Dic3—62no—
210C2×C3×Dic3—61no—
211C2×C3×Dic3—61no—
212C2×C3×Dic3—62no—
213C2×C3×Dic3—62no—
214C2×C3×Dic3—62no—
215C2×C3×Dic3—62no—
216C2×C3×Dic3—61no—
217C2×C3×Dic3—61no—
218C2×C3×Dic3—122no—
219C2×C3×Dic3—122no—
220C3×D12—62no—
221C3×D12—61no—
222C3×D12—61no—
223C3×C4×D3—64no—
224C3×C4×D3—62no—
225C3×C4×D3—62no—
226C3×Dic6—62no—
227C3×Dic6—61no—
228C3×Dic6—61no—
229#458—271noSylow
230C2×C33—21no—
231C2×C33—22no—
232C2×C33—41no—
233C2×C33#12011yescharacteristic
234C3×C3⋊D3—42no—
235C3×#41—182no—
236C3×#41—181no—
237C2×C3×D4—21no—
238C2×C3×D4—21no—
239C2×C3×D4—21no—
240C2×C3×D4—41no—
241C2×C3×D4—92no—
242C2×C3×D4—182no—
243C2×C3×D4—181no—
244C2×C3×D4D3211yescharacteristic
245C2×C3×D4D3211yescharacteristic
246C2×C3×D4D3211yescharacteristic
247C22×C3×C4—91no—
248C2×#67—32no—
249C2×#67—32no—
250C2×#67—62no—
251C2×#67—62no—
252C2×#67—61no—
253C2×#67—61no—
254C22×Dic3—31no—
255C22×Dic3—31no—
256C22×Dic3—61no—
257#237—62no—
258#237—62no—
259#237—64no—
260D3×D4—62no—
261D3×D4—62no—
262D3×D4—64no—
263#235—62no—
264#235—61no—
265#235—61no—
266C3×C4⋊C4—181no—
267C3×C22⋊C4—91no—
268C3×C22⋊C4—92no—
269C3×C22⋊C4—181no—
270#217—31no—
271#217—32no—
272#217—32no—
273#217—31no—
274#217—61no—
275#217—62no—
276#217—62no—
277#217—61no—
278#217—61no—
279#217—182no—
280(C2×D3)⋊C4—182no—
281(C2×D3)⋊C4—182no—
282C4⋊Dic3—182no—
283Dic3⋊C4—62no—
284Dic3⋊C4—61no—
285Dic3⋊C4—61no—
286Dic3⋊C4—182no—
287Dic3⋊C4—182no—
288C22×C32—21no—
289C22×C32—22no—
290C22×C32—21no—
291C22×C32—21no—
292C22×C32—21no—
293C22×C32—21no—
294C22×C32—21no—
295C22×C32—22no—
296C22×C32—22no—
297C22×C32—22no—
298C22×C32—41no—
299C22×C32—42no—
300C22×C32—41no—
301C22×C32—41no—
302C22×C32—42no—
303C22×C32—41no—
304C22×C32—41no—
305C22×C32—41no—
306C22×C32—41no—
307C22×C32—41no—
308C22×C32—41no—
309C22×C32—42no—
310C22×C32—42no—
311C22×C32—81no—
312C22×C32—81no—
313C22×C32—81no—
314C22×C32—81no—
315C22×C32—82no—
316C22×C32—82no—
317C22×C32C2×D1211yescharacteristic
318C22×C32D3×D412yes—
319C22×C32D3×D412yes—
320C22×C32C2×#6711yescharacteristic
321C22×C32C2×#6711yescharacteristic
322C22×C32C2×#6712yes—
323C2×C3⋊D3—22no—
324C2×C3⋊D3—42no—
325C2×C3⋊D3—81no—
326C2×C3×D3—64no—
327C2×C3×D3—62no—
328C2×C3×D3—62no—
329C2×C3×D3—122no—
330C2×C3×D3—122no—
331C2×C3×D3—242no—
332C2×C3×D3—242no—
333C2×C3×D3—241no—
334C2×C3×D3—241no—
335C32×C4—21no—
336C32×C4—22no—
337C32×C4—21no—
338C32×C4—21no—
339C32×C4—41no—
340C32×C4—41no—
341C32×C4—42no—
342C3⋊Dic3—22no—
343C3⋊Dic3—41no—
344C3⋊Dic3—41no—
345C3⋊Dic3—42no—
346C3×Dic3—64no—
347C3×Dic3—62no—
348C3×Dic3—62no—
349C3×Dic3—122no—
350C3×Dic3—121no—
351C3×Dic3—121no—
352C3×Dic3—122no—
353C3×Dic3—122no—
354C3×Dic3—122no—
355C3×Dic3—121no—
356C3×Dic3—121no—
357C3×Dic3—242no—
358H(5,𝔽2)—91no—
359#123—91no—
360#123—272no—
361#99—91no—
362#99—272no—
363C33#45811yescharacteristic, complemented, Sylow
364C23×C3—21no—
365C23×C3—21no—
366C23×C3—21no—
367C23×C3—21no—
368C23×C3—21no—
369C23×C3—21no—
370C23×C3—41no—
371C23×C3—41no—
372C23×C3—92no—
373C23×C3—181no—
374C23×C3#62511yescharacteristic
375C23×C3#62611yescharacteristic
376C23×C3#62611yescharacteristic
377C23×C3C2×D3211yescharacteristic
378C23×C3C2×D3211yescharacteristic
379C23×C3C2×D3211yescharacteristic
380C22×D3—32no—
381C22×D3—32no—
382C22×D3—62no—
383C22×D3—62no—
384C22×D3—61no—
385C22×D3—61no—
386C3×Q8—181no—
387C3×D4—24no—
388C3×D4—22no—
389C3×D4—22no—
390C3×D4—22no—
391C3×D4—42no—
392C3×D4—42no—
393C3×D4—44no—
394C3×D4—181no—
395C3×D4—182no—
396C3×D4—182no—
397C3×D4—182no—
398C2×C3×C4—21no—
399C2×C3×C4—21no—
400C2×C3×C4—21no—
401C2×C3×C4—41no—
402C2×C3×C4—92no—
403C2×C3×C4—182no—
404C2×C3×C4—181no—
405C2×C3×C4—181no—
406C2×C3×C4—181no—
407C2×C3×C4—182no—
408C2×C3×C4—182no—
409C2×C3×C4#62511yescharacteristic
410C2×C3×C4#62611yescharacteristic
411C2×C3×C4#62611yescharacteristic
412#67—64no—
413#67—64no—
414#67—62no—
415#67—62no—
416#67—64no—
417#67—62no—
418#67—62no—
419#67—62no—
420#67—62no—
421C2×Dic3—32no—
422C2×Dic3—32no—
423C2×Dic3—62no—
424C2×Dic3—62no—
425C2×Dic3—61no—
426C2×Dic3—61no—
427C2×Dic3—62no—
428C2×Dic3—62no—
429C2×Dic3—61no—
430C2×Dic3—61no—
431C2×Dic3—61no—
432C2×Dic3—61no—
433C2×Dic3—62no—
434C2×Dic3—62no—
435C2×Dic3—62no—
436C2×Dic3—62no—
437C2×Dic3—122no—
438C2×Dic3—122no—
439D12—62no—
440D12—61no—
441D12—61no—
442C4×D3—64no—
443C4×D3—62no—
444C4×D3—62no—
445Dic6—62no—
446Dic6—61no—
447Dic6—61no—
448C2×C32—21no—
449C2×C32—22no—
450C2×C32—22no—
451C2×C32—21no—
452C2×C32—21no—
453C2×C32—22no—
454C2×C32—22no—
455C2×C32—21no—
456C2×C32—21no—
457C2×C32—21no—
458C2×C32—41no—
459C2×C32—41no—
460C2×C32—41no—
461C2×C32—42no—
462C2×C32—42no—
463C2×C32—42no—
464C2×C32—42no—
465C2×C32—41no—
466C2×C32—42no—
467C2×C32—41no—
468C2×C32—41no—
469C2×C32—82no—
470C2×C32—81no—
471C2×C32—81no—
472C2×C32—82no—
473C2×C32#88211yescharacteristic
474C2×C32#93711yescharacteristic
475C2×C32#95311yescharacteristic
476C3⋊D3—42no—
477C3×D3—124no—
478C3×D3—122no—
479C3×D3—122no—
480#41—182no—
481#41—181no—
482C2×D4—92no—
483C2×D4—182no—
484C2×D4—181no—
485C2×D4#111311yescharacteristic
486C22×C4—91no—
487C4⋊C4—181no—
488C4⋊C4—542no—
489C22⋊C4—92no—
490C22⋊C4—91no—
491C22⋊C4—181no—
492C22⋊C4—542no—
493C22×C3—22no—
494C22×C3—21no—
495C22×C3—21no—
496C22×C3—21no—
497C22×C3—21no—
498C22×C3—21no—
499C22×C3—21no—
500C22×C3—22no—
501C22×C3—22no—
502C22×C3—22no—
503C22×C3—41no—
504C22×C3—41no—
505C22×C3—42no—
506C22×C3—41no—
507C22×C3—41no—
508C22×C3—41no—
509C22×C3—41no—
510C22×C3—41no—
511C22×C3—41no—
512C22×C3—42no—
513C22×C3—42no—
514C22×C3—41no—
515C22×C3—42no—
516C22×C3—81no—
517C22×C3—81no—
518C22×C3—81no—
519C22×C3—81no—
520C22×C3—82no—
521C22×C3—82no—
522C22×C3—182no—
523C22×C3—362no—
524C22×C3—721no—
525C22×C3C2×#62511yescharacteristic
526C22×C3C2×#62611yescharacteristic
527C22×C3C2×#62611yescharacteristic
528C22×C3D3×#6712yes—
529C22×C3D3×#6712yes—
530C22×C3#381112yes—
531C2×D3—64no—
532C2×D3—62no—
533C2×D3—62no—
534C2×D3—122no—
535C2×D3—122no—
536C2×D3—242no—
537C2×D3—242no—
538C2×D3—241no—
539C2×D3—241no—
540C3×C4—22no—
541C3×C4—21no—
542C3×C4—21no—
543C3×C4—21no—
544C3×C4—41no—
545C3×C4—41no—
546C3×C4—42no—
547C3×C4—182no—
548C3×C4—361no—
549C3×C4—362no—
550C3×C4—361no—
551Dic3—64no—
552Dic3—62no—
553Dic3—62no—
554Dic3—122no—
555Dic3—122no—
556Dic3—121no—
557Dic3—121no—
558Dic3—122no—
559Dic3—122no—
560Dic3—121no—
561Dic3—121no—
562Dic3—242no—
563C32—21no—
564C32—21no—
565C32—21no—
566C32—41no—
567C32#474211yescharacteristic, complemented
568C32#505311yescharacteristic, complemented
569C32#524311yescharacteristic, complemented
570C23—92no—
571C23—181no—
572C23#629511yescharacteristic
573C23C2×#111311yescharacteristic
574Q8—181no—
575D4—22no—
576D4—181no—
577D4—182no—
578D4—182no—
579D4—182no—
580C2×C4—92no—
581C2×C4—182no—
582C2×C4—181no—
583C2×C4—181no—
584C2×C4—181no—
585C2×C4—182no—
586C2×C4—182no—
587C2×C4#629511yescharacteristic
588C2×C3—22no—
589C2×C3—21no—
590C2×C3—22no—
591C2×C3—22no—
592C2×C3—21no—
593C2×C3—21no—
594C2×C3—22no—
595C2×C3—21no—
596C2×C3—21no—
597C2×C3—21no—
598C2×C3—42no—
599C2×C3—42no—
600C2×C3—42no—
601C2×C3—41no—
602C2×C3—41no—
603C2×C3—42no—
604C2×C3—42no—
605C2×C3—41no—
606C2×C3—41no—
607C2×C3—41no—
608C2×C3—41no—
609C2×C3—82no—
610C2×C3—81no—
611C2×C3—81no—
612C2×C3—82no—
613C2×C3—362no—
614C2×C3#6397611yescharacteristic
615C2×C3#6397711yescharacteristic
616C2×C3#6398111yescharacteristic
617D3—124no—
618D3—122no—
619D3—122no—
620C22—21no—
621C22—41no—
622C22—41no—
623C22—182no—
624C22—362no—
625C22—721no—
626C22C2×#629511yescharacteristic
627C22#8904912yes—
628C4—21no—
629C4—182no—
630C4—361no—
631C4—362no—
632C4—361no—
633C3—21no—
634C3—21no—
635C3—21no—
636C3—41no—
637C3#1059153311yescharacteristic, complemented
638C3#1059153411yescharacteristic, complemented
639C3#1059153511yescharacteristic, complemented
640C2—21no—
641C2—22no—
642C2—41no—
643C2—362no—
644C2#1172372711yescharacteristic
subgroupG/H
centerC2#11723727
commutator, cosocleC23×C33C23
FrattiniC23C2×#1113
FittingC2×C33×D4C22
socleC2×C33#120
prime radicalC3×#64361C2
hypercenterC2×D4#1113
hypocenterC33#458
LCSG ▷ C23×C33 ▷ C2×C33 ▷ C33
UCSC1 ◁ C2 ◁ C23 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
2. C3×#64144C2semidirect
3. #11723720C2semidirect
4. C3×#63661C2semidirect
5. #11722083C2semidirect
12. C22×C3×C3⋊Dic3C22semidirect
17. C3×#3757C22semidirect
18. C3×#3757C22semidirect
26. #64144D3semidirect
33. #63661D3semidirect
70. C22×C3⋊Dic3C2×D3semidirect
93. #3757C2×D3semidirect
94. #3757C2×D3semidirect
363. C33#458semidirect
567. C32#4742semidirect
568. C32#5053semidirect
569. C32#5243semidirect
637. C3#10591533semidirect
638. C3#10591534semidirect
639. C3#10591535semidirect

Maximal subgroups

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

subgroupindex
C3×#643612
C3×#641442
#117237202
C3×#636612
#117220832
#105915353
#105915343
#105915333