Finite Group Databasebeta

← #13155413

Subgroups — #13155413

Order 1728.

878 conjugacy classes of subgroups.

The subgroup lattice has 612 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×D3×#634C211yesmaximal, characteristic
2C3×#63841C211yesmaximal, characteristic
3C2×C3×#3711C211yesmaximal, characteristic
4#11722076C211yesmaximal, characteristic, complemented
5#11722042C211yesmaximal, characteristic, complemented
6#11722041C211yesmaximal, characteristic, complemented
7C3×#63567C211yesmaximal, characteristic, complemented
8#10591107—31nomaximal
9#10590948—31nomaximal
10#10590600—31nomaximal
11C2×#6308C2211yescharacteristic
12C2×C3×#634C2211yescharacteristic
13C2×C32×C4×D3C2211yescharacteristic
14C2×#6293—21no—
15C2×D3×C3⋊Dic3—21no—
16D3×#634—22no—
17D3×#634C414yescomplemented
18C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
19C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
20C32×C4⋊Dic3C2211yescharacteristic
21C2×C32×C3⋊C8C2211yescharacteristic
22C3×(C2×D3)⋊Dic3—21no—
23C3×Dic32—21no—
24C3×#3711—22no—
25C22×#634—31no—
26C2×D3×Dic6—31no—
27C2×D3×Dic6—31no—
28C2×D3×Dic6—61no—
29C3×#926—31no—
30C2×C3×#208—31no—
31C2×C3×ΓL(1,𝔽25)—31no—
32C3×#892—31no—
33#63841D311yescharacteristic
34C2×#3711D311yescharacteristic
35#63643—31no—
36#63637—31no—
37C3×#816—31no—
38C3×#810—31no—
39#63567—31no—
40#63567—31no—
41#63567—31no—
42#63567—31no—
43#63567—31no—
44#63567—61no—
45#63567—61no—
46#63567D311yescharacteristic, complemented
47#63566—31no—
48#63565—31no—
49C22×C32×D3D411yescharacteristic
50C2×C33×C4C2311yescharacteristic
51C2×C3⋊(C3⋊Dic3)—21no—
52#6308—21no—
53#6308C2×C412yes—
54C2×C3×C3⋊Dic3—21no—
55C2×C3×C3⋊Dic3—21no—
56C3×#634—21no—
57C3×#634C2×C412yes—
58C2×C32×Dic3—21no—
59C2×C32×Dic3D411yescharacteristic
60C32×C4×D3C2×C412yes—
61C32×C4×D3D412yes—
62#6293—24no—
63D3×C3⋊Dic3—24no—
64C3×C3⋊(C3⋊C8)—21no—
65C32×C3⋊C8—21no—
66#5120—91no—
67#5004—91no—
68#4802—91no—
69C22×C32×C4—31no—
70C22×C3⋊Dic3—61no—
71C2×#634—31no—
72C2×#634—31no—
73C2×#634—31no—
74C2×#634—34no—
75C2×#634—62no—
76C2×#634—61no—
77C2×#634—61no—
78C2×#634—61no—
79C2×#634—61no—
80C2×#634—121no—
81C2×#634C2×D311yescharacteristic
82C2×C3×C4×D3—21no—
83C2×C3×C4×D3C2×D311yescharacteristic
84C2×C3×C4×D3C2×D311yescharacteristic
85C2×C3×Dic6—31no—
86C2×C3×Dic6—31no—
87C2×C3×Dic6—61no—
88C2×#627—62no—
89C2×#627—61no—
90C2×#627—61no—
91C2×D3×Dic3—62no—
92C2×D3×Dic3—61no—
93C2×D3×Dic3—61no—
94D3×Dic6—34no—
95D3×Dic6—34no—
96D3×Dic6—64no—
97D3×Dic6—64no—
98D3×Dic6—62no—
99D3×Dic6—62no—
100C2×C32×C8—31no—
101C32×C4⋊C4—31no—
102C4⋊(C3⋊Dic3)C2×D311yescharacteristic
103C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
104C3×#217—61no—
105C3×(C2×D3)⋊C4—61no—
106C3×C4⋊Dic3—21no—
107C3×C4⋊Dic3—31no—
108C3×C4⋊Dic3—31no—
109C3×C4⋊Dic3—61no—
110C3×C4⋊Dic3C2×D311yescharacteristic
111C3×C4⋊Dic3C2×D311yescharacteristic
112C3×C4×Dic3—61no—
113C3×C4×Dic3—61no—
114C3×#208—62no—
115C2×C3×C3⋊C8—21no—
116C2×C3×C3⋊C8—31no—
117C2×C3×C3⋊C8—31no—
118C2×C3×C3⋊C8—61no—
119C2×C3×C3⋊C8C2×D311yescharacteristic
120C2×C3×C3⋊C8C2×D311yescharacteristic
121C3×ΓL(1,𝔽25)—62no—
122(C2×D3)⋊Dic3—21no—
123Dic32—21no—
124#3711—22no—
125C22×C33C2×D411yescharacteristic
126C2×C32×D3—21no—
127C2×C32×D3—21no—
128C2×C32×D3C22⋊C412yes—
129C33×C4C22×C411yescharacteristic
130C33×C4C2×D411yescharacteristic
131C3⋊(C3⋊Dic3)—22no—
132C3×C3⋊Dic3—22no—
133C3×C3⋊Dic3—41no—
134C32×Dic3—41no—
135C32×Dic3C22⋊C412yes—
136C2×D3×Q8—91no—
137C22×Dic6—91no—
138C22×Dic6—91no—
139C22×Dic6—181no—
140C2×C3×M16—91no—
141C3×#117—91no—
142#926—31no—
143C2×#208—31no—
144C2×ΓL(1,𝔽25)—31no—
145#892—31no—
146C3×#103—91no—
147#835—91no—
148#816—31no—
149#816—91no—
150#816—91no—
151#816—91no—
152#816—91no—
153#816—181no—
154#810—31no—
155#810—91no—
156#810—91no—
157#810—91no—
158#810—91no—
159#810—181no—
160C23×C32—31no—
161C22×C3×D3—21no—
162C22×C3×D3D1211yescharacteristic
163C22×C3×D3#6711yescharacteristic
164C2×C32×C4—21no—
165C2×C32×C4—21no—
166C2×C32×C4—21no—
167C2×C32×C4—31no—
168C2×C32×C4—32no—
169C2×C32×C4—32no—
170C2×C32×C4—41no—
171C2×C32×C4—61no—
172C2×C32×C4C22×D311yescharacteristic
173C2×C32×C4C22×D311yescharacteristic
174C2×C32×C4C22×D311yescharacteristic
175C2×C3⋊Dic3—21no—
176C2×C3⋊Dic3—21no—
177C2×C3⋊Dic3—64no—
178C2×C3⋊Dic3—61no—
179C2×C3⋊Dic3—62no—
180C2×C3⋊Dic3—61no—
181C2×C3⋊Dic3—61no—
182C2×C3⋊Dic3—121no—
183C2×C3⋊Dic3—121no—
184C2×C3⋊Dic3—122no—
185#634—21no—
186#634—32no—
187#634—32no—
188#634—32no—
189#634—62no—
190#634—62no—
191#634—62no—
192#634—64no—
193#634—61no—
194#634—62no—
195#634—61no—
196#634—61no—
197#634—122no—
198#634—121no—
199#634—121no—
200#634—122no—
201#634C4×D312yes—
202C2×C3×Dic3—21no—
203C2×C3×Dic3—21no—
204C2×C3×Dic3—21no—
205C2×C3×Dic3—41no—
206C2×C3×Dic3—62no—
207C2×C3×Dic3—61no—
208C2×C3×Dic3—61no—
209C2×C3×Dic3—61no—
210C2×C3×Dic3—61no—
211C2×C3×Dic3—121no—
212C2×C3×Dic3D1211yescharacteristic
213C2×C3×Dic3#6711yescharacteristic
214C3×C4×D3—22no—
215C3×C4×D3—22no—
216C3×C4×D3C4×D312yes—
217C3×C4×D3C4×D312yes—
218C3×C4×D3D1212yes—
219C3×C4×D3#6712yes—
220C3×Dic6—32no—
221C3×Dic6—32no—
222C3×Dic6—62no—
223C3×Dic6—62no—
224C3×Dic6—61no—
225C3×Dic6—61no—
226#627—68no—
227#627—64no—
228#627—64no—
229D3×Dic3—68no—
230D3×Dic3—64no—
231D3×Dic3—64no—
232C32×C8—61no—
233C3⋊(C3⋊C8)—21no—
234C3×C3⋊C8—21no—
235C3×C3⋊C8—21no—
236C3×C3⋊C8—41no—
237C3×C3⋊C8—61no—
238C3×C3⋊C8—61no—
239C3×C3⋊C8—121no—
240#419—271noSylow
241C2×C33C2×C22⋊C411yescharacteristic
242C2×C33#13711yescharacteristic
243C2×C33#13711yescharacteristic
244C32×D3—22no—
245C2×C3×Q8—91no—
246C22×C3×C4—31no—
247C22×C3×C4—31no—
248C22×C3×C4—61no—
249C22×Dic3—181no—
250C22×Dic3—181no—
251C22×Dic3—182no—
252D3×Q8—94no—
253D3×Q8—182no—
254C2×C4×D3—181no—
255C2×C4×D3D3211yescharacteristic
256C2×Dic6—31no—
257C2×Dic6—31no—
258C2×Dic6—61no—
259C2×Dic6—94no—
260C2×Dic6—94no—
261C2×Dic6—91no—
262C2×Dic6—91no—
263C2×Dic6—91no—
264C2×Dic6—184no—
265C2×Dic6—181no—
266C2×Dic6—181no—
267C2×Dic6—181no—
268C2×Dic6—182no—
269C2×Dic6—182no—
270C2×Dic6—184no—
271C2×Dic6—181no—
272C2×Dic6—181no—
273C2×Dic6—182no—
274C2×Dic6—181no—
275C2×Dic6—361no—
276C3×M16—182no—
277C2×C3×C8—31no—
278C2×C3×C8—31no—
279C2×C3×C8—61no—
280C2×C3×C8—91no—
281C3×C4⋊C4—31no—
282C3×C4⋊C4—31no—
283C3×C4⋊C4—61no—
284C3×C4⋊C4—91no—
285C3×C22⋊C4—181no—
286C3×C42—181no—
287#217—61no—
288(C2×D3)⋊C4—61no—
289C4⋊Dic3—31no—
290C4⋊Dic3—31no—
291C4⋊Dic3—61no—
292C4⋊Dic3D3211yescharacteristic
293C4×Dic3—61no—
294C4×Dic3—61no—
295#208—62no—
296C2×C3⋊C8—31no—
297C2×C3⋊C8—31no—
298C2×C3⋊C8—61no—
299C2×C3⋊C8D3211yescharacteristic
300ΓL(1,𝔽25)—62no—
301C22×C32—21no—
302C22×C32—21no—
303C22×C32—21no—
304C22×C32—32no—
305C22×C32—41no—
306C22×C32—61no—
307C22×C32—61no—
308C22×C32C2×D1211yescharacteristic
309C22×C32D3×D411yescharacteristic
310C22×C32C2×#6711yescharacteristic
311C2×C3×D3—22no—
312C2×C3×D3—22no—
313C2×C3×D3—21no—
314C2×C3×D3—21no—
315C2×C3×D3—21no—
316C2×C3×D3—21no—
317C2×C3×D3—22no—
318C2×C3×D3(C2×D3)⋊C412yes—
319C2×C3×D3(C2×D3)⋊C412yes—
320C32×C4—21no—
321C32×C4—21no—
322C32×C4—21no—
323C32×C4—21no—
324C32×C4—21no—
325C32×C4—21no—
326C32×C4—32no—
327C32×C4—41no—
328C32×C4—41no—
329C32×C4—121no—
330C32×C4C2×C4×D311yescharacteristic
331C32×C4C2×C4×D311yescharacteristic
332C32×C4C2×C4×D311yescharacteristic
333C32×C4C2×D1211yescharacteristic
334C32×C4D3×D411yescharacteristic
335C32×C4C2×#6711yescharacteristic
336C3⋊Dic3—22no—
337C3⋊Dic3—41no—
338C3⋊Dic3—62no—
339C3⋊Dic3—64no—
340C3⋊Dic3—62no—
341C3⋊Dic3—62no—
342C3⋊Dic3—122no—
343C3⋊Dic3—122no—
344C3⋊Dic3—124no—
345C3×Dic3—22no—
346C3×Dic3—41no—
347C3×Dic3—41no—
348C3×Dic3—64no—
349C3×Dic3—62no—
350C3×Dic3—62no—
351C3×Dic3—81no—
352C3×Dic3—121no—
353C3×Dic3—121no—
354C3×Dic3—241no—
355C3×Dic3(C2×D3)⋊C412yes—
356C3×Dic3(C2×D3)⋊C412yes—
357C22×Q8—271no—
358C2×M16—91no—
359#117—91no—
360#103—91no—
361#103—271no—
362#103—271no—
363#103—271no—
364C33#41911yescharacteristic, complemented, Sylow
365C23×C3—31no—
366C23×C3—31no—
367C23×C3—61no—
368C22×D3#62611yescharacteristic
369C3×Q8—92no—
370C3×Q8—181no—
371C2×C3×C4—21no—
372C2×C3×C4—21no—
373C2×C3×C4—21no—
374C2×C3×C4—31no—
375C2×C3×C4—31no—
376C2×C3×C4—32no—
377C2×C3×C4—32no—
378C2×C3×C4—32no—
379C2×C3×C4—32no—
380C2×C3×C4—41no—
381C2×C3×C4—62no—
382C2×C3×C4—62no—
383C2×C3×C4—61no—
384C2×C3×C4—61no—
385C2×C3×C4—61no—
386C2×C3×C4—121no—
387C2×C3×C4—181no—
388C2×C3×C4—181no—
389C2×C3×C4C2×D3211yescharacteristic
390C2×C3×C4C2×D3211yescharacteristic
391C2×C3×C4C2×D3211yescharacteristic
392C2×Dic3—21no—
393C2×Dic3—62no—
394C2×Dic3—61no—
395C2×Dic3—61no—
396C2×Dic3—61no—
397C2×Dic3—61no—
398C2×Dic3—121no—
399C2×Dic3—184no—
400C2×Dic3—184no—
401C2×Dic3—188no—
402C2×Dic3—182no—
403C2×Dic3—181no—
404C2×Dic3—181no—
405C2×Dic3—181no—
406C2×Dic3—361no—
407C2×Dic3—361no—
408C2×Dic3—362no—
409C2×Dic3#62611yescharacteristic
410C4×D3—184no—
411C4×D3#62412yes—
412C4×D3#62612yes—
413Dic6—32no—
414Dic6—32no—
415Dic6—62no—
416Dic6—62no—
417Dic6—61no—
418Dic6—61no—
419Dic6—92no—
420Dic6—92no—
421Dic6—92no—
422Dic6—182no—
423Dic6—182no—
424Dic6—182no—
425Dic6—184no—
426Dic6—184no—
427Dic6—188no—
428Dic6—182no—
429Dic6—181no—
430Dic6—181no—
431Dic6—181no—
432Dic6—184no—
433Dic6—362no—
434Dic6—361no—
435Dic6—361no—
436Dic6—362no—
437C3×C8—61no—
438C3×C8—61no—
439C3×C8—121no—
440C3×C8—181no—
441C3⋊C8—21no—
442C3⋊C8—61no—
443C3⋊C8—61no—
444C3⋊C8—121no—
445C2×C32—21no—
446C2×C32—21no—
447C2×C32—21no—
448C2×C32—21no—
449C2×C32—21no—
450C2×C32—21no—
451C2×C32—21no—
452C2×C32—21no—
453C2×C32—21no—
454C2×C32—41no—
455C2×C32—41no—
456C2×C32—41no—
457C2×C32—62no—
458C2×C32D3×C22⋊C411yescharacteristic
459C2×C32#90911yescharacteristic
460C2×C32#90911yescharacteristic
461C2×C32#91511yescharacteristic
462C2×C32#91811yescharacteristic
463C2×C32C2×(C2×D3)⋊C411yescharacteristic
464C2×C32C2×(C2×D3)⋊C411yescharacteristic
465C2×C32#95111yescharacteristic
466C2×C32#95111yescharacteristic
467C3×D3—24no—
468C3×D3—22no—
469C3×D3—22no—
470C2×Q8—91no—
471C2×Q8—274no—
472C2×Q8—271no—
473C2×Q8—541no—
474C2×Q8—542no—
475C22×C4—31no—
476C22×C4—541no—
477M16—182no—
478C2×C8—31no—
479C2×C8—91no—
480C4⋊C4—31no—
481C4⋊C4—91no—
482C22⋊C4—181no—
483C42—181no—
484C22×C3—21no—
485C22×C3—21no—
486C22×C3—21no—
487C22×C3—32no—
488C22×C3—32no—
489C22×C3—41no—
490C22×C3—62no—
491C22×C3—62no—
492C22×C3—62no—
493C22×C3—61no—
494C22×C3—61no—
495C22×C3—61no—
496C22×C3—61no—
497C22×C3D3×D1211yescharacteristic
498C22×C3C2×#62611yescharacteristic
499C22×C3D3×#6711yescharacteristic
500C2×D3—21no—
501C2×D3—21no—
502C2×D3#372212yes—
503C3×C4—21no—
504C3×C4—21no—
505C3×C4—21no—
506C3×C4—21no—
507C3×C4—21no—
508C3×C4—21no—
509C3×C4—32no—
510C3×C4—32no—
511C3×C4—41no—
512C3×C4—41no—
513C3×C4—62no—
514C3×C4—121no—
515C3×C4—121no—
516C3×C4—182no—
517C3×C4—241no—
518C3×C4—361no—
519C3×C4C4×D3211yescharacteristic
520C3×C4C4×D3211yescharacteristic
521C3×C4D3×D1211yescharacteristic
522C3×C4C2×#62411yescharacteristic
523C3×C4C2×#62611yescharacteristic
524C3×C4D3×#6711yescharacteristic
525Dic3—41no—
526Dic3—64no—
527Dic3—62no—
528Dic3—62no—
529Dic3—121no—
530Dic3—121no—
531Dic3—184no—
532Dic3—182no—
533Dic3—182no—
534Dic3—182no—
535Dic3—241no—
536Dic3—362no—
537Dic3—362no—
538Dic3—364no—
539Dic3#372212yes—
540C32—21no—
541C32—21no—
542C32—21no—
543C32—41no—
544C32#480211yescharacteristic, complemented
545C32#500411yescharacteristic, complemented
546C32#512011yescharacteristic, complemented
547C23—31no—
548Q8—92no—
549Q8—181no—
550Q8—272no—
551Q8—544no—
552Q8—541no—
553C2×C4—31no—
554C2×C4—32no—
555C2×C4—32no—
556C2×C4—61no—
557C2×C4—181no—
558C2×C4—181no—
559C2×C4—544no—
560C2×C4—541no—
561C2×C4D3311yescharacteristic
562C8—61no—
563C8—181no—
564C2×C3—21no—
565C2×C3—21no—
566C2×C3—21no—
567C2×C3—21no—
568C2×C3—21no—
569C2×C3—21no—
570C2×C3—21no—
571C2×C3—21no—
572C2×C3—21no—
573C2×C3—41no—
574C2×C3—41no—
575C2×C3—41no—
576C2×C3—64no—
577C2×C3—62no—
578C2×C3—62no—
579C2×C3#6380011yescharacteristic
580C2×C3#6380111yescharacteristic
581C2×C3#6383111yescharacteristic
582C2×C3#6383111yescharacteristic
583C2×C3D3×(C2×D3)⋊C411yescharacteristic
584C2×C3D3×(C2×D3)⋊C411yescharacteristic
585C2×C3#6392911yescharacteristic
586C2×C3#6394311yescharacteristic
587C2×C3C2×#372211yescharacteristic
588D3—22no—
589C22—32no—
590C22—61no—
591C22—61no—
592C22D3×#62611yescharacteristic
593C4—32no—
594C4—121no—
595C4—182no—
596C4—361no—
597C4—542no—
598C4D3×#62411yescharacteristic
599C4D3×#62611yescharacteristic
600C3—21no—
601C3—21no—
602C3—21no—
603C3—41no—
604C3#1059060011yescharacteristic, complemented
605C3#1059094811yescharacteristic, complemented
606C3#1059110711yescharacteristic, complemented
607C2—62no—
608C2#1172288611yescharacteristic
609C2#1172288711yescharacteristic
610C2D3×#372211yescharacteristic
subgroupG/H
centerC22D3×#626
commutatorC33×C4C22×C4
Frattini, hypercenterC2×C4D33
Fitting, cosocleC2×C33×C4C23
socleC22×C33C2×D4
prime radicalC22×C32×D3D4
hypocenterC33#419
LCSG ▷ C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
4. #11722076C2semidirect
5. #11722042C2semidirect
6. #11722041C2semidirect
7. C3×#63567C2semidirect
17. D3×#634C4semidirect
46. #63567D3semidirect
364. C33#419semidirect
544. C32#4802semidirect
545. C32#5004semidirect
546. C32#5120semidirect
604. C3#10590600semidirect
605. C3#10590948semidirect
606. C3#10591107semidirect

Maximal subgroups

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

subgroupindex
C2×D3×#6342
C3×#638412
C2×C3×#37112
#117220762
#117220422
#117220412
C3×#635672
#105911073
#105909483
#105906003