Finite Group Databasebeta

← #13161772

Subgroups — #13161772

Order 1728.

772 conjugacy classes of subgroups.

The subgroup lattice has 650 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#89041C211yesmaximal, characteristic
2C3×M16×C3⋊D3C211yesmaximal, characteristic, complemented
3#11723656C211yesmaximal, characteristic, complemented
4C3×#63651C211yesmaximal, characteristic, complemented
5C3×#63650C211yesmaximal, characteristic, complemented
6#11722074C211yesmaximal, characteristic, complemented
7#11722073C211yesmaximal, characteristic, complemented
8#10591010—31nomaximal
9#10591007—31nomaximal
10#10591006—31nomaximal
11C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
12C2×C3×C4×C3⋊D3C2211yescharacteristic
13C2×C3×#634C2211yescharacteristic
14C2×#6295C412yes—
15C2×#6294C412yes—
16#89041—21no—
17#89041—21no—
18#89041—21no—
19#89041—21no—
20C33×M16C2211yescharacteristic
21C3×#3748C2211yescharacteristic
22C3×#3748C2211yescharacteristic
23C3×#3748C2211yescharacteristic
24C3×#3743—21no—
25C3×C8×C3⋊D3—21no—
26#88809—21no—
27#88808—21no—
28C2×#3799—31no—
29C2×#3796—31no—
30C2×#3795—31no—
31M16×C3⋊D3D311yescharacteristic, complemented
32C3×D3×M16—31no—
33C3×D3×M16—31no—
34C3×D3×M16—61no—
35#63818—31no—
36D3×#208—31no—
37D3×#208—31no—
38#63651D311yescharacteristic, complemented
39#63650D311yescharacteristic, complemented
40C3×#824—31no—
41C3×#824—31no—
42C3×#824—61no—
43C3×#823—31no—
44C3×#823—31no—
45C3×#823—61no—
46#63561—31no—
47#63560—31no—
48#63560—31no—
49#63559—31no—
50#63558—31no—
51#63558—31no—
52C22×C3×C3⋊D3C2×C412yes—
53C22×C3×C3⋊D3C2×C411yescharacteristic
54C2×C33×C4C2311yescharacteristic
55C2×C3×C3⋊Dic3C2×C412yes—
56C2×C3×C3⋊Dic3C2×C411yescharacteristic
57C3×C4⋊(C3⋊D3)—21no—
58C3×C4⋊(C3⋊D3)—21no—
59C3×C4×C3⋊D3D411yescharacteristic
60C3×C4×C3⋊D3D411yescharacteristic
61C3×C4×C3⋊D3D411yescharacteristic
62C3×C4×C3⋊D3D411yescharacteristic
63C3×#634—21no—
64C3×#634—21no—
65#6295—22no—
66#6295—22no—
67#6294—22no—
68#6294—22no—
69C33×C8—21no—
70C3×C3⋊(C3⋊C8)—21no—
71C3×C3⋊(C3⋊C8)—21no—
72C3×C3⋊(C3⋊C8)—21no—
73#5132—91no—
74#4751—91no—
75#4745—91no—
76C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
77C2×C4×C3⋊D3C2×D311yescharacteristic
78C2×#634C2×D311yescharacteristic
79C2×C3×D12—31no—
80C2×C3×D12—31no—
81C2×C3×D12—61no—
82C2×C3×C4×D3—31no—
83C2×C3×C4×D3—31no—
84C2×C3×C4×D3—61no—
85C2×C3×Dic6—31no—
86C2×C3×Dic6—31no—
87C2×C3×Dic6—61no—
88C2×#626—32no—
89C2×#626—32no—
90C2×#625—32no—
91C2×#624—32no—
92C2×D3×Dic3—32no—
93C2×D3×Dic3—32no—
94#3799—61no—
95#3799—61no—
96#3799—61no—
97#3799—61no—
98#3796—61no—
99#3796—61no—
100#3796—61no—
101#3796—61no—
102#3795—61no—
103#3795—61no—
104#3795—61no—
105#3795—61no—
106C32×M16—21no—
107C32×M16—21no—
108C32×M16—21no—
109C32×M16—41no—
110C32×M16C2×D311yescharacteristic
111C32×M16C2×D311yescharacteristic
112C32×M16C2×D311yescharacteristic
113#3748C2×D311yescharacteristic
114#3748C2×D311yescharacteristic
115#3748C2×D311yescharacteristic
116#3743—21no—
117C8×C3⋊D3—21no—
118C3×#208—31no—
119C3×#208—31no—
120C3×#208—31no—
121C3×#208—31no—
122C3×#208—31no—
123C3×#208—31no—
124C3×#208—61no—
125C3×#208—61no—
126C3×#208—61no—
127C3×ΓL(1,𝔽25)—61no—
128C3×ΓL(1,𝔽25)—61no—
129C3×ΓL(1,𝔽25)—62no—
130C3×D3×C8—61no—
131C3×D3×C8—61no—
132C3×D3×C8—62no—
133#3712—61no—
134#3711—61no—
135#3711—61no—
136#3710—61no—
137D3×C3⋊C8—61no—
138D3×C3⋊C8—61no—
139C22×C33C22×C411yescharacteristic
140C2×C3×C3⋊D3—22no—
141C2×C3×C3⋊D3—41no—
142C2×C3×C3⋊D3—42no—
143C2×C3×C3⋊D3C22⋊C411yescharacteristic
144C2×C3×C3⋊D3C22⋊C411yescharacteristic
145C33×C4C2×D411yescharacteristic
146C33×C4C2×D411yescharacteristic
147C3×C3⋊Dic3—22no—
148C3×C3⋊Dic3C22⋊C411yescharacteristic
149C3×C3⋊Dic3C22⋊C411yescharacteristic
150C2×#239—91no—
151C2×#237—91no—
152C2×#235—91no—
153C2×C3×M16—91no—
154C2×#208—91no—
155D3×M16—31no—
156D3×M16—31no—
157D3×M16—61no—
158D3×M16—91no—
159D3×M16—91no—
160C3×#101—91no—
161C3×#100—91no—
162#836—91no—
163#833—91no—
164#824—31no—
165#824—31no—
166#824—61no—
167#824—91no—
168#824—91no—
169#823—31no—
170#823—31no—
171#823—61no—
172#823—91no—
173#823—91no—
174C22×C3⋊D3C4×D312yes—
175C22×C3⋊D3C4×D311yescharacteristic
176C22×C3×D3—31no—
177C22×C3×D3—31no—
178C22×C3×D3—32no—
179C22×C3×D3—32no—
180C22×C3×D3—61no—
181C22×C3×D3—62no—
182C2×C32×C4—21no—
183C2×C32×C4—21no—
184C2×C32×C4—21no—
185C2×C32×C4—41no—
186C2×C32×C4C22×D311yescharacteristic
187C2×C32×C4C22×D311yescharacteristic
188C2×C32×C4C22×D311yescharacteristic
189C2×C3⋊Dic3C4×D312yes—
190C2×C3⋊Dic3C4×D311yescharacteristic
191C4⋊(C3⋊D3)—21no—
192C4⋊(C3⋊D3)—21no—
193C4×C3⋊D3D1211yescharacteristic
194C4×C3⋊D3D1211yescharacteristic
195C4×C3⋊D3#6711yescharacteristic
196C4×C3⋊D3#6711yescharacteristic
197#634—21no—
198#634—21no—
199C2×C3×Dic3—31no—
200C2×C3×Dic3—31no—
201C2×C3×Dic3—32no—
202C2×C3×Dic3—32no—
203C2×C3×Dic3—61no—
204C2×C3×Dic3—62no—
205C3×D12—61no—
206C3×D12—61no—
207C3×D12—61no—
208C3×D12—61no—
209C3×D12—121no—
210C3×D12—121no—
211C3×C4×D3—31no—
212C3×C4×D3—31no—
213C3×C4×D3—31no—
214C3×C4×D3—31no—
215C3×C4×D3—31no—
216C3×C4×D3—31no—
217C3×C4×D3—31no—
218C3×C4×D3—31no—
219C3×C4×D3—61no—
220C3×C4×D3—61no—
221C3×C4×D3—61no—
222C3×C4×D3—61no—
223C3×Dic6—61no—
224C3×Dic6—61no—
225C3×Dic6—61no—
226C3×Dic6—61no—
227C3×Dic6—121no—
228C3×Dic6—121no—
229#626—62no—
230#626—62no—
231#626—62no—
232#626—62no—
233#625—62no—
234#625—62no—
235#624—62no—
236#624—62no—
237D3×Dic3—62no—
238D3×Dic3—62no—
239D3×Dic3—62no—
240D3×Dic3—62no—
241C32×C8—21no—
242C32×C8—22no—
243C32×C8—21no—
244C32×C8—21no—
245C32×C8—41no—
246C32×C8—41no—
247C32×C8—42no—
248C3⋊(C3⋊C8)—21no—
249C3⋊(C3⋊C8)—21no—
250C3⋊(C3⋊C8)—21no—
251C3×C3⋊C8—62no—
252C3×C3⋊C8—61no—
253C3×C3⋊C8—61no—
254C3×C3⋊C8—61no—
255C3×C3⋊C8—61no—
256C3×C3⋊C8—61no—
257C3×C3⋊C8—61no—
258C3×C3⋊C8—121no—
259C3×C3⋊C8—121no—
260#413—271noSylow
261C2×C33—21no—
262C2×C33C2×C22⋊C411yescharacteristic
263C3×C3⋊D3—21no—
264C3×C3⋊D3—21no—
265C3×C3⋊D3—42no—
266C2×C3×Q8—91no—
267C2×C3×D4—91no—
268C22×C3×C4—91no—
269C2×#67—92no—
270C2×#67—92no—
271C22×Dic3—92no—
272#239—181no—
273#239—181no—
274#239—181no—
275#239—181no—
276#237—181no—
277#237—181no—
278#237—181no—
279#237—181no—
280#235—181no—
281#235—181no—
282#235—181no—
283#235—181no—
284C2×D12—31no—
285C2×D12—31no—
286C2×D12—61no—
287C2×D12—92no—
288C2×C4×D3—31no—
289C2×C4×D3—31no—
290C2×C4×D3—61no—
291C2×C4×D3—92no—
292C2×C4×D3—92no—
293C2×Dic6—31no—
294C2×Dic6—31no—
295C2×Dic6—61no—
296C3×M16—21no—
297C3×M16—21no—
298C3×M16—21no—
299C3×M16—41no—
300C3×M16—91no—
301C3×M16—91no—
302C3×M16—91no—
303C3×M16—181no—
304C3×M16D3211yescharacteristic
305C3×M16D3211yescharacteristic
306C3×M16D3211yescharacteristic
307C2×C3×C8—181no—
308#208—31no—
309#208—31no—
310#208—31no—
311#208—31no—
312#208—31no—
313#208—31no—
314#208—61no—
315#208—61no—
316#208—61no—
317#208—181no—
318C2×C3⋊C8—181no—
319ΓL(1,𝔽25)—61no—
320ΓL(1,𝔽25)—61no—
321ΓL(1,𝔽25)—62no—
322ΓL(1,𝔽25)—181no—
323ΓL(1,𝔽25)—181no—
324D3×C8—61no—
325D3×C8—61no—
326D3×C8—62no—
327D3×C8—181no—
328D3×C8—181no—
329C22×C32—21no—
330C22×C32—21no—
331C22×C32—21no—
332C22×C32—41no—
333C22×C32C2×C4×D311yescharacteristic
334C22×C32C2×C4×D311yescharacteristic
335C22×C32C2×C4×D311yescharacteristic
336C2×C3⋊D3—22no—
337C2×C3⋊D3—41no—
338C2×C3⋊D3—42no—
339C2×C3⋊D3(C2×D3)⋊C411yescharacteristic
340C2×C3⋊D3(C2×D3)⋊C411yescharacteristic
341C2×C3×D3—31no—
342C2×C3×D3—31no—
343C2×C3×D3—31no—
344C2×C3×D3—31no—
345C2×C3×D3—61no—
346C2×C3×D3—61no—
347C2×C3×D3—62no—
348C2×C3×D3—62no—
349C2×C3×D3—122no—
350C2×C3×D3—121no—
351C2×C3×D3—121no—
352C2×C3×D3—122no—
353C2×C3×D3—122no—
354C2×C3×D3—122no—
355C2×C3×D3—242no—
356C32×C4—21no—
357C32×C4—21no—
358C32×C4—21no—
359C32×C4—21no—
360C32×C4—21no—
361C32×C4—21no—
362C32×C4—41no—
363C32×C4—41no—
364C32×C4C2×D1211yescharacteristic
365C32×C4D3×D411yescharacteristic
366C32×C4D3×D411yescharacteristic
367C32×C4D3×D411yescharacteristic
368C32×C4D3×D411yescharacteristic
369C32×C4C2×#6711yescharacteristic
370C3⋊Dic3—22no—
371C3⋊Dic3(C2×D3)⋊C411yescharacteristic
372C3⋊Dic3(C2×D3)⋊C411yescharacteristic
373C3×Dic3—31no—
374C3×Dic3—31no—
375C3×Dic3—31no—
376C3×Dic3—31no—
377C3×Dic3—61no—
378C3×Dic3—61no—
379C3×Dic3—62no—
380C3×Dic3—62no—
381C3×Dic3—122no—
382C2×#41—271no—
383C2×M16—91no—
384C2×M16—271no—
385#101—91no—
386#101—271no—
387#100—91no—
388#100—271no—
389C33#41311yescharacteristic, complemented, Sylow
390C23×C3—91no—
391C23×C3—92no—
392C22×D3—31no—
393C22×D3—31no—
394C22×D3—32no—
395C22×D3—32no—
396C22×D3—61no—
397C22×D3—62no—
398C3×Q8—181no—
399C3×Q8—181no—
400C3×D4—181no—
401C3×D4—181no—
402C2×C3×C4—21no—
403C2×C3×C4—21no—
404C2×C3×C4—21no—
405C2×C3×C4—41no—
406C2×C3×C4—91no—
407C2×C3×C4—91no—
408C2×C3×C4—91no—
409C2×C3×C4—91no—
410C2×C3×C4—91no—
411C2×C3×C4—92no—
412C2×C3×C4C2×D3211yescharacteristic
413C2×C3×C4C2×D3211yescharacteristic
414C2×C3×C4C2×D3211yescharacteristic
415#67—182no—
416#67—182no—
417#67—182no—
418#67—182no—
419C2×Dic3—31no—
420C2×Dic3—31no—
421C2×Dic3—32no—
422C2×Dic3—32no—
423C2×Dic3—61no—
424C2×Dic3—62no—
425C2×Dic3—182no—
426C2×Dic3—182no—
427D12—61no—
428D12—61no—
429D12—61no—
430D12—61no—
431D12—121no—
432D12—121no—
433D12—182no—
434D12—182no—
435C4×D3—31no—
436C4×D3—31no—
437C4×D3—31no—
438C4×D3—31no—
439C4×D3—31no—
440C4×D3—31no—
441C4×D3—31no—
442C4×D3—31no—
443C4×D3—61no—
444C4×D3—61no—
445C4×D3—61no—
446C4×D3—61no—
447C4×D3—182no—
448C4×D3—182no—
449C4×D3—182no—
450C4×D3—182no—
451Dic6—61no—
452Dic6—61no—
453Dic6—61no—
454Dic6—61no—
455Dic6—121no—
456Dic6—121no—
457C3×C8—22no—
458C3×C8—21no—
459C3×C8—21no—
460C3×C8—21no—
461C3×C8—41no—
462C3×C8—41no—
463C3×C8—42no—
464C3×C8—181no—
465C3×C8—181no—
466C3×C8—181no—
467C3⋊C8—62no—
468C3⋊C8—61no—
469C3⋊C8—61no—
470C3⋊C8—61no—
471C3⋊C8—61no—
472C3⋊C8—61no—
473C3⋊C8—61no—
474C3⋊C8—121no—
475C3⋊C8—121no—
476C2×C32—21no—
477C2×C32—21no—
478C2×C32—21no—
479C2×C32—21no—
480C2×C32—21no—
481C2×C32—21no—
482C2×C32—41no—
483C2×C32—41no—
484C2×C32—41no—
485C2×C32—41no—
486C2×C32—81no—
487C2×C32D3×C22⋊C411yescharacteristic
488C2×C32D3×C22⋊C411yescharacteristic
489C2×C32C2×(C2×D3)⋊C411yescharacteristic
490C3⋊D3—21no—
491C3⋊D3—21no—
492C3⋊D3—42no—
493C3×D3—62no—
494C3×D3—61no—
495C3×D3—61no—
496C3×D3—61no—
497C3×D3—61no—
498C3×D3—121no—
499C3×D3—122no—
500C3×D3—122no—
501C3×D3—242no—
502#41—541no—
503#41—541no—
504#41—541no—
505#41—541no—
506C2×Q8—91no—
507C2×D4—91no—
508C2×D4—272no—
509C22×C4—91no—
510C22×C4—272no—
511M16—91no—
512M16—91no—
513M16—91no—
514M16—181no—
515M16—541no—
516M16#111311yescharacteristic
517C2×C8—181no—
518C2×C8—541no—
519C22×C3—21no—
520C22×C3—21no—
521C22×C3—21no—
522C22×C3—41no—
523C22×C3—91no—
524C22×C3—91no—
525C22×C3—182no—
526C22×C3—361no—
527C22×C3—362no—
528C22×C3C4×D3211yescharacteristic
529C22×C3C4×D3211yescharacteristic
530C22×C3C4×D3211yescharacteristic
531C2×D3—31no—
532C2×D3—31no—
533C2×D3—31no—
534C2×D3—31no—
535C2×D3—61no—
536C2×D3—61no—
537C2×D3—62no—
538C2×D3—62no—
539C2×D3—122no—
540C2×D3—121no—
541C2×D3—121no—
542C2×D3—122no—
543C2×D3—122no—
544C2×D3—122no—
545C2×D3—242no—
546C3×C4—21no—
547C3×C4—21no—
548C3×C4—21no—
549C3×C4—21no—
550C3×C4—21no—
551C3×C4—21no—
552C3×C4—41no—
553C3×C4—41no—
554C3×C4—91no—
555C3×C4—91no—
556C3×C4—182no—
557C3×C4D3×D1211yescharacteristic
558C3×C4D3×D1211yescharacteristic
559C3×C4#380211yescharacteristic
560C3×C4D3×#6711yescharacteristic
561C3×C4D3×#6711yescharacteristic
562C3×C4#381111yescharacteristic
563Dic3—31no—
564Dic3—31no—
565Dic3—31no—
566Dic3—31no—
567Dic3—61no—
568Dic3—61no—
569Dic3—62no—
570Dic3—62no—
571Dic3—122no—
572C32—21no—
573C32—21no—
574C32—21no—
575C32—41no—
576C32#474511yescharacteristic, complemented
577C32#475111yescharacteristic, complemented
578C32#513211yescharacteristic, complemented
579C23—91no—
580C23—92no—
581Q8—181no—
582Q8—181no—
583D4—181no—
584D4—181no—
585D4—542no—
586D4—542no—
587C2×C4—91no—
588C2×C4—91no—
589C2×C4—91no—
590C2×C4—91no—
591C2×C4—91no—
592C2×C4—92no—
593C2×C4—542no—
594C2×C4—542no—
595C2×C4C2×#111311yescharacteristic
596C8—21no—
597C8—181no—
598C8—181no—
599C8—181no—
600C2×C3—21no—
601C2×C3—21no—
602C2×C3—21no—
603C2×C3—21no—
604C2×C3—21no—
605C2×C3—21no—
606C2×C3—41no—
607C2×C3—41no—
608C2×C3—41no—
609C2×C3—41no—
610C2×C3—81no—
611C2×C3—181no—
612C2×C3—181no—
613C2×C3—362no—
614C2×C3D3×(C2×D3)⋊C411yescharacteristic
615C2×C3D3×(C2×D3)⋊C411yescharacteristic
616C2×C3#6392111yescharacteristic
617D3—62no—
618D3—61no—
619D3—61no—
620D3—61no—
621D3—61no—
622D3—121no—
623D3—122no—
624D3—122no—
625D3—242no—
626C22—91no—
627C22—91no—
628C22—182no—
629C22—361no—
630C22—362no—
631C22C4×#111311yescharacteristic
632C4—91no—
633C4—91no—
634C4—182no—
635C4C4⋊#111311yescharacteristic
636C4#8904911yescharacteristic
637C3—21no—
638C3—21no—
639C3—21no—
640C3—41no—
641C3#1059100611yescharacteristic, complemented
642C3#1059100711yescharacteristic, complemented
643C3#1059101011yescharacteristic, complemented
644C2—21no—
645C2—181no—
646C2—181no—
647C2—362no—
648C2#1172370011yescharacteristic
subgroupG/H
centerC2#11723700
commutatorC22×C33C22×C4
FrattiniC2×C4C2×#1113
FittingC33×M16C22
socleC2×C33C2×C22⋊C4
prime radicalC2×#89041C2
cosocleC2×C33×C4C23
hypercenterM16#1113
hypocenterC33#413
LCSG ▷ C22×C33 ▷ C2×C33 ▷ C33
UCSC1 ◁ C2 ◁ C2×C4 ◁ M16

Complements

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

NH ≅ G/Ntype
2. C3×M16×C3⋊D3C2semidirect
3. #11723656C2semidirect
4. C3×#63651C2semidirect
5. C3×#63650C2semidirect
6. #11722074C2semidirect
7. #11722073C2semidirect
31. M16×C3⋊D3D3semidirect
38. #63651D3semidirect
39. #63650D3semidirect
389. C33#413semidirect
576. C32#4745semidirect
577. C32#4751semidirect
578. C32#5132semidirect
641. C3#10591006semidirect
642. C3#10591007semidirect
643. C3#10591010semidirect

Maximal subgroups

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

subgroupindex
C2×#890412
C3×M16×C3⋊D32
#117236562
C3×#636512
C3×#636502
#117220742
#117220732
#105910103
#105910073
#105910063