Finite Group Databasebeta

← #13161750

Subgroups — #13161750

Order 1728.

822 conjugacy classes of subgroups.

The subgroup lattice has 614 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#89041C211yesmaximal, characteristic
2C2×C3×C8×C3⋊D3C211yesmaximal, characteristic, complemented
3#11723667C211yesmaximal, characteristic, complemented
4C3×#63648C211yesmaximal, characteristic, complemented
5C3×#63643C211yesmaximal, characteristic, complemented
6#11722076C211yesmaximal, characteristic, complemented
7#11722075C211yesmaximal, characteristic, complemented
8#10590956—31nomaximal
9#10590954—31nomaximal
10#10590690—31nomaximal
11C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
12C2×C3×C4×C3⋊D3C2211yescharacteristic
13C2×C3×#634C2211yescharacteristic
14C2×#6295—21no—
15C2×#6294—21no—
16#89041—22no—
17#89041C414yescomplemented
18C2×C33×C8C2211yescharacteristic
19C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
20C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
21C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
22C3×C8×C3⋊D3—22no—
23#88814—21no—
24#88813—21no—
25C2×#3799—31no—
26C2×#3796—31no—
27C2×#3795—31no—
28C2×C8×C3⋊D3D311yescharacteristic, complemented
29C2×C3×D3×C8—31no—
30C2×C3×D3×C8—31no—
31C2×C3×D3×C8—61no—
32#63849—31no—
33#63841—31no—
34#63841—31no—
35#63648D311yescharacteristic, complemented
36#63643D311yescharacteristic, complemented
37C3×#821—31no—
38C3×#821—31no—
39C3×#821—61no—
40C3×#816—31no—
41C3×#816—31no—
42C3×#816—61no—
43#63567—31no—
44#63566—31no—
45#63565—31no—
46#63564—31no—
47#63563—31no—
48#63562—31no—
49C22×C3×C3⋊D3—21no—
50C22×C3×C3⋊D3D411yescharacteristic
51C2×C33×C4C2311yescharacteristic
52C2×C3×C3⋊Dic3—21no—
53C2×C3×C3⋊Dic3—21no—
54C2×C3×C3⋊Dic3—21no—
55C2×C3×C3⋊Dic3D411yescharacteristic
56C3×C4⋊(C3⋊D3)—21no—
57C3×C4⋊(C3⋊D3)C2×C412yes—
58C3×C4×C3⋊D3C2×C412yes—
59C3×C4×C3⋊D3D412yes—
60C3×#634—21no—
61C3×#634C2×C412yes—
62#6295—24no—
63#6294—24no—
64C33×C8—21no—
65C3×C3⋊(C3⋊C8)—21no—
66#5113—91no—
67#4804—91no—
68#4772—91no—
69C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
70C2×C4×C3⋊D3C2×D311yescharacteristic
71C2×#634C2×D311yescharacteristic
72C2×C3×D12—31no—
73C2×C3×D12—31no—
74C2×C3×D12—61no—
75C2×C3×C4×D3—31no—
76C2×C3×C4×D3—31no—
77C2×C3×C4×D3—61no—
78C2×C3×Dic6—31no—
79C2×C3×Dic6—31no—
80C2×C3×Dic6—61no—
81C2×#626—61no—
82C2×#626—61no—
83C2×#625—61no—
84C2×#624—61no—
85C2×D3×Dic3—61no—
86C2×D3×Dic3—61no—
87#3799—34no—
88#3799—62no—
89#3796—34no—
90#3796—62no—
91#3795—34no—
92#3795—62no—
93C2×C32×C8—21no—
94C2×C32×C8—21no—
95C2×C32×C8—21no—
96C2×C32×C8—41no—
97C2×C32×C8C2×D311yescharacteristic
98C2×C32×C8C2×D311yescharacteristic
99C2×C32×C8C2×D311yescharacteristic
100C4⋊(C3⋊Dic3)C2×D311yescharacteristic
101C4⋊(C3⋊Dic3)C2×D311yescharacteristic
102C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
103C8×C3⋊D3—22no—
104C3×C4⋊Dic3—31no—
105C3×C4⋊Dic3—31no—
106C3×C4⋊Dic3—31no—
107C3×C4⋊Dic3—31no—
108C3×C4⋊Dic3—61no—
109C3×C4⋊Dic3—61no—
110C2×C3×C3⋊C8—31no—
111C2×C3×C3⋊C8—31no—
112C2×C3×C3⋊C8—61no—
113C3×D3×C8—62no—
114C3×D3×C8—62no—
115C3×D3×C8—64no—
116#3722—61no—
117(C2×D3)⋊Dic3—61no—
118(C2×D3)⋊Dic3—61no—
119Dic32—61no—
120Dic32—61no—
121Dic32—61no—
122C22×C33C2×D411yescharacteristic
123C2×C3×C3⋊D3—21no—
124C2×C3×C3⋊D3—21no—
125C2×C3×C3⋊D3—22no—
126C2×C3×C3⋊D3—41no—
127C2×C3×C3⋊D3—41no—
128C2×C3×C3⋊D3C22⋊C412yes—
129C33×C4C22×C411yescharacteristic
130C33×C4C2×D411yescharacteristic
131C3×C3⋊Dic3—22no—
132C3×C3⋊Dic3—41no—
133C3×C3⋊Dic3—41no—
134C3×C3⋊Dic3C22⋊C412yes—
135C2×#239—91no—
136C2×#237—91no—
137C2×#235—91no—
138C22×C3×C8—91no—
139#926—91no—
140C2×D3×C8—31no—
141C2×D3×C8—31no—
142C2×D3×C8—61no—
143#892—91no—
144#892—91no—
145C3×#103—91no—
146C3×#102—91no—
147#835—91no—
148#832—91no—
149#821—31no—
150#821—31no—
151#821—61no—
152#821—91no—
153#816—31no—
154#816—31no—
155#816—61no—
156#816—91no—
157#810—91no—
158#809—91no—
159C22×C3⋊D3—21no—
160C22×C3⋊D3D1211yescharacteristic
161C22×C3×D3—31no—
162C22×C3×D3—31no—
163C22×C3×D3—61no—
164C22×C3×D3—61no—
165C22×C3×D3—61no—
166C22×C3×D3—121no—
167C2×C32×C4—21no—
168C2×C32×C4—21no—
169C2×C32×C4—21no—
170C2×C32×C4—41no—
171C2×C32×C4C22×D311yescharacteristic
172C2×C32×C4C22×D311yescharacteristic
173C2×C32×C4C22×D311yescharacteristic
174C2×C3⋊Dic3—21no—
175C2×C3⋊Dic3—21no—
176C2×C3⋊Dic3—21no—
177C2×C3⋊Dic3D1211yescharacteristic
178C4⋊(C3⋊D3)—21no—
179C4⋊(C3⋊D3)C4×D312yes—
180C4×C3⋊D3C4×D312yes—
181C4×C3⋊D3#6712yes—
182#634—21no—
183#634C4×D312yes—
184C2×C3×Dic3—31no—
185C2×C3×Dic3—31no—
186C2×C3×Dic3—61no—
187C2×C3×Dic3—61no—
188C2×C3×Dic3—61no—
189C2×C3×Dic3—61no—
190C2×C3×Dic3—61no—
191C2×C3×Dic3—61no—
192C2×C3×Dic3—61no—
193C2×C3×Dic3—121no—
194C2×C3×Dic3—121no—
195C2×C3×Dic3—121no—
196C3×D12—32no—
197C3×D12—32no—
198C3×D12—62no—
199C3×D12—61no—
200C3×D12—61no—
201C3×D12—121no—
202C3×C4×D3—32no—
203C3×C4×D3—32no—
204C3×C4×D3—32no—
205C3×C4×D3—32no—
206C3×C4×D3—62no—
207C3×C4×D3—62no—
208C3×Dic6—32no—
209C3×Dic6—32no—
210C3×Dic6—62no—
211C3×Dic6—61no—
212C3×Dic6—61no—
213C3×Dic6—121no—
214#626—64no—
215#626—64no—
216#625—64no—
217#624—64no—
218D3×Dic3—64no—
219D3×Dic3—64no—
220C32×C8—21no—
221C32×C8—22no—
222C32×C8—21no—
223C32×C8—21no—
224C32×C8—41no—
225C32×C8—41no—
226C32×C8—42no—
227C3⋊(C3⋊C8)—21no—
228C3×C3⋊C8—62no—
229C3×C3⋊C8—61no—
230C3×C3⋊C8—61no—
231#416—271noSylow
232C2×C33C2×C22⋊C411yescharacteristic
233C2×C33#13511yescharacteristic
234C2×C33#13511yescharacteristic
235C3×C3⋊D3—22no—
236C3×C3⋊D3—42no—
237C2×C3×Q8—91no—
238C2×C3×D4—91no—
239C22×C3×C4—91no—
240C2×#67—181no—
241C2×#67—181no—
242C22×Dic3—181no—
243#239—94no—
244#239—182no—
245#237—94no—
246#237—182no—
247#235—94no—
248#235—182no—
249C2×D12—31no—
250C2×D12—31no—
251C2×D12—61no—
252C2×D12—181no—
253C2×C4×D3—31no—
254C2×C4×D3—31no—
255C2×C4×D3—61no—
256C2×C4×D3—181no—
257C2×C4×D3—181no—
258C2×Dic6—31no—
259C2×Dic6—31no—
260C2×Dic6—61no—
261C2×C3×C8—21no—
262C2×C3×C8—21no—
263C2×C3×C8—21no—
264C2×C3×C8—41no—
265C2×C3×C8—91no—
266C2×C3×C8—182no—
267C2×C3×C8D3211yescharacteristic
268C2×C3×C8D3211yescharacteristic
269C2×C3×C8D3211yescharacteristic
270C3×C4⋊C4—91no—
271C3×C4⋊C4—91no—
272#217—181no—
273(C2×D3)⋊C4—181no—
274(C2×D3)⋊C4—181no—
275C4⋊Dic3—31no—
276C4⋊Dic3—31no—
277C4⋊Dic3—31no—
278C4⋊Dic3—31no—
279C4⋊Dic3—61no—
280C4⋊Dic3—61no—
281C4×Dic3—181no—
282C4×Dic3—181no—
283C4×Dic3—181no—
284C2×C3⋊C8—31no—
285C2×C3⋊C8—31no—
286C2×C3⋊C8—61no—
287D3×C8—62no—
288D3×C8—62no—
289D3×C8—64no—
290C22×C32—21no—
291C22×C32—21no—
292C22×C32—21no—
293C22×C32—41no—
294C22×C32C2×D1211yescharacteristic
295C22×C32D3×D411yescharacteristic
296C22×C32D3×D411yescharacteristic
297C2×C3⋊D3—21no—
298C2×C3⋊D3—21no—
299C2×C3⋊D3—22no—
300C2×C3⋊D3—41no—
301C2×C3⋊D3—41no—
302C2×C3⋊D3(C2×D3)⋊C412yes—
303C2×C3×D3—32no—
304C2×C3×D3—32no—
305C2×C3×D3—62no—
306C2×C3×D3—62no—
307C2×C3×D3—62no—
308C2×C3×D3—61no—
309C2×C3×D3—61no—
310C2×C3×D3—61no—
311C2×C3×D3—61no—
312C2×C3×D3—62no—
313C2×C3×D3—62no—
314C2×C3×D3—122no—
315C2×C3×D3—121no—
316C2×C3×D3—121no—
317C2×C3×D3—121no—
318C2×C3×D3—121no—
319C2×C3×D3—241no—
320C2×C3×D3—241no—
321C32×C4—21no—
322C32×C4—21no—
323C32×C4—21no—
324C32×C4—21no—
325C32×C4—21no—
326C32×C4—21no—
327C32×C4—41no—
328C32×C4—41no—
329C32×C4C2×C4×D311yescharacteristic
330C32×C4C2×C4×D311yescharacteristic
331C32×C4C2×C4×D311yescharacteristic
332C32×C4D3×D411yescharacteristic
333C32×C4D3×D411yescharacteristic
334C32×C4C2×#6711yescharacteristic
335C3⋊Dic3—22no—
336C3⋊Dic3—41no—
337C3⋊Dic3—41no—
338C3⋊Dic3(C2×D3)⋊C412yes—
339C3×Dic3—32no—
340C3×Dic3—32no—
341C3×Dic3—62no—
342C3×Dic3—62no—
343C3×Dic3—62no—
344C3×Dic3—122no—
345C3×Dic3—121no—
346C3×Dic3—121no—
347C3×Dic3—121no—
348C3×Dic3—121no—
349C3×Dic3—241no—
350C3×Dic3—241no—
351C2×#41—271no—
352C22×C8—91no—
353#117—271no—
354#103—91no—
355#103—271no—
356#102—91no—
357#102—271no—
358C33#41611yescharacteristic, complemented, Sylow
359C23×C3—91no—
360C23×C3—181no—
361C22×D3—31no—
362C22×D3—31no—
363C22×D3—61no—
364C22×D3—61no—
365C22×D3—61no—
366C22×D3—121no—
367C3×Q8—92no—
368C3×Q8—181no—
369C3×D4—92no—
370C3×D4—181no—
371C2×C3×C4—21no—
372C2×C3×C4—21no—
373C2×C3×C4—21no—
374C2×C3×C4—41no—
375C2×C3×C4—91no—
376C2×C3×C4—92no—
377C2×C3×C4—92no—
378C2×C3×C4—181no—
379C2×C3×C4—181no—
380C2×C3×C4—181no—
381C2×C3×C4C2×D3211yescharacteristic
382C2×C3×C4C2×D3211yescharacteristic
383C2×C3×C4C2×D3211yescharacteristic
384#67—184no—
385#67—184no—
386C2×Dic3—31no—
387C2×Dic3—31no—
388C2×Dic3—61no—
389C2×Dic3—61no—
390C2×Dic3—61no—
391C2×Dic3—61no—
392C2×Dic3—61no—
393C2×Dic3—61no—
394C2×Dic3—61no—
395C2×Dic3—121no—
396C2×Dic3—121no—
397C2×Dic3—121no—
398C2×Dic3—184no—
399D12—32no—
400D12—32no—
401D12—62no—
402D12—61no—
403D12—61no—
404D12—121no—
405D12—184no—
406C4×D3—32no—
407C4×D3—32no—
408C4×D3—32no—
409C4×D3—32no—
410C4×D3—62no—
411C4×D3—62no—
412C4×D3—184no—
413C4×D3—184no—
414Dic6—32no—
415Dic6—32no—
416Dic6—62no—
417Dic6—61no—
418Dic6—61no—
419Dic6—121no—
420C3×C8—22no—
421C3×C8—21no—
422C3×C8—21no—
423C3×C8—21no—
424C3×C8—41no—
425C3×C8—41no—
426C3×C8—42no—
427C3×C8—181no—
428C3⋊C8—62no—
429C3⋊C8—61no—
430C3⋊C8—61no—
431C2×C32—21no—
432C2×C32—21no—
433C2×C32—21no—
434C2×C32—21no—
435C2×C32—21no—
436C2×C32—21no—
437C2×C32—21no—
438C2×C32—21no—
439C2×C32—21no—
440C2×C32—41no—
441C2×C32—41no—
442C2×C32—41no—
443C2×C32D3×C22⋊C411yescharacteristic
444C2×C32D3×C22⋊C411yescharacteristic
445C2×C32#90411yescharacteristic
446C2×C32#90411yescharacteristic
447C2×C32#91211yescharacteristic
448C2×C32#91611yescharacteristic
449C2×C32#91611yescharacteristic
450C2×C32#91911yescharacteristic
451C2×C32C2×(C2×D3)⋊C411yescharacteristic
452C3⋊D3—22no—
453C3⋊D3—42no—
454C3×D3—64no—
455C3×D3—62no—
456C3×D3—62no—
457C3×D3—122no—
458C3×D3—122no—
459C3×D3—242no—
460#41—274no—
461#41—542no—
462C2×Q8—91no—
463C2×D4—91no—
464C2×D4—541no—
465C22×C4—91no—
466C22×C4—541no—
467C2×C8—91no—
468C2×C8—182no—
469C2×C8#111311yescharacteristic
470C4⋊C4—91no—
471C4⋊C4—91no—
472C22⋊C4—541no—
473C42—541no—
474C22×C3—21no—
475C22×C3—21no—
476C22×C3—21no—
477C22×C3—41no—
478C22×C3—92no—
479C22×C3—181no—
480C22×C3—181no—
481C22×C3—182no—
482C22×C3—361no—
483C22×C3—361no—
484C22×C3D3×D1211yescharacteristic
485C22×C3D3×D1211yescharacteristic
486C22×C3#380211yescharacteristic
487C2×D3—32no—
488C2×D3—32no—
489C2×D3—62no—
490C2×D3—62no—
491C2×D3—62no—
492C2×D3—61no—
493C2×D3—61no—
494C2×D3—61no—
495C2×D3—61no—
496C2×D3—62no—
497C2×D3—62no—
498C2×D3—122no—
499C2×D3—121no—
500C2×D3—121no—
501C2×D3—121no—
502C2×D3—121no—
503C2×D3—241no—
504C2×D3—241no—
505C3×C4—21no—
506C3×C4—21no—
507C3×C4—21no—
508C3×C4—21no—
509C3×C4—21no—
510C3×C4—21no—
511C3×C4—41no—
512C3×C4—41no—
513C3×C4—92no—
514C3×C4—182no—
515C3×C4—361no—
516C3×C4—361no—
517C3×C4C4×D3211yescharacteristic
518C3×C4C4×D3211yescharacteristic
519C3×C4C4×D3211yescharacteristic
520C3×C4D3×#6711yescharacteristic
521C3×C4D3×#6711yescharacteristic
522C3×C4#381111yescharacteristic
523Dic3—32no—
524Dic3—32no—
525Dic3—62no—
526Dic3—62no—
527Dic3—62no—
528Dic3—122no—
529Dic3—121no—
530Dic3—121no—
531Dic3—121no—
532Dic3—121no—
533Dic3—241no—
534Dic3—241no—
535C32—21no—
536C32—21no—
537C32—21no—
538C32—41no—
539C32#477211yescharacteristic, complemented
540C32#480411yescharacteristic, complemented
541C32#511311yescharacteristic, complemented
542C23—91no—
543C23—181no—
544Q8—92no—
545Q8—181no—
546D4—92no—
547D4—181no—
548D4—544no—
549C2×C4—91no—
550C2×C4—92no—
551C2×C4—92no—
552C2×C4—181no—
553C2×C4—181no—
554C2×C4—181no—
555C2×C4—544no—
556C2×C4C2×#111311yescharacteristic
557C8—21no—
558C8—181no—
559C2×C3—21no—
560C2×C3—21no—
561C2×C3—21no—
562C2×C3—21no—
563C2×C3—21no—
564C2×C3—21no—
565C2×C3—21no—
566C2×C3—21no—
567C2×C3—21no—
568C2×C3—41no—
569C2×C3—41no—
570C2×C3—41no—
571C2×C3—182no—
572C2×C3—362no—
573C2×C3#6380611yescharacteristic
574C2×C3#6380611yescharacteristic
575C2×C3#6380711yescharacteristic
576C2×C3#6380811yescharacteristic
577C2×C3#6380911yescharacteristic
578C2×C3#6381011yescharacteristic
579C2×C3D3×(C2×D3)⋊C411yescharacteristic
580C2×C3D3×(C2×D3)⋊C411yescharacteristic
581C2×C3#6392111yescharacteristic
582D3—64no—
583D3—62no—
584D3—62no—
585D3—122no—
586D3—122no—
587D3—242no—
588C22—92no—
589C22—181no—
590C22—181no—
591C22—182no—
592C22—361no—
593C22—361no—
594C22C4⋊#111311yescharacteristic
595C4—92no—
596C4—182no—
597C4—361no—
598C4—361no—
599C4C4×#111311yescharacteristic
600C4#8904911yescharacteristic
601C3—21no—
602C3—21no—
603C3—21no—
604C3—41no—
605C3#1059069011yescharacteristic, complemented
606C3#1059095411yescharacteristic, complemented
607C3#1059095611yescharacteristic, complemented
608C2—182no—
609C2—362no—
610C2#1172365311yescharacteristic
611C2#1172365411yescharacteristic
612C2#1172370011yescharacteristic
subgroupG/H
centerC22C4⋊#1113
commutatorC33×C4C22×C4
FrattiniC2×C4C2×#1113
FittingC2×C33×C8C22
socleC22×C33C2×D4
prime radicalC2×#89041C2
cosocleC2×C33×C4C23
hypercenterC2×C8#1113
hypocenterC33#416
LCSG ▷ C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×C8

Complements

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

NH ≅ G/Ntype
2. C2×C3×C8×C3⋊D3C2semidirect
3. #11723667C2semidirect
4. C3×#63648C2semidirect
5. C3×#63643C2semidirect
6. #11722076C2semidirect
7. #11722075C2semidirect
17. #89041C4semidirect
28. C2×C8×C3⋊D3D3semidirect
35. #63648D3semidirect
36. #63643D3semidirect
358. C33#416semidirect
539. C32#4772semidirect
540. C32#4804semidirect
541. C32#5113semidirect
605. C3#10590690semidirect
606. C3#10590954semidirect
607. C3#10590956semidirect

Maximal subgroups

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

subgroupindex
C2×#890412
C2×C3×C8×C3⋊D32
#117236672
C3×#636482
C3×#636432
#117220762
#117220752
#105909563
#105909543
#105906903