Finite Group Databasebeta

← #13156006

Subgroups — #13156006

Order 1728.

764 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
1#11723484C211yesmaximal, characteristic
2C2×#88797C211yesmaximal, characteristic
3C2×#88795C211yesmaximal, characteristic
4C3×#63866C211yesmaximal, characteristic, complemented
5#11722076C211yesmaximal, characteristic, complemented
6C3×Dic3⋊(C3⋊C8)C211yesmaximal, characteristic, complemented
7C3×#63566C211yesmaximal, characteristic, complemented
8#10591565—31nomaximal
9#10590760—31nomaximal
10#10590379—31nomaximal
11C2×C4⋊(C3⋊(C3⋊D3))C2211yescharacteristic
12C2×C3×#634C2211yescharacteristic
13C2×C32×Dic6C2211yescharacteristic
14C2×#6289—21no—
15C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
16C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
17C32×C4×Dic3C2211yescharacteristic
18C2×C32×C3⋊C8C2211yescharacteristic
19#88804—21no—
20#88797—22no—
21#88795—22no—
22C3×Dic3⋊Dic3—21no—
23C3×Dic3⋊Dic3—21no—
24C2×#3755—31no—
25C2×#3744—31no—
26C4×C4⋊(C3⋊D3)—31no—
27C3×#944—31no—
28C3×#869—31no—
29#63894—31no—
30#63894—31no—
31#63894—61no—
32#63866D311yescharacteristic, complemented
33C2×#3717—31no—
34C2×#3717—31no—
35C2×#3717—31no—
36C2×#3717—31no—
37C2×#3717—61no—
38C2×#3717—61no—
39C3×#835—31no—
40C3×#816—31no—
41C3×Dic3⋊C8—31no—
42C3×C4⋊(C3⋊C8)—31no—
43Dic3⋊(C3⋊C8)D311yescharacteristic, complemented
44#63567—31no—
45#63566—31no—
46#63566D311yescharacteristic, complemented
47#63565—31no—
48C22×C3⋊(C3⋊D3)—21no—
49C2×C33×C4C2311yescharacteristic
50C4⋊(C3⋊(C3⋊D3))—21no—
51C4⋊(C3⋊(C3⋊D3))D412yes—
52C2×C3×C3⋊Dic3—21no—
53C2×C3×C3⋊Dic3—21no—
54C3×#634—21no—
55C3×#634—21no—
56C2×C32×Dic3—21no—
57C2×C32×Dic3D411yescharacteristic
58C2×C32×Dic3D411yescharacteristic
59C32×Dic6—21no—
60C32×Dic6—21no—
61#6289—42no—
62C3×C3⋊(C3⋊C8)—21no—
63C32×C3⋊C8—21no—
64#5083—91no—
65#4841—91no—
66#4818—91no—
67C2×C32×Q8—31no—
68C2×C4⋊(C3⋊D3)—31no—
69C2×C4⋊(C3⋊D3)—31no—
70C2×C4⋊(C3⋊D3)—31no—
71C2×C4⋊(C3⋊D3)—61no—
72C2×C4⋊(C3⋊D3)—61no—
73C2×C4⋊(C3⋊D3)—61no—
74C2×C4⋊(C3⋊D3)—121no—
75C2×C4×C3⋊D3—61no—
76C2×#634C2×D311yescharacteristic
77C2×C3×Dic6—21no—
78C2×C3×Dic6—31no—
79C2×C3×Dic6—31no—
80C2×C3×Dic6—61no—
81C2×C3×Dic6C2×D311yescharacteristic
82C2×C3×Dic6C2×D311yescharacteristic
83C2×#624—62no—
84C2×#624—61no—
85C2×#624—61no—
86C2×C32×C8—31no—
87C32×C42—31no—
88#3755—62no—
89#3752—61no—
90C4⋊(C3⋊Dic3)C2×D311yescharacteristic
91C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
92#3744—62no—
93C3×C4⋊Dic3—31no—
94C3×C4⋊Dic3—31no—
95C3×C4⋊Dic3—61no—
96C3×C4⋊Dic3—61no—
97C3×C4⋊Dic3—61no—
98C3×Dic3⋊C4—61no—
99C3×Dic3⋊C4—61no—
100C3×C4×Dic3—21no—
101C3×C4×Dic3C2×D311yescharacteristic
102C3×C4×Dic3C2×D311yescharacteristic
103C2×C3×C3⋊C8—21no—
104C2×C3×C3⋊C8—31no—
105C2×C3×C3⋊C8—31no—
106C2×C3×C3⋊C8—61no—
107C2×C3×C3⋊C8C2×D311yescharacteristic
108C2×C3×C3⋊C8C2×D311yescharacteristic
109Dic3⋊Dic3—21no—
110Dic3⋊Dic3—21no—
111#3722—62no—
112#3722—61no—
113#3722—61no—
114#3717—62no—
115#3717—62no—
116#3717—64no—
117#3717—62no—
118#3717—62no—
119#3717—64no—
120C22×C33C2×D411yescharacteristic
121C2×C3⋊(C3⋊D3)—22no—
122C2×C3⋊(C3⋊D3)—41no—
123C2×C3⋊(C3⋊D3)—41no—
124C33×C4C2×D411yescharacteristic
125C33×C4#4111yescharacteristic
126C3×C3⋊Dic3—41no—
127C3×C3⋊Dic3—41no—
128C32×Dic3—21no—
129C32×Dic3—41no—
130C32×Dic3SD1612yescomplemented
131C3×C4⋊Q8—91no—
132#944—31no—
133C2×#215—91no—
134C2×#215—91no—
135C2×#215—91no—
136C2×#215—181no—
137C2×#204—91no—
138C2×#204—91no—
139C2×#204—91no—
140C2×#204—181no—
141#893—91no—
142C4×D12—91no—
143C4×D12—91no—
144C4×D12—181no—
145#869—31no—
146C3×C4⋊C8—91no—
147C3×#103—91no—
148#835—31no—
149#835—91no—
150#816—31no—
151#816—91no—
152Dic3⋊C8—31no—
153#810—91no—
154C4⋊(C3⋊C8)—31no—
155C22×C3⋊D3—61no—
156C22×C3⋊D3—62no—
157C22×C3⋊D3—61no—
158C22×C3⋊D3—61no—
159C22×C3⋊D3—121no—
160C22×C3⋊D3—121no—
161C22×C3⋊D3—122no—
162C32×Q8—61no—
163C32×Q8—61no—
164C2×C32×C4—21no—
165C2×C32×C4—21no—
166C2×C32×C4—21no—
167C2×C32×C4—31no—
168C2×C32×C4—31no—
169C2×C32×C4—41no—
170C2×C32×C4—61no—
171C2×C32×C4C22×D311yescharacteristic
172C2×C32×C4C22×D311yescharacteristic
173C2×C32×C4C22×D311yescharacteristic
174C2×C3⋊Dic3—21no—
175C2×C3⋊Dic3—21no—
176C4⋊(C3⋊D3)—32no—
177C4⋊(C3⋊D3)—32no—
178C4⋊(C3⋊D3)—32no—
179C4⋊(C3⋊D3)—62no—
180C4⋊(C3⋊D3)—62no—
181C4⋊(C3⋊D3)—62no—
182C4⋊(C3⋊D3)—61no—
183C4⋊(C3⋊D3)—62no—
184C4⋊(C3⋊D3)—61no—
185C4⋊(C3⋊D3)—61no—
186C4⋊(C3⋊D3)—122no—
187C4⋊(C3⋊D3)—121no—
188C4⋊(C3⋊D3)—121no—
189C4⋊(C3⋊D3)—122no—
190C4×C3⋊D3—122no—
191#634—21no—
192#634—21no—
193C2×C3×Dic3—21no—
194C2×C3×Dic3—21no—
195C2×C3×Dic3—21no—
196C2×C3×Dic3—21no—
197C2×C3×Dic3—41no—
198C2×C3×Dic3—62no—
199C2×C3×Dic3—61no—
200C2×C3×Dic3—61no—
201C2×C3×Dic3—61no—
202C2×C3×Dic3—61no—
203C2×C3×Dic3—121no—
204C2×C3×Dic3D1211yescharacteristic
205C2×C3×Dic3D1211yescharacteristic
206C2×C3×Dic3#6711yescharacteristic
207C2×C3×Dic3#6711yescharacteristic
208C3×Dic6—22no—
209C3×Dic6—21no—
210C3×Dic6—21no—
211C3×Dic6—21no—
212C3×Dic6—21no—
213C3×Dic6—41no—
214C3×Dic6—62no—
215C3×Dic6—61no—
216C3×Dic6—61no—
217C3×Dic6—61no—
218C3×Dic6—61no—
219C3×Dic6—121no—
220#624—124no—
221#624—122no—
222#624—122no—
223C32×C8—61no—
224C3⋊(C3⋊C8)—21no—
225C3×C3⋊C8—22no—
226C3×C3⋊C8—21no—
227C3×C3⋊C8—21no—
228C3×C3⋊C8—62no—
229C3×C3⋊C8—61no—
230C3×C3⋊C8—61no—
231#461—271noSylow
232C2×C33#12111yescharacteristic
233C2×C33C2×SD1611yescharacteristic
234C2×C33#13711yescharacteristic
235C3⋊(C3⋊D3)—42no—
236C2×C3×Q8—31no—
237C2×C3×Q8—31no—
238C2×C3×Q8—61no—
239C2×C3×Q8—91no—
240C2×D12—91no—
241C2×D12—91no—
242C2×D12—91no—
243C2×D12—181no—
244C2×D12—181no—
245C2×D12—181no—
246C2×D12—361no—
247C2×C4×D3—181no—
248C2×C4×D3—181no—
249C2×C4×D3—182no—
250C2×C4×D3—181no—
251C2×Dic6—31no—
252C2×Dic6—31no—
253C2×Dic6—61no—
254C2×Dic6D3211yescharacteristic
255C2×C3×C8—31no—
256C2×C3×C8—31no—
257C2×C3×C8—61no—
258C2×C3×C8—91no—
259C3×C4⋊C4—91no—
260C3×C4⋊C4—181no—
261C3×C4⋊C4—181no—
262C3×C42—31no—
263C3×C42—31no—
264C3×C42—61no—
265#215—182no—
266#215—182no—
267#215—182no—
268#215—184no—
269(C2×D3)⋊C4—181no—
270(C2×D3)⋊C4—181no—
271(C2×D3)⋊C4—182no—
272(C2×D3)⋊C4—181no—
273C4⋊Dic3—31no—
274C4⋊Dic3—31no—
275C4⋊Dic3—61no—
276C4⋊Dic3—61no—
277C4⋊Dic3—61no—
278Dic3⋊C4—61no—
279Dic3⋊C4—61no—
280C4×Dic3D3211yescharacteristic
281C2×C3⋊C8—31no—
282C2×C3⋊C8—31no—
283C2×C3⋊C8—61no—
284C2×C3⋊C8D3211yescharacteristic
285#204—182no—
286#204—182no—
287#204—184no—
288#204—182no—
289C22×C32—21no—
290C22×C32—21no—
291C22×C32—21no—
292C22×C32—41no—
293C22×C32C2×D1211yescharacteristic
294C22×C32D3×D411yescharacteristic
295C22×C32C2×#6711yescharacteristic
296C2×C3⋊D3—62no—
297C2×C3⋊D3—64no—
298C2×C3⋊D3—62no—
299C2×C3⋊D3—62no—
300C2×C3⋊D3—122no—
301C2×C3⋊D3—122no—
302C2×C3⋊D3—124no—
303C2×C3⋊D3—121no—
304C2×C3⋊D3—121no—
305C2×C3⋊D3—122no—
306C2×C3⋊D3—122no—
307C2×C3⋊D3—121no—
308C2×C3⋊D3—121no—
309C2×C3⋊D3—121no—
310C2×C3⋊D3—121no—
311C2×C3⋊D3—241no—
312C2×C3⋊D3—241no—
313C2×C3⋊D3—241no—
314C2×C3⋊D3—241no—
315C2×C3⋊D3—242no—
316C2×C3⋊D3—242no—
317C32×C4—21no—
318C32×C4—21no—
319C32×C4—21no—
320C32×C4—21no—
321C32×C4—21no—
322C32×C4—21no—
323C32×C4—32no—
324C32×C4—41no—
325C32×C4—41no—
326C32×C4—61no—
327C32×C4—121no—
328C32×C4#23511yescharacteristic
329C32×C4D3×D411yescharacteristic
330C32×C4D3×D411yescharacteristic
331C32×C4D3×D411yescharacteristic
332C32×C4#23711yescharacteristic
333C32×C4#23911yescharacteristic
334C3⋊Dic3—41no—
335C3⋊Dic3—41no—
336C3×Dic3—21no—
337C3×Dic3—21no—
338C3×Dic3—22no—
339C3×Dic3—41no—
340C3×Dic3—42no—
341C3×Dic3—41no—
342C3×Dic3—41no—
343C3×Dic3—122no—
344C3×Dic3—122no—
345C3×Dic3—121no—
346C3×Dic3—121no—
347C3×Dic3—121no—
348C3×Dic3—121no—
349C3×Dic3#20412yescomplemented
350C3×Dic3#21512yescomplemented
351C2×SD16—271no—
352C2×SD16—271no—
353C4⋊Q8—91no—
354C4×D4—271no—
355C4⋊C8—91no—
356#103—91no—
357#103—271no—
358C33#46111yescharacteristic, complemented, Sylow
359C22×D3—182no—
360C22×D3—181no—
361C22×D3—181no—
362C22×D3—181no—
363C22×D3—361no—
364C22×D3—361no—
365C22×D3—362no—
366C3×Q8—62no—
367C3×Q8—61no—
368C3×Q8—61no—
369C3×Q8—61no—
370C3×Q8—61no—
371C3×Q8—121no—
372C3×Q8—181no—
373C3×Q8—181no—
374C2×C3×C4—21no—
375C2×C3×C4—21no—
376C2×C3×C4—21no—
377C2×C3×C4—31no—
378C2×C3×C4—31no—
379C2×C3×C4—31no—
380C2×C3×C4—31no—
381C2×C3×C4—41no—
382C2×C3×C4—61no—
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#62611yescharacteristic
400C2×Dic3#62611yescharacteristic
401D12—92no—
402D12—92no—
403D12—92no—
404D12—182no—
405D12—182no—
406D12—182no—
407D12—182no—
408D12—181no—
409D12—181no—
410D12—181no—
411D12—362no—
412D12—361no—
413D12—361no—
414D12—362no—
415C4×D3—362no—
416C4×D3—362no—
417C4×D3—364no—
418C4×D3—362no—
419Dic6—21no—
420Dic6—21no—
421Dic6—62no—
422Dic6—61no—
423Dic6—61no—
424Dic6—61no—
425Dic6—61no—
426Dic6—121no—
427C3×C8—62no—
428C3×C8—61no—
429C3×C8—61no—
430C3×C8—181no—
431C3⋊C8—21no—
432C3⋊C8—62no—
433C3⋊C8—61no—
434C3⋊C8—61no—
435C2×C32—21no—
436C2×C32—21no—
437C2×C32—21no—
438C2×C32—21no—
439C2×C32—21no—
440C2×C32—21no—
441C2×C32—21no—
442C2×C32—21no—
443C2×C32—21no—
444C2×C32—41no—
445C2×C32—41no—
446C2×C32—41no—
447C2×C32#88411yescharacteristic
448C2×C32#89511yescharacteristic
449C2×C32C2×#20411yescharacteristic
450C2×C32#90911yescharacteristic
451C2×C32D3×SD1611yescharacteristic
452C2×C32#91811yescharacteristic
453C2×C32#93811yescharacteristic
454C2×C32C2×#21511yescharacteristic
455C2×C32#94211yescharacteristic
456C3⋊D3—122no—
457C3⋊D3—124no—
458C3⋊D3—122no—
459C3⋊D3—122no—
460C3⋊D3—242no—
461C3⋊D3—242no—
462C3⋊D3—244no—
463C2×Q8—31no—
464C2×Q8—91no—
465C2×D4—271no—
466C22×C4—541no—
467SD16—542no—
468SD16—542no—
469C2×C8—31no—
470C2×C8—91no—
471C4⋊C4—91no—
472C4⋊C4—181no—
473C4⋊C4—181no—
474C22⋊C4—541no—
475C42—31no—
476C22×C3—21no—
477C22×C3—21no—
478C22×C3—21no—
479C22×C3—41no—
480C22×C3D3×D1211yescharacteristic
481C22×C3C2×#62611yescharacteristic
482C22×C3D3×#6711yescharacteristic
483C2×D3—184no—
484C2×D3—182no—
485C2×D3—182no—
486C2×D3—182no—
487C2×D3—362no—
488C2×D3—362no—
489C2×D3—364no—
490C2×D3—362no—
491C2×D3—362no—
492C2×D3—361no—
493C2×D3—361no—
494C2×D3—361no—
495C2×D3—361no—
496C2×D3—361no—
497C2×D3—361no—
498C2×D3—721no—
499C2×D3—721no—
500C2×D3—721no—
501C2×D3—721no—
502C2×D3—722no—
503C2×D3—722no—
504C3×C4—21no—
505C3×C4—21no—
506C3×C4—21no—
507C3×C4—21no—
508C3×C4—21no—
509C3×C4—21no—
510C3×C4—32no—
511C3×C4—32no—
512C3×C4—41no—
513C3×C4—41no—
514C3×C4—62no—
515C3×C4—61no—
516C3×C4—61no—
517C3×C4—121no—
518C3×C4—122no—
519C3×C4—121no—
520C3×C4—121no—
521C3×C4—361no—
522C3×C4—361no—
523C3×C4#379611yescharacteristic
524C3×C4#379911yescharacteristic
525C3×C4#380211yescharacteristic
526C3×C4#380411yescharacteristic
527C3×C4#381111yescharacteristic
528C3×C4#381111yescharacteristic
529Dic3—21no—
530Dic3—41no—
531Dic3—122no—
532Dic3—122no—
533Dic3—121no—
534Dic3—121no—
535Dic3—121no—
536Dic3—121no—
537Dic3#371712yescomplemented
538C32—21no—
539C32—21no—
540C32—21no—
541C32—41no—
542C32#481811yescharacteristic, complemented
543C32#484111yescharacteristic, complemented
544C32#508311yescharacteristic, complemented
545C23—541no—
546Q8—61no—
547Q8—61no—
548Q8—181no—
549Q8—181no—
550D4—272no—
551D4—541no—
552C2×C4—31no—
553C2×C4—31no—
554C2×C4—61no—
555C2×C4—181no—
556C2×C4—181no—
557C2×C4—1082no—
558C2×C4D3311yescharacteristic
559C8—61no—
560C8—181no—
561C2×C3—21no—
562C2×C3—21no—
563C2×C3—21no—
564C2×C3—21no—
565C2×C3—21no—
566C2×C3—21no—
567C2×C3—21no—
568C2×C3—21no—
569C2×C3—21no—
570C2×C3—41no—
571C2×C3—41no—
572C2×C3—41no—
573C2×C3D3×#20411yescharacteristic
574C2×C3#6380111yescharacteristic
575C2×C3C2×#371711yescharacteristic
576C2×C3#6383311yescharacteristic
577C2×C3#6391011yescharacteristic
578C2×C3#6391611yescharacteristic
579C2×C3D3×#21511yescharacteristic
580C2×C3#6394811yescharacteristic
581C2×C3#6395811yescharacteristic
582D3—364no—
583D3—362no—
584D3—362no—
585D3—362no—
586D3—722no—
587D3—722no—
588D3—724no—
589C22—542no—
590C22—1081no—
591C22—1081no—
592C22D3×#62611yescharacteristic
593C4—32no—
594C4—61no—
595C4—121no—
596C4—361no—
597C4—361no—
598C4#8895411yescharacteristic
599C4#8896211yescharacteristic
600C3—21no—
601C3—21no—
602C3—21no—
603C3—41no—
604C3#1059037911yescharacteristic, complemented
605C3#1059076011yescharacteristic, complemented
606C3#1059156511yescharacteristic, complemented
607C2—1082no—
608C2D3×#371711yescharacteristic
609C2#1172288511yescharacteristic
610C2#1172296311yescharacteristic
subgroupG/H
centerC22D3×#626
commutator, Fitting, cosocleC2×C33×C4C23
Frattini, hypercenterC2×C4D33
socleC22×C33C2×D4
prime radicalC2×C4⋊(C3⋊(C3⋊D3))C22
hypocenterC33#461
LCSG ▷ C2×C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
4. C3×#63866C2semidirect
5. #11722076C2semidirect
6. C3×Dic3⋊(C3⋊C8)C2semidirect
7. C3×#63566C2semidirect
32. #63866D3semidirect
43. Dic3⋊(C3⋊C8)D3semidirect
46. #63566D3semidirect
130. C32×Dic3SD16semidirect
349. C3×Dic3#204semidirect
350. C3×Dic3#215semidirect
358. C33#461semidirect
537. Dic3#3717semidirect
542. C32#4818semidirect
543. C32#4841semidirect
544. C32#5083semidirect
604. C3#10590379semidirect
605. C3#10590760semidirect
606. C3#10591565semidirect

Maximal subgroups

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

subgroupindex
#117234842
C2×#887972
C2×#887952
C3×#638662
#117220762
C3×Dic3⋊(C3⋊C8)2
C3×#635662
#105915653
#105907603
#105903793