Finite Group Databasebeta

← #13156583

Subgroups — #13156583

Order 1728.

866 conjugacy classes of subgroups.

The subgroup lattice has 598 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#88811C211yesmaximal, characteristic, complemented
2#11723517C211yesmaximal, characteristic
3C2×#88797C211yesmaximal, characteristic, complemented
4C2×#88795C211yesmaximal, characteristic, complemented
5C3×#63866C211yesmaximal, characteristic, complemented
6C2×C3×#3717C211yesmaximal, characteristic, complemented
7C3×Dic3×C3⋊C8C211yesmaximal, characteristic, complemented
8#10591618—31nomaximal
9#10590877—31nomaximal
10#10590373—31nomaximal
11C2×C4⋊(C3⋊(C3⋊D3))C2211yescharacteristic
12C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
13C2×C3×#634C2211yescharacteristic
14C2×C32×Dic6C2211yescharacteristic
15C2×#6292—22no—
16C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
17C32×C4×Dic3C2211yescharacteristic
18C2×C32×C3⋊C8C2211yescharacteristic
19#88811—22no—
20#88797—22no—
21#88795—22no—
22C3×Dic3⋊Dic3—22no—
23C3×#3717—22no—
24C2×#3755—31no—
25C2×#3744—31no—
26#64083—31no—
27C3×#944—31no—
28C2×C3×#215—31no—
29C2×C3×#204—31no—
30C3×#869—31no—
31#63911—31no—
32#63911—31no—
33#63911—61no—
34#63866D311yescharacteristic, complemented
35C2×#3717—31no—
36C2×#3717—31no—
37C2×#3717—31no—
38C2×#3717—31no—
39C2×#3717—31no—
40C2×#3717—61no—
41C2×#3717—61no—
42C2×#3717D311yescharacteristic, complemented
43C2×#3716—31no—
44C2×#3715—31no—
45C3×C8×Dic3—31no—
46C3×C4×C3⋊C8—31no—
47Dic3×C3⋊C8D311yescharacteristic, complemented
48C22×C3⋊(C3⋊D3)—21no—
49C22×C3×C3⋊D3—21no—
50C2×C33×C4C2311yescharacteristic
51C4⋊(C3⋊(C3⋊D3))—21no—
52C4⋊(C3⋊(C3⋊D3))—21no—
53C2×C3×C3⋊Dic3—21no—
54C3×C4⋊(C3⋊D3)—21no—
55C3×C4⋊(C3⋊D3)—21no—
56C3×#634—21no—
57C3×#634—21no—
58C2×C32×Dic3—21no—
59C2×C32×Dic3D412yes—
60C32×Dic6—21no—
61C32×Dic6—21no—
62#6292—44no—
63C3×C3⋊(C3⋊C8)D412yescomplemented
64C32×C3⋊C8D412yescomplemented
65#5175—91no—
66#5085—91no—
67#4693—91no—
68C2×C32×Q8—31no—
69C2×C4⋊(C3⋊D3)—31no—
70C2×C4⋊(C3⋊D3)—31no—
71C2×C4⋊(C3⋊D3)—31no—
72C2×C4⋊(C3⋊D3)—61no—
73C2×C4⋊(C3⋊D3)—61no—
74C2×C4⋊(C3⋊D3)—61no—
75C2×C4⋊(C3⋊D3)—62no—
76C2×C4⋊(C3⋊D3)—121no—
77C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
78C2×#634C2×D311yescharacteristic
79C2×C3×D12—31no—
80C2×C3×D12—31no—
81C2×C3×D12—61no—
82C2×C3×Dic6—21no—
83C2×C3×Dic6—31no—
84C2×C3×Dic6—31no—
85C2×C3×Dic6—61no—
86C2×C3×Dic6C2×D311yescharacteristic
87C2×C3×Dic6C2×D311yescharacteristic
88C2×#626—64no—
89C2×#626—62no—
90C2×#626—62no—
91C2×C32×C8—31no—
92C32×C42—31no—
93#3755—62no—
94C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
95#3744—62no—
96C3×#215—62no—
97C3×C4⋊Dic3—62no—
98C3×Dic3⋊C4—62no—
99C3×C4×Dic3—21no—
100C3×C4×Dic3C2×D311yescharacteristic
101C3×C4×Dic3C2×D311yescharacteristic
102C2×C3×C3⋊C8—21no—
103C2×C3×C3⋊C8—31no—
104C2×C3×C3⋊C8—31no—
105C2×C3×C3⋊C8—61no—
106C2×C3×C3⋊C8C2×D311yescharacteristic
107C2×C3×C3⋊C8C2×D311yescharacteristic
108C3×#204—62no—
109Dic3⋊Dic3—22no—
110#3717—22no—
111#3717—64no—
112#3717—62no—
113#3717—62no—
114#3717—62no—
115#3717—64no—
116#3717—62no—
117#3717—62no—
118#3716—62no—
119#3715—62no—
120C22×C33C2×D411yescharacteristic
121C2×C3⋊(C3⋊D3)—41no—
122C2×C3⋊(C3⋊D3)—42no—
123C2×C3×C3⋊D3—41no—
124C2×C3×C3⋊D3—42no—
125C33×C4C2×D411yescharacteristic
126C33×C4C2×D411yescharacteristic
127C3×C3⋊Dic3—41no—
128C32×Dic3—41no—
129C32×Dic3SD1614yescomplemented
130C2×C3×SD16—91no—
131C3×C4⋊Q8—91no—
132#944—31no—
133C2×#215—31no—
134C2×#215—91no—
135C2×#215—91no—
136C2×#215—91no—
137C2×#215—91no—
138C2×#215—181no—
139#940—91no—
140C2×#214—91no—
141C2×#204—31no—
142C2×#204—91no—
143C2×#204—91no—
144C2×#204—91no—
145C2×#204—91no—
146C2×#204—181no—
147#874—91no—
148#874—91no—
149#874—181no—
150#869—31no—
151C3×C4×C8—91no—
152C8×Dic3—31no—
153C4×C3⋊C8—31no—
154C22×C3⋊D3—21no—
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—
162C22×C3×D3—62no—
163C22×C3×D3—61no—
164C22×C3×D3—61no—
165C32×Q8—61no—
166C32×Q8—61no—
167C2×C32×C4—21no—
168C2×C32×C4—21no—
169C2×C32×C4—21no—
170C2×C32×C4—32no—
171C2×C32×C4—41no—
172C2×C32×C4—61no—
173C2×C32×C4C22×D311yescharacteristic
174C2×C32×C4C22×D311yescharacteristic
175C2×C32×C4C22×D311yescharacteristic
176C2×C3⋊Dic3—21no—
177C4⋊(C3⋊D3)—21no—
178C4⋊(C3⋊D3)—21no—
179C4⋊(C3⋊D3)—61no—
180C4⋊(C3⋊D3)—61no—
181C4⋊(C3⋊D3)—62no—
182C4⋊(C3⋊D3)—62no—
183C4⋊(C3⋊D3)—61no—
184C4⋊(C3⋊D3)—61no—
185C4⋊(C3⋊D3)—61no—
186C4⋊(C3⋊D3)—61no—
187C4⋊(C3⋊D3)—121no—
188C4⋊(C3⋊D3)—121no—
189C4⋊(C3⋊D3)—121no—
190C4⋊(C3⋊D3)—121no—
191C4⋊(C3⋊D3)—122no—
192C4⋊(C3⋊D3)—122no—
193C4⋊(C3⋊D3)—124no—
194#634—21no—
195#634—21no—
196C2×C3×Dic3—21no—
197C2×C3×Dic3—21no—
198C2×C3×Dic3—22no—
199C2×C3×Dic3—41no—
200C2×C3×Dic3—61no—
201C2×C3×Dic3—61no—
202C2×C3×Dic3—121no—
203C2×C3×Dic3D1212yes—
204C2×C3×Dic3#6712yes—
205C3×D12—61no—
206C3×D12—61no—
207C3×D12—61no—
208C3×D12—61no—
209C3×D12—121no—
210C3×D12—121no—
211C3×Dic6—22no—
212C3×Dic6—21no—
213C3×Dic6—21no—
214C3×Dic6—21no—
215C3×Dic6—21no—
216C3×Dic6—41no—
217C3×Dic6—62no—
218C3×Dic6—61no—
219C3×Dic6—61no—
220C3×Dic6—61no—
221C3×Dic6—61no—
222C3×Dic6—121no—
223#626—128no—
224#626—124no—
225#626—124no—
226C32×C8—32no—
227C3⋊(C3⋊C8)D1212yescomplemented
228C3×C3⋊C8—22no—
229C3×C3⋊C8—32no—
230C3×C3⋊C8—32no—
231C3×C3⋊C8—62no—
232C3×C3⋊C8D1212yescomplemented
233C3×C3⋊C8#6712yescomplemented
234#492—271noSylow
235C2×C33#12711yescharacteristic
236C2×C33C2×SD1612yes—
237C3⋊(C3⋊D3)—81no—
238C3×C3⋊D3—81no—
239C2×C3×Q8—31no—
240C2×C3×Q8—31no—
241C2×C3×Q8—61no—
242C2×C3×Q8—91no—
243C2×C3×D4—91no—
244C2×#67—182no—
245C2×D12—31no—
246C2×D12—31no—
247C2×D12—61no—
248C2×D12—91no—
249C2×D12—91no—
250C2×D12—91no—
251C2×D12—181no—
252C2×D12—181no—
253C2×D12—181no—
254C2×D12—182no—
255C2×D12—182no—
256C2×D12—184no—
257C2×D12—361no—
258C2×Dic6—31no—
259C2×Dic6—31no—
260C2×Dic6—61no—
261C2×Dic6D3211yescharacteristic
262C3×SD16—182no—
263C2×C3×C8—31no—
264C2×C3×C8—31no—
265C2×C3×C8—61no—
266C2×C3×C8—91no—
267C3×C4⋊C4—182no—
268C3×C42—31no—
269C3×C42—31no—
270C3×C42—61no—
271#215—62no—
272#215—182no—
273#215—182no—
274#215—182no—
275#215—184no—
276#215—182no—
277#214—182no—
278C4⋊Dic3—62no—
279Dic3⋊C4—62no—
280C4×Dic3D3211yescharacteristic
281C2×C3⋊C8—31no—
282C2×C3⋊C8—31no—
283C2×C3⋊C8—61no—
284C2×C3⋊C8D3211yescharacteristic
285#204—62no—
286#204—182no—
287#204—182no—
288#204—184no—
289#204—182no—
290#204—182no—
291C22×C32—21no—
292C22×C32—21no—
293C22×C32—21no—
294C22×C32—41no—
295C22×C32C2×D1211yescharacteristic
296C22×C32D3×D411yescharacteristic
297C22×C32C2×#6711yescharacteristic
298C2×C3⋊D3—42no—
299C2×C3⋊D3—41no—
300C2×C3⋊D3—122no—
301C2×C3⋊D3—122no—
302C2×C3⋊D3—122no—
303C2×C3⋊D3—124no—
304C2×C3⋊D3—121no—
305C2×C3⋊D3—122no—
306C2×C3⋊D3—121no—
307C2×C3⋊D3—121no—
308C2×C3⋊D3—124no—
309C2×C3⋊D3—122no—
310C2×C3⋊D3—122no—
311C2×C3⋊D3—242no—
312C2×C3⋊D3—242no—
313C2×C3⋊D3—244no—
314C2×C3×D3—124no—
315C2×C3×D3—122no—
316C2×C3×D3—121no—
317C2×C3×D3—122no—
318C2×C3×D3—121no—
319C2×C3×D3—122no—
320C32×C4—21no—
321C32×C4—21no—
322C32×C4—21no—
323C32×C4—21no—
324C32×C4—21no—
325C32×C4—21no—
326C32×C4—34no—
327C32×C4—41no—
328C32×C4—41no—
329C32×C4—121no—
330C32×C4C2×D1211yescharacteristic
331C32×C4C2×D1211yescharacteristic
332C32×C4D3×D411yescharacteristic
333C32×C4D3×D411yescharacteristic
334C32×C4D3×D411yescharacteristic
335C32×C4C2×#6711yescharacteristic
336C3⋊Dic3—41no—
337C3×Dic3—24no—
338C3×Dic3—42no—
339C3×Dic3—41no—
340C3×Dic3—41no—
341C3×Dic3—122no—
342C3×Dic3—121no—
343C3×Dic3—121no—
344C3×Dic3#20414yescomplemented
345C3×Dic3#21514yescomplemented
346C2×SD16—91no—
347C2×SD16—271no—
348C2×SD16—271no—
349C2×SD16—271no—
350C4⋊Q8—91no—
351#127—271no—
352C4×C8—91no—
353C33#49211yescharacteristic, complemented, Sylow
354C23×C3—181no—
355C22×D3—62no—
356C22×D3—61no—
357C22×D3—61no—
358C22×D3—182no—
359C22×D3—181no—
360C22×D3—181no—
361C22×D3—181no—
362C22×D3—361no—
363C22×D3—361no—
364C22×D3—362no—
365C3×Q8—62no—
366C3×Q8—61no—
367C3×Q8—61no—
368C3×Q8—61no—
369C3×Q8—61no—
370C3×Q8—121no—
371C3×Q8—181no—
372C3×Q8—181no—
373C3×D4—181no—
374C3×D4—181no—
375C2×C3×C4—21no—
376C2×C3×C4—21no—
377C2×C3×C4—21no—
378C2×C3×C4—32no—
379C2×C3×C4—32no—
380C2×C3×C4—41no—
381C2×C3×C4—62no—
382C2×C3×C4—61no—
383C2×C3×C4—61no—
384C2×C3×C4—121no—
385C2×C3×C4—181no—
386C2×C3×C4C2×D3211yescharacteristic
387C2×C3×C4C2×D3211yescharacteristic
388C2×C3×C4C2×D3211yescharacteristic
389#67—364no—
390C2×Dic3—21no—
391C2×Dic3—61no—
392C2×Dic3—61no—
393C2×Dic3—121no—
394C2×Dic3#62612yes—
395D12—61no—
396D12—61no—
397D12—61no—
398D12—61no—
399D12—121no—
400D12—121no—
401D12—182no—
402D12—182no—
403D12—181no—
404D12—181no—
405D12—181no—
406D12—181no—
407D12—181no—
408D12—181no—
409D12—361no—
410D12—361no—
411D12—361no—
412D12—361no—
413D12—362no—
414D12—362no—
415D12—364no—
416D12—364no—
417D12—368no—
418Dic6—21no—
419Dic6—21no—
420Dic6—62no—
421Dic6—61no—
422Dic6—61no—
423Dic6—61no—
424Dic6—61no—
425Dic6—121no—
426C3×C8—32no—
427C3×C8—32no—
428C3×C8—62no—
429C3×C8—92no—
430C3⋊C8—32no—
431C3⋊C8—32no—
432C3⋊C8—62no—
433C3⋊C8#62612yescomplemented
434C2×C32—22no—
435C2×C32—22no—
436C2×C32—21no—
437C2×C32—21no—
438C2×C32—22no—
439C2×C32—21no—
440C2×C32—41no—
441C2×C32—42no—
442C2×C32#87411yescharacteristic
443C2×C32C2×#20412yes—
444C2×C32D3×SD1612yes—
445C2×C32#94011yescharacteristic
446C2×C32#94011yescharacteristic
447C2×C32C2×#21512yes—
448C3⋊D3—81no—
449C3⋊D3—242no—
450C3⋊D3—242no—
451C3⋊D3—242no—
452C3⋊D3—244no—
453C3⋊D3—241no—
454C3⋊D3—241no—
455C3⋊D3—241no—
456C3×D3—242no—
457C3×D3—241no—
458C3×D3—241no—
459C2×Q8—31no—
460C2×Q8—91no—
461C2×D4—91no—
462C2×D4—271no—
463C2×D4—542no—
464SD16—182no—
465SD16—542no—
466SD16—542no—
467SD16—542no—
468C2×C8—31no—
469C2×C8—91no—
470C4⋊C4—182no—
471C42—31no—
472C22×C3—21no—
473C22×C3—21no—
474C22×C3—21no—
475C22×C3—41no—
476C22×C3—362no—
477C22×C3—361no—
478C22×C3D3×D1211yescharacteristic
479C22×C3C2×#62611yescharacteristic
480C22×C3D3×#6711yescharacteristic
481C2×D3—122no—
482C2×D3—124no—
483C2×D3—122no—
484C2×D3—121no—
485C2×D3—122no—
486C2×D3—121no—
487C2×D3—362no—
488C2×D3—362no—
489C2×D3—362no—
490C2×D3—364no—
491C2×D3—364no—
492C2×D3—362no—
493C2×D3—362no—
494C2×D3—361no—
495C2×D3—361no—
496C2×D3—362no—
497C2×D3—361no—
498C2×D3—722no—
499C2×D3—722no—
500C2×D3—724no—
501C3×C4—21no—
502C3×C4—21no—
503C3×C4—21no—
504C3×C4—21no—
505C3×C4—21no—
506C3×C4—21no—
507C3×C4—34no—
508C3×C4—34no—
509C3×C4—41no—
510C3×C4—41no—
511C3×C4—64no—
512C3×C4—122no—
513C3×C4—121no—
514C3×C4—121no—
515C3×C4—361no—
516C3×C4D3×D1211yescharacteristic
517C3×C4D3×D1211yescharacteristic
518C3×C4D3×D1211yescharacteristic
519C3×C4C2×#62611yescharacteristic
520C3×C4D3×#6711yescharacteristic
521C3×C4#381111yescharacteristic
522Dic3—41no—
523Dic3—122no—
524Dic3—121no—
525Dic3—121no—
526Dic3#371714yescomplemented
527C32—21no—
528C32—21no—
529C32—21no—
530C32—41no—
531C32#469311yescharacteristic, complemented
532C32#508511yescharacteristic, complemented
533C32#517511yescharacteristic, complemented
534C23—181no—
535C23—541no—
536Q8—61no—
537Q8—61no—
538Q8—181no—
539Q8—181no—
540D4—181no—
541D4—181no—
542D4—541no—
543D4—541no—
544D4—1084no—
545C2×C4—32no—
546C2×C4—61no—
547C2×C4—181no—
548C2×C4D3311yescharacteristic
549C8—32no—
550C8—92no—
551C2×C3—21no—
552C2×C3—21no—
553C2×C3—21no—
554C2×C3—22no—
555C2×C3—22no—
556C2×C3—22no—
557C2×C3—41no—
558C2×C3—42no—
559C2×C3—721no—
560C2×C3D3×#20412yes—
561C2×C3C2×#371712yes—
562C2×C3#6391111yescharacteristic
563C2×C3#6391111yescharacteristic
564C2×C3D3×#21512yes—
565C2×C3#6397911yescharacteristic
566D3—242no—
567D3—241no—
568D3—241no—
569D3—722no—
570D3—722no—
571D3—722no—
572D3—724no—
573D3—721no—
574D3—721no—
575D3—721no—
576C22—361no—
577C22—362no—
578C22—1081no—
579C22—1082no—
580C22D3×#62611yescharacteristic
581C4—34no—
582C4—121no—
583C4—361no—
584C4D3×#62611yescharacteristic
585C4#8895511yescharacteristic
586C3—21no—
587C3—21no—
588C3—21no—
589C3—41no—
590C3#1059037311yescharacteristic, complemented
591C3#1059087711yescharacteristic, complemented
592C3#1059161811yescharacteristic, complemented
593C2—721no—
594C2—2161no—
595C2D3×#371712yes—
596C2#1172303011yescharacteristic
subgroupG/H
centerC22D3×#626
commutator, Fitting, cosocleC2×C33×C4C23
Frattini, hypercenterC2×C4D33
socleC22×C33C2×D4
prime radical#11723517C2
hypocenterC33#492
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
1. C2×#88811C2semidirect
3. C2×#88797C2semidirect
4. C2×#88795C2semidirect
5. C3×#63866C2semidirect
6. C2×C3×#3717C2semidirect
7. C3×Dic3×C3⋊C8C2semidirect
34. #63866D3semidirect
42. C2×#3717D3semidirect
47. Dic3×C3⋊C8D3semidirect
63. C3×C3⋊(C3⋊C8)D4semidirect
64. C32×C3⋊C8D4semidirect
129. C32×Dic3SD16semidirect
227. C3⋊(C3⋊C8)D12semidirect
232. C3×C3⋊C8D12semidirect
233. C3×C3⋊C8#67semidirect
344. C3×Dic3#204semidirect
345. C3×Dic3#215semidirect
353. C33#492semidirect
433. C3⋊C8#626semidirect
526. Dic3#3717semidirect
531. C32#4693semidirect
532. C32#5085semidirect
533. C32#5175semidirect
590. C3#10590373semidirect
591. C3#10590877semidirect
592. C3#10591618semidirect

Maximal subgroups

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

subgroupindex
C2×#888112
#117235172
C2×#887972
C2×#887952
C3×#638662
C2×C3×#37172
C3×Dic3×C3⋊C82
#105916183
#105908773
#105903733