Finite Group Databasebeta

← #13156619

Subgroups — #13156619

Order 1728.

830 conjugacy classes of subgroups; only the 108 normal subgroups are shown.

normal subgroups only

This group has 830 conjugacy classes of subgroups — too many to draw. Only its 108 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#88811C211yesmaximal, characteristic, complemented
2#11723529C211yesmaximal, characteristic, complemented
3#11723434C211yesmaximal, characteristic, complemented
4C2×C3×#3716C211yesmaximal, characteristic, complemented
5C2×C3×#3714C211yesmaximal, characteristic, complemented
6C2×C3×#3713C211yesmaximal, characteristic, complemented
7#11722021C211yesmaximal, characteristic, complemented
8#10591485—31nomaximal
9#10591475—31nomaximal
10#10590887—31nomaximal
11C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
12C2×C3×#634C2211yescharacteristic
13C2×C32×D12C2211yescharacteristic
14C2×C32×D12C2211yescharacteristic
15C2×#6290—22no—
16C4×C3⋊(C3⋊Dic3)C2211yescharacteristic
17C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
18C2×C32×C3⋊C8C2211yescharacteristic
19#88811—22no—
20(C2×D3)⋊(C3⋊Dic3)—22no—
21C3×#3716—22no—
22C3×#3714—22no—
23C3×#3713—22no—
24#64149—31no—
25#64145—31no—
26C2×C3×#214—31no—
27C2×C3×#213—31no—
28C2×C3×#213—31no—
29C2×C3×#213—31no—
30C2×C3×D24—31no—
31C2×C3×#204—31no—
32#63914—31no—
33#63914—31no—
34#63914—61no—
35#63863—31no—
36#63863—31no—
37#63863—61no—
38C2×#3717—31no—
39C2×#3716—31no—
40C2×#3716D311yescharacteristic, complemented
41C2×#3715—31no—
42C2×#3714D311yescharacteristic, complemented
43C2×#3713D311yescharacteristic, complemented
44#63642—31no—
45#63555—31no—
46#63555—31no—
47#63555—61no—
48C22×C3×C3⋊D3—21no—
49C22×C32×D3—21no—
50C22×C32×D3—21no—
51C2×C33×C4C2311yescharacteristic
52C2×C3⋊(C3⋊Dic3)D412yes—
53C2×C3×C3⋊Dic3—21no—
54C3×C4⋊(C3⋊D3)—21no—
55C3×C4⋊(C3⋊D3)—21no—
56C3×#634—21no—
57C3×#634—21no—
58C32×D12—21no—
59C32×D12—21no—
60C32×D12—21no—
61C32×D12—21no—
62#6290—44no—
63C3×C3⋊(C3⋊C8)D412yescomplemented
64C32×C3⋊C8D412yescomplemented
65#5176—91no—
66#5154—91no—
67#5154—91no—
68C2×C32×D4—31no—
69C2×C32×D4—31no—
70C2×#638—62no—
71C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
72C2×#634C2×D311yescharacteristic
73C2×C3×D12—21no—
74C2×C3×D12—21no—
75C2×C3×D12—31no—
76C2×C3×D12—31no—
77C2×C3×D12—61no—
78C2×C3×D12C2×D311yescharacteristic
79C2×C3×D12C2×D311yescharacteristic
80C2×C3×D12C2×D311yescharacteristic
81C2×C3×D12C2×D311yescharacteristic
82C2×C3×Dic6—31no—
83C2×C3×Dic6—31no—
84C2×C3×Dic6—61no—
85C2×#625—62no—
86C2×#625—62no—
87C2×#625—64no—
88C2×C32×C8—31no—
89#3757—62no—
90C4×C3⋊Dic3—31no—
91C4×C3⋊Dic3—31no—
92C4×C3⋊Dic3—31no—
93C4×C3⋊Dic3—61no—
94C4×C3⋊Dic3—61no—
95C4×C3⋊Dic3—61no—
96C4×C3⋊Dic3—121no—
97C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
98C3×#214—62no—
99C3×#213—62no—
100C3×#213—62no—
101C3×#213—62no—
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×D24—62no—
109C3×#204—62no—
110(C2×D3)⋊Dic3—62no—
111(C2×D3)⋊Dic3—62no—
112(C2×D3)⋊Dic3—64no—
113#3717—62no—
114#3716—22no—
115#3716—62no—
116#3715—62no—
117#3714—22no—
118#3713—22no—
119C22×C33C2×D411yescharacteristic
120C2×C3×C3⋊D3—41no—
121C2×C3×C3⋊D3—42no—
122C2×C32×D3—41no—
123C2×C32×D3—41no—
124C2×C32×D3—42no—
125C2×C32×D3—42no—
126C33×C4C2×D411yescharacteristic
127C33×C4C2×D411yescharacteristic
128C3⋊(C3⋊Dic3)—22no—
129C3×C3⋊Dic3—41no—
130C2×C3×SD16—91no—
131C2×C3×D8—91no—
132C2×C3×D8—91no—
133#947—91no—
134C2×#215—91no—
135#940—91no—
136#940—91no—
137#940—91no—
138#940—181no—
139#936—91no—
140#936—91no—
141#936—181no—
142C2×#214—31no—
143C2×#214—91no—
144C2×#213—31no—
145C2×#213—31no—
146C2×#213—31no—
147C2×D24—31no—
148C2×#204—31no—
149C2×#204—91no—
150#815—91no—
151#815—91no—
152#815—91no—
153#815—181no—
154C23×C32—61no—
155C23×C32—61no—
156C22×C3⋊D3—21no—
157C22×C3×D3—22no—
158C22×C3×D3—22no—
159C22×C3×D3—21no—
160C22×C3×D3—21no—
161C22×C3×D3—21no—
162C22×C3×D3—21no—
163C22×C3×D3—62no—
164C22×C3×D3—61no—
165C22×C3×D3—61no—
166C32×D4—61no—
167C32×D4—61no—
168C32×D4—61no—
169C32×D4—61no—
170C2×C32×C4—21no—
171C2×C32×C4—21no—
172C2×C32×C4—21no—
173C2×C32×C4—41no—
174C2×C32×C4C22×D311yescharacteristic
175C2×C32×C4C22×D311yescharacteristic
176C2×C32×C4C22×D311yescharacteristic
177#638—124no—
178C2×C3⋊Dic3—21no—
179C2×C3⋊Dic3—32no—
180C2×C3⋊Dic3—32no—
181C2×C3⋊Dic3—32no—
182C2×C3⋊Dic3—62no—
183C2×C3⋊Dic3—62no—
184C2×C3⋊Dic3—62no—
185C2×C3⋊Dic3—122no—
186C4⋊(C3⋊D3)—21no—
187C4⋊(C3⋊D3)—21no—
188#634—21no—
189#634—21no—
190C2×C3×Dic3—62no—
191C2×C3×Dic3—61no—
192C2×C3×Dic3—61no—
193C3×D12—21no—
194C3×D12—21no—
195C3×D12—21no—
196C3×D12—21no—
197C3×D12—21no—
198C3×D12—21no—
199C3×D12—21no—
200C3×D12—21no—
201C3×D12—41no—
202C3×D12—41no—
203C3×D12—41no—
204C3×D12—41no—
205C3×D12—61no—
206C3×D12—61no—
207C3×D12—61no—
208C3×D12—61no—
209C3×D12—121no—
210C3×D12—121no—
211C3×Dic6—61no—
212C3×Dic6—61no—
213C3×Dic6—61no—
214C3×Dic6—61no—
215C3×Dic6—121no—
216C3×Dic6—121no—
217#625—128no—
218#625—124no—
219#625—124no—
220C32×C8—32no—
221C3⋊(C3⋊C8)#6712yescomplemented
222C3×C3⋊C8—22no—
223C3×C3⋊C8—32no—
224C3×C3⋊C8—32no—
225C3×C3⋊C8—62no—
226C3×C3⋊C8#6712yescomplemented
227C3×C3⋊C8#6712yescomplemented
228#496—271noSylow
229C2×C33#12711yescharacteristic
230C2×C33AGL(1,ℤ/8)12yes—
231C3×C3⋊D3—81no—
232C32×D3—81no—
233C32×D3—81no—
234C2×C3×Q8—91no—
235C2×C3×D4—31no—
236C2×C3×D4—31no—
237C2×C3×D4—31no—
238C2×C3×D4—31no—
239C2×C3×D4—61no—
240C2×C3×D4—61no—
241C2×C3×D4—91no—
242C2×#67—182no—
243C2×#67—182no—
244C2×#67—182no—
245C2×#67—184no—
246C2×D12—31no—
247C2×D12—31no—
248C2×D12—61no—
249C2×D12D3211yescharacteristic
250C2×D12D3211yescharacteristic
251C2×Dic6—31no—
252C2×Dic6—31no—
253C2×Dic6—61no—
254C3×SD16—182no—
255C3×D8—182no—
256C3×D8—182no—
257C2×C3×C8—31no—
258C2×C3×C8—31no—
259C2×C3×C8—61no—
260C2×C3×C8—91no—
261#217—182no—
262#217—182no—
263#217—184no—
264#215—182no—
265#214—62no—
266#214—182no—
267#213—62no—
268#213—62no—
269#213—62no—
270(C2×D3)⋊C4—182no—
271C4×Dic3—91no—
272C4×Dic3—91no—
273C4×Dic3—91no—
274C4×Dic3—181no—
275C4×Dic3—181no—
276C4×Dic3—181no—
277C4×Dic3—361no—
278C2×C3⋊C8—31no—
279C2×C3⋊C8—31no—
280C2×C3⋊C8—61no—
281C2×C3⋊C8D3211yescharacteristic
282D24—62no—
283#204—62no—
284#204—182no—
285C22×C32—21no—
286C22×C32—21no—
287C22×C32—21no—
288C22×C32—41no—
289C22×C32—122no—
290C22×C32—122no—
291C22×C32—121no—
292C22×C32—121no—
293C22×C32D3×D411yescharacteristic
294C22×C32D3×D411yescharacteristic
295C22×C32D3×D411yescharacteristic
296C2×C3⋊D3—42no—
297C2×C3⋊D3—41no—
298C2×C3×D3—44no—
299C2×C3×D3—42no—
300C2×C3×D3—42no—
301C2×C3×D3—44no—
302C2×C3×D3—42no—
303C2×C3×D3—42no—
304C2×C3×D3—42no—
305C2×C3×D3—42no—
306C2×C3×D3—41no—
307C2×C3×D3—41no—
308C2×C3×D3—41no—
309C2×C3×D3—41no—
310C2×C3×D3—124no—
311C2×C3×D3—122no—
312C2×C3×D3—122no—
313C2×C3×D3—122no—
314C2×C3×D3—121no—
315C2×C3×D3—121no—
316C32×C4—21no—
317C32×C4—21no—
318C32×C4—21no—
319C32×C4—21no—
320C32×C4—21no—
321C32×C4—21no—
322C32×C4—41no—
323C32×C4—41no—
324C32×C4D3×D411yescharacteristic
325C32×C4D3×D411yescharacteristic
326C32×C4D3×D411yescharacteristic
327C32×C4C2×#6711yescharacteristic
328C32×C4C2×#6711yescharacteristic
329C32×C4C2×#6711yescharacteristic
330C3⋊Dic3—41no—
331C3⋊Dic3—62no—
332C3⋊Dic3—64no—
333C3⋊Dic3—62no—
334C3⋊Dic3—62no—
335C3⋊Dic3—122no—
336C3⋊Dic3—122no—
337C3⋊Dic3—124no—
338C3×Dic3—122no—
339C3×Dic3—121no—
340C3×Dic3—121no—
341C2×SD16—91no—
342C2×SD16—271no—
343C2×D8—91no—
344C2×D8—91no—
345#127—271no—
346#124—271no—
347#97—271no—
348C33#49611yescharacteristic, complemented, Sylow
349C23×C3—62no—
350C23×C3—62no—
351C23×C3—61no—
352C23×C3—61no—
353C23×C3—61no—
354C23×C3—61no—
355C23×C3—181no—
356C22×D3—21no—
357C22×D3—21no—
358C22×D3—62no—
359C22×D3—61no—
360C22×D3—61no—
361C3×Q8—181no—
362C3×Q8—181no—
363C3×D4—61no—
364C3×D4—61no—
365C3×D4—61no—
366C3×D4—61no—
367C3×D4—61no—
368C3×D4—61no—
369C3×D4—61no—
370C3×D4—61no—
371C3×D4—121no—
372C3×D4—121no—
373C3×D4—121no—
374C3×D4—121no—
375C3×D4—181no—
376C3×D4—181no—
377C2×C3×C4—21no—
378C2×C3×C4—21no—
379C2×C3×C4—21no—
380C2×C3×C4—41no—
381C2×C3×C4—181no—
382C2×C3×C4C2×D3211yescharacteristic
383C2×C3×C4C2×D3211yescharacteristic
384C2×C3×C4C2×D3211yescharacteristic
385#67—364no—
386#67—368no—
387#67—364no—
388#67—364no—
389C2×Dic3—62no—
390C2×Dic3—61no—
391C2×Dic3—61no—
392C2×Dic3—92no—
393C2×Dic3—92no—
394C2×Dic3—92no—
395C2×Dic3—182no—
396C2×Dic3—182no—
397C2×Dic3—182no—
398C2×Dic3—362no—
399D12—21no—
400D12—21no—
401D12—21no—
402D12—21no—
403D12—61no—
404D12—61no—
405D12—61no—
406D12—61no—
407D12—121no—
408D12—121no—
409Dic6—61no—
410Dic6—61no—
411Dic6—61no—
412Dic6—61no—
413Dic6—121no—
414Dic6—121no—
415C3×C8—32no—
416C3×C8—32no—
417C3×C8—62no—
418C3×C8—92no—
419C3⋊C8—32no—
420C3⋊C8—32no—
421C3⋊C8—62no—
422C3⋊C8#62512yescomplemented
423C2×C32—22no—
424C2×C32—21no—
425C2×C32—22no—
426C2×C32—22no—
427C2×C32—21no—
428C2×C32—21no—
429C2×C32—41no—
430C2×C32—42no—
431C2×C32—241no—
432C2×C32—241no—
433C2×C32#91112yes—
434C2×C32#91112yes—
435C2×C32#91412yes—
436C2×C32#94011yescharacteristic
437C2×C32#94011yescharacteristic
438C2×C32#94011yescharacteristic
439C3⋊D3—81no—
440C3×D3—82no—
441C3×D3—82no—
442C3×D3—81no—
443C3×D3—81no—
444C3×D3—81no—
445C3×D3—81no—
446C3×D3—242no—
447C3×D3—241no—
448C3×D3—241no—
449C2×Q8—91no—
450C2×D4—31no—
451C2×D4—31no—
452C2×D4—91no—
453C2×D4—542no—
454SD16—182no—
455SD16—542no—
456D8—182no—
457D8—182no—
458C2×C8—31no—
459C2×C8—91no—
460C22⋊C4—542no—
461C42—271no—
462C22×C3—21no—
463C22×C3—21no—
464C22×C3—21no—
465C22×C3—41no—
466C22×C3—124no—
467C22×C3—122no—
468C22×C3—122no—
469C22×C3—124no—
470C22×C3—122no—
471C22×C3—122no—
472C22×C3—121no—
473C22×C3—121no—
474C22×C3—122no—
475C22×C3—122no—
476C22×C3—121no—
477C22×C3—121no—
478C22×C3—362no—
479C22×C3—361no—
480C22×C3#380211yescharacteristic
481C22×C3#381111yescharacteristic
482C22×C3#381111yescharacteristic
483C2×D3—42no—
484C2×D3—42no—
485C2×D3—41no—
486C2×D3—41no—
487C2×D3—122no—
488C2×D3—124no—
489C2×D3—122no—
490C2×D3—121no—
491C2×D3—122no—
492C2×D3—121no—
493C3×C4—21no—
494C3×C4—21no—
495C3×C4—21no—
496C3×C4—21no—
497C3×C4—21no—
498C3×C4—21no—
499C3×C4—41no—
500C3×C4—41no—
501C3×C4—361no—
502C3×C4#380211yescharacteristic
503C3×C4C2×#62511yescharacteristic
504C3×C4D3×#6711yescharacteristic
505C3×C4D3×#6711yescharacteristic
506C3×C4D3×#6711yescharacteristic
507C3×C4D3×#6711yescharacteristic
508Dic3—122no—
509Dic3—121no—
510Dic3—121no—
511Dic3—184no—
512Dic3—182no—
513Dic3—182no—
514Dic3—182no—
515Dic3—362no—
516Dic3—362no—
517Dic3—364no—
518C32—21no—
519C32—21no—
520C32—21no—
521C32—41no—
522C32#515411yescharacteristic, complemented
523C32#515411yescharacteristic, complemented
524C32#517611yescharacteristic, complemented
525C23—61no—
526C23—61no—
527C23—181no—
528Q8—181no—
529Q8—181no—
530D4—61no—
531D4—61no—
532D4—61no—
533D4—61no—
534D4—181no—
535D4—181no—
536D4—1084no—
537C2×C4—181no—
538C2×C4—272no—
539C2×C4D3311yescharacteristic
540C8—32no—
541C8—92no—
542C2×C3—21no—
543C2×C3—21no—
544C2×C3—21no—
545C2×C3—22no—
546C2×C3—22no—
547C2×C3—22no—
548C2×C3—41no—
549C2×C3—42no—
550C2×C3—242no—
551C2×C3—242no—
552C2×C3—241no—
553C2×C3—241no—
554C2×C3—241no—
555C2×C3—241no—
556C2×C3—721no—
557C2×C3#6379812yes—
558C2×C3#6391411yescharacteristic
559C2×C3#6392712yes—
560C2×C3#6393712yes—
561C2×C3#6397911yescharacteristic
562C2×C3#6397911yescharacteristic
563D3—81no—
564D3—81no—
565D3—242no—
566D3—241no—
567D3—241no—
568C22—121no—
569C22—121no—
570C22—122no—
571C22—122no—
572C22—361no—
573C22—362no—
574C22#8895411yescharacteristic
575C4—361no—
576C4—542no—
577C4D3×#62511yescharacteristic
578C4#8895311yescharacteristic
579C3—21no—
580C3—21no—
581C3—21no—
582C3—41no—
583C3#1059088711yescharacteristic, complemented
584C3#1059147511yescharacteristic, complemented
585C3#1059148511yescharacteristic, complemented
586C2—241no—
587C2—241no—
588C2—721no—
589C2#1172287712yes—
590C2#1172302911yescharacteristic
subgroupG/H
centerC22#88954
commutator, Fitting, cosocleC2×C33×C4C23
Frattini, hypercenterC2×C4D33
socleC22×C33C2×D4
hypocenterC33#496
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
2. #11723529C2semidirect
3. #11723434C2semidirect
4. C2×C3×#3716C2semidirect
5. C2×C3×#3714C2semidirect
6. C2×C3×#3713C2semidirect
7. #11722021C2semidirect
40. C2×#3716D3semidirect
42. C2×#3714D3semidirect
43. C2×#3713D3semidirect
63. C3×C3⋊(C3⋊C8)D4semidirect
64. C32×C3⋊C8D4semidirect
221. C3⋊(C3⋊C8)#67semidirect
226. C3×C3⋊C8#67semidirect
227. C3×C3⋊C8#67semidirect
348. C33#496semidirect
422. C3⋊C8#625semidirect
522. C32#5154semidirect
523. C32#5154semidirect
524. C32#5176semidirect
583. C3#10590887semidirect
584. C3#10591475semidirect
585. C3#10591485semidirect

Maximal subgroups

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

subgroupindex
C2×#888112
#117235292
#117234342
C2×C3×#37162
C2×C3×#37142
C2×C3×#37132
#117220212
#105914853
#105914753
#105908873