Finite Group Databasebeta

← #13156626

Subgroups — #13156626

Order 1728.

882 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 882 conjugacy classes of subgroups — too many to draw. Only its 114 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
1#11723677C211yesmaximal, characteristic, complemented
2#11723524C211yesmaximal, characteristic, complemented
3C2×#88797C211yesmaximal, characteristic, complemented
4C2×#88791C211yesmaximal, characteristic, complemented
5C2×C3×#3716C211yesmaximal, characteristic, complemented
6C2×C3×#3714C211yesmaximal, characteristic, complemented
7C3×#63555C211yesmaximal, characteristic, complemented
8#10591475—31nomaximal
9#10590883—31nomaximal
10#10590881—31nomaximal
11C2×C4⋊(C3⋊(C3⋊D3))C2211yescharacteristic
12C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
13C2×C3×#634C2211yescharacteristic
14C2×C32×D12C2211yescharacteristic
15C2×#6291—22no—
16C3×C4×C3⋊Dic3C2211yescharacteristic
17C2×C32×C3⋊C8C2211yescharacteristic
18C2×C32×C3⋊C8C2211yescharacteristic
19#88814—22no—
20#88797—22no—
21#88791—22no—
22C3×#3716—22no—
23C3×#3714—22no—
24#64149—31no—
25C2×#3745—31no—
26C2×#3744—31no—
27C2×C3×#214—31no—
28C2×C3×#213—31no—
29C2×C3×D24—31no—
30C2×C3×#204—31no—
31#63911—31no—
32#63911—31no—
33#63911—61no—
34#63864—31no—
35#63863—31no—
36#63860—31no—
37C2×#3717—31no—
38C2×#3717—31no—
39C2×#3717—61no—
40C2×#3716D311yescharacteristic, complemented
41C2×#3714—31no—
42C2×#3714—31no—
43C2×#3714—61no—
44C2×#3714D311yescharacteristic, complemented
45C3×#815—31no—
46C3×#815—31no—
47#63555D311yescharacteristic, complemented
48C22×C3⋊(C3⋊D3)—21no—
49C22×C3×C3⋊D3—21no—
50C22×C32×D3—21no—
51C2×C33×C4C2311yescharacteristic
52C4⋊(C3⋊(C3⋊D3))—21no—
53C4⋊(C3⋊(C3⋊D3))—21no—
54C2×C3×C3⋊Dic3—21no—
55C2×C3×C3⋊Dic3D412yes—
56C3×C4⋊(C3⋊D3)—21no—
57C3×C4⋊(C3⋊D3)—21no—
58C3×#634—21no—
59C3×#634—21no—
60C32×D12—21no—
61C32×D12—21no—
62#6291—44no—
63C32×C3⋊C8D412yescomplemented
64C32×C3⋊C8D412yescomplemented
65#5176—91no—
66#5154—91no—
67#4712—91no—
68C2×C32×D4—31no—
69C2×#638—62no—
70C2×C4⋊(C3⋊D3)—31no—
71C2×C4⋊(C3⋊D3)—31no—
72C2×C4⋊(C3⋊D3)—31no—
73C2×C4⋊(C3⋊D3)—61no—
74C2×C4⋊(C3⋊D3)—61no—
75C2×C4⋊(C3⋊D3)—61no—
76C2×C4⋊(C3⋊D3)—121no—
77C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
78C2×#634C2×D311yescharacteristic
79C2×C3×D12—21no—
80C2×C3×D12—31no—
81C2×C3×D12—31no—
82C2×C3×D12—61no—
83C2×C3×D12C2×D311yescharacteristic
84C2×C3×D12C2×D311yescharacteristic
85C2×C3×Dic6—31no—
86C2×C3×Dic6—31no—
87C2×C3×Dic6—61no—
88C2×#626—62no—
89C2×#626—62no—
90C2×#626—64no—
91C2×C32×C8—31no—
92C2×C32×C8—31no—
93C4×C3⋊Dic3C2×D311yescharacteristic
94#3745—62no—
95#3744—62no—
96C3×#214—62no—
97C3×#213—62no—
98C3×C4×Dic3—31no—
99C3×C4×Dic3—31no—
100C3×C4×Dic3—61no—
101C2×C3×C3⋊C8—21no—
102C2×C3×C3⋊C8—21no—
103C2×C3×C3⋊C8C2×D311yescharacteristic
104C2×C3×C3⋊C8C2×D311yescharacteristic
105C2×C3×C3⋊C8C2×D311yescharacteristic
106C2×C3×C3⋊C8C2×D311yescharacteristic
107C3×D24—62no—
108C3×#204—62no—
109#3722—62no—
110(C2×D3)⋊Dic3—62no—
111(C2×D3)⋊Dic3—62no—
112#3717—64no—
113#3717—62no—
114#3717—62no—
115#3716—22no—
116#3714—22no—
117#3714—64no—
118#3714—62no—
119#3714—62no—
120C22×C33C2×D411yescharacteristic
121C2×C3⋊(C3⋊D3)—41no—
122C2×C3⋊(C3⋊D3)—42no—
123C2×C3×C3⋊D3—41no—
124C2×C3×C3⋊D3—42no—
125C2×C32×D3—41no—
126C2×C32×D3—42no—
127C33×C4C2×D411yescharacteristic
128C33×C4C2×D411yescharacteristic
129C3×C3⋊Dic3—22no—
130C3×C3⋊Dic3—41no—
131C2×C3×SD16—91no—
132C2×C3×D8—91no—
133#947—91no—
134C2×#215—91no—
135#940—91no—
136#940—91no—
137#940—181no—
138#936—91no—
139C2×#214—31no—
140C2×#213—31no—
141C2×#213—91no—
142C2×D24—31no—
143C2×D24—91no—
144C2×D24—91no—
145C2×D24—181no—
146C2×#204—31no—
147C2×#204—91no—
148C2×#204—91no—
149C2×#204—181no—
150#875—91no—
151#874—91no—
152C3×#97—91no—
153#815—31no—
154#815—31no—
155C23×C32—61no—
156C22×C3⋊D3—21no—
157C22×C3⋊D3—61no—
158C22×C3⋊D3—62no—
159C22×C3⋊D3—61no—
160C22×C3⋊D3—61no—
161C22×C3⋊D3—121no—
162C22×C3⋊D3—121no—
163C22×C3⋊D3—122no—
164C22×C3×D3—22no—
165C22×C3×D3—21no—
166C22×C3×D3—21no—
167C22×C3×D3—61no—
168C22×C3×D3—61no—
169C22×C3×D3—121no—
170C32×D4—61no—
171C32×D4—61no—
172C2×C32×C4—21no—
173C2×C32×C4—21no—
174C2×C32×C4—21no—
175C2×C32×C4—41no—
176C2×C32×C4C22×D311yescharacteristic
177C2×C32×C4C22×D311yescharacteristic
178C2×C32×C4C22×D311yescharacteristic
179#638—124no—
180C2×C3⋊Dic3—21no—
181C2×C3⋊Dic3D1212yes—
182C4⋊(C3⋊D3)—21no—
183C4⋊(C3⋊D3)—21no—
184C4⋊(C3⋊D3)—61no—
185C4⋊(C3⋊D3)—61no—
186C4⋊(C3⋊D3)—62no—
187C4⋊(C3⋊D3)—62no—
188C4⋊(C3⋊D3)—61no—
189C4⋊(C3⋊D3)—61no—
190C4⋊(C3⋊D3)—61no—
191C4⋊(C3⋊D3)—61no—
192C4⋊(C3⋊D3)—121no—
193C4⋊(C3⋊D3)—121no—
194C4⋊(C3⋊D3)—121no—
195C4⋊(C3⋊D3)—121no—
196C4⋊(C3⋊D3)—122no—
197C4⋊(C3⋊D3)—122no—
198#634—21no—
199#634—21no—
200C2×C3×Dic3—32no—
201C2×C3×Dic3—32no—
202C2×C3×Dic3—62no—
203C2×C3×Dic3—61no—
204C2×C3×Dic3—61no—
205C2×C3×Dic3—121no—
206C3×D12—21no—
207C3×D12—21no—
208C3×D12—21no—
209C3×D12—21no—
210C3×D12—41no—
211C3×D12—41no—
212C3×D12—62no—
213C3×D12—61no—
214C3×D12—61no—
215C3×D12—61no—
216C3×D12—61no—
217C3×D12—121no—
218C3×Dic6—62no—
219C3×Dic6—61no—
220C3×Dic6—61no—
221C3×Dic6—61no—
222C3×Dic6—61no—
223C3×Dic6—121no—
224#626—128no—
225#626—124no—
226#626—124no—
227C32×C8—32no—
228C32×C8—32no—
229C3×C3⋊C8—22no—
230C3×C3⋊C8—22no—
231C3×C3⋊C8D1212yescomplemented
232C3×C3⋊C8D1212yescomplemented
233C3×C3⋊C8#6712yescomplemented
234C3×C3⋊C8#6712yescomplemented
235#496—271noSylow
236C2×C33#12711yescharacteristic
237C2×C33AGL(1,ℤ/8)12yes—
238C3⋊(C3⋊D3)—81no—
239C3×C3⋊D3—81no—
240C32×D3—81no—
241C2×C3×Q8—91no—
242C2×C3×D4—31no—
243C2×C3×D4—31no—
244C2×C3×D4—61no—
245C2×C3×D4—91no—
246C2×#67—182no—
247C2×#67—182no—
248C2×#67—184no—
249C2×D12—31no—
250C2×D12—31no—
251C2×D12—61no—
252C2×D12—91no—
253C2×D12—91no—
254C2×D12—91no—
255C2×D12—181no—
256C2×D12—181no—
257C2×D12—181no—
258C2×D12—182no—
259C2×D12—361no—
260C2×D12D3211yescharacteristic
261C2×Dic6—31no—
262C2×Dic6—31no—
263C2×Dic6—61no—
264C3×SD16—182no—
265C3×D8—182no—
266C2×C3×C8—31no—
267C2×C3×C8—31no—
268C2×C3×C8—31no—
269C2×C3×C8—31no—
270C2×C3×C8—61no—
271C2×C3×C8—61no—
272C3×C42—91no—
273#217—182no—
274#215—182no—
275#214—62no—
276#213—62no—
277#213—182no—
278(C2×D3)⋊C4—182no—
279(C2×D3)⋊C4—182no—
280C4×Dic3—31no—
281C4×Dic3—31no—
282C4×Dic3—61no—
283C2×C3⋊C8D3211yescharacteristic
284C2×C3⋊C8D3211yescharacteristic
285D24—62no—
286D24—182no—
287D24—182no—
288D24—184no—
289#204—62no—
290#204—182no—
291#204—182no—
292#204—184no—
293C22×C32—21no—
294C22×C32—21no—
295C22×C32—21no—
296C22×C32—41no—
297C22×C32—121no—
298C22×C32—122no—
299C22×C32C2×D1211yescharacteristic
300C22×C32D3×D411yescharacteristic
301C22×C32D3×D411yescharacteristic
302C2×C3⋊D3—41no—
303C2×C3⋊D3—42no—
304C2×C3⋊D3—122no—
305C2×C3⋊D3—122no—
306C2×C3⋊D3—122no—
307C2×C3⋊D3—124no—
308C2×C3⋊D3—121no—
309C2×C3⋊D3—122no—
310C2×C3⋊D3—121no—
311C2×C3⋊D3—121no—
312C2×C3⋊D3—124no—
313C2×C3⋊D3—122no—
314C2×C3⋊D3—122no—
315C2×C3⋊D3—242no—
316C2×C3⋊D3—242no—
317C2×C3⋊D3—244no—
318C2×C3×D3—44no—
319C2×C3×D3—42no—
320C2×C3×D3—42no—
321C2×C3×D3—42no—
322C2×C3×D3—41no—
323C2×C3×D3—41no—
324C2×C3×D3—122no—
325C2×C3×D3—121no—
326C2×C3×D3—122no—
327C2×C3×D3—122no—
328C2×C3×D3—121no—
329C2×C3×D3—242no—
330C32×C4—21no—
331C32×C4—21no—
332C32×C4—21no—
333C32×C4—21no—
334C32×C4—21no—
335C32×C4—21no—
336C32×C4—41no—
337C32×C4—41no—
338C32×C4C2×D1211yescharacteristic
339C32×C4C2×D1211yescharacteristic
340C32×C4D3×D411yescharacteristic
341C32×C4D3×D411yescharacteristic
342C32×C4C2×#6711yescharacteristic
343C32×C4C2×#6711yescharacteristic
344C3⋊Dic3—22no—
345C3⋊Dic3—41no—
346C3×Dic3—64no—
347C3×Dic3—62no—
348C3×Dic3—62no—
349C3×Dic3—122no—
350C3×Dic3—121no—
351C3×Dic3—121no—
352C2×SD16—91no—
353C2×SD16—271no—
354C2×D8—91no—
355C2×D8—271no—
356#127—271no—
357#124—271no—
358#97—91no—
359C33#49611yescharacteristic, complemented, Sylow
360C23×C3—62no—
361C23×C3—61no—
362C23×C3—61no—
363C23×C3—181no—
364C22×D3—21no—
365C22×D3—61no—
366C22×D3—61no—
367C22×D3—121no—
368C22×D3—182no—
369C22×D3—181no—
370C22×D3—181no—
371C22×D3—181no—
372C22×D3—361no—
373C22×D3—361no—
374C22×D3—362no—
375C3×Q8—181no—
376C3×Q8—181no—
377C3×D4—61no—
378C3×D4—61no—
379C3×D4—61no—
380C3×D4—61no—
381C3×D4—121no—
382C3×D4—121no—
383C3×D4—181no—
384C3×D4—181no—
385C2×C3×C4—21no—
386C2×C3×C4—21no—
387C2×C3×C4—21no—
388C2×C3×C4—41no—
389C2×C3×C4—92no—
390C2×C3×C4—181no—
391C2×C3×C4C2×D3211yescharacteristic
392C2×C3×C4C2×D3211yescharacteristic
393C2×C3×C4C2×D3211yescharacteristic
394#67—368no—
395#67—364no—
396#67—364no—
397C2×Dic3—32no—
398C2×Dic3—32no—
399C2×Dic3—62no—
400C2×Dic3—61no—
401C2×Dic3—61no—
402C2×Dic3—121no—
403D12—21no—
404D12—21no—
405D12—62no—
406D12—61no—
407D12—61no—
408D12—61no—
409D12—61no—
410D12—121no—
411D12—182no—
412D12—182no—
413D12—181no—
414D12—181no—
415D12—181no—
416D12—181no—
417D12—181no—
418D12—181no—
419D12—361no—
420D12—361no—
421D12—361no—
422D12—361no—
423D12—362no—
424D12—362no—
425D12—364no—
426Dic6—62no—
427Dic6—61no—
428Dic6—61no—
429Dic6—61no—
430Dic6—61no—
431Dic6—121no—
432C3×C8—32no—
433C3×C8—32no—
434C3×C8—32no—
435C3×C8—32no—
436C3×C8—62no—
437C3×C8—62no—
438C3⋊C8#62612yescomplemented
439C3⋊C8#62612yescomplemented
440C2×C32—22no—
441C2×C32—22no—
442C2×C32—21no—
443C2×C32—21no—
444C2×C32—22no—
445C2×C32—21no—
446C2×C32—41no—
447C2×C32—42no—
448C2×C32—241no—
449C2×C32#87411yescharacteristic
450C2×C32#90812yes—
451C2×C32#91112yes—
452C2×C32#91412yes—
453C2×C32#94011yescharacteristic
454C2×C32#94011yescharacteristic
455C3⋊D3—81no—
456C3⋊D3—242no—
457C3⋊D3—242no—
458C3⋊D3—242no—
459C3⋊D3—244no—
460C3⋊D3—241no—
461C3⋊D3—241no—
462C3⋊D3—241no—
463C3×D3—82no—
464C3×D3—81no—
465C3×D3—81no—
466C3×D3—242no—
467C3×D3—241no—
468C3×D3—241no—
469C2×Q8—91no—
470C2×D4—31no—
471C2×D4—91no—
472C2×D4—271no—
473C2×D4—542no—
474SD16—182no—
475SD16—542no—
476D8—182no—
477D8—542no—
478C2×C8—31no—
479C2×C8—31no—
480C22⋊C4—542no—
481C42—91no—
482C22×C3—21no—
483C22×C3—21no—
484C22×C3—21no—
485C22×C3—41no—
486C22×C3—124no—
487C22×C3—122no—
488C22×C3—122no—
489C22×C3—122no—
490C22×C3—121no—
491C22×C3—121no—
492C22×C3—362no—
493C22×C3—361no—
494C22×C3D3×D1211yescharacteristic
495C22×C3D3×D1211yescharacteristic
496C22×C3#381111yescharacteristic
497C2×D3—42no—
498C2×D3—41no—
499C2×D3—122no—
500C2×D3—122no—
501C2×D3—121no—
502C2×D3—122no—
503C2×D3—121no—
504C2×D3—242no—
505C2×D3—362no—
506C2×D3—362no—
507C2×D3—362no—
508C2×D3—364no—
509C2×D3—364no—
510C2×D3—362no—
511C2×D3—362no—
512C2×D3—361no—
513C2×D3—361no—
514C2×D3—362no—
515C2×D3—361no—
516C2×D3—722no—
517C2×D3—722no—
518C2×D3—724no—
519C3×C4—21no—
520C3×C4—21no—
521C3×C4—21no—
522C3×C4—21no—
523C3×C4—21no—
524C3×C4—21no—
525C3×C4—41no—
526C3×C4—41no—
527C3×C4—182no—
528C3×C4—361no—
529C3×C4D3×D1211yescharacteristic
530C3×C4D3×D1211yescharacteristic
531C3×C4C2×#62611yescharacteristic
532C3×C4C2×#62611yescharacteristic
533C3×C4D3×#6711yescharacteristic
534C3×C4D3×#6711yescharacteristic
535Dic3—64no—
536Dic3—62no—
537Dic3—62no—
538Dic3—122no—
539Dic3—121no—
540Dic3—121no—
541C32—21no—
542C32—21no—
543C32—21no—
544C32—41no—
545C32#471211yescharacteristic, complemented
546C32#515411yescharacteristic, complemented
547C32#517611yescharacteristic, complemented
548C23—61no—
549C23—181no—
550C23—541no—
551Q8—181no—
552Q8—181no—
553D4—61no—
554D4—61no—
555D4—181no—
556D4—181no—
557D4—541no—
558D4—541no—
559D4—1084no—
560C2×C4—92no—
561C2×C4—181no—
562C2×C4D3311yescharacteristic
563C8—32no—
564C8—32no—
565C2×C3—21no—
566C2×C3—21no—
567C2×C3—21no—
568C2×C3—22no—
569C2×C3—22no—
570C2×C3—22no—
571C2×C3—41no—
572C2×C3—42no—
573C2×C3—242no—
574C2×C3—241no—
575C2×C3—241no—
576C2×C3—721no—
577C2×C3#6379412yes—
578C2×C3#6379512yes—
579C2×C3#6391111yescharacteristic
580C2×C3#6391111yescharacteristic
581C2×C3#6393712yes—
582C2×C3#6397911yescharacteristic
583D3—81no—
584D3—242no—
585D3—241no—
586D3—241no—
587D3—722no—
588D3—722no—
589D3—722no—
590D3—724no—
591D3—721no—
592D3—721no—
593D3—721no—
594C22—121no—
595C22—122no—
596C22—361no—
597C22—362no—
598C22—1081no—
599C22—1082no—
600C22#8895511yescharacteristic
601C4—182no—
602C4—361no—
603C4D3×#62611yescharacteristic
604C4D3×#62611yescharacteristic
605C3—21no—
606C3—21no—
607C3—21no—
608C3—41no—
609C3#1059088111yescharacteristic, complemented
610C3#1059088311yescharacteristic, complemented
611C3#1059147511yescharacteristic, complemented
612C2—241no—
613C2—721no—
614C2—2161no—
615C2#1172287912yes—
616C2#1172303011yescharacteristic
subgroupG/H
centerC22#88955
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. #11723677C2semidirect
2. #11723524C2semidirect
3. C2×#88797C2semidirect
4. C2×#88791C2semidirect
5. C2×C3×#3716C2semidirect
6. C2×C3×#3714C2semidirect
7. C3×#63555C2semidirect
40. C2×#3716D3semidirect
44. C2×#3714D3semidirect
47. #63555D3semidirect
63. C32×C3⋊C8D4semidirect
64. C32×C3⋊C8D4semidirect
231. C3×C3⋊C8D12semidirect
232. C3×C3⋊C8D12semidirect
233. C3×C3⋊C8#67semidirect
234. C3×C3⋊C8#67semidirect
359. C33#496semidirect
438. C3⋊C8#626semidirect
439. C3⋊C8#626semidirect
545. C32#4712semidirect
546. C32#5154semidirect
547. C32#5176semidirect
609. C3#10590881semidirect
610. C3#10590883semidirect
611. C3#10591475semidirect

Maximal subgroups

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

subgroupindex
#117236772
#117235242
C2×#887972
C2×#887912
C2×C3×#37162
C2×C3×#37142
C3×#635552
#105914753
#105908833
#105908813