Finite Group Databasebeta

← #13156631

Subgroups — #13156631

Order 1728.

882 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 882 conjugacy classes of subgroups — too many to draw. Only its 106 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#11723524C211yesmaximal, characteristic, complemented
2C2×#88795C211yesmaximal, characteristic, complemented
3C2×#88790C211yesmaximal, characteristic, complemented
4C3×#63863C211yesmaximal, characteristic, complemented
5C2×C3×#3715C211yesmaximal, characteristic, complemented
6C2×C3×#3713C211yesmaximal, characteristic, complemented
7#11722071C211yesmaximal, characteristic, complemented
8#10591619—31nomaximal
9#10591483—31nomaximal
10#10591475—31nomaximal
11C2×C4⋊(C3⋊(C3⋊D3))C2211yescharacteristic
12C2×C32×D12C2211yescharacteristic
13C2×C32×D12C2211yescharacteristic
14C2×C32×Dic6C2211yescharacteristic
15C2×#6291—22no—
16C3×C4×C3⋊Dic3C2211yescharacteristic
17C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
18C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
19#88795—22no—
20#88790—22no—
21C3×(C2×D3)⋊Dic3—22no—
22C3×#3715—22no—
23C3×#3713—22no—
24C2×#3755—31no—
25#64149—31no—
26C2×#3753—31no—
27C3×#947—31no—
28C2×C3×#215—31no—
29C3×#936—31no—
30C2×C3×#214—31no—
31C2×C3×#213—31no—
32C2×C3×#213—31no—
33#63911—31no—
34#63911—31no—
35#63911—61no—
36#63863D311yescharacteristic, complemented
37C2×#3717—31no—
38C2×#3717—31no—
39C2×#3717—61no—
40C2×#3715D311yescharacteristic, complemented
41C2×#3714—31no—
42C2×#3714—31no—
43C2×#3714—61no—
44C2×#3713D311yescharacteristic, complemented
45#63555—31no—
46#63554—31no—
47#63554—31no—
48C22×C3⋊(C3⋊D3)—21no—
49C22×C32×D3—21no—
50C22×C32×D3—21no—
51C2×C33×C4C2311yescharacteristic
52C4⋊(C3⋊(C3⋊D3))—21no—
53C4⋊(C3⋊(C3⋊D3))—21no—
54C2×C3×C3⋊Dic3D412yes—
55C2×C32×Dic3—21no—
56C32×D12—21no—
57C32×D12—21no—
58C32×D12—21no—
59C32×D12—21no—
60C32×Dic6—21no—
61C32×Dic6—21no—
62#6291—44no—
63C3×C3⋊(C3⋊C8)D412yescomplemented
64C3×C3⋊(C3⋊C8)D412yescomplemented
65#5176—91no—
66#5154—91no—
67#5058—91no—
68C2×C32×Q8—31no—
69C2×C32×D4—31no—
70C2×C32×D4—31no—
71C2×#638—62no—
72C2×C4⋊(C3⋊D3)—31no—
73C2×C4⋊(C3⋊D3)—31no—
74C2×C4⋊(C3⋊D3)—31no—
75C2×C4⋊(C3⋊D3)—61no—
76C2×C4⋊(C3⋊D3)—61no—
77C2×C4⋊(C3⋊D3)—61no—
78C2×C4⋊(C3⋊D3)—121no—
79C2×C3×D12—21no—
80C2×C3×D12—21no—
81C2×C3×D12C2×D311yescharacteristic
82C2×C3×D12C2×D311yescharacteristic
83C2×C3×D12C2×D311yescharacteristic
84C2×C3×D12C2×D311yescharacteristic
85C2×C3×Dic6—21no—
86C2×C3×Dic6C2×D311yescharacteristic
87C2×C3×Dic6C2×D311yescharacteristic
88C2×#626—62no—
89C2×#626—62no—
90C2×#626—64no—
91#3755—62no—
92#3753—62no—
93C4×C3⋊Dic3C2×D311yescharacteristic
94C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
95C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
96C3×#217—62no—
97C3×#215—62no—
98C3×#214—62no—
99C3×#213—62no—
100C3×#213—62no—
101C3×(C2×D3)⋊C4—62no—
102C3×C4×Dic3—31no—
103C3×C4×Dic3—31no—
104C3×C4×Dic3—61no—
105C2×C3×C3⋊C8—31no—
106C2×C3×C3⋊C8—31no—
107C2×C3×C3⋊C8—31no—
108C2×C3×C3⋊C8—31no—
109C2×C3×C3⋊C8—61no—
110C2×C3×C3⋊C8—61no—
111(C2×D3)⋊Dic3—22no—
112#3717—62no—
113#3717—62no—
114#3717—64no—
115#3715—22no—
116#3714—62no—
117#3714—62no—
118#3714—64no—
119#3713—22no—
120C22×C33C2×D411yescharacteristic
121C2×C3⋊(C3⋊D3)—41no—
122C2×C3⋊(C3⋊D3)—42no—
123C2×C32×D3—41no—
124C2×C32×D3—41no—
125C2×C32×D3—42no—
126C2×C32×D3—42no—
127C33×C4C2×D411yescharacteristic
128C33×C4C2×D411yescharacteristic
129C3×C3⋊Dic3—22no—
130C32×Dic3—41no—
131C2×C3×SD16—91no—
132C2×C3×D8—91no—
133C3×#124—91no—
134#947—31no—
135C2×#215—31no—
136C2×#215—91no—
137C2×#215—91no—
138C2×#215—181no—
139#940—91no—
140#940—91no—
141#940—181no—
142#936—31no—
143C2×#214—31no—
144C2×#213—31no—
145C2×#213—31no—
146C2×#213—91no—
147C2×#213—91no—
148C2×#213—181no—
149C2×D24—91no—
150C2×#204—91no—
151#874—91no—
152#815—91no—
153#815—91no—
154#803—91no—
155C23×C32—61no—
156C23×C32—61no—
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—21no—
168C22×C3×D3—21no—
169C22×C3×D3—41no—
170C32×Q8—61no—
171C32×Q8—61no—
172C32×D4—61no—
173C32×D4—61no—
174C32×D4—61no—
175C32×D4—61no—
176C2×C32×C4—21no—
177C2×C32×C4—21no—
178C2×C32×C4—21no—
179C2×C32×C4—41no—
180C2×C32×C4—61no—
181C2×C32×C4C22×D311yescharacteristic
182C2×C32×C4C22×D311yescharacteristic
183C2×C32×C4C22×D311yescharacteristic
184#638—124no—
185C2×C3⋊Dic3#6712yes—
186C4⋊(C3⋊D3)—61no—
187C4⋊(C3⋊D3)—61no—
188C4⋊(C3⋊D3)—62no—
189C4⋊(C3⋊D3)—62no—
190C4⋊(C3⋊D3)—61no—
191C4⋊(C3⋊D3)—61no—
192C4⋊(C3⋊D3)—61no—
193C4⋊(C3⋊D3)—61no—
194C4⋊(C3⋊D3)—121no—
195C4⋊(C3⋊D3)—121no—
196C4⋊(C3⋊D3)—121no—
197C4⋊(C3⋊D3)—121no—
198C4⋊(C3⋊D3)—122no—
199C4⋊(C3⋊D3)—122no—
200C2×C3×Dic3—21no—
201C2×C3×Dic3—21no—
202C2×C3×Dic3—32no—
203C2×C3×Dic3—32no—
204C2×C3×Dic3—41no—
205C2×C3×Dic3—62no—
206C3×D12—22no—
207C3×D12—21no—
208C3×D12—21no—
209C3×D12—21no—
210C3×D12—21no—
211C3×D12—21no—
212C3×D12—21no—
213C3×D12—21no—
214C3×D12—21no—
215C3×D12—41no—
216C3×D12—41no—
217C3×D12—41no—
218C3×Dic6—22no—
219C3×Dic6—21no—
220C3×Dic6—21no—
221C3×Dic6—21no—
222C3×Dic6—21no—
223C3×Dic6—41no—
224#626—128no—
225#626—124no—
226#626—124no—
227C3⋊(C3⋊C8)#6712yescomplemented
228C3⋊(C3⋊C8)#6712yescomplemented
229C3×C3⋊C8—32no—
230C3×C3⋊C8—32no—
231C3×C3⋊C8—32no—
232C3×C3⋊C8—32no—
233C3×C3⋊C8—62no—
234C3×C3⋊C8—62no—
235#496—271noSylow
236C2×C33#12711yescharacteristic
237C2×C33AGL(1,ℤ/8)12yes—
238C3⋊(C3⋊D3)—81no—
239C32×D3—81no—
240C32×D3—81no—
241C2×C3×Q8—31no—
242C2×C3×Q8—31no—
243C2×C3×Q8—61no—
244C2×C3×D4—31no—
245C2×C3×D4—31no—
246C2×C3×D4—31no—
247C2×C3×D4—31no—
248C2×C3×D4—61no—
249C2×C3×D4—61no—
250C2×#67—182no—
251C2×#67—182no—
252C2×#67—184no—
253C2×D12—91no—
254C2×D12—91no—
255C2×D12—91no—
256C2×D12—181no—
257C2×D12—181no—
258C2×D12—181no—
259C2×D12—182no—
260C2×D12—361no—
261C2×D12D3211yescharacteristic
262C2×D12D3211yescharacteristic
263C2×Dic6D3211yescharacteristic
264C3×SD16—182no—
265C3×D8—182no—
266C2×C3×C8—91no—
267C2×C3×C8—91no—
268C3×C22⋊C4—182no—
269C3×C42—91no—
270#217—62no—
271#215—62no—
272#215—182no—
273#215—182no—
274#215—184no—
275#214—62no—
276#213—62no—
277#213—62no—
278#213—182no—
279#213—182no—
280#213—184no—
281(C2×D3)⋊C4—62no—
282C4×Dic3—31no—
283C4×Dic3—31no—
284C4×Dic3—61no—
285C2×C3⋊C8—31no—
286C2×C3⋊C8—31no—
287C2×C3⋊C8—31no—
288C2×C3⋊C8—31no—
289C2×C3⋊C8—61no—
290C2×C3⋊C8—61no—
291D24—182no—
292#204—182no—
293C22×C32—21no—
294C22×C32—21no—
295C22×C32—21no—
296C22×C32—41no—
297C22×C32—121no—
298C22×C32—121no—
299C22×C32—122no—
300C22×C32—122no—
301C22×C32D3×D411yescharacteristic
302C22×C32D3×D411yescharacteristic
303C22×C32C2×#6711yescharacteristic
304C2×C3⋊D3—122no—
305C2×C3⋊D3—122no—
306C2×C3⋊D3—122no—
307C2×C3⋊D3—124no—
308C2×C3⋊D3—121no—
309C2×C3⋊D3—121no—
310C2×C3⋊D3—121no—
311C2×C3⋊D3—122no—
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—42no—
323C2×C3×D3—42no—
324C2×C3×D3—41no—
325C2×C3×D3—41no—
326C2×C3×D3—41no—
327C2×C3×D3—41no—
328C2×C3×D3—42no—
329C2×C3×D3—82no—
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×C4—121no—
339C32×C4D3×D411yescharacteristic
340C32×C4D3×D411yescharacteristic
341C32×C4D3×D411yescharacteristic
342C32×C4D3×D411yescharacteristic
343C32×C4C2×#6711yescharacteristic
344C32×C4C2×#6711yescharacteristic
345C3⋊Dic3—22no—
346C3×Dic3—42no—
347C3×Dic3—41no—
348C3×Dic3—41no—
349C3×Dic3—64no—
350C3×Dic3—62no—
351C3×Dic3—62no—
352C2×SD16—91no—
353C2×SD16—271no—
354C2×D8—91no—
355C2×D8—271no—
356#127—271no—
357#124—91no—
358#97—271no—
359C33#49611yescharacteristic, complemented, Sylow
360C23×C3—62no—
361C23×C3—61no—
362C23×C3—61no—
363C23×C3—61no—
364C23×C3—61no—
365C23×C3—121no—
366C22×D3—21no—
367C22×D3—21no—
368C22×D3—182no—
369C22×D3—181no—
370C22×D3—181no—
371C22×D3—181no—
372C22×D3—361no—
373C22×D3—361no—
374C22×D3—362no—
375C3×Q8—62no—
376C3×Q8—61no—
377C3×Q8—61no—
378C3×Q8—61no—
379C3×Q8—61no—
380C3×Q8—121no—
381C3×D4—62no—
382C3×D4—61no—
383C3×D4—61no—
384C3×D4—61no—
385C3×D4—61no—
386C3×D4—61no—
387C3×D4—61no—
388C3×D4—61no—
389C3×D4—61no—
390C3×D4—121no—
391C3×D4—121no—
392C3×D4—121no—
393C2×C3×C4—21no—
394C2×C3×C4—21no—
395C2×C3×C4—21no—
396C2×C3×C4—41no—
397C2×C3×C4—61no—
398C2×C3×C4—61no—
399C2×C3×C4—92no—
400C2×C3×C4—121no—
401C2×C3×C4C2×D3211yescharacteristic
402C2×C3×C4C2×D3211yescharacteristic
403C2×C3×C4C2×D3211yescharacteristic
404#67—368no—
405#67—364no—
406#67—364no—
407C2×Dic3—21no—
408C2×Dic3—32no—
409C2×Dic3—32no—
410C2×Dic3—62no—
411D12—21no—
412D12—21no—
413D12—21no—
414D12—21no—
415D12—182no—
416D12—182no—
417D12—181no—
418D12—181no—
419D12—181no—
420D12—181no—
421D12—181no—
422D12—181no—
423D12—361no—
424D12—361no—
425D12—361no—
426D12—361no—
427D12—362no—
428D12—362no—
429D12—364no—
430Dic6—21no—
431Dic6—21no—
432C3×C8—92no—
433C3×C8—92no—
434C3⋊C8—32no—
435C3⋊C8—32no—
436C3⋊C8—32no—
437C3⋊C8—32no—
438C3⋊C8—62no—
439C3⋊C8—62no—
440C2×C32—22no—
441C2×C32—22no—
442C2×C32—22no—
443C2×C32—21no—
444C2×C32—21no—
445C2×C32—21no—
446C2×C32—41no—
447C2×C32—42no—
448C2×C32—241no—
449C2×C32—241no—
450C2×C32#91112yes—
451C2×C32#91412yes—
452C2×C32#94011yescharacteristic
453C2×C32#94011yescharacteristic
454C2×C32#94011yescharacteristic
455C2×C32#94912yes—
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—82no—
465C3×D3—81no—
466C3×D3—81no—
467C3×D3—81no—
468C3×D3—81no—
469C2×Q8—31no—
470C2×D4—31no—
471C2×D4—31no—
472C2×D4—271no—
473C2×D4—542no—
474SD16—182no—
475SD16—542no—
476D8—182no—
477D8—542no—
478C2×C8—91no—
479C2×C8—91no—
480C22⋊C4—182no—
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—122no—
491C22×C3—122no—
492C22×C3—122no—
493C22×C3—121no—
494C22×C3—121no—
495C22×C3—121no—
496C22×C3—121no—
497C22×C3—242no—
498C22×C3D3×#6711yescharacteristic
499C22×C3D3×#6711yescharacteristic
500C22×C3#381111yescharacteristic
501C2×D3—42no—
502C2×D3—42no—
503C2×D3—41no—
504C2×D3—41no—
505C2×D3—362no—
506C2×D3—362no—
507C2×D3—362no—
508C2×D3—364no—
509C2×D3—364no—
510C2×D3—362no—
511C2×D3—362no—
512C2×D3—362no—
513C2×D3—361no—
514C2×D3—361no—
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—122no—
528C3×C4—121no—
529C3×C4—121no—
530C3×C4—182no—
531C3×C4D3×#6711yescharacteristic
532C3×C4D3×#6711yescharacteristic
533C3×C4D3×#6711yescharacteristic
534C3×C4D3×#6711yescharacteristic
535C3×C4#381111yescharacteristic
536C3×C4#381111yescharacteristic
537Dic3—41no—
538Dic3—64no—
539Dic3—62no—
540Dic3—62no—
541C32—21no—
542C32—21no—
543C32—21no—
544C32—41no—
545C32#505811yescharacteristic, complemented
546C32#515411yescharacteristic, complemented
547C32#517611yescharacteristic, complemented
548C23—61no—
549C23—61no—
550C23—541no—
551Q8—61no—
552Q8—61no—
553D4—61no—
554D4—61no—
555D4—61no—
556D4—61no—
557D4—541no—
558D4—541no—
559D4—1084no—
560C2×C4—61no—
561C2×C4—92no—
562C2×C4D3311yescharacteristic
563C8—92no—
564C8—92no—
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—242no—
575C2×C3—241no—
576C2×C3—241no—
577C2×C3—241no—
578C2×C3—241no—
579C2×C3#6392512yes—
580C2×C3#6393712yes—
581C2×C3#6393912yes—
582C2×C3#6397911yescharacteristic
583C2×C3#6397911yescharacteristic
584C2×C3#6397911yescharacteristic
585D3—81no—
586D3—81no—
587D3—722no—
588D3—722no—
589D3—722no—
590D3—724no—
591D3—721no—
592D3—721no—
593D3—721no—
594C22—121no—
595C22—121no—
596C22—122no—
597C22—122no—
598C22—1081no—
599C22—1082no—
600C22#8895211yescharacteristic
601C4—121no—
602C4—182no—
603C4#8895211yescharacteristic
604C4#8895211yescharacteristic
605C3—21no—
606C3—21no—
607C3—21no—
608C3—41no—
609C3#1059147511yescharacteristic, complemented
610C3#1059148311yescharacteristic, complemented
611C3#1059161911yescharacteristic, complemented
612C2—241no—
613C2—241no—
614C2—2161no—
615C2#1172287412yes—
616C2#1172303111yescharacteristic
subgroupG/H
centerC22#88952
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. #11723524C2semidirect
2. C2×#88795C2semidirect
3. C2×#88790C2semidirect
4. C3×#63863C2semidirect
5. C2×C3×#3715C2semidirect
6. C2×C3×#3713C2semidirect
7. #11722071C2semidirect
36. #63863D3semidirect
40. C2×#3715D3semidirect
44. C2×#3713D3semidirect
63. C3×C3⋊(C3⋊C8)D4semidirect
64. C3×C3⋊(C3⋊C8)D4semidirect
227. C3⋊(C3⋊C8)#67semidirect
228. C3⋊(C3⋊C8)#67semidirect
359. C33#496semidirect
545. C32#5058semidirect
546. C32#5154semidirect
547. C32#5176semidirect
609. C3#10591475semidirect
610. C3#10591483semidirect
611. C3#10591619semidirect

Maximal subgroups

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

subgroupindex
#117235242
C2×#887952
C2×#887902
C3×#638632
C2×C3×#37152
C2×C3×#37132
#117220712
#105916193
#105914833
#105914753