Finite Group Databasebeta

← #13181087

Subgroups — #13181087

Order 1728.

1154 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 1154 conjugacy classes of subgroups — too many to draw. Only its 131 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#11724618C211yesmaximal, characteristic, complemented
2C2×C3×C4×D32C211yesmaximal, characteristic, complemented
3#11724514C211yesmaximal, characteristic, complemented
4#11724331C212yesmaximal, complemented
5#11724317C212yesmaximal, complemented
6#11724314C212yesmaximal, complemented
7C4×D32⋊D3C214yesmaximal, complemented
8#11724300C212yesmaximal, complemented
9#10596510—31nomaximal
10C22×C3×D32C2211yescharacteristic
11C2×D32⋊D3C2214yes—
12C3×#3828—21no—
13C2×C3×C4×C3⋊D3C2211yescharacteristic
14C2×C32×C4×D3—21no—
15#89049C2212yes—
16#89046—21no—
17C4⋊#1113C2211yescharacteristic, complemented
18C4×#1113C2212yescomplemented
19#89042—21no—
20#89041—21no—
21#89040C2211yescharacteristic, complemented
22C2×C3×#624C2211yescharacteristic
23C2×C3×D3×Dic3—21no—
24C3×C4×D32—21no—
25C3×C4×D32—21no—
26C3×C4×D32C2212yes—
27C3×C4×D32C2212yes—
28#88994C2212yes—
29C4⋊(C3⋊#157)C2212yescomplemented
30C4×C3⋊#157C2212yescomplemented
31#88934C2214yescomplemented
32D32⋊Dic3C2214yes—
33#88932C2214yescomplemented
34C22×C3×C4×D3—61no—
35#64306—31no—
36D3×#235—61no—
37C2×C4×D32D311yescharacteristic, complemented
38#64286—31no—
39#64241—32no—
40#64233—32no—
41#64230—32no—
42C4×GO+(4,𝔽2)—34no—
43D32⋊Q8—32no—
44C22×C3×C3⋊D3C2311yescharacteristic
45C22×C32×D3—21no—
46C2×#1113C2312yes—
47C2×C3×D32—21no—
48C2×C3×D32—24no—
49C2×C3×D32C2312yes—
50C2×C3⋊#157C2314yes—
51D32⋊D3—28no—
52C2×C33×C4D411yescharacteristic, complemented
53C3×#638—22no—
54C2×C3×C3⋊Dic3D411yescharacteristic, complemented
55C3×C4⋊(C3⋊D3)—21no—
56C3×C4×C3⋊D3—22no—
57C3×C4×C3⋊D3D411yescharacteristic, complemented
58C3×C4×C3⋊D3D411yescharacteristic, complemented
59C3×C4×C3⋊D3C2311yescharacteristic
60C3×C4×C3⋊D3C2311yescharacteristic
61C3×#634—21no—
62C2×C32×Dic3—21no—
63C32×C4×D3—22no—
64C32×C4×D3—22no—
65#6296—21no—
66#6295—22no—
67#6295—21no—
68#6294—22no—
69#6294C2312yes—
70C3×#624—21no—
71C3×#624C2312yes—
72C3×D3×Dic3—22no—
73C3×D3×Dic3—22no—
74#5791—91nomaximal
75C23×C3×D3—61no—
76C22×D32C2×D311yescharacteristic
77C2×GO+(4,𝔽2)—34no—
78C22×C32×C4—61no—
79#3828—21no—
80C2×C4×C3⋊D3C2×D311yescharacteristic
81C22×C3×Dic3—61no—
82C3×#235—61no—
83C3×#235—61no—
84C3×#235—121no—
85C2×C3×C4×D3—21no—
86C2×C3×C4×D3—21no—
87C2×C3×C4×D3—41no—
88C2×C3×C4×D3—62no—
89C2×C3×C4×D3—62no—
90C2×C3×C4×D3—62no—
91C2×C3×C4×D3—62no—
92C2×C3×C4×D3—61no—
93C2×C3×C4×D3—61no—
94C2×C3×C4×D3—61no—
95C2×C3×C4×D3—61no—
96#3811—32no—
97D3×#67—62no—
98C2×#624C2×D311yescharacteristic
99#3805—61no—
100#3804—62no—
101C2×D3×Dic3—21no—
102#3802—31no—
103D3×D12—61no—
104C4×D32—21no—
105C4×D32—21no—
106C4×D32—32no—
107C4×D32—62no—
108C4×D32C2×D312yes—
109C4×D32C2×D312yes—
110#3799—61no—
111#3798—62no—
112#3798—61no—
113#3797—31no—
114#3796—61no—
115#3795—61no—
116D3×Dic6—61no—
117C22⋊#157—32no—
118C4⋊#157—32no—
119C4×#157—32no—
120#3773—34no—
121D32⋊C4—34no—
122D32⋊C4—34no—
123C22×C33C2×D411yescharacteristic
124C2×C3×C3⋊D3—22no—
125C2×C3×C3⋊D3—22no—
126C2×C3×C3⋊D3C2×D411yescharacteristic
127C2×C3×C3⋊D3C2411yescharacteristic
128C2×C32×D3—24no—
129C2×C32×D3—22no—
130#1113—24no—
131C3×D32—24no—
132C3×D32—22no—
133C3×D32#4114yescomplemented
134C3⋊#157—24no—
135C33×C4C2×D411yescharacteristic
136C33×C4C2×D411yescharacteristic
137C3×C3⋊Dic3—22no—
138C3×C3⋊Dic3C2×D411yescharacteristic
139C3×C3⋊Dic3C2×D411yescharacteristic
140C32×Dic3—22no—
141C23×C3×C4—91no—
142D3×#41—181no—
143C2×#235—91no—
144C22×C4×D3—61no—
145#953—92no—
146#930—92no—
147#929—92no—
148C4×#67—94no—
149#926—91no—
150#924—92no—
151C23×C32—61no—
152C22×C3⋊D3C22×D311yescharacteristic
153C22×C3×D3—21no—
154C22×C3×D3—21no—
155C22×C3×D3—41no—
156C22×C3×D3—62no—
157C22×C3×D3—61no—
158C22×C3×D3—61no—
159C22×C3×D3—68no—
160C22×C3×D3—62no—
161C2×D32—21no—
162C2×D32—24no—
163C2×D32—32no—
164C2×D32—62no—
165C2×D32C22×D312yes—
166C2×#157—34no—
167GO+(4,𝔽2)—68no—
168C2×C32×C4—21no—
169C2×C32×C4—21no—
170C2×C32×C4—41no—
171C2×C32×C4—41no—
172C2×C32×C4—62no—
173C2×C32×C4—62no—
174C2×C32×C4—61no—
175C2×C32×C4#6711yescharacteristic, complemented
176#638—22no—
177C2×C3⋊Dic3#6711yescharacteristic, complemented
178C4⋊(C3⋊D3)—21no—
179C4×C3⋊D3—22no—
180C4×C3⋊D3#6711yescharacteristic, complemented
181C4×C3⋊D3#6711yescharacteristic, complemented
182C4×C3⋊D3C22×D311yescharacteristic
183C4×C3⋊D3C22×D311yescharacteristic
184#634—21no—
185C3×#67—62no—
186C3×#67—62no—
187C3×#67—122no—
188C2×C3×Dic3—21no—
189C2×C3×Dic3—21no—
190C2×C3×Dic3—41no—
191C2×C3×Dic3—61no—
192C2×C3×Dic3—61no—
193C2×C3×Dic3—62no—
194C2×C3×Dic3—62no—
195C3×D12—61no—
196C3×D12—61no—
197C3×D12—121no—
198C3×C4×D3—22no—
199C3×C4×D3—22no—
200C3×C4×D3—22no—
201C3×C4×D3—22no—
202C3×C4×D3—42no—
203C3×C4×D3—42no—
204C3×C4×D3—62no—
205C3×C4×D3—61no—
206C3×C4×D3—61no—
207C3×C4×D3—61no—
208C3×C4×D3—61no—
209C3×C4×D3—61no—
210C3×C4×D3—61no—
211C3×C4×D3—61no—
212C3×C4×D3—61no—
213C3×C4×D3—62no—
214C3×C4×D3—62no—
215C3×C4×D3—62no—
216C3×C4×D3—62no—
217C3×C4×D3—62no—
218C3×C4×D3—122no—
219C3×Dic6—61no—
220C3×Dic6—61no—
221C3×Dic6—121no—
222#627—62no—
223#627—61no—
224#626—62no—
225#626—62no—
226#626—62no—
227#625—62no—
228#625—61no—
229#624—21no—
230#624—32no—
231#624—62no—
232#624C22×D312yes—
233D3×Dic3—22no—
234D3×Dic3—22no—
235D3×Dic3—62no—
236D3×Dic3—62no—
237D3×Dic3—62no—
238#525—271noSylow
239C2×C33—21no—
240C2×C33C22×D411yescharacteristic
241C3×C3⋊D3—21no—
242C3×C3⋊D3—42no—
243C3×C3⋊D3C2×#4112yes—
244C32×D3—24no—
245C24×C3—91no—
246C23×D3—61no—
247C3×#41—181no—
248C22×C3×C4—61no—
249C22×C3×C4—61no—
250C22×C3×C4—92no—
251C22×C3×C4—92no—
252C22×C3×C4—91no—
253C22×C3×C4—91no—
254C22×C3×C4—121no—
255C22×C3×C4—181no—
256C22×C3×C4—181no—
257C22×C3×C4—181no—
258C2×#67—92no—
259C2×#67—94no—
260C22×Dic3—61no—
261#239—181no—
262D3×Q8—181no—
263#237—181no—
264#237—182no—
265D3×D4—181no—
266D3×D4—182no—
267#235—61no—
268#235—61no—
269#235—121no—
270#235—181no—
271#235—181no—
272#235—182no—
273#235—181no—
274C2×D12—91no—
275C2×C4×D3—21no—
276C2×C4×D3—62no—
277C2×C4×D3—62no—
278C2×C4×D3—61no—
279C2×C4×D3—61no—
280C2×C4×D3—61no—
281C2×C4×D3—62no—
282C2×C4×D3—62no—
283C2×C4×D3—61no—
284C2×C4×D3—92no—
285C2×C4×D3—182no—
286C2×Dic6—91no—
287#217—94no—
288#217—92no—
289(C2×D3)⋊C4—94no—
290C4⋊Dic3—92no—
291Dic3⋊C4—94no—
292C4×Dic3—92no—
293C22×C32—21no—
294C22×C32—21no—
295C22×C32—41no—
296C22×C32—41no—
297C22×C32—62no—
298C22×C32—64no—
299C22×C32C2×#6711yescharacteristic
300C2×C3⋊D3—22no—
301C2×C3⋊D3—22no—
302C2×C3⋊D3C2×#6711yescharacteristic
303C2×C3⋊D3C23×D311yescharacteristic
304C2×C3×D3—22no—
305C2×C3×D3—24no—
306C2×C3×D3—24no—
307C2×C3×D3—22no—
308C2×C3×D3—42no—
309C2×C3×D3—44no—
310C2×C3×D3—64no—
311C2×C3×D3—61no—
312C2×C3×D3—61no—
313C2×C3×D3—61no—
314C2×C3×D3—61no—
315C2×C3×D3—64no—
316C2×C3×D3—68no—
317C2×C3×D3—64no—
318C2×C3×D3—62no—
319C2×C3×D3—62no—
320C2×C3×D3—122no—
321C2×C3×D3—122no—
322D32—24no—
323D32—22no—
324D32—64no—
325D32—124no—
326D32#23514yescomplemented
327#157—64no—
328C32×C4—21no—
329C32×C4—21no—
330C32×C4—21no—
331C32×C4—21no—
332C32×C4—41no—
333C32×C4—41no—
334C32×C4—41no—
335C32×C4—41no—
336C32×C4—62no—
337C32×C4C2×#6711yescharacteristic
338C32×C4C2×#6711yescharacteristic
339C3⋊Dic3—22no—
340C3⋊Dic3C2×#6711yescharacteristic
341C3⋊Dic3C2×#6711yescharacteristic
342C3×Dic3—22no—
343C3×Dic3—22no—
344C3×Dic3—42no—
345C3×Dic3—61no—
346C3×Dic3—61no—
347C3×Dic3—61no—
348C3×Dic3—61no—
349C3×Dic3—62no—
350C3×Dic3—62no—
351C3×Dic3—122no—
352C2×#41—271no—
353C23×C4—91no—
354#123—272no—
355C22⋊Q8—272no—
356#121—272no—
357#120—272no—
358C4×D4—274no—
359#117—271no—
360C33#52511yescharacteristic, complemented, Sylow
361C23×C3—61no—
362C23×C3—61no—
363C23×C3—92no—
364C23×C3—91no—
365C23×C3—121no—
366C23×C3—181no—
367C23×C3—184no—
368C22×D3—21no—
369C22×D3—62no—
370C22×D3—61no—
371C22×D3—61no—
372C22×D3—62no—
373C22×D3—68no—
374C22×D3—92no—
375C22×D3—182no—
376C3×Q8—181no—
377C3×D4—181no—
378C3×D4—182no—
379C2×C3×C4—21no—
380C2×C3×C4—21no—
381C2×C3×C4—41no—
382C2×C3×C4—41no—
383C2×C3×C4—62no—
384C2×C3×C4—62no—
385C2×C3×C4—61no—
386C2×C3×C4—61no—
387C2×C3×C4—62no—
388C2×C3×C4—62no—
389C2×C3×C4—92no—
390C2×C3×C4—91no—
391C2×C3×C4—91no—
392C2×C3×C4—91no—
393C2×C3×C4—91no—
394C2×C3×C4—91no—
395C2×C3×C4—122no—
396C2×C3×C4—122no—
397C2×C3×C4—121no—
398C2×C3×C4—182no—
399C2×C3×C4—181no—
400C2×C3×C4—182no—
401C2×C3×C4—182no—
402C2×C3×C4GO+(4,𝔽2)11yescharacteristic, complemented
403#67—62no—
404#67—62no—
405#67—122no—
406#67—182no—
407#67—182no—
408#67—182no—
409#67—188no—
410C2×Dic3—21no—
411C2×Dic3—61no—
412C2×Dic3—61no—
413C2×Dic3—62no—
414C2×Dic3—62no—
415C2×Dic3—92no—
416C2×Dic3—94no—
417C2×Dic3—182no—
418D12—61no—
419D12—61no—
420D12—121no—
421D12—181no—
422D12—182no—
423C4×D3—22no—
424C4×D3—22no—
425C4×D3—62no—
426C4×D3—61no—
427C4×D3—61no—
428C4×D3—61no—
429C4×D3—61no—
430C4×D3—61no—
431C4×D3—61no—
432C4×D3—61no—
433C4×D3—61no—
434C4×D3—62no—
435C4×D3—62no—
436C4×D3—62no—
437C4×D3—62no—
438C4×D3—62no—
439C4×D3—122no—
440C4×D3—182no—
441C4×D3—182no—
442C4×D3—182no—
443Dic6—61no—
444Dic6—61no—
445Dic6—121no—
446Dic6—181no—
447Dic6—182no—
448C2×C32—22no—
449C2×C32—21no—
450C2×C32—21no—
451C2×C32—21no—
452C2×C32—41no—
453C2×C32—41no—
454C2×C32—41no—
455C2×C32—42no—
456C2×C32—64no—
457C2×C32—81no—
458C2×C32C22×#6711yescharacteristic
459C3⋊D3—21no—
460C3⋊D3—42no—
461C3⋊D3C2×#23512yes—
462C3×D3—24no—
463C3×D3—24no—
464C3×D3—44no—
465C3×D3—62no—
466C3×D3—62no—
467C3×D3—62no—
468C3×D3—121no—
469C3×D3—122no—
470C3×D3—122no—
471C3×D3—242no—
472C24—91no—
473#41—181no—
474#41—541no—
475#41—541no—
476#41—541no—
477C2×Q8—271no—
478C2×D4—271no—
479C2×D4—272no—
480C2×D4—274no—
481C22×C4—61no—
482C22×C4—92no—
483C22×C4—92no—
484C22×C4—91no—
485C22×C4—91no—
486C22×C4—181no—
487C22×C4—181no—
488C22×C4—181no—
489C22×C4—272no—
490C4⋊C4—272no—
491C4⋊C4—274no—
492C22⋊C4—274no—
493C22⋊C4—272no—
494C22⋊C4—274no—
495C42—272no—
496C22×C3—21no—
497C22×C3—21no—
498C22×C3—41no—
499C22×C3—41no—
500C22×C3—62no—
501C22×C3—64no—
502C22×C3—64no—
503C22×C3—62no—
504C22×C3—94no—
505C22×C3—91no—
506C22×C3—91no—
507C22×C3—122no—
508C22×C3—124no—
509C22×C3—182no—
510C22×C3—182no—
511C22×C3—182no—
512C22×C3—184no—
513C22×C3C2×GO+(4,𝔽2)11yescharacteristic
514C2×D3—22no—
515C2×D3—24no—
516C2×D3—64no—
517C2×D3—61no—
518C2×D3—61no—
519C2×D3—61no—
520C2×D3—61no—
521C2×D3—64no—
522C2×D3—68no—
523C2×D3—64no—
524C2×D3—62no—
525C2×D3—62no—
526C2×D3—122no—
527C2×D3—122no—
528C2×D3—184no—
529C2×D3—364no—
530C3×C4—21no—
531C3×C4—21no—
532C3×C4—21no—
533C3×C4—21no—
534C3×C4—41no—
535C3×C4—41no—
536C3×C4—41no—
537C3×C4—41no—
538C3×C4—62no—
539C3×C4—62no—
540C3×C4—91no—
541C3×C4—91no—
542C3×C4—122no—
543C3×C4—182no—
544C3×C4C2×GO+(4,𝔽2)11yescharacteristic
545C3×C4C2×GO+(4,𝔽2)11yescharacteristic
546Dic3—22no—
547Dic3—61no—
548Dic3—61no—
549Dic3—61no—
550Dic3—61no—
551Dic3—62no—
552Dic3—62no—
553Dic3—122no—
554Dic3—184no—
555C32—21no—
556C32—21no—
557C32—41no—
558C32—41no—
559C32#579111yescharacteristic, complemented
560C23—61no—
561C23—92no—
562C23—91no—
563C23—181no—
564C23—184no—
565C23—272no—
566Q8—181no—
567Q8—541no—
568D4—182no—
569D4—181no—
570D4—541no—
571D4—542no—
572D4—548no—
573C2×C4—61no—
574C2×C4—62no—
575C2×C4—62no—
576C2×C4—92no—
577C2×C4—91no—
578C2×C4—91no—
579C2×C4—91no—
580C2×C4—91no—
581C2×C4—91no—
582C2×C4—181no—
583C2×C4—182no—
584C2×C4—182no—
585C2×C4—182no—
586C2×C4—272no—
587C2×C4—274no—
588C2×C4—542no—
589C2×C4D32⋊D311yescharacteristic, complemented
590C2×C3—22no—
591C2×C3—21no—
592C2×C3—21no—
593C2×C3—21no—
594C2×C3—41no—
595C2×C3—42no—
596C2×C3—41no—
597C2×C3—41no—
598C2×C3—64no—
599C2×C3—64no—
600C2×C3—81no—
601C2×C3—92no—
602C2×C3—124no—
603C2×C3—181no—
604C2×C3—362no—
605C2×C3C22×GO+(4,𝔽2)11yescharacteristic
606D3—24no—
607D3—62no—
608D3—62no—
609D3—62no—
610D3—121no—
611D3—122no—
612D3—122no—
613D3—242no—
614C22—64no—
615C22—62no—
616C22—94no—
617C22—91no—
618C22—91no—
619C22—182no—
620C22—182no—
621C22—184no—
622C22—182no—
623C22—544no—
624C22C2×D32⋊D311yescharacteristic
625C4—62no—
626C4—91no—
627C4—91no—
628C4—182no—
629C4—544no—
630C4C2×D32⋊D311yescharacteristic
631C4C2×D32⋊D311yescharacteristic
632C3—21no—
633C3—21no—
634C3—41no—
635C3—41no—
636C3#1059651011yescharacteristic, complemented
637C2—21no—
638C2—64no—
639C2—92no—
640C2—181no—
641C2—362no—
642C2C22×D32⋊D311yescharacteristic
subgroupG/H
centerC4C2×D32⋊D3
commutator, cosocleC2×C3×C3⋊D3C24
FrattiniC2C22×D32⋊D3
FittingC2×C33×C4D4
socleC2×C33C22×D4
hypercenterC2×C4D32⋊D3
hypocenterC33#525
LCSG ▷ C2×C3×C3⋊D3 ▷ C33
UCSC1 ◁ C4 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
1. #11724618C2semidirect
2. C2×C3×C4×D32C2semidirect
3. #11724514C2semidirect
4. #11724331C2semidirect
5. #11724317C2semidirect
6. #11724314C2semidirect
7. C4×D32⋊D3C2semidirect
8. #11724300C2semidirect
17. C4⋊#1113C22semidirect
18. C4×#1113C22semidirect
21. #89040C22semidirect
29. C4⋊(C3⋊#157)C22semidirect
30. C4×C3⋊#157C22semidirect
31. #88934C22semidirect
33. #88932C22semidirect
37. C2×C4×D32D3semidirect
52. C2×C33×C4D4semidirect
54. C2×C3×C3⋊Dic3D4semidirect
57. C3×C4×C3⋊D3D4semidirect
58. C3×C4×C3⋊D3D4semidirect
133. C3×D32#41semidirect
175. C2×C32×C4#67semidirect
177. C2×C3⋊Dic3#67semidirect
180. C4×C3⋊D3#67semidirect
181. C4×C3⋊D3#67semidirect
326. D32#235semidirect
360. C33#525semidirect
402. C2×C3×C4GO+(4,𝔽2)semidirect
559. C32#5791semidirect
589. C2×C4D32⋊D3semidirect
636. C3#10596510semidirect

Maximal subgroups

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

subgroupindex
#117246182
C2×C3×C4×D322
#117245142
#117243312
#117243172
#117243142
C4×D32⋊D32
#117243002
#105965103
#57919