Finite Group Databasebeta

← #13151490

Subgroups — #13151490

Order 1728.

1012 conjugacy classes of subgroups; only the 129 normal subgroups are shown.

normal subgroups only

This group has 1012 conjugacy classes of subgroups — too many to draw. Only its 129 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#11721847C211yesmaximal, characteristic, complemented
2#64149⋊C3C211yesmaximal, characteristic, complemented
3#64144⋊C3C211yesmaximal, characteristic, complemented
4#11721323C212yesmaximal, complemented
5#11721322C212yesmaximal, complemented
6#11721312C212yesmaximal, complemented
7#11721311C212yesmaximal, complemented
8#11721127C212yesmaximal, complemented
9#11721092C212yesmaximal, complemented
10#10595261—31nomaximal
11#10595259—31nomaximal
12C23×#272C2211yescharacteristic
13C2×D4×H(3,𝔽3)C2211yescharacteristic
14C2×#6235C2212yes—
15C2×#638⋊C3C2212yes—
16C2×#638⋊C3C2212yescomplemented
17C22×(C3⋊Dic3)⋊C3C2211yescharacteristic, complemented
18C2×C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic, complemented
19C2×#6200C2212yes—
20C2×#6200C2212yes—
21C2×((C3⋊D3)⋊C3)⋊C4C2212yescomplemented
22#88552C2211yescharacteristic
23#88542C2212yes—
24#3757⋊C3C2212yes—
25#3757⋊C3C2211yescharacteristic, complemented
26#3750⋊C3C2212yescomplemented
27C4×(C3⋊Dic3)⋊C3C2211yescharacteristic, complemented
28#88451C2212yes—
29#88451C2212yes—
30#88450C2212yescomplemented
31#88449C2212yescomplemented
32#88448C2212yescomplemented
33#64149D311yescharacteristic, complemented
34#64144D311yescharacteristic, complemented
35C3×#940—31no—
36C3×#937—31no—
37C3×#937—31no—
38C3×#937—61no—
39C3×#935—31no—
40#63980—32no—
41#63978—32no—
42#63977—32no—
43#63976—32no—
44#63975—32no—
45#63970—32no—
46#63969—32no—
47#63958—32no—
48#63910—32no—
49#63885—32no—
50#63862—32no—
51#63859—32no—
52C23×H(3,𝔽3)C2311yescharacteristic
53C23×H(3,𝔽3)C2311yescharacteristic
54C22×#272—42no—
55C22×#272—41no—
56C22×#272C2312yes—
57C22×(C3⋊D3)⋊C3C2312yes—
58D4×H(3,𝔽3)—41no—
59C2×C4×H(3,𝔽3)C2311yescharacteristic
60#6235—42no—
61C2×#1084C2312yes—
62#638⋊C3—42no—
63#638⋊C3—42no—
64C2×(C3⋊Dic3)⋊C3—41no—
65C2×(C3⋊Dic3)⋊C3C2312yes—
66C2×(C3⋊Dic3)⋊C3C2312yes—
67C2×(C3⋊Dic3)⋊C3C2312yes—
68C4⋊((C3⋊D3)⋊C3)—41no—
69#6200—42no—
70#6200—42no—
71((C3⋊D3)⋊C3)⋊C4—42no—
72#5720—91no—
73#5666—91no—
74#5595—91no—
75C23×C3×D3—31no—
76C23×C3×D3—31no—
77C23×C3×D3—61no—
78C2×C32×D4—21no—
79C2×C32×D4C2×D311yescharacteristic
80C2×C32×D4C2×D311yescharacteristic
81C2×#638C2×D312yes—
82C2×#638C2×D312yescomplemented
83C22×C3⋊Dic3C2×D311yescharacteristic, complemented
84C2×C4⋊(C3⋊D3)C2×D311yescharacteristic, complemented
85C2×C3×#67—32no—
86C2×C3×#67—32no—
87C2×C3×#67—32no—
88C2×C3×#67—32no—
89C2×C3×#67—62no—
90C22×C3×Dic3—31no—
91C2×C3×D12—31no—
92C2×#626—32no—
93C2×#626—32no—
94C2×#625—32no—
95C2×#625—32no—
96C2×#624—32no—
97C2×D3×Dic3—32no—
98#3757C2×D312yes—
99#3757C2×D311yescharacteristic, complemented
100#3750C2×D312yescomplemented
101C4×C3⋊Dic3C2×D311yescharacteristic, complemented
102C3×#217—32no—
103C3×#217—31no—
104C3×#217—31no—
105C3×#217—31no—
106C3×#217—61no—
107C3×(C2×D3)⋊C4—32no—
108C3×(C2×D3)⋊C4—32no—
109C3×(C2×D3)⋊C4—62no—
110C3×Dic3⋊C4—32no—
111C3×C4×Dic3—31no—
112#3724—32no—
113Dic3⋊Dic3—32no—
114Dic3⋊Dic3—32no—
115Dic3⋊Dic3—32no—
116#3722—32no—
117(C2×D3)⋊Dic3—32no—
118(C2×D3)⋊Dic3—32no—
119(C2×D3)⋊Dic3—32no—
120(C2×D3)⋊Dic3—32no—
121(C2×D3)⋊Dic3—32no—
122C22×H(3,𝔽3)—21no—
123C22×H(3,𝔽3)—21no—
124C22×H(3,𝔽3)—22no—
125C22×H(3,𝔽3)—22no—
126C22×H(3,𝔽3)C2411yescharacteristic
127C2×#272—22no—
128C2×#272—22no—
129C2×#272—22no—
130C2×#272—44no—
131C2×(C3⋊D3)⋊C3—22no—
132C2×(C3⋊D3)⋊C3—22no—
133C2×(C3⋊D3)⋊C3—22no—
134C4×H(3,𝔽3)—21no—
135#1084—22no—
136(C3⋊Dic3)⋊C3—22no—
137(C3⋊Dic3)⋊C3—22no—
138(C3⋊Dic3)⋊C3—22no—
139C3×#127—91no—
140C3×#123—91no—
141C3×#120—91no—
142#953—92no—
143#940—31no—
144#940—91no—
145#939—92no—
146#939—92no—
147#939—92no—
148#939—92no—
149#938—92no—
150#937—31no—
151#937—31no—
152#937—61no—
153#937—92no—
154#935—31no—
155#935—91no—
156#935—92no—
157#930—92no—
158#898—92no—
159#895—92no—
160#884—92no—
161#884—92no—
162#883—92no—
163#883—92no—
164#882—92no—
165#879—92no—
166#876—92no—
167C23×C32—21no—
168C23×C32—21no—
169C23×C32C22×D311yescharacteristic
170C23×C32C22×D311yescharacteristic
171C23×C32C22×D311yescharacteristic
172C23×C32C22×D311yescharacteristic
173C22×C3⋊D3C22×D312yes—
174C22×C3×D3—32no—
175C22×C3×D3—32no—
176C22×C3×D3—32no—
177C22×C3×D3—62no—
178C22×C3×D3—121no—
179C22×C3×D3—121no—
180C22×C3×D3—122no—
181C22×C3×D3—122no—
182C22×C3×D3—124no—
183C22×C3×D3—122no—
184C32×D4—41no—
185C32×D4—41no—
186C32×D4—42no—
187C2×C32×C4—21no—
188C2×C32×C4C22×D311yescharacteristic
189C2×C32×C4C22×D311yescharacteristic
190#638—42no—
191#638—42no—
192C2×C3⋊Dic3—41no—
193C2×C3⋊Dic3C22×D312yes—
194C2×C3⋊Dic3C22×D312yes—
195C2×C3⋊Dic3C22×D312yes—
196C4⋊(C3⋊D3)—41no—
197C3×#67—122no—
198C3×#67—122no—
199C3×#67—122no—
200C3×#67—122no—
201C3×#67—124no—
202C2×C3×Dic3—32no—
203C2×C3×Dic3—32no—
204C2×C3×Dic3—32no—
205C2×C3×Dic3—32no—
206C2×C3×Dic3—32no—
207C2×C3×Dic3—62no—
208C2×C3×Dic3—121no—
209C3×D12—121no—
210#626—122no—
211#626—122no—
212#625—122no—
213#625—122no—
214#624—122no—
215D3×Dic3—122no—
216#560—271noSylow
217C2×H(3,𝔽3)—41no—
218C2×H(3,𝔽3)—41no—
219C2×H(3,𝔽3)H(5,𝔽2)11yescharacteristic
220C2×H(3,𝔽3)H(5,𝔽2)12yes—
221#272—42no—
222(C3⋊D3)⋊C3—42no—
223C24×C3—91no—
224C23×D3—31no—
225C23×D3—31no—
226C23×D3—61no—
227C2×C3×D4—31no—
228C2×C3×D4—31no—
229C2×C3×D4—61no—
230C2×C3×D4—91no—
231C2×C3×D4—92no—
232C2×C3×D4—92no—
233C2×C3×D4—92no—
234C2×C3×D4D3211yescharacteristic
235C22×C3×C4—91no—
236C2×#67—32no—
237C2×#67—32no—
238C2×#67—32no—
239C2×#67—32no—
240C2×#67—62no—
241C2×#67—92no—
242C2×#67—92no—
243C2×#67—92no—
244C2×#67—92no—
245C2×#67—92no—
246C2×#67—92no—
247C22×Dic3—31no—
248C22×Dic3—91no—
249C22×Dic3—92no—
250C2×D12—31no—
251C2×D12—91no—
252C2×D12—92no—
253C2×D12—92no—
254C2×C4×D3—92no—
255C2×C4×D3—92no—
256C3×C4⋊C4—92no—
257C3×C22⋊C4—92no—
258C3×C22⋊C4—91no—
259C3×C22⋊C4—91no—
260C3×C22⋊C4—92no—
261C3×C42—91no—
262#217—31no—
263#217—31no—
264#217—32no—
265#217—31no—
266#217—61no—
267#217—92no—
268#217—91no—
269#217—92no—
270#217—92no—
271#217—92no—
272(C2×D3)⋊C4—32no—
273(C2×D3)⋊C4—32no—
274(C2×D3)⋊C4—62no—
275(C2×D3)⋊C4—92no—
276(C2×D3)⋊C4—92no—
277(C2×D3)⋊C4—92no—
278(C2×D3)⋊C4—92no—
279(C2×D3)⋊C4—92no—
280(C2×D3)⋊C4—92no—
281C4⋊Dic3—92no—
282C4⋊Dic3—92no—
283Dic3⋊C4—32no—
284Dic3⋊C4—92no—
285Dic3⋊C4—92no—
286Dic3⋊C4—92no—
287Dic3⋊C4—92no—
288Dic3⋊C4—92no—
289C4×Dic3—31no—
290C4×Dic3—91no—
291C22×C32—22no—
292C22×C32—21no—
293C22×C32—22no—
294C22×C32—21no—
295C22×C32—22no—
296C22×C32—21no—
297C22×C32—21no—
298C22×C32—22no—
299C22×C32—22no—
300C22×C32—21no—
301C22×C32—42no—
302C22×C32—42no—
303C22×C32—41no—
304C22×C32C23×D311yescharacteristic
305C22×C32C23×D311yescharacteristic
306C2×C3⋊D3—22no—
307C2×C3⋊D3—22no—
308C2×C3⋊D3—22no—
309C2×C3×D3—64no—
310C2×C3×D3—62no—
311C2×C3×D3—62no—
312C2×C3×D3—62no—
313C2×C3×D3—62no—
314C2×C3×D3—62no—
315C2×C3×D3—62no—
316C2×C3×D3—62no—
317C2×C3×D3—62no—
318C2×C3×D3—62no—
319C2×C3×D3—122no—
320C2×C3×D3—122no—
321C2×C3×D3—128no—
322C2×C3×D3—124no—
323C2×C3×D3—124no—
324C32×C4—21no—
325C32×C4—21no—
326C32×C4—41no—
327C3⋊Dic3—22no—
328C3⋊Dic3—22no—
329C3⋊Dic3—22no—
330C3×Dic3—62no—
331C3×Dic3—62no—
332C3×Dic3—62no—
333C3×Dic3—62no—
334C3×Dic3—62no—
335C3×Dic3—122no—
336#127—91no—
337#126—272no—
338#123—91no—
339#123—272no—
340#121—272no—
341#121—272no—
342#121—272no—
343#120—91no—
344#120—272no—
345H(3,𝔽3)#56011yescharacteristic, complemented, Sylow
346C23×C3—31no—
347C23×C3—31no—
348C23×C3—31no—
349C23×C3—31no—
350C23×C3—61no—
351C23×C3—61no—
352C23×C3—92no—
353C23×C3—92no—
354C23×C3—362no—
355C23×C3—361no—
356C23×C3C2×D3211yescharacteristic
357C23×C3C2×D3211yescharacteristic
358C22×D3—32no—
359C22×D3—32no—
360C22×D3—32no—
361C22×D3—62no—
362C22×D3—92no—
363C22×D3—121no—
364C22×D3—122no—
365C22×D3—122no—
366C22×D3—121no—
367C22×D3—124no—
368C22×D3—122no—
369C3×D4—41no—
370C3×D4—121no—
371C3×D4—121no—
372C3×D4—122no—
373C3×D4—361no—
374C3×D4—362no—
375C3×D4—362no—
376C3×D4—362no—
377C2×C3×C4—31no—
378C2×C3×C4—31no—
379C2×C3×C4—61no—
380C2×C3×C4—92no—
381C2×C3×C4—92no—
382C2×C3×C4—92no—
383C2×C3×C4—92no—
384C2×C3×C4—361no—
385C2×C3×C4C2×D3211yescharacteristic
386#67—122no—
387#67—122no—
388#67—122no—
389#67—122no—
390#67—124no—
391#67—362no—
392#67—362no—
393#67—362no—
394#67—362no—
395#67—362no—
396#67—362no—
397C2×Dic3—32no—
398C2×Dic3—32no—
399C2×Dic3—32no—
400C2×Dic3—32no—
401C2×Dic3—32no—
402C2×Dic3—62no—
403C2×Dic3—92no—
404C2×Dic3—92no—
405C2×Dic3—92no—
406C2×Dic3—121no—
407C2×Dic3—361no—
408C2×Dic3—362no—
409D12—121no—
410D12—361no—
411D12—362no—
412D12—362no—
413C4×D3—362no—
414C4×D3—362no—
415C2×C32—22no—
416C2×C32—21no—
417C2×C32—42no—
418C2×C32—41no—
419C2×C32—41no—
420C2×C32—41no—
421C2×C32—41no—
422C2×C32—81no—
423C2×C32#100412yes—
424C2×C32#100411yescharacteristic
425C2×C32#100411yescharacteristic
426C2×C32#100912yes—
427C3⋊D3—42no—
428C3×D3—124no—
429C3×D3—122no—
430C3×D3—122no—
431C3×D3—122no—
432C24—91no—
433C2×D4—92no—
434C2×D4—92no—
435C2×D4—92no—
436C2×D4—91no—
437C2×D4—272no—
438C2×D4—272no—
439C2×D4((C3⋊D3)⋊C3)⋊C211yescharacteristic
440C22×C4—91no—
441C22×C4—272no—
442C4⋊C4—92no—
443C4⋊C4—272no—
444C4⋊C4—272no—
445C22⋊C4—91no—
446C22⋊C4—92no—
447C22⋊C4—91no—
448C22⋊C4—92no—
449C22⋊C4—272no—
450C22⋊C4—272no—
451C22⋊C4—272no—
452C42—91no—
453C22×C3—22no—
454C22×C3—22no—
455C22×C3—21no—
456C22×C3—21no—
457C22×C3—31no—
458C22×C3—31no—
459C22×C3—61no—
460C22×C3—62no—
461C22×C3—62no—
462C22×C3—62no—
463C22×C3—61no—
464C22×C3—61no—
465C22×C3—61no—
466C22×C3—62no—
467C22×C3—62no—
468C22×C3—61no—
469C22×C3—122no—
470C22×C3—122no—
471C22×C3—121no—
472C22×C3—182no—
473C22×C3—182no—
474C22×C3—182no—
475C22×C3—182no—
476C22×C3—182no—
477C22×C3—182no—
478C22×C3—364no—
479C22×C3C22×D3211yescharacteristic
480C2×D3—62no—
481C2×D3—62no—
482C2×D3—62no—
483C2×D3—62no—
484C2×D3—62no—
485C2×D3—62no—
486C2×D3—64no—
487C2×D3—62no—
488C2×D3—62no—
489C2×D3—62no—
490C2×D3—122no—
491C2×D3—122no—
492C2×D3—124no—
493C2×D3—124no—
494C2×D3—128no—
495C2×D3—182no—
496C2×D3—182no—
497C2×D3—182no—
498C3×C4—21no—
499C3×C4—61no—
500C3×C4—61no—
501C3×C4—121no—
502C3×C4—182no—
503C3×C4—182no—
504C3×C4—182no—
505C3×C4—182no—
506Dic3—62no—
507Dic3—62no—
508Dic3—62no—
509Dic3—62no—
510Dic3—62no—
511Dic3—122no—
512Dic3—182no—
513Dic3—182no—
514Dic3—182no—
515C32—21no—
516C32#566611yescharacteristic, complemented
517C32#572011yescharacteristic, complemented
518C23—92no—
519C23—92no—
520C23—362no—
521C23—361no—
522C23C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
523C23C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
524D4—41no—
525D4—361no—
526D4—362no—
527D4—362no—
528D4—362no—
529D4—1082no—
530D4—1082no—
531C2×C4—92no—
532C2×C4—92no—
533C2×C4—92no—
534C2×C4—92no—
535C2×C4—361no—
536C2×C4—1082no—
537C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
538C2×C3—32no—
539C2×C3—32no—
540C2×C3—31no—
541C2×C3—31no—
542C2×C3—41no—
543C2×C3—41no—
544C2×C3—62no—
545C2×C3—61no—
546C2×C3—121no—
547C2×C3—121no—
548C2×C3—121no—
549C2×C3—121no—
550C2×C3—122no—
551C2×C3—241no—
552C2×C3—362no—
553C2×C3—362no—
554C2×C3#6431412yes—
555C2×C3#6433011yescharacteristic
556D3—122no—
557D3—122no—
558D3—124no—
559D3—122no—
560D3—362no—
561C22—22no—
562C22—22no—
563C22—21no—
564C22—21no—
565C22—182no—
566C22—182no—
567C22—182no—
568C22—182no—
569C22—182no—
570C22—182no—
571C22—364no—
572C22C22×((C3⋊D3)⋊C3)⋊C211yescharacteristic
573C4—21no—
574C4—182no—
575C4—182no—
576C4—182no—
577C4—182no—
578C3—31no—
579C3—31no—
580C3—61no—
581C3#1059526611yescharacteristic
582C2—41no—
583C2—41no—
584C2—362no—
585C2—362no—
586C2#1172250712yes—
587C2#1172254511yescharacteristic
subgroupG/H
centerC22C22×((C3⋊D3)⋊C3)⋊C2
commutator, cosocleC22×H(3,𝔽3)C24
Frattini, socleC22×C3C22×D32
FittingC2×D4×H(3,𝔽3)C22
hypercenterC2×D4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#560
LCSG ▷ C22×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
1. #11721847C2semidirect
2. #64149⋊C3C2semidirect
3. #64144⋊C3C2semidirect
4. #11721323C2semidirect
5. #11721322C2semidirect
6. #11721312C2semidirect
7. #11721311C2semidirect
8. #11721127C2semidirect
9. #11721092C2semidirect
16. C2×#638⋊C3C22semidirect
17. C22×(C3⋊Dic3)⋊C3C22semidirect
18. C2×C4⋊((C3⋊D3)⋊C3)C22semidirect
21. C2×((C3⋊D3)⋊C3)⋊C4C22semidirect
25. #3757⋊C3C22semidirect
26. #3750⋊C3C22semidirect
27. C4×(C3⋊Dic3)⋊C3C22semidirect
30. #88450C22semidirect
31. #88449C22semidirect
32. #88448C22semidirect
33. #64149D3semidirect
34. #64144D3semidirect
82. C2×#638C2×D3semidirect
83. C22×C3⋊Dic3C2×D3semidirect
84. C2×C4⋊(C3⋊D3)C2×D3semidirect
99. #3757C2×D3semidirect
100. #3750C2×D3semidirect
101. C4×C3⋊Dic3C2×D3semidirect
345. H(3,𝔽3)#560semidirect
516. C32#5666semidirect
517. C32#5720semidirect

Maximal subgroups

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

subgroupindex
#117218472
#64149⋊C32
#64144⋊C32
#117213232
#117213222
#117213122
#117213112
#117211272
#117210922
#105952613
#105952593