Finite Group Databasebeta

← #13151492

Subgroups — #13151492

Order 1728.

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

normal subgroups only

This group has 990 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#11721850C211yesmaximal, characteristic, complemented
2#64146⋊C3C211yesmaximal, characteristic, complemented
3#64144⋊C3C211yesmaximal, characteristic, complemented
4#11721322C212yesmaximal, complemented
5#11721321C212yesmaximal, complemented
6#11721319C212yesmaximal, complemented
7#11721303C212yesmaximal, complemented
8#11721157C212yesmaximal, complemented
9#11721112C212yesmaximal, complemented
10#10595266—31nomaximal
11#10595259—31nomaximal
12C23×(C3⋊D3)⋊C3C2211yescharacteristic
13C2×D4×H(3,𝔽3)C2211yescharacteristic
14C2×#6235C2212yes—
15C2×#6235C2212yescomplemented
16C2×C4⋊#272C2211yescharacteristic, complemented
17C2×#638⋊C3C2212yes—
18C22×(C3⋊Dic3)⋊C3C2211yescharacteristic, complemented
19C2×#6200C2212yes—
20C2×#6200C2212yes—
21C2×#6199C2212yescomplemented
22C4×#1084C2211yescharacteristic, complemented
23#3757⋊C3C2211yescharacteristic
24#3757⋊C3C2212yes—
25#3757⋊C3C2211yescharacteristic, complemented
26#3752⋊C3C2212yes—
27#3750⋊C3C2212yescomplemented
28#88451C2212yescomplemented
29#88450C2212yes—
30#88450C2212yes—
31#88448C2212yescomplemented
32#88448C2212yescomplemented
33#64146D311yescharacteristic, complemented
34#64144D311yescharacteristic, complemented
35C3×#940—31no—
36C3×#940—31no—
37C3×#940—61no—
38C3×#937—31no—
39C3×#935—31no—
40#63981—32no—
41#63980—32no—
42#63978—32no—
43#63976—32no—
44#63971—32no—
45#63970—32no—
46#63969—32no—
47#63969—32no—
48#63886—32no—
49#63885—32no—
50#63868—32no—
51#63859—32no—
52C23×H(3,𝔽3)C2311yescharacteristic
53C23×H(3,𝔽3)C2311yescharacteristic
54C22×#272C2312yes—
55C22×(C3⋊D3)⋊C3—41no—
56C22×(C3⋊D3)⋊C3—42no—
57C22×(C3⋊D3)⋊C3C2312yes—
58D4×H(3,𝔽3)—41no—
59C2×C4×H(3,𝔽3)C2311yescharacteristic
60#6235—42no—
61#6235—42no—
62C2×#1084C2312yes—
63C4⋊#272—41no—
64#638⋊C3—42no—
65C2×(C3⋊Dic3)⋊C3—41no—
66C2×(C3⋊Dic3)⋊C3C2312yes—
67C2×(C3⋊Dic3)⋊C3C2312yes—
68C2×(C3⋊Dic3)⋊C3C2312yes—
69#6200—42no—
70#6200—42no—
71#6199—42no—
72#5720—91no—
73#5666—91no—
74#5595—91no—
75C23×C3⋊D3C2×D311yescharacteristic
76C23×C3×D3—31no—
77C2×C32×D4—21no—
78C2×C32×D4C2×D311yescharacteristic
79C2×C32×D4C2×D311yescharacteristic
80C2×#638C2×D312yes—
81C22×C3⋊Dic3C2×D311yescharacteristic, complemented
82C2×C3×#67—32no—
83C2×C3×#67—32no—
84C2×C3×#67—32no—
85C2×C3×#67—32no—
86C2×C3×#67—32no—
87C2×C3×#67—62no—
88C2×C3×#67—62no—
89C22×C3×Dic3—31no—
90C2×C3×D12—31no—
91C2×C3×D12—31no—
92C2×C3×D12—61no—
93C2×#626—32no—
94C2×#626—32no—
95C2×#625—32no—
96C2×#625—32no—
97C2×D3×Dic3—32no—
98C2×D3×Dic3—32no—
99#3757C2×D312yes—
100#3757C2×D311yescharacteristic
101#3757C2×D311yescharacteristic, complemented
102#3752C2×D312yes—
103#3750C2×D312yescomplemented
104C3×#217—32no—
105C3×#217—31no—
106C3×#217—31no—
107C3×(C2×D3)⋊C4—32no—
108C3×Dic3⋊C4—32no—
109C3×C4×Dic3—31no—
110C3×C4×Dic3—31no—
111C3×C4×Dic3—61no—
112#3724—32no—
113#3724—32no—
114Dic3⋊Dic3—32no—
115Dic3⋊Dic3—32no—
116#3722—32no—
117#3722—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)—22no—
124C22×H(3,𝔽3)—22no—
125C22×H(3,𝔽3)—21no—
126C22×H(3,𝔽3)C2411yescharacteristic
127C2×#272—22no—
128C2×#272—22no—
129C2×#272—22no—
130C2×(C3⋊D3)⋊C3—22no—
131C2×(C3⋊D3)⋊C3—22no—
132C2×(C3⋊D3)⋊C3—22no—
133C2×(C3⋊D3)⋊C3—44no—
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—31no—
145#940—61no—
146#939—92no—
147#939—92no—
148#939—92no—
149#939—92no—
150#938—92no—
151#937—31no—
152#937—91no—
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⋊D3—41no—
174C22×C3⋊D3—42no—
175C22×C3⋊D3C22×D312yes—
176C22×C3×D3—32no—
177C22×C3×D3—32no—
178C22×C3×D3—32no—
179C22×C3×D3—62no—
180C22×C3×D3—121no—
181C22×C3×D3—122no—
182C32×D4—41no—
183C32×D4—41no—
184C32×D4—42no—
185C2×C32×C4—21no—
186C2×C32×C4C22×D311yescharacteristic
187C2×C32×C4C22×D311yescharacteristic
188#638—42no—
189C2×C3⋊Dic3—41no—
190C2×C3⋊Dic3C22×D312yes—
191C2×C3⋊Dic3C22×D312yes—
192C2×C3⋊Dic3C22×D312yes—
193C3×#67—122no—
194C3×#67—122no—
195C3×#67—122no—
196C3×#67—122no—
197C3×#67—122no—
198C3×#67—124no—
199C3×#67—124no—
200C2×C3×Dic3—32no—
201C2×C3×Dic3—32no—
202C2×C3×Dic3—32no—
203C2×C3×Dic3—32no—
204C2×C3×Dic3—32no—
205C2×C3×Dic3—62no—
206C2×C3×Dic3—121no—
207C3×D12—121no—
208C3×D12—121no—
209C3×D12—122no—
210#626—122no—
211#626—122no—
212#625—122no—
213#625—122no—
214D3×Dic3—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—91no—
226C2×C3×D4—31no—
227C2×C3×D4—31no—
228C2×C3×D4—61no—
229C2×C3×D4—92no—
230C2×C3×D4—92no—
231C2×C3×D4—92no—
232C2×C3×D4—91no—
233C2×C3×D4D3211yescharacteristic
234C22×C3×C4—91no—
235C2×#67—32no—
236C2×#67—32no—
237C2×#67—32no—
238C2×#67—32no—
239C2×#67—32no—
240C2×#67—62no—
241C2×#67—62no—
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—31no—
252C2×D12—61no—
253C2×D12—92no—
254C2×D12—92no—
255C2×C4×D3—92no—
256C2×C4×D3—92no—
257C3×C4⋊C4—92no—
258C3×C22⋊C4—92no—
259C3×C22⋊C4—92no—
260C3×C22⋊C4—91no—
261C3×C22⋊C4—91no—
262C3×C42—91no—
263#217—32no—
264#217—31no—
265#217—31no—
266#217—91no—
267#217—92no—
268#217—91no—
269#217—92no—
270#217—92no—
271#217—92no—
272(C2×D3)⋊C4—32no—
273(C2×D3)⋊C4—92no—
274(C2×D3)⋊C4—92no—
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—
280C4⋊Dic3—92no—
281C4⋊Dic3—92no—
282Dic3⋊C4—32no—
283Dic3⋊C4—92no—
284Dic3⋊C4—92no—
285Dic3⋊C4—92no—
286Dic3⋊C4—92no—
287Dic3⋊C4—92no—
288C4×Dic3—31no—
289C4×Dic3—31no—
290C4×Dic3—61no—
291C22×C32—21no—
292C22×C32—22no—
293C22×C32—22no—
294C22×C32—21no—
295C22×C32—21no—
296C22×C32—21no—
297C22×C32—22no—
298C22×C32—22no—
299C22×C32—21no—
300C22×C32—42no—
301C22×C32—42no—
302C22×C32—41no—
303C22×C32—41no—
304C22×C32C23×D311yescharacteristic
305C22×C32C23×D311yescharacteristic
306C2×C3⋊D3—22no—
307C2×C3⋊D3—22no—
308C2×C3⋊D3—22no—
309C2×C3⋊D3—44no—
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—122no—
322C2×C3×D3—124no—
323C32×C4—22no—
324C32×C4—21no—
325C32×C4—21no—
326C3⋊Dic3—22no—
327C3⋊Dic3—22no—
328C3⋊Dic3—22no—
329C3×Dic3—64no—
330C3×Dic3—62no—
331C3×Dic3—62no—
332C3×Dic3—62no—
333C3×Dic3—62no—
334C3×Dic3—62no—
335#127—91no—
336#126—272no—
337#123—91no—
338#123—272no—
339#121—272no—
340#121—272no—
341#121—272no—
342#120—91no—
343#120—272no—
344H(3,𝔽3)#56011yescharacteristic, complemented, Sylow
345C23×C3—31no—
346C23×C3—31no—
347C23×C3—31no—
348C23×C3—31no—
349C23×C3—61no—
350C23×C3—61no—
351C23×C3—92no—
352C23×C3—92no—
353C23×C3—362no—
354C23×C3—361no—
355C23×C3C2×D3211yescharacteristic
356C23×C3C2×D3211yescharacteristic
357C22×D3—32no—
358C22×D3—32no—
359C22×D3—32no—
360C22×D3—62no—
361C22×D3—92no—
362C22×D3—122no—
363C22×D3—121no—
364C22×D3—361no—
365C22×D3—362no—
366C3×D4—41no—
367C3×D4—121no—
368C3×D4—121no—
369C3×D4—122no—
370C3×D4—361no—
371C3×D4—362no—
372C3×D4—362no—
373C3×D4—362no—
374C2×C3×C4—31no—
375C2×C3×C4—31no—
376C2×C3×C4—61no—
377C2×C3×C4—92no—
378C2×C3×C4—92no—
379C2×C3×C4—92no—
380C2×C3×C4—92no—
381C2×C3×C4—361no—
382C2×C3×C4C2×D3211yescharacteristic
383#67—122no—
384#67—122no—
385#67—122no—
386#67—122no—
387#67—122no—
388#67—124no—
389#67—124no—
390#67—362no—
391#67—362no—
392#67—362no—
393#67—362no—
394#67—362no—
395C2×Dic3—32no—
396C2×Dic3—32no—
397C2×Dic3—32no—
398C2×Dic3—32no—
399C2×Dic3—32no—
400C2×Dic3—62no—
401C2×Dic3—92no—
402C2×Dic3—92no—
403C2×Dic3—92no—
404C2×Dic3—121no—
405C2×Dic3—361no—
406C2×Dic3—362no—
407D12—121no—
408D12—121no—
409D12—122no—
410D12—362no—
411D12—362no—
412C4×D3—362no—
413C4×D3—362no—
414C2×C32—22no—
415C2×C32—21no—
416C2×C32—41no—
417C2×C32—41no—
418C2×C32—41no—
419C2×C32—41no—
420C2×C32—81no—
421C2×C32—81no—
422C2×C32#100412yes—
423C2×C32#100411yescharacteristic
424C2×C32#100411yescharacteristic
425C2×C32#100912yes—
426C3⋊D3—42no—
427C3×D3—122no—
428C3×D3—122no—
429C3×D3—122no—
430C3×D3—242no—
431C24—91no—
432C2×D4—92no—
433C2×D4—92no—
434C2×D4—92no—
435C2×D4—91no—
436C2×D4—272no—
437C2×D4—272no—
438C2×D4((C3⋊D3)⋊C3)⋊C211yescharacteristic
439C22×C4—91no—
440C22×C4—272no—
441C4⋊C4—92no—
442C4⋊C4—272no—
443C4⋊C4—272no—
444C22⋊C4—91no—
445C22⋊C4—92no—
446C22⋊C4—91no—
447C22⋊C4—92no—
448C22⋊C4—272no—
449C22⋊C4—272no—
450C22⋊C4—272no—
451C42—91no—
452C22×C3—22no—
453C22×C3—22no—
454C22×C3—21no—
455C22×C3—21no—
456C22×C3—31no—
457C22×C3—31no—
458C22×C3—61no—
459C22×C3—62no—
460C22×C3—62no—
461C22×C3—62no—
462C22×C3—62no—
463C22×C3—61no—
464C22×C3—61no—
465C22×C3—61no—
466C22×C3—61no—
467C22×C3—122no—
468C22×C3—122no—
469C22×C3—121no—
470C22×C3—121no—
471C22×C3—182no—
472C22×C3—182no—
473C22×C3—182no—
474C22×C3—182no—
475C22×C3—182no—
476C22×C3—182no—
477C22×C3—364no—
478C22×C3C22×D3211yescharacteristic
479C2×D3—62no—
480C2×D3—62no—
481C2×D3—62no—
482C2×D3—62no—
483C2×D3—62no—
484C2×D3—62no—
485C2×D3—62no—
486C2×D3—62no—
487C2×D3—62no—
488C2×D3—122no—
489C2×D3—122no—
490C2×D3—122no—
491C2×D3—124no—
492C2×D3—182no—
493C2×D3—182no—
494C2×D3—182no—
495C2×D3—364no—
496C3×C4—21no—
497C3×C4—61no—
498C3×C4—61no—
499C3×C4—62no—
500C3×C4—182no—
501C3×C4—182no—
502C3×C4—182no—
503C3×C4—182no—
504Dic3—62no—
505Dic3—62no—
506Dic3—64no—
507Dic3—62no—
508Dic3—62no—
509Dic3—62no—
510Dic3—182no—
511Dic3—182no—
512Dic3—182no—
513C32—21no—
514C32#559511yescharacteristic, complemented
515C32#566611yescharacteristic, complemented
516C23—92no—
517C23—92no—
518C23—362no—
519C23—361no—
520C23C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
521C23C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
522D4—41no—
523D4—361no—
524D4—362no—
525D4—362no—
526D4—362no—
527D4—1082no—
528D4—1082no—
529C2×C4—92no—
530C2×C4—92no—
531C2×C4—92no—
532C2×C4—92no—
533C2×C4—361no—
534C2×C4—1082no—
535C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
536C2×C3—32no—
537C2×C3—32no—
538C2×C3—31no—
539C2×C3—31no—
540C2×C3—41no—
541C2×C3—41no—
542C2×C3—62no—
543C2×C3—61no—
544C2×C3—121no—
545C2×C3—121no—
546C2×C3—121no—
547C2×C3—121no—
548C2×C3—241no—
549C2×C3—241no—
550C2×C3—362no—
551C2×C3—362no—
552C2×C3#6431412yes—
553C2×C3#6433011yescharacteristic
554D3—122no—
555D3—122no—
556D3—122no—
557D3—242no—
558D3—362no—
559C22—22no—
560C22—22no—
561C22—21no—
562C22—21no—
563C22—182no—
564C22—182no—
565C22—182no—
566C22—182no—
567C22—182no—
568C22—182no—
569C22—364no—
570C22C22×((C3⋊D3)⋊C3)⋊C211yescharacteristic
571C4—21no—
572C4—182no—
573C4—182no—
574C4—182no—
575C4—182no—
576C3—31no—
577C3—31no—
578C3—61no—
579C3#1059526111yescharacteristic
580C2—41no—
581C2—41no—
582C2—362no—
583C2—362no—
584C2#1172250712yes—
585C2#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. #11721850C2semidirect
2. #64146⋊C3C2semidirect
3. #64144⋊C3C2semidirect
4. #11721322C2semidirect
5. #11721321C2semidirect
6. #11721319C2semidirect
7. #11721303C2semidirect
8. #11721157C2semidirect
9. #11721112C2semidirect
15. C2×#6235C22semidirect
16. C2×C4⋊#272C22semidirect
18. C22×(C3⋊Dic3)⋊C3C22semidirect
21. C2×#6199C22semidirect
22. C4×#1084C22semidirect
25. #3757⋊C3C22semidirect
27. #3750⋊C3C22semidirect
28. #88451C22semidirect
31. #88448C22semidirect
32. #88448C22semidirect
33. #64146D3semidirect
34. #64144D3semidirect
81. C22×C3⋊Dic3C2×D3semidirect
101. #3757C2×D3semidirect
103. #3750C2×D3semidirect
344. H(3,𝔽3)#560semidirect
514. C32#5595semidirect
515. C32#5666semidirect

Maximal subgroups

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

subgroupindex
#117218502
#64146⋊C32
#64144⋊C32
#117213222
#117213212
#117213192
#117213032
#117211572
#117211122
#105952663
#105952593