Finite Group Databasebeta

← #13152090

Subgroups — #13152090

Order 1728.

1447 conjugacy classes of subgroups.

The subgroup lattice has 594 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#6235C211yesmaximal, characteristic, complemented
2C22×#638⋊C3C211yesmaximal, characteristic, complemented
3C2×#3757⋊C3C211yesmaximal, characteristic, complemented
4#11721323C212yesmaximal, complemented
5#11721322C214yesmaximal, complemented
6#11721321C212yesmaximal, complemented
7#11721307C212yesmaximal, complemented
8#11721301C212yesmaximal, complemented
9#10595519—31nomaximal
10#10595518—31nomaximal
11C24×H(3,𝔽3)C2211yescharacteristic
12C23×#272C2211yescharacteristic, complemented
13C23×(C3⋊D3)⋊C3C2211yescharacteristic, complemented
14C2×#6235—24no—
15C2×#6235C2212yes—
16C2×#6235C2212yescomplemented
17C22×#1084C2211yescharacteristic, complemented
18C2×#638⋊C3—24no—
19C2×#638⋊C3C2212yes—
20C2×#638⋊C3C2212yescomplemented
21C22×(C3⋊Dic3)⋊C3C2211yescharacteristic, complemented
22C22×(C3⋊Dic3)⋊C3C2212yescomplemented
23C2×#6200C2214yescomplemented
24#3757⋊C3C2212yescomplemented
25#3757⋊C3C2212yescomplemented
26#88451C2214yescomplemented
27#88450C2214yescomplemented
28#88448C2214yescomplemented
29C22×#638D311yescharacteristic, complemented
30C22×C3×#67—31no—
31C22×C3×#67—31no—
32C22×C3×#67—31no—
33C22×C3×#67—61no—
34C2×#3757D311yescharacteristic, complemented
35C2×C3×#217—31no—
36#63981—32no—
37#63980—34no—
38#63978—34no—
39#63977—32no—
40#63976—32no—
41#63976—32no—
42#63966—32no—
43#63962—32no—
44#63961—32no—
45#63961—32no—
46C23×H(3,𝔽3)—42no—
47C23×H(3,𝔽3)D414yescomplemented
48C23×H(3,𝔽3)C2311yescharacteristic
49C23×H(3,𝔽3)C2311yescharacteristic
50C23×H(3,𝔽3)C2311yescharacteristic
51C22×#272—22no—
52C22×#272—42no—
53C22×#272C2312yes—
54C22×(C3⋊D3)⋊C3—22no—
55C22×(C3⋊D3)⋊C3—42no—
56C22×(C3⋊D3)⋊C3C2312yes—
57#6235—24no—
58#6235—24no—
59C2×#1084—22no—
60C2×#1084C2312yes—
61#638⋊C3—24no—
62#638⋊C3—24no—
63C2×(C3⋊Dic3)⋊C3—22no—
64C2×(C3⋊Dic3)⋊C3—42no—
65C2×(C3⋊Dic3)⋊C3C2314yes—
66C2×(C3⋊Dic3)⋊C3C2312yes—
67#6200—44no—
68#5804—91no—
69#5804—91no—
70#5493—91no—
71C24×C32—21no—
72C24×C32C2×D311yescharacteristic
73C24×C32C2×D311yescharacteristic
74C23×C3⋊D3C2×D311yescharacteristic, complemented
75C23×C3×D3—31no—
76C23×C3×D3—31no—
77C23×C3×D3—31no—
78C23×C3×D3—61no—
79C2×#638—24no—
80C2×#638C2×D312yes—
81C2×#638C2×D312yescomplemented
82C22×C3⋊Dic3C2×D311yescharacteristic, complemented
83C22×C3⋊Dic3C2×D312yescomplemented
84C2×C3×#67—32no—
85C2×C3×#67—32no—
86C2×C3×#67—32no—
87C2×C3×#67—32no—
88C2×C3×#67—32no—
89C2×C3×#67—32no—
90C2×C3×#67—62no—
91C2×C3×#67—62no—
92C2×C3×#67—68no—
93C2×C3×#67—64no—
94C2×C3×#67—64no—
95C2×C3×#67—64no—
96C22×C3×Dic3—31no—
97C22×C3×Dic3—31no—
98C22×C3×Dic3—31no—
99C22×C3×Dic3—32no—
100C22×C3×Dic3—61no—
101C2×#626—34no—
102C2×#625—34no—
103#3757C2×D312yescomplemented
104#3757C2×D312yescomplemented
105C3×#217—32no—
106C3×#217—32no—
107#3724—34no—
108Dic3⋊Dic3—34no—
109#3722—34no—
110(C2×D3)⋊Dic3—34no—
111(C2×D3)⋊Dic3—34no—
112(C2×D3)⋊Dic3—34no—
113C22×H(3,𝔽3)—22no—
114C22×H(3,𝔽3)—22no—
115C22×H(3,𝔽3)—22no—
116C22×H(3,𝔽3)—44no—
117C22×H(3,𝔽3)C2×D412yes—
118C22×H(3,𝔽3)C2×D412yes—
119C22×H(3,𝔽3)C2×D412yes—
120C22×H(3,𝔽3)C2411yescharacteristic
121C2×#272—22no—
122C2×#272—24no—
123C2×#272—44no—
124C2×(C3⋊D3)⋊C3—22no—
125C2×(C3⋊D3)⋊C3—24no—
126C2×(C3⋊D3)⋊C3—44no—
127#1084—22no—
128(C3⋊Dic3)⋊C3—22no—
129(C3⋊Dic3)⋊C3—24no—
130C22×C3×D4—91no—
131C22×C3×D4—91no—
132C22×#67—31no—
133C22×#67—31no—
134C22×#67—31no—
135C22×#67—61no—
136C22×#67—91no—
137C2×C3×C22⋊C4—91no—
138#953—92no—
139#953—92no—
140C2×#217—31no—
141C2×#217—91no—
142#939—94no—
143#939—94no—
144#937—92no—
145#937—92no—
146#935—92no—
147#935—92no—
148#930—94no—
149#929—92no—
150#929—92no—
151#886—92no—
152#886—92no—
153#882—92no—
154#882—92no—
155C23×C32—24no—
156C23×C32—21no—
157C23×C32—21no—
158C23×C32—21no—
159C23×C32—42no—
160C23×C32—42no—
161C23×C32—44no—
162C23×C32D1214yescomplemented
163C23×C32#6714yescomplemented
164C23×C32C22×D311yescharacteristic
165C23×C32C22×D311yescharacteristic
166C23×C32C22×D311yescharacteristic
167C23×C32C22×D311yescharacteristic
168C23×C32C22×D311yescharacteristic
169C23×C32C22×D311yescharacteristic
170C22×C3⋊D3—22no—
171C22×C3⋊D3—42no—
172C22×C3⋊D3C22×D312yes—
173C22×C3×D3—32no—
174C22×C3×D3—32no—
175C22×C3×D3—32no—
176C22×C3×D3—62no—
177C22×C3×D3—64no—
178C22×C3×D3—62no—
179C22×C3×D3—62no—
180C22×C3×D3—62no—
181C22×C3×D3—122no—
182C22×C3×D3—122no—
183C22×C3×D3—122no—
184C22×C3×D3—124no—
185#638—24no—
186#638—24no—
187C2×C3⋊Dic3—22no—
188C2×C3⋊Dic3—42no—
189C2×C3⋊Dic3C22×D312yes—
190C2×C3⋊Dic3C22×D314yes—
191C3×#67—68no—
192C3×#67—68no—
193C3×#67—64no—
194C3×#67—64no—
195C3×#67—64no—
196C3×#67—64no—
197C3×#67—64no—
198C3×#67—64no—
199C2×C3×Dic3—32no—
200C2×C3×Dic3—34no—
201C2×C3×Dic3—32no—
202C2×C3×Dic3—32no—
203C2×C3×Dic3—62no—
204C2×C3×Dic3—64no—
205C2×C3×Dic3—62no—
206C2×C3×Dic3—62no—
207C2×C3×Dic3—62no—
208C2×C3×Dic3—122no—
209#626—124no—
210#625—124no—
211#534—271noSylow
212C2×H(3,𝔽3)—22no—
213C2×H(3,𝔽3)—22no—
214C2×H(3,𝔽3)—22no—
215C2×H(3,𝔽3)C22×D411yescharacteristic
216C2×H(3,𝔽3)H(5,𝔽2)12yes—
217#272—42no—
218(C3⋊D3)⋊C3—42no—
219C24×C3—31no—
220C24×C3—31no—
221C24×C3—61no—
222C24×C3—91no—
223C24×C3—91no—
224C24×C3D3211yescharacteristic
225C23×D3—31no—
226C23×D3—31no—
227C23×D3—31no—
228C23×D3—61no—
229C23×D3—91no—
230C2×C3×D4—92no—
231C2×C3×D4—92no—
232C2×C3×D4—92no—
233C2×C3×D4—92no—
234C2×C3×D4—184no—
235C2×C3×D4—184no—
236C22×C3×C4—91no—
237C22×C3×C4—91no—
238C22×C3×C4—92no—
239C2×#67—32no—
240C2×#67—32no—
241C2×#67—32no—
242C2×#67—32no—
243C2×#67—32no—
244C2×#67—32no—
245C2×#67—62no—
246C2×#67—62no—
247C2×#67—64no—
248C2×#67—64no—
249C2×#67—68no—
250C2×#67—64no—
251C2×#67—92no—
252C2×#67—92no—
253C2×#67—94no—
254C2×#67—94no—
255C2×#67—184no—
256C22×Dic3—31no—
257C22×Dic3—31no—
258C22×Dic3—31no—
259C22×Dic3—32no—
260C22×Dic3—61no—
261C22×Dic3—91no—
262C22×Dic3—92no—
263C2×D12—94no—
264C3×C22⋊C4—92no—
265C3×C22⋊C4—92no—
266#217—32no—
267#217—32no—
268#217—92no—
269#217—92no—
270#217—94no—
271#217—94no—
272(C2×D3)⋊C4—94no—
273(C2×D3)⋊C4—94no—
274(C2×D3)⋊C4—94no—
275(C2×D3)⋊C4—94no—
276C4⋊Dic3—94no—
277Dic3⋊C4—94no—
278Dic3⋊C4—94no—
279C22×C32—24no—
280C22×C32—22no—
281C22×C32—22no—
282C22×C32—22no—
283C22×C32—22no—
284C22×C32—22no—
285C22×C32—22no—
286C22×C32—21no—
287C22×C32—22no—
288C22×C32—22no—
289C22×C32—22no—
290C22×C32—42no—
291C22×C32—42no—
292C22×C32—44no—
293C22×C32—44no—
294C22×C32—48no—
295C22×C32C2×D1212yes—
296C22×C32C2×D1212yes—
297C22×C32C2×D1212yes—
298C22×C32C2×#6712yes—
299C22×C32C2×#6712yes—
300C22×C32C2×#6712yes—
301C22×C32C23×D311yescharacteristic
302C22×C32C23×D311yescharacteristic
303C2×C3⋊D3—24no—
304C2×C3⋊D3—22no—
305C2×C3⋊D3—44no—
306C2×C3×D3—64no—
307C2×C3×D3—62no—
308C2×C3×D3—62no—
309C2×C3×D3—64no—
310C2×C3×D3—64no—
311C2×C3×D3—64no—
312C2×C3×D3—62no—
313C2×C3×D3—124no—
314C2×C3×D3—124no—
315C2×C3×D3—128no—
316C2×C3×D3—124no—
317C2×C3×D3—124no—
318C3⋊Dic3—24no—
319C3⋊Dic3—22no—
320C3×Dic3—64no—
321C3×Dic3—62no—
322C3×Dic3—62no—
323C3×Dic3—64no—
324C3×Dic3—62no—
325C22×D4—91no—
326C22×D4—91no—
327#123—272no—
328#123—272no—
329#121—274no—
330#120—272no—
331#120—272no—
332C2×C22⋊C4—91no—
333H(3,𝔽3)#53411yescharacteristic, complemented, Sylow
334C23×C3—34no—
335C23×C3—34no—
336C23×C3—31no—
337C23×C3—31no—
338C23×C3—31no—
339C23×C3—31no—
340C23×C3—31no—
341C23×C3—31no—
342C23×C3—42no—
343C23×C3—64no—
344C23×C3—61no—
345C23×C3—61no—
346C23×C3—61no—
347C23×C3—92no—
348C23×C3—92no—
349C23×C3—122no—
350C23×C3—122no—
351C23×C3—124no—
352C23×C3—182no—
353C23×C3—182no—
354C23×C3—362no—
355C23×C3—362no—
356C23×C3#62614yes—
357C23×C3C2×D3211yescharacteristic
358C23×C3C2×D3211yescharacteristic
359C23×C3C2×D3211yescharacteristic
360C22×D3—32no—
361C22×D3—32no—
362C22×D3—32no—
363C22×D3—62no—
364C22×D3—62no—
365C22×D3—62no—
366C22×D3—64no—
367C22×D3—62no—
368C22×D3—92no—
369C22×D3—122no—
370C22×D3—122no—
371C22×D3—122no—
372C22×D3—124no—
373C22×D3—182no—
374C22×D3—362no—
375C3×D4—184no—
376C3×D4—184no—
377C3×D4—184no—
378C3×D4—184no—
379C2×C3×C4—92no—
380C2×C3×C4—94no—
381C2×C3×C4—92no—
382C2×C3×C4—182no—
383C2×C3×C4—182no—
384C2×C3×C4—362no—
385#67—64no—
386#67—64no—
387#67—64no—
388#67—64no—
389#67—68no—
390#67—68no—
391#67—64no—
392#67—64no—
393#67—184no—
394#67—184no—
395#67—364no—
396#67—364no—
397C2×Dic3—32no—
398C2×Dic3—32no—
399C2×Dic3—32no—
400C2×Dic3—34no—
401C2×Dic3—62no—
402C2×Dic3—62no—
403C2×Dic3—62no—
404C2×Dic3—64no—
405C2×Dic3—62no—
406C2×Dic3—94no—
407C2×Dic3—92no—
408C2×Dic3—122no—
409C2×Dic3—182no—
410C2×Dic3—362no—
411D12—364no—
412C2×C32—24no—
413C2×C32—22no—
414C2×C32—22no—
415C2×C32—22no—
416C2×C32—22no—
417C2×C32—22no—
418C2×C32—22no—
419C2×C32—21no—
420C2×C32—22no—
421C2×C32—42no—
422C2×C32—42no—
423C2×C32C22×D1211yescharacteristic
424C2×C32#100412yes—
425C2×C32#100412yes—
426C2×C32C22×#6711yescharacteristic
427C3⋊D3—42no—
428C3×D3—124no—
429C3×D3—122no—
430C3×D3—122no—
431C3×D3—122no—
432C24—91no—
433C24—91no—
434C24((C3⋊D3)⋊C3)⋊C211yescharacteristic
435C2×D4—92no—
436C2×D4—92no—
437C2×D4—92no—
438C2×D4—92no—
439C2×D4—184no—
440C2×D4—184no—
441C2×D4—274no—
442C22×C4—91no—
443C22×C4—91no—
444C22×C4—92no—
445C4⋊C4—274no—
446C22⋊C4—92no—
447C22⋊C4—92no—
448C22⋊C4—274no—
449C22⋊C4—274no—
450C22×C3—22no—
451C22×C3—22no—
452C22×C3—22no—
453C22×C3—31no—
454C22×C3—31no—
455C22×C3—32no—
456C22×C3—32no—
457C22×C3—32no—
458C22×C3—32no—
459C22×C3—32no—
460C22×C3—32no—
461C22×C3—44no—
462C22×C3—62no—
463C22×C3—62no—
464C22×C3—62no—
465C22×C3—62no—
466C22×C3—62no—
467C22×C3—62no—
468C22×C3—64no—
469C22×C3—61no—
470C22×C3—62no—
471C22×C3—62no—
472C22×C3—62no—
473C22×C3—122no—
474C22×C3—122no—
475C22×C3—124no—
476C22×C3—124no—
477C22×C3—128no—
478C22×C3—184no—
479C22×C3—182no—
480C22×C3—182no—
481C22×C3—184no—
482C22×C3—364no—
483C22×C3—364no—
484C22×C3C2×#62612yes—
485C22×C3C2×#62612yes—
486C22×C3C2×#62612yes—
487C22×C3C22×D3211yescharacteristic
488C2×D3—64no—
489C2×D3—62no—
490C2×D3—62no—
491C2×D3—64no—
492C2×D3—64no—
493C2×D3—62no—
494C2×D3—64no—
495C2×D3—124no—
496C2×D3—124no—
497C2×D3—124no—
498C2×D3—124no—
499C2×D3—128no—
500C2×D3—184no—
501C2×D3—182no—
502C2×D3—364no—
503C3×C4—184no—
504C3×C4—182no—
505C3×C4—182no—
506Dic3—62no—
507Dic3—62no—
508Dic3—64no—
509Dic3—62no—
510Dic3—64no—
511Dic3—182no—
512Dic3—184no—
513C32—21no—
514C32#549311yescharacteristic, complemented
515C32#580411yescharacteristic, complemented
516C23—42no—
517C23—92no—
518C23—92no—
519C23—182no—
520C23—182no—
521C23—362no—
522C23—362no—
523C23#620014yescomplemented
524C23C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
525C23C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
526C23C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
527D4—184no—
528D4—184no—
529D4—184no—
530D4—184no—
531D4—1084no—
532C2×C4—94no—
533C2×C4—92no—
534C2×C4—92no—
535C2×C4—182no—
536C2×C4—182no—
537C2×C4—362no—
538C2×C3—22no—
539C2×C3—22no—
540C2×C3—22no—
541C2×C3—31no—
542C2×C3—31no—
543C2×C3—32no—
544C2×C3—32no—
545C2×C3—62no—
546C2×C3—62no—
547C2×C3—62no—
548C2×C3—62no—
549C2×C3—62no—
550C2×C3—62no—
551C2×C3—64no—
552C2×C3—61no—
553C2×C3—62no—
554C2×C3—122no—
555C2×C3—122no—
556C2×C3—362no—
557C2×C3—362no—
558C2×C3C22×#62611yescharacteristic
559C2×C3#6433012yes—
560D3—122no—
561D3—122no—
562D3—124no—
563D3—122no—
564D3—362no—
565C22—22no—
566C22—22no—
567C22—22no—
568C22—44no—
569C22—182no—
570C22—182no—
571C22—184no—
572C22—184no—
573C22—364no—
574C22—364no—
575C22C2×#620012yes—
576C22C2×#620012yes—
577C22C2×#620012yes—
578C22C22×((C3⋊D3)⋊C3)⋊C211yescharacteristic
579C4—182no—
580C4—182no—
581C4—184no—
582C3—31no—
583C3—31no—
584C3—61no—
585C3#1059551911yescharacteristic
586C2—22no—
587C2—22no—
588C2—22no—
589C2—362no—
590C2—362no—
591C2C22×#620011yescharacteristic
592C2#1172254512yes—
subgroupG/H
centerC22C22×((C3⋊D3)⋊C3)⋊C2
commutator, cosocleC22×H(3,𝔽3)C24
Frattini, socleC22×C3C22×D32
FittingC24×H(3,𝔽3)C22
hypercenterC24((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#534
LCSG ▷ C22×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C24

Complements

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

NH ≅ G/Ntype
1. C22×#6235C2semidirect
2. C22×#638⋊C3C2semidirect
3. C2×#3757⋊C3C2semidirect
4. #11721323C2semidirect
5. #11721322C2semidirect
6. #11721321C2semidirect
7. #11721307C2semidirect
8. #11721301C2semidirect
12. C23×#272C22semidirect
13. C23×(C3⋊D3)⋊C3C22semidirect
16. C2×#6235C22semidirect
17. C22×#1084C22semidirect
20. C2×#638⋊C3C22semidirect
21. C22×(C3⋊Dic3)⋊C3C22semidirect
22. C22×(C3⋊Dic3)⋊C3C22semidirect
23. C2×#6200C22semidirect
24. #3757⋊C3C22semidirect
25. #3757⋊C3C22semidirect
26. #88451C22semidirect
27. #88450C22semidirect
28. #88448C22semidirect
29. C22×#638D3semidirect
34. C2×#3757D3semidirect
47. C23×H(3,𝔽3)D4semidirect
74. C23×C3⋊D3C2×D3semidirect
81. C2×#638C2×D3semidirect
82. C22×C3⋊Dic3C2×D3semidirect
83. C22×C3⋊Dic3C2×D3semidirect
103. #3757C2×D3semidirect
104. #3757C2×D3semidirect
162. C23×C32D12semidirect
163. C23×C32#67semidirect
333. H(3,𝔽3)#534semidirect
514. C32#5493semidirect
515. C32#5804semidirect
523. C23#6200semidirect

Maximal subgroups

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

subgroupindex
C22×#62352
C22×#638⋊C32
C2×#3757⋊C32
#117213232
#117213222
#117213212
#117213072
#117213012
#105955193
#105955183