Finite Group Databasebeta

← #13172083

Subgroups — #13172083

Order 1728.

1766 conjugacy classes of subgroups; only the 212 normal subgroups are shown.

normal subgroups only

This group has 1766 conjugacy classes of subgroups — too many to draw. Only its 212 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
1C2×#89046C211yesmaximal, characteristic, complemented
2C22×D3×C3⋊Dic3C211yesmaximal, characteristic, complemented
3#11723609C212yesmaximal, complemented
4C3×#63977C212yesmaximal, complemented
5C3×#63957C211yesmaximal, characteristic, complemented
6#11722981C212yesmaximal, complemented
7#11722955C212yesmaximal, complemented
8#11722928C214yesmaximal, complemented
9#10595501—31nomaximal
10#10594078—32nomaximal
11C23×C32×D3C2211yescharacteristic
12C22×C3⋊(C3⋊Dic3)C2211yescharacteristic
13C2×C3×#638C2212yes—
14C22×C3×C3⋊Dic3C2211yescharacteristic
15C2×#6296C2211yescharacteristic, complemented
16C2×#6295C2211yescharacteristic, complemented
17#89046—24no—
18C2×#6294C2212yescomplemented
19C2×D3×C3⋊Dic3—24no—
20C2×D3×C3⋊Dic3C2212yes—
21C2×D3×C3⋊Dic3C2212yes—
22C2×C3×#626C2214yes—
23C32×#217C2212yes—
24Dic3⋊(C3⋊Dic3)C2214yescomplemented
25#88803C2214yescomplemented
26C3×#3724C2212yescomplemented
27C3×(C2×D3)⋊Dic3C2214yes—
28C3×Dic32C2212yescomplemented
29C23×C3⋊Dic3—31no—
30C2×#3805—31no—
31C2×#3804—32no—
32C22×D3×Dic3—32no—
33C22×D3×Dic3—61no—
34#64095—32no—
35C3×#953—32no—
36C3×#882—32no—
37C3×#877—32no—
38#63982—32no—
39#63980—32no—
40#63977D312yescomplemented
41#67⋊Dic3—34no—
42#63962—32no—
43#63962—32no—
44#63962—62no—
45#63957D311yescharacteristic, complemented
46#63893—34no—
47Dic3×D12—34no—
48#63886—34no—
49#63880—34no—
50C23×C33C2311yescharacteristic
51C22×C3×C3⋊D3C2312yes—
52C22×C32×D3—22no—
53C22×C32×D3—24no—
54C22×C32×D3C2312yes—
55C2×C3⋊(C3⋊Dic3)—22no—
56C2×C3⋊(C3⋊Dic3)C2312yes—
57C3×#638—24no—
58C2×C3×C3⋊Dic3D414yescomplemented
59C2×C3×C3⋊Dic3C2312yes—
60C2×C3×C3⋊Dic3C2311yescharacteristic
61C2×C3×C3⋊Dic3C2311yescharacteristic
62C2×C32×Dic3C2314yes—
63#6296—22no—
64#6295—22no—
65#6294—24no—
66D3×C3⋊Dic3—24no—
67D3×C3⋊Dic3—24no—
68C3×#626—28no—
69#5586—92no—
70#5524—91no—
71C24×C32—31no—
72C23×C3×D3—21no—
73C23×C3×D3C2×D312yes—
74C2×#638C2×D312yes—
75C22×C3⋊Dic3—32no—
76C22×C3⋊Dic3—31no—
77C22×C3⋊Dic3—32no—
78C22×C3⋊Dic3—32no—
79C22×C3⋊Dic3—61no—
80C22×C3⋊Dic3—64no—
81C22×C3⋊Dic3—62no—
82C22×C3⋊Dic3—121no—
83C22×C3⋊Dic3C2×D311yescharacteristic
84C2×C3×#67—32no—
85C2×C3×#67—32no—
86C2×C3×#67—34no—
87C2×C3×#67—62no—
88C22×C3×Dic3—32no—
89C22×C3×Dic3—61no—
90C2×C3×D12—34no—
91C2×#627—32no—
92C2×#627—31no—
93C2×#626—32no—
94C2×#626C2×D314yes—
95C2×#625—31no—
96C2×#624—32no—
97#3805—64no—
98#3804—68no—
99C2×D3×Dic3—34no—
100C2×D3×Dic3—34no—
101C2×D3×Dic3—32no—
102C2×D3×Dic3—32no—
103C2×D3×Dic3—68no—
104C2×D3×Dic3—62no—
105C2×D3×Dic3—62no—
106C2×D3×Dic3—68no—
107C32×C22⋊C4—32no—
108#3752—34no—
109C4⋊(C3⋊Dic3)—34no—
110C3×#217—22no—
111C3×#217—34no—
112C3×#217C2×D312yes—
113C3×#217C2×D312yes—
114C3×(C2×D3)⋊C4—34no—
115C3×Dic3⋊C4—34no—
116C3×C4×Dic3—34no—
117#3724C2×D312yescomplemented
118Dic3⋊Dic3—34no—
119Dic3⋊Dic3—34no—
120Dic3⋊Dic3—64no—
121(C2×D3)⋊Dic3—34no—
122(C2×D3)⋊Dic3—34no—
123(C2×D3)⋊Dic3—64no—
124(C2×D3)⋊Dic3C2×D314yes—
125Dic32C2×D312yescomplemented
126C22×C33—22no—
127C22×C33C2×D412yes—
128C22×C33C2411yescharacteristic
129C2×C3×C3⋊D3—22no—
130C2×C3×C3⋊D3—24no—
131C2×C32×D3—28no—
132C2×C32×D3—22no—
133C2×C32×D3—22no—
134C2×C32×D3#4114yescomplemented
135C3⋊(C3⋊Dic3)—22no—
136C3×C3⋊Dic3—22no—
137C3×C3⋊Dic3C2×D412yes—
138C3×C3⋊Dic3C2×D412yes—
139C32×Dic3—24no—
140C23×Dic3—92no—
141C23×Dic3—181no—
142C2×#237—92no—
143C2×#235—91no—
144C22×C4×D3—91no—
145C3×#120—92no—
146C3×#117—91no—
147#953—32no—
148#939—94no—
149#935—92no—
150D4×Dic3—94no—
151#897—92no—
152#895—92no—
153#886—92no—
154#886—92no—
155#886—182no—
156#882—32no—
157#67⋊C4—94no—
158#878—94no—
159#877—32no—
160C4×D12—94no—
161C23×C32—21no—
162C23×C32—22no—
163C23×C32—32no—
164C23×C32—41no—
165C23×C32—64no—
166C23×C32—62no—
167C23×C32C22×D311yescharacteristic
168C23×C32C22×D312yes—
169C22×C3⋊D3C22×D312yes—
170C22×C3×D3—28no—
171C22×C3×D3—24no—
172C22×C3×D3—24no—
173C22×C3×D3—28no—
174C22×C3×D3—22no—
175C22×C3×D3—32no—
176C22×C3×D3—32no—
177C22×C3×D3—62no—
178C22×C3×D3C22×D314yes—
179C2×C32×C4—34no—
180#638—24no—
181C2×C3⋊Dic3—32no—
182C2×C3⋊Dic3—34no—
183C2×C3⋊Dic3—62no—
184C2×C3⋊Dic3—64no—
185C2×C3⋊Dic3—62no—
186C2×C3⋊Dic3—64no—
187C2×C3⋊Dic3—64no—
188C2×C3⋊Dic3—64no—
189C2×C3⋊Dic3—64no—
190C2×C3⋊Dic3—122no—
191C2×C3⋊Dic3—124no—
192C2×C3⋊Dic3—124no—
193C2×C3⋊Dic3D1214yescomplemented
194C2×C3⋊Dic3C22×D312yes—
195C2×C3⋊Dic3C22×D311yescharacteristic
196C2×C3⋊Dic3C22×D311yescharacteristic
197C3×#67—68no—
198C3×#67—64no—
199C3×#67—64no—
200C3×#67—124no—
201C2×C3×Dic3—24no—
202C2×C3×Dic3—38no—
203C2×C3×Dic3—32no—
204C2×C3×Dic3—32no—
205C2×C3×Dic3—32no—
206C2×C3×Dic3—32no—
207C2×C3×Dic3—64no—
208C2×C3×Dic3—61no—
209C2×C3×Dic3—61no—
210C2×C3×Dic3—62no—
211C2×C3×Dic3C22×D314yes—
212C2×C3×Dic3C22×D314yes—
213C3×D12—68no—
214#627—64no—
215#627—62no—
216#626—28no—
217#626—64no—
218#625—62no—
219#624—64no—
220D3×Dic3—64no—
221D3×Dic3—68no—
222D3×Dic3—68no—
223D3×Dic3—68no—
224D3×Dic3—68no—
225D3×Dic3—64no—
226#525—271noSylow
227C2×C33—22no—
228C2×C33C22×D411yescharacteristic
229C2×C33C2×#4112yes—
230C3×C3⋊D3—42no—
231C32×D3—24no—
232C24×C3—32no—
233C24×C3—61no—
234C23×D3D3211yescharacteristic
235C2×C3×D4—94no—
236C2×C3×D4—92no—
237C22×C3×C4—91no—
238C2×#67—32no—
239C2×#67—32no—
240C2×#67—34no—
241C2×#67—62no—
242C2×#67—92no—
243C22×Dic3—32no—
244C22×Dic3—61no—
245C22×Dic3—92no—
246C22×Dic3—94no—
247C22×Dic3—94no—
248C22×Dic3—91no—
249C22×Dic3—92no—
250C22×Dic3—182no—
251C22×Dic3—182no—
252C22×Dic3—181no—
253C22×Dic3—182no—
254C22×Dic3—188no—
255C22×Dic3—188no—
256C22×Dic3—361no—
257#237—188no—
258#235—184no—
259C2×D12—34no—
260C2×D12—91no—
261C2×C4×D3—92no—
262C2×C4×D3—92no—
263C2×C4×D3—92no—
264C2×C4×D3—92no—
265C2×C4×D3—184no—
266C2×Dic6—92no—
267C2×Dic6—91no—
268C3×C4⋊C4—92no—
269C3×C22⋊C4—32no—
270C3×C22⋊C4—32no—
271C3×C22⋊C4—62no—
272C3×C22⋊C4—94no—
273C3×C42—92no—
274#217—34no—
275#217—94no—
276#217D3212yes—
277(C2×D3)⋊C4—34no—
278(C2×D3)⋊C4—94no—
279(C2×D3)⋊C4—94no—
280(C2×D3)⋊C4—184no—
281C4⋊Dic3—94no—
282C4⋊Dic3—94no—
283C4⋊Dic3—184no—
284Dic3⋊C4—34no—
285Dic3⋊C4—94no—
286C4×Dic3—34no—
287C22×C32—22no—
288C22×C32—24no—
289C22×C32—24no—
290C22×C32—22no—
291C22×C32—21no—
292C22×C32—24no—
293C22×C32—22no—
294C22×C32—34no—
295C22×C32—42no—
296C22×C32—41no—
297C22×C32—44no—
298C22×C32—44no—
299C22×C32—68no—
300C22×C32—62no—
301C22×C32—62no—
302C22×C32C2×D1212yes—
303C22×C32D3×D414yes—
304C22×C32C23×D311yescharacteristic
305C22×C32C23×D312yes—
306C2×C3⋊D3—22no—
307C2×C3⋊D3—24no—
308C2×C3×D3—24no—
309C2×C3×D3—216no—
310C2×C3×D3—216no—
311C2×C3×D3—24no—
312C2×C3×D3—24no—
313C2×C3×D3—24no—
314C2×C3×D3—24no—
315C2×C3×D3—62no—
316C2×C3×D3—62no—
317C2×C3×D3—64no—
318C2×C3×D3—64no—
319C2×C3×D3—122no—
320C2×C3×D3—124no—
321C2×C3×D3#23718yescomplemented
322C32×C4—64no—
323C3⋊Dic3—22no—
324C3⋊Dic3—62no—
325C3⋊Dic3—64no—
326C3⋊Dic3—64no—
327C3⋊Dic3—124no—
328C3⋊Dic3—124no—
329C3⋊Dic3C2×D1212yes—
330C3⋊Dic3C2×D1212yes—
331C3×Dic3—24no—
332C3×Dic3—24no—
333C3×Dic3—34no—
334C3×Dic3—34no—
335C3×Dic3—44no—
336C3×Dic3—62no—
337C3×Dic3—62no—
338C3×Dic3—62no—
339C3×Dic3—62no—
340C3×Dic3—122no—
341C2×#41—271no—
342C23×C4—271no—
343#123—272no—
344C22⋊Q8—272no—
345#121—272no—
346#120—92no—
347C4×D4—274no—
348#117—91no—
349C33#52511yescharacteristic, complemented, Sylow
350C23×C3—21no—
351C23×C3—22no—
352C23×C3—34no—
353C23×C3—41no—
354C23×C3—62no—
355C23×C3—68no—
356C23×C3—64no—
357C23×C3—68no—
358C23×C3—64no—
359C23×C3—92no—
360C23×C3C2×D3212yes—
361C23×C3C2×D3211yescharacteristic
362C22×D3—22no—
363C22×D3—24no—
364C22×D3—32no—
365C22×D3—32no—
366C22×D3—62no—
367C22×D3C2×D3212yes—
368C3×D4—188no—
369C3×D4—184no—
370C2×C3×C4—34no—
371C2×C3×C4—34no—
372C2×C3×C4—64no—
373C2×C3×C4—91no—
374C2×C3×C4—91no—
375C2×C3×C4—94no—
376C2×C3×C4—92no—
377#67—68no—
378#67—64no—
379#67—64no—
380#67—124no—
381#67—184no—
382C2×Dic3—32no—
383C2×Dic3—32no—
384C2×Dic3—38no—
385C2×Dic3—32no—
386C2×Dic3—32no—
387C2×Dic3—64no—
388C2×Dic3—61no—
389C2×Dic3—61no—
390C2×Dic3—62no—
391C2×Dic3—94no—
392C2×Dic3—92no—
393C2×Dic3—182no—
394C2×Dic3—184no—
395C2×Dic3—184no—
396C2×Dic3—188no—
397C2×Dic3—188no—
398C2×Dic3—188no—
399C2×Dic3—188no—
400C2×Dic3—184no—
401C2×Dic3—184no—
402C2×Dic3—182no—
403C2×Dic3—362no—
404C2×Dic3—364no—
405C2×Dic3—364no—
406C2×Dic3C2×D3214yes—
407D12—68no—
408D12—182no—
409C4×D3—184no—
410C4×D3—184no—
411C4×D3—184no—
412C4×D3—184no—
413Dic6—184no—
414Dic6—182no—
415C2×C32—22no—
416C2×C32—24no—
417C2×C32—24no—
418C2×C32—22no—
419C2×C32—24no—
420C2×C32—22no—
421C2×C32—21no—
422C2×C32—41no—
423C2×C32—42no—
424C2×C32—44no—
425C2×C32—44no—
426C2×C32—64no—
427C2×C32C22×D1211yescharacteristic
428C2×C32C2×D3×D412yes—
429C2×C32C2×#23714yes—
430C2×C32D3×#4112yes—
431C3⋊D3—42no—
432C3×D3—28no—
433C3×D3—28no—
434C3×D3—122no—
435C3×D3—122no—
436C3×D3—242no—
437C24—31no—
438#41—544no—
439C2×Q8—271no—
440C2×D4—94no—
441C2×D4—92no—
442C2×D4—271no—
443C22×C4—91no—
444C22×C4—272no—
445C22×C4—272no—
446C22×C4—272no—
447C22×C4—271no—
448C22×C4—544no—
449C4⋊C4—92no—
450C4⋊C4—274no—
451C22⋊C4—32no—
452C22⋊C4—94no—
453C22⋊C4—274no—
454C42—92no—
455C22×C3—24no—
456C22×C3—24no—
457C22×C3—22no—
458C22×C3—21no—
459C22×C3—22no—
460C22×C3—22no—
461C22×C3—24no—
462C22×C3—38no—
463C22×C3—44no—
464C22×C3—44no—
465C22×C3—41no—
466C22×C3—42no—
467C22×C3—64no—
468C22×C3—64no—
469C22×C3—616no—
470C22×C3—616no—
471C22×C3—64no—
472C22×C3—64no—
473C22×C3—64no—
474C22×C3—182no—
475C22×C3—184no—
476C22×C3D3×D1214yes—
477C22×C3#381112yes—
478C22×C3C22×D3212yes—
479C22×C3C22×D3211yescharacteristic
480C2×D3—28no—
481C2×D3—22no—
482C2×D3—22no—
483C2×D3—62no—
484C2×D3—64no—
485C2×D3—64no—
486C2×D3—62no—
487C2×D3—122no—
488C2×D3—124no—
489C2×D3#380514yescomplemented
490C3×C4—64no—
491C3×C4—64no—
492C3×C4—92no—
493C3×C4—92no—
494C3×C4—124no—
495C3×C4—182no—
496Dic3—24no—
497Dic3—34no—
498Dic3—34no—
499Dic3—62no—
500Dic3—62no—
501Dic3—62no—
502Dic3—62no—
503Dic3—122no—
504Dic3—184no—
505Dic3—184no—
506Dic3—182no—
507Dic3—364no—
508Dic3—364no—
509C32—22no—
510C32—21no—
511C32—41no—
512C32#552411yescharacteristic, complemented
513C32#558612yescomplemented
514C23—32no—
515C23—64no—
516C23—62no—
517C23—92no—
518C23D3311yescharacteristic
519Q8—542no—
520D4—188no—
521D4—184no—
522D4—542no—
523C2×C4—34no—
524C2×C4—91no—
525C2×C4—91no—
526C2×C4—94no—
527C2×C4—92no—
528C2×C4—272no—
529C2×C4—544no—
530C2×C4—544no—
531C2×C4—544no—
532C2×C4—542no—
533C2×C3—24no—
534C2×C3—24no—
535C2×C3—22no—
536C2×C3—24no—
537C2×C3—22no—
538C2×C3—21no—
539C2×C3—22no—
540C2×C3—44no—
541C2×C3—44no—
542C2×C3—42no—
543C2×C3—41no—
544C2×C3—68no—
545C2×C3—68no—
546C2×C3—362no—
547C2×C3C2×D3×D1212yes—
548C2×C3D3×#23714yes—
549C2×C3C2×#380512yes—
550C2×C3C2×#381111yescharacteristic
551D3—24no—
552D3—122no—
553D3—122no—
554D3—242no—
555C22—22no—
556C22—34no—
557C22—68no—
558C22—62no—
559C22—62no—
560C22—184no—
561C22—182no—
562C22#8895512yes—
563C22C2×D3311yescharacteristic
564C4—64no—
565C4—92no—
566C4—92no—
567C4—182no—
568C4—542no—
569C3—22no—
570C3—21no—
571C3—41no—
572C3#1059407812yescomplemented
573C3#1059550111yescharacteristic, complemented
574C2—22no—
575C2—64no—
576C2—362no—
577C2D3×#380512yes—
578C2C2×#8895511yescharacteristic
subgroupG/H
center, FrattiniC22C2×D33
commutator, socle, cosocleC22×C33C24
FittingC23×C33C23
hypercenterC23D33
hypocenterC33#525
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22 ◁ C23

Complements

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

NH ≅ G/Ntype
1. C2×#89046C2semidirect
2. C22×D3×C3⋊Dic3C2semidirect
3. #11723609C2semidirect
4. C3×#63977C2semidirect
5. C3×#63957C2semidirect
6. #11722981C2semidirect
7. #11722955C2semidirect
8. #11722928C2semidirect
15. C2×#6296C22semidirect
16. C2×#6295C22semidirect
18. C2×#6294C22semidirect
24. Dic3⋊(C3⋊Dic3)C22semidirect
25. #88803C22semidirect
26. C3×#3724C22semidirect
28. C3×Dic32C22semidirect
40. #63977D3semidirect
45. #63957D3semidirect
58. C2×C3×C3⋊Dic3D4semidirect
117. #3724C2×D3semidirect
125. Dic32C2×D3semidirect
134. C2×C32×D3#41semidirect
193. C2×C3⋊Dic3D12semidirect
321. C2×C3×D3#237semidirect
349. C33#525semidirect
489. C2×D3#3805semidirect
512. C32#5524semidirect
513. C32#5586semidirect
572. C3#10594078semidirect
573. C3#10595501semidirect

Maximal subgroups

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

subgroupindex
C2×#890462
C22×D3×C3⋊Dic32
#117236092
C3×#639772
C3×#639572
#117229812
#117229552
#117229282
#105955013
#105940783