Finite Group Databasebeta

← #13178061

Subgroups — #13178061

Order 1728.

2924 conjugacy classes of subgroups.

The subgroup lattice has 588 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×C3⋊(C3⋊Dic3)C211yesmaximal, characteristic, complemented
2C2×#89029C211yesmaximal, characteristic, complemented
3C3×#64162C211yesmaximal, characteristic, complemented
4C32×#953C211yesmaximal, characteristic, complemented
5#11723642C212yesmaximal, complemented
6#11723637C212yesmaximal, complemented
7#11723626C212yesmaximal, complemented
8#67⋊(C3⋊Dic3)C212yesmaximal, complemented
9#11723609C211yesmaximal, characteristic, complemented
10#11723603C211yesmaximal, characteristic, complemented
11#11723598C211yesmaximal, characteristic, complemented
12#10596057—31nomaximal
13#10595501—34nomaximal
14C24×C33C2211yescharacteristic
15C22×C3⋊(C3⋊Dic3)—24no—
16C22×C3⋊(C3⋊Dic3)C2211yescharacteristic
17C22×C3⋊(C3⋊Dic3)C2212yes—
18C22×C3⋊(C3⋊Dic3)C2212yes—
19C22×C3⋊(C3⋊Dic3)C2211yescharacteristic
20C2×C3×#638C2211yescharacteristic
21C2×C3×#638C2212yes—
22C2×C32×#67C2211yescharacteristic
23C2×C32×#67C2212yes—
24C2×#6293C2211yescharacteristic, complemented
25C2×#6290C2211yescharacteristic, complemented
26C2×Dic3×C3⋊D3C2211yescharacteristic, complemented
27#89029—24no—
28C2×D3×C3⋊Dic3C2211yescharacteristic, complemented
29C3×#3757C2211yescharacteristic
30C3×#3757C2212yes—
31C32×#217C2211yescharacteristic
32C32×#217C2212yes—
33#88807C2212yescomplemented
34#88806C2212yescomplemented
35Dic3⋊(C3⋊Dic3)C2212yescomplemented
36#88803C2212yescomplemented
37(C2×D3)⋊(C3⋊Dic3)C2212yescomplemented
38Dic3×C3⋊Dic3C2212yescomplemented
39C23×C3⋊Dic3—31no—
40C23×C3⋊Dic3—34no—
41C23×C3⋊Dic3—64no—
42C2×#3830—31no—
43C2×#3805—34no—
44C32×#120—31no—
45#64162D311yescharacteristic, complemented
46#64148—32no—
47#64144—31no—
48D4×C3⋊Dic3—32no—
49#64095—31no—
50#64090—32no—
51#64087—32no—
52#64086—31no—
53C3×#953—34no—
54C3×#953D314yescomplemented
55#63982—38no—
56#63980—38no—
57#67⋊Dic3—38no—
58#67⋊Dic3—38no—
59#63962—34no—
60#63962—34no—
61#63957—34no—
62C23×C33—22no—
63C23×C33—24no—
64C23×C33C2311yescharacteristic
65C23×C33C2312yes—
66C22×C3×C3⋊D3C2311yescharacteristic
67C22×C32×D3C2311yescharacteristic
68C2×C3⋊(C3⋊Dic3)—22no—
69C2×C3⋊(C3⋊Dic3)—24no—
70C2×C3⋊(C3⋊Dic3)—24no—
71C2×C3⋊(C3⋊Dic3)D414yescomplemented
72C2×C3⋊(C3⋊Dic3)C2312yes—
73C2×C3⋊(C3⋊Dic3)C2311yescharacteristic
74C2×C3⋊(C3⋊Dic3)C2311yescharacteristic
75C3×#638—22no—
76C3×#638—24no—
77C2×C3×C3⋊Dic3C2311yescharacteristic
78C2×C3×C3⋊Dic3C2312yes—
79C32×#67—22no—
80C32×#67—24no—
81C2×C32×Dic3C2311yescharacteristic
82C2×C32×Dic3C2312yes—
83#6293—22no—
84#6290—22no—
85Dic3×C3⋊D3—22no—
86D3×C3⋊Dic3—22no—
87#5586—94no—
88#5586—91no—
89C24×C32—24no—
90C24×C32C2×D314yes—
91C24×C32C2×D311yescharacteristic
92C2×C32×D4—31no—
93C2×C32×D4—32no—
94C2×#638—31no—
95C2×#638C2×D312yes—
96C2×#638C2×D311yescharacteristic
97C22×C3⋊Dic3—31no—
98C22×C3⋊Dic3—34no—
99C22×C3⋊Dic3—32no—
100C22×C3⋊Dic3—32no—
101C22×C3⋊Dic3—31no—
102C22×C3⋊Dic3—34no—
103C22×C3⋊Dic3—38no—
104C22×C3⋊Dic3—38no—
105C22×C3⋊Dic3—31no—
106C22×C3⋊Dic3—64no—
107C22×C3⋊Dic3—616no—
108C22×C3⋊Dic3—64no—
109C22×C3⋊Dic3—68no—
110C22×C3⋊Dic3—68no—
111C22×C3⋊Dic3—64no—
112C22×C3⋊Dic3—632no—
113#3830—64no—
114C2×C4×C3⋊D3—31no—
115C2×#634—31no—
116C2×C3×#67—34no—
117C2×C3×#67—38no—
118C2×C3×#67C2×D314yes—
119C2×C3×#67C2×D318yes—
120C2×#627—34no—
121C2×#625—34no—
122#3805—616no—
123C2×D3×Dic3—34no—
124C2×D3×Dic3—34no—
125C32×C22⋊C4—31no—
126C32×C22⋊C4—32no—
127#3757—32no—
128#3757C2×D311yescharacteristic
129#3757C2×D312yes—
130#3752—32no—
131C4⋊(C3⋊Dic3)—32no—
132#3750—32no—
133#3750—32no—
134C4×C3⋊Dic3—32no—
135C3×#217—34no—
136C3×#217—38no—
137C3×#217C2×D314yes—
138C3×#217C2×D318yes—
139#3724—38no—
140Dic3⋊Dic3—38no—
141Dic3⋊Dic3—38no—
142(C2×D3)⋊Dic3—38no—
143(C2×D3)⋊Dic3—38no—
144Dic32—38no—
145C22×C33—28no—
146C22×C33—22no—
147C22×C33—22no—
148C22×C33—22no—
149C22×C33C2×D412yes—
150C22×C33#4114yescomplemented
151C22×C33C2411yescharacteristic
152C2×C3×C3⋊D3—21no—
153C2×C3×C3⋊D3—22no—
154C2×C32×D3—21no—
155C2×C32×D3—22no—
156C3⋊(C3⋊Dic3)—22no—
157C3⋊(C3⋊Dic3)C2×D412yes—
158C3⋊(C3⋊Dic3)C2×D412yes—
159C3×C3⋊Dic3—21no—
160C3×C3⋊Dic3—22no—
161C32×Dic3—21no—
162C32×Dic3—22no—
163C23×Dic3—94no—
164C23×Dic3—91no—
165C23×Dic3—184no—
166C2×#237—94no—
167C2×#237—91no—
168C3×#120—34no—
169C3×#120—91no—
170#953—34no—
171#953C3⋊D311yescharacteristic, complemented
172#939—98no—
173#939—92no—
174#935—94no—
175#935—91no—
176D4×Dic3—98no—
177D4×Dic3—92no—
178#886—94no—
179#886—91no—
180#67⋊C4—98no—
181#67⋊C4—92no—
182#878—98no—
183#878—92no—
184#877—94no—
185#877—91no—
186C23×C32—232no—
187C23×C32—216no—
188C23×C32—28no—
189C23×C32—22no—
190C23×C32—216no—
191C23×C32—24no—
192C23×C32—28no—
193C23×C32—24no—
194C23×C32—31no—
195C23×C32C22×D314yes—
196C23×C32C22×D318yes—
197C23×C32C22×D311yescharacteristic
198C23×C32C22×D312yes—
199C22×C3⋊D3C22×D311yescharacteristic
200C22×C3×D3—34no—
201C22×C3×D3C22×D314yes—
202C32×D4—62no—
203C32×D4—64no—
204C2×C32×C4—32no—
205C2×C32×C4—31no—
206#638—22no—
207#638—24no—
208#638—62no—
209C2×C3⋊Dic3—34no—
210C2×C3⋊Dic3—31no—
211C2×C3⋊Dic3—31no—
212C2×C3⋊Dic3—32no—
213C2×C3⋊Dic3—316no—
214C2×C3⋊Dic3—34no—
215C2×C3⋊Dic3—34no—
216C2×C3⋊Dic3—38no—
217C2×C3⋊Dic3—616no—
218C2×C3⋊Dic3—64no—
219C2×C3⋊Dic3—64no—
220C2×C3⋊Dic3—68no—
221C2×C3⋊Dic3—62no—
222C2×C3⋊Dic3—64no—
223C2×C3⋊Dic3—64no—
224C2×C3⋊Dic3—616no—
225C2×C3⋊Dic3—616no—
226C2×C3⋊Dic3—632no—
227C2×C3⋊Dic3—632no—
228C2×C3⋊Dic3—616no—
229C2×C3⋊Dic3—68no—
230C2×C3⋊Dic3—62no—
231C2×C3⋊Dic3C22×D312yes—
232C2×C3⋊Dic3C22×D311yescharacteristic
233C4×C3⋊D3—62no—
234#634—62no—
235C3×#67—28no—
236C3×#67—216no—
237C3×#67—68no—
238C3×#67—616no—
239C2×C3×Dic3—38no—
240C2×C3×Dic3—34no—
241C2×C3×Dic3C22×D318yes—
242C2×C3×Dic3C22×D314yes—
243#627—68no—
244#625—68no—
245D3×Dic3—68no—
246D3×Dic3—68no—
247#525—271noSylow
248C2×C33—22no—
249C2×C33—24no—
250C2×C33C22×D411yescharacteristic
251C2×C33C2×#4112yes—
252C3×C3⋊D3—41no—
253C32×D3—41no—
254C24×C3—24no—
255C24×C3D3214yes—
256C24×C3C2×C3⋊D311yescharacteristic
257C2×C3×D4—38no—
258C2×C3×D4—34no—
259C2×C3×D4—92no—
260C2×C3×D4—91no—
261C2×#67—38no—
262C2×#67—34no—
263C2×#67—94no—
264C2×#67—91no—
265C2×#67C2×C3⋊D312yes—
266C2×#67C2×C3⋊D311yescharacteristic
267C22×Dic3—94no—
268C22×Dic3—94no—
269C22×Dic3—91no—
270C22×Dic3—98no—
271C22×Dic3—98no—
272C22×Dic3—94no—
273C22×Dic3—92no—
274C22×Dic3—92no—
275C22×Dic3—91no—
276C22×Dic3—91no—
277C22×Dic3—184no—
278C22×Dic3—188no—
279C22×Dic3—188no—
280C22×Dic3—184no—
281C22×Dic3—1816no—
282C22×Dic3—184no—
283C22×Dic3—1832no—
284#237—184no—
285#237—1816no—
286C2×C4×D3—94no—
287C2×C4×D3—91no—
288C2×Dic6—94no—
289C2×Dic6—91no—
290C3×C22⋊C4—34no—
291C3×C22⋊C4—38no—
292C3×C22⋊C4—91no—
293C3×C22⋊C4—92no—
294#217—34no—
295#217—38no—
296#217—98no—
297#217—92no—
298#217C2×C3⋊D311yescharacteristic
299#217C2×C3⋊D312yes—
300(C2×D3)⋊C4—98no—
301(C2×D3)⋊C4—92no—
302C4⋊Dic3—98no—
303C4⋊Dic3—92no—
304Dic3⋊C4—98no—
305Dic3⋊C4—98no—
306Dic3⋊C4—92no—
307Dic3⋊C4—92no—
308C4×Dic3—98no—
309C4×Dic3—92no—
310C22×C32—216no—
311C22×C32—264no—
312C22×C32—232no—
313C22×C32—28no—
314C22×C32—216no—
315C22×C32—216no—
316C22×C32—28no—
317C22×C32—22no—
318C22×C32—28no—
319C22×C32—28no—
320C22×C32—22no—
321C22×C32—22no—
322C22×C32—24no—
323C22×C32—216no—
324C22×C32—28no—
325C22×C32—61no—
326C22×C32—62no—
327C22×C32D3×D418yes—
328C22×C32D3×D412yes—
329C22×C32#237116yescomplemented
330C22×C32#23714yescomplemented
331C22×C32C23×D314yes—
332C22×C32C23×D311yescharacteristic
333C2×C3⋊D3—21no—
334C2×C3⋊D3—22no—
335C2×C3×D3—24no—
336C2×C3×D3—28no—
337C2×C3×D3—64no—
338C2×C3×D3—68no—
339C32×C4—61no—
340C32×C4—62no—
341C3⋊Dic3—21no—
342C3⋊Dic3—22no—
343C3⋊Dic3—32no—
344C3⋊Dic3—32no—
345C3⋊Dic3—38no—
346C3⋊Dic3—38no—
347C3⋊Dic3—68no—
348C3⋊Dic3—68no—
349C3⋊Dic3—616no—
350C3⋊Dic3—68no—
351C3⋊Dic3—62no—
352C3×Dic3—24no—
353C3×Dic3—28no—
354C3×Dic3—64no—
355C3×Dic3—68no—
356C2×#41—271no—
357C23×C4—271no—
358#123—271no—
359#123—271no—
360C22⋊Q8—272no—
361#121—272no—
362#120—31no—
363#120—91no—
364C4×D4—272no—
365C4×D4—272no—
366#117—271no—
367C33#52511yescharacteristic, complemented, Sylow
368C23×C3—216no—
369C23×C3—232no—
370C23×C3—216no—
371C23×C3—28no—
372C23×C3—22no—
373C23×C3—24no—
374C23×C3—28no—
375C23×C3—24no—
376C23×C3—34no—
377C23×C3—91no—
378C23×C3C2×D3218yes—
379C23×C3C2×D3214yes—
380C23×C3C22×C3⋊D311yescharacteristic
381C23×C3C22×C3⋊D312yes—
382C22×D3—34no—
383C22×D3C22×C3⋊D311yescharacteristic
384C3×D4—616no—
385C3×D4—68no—
386C3×D4—182no—
387C3×D4—184no—
388C2×C3×C4—38no—
389C2×C3×C4—34no—
390C2×C3×C4—92no—
391C2×C3×C4—91no—
392#67—22no—
393#67—24no—
394#67—616no—
395#67—68no—
396#67—182no—
397#67—188no—
398C2×Dic3—38no—
399C2×Dic3—34no—
400C2×Dic3—94no—
401C2×Dic3—94no—
402C2×Dic3—91no—
403C2×Dic3—91no—
404C2×Dic3—916no—
405C2×Dic3—94no—
406C2×Dic3—98no—
407C2×Dic3—92no—
408C2×Dic3—184no—
409C2×Dic3—184no—
410C2×Dic3—1816no—
411C2×Dic3—188no—
412C2×Dic3—182no—
413C2×Dic3—188no—
414C2×Dic3—1816no—
415C2×Dic3—1816no—
416C2×Dic3—184no—
417C2×Dic3—184no—
418C2×Dic3—1832no—
419C2×Dic3—1832no—
420C2×Dic3—1816no—
421C2×Dic3—188no—
422C2×Dic3—182no—
423C2×Dic3C22×C3⋊D312yes—
424C2×Dic3C22×C3⋊D311yescharacteristic
425C4×D3—188no—
426C4×D3—182no—
427Dic6—182no—
428Dic6—188no—
429C2×C32—232no—
430C2×C32—216no—
431C2×C32—28no—
432C2×C32—22no—
433C2×C32—216no—
434C2×C32—24no—
435C2×C32—28no—
436C2×C32—24no—
437C2×C32—121no—
438C2×C32C2×D3×D414yes—
439C2×C32C2×D3×D411yescharacteristic
440C2×C32C2×#23718yes—
441C2×C32C2×#23712yes—
442C3⋊D3—41no—
443C3×D3—44no—
444C3×D3—124no—
445C24D3×C3⋊D311yescharacteristic
446#41—544no—
447C2×Q8—271no—
448C2×D4—32no—
449C2×D4—31no—
450C2×D4—92no—
451C2×D4—91no—
452C2×D4—271no—
453C22×C4—271no—
454C22×C4—271no—
455C22×C4—271no—
456C22×C4—272no—
457C22×C4—272no—
458C22×C4—271no—
459C22×C4—544no—
460C4⋊C4—272no—
461C4⋊C4—272no—
462C4⋊C4—272no—
463C22⋊C4—31no—
464C22⋊C4—32no—
465C22⋊C4—91no—
466C22⋊C4—92no—
467C22⋊C4—272no—
468C22⋊C4—272no—
469C42—272no—
470C22×C3—216no—
471C22×C3—264no—
472C22×C3—232no—
473C22×C3—28no—
474C22×C3—216no—
475C22×C3—216no—
476C22×C3—28no—
477C22×C3—28no—
478C22×C3—28no—
479C22×C3—22no—
480C22×C3—22no—
481C22×C3—22no—
482C22×C3—24no—
483C22×C3—216no—
484C22×C3—28no—
485C22×C3—68no—
486C22×C3—64no—
487C22×C3—181no—
488C22×C3—182no—
489C22×C3#3805116yescomplemented
490C22×C3#381118yes—
491C22×C3D4×C3⋊D312yes—
492C22×C3#383014yescomplemented
493C22×C3C22×D3214yes—
494C22×C3C23×C3⋊D311yescharacteristic
495C2×D3—21no—
496C2×D3—22no—
497C2×D3—68no—
498C2×D3—64no—
499C3×C4—68no—
500C3×C4—64no—
501C3×C4—181no—
502C3×C4—182no—
503Dic3—21no—
504Dic3—22no—
505Dic3—68no—
506Dic3—64no—
507Dic3—98no—
508Dic3—98no—
509Dic3—92no—
510Dic3—92no—
511Dic3—188no—
512Dic3—188no—
513Dic3—1816no—
514Dic3—188no—
515Dic3—182no—
516C32—24no—
517C32#558614yescomplemented
518C32#558611yescharacteristic, complemented
519C23—24no—
520C23—22no—
521C23—31no—
522C23—91no—
523C23C2×D3×C3⋊D312yes—
524C23C2×D3×C3⋊D311yescharacteristic
525Q8—542no—
526D4—64no—
527D4—62no—
528D4—184no—
529D4—182no—
530D4—542no—
531C2×C4—32no—
532C2×C4—31no—
533C2×C4—92no—
534C2×C4—91no—
535C2×C4—271no—
536C2×C4—271no—
537C2×C4—274no—
538C2×C4—272no—
539C2×C4—542no—
540C2×C4—542no—
541C2×C4—544no—
542C2×C4—544no—
543C2×C4—542no—
544C2×C3—232no—
545C2×C3—216no—
546C2×C3—216no—
547C2×C3—28no—
548C2×C3—22no—
549C2×C3—24no—
550C2×C3—28no—
551C2×C3—24no—
552C2×C3—124no—
553C2×C3—361no—
554C2×C3C2×#380518yes—
555C2×C3C2×#381114yes—
556C2×C3C2×D4×C3⋊D311yescharacteristic
557C2×C3C2×#383012yes—
558D3—41no—
559D3—124no—
560C22—28no—
561C22—22no—
562C22—22no—
563C22—22no—
564C22—62no—
565C22—61no—
566C22—182no—
567C22—181no—
568C22#8902914yescomplemented
569C22#8903912yes—
570C22C22×D3×C3⋊D311yescharacteristic
571C4—62no—
572C4—61no—
573C4—182no—
574C4—181no—
575C4—272no—
576C4—272no—
577C4—542no—
578C3—24no—
579C3#1059550114yescomplemented
580C3#1059605711yescharacteristic, complemented
581C2—24no—
582C2—22no—
583C2—121no—
584C2—361no—
585C2C2×#8902912yes—
586C2C2×#8903911yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×C3⋊D3
commutator, socle, cosocleC22×C33C24
FittingC24×C33C22
hypercenterC24D3×C3⋊D3
hypocenterC33#525
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22 ◁ C24

Complements

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

NH ≅ G/Ntype
1. C23×C3⋊(C3⋊Dic3)C2semidirect
2. C2×#89029C2semidirect
3. C3×#64162C2semidirect
4. C32×#953C2semidirect
5. #11723642C2semidirect
6. #11723637C2semidirect
7. #11723626C2semidirect
8. #67⋊(C3⋊Dic3)C2semidirect
9. #11723609C2semidirect
10. #11723603C2semidirect
11. #11723598C2semidirect
24. C2×#6293C22semidirect
25. C2×#6290C22semidirect
26. C2×Dic3×C3⋊D3C22semidirect
28. C2×D3×C3⋊Dic3C22semidirect
33. #88807C22semidirect
34. #88806C22semidirect
35. Dic3⋊(C3⋊Dic3)C22semidirect
36. #88803C22semidirect
37. (C2×D3)⋊(C3⋊Dic3)C22semidirect
38. Dic3×C3⋊Dic3C22semidirect
45. #64162D3semidirect
54. C3×#953D3semidirect
71. C2×C3⋊(C3⋊Dic3)D4semidirect
150. C22×C33#41semidirect
171. #953C3⋊D3semidirect
329. C22×C32#237semidirect
330. C22×C32#237semidirect
367. C33#525semidirect
489. C22×C3#3805semidirect
492. C22×C3#3830semidirect
517. C32#5586semidirect
518. C32#5586semidirect
568. C22#89029semidirect
579. C3#10595501semidirect
580. C3#10596057semidirect

Maximal subgroups

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

subgroupindex
C23×C3⋊(C3⋊Dic3)2
C2×#890292
C3×#641622
C32×#9532
#117236422
#117236372
#117236262
#67⋊(C3⋊Dic3)2
#117236092
#117236032
#117235982
#105960573
#105955013