Finite Group Databasebeta

← #13178276

Subgroups — #13178276

Order 1728.

1146 conjugacy classes of subgroups.

The subgroup lattice has 560 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#64087C212yesmaximal
2C3×#64086C211yesmaximal, characteristic
3#11723721C212yesmaximal
4#11723715C211yesmaximal, characteristic
5C2×#88813C211yesmaximal, characteristic
6#11723669C212yesmaximal, complemented
7#11723668C212yesmaximal, complemented
8#11723666C212yesmaximal, complemented
9#11723665C212yesmaximal, complemented
10#10594030—31nomaximal
11#10594029—32nomaximal
12C22×C3×C3⋊Dic3C2212yes—
13C22×C3×C3⋊Dic3C2211yescharacteristic
14C2×C3×#634C2212yes—
15C2×#6296C2212yes—
16C33×C22⋊C4C2211yescharacteristic
17C3×#3757C2211yescharacteristic
18C3×#3757C2212yes—
19C3×C4⋊(C3⋊Dic3)C2212yes—
20C3×#3750C2212yes—
21C3×#3750C2212yes—
22C3×#3750C2212yes—
23C3×C4×C3⋊Dic3C2212yes—
24#88815C2212yes—
25#88815C2212yes—
26#88815C2212yes—
27#88815C2212yes—
28#88813C2211yescharacteristic
29#88813C2212yes—
30#88813C2211yescharacteristic
31#88813C2211yescharacteristic
32#88813C2211yescharacteristic
33#64087D312yes—
34#64086D311yescharacteristic
35C3×#878—32no—
36C3×#878—32no—
37C3×#878—62no—
38C3×#877—32no—
39C3×#877—61no—
40#63982—32no—
41#63964—32no—
42#63964—32no—
43#63957—31no—
44#63955—32no—
45C2×Dic32—32no—
46C2×Dic32—31no—
47#63847—32no—
48#63846—32no—
49#63845—32no—
50#63844—32no—
51#63843—32no—
52Dic3×Dic6—32no—
53#63840—32no—
54#63839—32no—
55#63839—32no—
56#63838—32no—
57#63837—32no—
58#63837—32no—
59C23×C33C2311yescharacteristic
60C2×C33×C4C2312yes—
61C2×C3×C3⋊Dic3—24no—
62C2×C3×C3⋊Dic3Q814yescomplemented
63C2×C3×C3⋊Dic3C2312yes—
64C2×C3×C3⋊Dic3C2312yes—
65C2×C3×C3⋊Dic3C2312yes—
66C2×C3×C3⋊Dic3C2312yes—
67C2×C3×C3⋊Dic3C2311yescharacteristic
68C2×C3×C3⋊Dic3C2311yescharacteristic
69C2×C3×C3⋊Dic3C2312yes—
70C3×#634—24no—
71#6296—24no—
72#5622—92no—
73#5517—91no—
74C22×C3⋊Dic3C2×D312yes—
75C22×C3⋊Dic3C2×D311yescharacteristic
76C2×#634C2×D312yes—
77C22×C3×Dic3—32no—
78C22×C3×Dic3—32no—
79C22×C3×Dic3—32no—
80C22×C3×Dic3—61no—
81C22×C3×Dic3—62no—
82C2×C3×Dic6—32no—
83C2×C3×Dic6—32no—
84C2×C3×Dic6—62no—
85C2×#627—32no—
86C2×#627—32no—
87C2×#627—32no—
88C32×C22⋊C4—21no—
89C32×C22⋊C4—22no—
90C32×C22⋊C4—41no—
91C32×C22⋊C4C2×D312yes—
92C32×C22⋊C4C2×D311yescharacteristic
93#3757C2×D311yescharacteristic
94#3757C2×D312yes—
95C4⋊(C3⋊Dic3)C2×D312yes—
96#3750C2×D312yes—
97#3750C2×D312yes—
98#3750C2×D312yes—
99C4×C3⋊Dic3C2×D312yes—
100C3×#217—32no—
101C3×#217—32no—
102C3×#217—32no—
103C3×#217—61no—
104C3×#217—62no—
105C3×C4⋊Dic3—32no—
106C3×C4⋊Dic3—32no—
107C3×C4⋊Dic3—62no—
108C3×Dic3⋊C4—32no—
109C3×Dic3⋊C4—32no—
110C3×Dic3⋊C4—32no—
111C3×Dic3⋊C4—32no—
112C3×Dic3⋊C4—32no—
113C3×Dic3⋊C4—32no—
114C3×Dic3⋊C4—62no—
115C3×Dic3⋊C4—62no—
116C3×Dic3⋊C4—62no—
117C3×C4×Dic3—32no—
118C3×C4×Dic3—32no—
119C3×C4×Dic3—62no—
120#3724—32no—
121#3724—32no—
122#3724—32no—
123#3724—32no—
124Dic3⋊Dic3—32no—
125Dic3⋊Dic3—32no—
126Dic3⋊Dic3—32no—
127Dic3⋊Dic3—32no—
128Dic3⋊Dic3—32no—
129Dic3⋊Dic3—32no—
130Dic3⋊Dic3—32no—
131Dic3⋊Dic3—32no—
132Dic32—32no—
133Dic32—32no—
134Dic32—32no—
135Dic32—31no—
136Dic32—32no—
137Dic32—31no—
138Dic32—32no—
139Dic32—32no—
140Dic32—31no—
141Dic32—32no—
142Dic32—31no—
143C22×C33—22no—
144C22×C33C2×Q812yes—
145C22×C33C2411yescharacteristic
146C33×C4—22no—
147C3×C3⋊Dic3—22no—
148C3×C3⋊Dic3—22no—
149C3×C3⋊Dic3—22no—
150C3×C3⋊Dic3—22no—
151C3×C3⋊Dic3C2×Q812yes—
152C3×C3⋊Dic3C2×Q812yes—
153C3×C3⋊Dic3#4114yes—
154C3×C22⋊Q8—92no—
155C3×#117—91no—
156Q8×Dic3—92no—
157#944—92no—
158#926—91no—
159#924—92no—
160C2×C4×Dic3—91no—
161C2×C4×Dic3—92no—
162#890—92no—
163#889—92no—
164#889—92no—
165#888—92no—
166#888—92no—
167#887—92no—
168#887—92no—
169#887—92no—
170#878—32no—
171#878—32no—
172#878—62no—
173#878—92no—
174#878—92no—
175#877—32no—
176#877—61no—
177#877—92no—
178C4×Dic6—92no—
179C4×Dic6—92no—
180C23×C32—22no—
181C23×C32—21no—
182C23×C32—41no—
183C23×C32C22×D311yescharacteristic
184C23×C32C22×D312yes—
185C2×C32×C4—22no—
186C2×C32×C4—22no—
187C2×C32×C4—22no—
188C2×C32×C4—42no—
189C2×C32×C4C22×D312yes—
190C2×C32×C4C22×D312yes—
191C2×C32×C4C22×D312yes—
192C2×C3⋊Dic3—24no—
193C2×C3⋊Dic3Dic614yescomplemented
194C2×C3⋊Dic3C22×D312yes—
195C2×C3⋊Dic3C22×D312yes—
196C2×C3⋊Dic3C22×D312yes—
197C2×C3⋊Dic3C22×D312yes—
198C2×C3⋊Dic3C22×D311yescharacteristic
199C2×C3⋊Dic3C22×D311yescharacteristic
200C2×C3⋊Dic3C22×D312yes—
201#634—24no—
202C2×C3×Dic3—38no—
203C2×C3×Dic3—32no—
204C2×C3×Dic3—32no—
205C2×C3×Dic3—32no—
206C2×C3×Dic3—32no—
207C2×C3×Dic3—32no—
208C2×C3×Dic3—32no—
209C2×C3×Dic3—32no—
210C2×C3×Dic3—32no—
211C2×C3×Dic3—32no—
212C2×C3×Dic3—32no—
213C2×C3×Dic3—32no—
214C2×C3×Dic3—32no—
215C2×C3×Dic3—61no—
216C2×C3×Dic3—61no—
217C2×C3×Dic3—62no—
218C2×C3×Dic3—64no—
219C2×C3×Dic3—62no—
220C2×C3×Dic3—62no—
221C2×C3×Dic3—62no—
222C2×C3×Dic3—62no—
223C2×C3×Dic3—64no—
224C2×C3×Dic3—64no—
225C2×C3×Dic3—124no—
226C3×Dic6—64no—
227C3×Dic6—64no—
228C3×Dic6—124no—
229#627—64no—
230#627—64no—
231#627—64no—
232#533—271noSylow
233C2×C33—22no—
234C2×C33C22×Q811yescharacteristic
235C2×C33C2×#4112yes—
236C2×C3×Q8—92no—
237C22×C3×C4—91no—
238C22×C3×C4—92no—
239C22×Dic3—32no—
240C22×Dic3—32no—
241C22×Dic3—32no—
242C22×Dic3—61no—
243C22×Dic3—62no—
244C2×Dic6—32no—
245C2×Dic6—32no—
246C2×Dic6—62no—
247C2×Dic6—92no—
248C2×Dic6—92no—
249C2×Dic6—92no—
250C3×C4⋊C4—92no—
251C3×C4⋊C4—92no—
252C3×C4⋊C4—92no—
253C3×C4⋊C4—92no—
254C3×C22⋊C4—21no—
255C3×C22⋊C4—22no—
256C3×C22⋊C4—41no—
257C3×C22⋊C4—92no—
258C3×C22⋊C4—91no—
259C3×C22⋊C4D3212yes—
260C3×C22⋊C4D3211yescharacteristic
261C3×C42—92no—
262#217—32no—
263#217—32no—
264#217—32no—
265#217—61no—
266#217—62no—
267C4⋊Dic3—32no—
268C4⋊Dic3—32no—
269C4⋊Dic3—62no—
270C4⋊Dic3—92no—
271C4⋊Dic3—92no—
272C4⋊Dic3—92no—
273C4⋊Dic3—92no—
274Dic3⋊C4—32no—
275Dic3⋊C4—32no—
276Dic3⋊C4—32no—
277Dic3⋊C4—32no—
278Dic3⋊C4—32no—
279Dic3⋊C4—32no—
280Dic3⋊C4—62no—
281Dic3⋊C4—62no—
282Dic3⋊C4—62no—
283Dic3⋊C4—92no—
284Dic3⋊C4—92no—
285Dic3⋊C4—92no—
286Dic3⋊C4—92no—
287Dic3⋊C4—92no—
288Dic3⋊C4—92no—
289Dic3⋊C4—92no—
290Dic3⋊C4—92no—
291C4×Dic3—32no—
292C4×Dic3—32no—
293C4×Dic3—62no—
294C4×Dic3—92no—
295C4×Dic3—91no—
296C4×Dic3—91no—
297C4×Dic3—92no—
298C4×Dic3—92no—
299C4×Dic3—92no—
300C4×Dic3—92no—
301C4×Dic3—91no—
302C4×Dic3—92no—
303C4×Dic3—92no—
304C4×Dic3—91no—
305C22×C32—22no—
306C22×C32—22no—
307C22×C32—22no—
308C22×C32—22no—
309C22×C32—21no—
310C22×C32—24no—
311C22×C32—22no—
312C22×C32—41no—
313C22×C32—42no—
314C22×C32—42no—
315C22×C32—42no—
316C22×C32—42no—
317C22×C32—82no—
318C22×C32C2×Dic612yes—
319C22×C32D3×Q814yes—
320C22×C32C23×D311yescharacteristic
321C22×C32C23×D312yes—
322C32×C4—22no—
323C32×C4—22no—
324C32×C4—22no—
325C32×C4—42no—
326C32×C4—42no—
327C32×C4—42no—
328C32×C4—82no—
329C3⋊Dic3—22no—
330C3⋊Dic3—22no—
331C3⋊Dic3—22no—
332C3⋊Dic3—22no—
333C3⋊Dic3C2×Dic612yes—
334C3⋊Dic3C2×Dic612yes—
335C3⋊Dic3#23714yes—
336C3×Dic3—34no—
337C3×Dic3—34no—
338C3×Dic3—34no—
339C3×Dic3—34no—
340C3×Dic3—62no—
341C3×Dic3—62no—
342C3×Dic3—64no—
343C3×Dic3—62no—
344C3×Dic3—62no—
345C3×Dic3—62no—
346C3×Dic3—62no—
347C3×Dic3—62no—
348C3×Dic3—62no—
349C3×Dic3—62no—
350C3×Dic3—62no—
351C3×Dic3—122no—
352C3×Dic3—122no—
353C3×Dic3—122no—
354C3×Dic3—122no—
355C4⋊Q8—272no—
356#125—272no—
357C22⋊Q8—92no—
358C22⋊Q8—272no—
359C4×Q8—272no—
360C4×Q8—272no—
361#117—91no—
362#117—271no—
363C2×C42—271no—
364C33#53311yescharacteristic, complemented, Sylow
365C23×C3—21no—
366C23×C3—22no—
367C23×C3—41no—
368C23×C3C2×D3212yes—
369C23×C3C2×D3211yescharacteristic
370C3×Q8—184no—
371C2×C3×C4—22no—
372C2×C3×C4—22no—
373C2×C3×C4—22no—
374C2×C3×C4—42no—
375C2×C3×C4—91no—
376C2×C3×C4—91no—
377C2×C3×C4—92no—
378C2×C3×C4—94no—
379C2×C3×C4—92no—
380C2×C3×C4—92no—
381C2×C3×C4—92no—
382C2×C3×C4—92no—
383C2×C3×C4—184no—
384C2×C3×C4C2×D3212yes—
385C2×C3×C4C2×D3212yes—
386C2×C3×C4C2×D3212yes—
387C2×Dic3—32no—
388C2×Dic3—32no—
389C2×Dic3—32no—
390C2×Dic3—32no—
391C2×Dic3—38no—
392C2×Dic3—32no—
393C2×Dic3—32no—
394C2×Dic3—32no—
395C2×Dic3—32no—
396C2×Dic3—32no—
397C2×Dic3—32no—
398C2×Dic3—32no—
399C2×Dic3—32no—
400C2×Dic3—64no—
401C2×Dic3—61no—
402C2×Dic3—61no—
403C2×Dic3—62no—
404C2×Dic3—62no—
405C2×Dic3—62no—
406C2×Dic3—62no—
407C2×Dic3—62no—
408C2×Dic3—64no—
409C2×Dic3—64no—
410C2×Dic3—124no—
411Dic6—64no—
412Dic6—64no—
413Dic6—124no—
414Dic6—184no—
415Dic6—184no—
416Dic6—184no—
417C2×C32—22no—
418C2×C32—24no—
419C2×C32—22no—
420C2×C32—21no—
421C2×C32—22no—
422C2×C32—22no—
423C2×C32—22no—
424C2×C32—41no—
425C2×C32—42no—
426C2×C32—44no—
427C2×C32—42no—
428C2×C32—82no—
429C2×C32C22×Dic611yescharacteristic
430C2×C32C2×#23712yes—
431C2×C32C2×D3×Q812yes—
432C2×C32D3×#4112yes—
433C2×C32D3×#4112yes—
434C2×Q8—92no—
435C2×Q8—272no—
436C22×C4—91no—
437C22×C4—92no—
438C4⋊C4—92no—
439C4⋊C4—92no—
440C4⋊C4—92no—
441C4⋊C4—92no—
442C4⋊C4—272no—
443C4⋊C4—272no—
444C4⋊C4—272no—
445C4⋊C4—272no—
446C22⋊C4—91no—
447C22⋊C4—92no—
448C22⋊C4#111311yescharacteristic
449C42—92no—
450C42—272no—
451C42—271no—
452C42—271no—
453C42—271no—
454C42—271no—
455C22×C3—22no—
456C22×C3—22no—
457C22×C3—22no—
458C22×C3—21no—
459C22×C3—22no—
460C22×C3—22no—
461C22×C3—24no—
462C22×C3—42no—
463C22×C3—42no—
464C22×C3—42no—
465C22×C3—41no—
466C22×C3—42no—
467C22×C3—82no—
468C22×C3D3×Dic614yes—
469C22×C3#379712yes—
470C22×C3C22×D3212yes—
471C22×C3C22×D3211yescharacteristic
472C3×C4—22no—
473C3×C4—22no—
474C3×C4—22no—
475C3×C4—42no—
476C3×C4—42no—
477C3×C4—42no—
478C3×C4—82no—
479C3×C4—92no—
480C3×C4—92no—
481C3×C4—94no—
482C3×C4—182no—
483C3×C4—182no—
484C3×C4—182no—
485C3×C4—182no—
486Dic3—34no—
487Dic3—34no—
488Dic3—34no—
489Dic3—34no—
490Dic3—64no—
491Dic3—62no—
492Dic3—62no—
493Dic3—62no—
494Dic3—62no—
495Dic3—62no—
496Dic3—62no—
497Dic3—62no—
498Dic3—62no—
499Dic3—62no—
500Dic3—62no—
501Dic3—122no—
502Dic3—122no—
503Dic3—122no—
504Dic3—122no—
505C32—22no—
506C32—21no—
507C32—41no—
508C32#551711yescharacteristic, complemented
509C32#562212yescomplemented
510C23C2×#111311yescharacteristic
511Q8—184no—
512Q8—544no—
513C2×C4—91no—
514C2×C4—91no—
515C2×C4—92no—
516C2×C4—94no—
517C2×C4—92no—
518C2×C4—92no—
519C2×C4—92no—
520C2×C4—92no—
521C2×C4—184no—
522C2×C4C2×#111312yes—
523C2×C3—24no—
524C2×C3—22no—
525C2×C3—21no—
526C2×C3—22no—
527C2×C3—22no—
528C2×C3—22no—
529C2×C3—22no—
530C2×C3—42no—
531C2×C3—44no—
532C2×C3—41no—
533C2×C3—42no—
534C2×C3—82no—
535C2×C3C2×D3×Dic612yes—
536C2×C3C2×#379711yescharacteristic
537C2×C3D3×#23712yes—
538C2×C3D3×#23712yes—
539C2×C3#6431212yes—
540C22—22no—
541C22#8904012yes—
542C22C22×#111311yescharacteristic
543C4—22no—
544C4—92no—
545C4—92no—
546C4—94no—
547C4—182no—
548C4—182no—
549C4—182no—
550C4—182no—
551C3—21no—
552C3—22no—
553C3—41no—
554C3#1059402912yescomplemented
555C3#1059403011yescharacteristic, complemented
556C2—22no—
557C2C2×#8904011yescharacteristic
558C2#1172462212yes—
subgroupG/H
center, FrattiniC22C22×#1113
commutator, socle, cosocleC22×C33C24
FittingC33×C22⋊C4C22
prime radicalC23×C33C23
hypercenterC22⋊C4#1113
hypocenterC33#533
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22 ◁ C22⋊C4

Complements

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

NH ≅ G/Ntype
6. #11723669C2semidirect
7. #11723668C2semidirect
8. #11723666C2semidirect
9. #11723665C2semidirect
62. C2×C3×C3⋊Dic3Q8semidirect
193. C2×C3⋊Dic3Dic6semidirect
364. C33#533semidirect
508. C32#5517semidirect
509. C32#5622semidirect
554. C3#10594029semidirect
555. C3#10594030semidirect

Maximal subgroups

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

subgroupindex
C3×#640872
C3×#640862
#117237212
#117237152
C2×#888132
#117236692
#117236682
#117236662
#117236652
#105940303
#105940293