Finite Group Databasebeta

← #13178742

Subgroups — #13178742

Order 1728.

1166 conjugacy classes of subgroups.

The subgroup lattice has 592 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#64155C212yesmaximal, complemented
2C3×#64153C211yesmaximal, characteristic, complemented
3#11723692C212yesmaximal, complemented
4#11723688C212yesmaximal, complemented
5#11723685C211yesmaximal, characteristic
6#11723684C212yesmaximal
7#11723682C212yesmaximal, complemented
8#11723669C212yesmaximal, complemented
9#11723665C211yesmaximal, characteristic, complemented
10#10595372—32nomaximal
11#10595370—31nomaximal
12C2×C33×Q8C2211yescharacteristic
13C2×C3×C4×C3⋊D3C2212yes—
14C2×C3×#634C2211yescharacteristic
15C2×#6296C2211yescharacteristic
16C2×#6295C2211yescharacteristic
17C2×#6294C2212yes—
18C2×#6294C2212yes—
19C3×#3752C2212yes—
20C3×#3752C2212yes—
21C3×C4⋊(C3⋊Dic3)C2212yes—
22C3×#3750C2212yes—
23C3×#3750C2212yes—
24C3×#3750C2212yes—
25C3×#3750C2212yes—
26C3×C4×C3⋊Dic3C2211yescharacteristic
27#88815C2211yescharacteristic
28#88815C2211yescharacteristic
29#88815C2212yes—
30#88815C2212yes—
31#88814C2212yes—
32#88813C2212yes—
33#64155D312yescomplemented
34#64153D311yescharacteristic, complemented
35C3×#946—32no—
36C3×#946—32no—
37C3×#946—62no—
38C3×#944—32no—
39C3×#944—61no—
40#63904—32no—
41#63902—31no—
42#63899—32no—
43#63896—32no—
44#63895—32no—
45#63893—32no—
46#63888—32no—
47#63887—32no—
48#63883—32no—
49#63881—32no—
50#63880—32no—
51#63879—32no—
52D3×Dic3⋊C4—32no—
53D3×Dic3⋊C4—32no—
54#63847—32no—
55#63846—32no—
56#63845—32no—
57#63838—31no—
58#63837—32no—
59C22×C3×C3⋊D3C2312yes—
60C33×Q8—41no—
61C2×C33×C4C2311yescharacteristic
62C2×C33×C4C2312yes—
63C2×C3×C3⋊Dic3C2312yes—
64C2×C3×C3⋊Dic3C2312yes—
65C2×C3×C3⋊Dic3C2312yes—
66C2×C3×C3⋊Dic3C2312yes—
67C2×C3×C3⋊Dic3C2311yescharacteristic
68C2×C3×C3⋊Dic3C2311yescharacteristic
69C3×C4×C3⋊D3—42no—
70C3×#634—41no—
71#6296—22no—
72#6295Q814yescomplemented
73#6294—24no—
74#6294—42no—
75#5730—91no—
76#5636—92no—
77C2×C32×Q8—21no—
78C2×C32×Q8—22no—
79C2×C32×Q8—41no—
80C2×C32×Q8C2×D312yes—
81C2×C32×Q8C2×D311yescharacteristic
82C2×C4×C3⋊D3C2×D312yes—
83C2×#634C2×D311yescharacteristic
84C2×C3×C4×D3—32no—
85C2×C3×C4×D3—32no—
86C2×C3×C4×D3—62no—
87C2×C3×Dic6—32no—
88C2×C3×Dic6—61no—
89C2×#627—32no—
90C2×#627—31no—
91C2×#626—32no—
92C2×#625—31no—
93C2×#624—32no—
94C2×#624—32no—
95C2×D3×Dic3—32no—
96C2×D3×Dic3—32no—
97C2×D3×Dic3—32no—
98C2×D3×Dic3—32no—
99#3752C2×D312yes—
100#3752C2×D312yes—
101C4⋊(C3⋊Dic3)C2×D312yes—
102#3750C2×D312yes—
103#3750C2×D312yes—
104#3750C2×D312yes—
105#3750C2×D312yes—
106C4×C3⋊Dic3C2×D311yescharacteristic
107C3×(C2×D3)⋊C4—32no—
108C3×(C2×D3)⋊C4—32no—
109C3×(C2×D3)⋊C4—32no—
110C3×(C2×D3)⋊C4—32no—
111C3×(C2×D3)⋊C4—62no—
112C3×(C2×D3)⋊C4—62no—
113C3×C4⋊Dic3—32no—
114C3×C4⋊Dic3—32no—
115C3×C4⋊Dic3—62no—
116C3×Dic3⋊C4—32no—
117C3×Dic3⋊C4—32no—
118C3×Dic3⋊C4—32no—
119C3×Dic3⋊C4—32no—
120C3×Dic3⋊C4—32no—
121C3×Dic3⋊C4—32no—
122C3×Dic3⋊C4—32no—
123C3×Dic3⋊C4—32no—
124C3×Dic3⋊C4—62no—
125C3×Dic3⋊C4—62no—
126C3×Dic3⋊C4—62no—
127C3×Dic3⋊C4—62no—
128C3×C4×Dic3—32no—
129C3×C4×Dic3—61no—
130#3724—31no—
131#3724—32no—
132#3724—31no—
133#3724—32no—
134Dic3⋊Dic3—32no—
135Dic3⋊Dic3—32no—
136Dic3⋊Dic3—32no—
137Dic3⋊Dic3—32no—
138Dic3⋊Dic3—32no—
139Dic3⋊Dic3—32no—
140#3722—32no—
141(C2×D3)⋊Dic3—32no—
142(C2×D3)⋊Dic3—32no—
143Dic32—32no—
144Dic32—32no—
145Dic32—32no—
146C22×C33C2411yescharacteristic
147C2×C3×C3⋊D3—22no—
148C2×C3×C3⋊D3—22no—
149C2×C3×C3⋊D3C2×Q814yes—
150C33×C4—21no—
151C33×C4—22no—
152C3×C3⋊Dic3—22no—
153C3×C3⋊Dic3—22no—
154C3×C3⋊Dic3—22no—
155C3×C3⋊Dic3—22no—
156C3×C3⋊Dic3C2×Q812yes—
157C3×C3⋊Dic3#4112yes—
158C3×C4⋊Q8—91no—
159C3×C22⋊Q8—92no—
160#946—32no—
161#946—32no—
162#946—62no—
163#946—92no—
164#944—32no—
165#944—61no—
166C4×#67—92no—
167C2×Dic3⋊C4—92no—
168#897—92no—
169#896—92no—
170#896—92no—
171#893—92no—
172D3×C4⋊C4—92no—
173D3×C4⋊C4—92no—
174#890—92no—
175#889—92no—
176#889—92no—
177#887—92no—
178#67⋊C4—92no—
179#67⋊C4—92no—
180#878—92no—
181#878—92no—
182C4×D12—91no—
183C4×Dic6—91no—
184C22×C3⋊D3C22×D312yes—
185C22×C3×D3—32no—
186C22×C3×D3—32no—
187C22×C3×D3—62no—
188C32×Q8—41no—
189C32×Q8—42no—
190C32×Q8—42no—
191C32×Q8—44no—
192C32×Q8—44no—
193C2×C32×C4—21no—
194C2×C32×C4—22no—
195C2×C32×C4—22no—
196C2×C32×C4—22no—
197C2×C32×C4—22no—
198C2×C32×C4—41no—
199C2×C32×C4—42no—
200C2×C32×C4C22×D312yes—
201C2×C32×C4C22×D312yes—
202C2×C32×C4C22×D312yes—
203C2×C32×C4C22×D311yescharacteristic
204C2×C32×C4C22×D312yes—
205C2×C3⋊Dic3C22×D312yes—
206C2×C3⋊Dic3C22×D312yes—
207C2×C3⋊Dic3C22×D312yes—
208C2×C3⋊Dic3C22×D312yes—
209C2×C3⋊Dic3C22×D311yescharacteristic
210C2×C3⋊Dic3C22×D311yescharacteristic
211C4×C3⋊D3—42no—
212#634—41no—
213C2×C3×Dic3—32no—
214C2×C3×Dic3—32no—
215C2×C3×Dic3—32no—
216C2×C3×Dic3—32no—
217C2×C3×Dic3—32no—
218C2×C3×Dic3—32no—
219C2×C3×Dic3—32no—
220C2×C3×Dic3—32no—
221C2×C3×Dic3—32no—
222C2×C3×Dic3—32no—
223C2×C3×Dic3—61no—
224C2×C3×Dic3—61no—
225C2×C3×Dic3—62no—
226C2×C3×Dic3—62no—
227C2×C3×Dic3—62no—
228C2×C3×Dic3—62no—
229C3×C4×D3—122no—
230C3×C4×D3—122no—
231C3×C4×D3—124no—
232C3×Dic6—122no—
233C3×Dic6—122no—
234#627—64no—
235#627—62no—
236#626—38no—
237#625—34no—
238#624—64no—
239#624—122no—
240D3×Dic3—64no—
241D3×Dic3—64no—
242D3×Dic3—122no—
243D3×Dic3—122no—
244#554—271noSylow
245C2×C33C22×Q811yescharacteristic
246C2×C33C2×#4111yescharacteristic
247C2×C33H(5,𝔽2)11yescharacteristic
248C3×C3⋊D3—24no—
249C2×C3×Q8—21no—
250C2×C3×Q8—22no—
251C2×C3×Q8—41no—
252C2×C3×Q8—91no—
253C2×C3×Q8D3212yes—
254C2×C3×Q8D3211yescharacteristic
255C22×C3×C4—92no—
256C2×#67—92no—
257C22×Dic3—92no—
258C22×Dic3—92no—
259C2×D12—91no—
260C2×C4×D3—32no—
261C2×C4×D3—32no—
262C2×C4×D3—62no—
263C2×C4×D3—92no—
264C2×C4×D3—92no—
265C2×C4×D3—92no—
266C2×C4×D3—92no—
267C2×Dic6—32no—
268C2×Dic6—61no—
269C2×Dic6—91no—
270C2×Dic6—92no—
271C3×C4⋊C4—92no—
272C3×C4⋊C4—92no—
273C3×C4⋊C4—92no—
274C3×C4⋊C4—92no—
275C3×C4⋊C4—92no—
276C3×C22⋊C4—92no—
277C3×C22⋊C4—92no—
278C3×C42—91no—
279#217—92no—
280(C2×D3)⋊C4—32no—
281(C2×D3)⋊C4—32no—
282(C2×D3)⋊C4—32no—
283(C2×D3)⋊C4—32no—
284(C2×D3)⋊C4—62no—
285(C2×D3)⋊C4—62no—
286(C2×D3)⋊C4—92no—
287(C2×D3)⋊C4—92no—
288C4⋊Dic3—32no—
289C4⋊Dic3—32no—
290C4⋊Dic3—62no—
291C4⋊Dic3—91no—
292C4⋊Dic3—91no—
293C4⋊Dic3—92no—
294C4⋊Dic3—92no—
295Dic3⋊C4—32no—
296Dic3⋊C4—32no—
297Dic3⋊C4—32no—
298Dic3⋊C4—32no—
299Dic3⋊C4—32no—
300Dic3⋊C4—32no—
301Dic3⋊C4—32no—
302Dic3⋊C4—32no—
303Dic3⋊C4—62no—
304Dic3⋊C4—62no—
305Dic3⋊C4—62no—
306Dic3⋊C4—62no—
307Dic3⋊C4—92no—
308Dic3⋊C4—92no—
309Dic3⋊C4—92no—
310Dic3⋊C4—92no—
311Dic3⋊C4—92no—
312Dic3⋊C4—92no—
313C4×Dic3—32no—
314C4×Dic3—61no—
315C4×Dic3—92no—
316C4×Dic3—92no—
317C4×Dic3—92no—
318C22×C32—21no—
319C22×C32—22no—
320C22×C32—41no—
321C22×C32C23×D311yescharacteristic
322C22×C32C23×D312yes—
323C2×C3⋊D3—22no—
324C2×C3⋊D3—22no—
325C2×C3⋊D3D3×Q814yes—
326C2×C3×D3—34no—
327C2×C3×D3—34no—
328C2×C3×D3—64no—
329C2×C3×D3—62no—
330C2×C3×D3—62no—
331C2×C3×D3—62no—
332C2×C3×D3—62no—
333C2×C3×D3—122no—
334C2×C3×D3—122no—
335C32×C4—21no—
336C32×C4—22no—
337C32×C4—22no—
338C32×C4—24no—
339C32×C4—22no—
340C32×C4—22no—
341C32×C4—22no—
342C32×C4—42no—
343C32×C4—42no—
344C32×C4—42no—
345C32×C4—42no—
346C32×C4—44no—
347C3⋊Dic3—22no—
348C3⋊Dic3—22no—
349C3⋊Dic3—22no—
350C3⋊Dic3—22no—
351C3⋊Dic3D3×Q812yes—
352C3⋊Dic3#23912yes—
353C3×Dic3—34no—
354C3×Dic3—34no—
355C3×Dic3—62no—
356C3×Dic3—62no—
357C3×Dic3—62no—
358C3×Dic3—62no—
359C3×Dic3—62no—
360C3×Dic3—62no—
361C3×Dic3—62no—
362C3×Dic3—62no—
363C3×Dic3—62no—
364C3×Dic3—62no—
365C3×Dic3—122no—
366C3×Dic3—122no—
367C3×Dic3—122no—
368C3×Dic3—122no—
369C4⋊Q8—91no—
370#125—272no—
371C22⋊Q8—92no—
372C22⋊Q8—272no—
373C22⋊Q8—272no—
374C4×Q8—271no—
375C4×D4—271no—
376C4×D4—272no—
377C2×C4⋊C4—272no—
378C33#55411yescharacteristic, complemented, Sylow
379C23×C3—92no—
380C22×D3—32no—
381C22×D3—32no—
382C22×D3—62no—
383C3×Q8—42no—
384C3×Q8—41no—
385C3×Q8—42no—
386C3×Q8—44no—
387C3×Q8—44no—
388C3×Q8—361no—
389C2×C3×C4—21no—
390C2×C3×C4—22no—
391C2×C3×C4—22no—
392C2×C3×C4—22no—
393C2×C3×C4—22no—
394C2×C3×C4—41no—
395C2×C3×C4—42no—
396C2×C3×C4—91no—
397C2×C3×C4—91no—
398C2×C3×C4—92no—
399C2×C3×C4—92no—
400C2×C3×C4—92no—
401C2×C3×C4—92no—
402C2×C3×C4—362no—
403C2×C3×C4C2×D3212yes—
404C2×C3×C4C2×D3212yes—
405C2×C3×C4C2×D3212yes—
406C2×C3×C4C2×D3211yescharacteristic
407C2×C3×C4C2×D3212yes—
408#67—98no—
409C2×Dic3—32no—
410C2×Dic3—32no—
411C2×Dic3—32no—
412C2×Dic3—32no—
413C2×Dic3—32no—
414C2×Dic3—32no—
415C2×Dic3—32no—
416C2×Dic3—32no—
417C2×Dic3—32no—
418C2×Dic3—32no—
419C2×Dic3—61no—
420C2×Dic3—61no—
421C2×Dic3—62no—
422C2×Dic3—62no—
423C2×Dic3—62no—
424C2×Dic3—62no—
425C2×Dic3—184no—
426C2×Dic3—362no—
427D12—94no—
428C4×D3—122no—
429C4×D3—122no—
430C4×D3—124no—
431C4×D3—184no—
432C4×D3—184no—
433C4×D3—362no—
434C4×D3—362no—
435Dic6—122no—
436Dic6—122no—
437Dic6—184no—
438Dic6—182no—
439C2×C32—21no—
440C2×C32—22no—
441C2×C32—22no—
442C2×C32—21no—
443C2×C32—21no—
444C2×C32—22no—
445C2×C32—41no—
446C2×C32—41no—
447C2×C32—41no—
448C2×C32#100412yes—
449C2×C32#100411yescharacteristic
450C2×C32C2×D3×Q812yes—
451C2×C32C2×D3×Q811yescharacteristic
452C2×C32C2×#23911yescharacteristic
453C2×C32D3×#4112yes—
454C3⋊D3—24no—
455C3×D3—64no—
456C3×D3—64no—
457C3×D3—124no—
458C2×Q8—91no—
459C2×Q8—271no—
460C2×Q8#111311yescharacteristic
461C2×D4—271no—
462C22×C4—92no—
463C22×C4—272no—
464C22×C4—272no—
465C4⋊C4—92no—
466C4⋊C4—92no—
467C4⋊C4—92no—
468C4⋊C4—92no—
469C4⋊C4—92no—
470C4⋊C4—271no—
471C4⋊C4—272no—
472C4⋊C4—271no—
473C4⋊C4—272no—
474C22⋊C4—92no—
475C22⋊C4—92no—
476C22⋊C4—272no—
477C42—91no—
478C42—272no—
479C22×C3—21no—
480C22×C3—22no—
481C22×C3—41no—
482C22×C3—94no—
483C22×C3—182no—
484C22×C3—182no—
485C22×C3C22×D3212yes—
486C22×C3C22×D3211yescharacteristic
487C2×D3—34no—
488C2×D3—34no—
489C2×D3—64no—
490C2×D3—62no—
491C2×D3—62no—
492C2×D3—62no—
493C2×D3—62no—
494C2×D3—122no—
495C2×D3—122no—
496C3×C4—22no—
497C3×C4—24no—
498C3×C4—22no—
499C3×C4—22no—
500C3×C4—22no—
501C3×C4—22no—
502C3×C4—21no—
503C3×C4—42no—
504C3×C4—42no—
505C3×C4—42no—
506C3×C4—44no—
507C3×C4—42no—
508C3×C4—92no—
509C3×C4—92no—
510C3×C4—182no—
511C3×C4—182no—
512C3×C4—182no—
513C3×C4—182no—
514Dic3—34no—
515Dic3—34no—
516Dic3—62no—
517Dic3—62no—
518Dic3—62no—
519Dic3—62no—
520Dic3—62no—
521Dic3—62no—
522Dic3—62no—
523Dic3—62no—
524Dic3—62no—
525Dic3—62no—
526Dic3—122no—
527Dic3—122no—
528Dic3—122no—
529Dic3—122no—
530C32—21no—
531C32—22no—
532C32—41no—
533C32#563612yescomplemented
534C32#573011yescharacteristic, complemented
535C23—92no—
536Q8—41no—
537Q8—361no—
538Q8—542no—
539D4—274no—
540C2×C4—91no—
541C2×C4—91no—
542C2×C4—92no—
543C2×C4—92no—
544C2×C4—92no—
545C2×C4—92no—
546C2×C4—362no—
547C2×C4—544no—
548C2×C4—1082no—
549C2×C4C2×#111311yescharacteristic
550C2×C4C2×#111312yes—
551C2×C3—21no—
552C2×C3—22no—
553C2×C3—21no—
554C2×C3—22no—
555C2×C3—21no—
556C2×C3—22no—
557C2×C3—41no—
558C2×C3—41no—
559C2×C3—41no—
560C2×C3—184no—
561C2×C3D32×Q811yescharacteristic
562C2×C3D32×Q812yes—
563C2×C3D3×#23912yes—
564C2×C3#6431911yescharacteristic
565C2×C3#6433011yescharacteristic
566C2×C3#6433012yes—
567D3—64no—
568D3—64no—
569D3—124no—
570C22—94no—
571C22—182no—
572C22—182no—
573C22C22×#111311yescharacteristic
574C4—22no—
575C4—21no—
576C4—92no—
577C4—92no—
578C4—182no—
579C4—182no—
580C4—182no—
581C4—182no—
582C3—22no—
583C3—21no—
584C3—41no—
585C3#1059537011yescharacteristic, complemented
586C3#1059537212yescomplemented
587C2—184no—
588C2Q8×#111311yescharacteristic
589C2#1172462611yescharacteristic
590C2#1172463311yescharacteristic
subgroupG/H
center, FrattiniC22C22×#1113
commutator, socle, cosocleC22×C33C24
FittingC2×C33×Q8C22
prime radicalC2×#6295C22
hypercenterC2×Q8#1113
hypocenterC33#554
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×Q8

Complements

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

NH ≅ G/Ntype
1. C3×#64155C2semidirect
2. C3×#64153C2semidirect
3. #11723692C2semidirect
4. #11723688C2semidirect
7. #11723682C2semidirect
8. #11723669C2semidirect
9. #11723665C2semidirect
33. #64155D3semidirect
34. #64153D3semidirect
72. #6295Q8semidirect
378. C33#554semidirect
533. C32#5636semidirect
534. C32#5730semidirect
585. C3#10595370semidirect
586. C3#10595372semidirect

Maximal subgroups

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

subgroupindex
C3×#641552
C3×#641532
#117236922
#117236882
#117236852
#117236842
#117236822
#117236692
#117236652
#105953723
#105953703