Finite Group Databasebeta

← #13150665

Subgroups — #13150665

Order 1728.

836 conjugacy classes of subgroups.

The subgroup lattice has 583 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#6235C211yesmaximal, characteristic
2#64135⋊C3C211yesmaximal, characteristic, complemented
3#64133⋊C3C211yesmaximal, characteristic, complemented
4#11721297C212yesmaximal, complemented
5#11721294C212yesmaximal
6#11721117C211yesmaximal, characteristic, complemented
7#11721094C211yesmaximal, characteristic, complemented
8#11721092C212yesmaximal, complemented
9#11721089C211yesmaximal, characteristic, complemented
10#11721084C211yesmaximal, characteristic, complemented
11#11721077C212yesmaximal, complemented
12#10594841—31nomaximal
13#10594840—31nomaximal
14C22×C4×H(3,𝔽3)C2211yescharacteristic
15C2×#6235C2211yescharacteristic
16C2×C4×#272C2211yescharacteristic
17C2×#634⋊C3C2211yescharacteristic
18C2×#6196C2212yes—
19#88552C2211yescharacteristic
20#88542C2211yescharacteristic
21#88540C2211yescharacteristic
22C4×#1084C2211yescharacteristic
23#3757⋊C3C2212yes—
24#3757⋊C3C2212yes—
25C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
26C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
27C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
28#3750⋊C3C2212yes—
29C4×(C3⋊Dic3)⋊C3C2212yes—
30#88451C2212yes—
31#88451C2212yes—
32#88449C2212yes—
33#88448C2212yes—
34#88448C2212yes—
35#88447C2212yes—
36#88447C2212yes—
37#64135D311yescharacteristic, complemented
38#64133D311yescharacteristic, complemented
39C3×C4×#67—31no—
40C3×C4×#67—31no—
41C3×C4×#67—61no—
42C3×#926—31no—
43C3×#924—31no—
44#63959—32no—
45#63959—32no—
46#63956—32no—
47#63956—32no—
48#63918—31no—
49#63918—31no—
50#63862—32no—
51#63859—32no—
52#63857—31no—
53#63856—31no—
54#63844—31no—
55Dic3×Dic6—31no—
56#63841—31no—
57#63841—31no—
58#63838—32no—
59#63837—32no—
60C23×H(3,𝔽3)C2311yescharacteristic
61C22×#272C2311yescharacteristic
62C2×C4×H(3,𝔽3)—22no—
63C2×C4×H(3,𝔽3)C2311yescharacteristic
64C2×C4×H(3,𝔽3)C2311yescharacteristic
65#6235—22no—
66C2×#1084C2311yescharacteristic
67C2×#1084C2311yescharacteristic
68C2×#1084C2311yescharacteristic
69C4×#272—22no—
70C2×(C3⋊Dic3)⋊C3C2312yes—
71C2×(C3⋊Dic3)⋊C3C2312yes—
72C2×(C3⋊Dic3)⋊C3C2312yes—
73C2×(C3⋊Dic3)⋊C3C2312yes—
74#634⋊C3—22no—
75#6196—24no—
76#5641—91no—
77#5542—91no—
78#5533—91no—
79C22×C32×C4—21no—
80C22×C32×C4C2×D311yescharacteristic
81C22×C32×C4C2×D311yescharacteristic
82C2×#634C2×D311yescharacteristic
83C2×C3×#67—31no—
84C2×C3×#67—31no—
85C2×C3×#67—61no—
86C2×C3×C4×D3—31no—
87C2×C3×C4×D3—31no—
88C2×C3×C4×D3—61no—
89C2×C3×Dic6—31no—
90C2×#627—32no—
91C2×#627—32no—
92#3757C2×D312yes—
93#3757C2×D312yes—
94C4⋊(C3⋊Dic3)C2×D311yescharacteristic
95C4⋊(C3⋊Dic3)C2×D311yescharacteristic
96C4⋊(C3⋊Dic3)C2×D311yescharacteristic
97#3750C2×D312yes—
98C4×C3⋊Dic3C2×D312yes—
99C3×#217—32no—
100C3×#217—32no—
101C3×#217—31no—
102C3×#217—31no—
103C3×#217—61no—
104C3×(C2×D3)⋊C4—31no—
105C3×(C2×D3)⋊C4—31no—
106C3×(C2×D3)⋊C4—61no—
107C3×C4⋊Dic3—31no—
108C3×C4⋊Dic3—31no—
109C3×C4⋊Dic3—31no—
110C3×Dic3⋊C4—32no—
111C3×Dic3⋊C4—31no—
112C3×Dic3⋊C4—31no—
113C3×Dic3⋊C4—61no—
114C3×C4×Dic3—32no—
115C3×C4×Dic3—31no—
116C3×C4×Dic3—31no—
117C3×C4×Dic3—61no—
118#3724—32no—
119#3724—32no—
120Dic3⋊Dic3—32no—
121Dic3⋊Dic3—32no—
122Dic3⋊Dic3—32no—
123Dic3⋊Dic3—32no—
124(C2×D3)⋊Dic3—32no—
125(C2×D3)⋊Dic3—32no—
126(C2×D3)⋊Dic3—32no—
127(C2×D3)⋊Dic3—32no—
128Dic32—32no—
129Dic32—32no—
130Dic32—32no—
131Dic32—32no—
132C22×H(3,𝔽3)—21no—
133C22×H(3,𝔽3)—21no—
134C22×H(3,𝔽3)—21no—
135C22×H(3,𝔽3)C2411yescharacteristic
136C2×#272—21no—
137C2×#272—21no—
138C2×#272—21no—
139C4×H(3,𝔽3)—21no—
140C4×H(3,𝔽3)#4112yes—
141#1084—21no—
142#1084—21no—
143#1084#4112yes—
144(C3⋊Dic3)⋊C3—22no—
145(C3⋊Dic3)⋊C3—22no—
146(C3⋊Dic3)⋊C3—22no—
147(C3⋊Dic3)⋊C3—22no—
148C3×C22⋊Q8—91no—
149C3×C4×D4—91no—
150C3×#117—91no—
151#946—91no—
152Q8×Dic3—91no—
153#936—92no—
154C4×#67—31no—
155C4×#67—31no—
156C4×#67—61no—
157#926—31no—
158#926—91no—
159#926—91no—
160#924—31no—
161#924—91no—
162#924—91no—
163#898—92no—
164#897—91no—
165#892—91no—
166#892—91no—
167#888—91no—
168#888—91no—
169#887—92no—
170#887—91no—
171#887—92no—
172#885—92no—
173#885—92no—
174#879—92no—
175#879—92no—
176#879—92no—
177#879—92no—
178#876—92no—
179#869—91no—
180C4×Dic6—92no—
181C4×Dic6—91no—
182C23×C32—21no—
183C23×C32C22×D311yescharacteristic
184C23×C32C22×D311yescharacteristic
185C22×C3×D3—31no—
186C22×C3×D3—31no—
187C22×C3×D3—61no—
188C2×C32×C4—22no—
189C2×C32×C4—22no—
190C2×C32×C4—21no—
191C2×C32×C4—21no—
192C2×C32×C4—42no—
193C2×C32×C4C22×D311yescharacteristic
194C2×C32×C4C22×D311yescharacteristic
195C2×C32×C4C22×D311yescharacteristic
196C2×C32×C4C22×D311yescharacteristic
197C2×C3⋊Dic3C22×D312yes—
198C2×C3⋊Dic3C22×D312yes—
199C2×C3⋊Dic3C22×D312yes—
200C2×C3⋊Dic3C22×D312yes—
201#634—22no—
202C3×#67—62no—
203C3×#67—62no—
204C3×#67—64no—
205C2×C3×Dic3—31no—
206C2×C3×Dic3—31no—
207C2×C3×Dic3—32no—
208C2×C3×Dic3—32no—
209C2×C3×Dic3—32no—
210C2×C3×Dic3—32no—
211C2×C3×Dic3—31no—
212C2×C3×Dic3—31no—
213C2×C3×Dic3—31no—
214C2×C3×Dic3—31no—
215C2×C3×Dic3—61no—
216C2×C3×Dic3—61no—
217C2×C3×Dic3—61no—
218C3×C4×D3—62no—
219C3×C4×D3—62no—
220C3×C4×D3—122no—
221C3×Dic6—62no—
222#627—64no—
223#627—64no—
224#556—271noSylow
225C2×H(3,𝔽3)—41no—
226C2×H(3,𝔽3)C2×#4111yescharacteristic
227C2×H(3,𝔽3)C2×#4111yescharacteristic
228C2×H(3,𝔽3)2−1+411yescharacteristic
229#272—41no—
230C2×C3×Q8—91no—
231C2×C3×D4—91no—
232C22×C3×C4—31no—
233C22×C3×C4—31no—
234C22×C3×C4—61no—
235C22×C3×C4—91no—
236C22×C3×C4D3211yescharacteristic
237C2×#67—31no—
238C2×#67—31no—
239C2×#67—61no—
240C2×C4×D3—31no—
241C2×C4×D3—31no—
242C2×C4×D3—61no—
243C2×Dic6—31no—
244C2×Dic6—91no—
245C2×Dic6—92no—
246C2×Dic6—92no—
247C2×Dic6—92no—
248C3×C4⋊C4—91no—
249C3×C4⋊C4—92no—
250C3×C4⋊C4—91no—
251C3×C4⋊C4—91no—
252C3×C4⋊C4—91no—
253C3×C22⋊C4—92no—
254C3×C22⋊C4—92no—
255C3×C22⋊C4—91no—
256C3×C22⋊C4—91no—
257C3×C42—92no—
258C3×C42—91no—
259#217—31no—
260#217—31no—
261#217—32no—
262#217—32no—
263#217—61no—
264#217—92no—
265#217—92no—
266#217—92no—
267#217—92no—
268(C2×D3)⋊C4—31no—
269(C2×D3)⋊C4—31no—
270(C2×D3)⋊C4—61no—
271(C2×D3)⋊C4—92no—
272(C2×D3)⋊C4—92no—
273(C2×D3)⋊C4—92no—
274(C2×D3)⋊C4—92no—
275C4⋊Dic3—31no—
276C4⋊Dic3—31no—
277C4⋊Dic3—31no—
278C4⋊Dic3—91no—
279C4⋊Dic3—91no—
280C4⋊Dic3—91no—
281C4⋊Dic3—92no—
282C4⋊Dic3—92no—
283C4⋊Dic3—92no—
284Dic3⋊C4—31no—
285Dic3⋊C4—31no—
286Dic3⋊C4—32no—
287Dic3⋊C4—61no—
288Dic3⋊C4—92no—
289Dic3⋊C4—92no—
290Dic3⋊C4—92no—
291Dic3⋊C4—92no—
292Dic3⋊C4—92no—
293Dic3⋊C4—92no—
294Dic3⋊C4—92no—
295C4×Dic3—31no—
296C4×Dic3—31no—
297C4×Dic3—32no—
298C4×Dic3—61no—
299C4×Dic3—92no—
300C4×Dic3—92no—
301C4×Dic3—92no—
302C4×Dic3—92no—
303C4×Dic3—92no—
304C4×Dic3—92no—
305C4×Dic3—92no—
306C22×C32—22no—
307C22×C32—21no—
308C22×C32—21no—
309C22×C32—21no—
310C22×C32—21no—
311C22×C32—21no—
312C22×C32—21no—
313C22×C32—21no—
314C22×C32—41no—
315C22×C32—41no—
316C22×C32C23×D311yescharacteristic
317C22×C32C23×D311yescharacteristic
318C2×C3×D3—62no—
319C2×C3×D3—61no—
320C2×C3×D3—61no—
321C2×C3×D3—61no—
322C2×C3×D3—61no—
323C2×C3×D3—61no—
324C2×C3×D3—61no—
325C2×C3×D3—121no—
326C2×C3×D3—121no—
327C32×C4—21no—
328C32×C4—21no—
329C32×C4—22no—
330C32×C4—41no—
331C32×C4#23512yes—
332C32×C4#23712yes—
333C3⋊Dic3—22no—
334C3⋊Dic3—22no—
335C3⋊Dic3—22no—
336C3⋊Dic3—22no—
337C3×Dic3—32no—
338C3×Dic3—32no—
339C3×Dic3—62no—
340C3×Dic3—62no—
341C3×Dic3—61no—
342C3×Dic3—61no—
343C3×Dic3—62no—
344C3×Dic3—61no—
345C3×Dic3—62no—
346C3×Dic3—62no—
347C3×Dic3—61no—
348C3×Dic3—121no—
349C3×Dic3—121no—
350C4⋊Q8—271no—
351#126—272no—
352#126—272no—
353#124—272no—
354C22⋊Q8—91no—
355C22⋊Q8—271no—
356C4×Q8—272no—
357C4×Q8—271no—
358C4×D4—91no—
359#117—91no—
360#117—271no—
361H(3,𝔽3)#55611yescharacteristic, complemented, Sylow
362C23×C3—31no—
363C23×C3—31no—
364C23×C3—61no—
365C23×C3—91no—
366C23×C3C2×D3211yescharacteristic
367C22×D3—31no—
368C22×D3—31no—
369C22×D3—61no—
370C3×Q8—182no—
371C3×D4—182no—
372C2×C3×C4—22no—
373C2×C3×C4—31no—
374C2×C3×C4—31no—
375C2×C3×C4—31no—
376C2×C3×C4—31no—
377C2×C3×C4—62no—
378C2×C3×C4—62no—
379C2×C3×C4—61no—
380C2×C3×C4—61no—
381C2×C3×C4—91no—
382C2×C3×C4—92no—
383C2×C3×C4—92no—
384C2×C3×C4—92no—
385C2×C3×C4—92no—
386C2×C3×C4—91no—
387C2×C3×C4—91no—
388C2×C3×C4—122no—
389C2×C3×C4—182no—
390C2×C3×C4C2×D3211yescharacteristic
391C2×C3×C4C2×D3211yescharacteristic
392#67—62no—
393#67—62no—
394#67—64no—
395C2×Dic3—31no—
396C2×Dic3—31no—
397C2×Dic3—31no—
398C2×Dic3—31no—
399C2×Dic3—31no—
400C2×Dic3—31no—
401C2×Dic3—32no—
402C2×Dic3—32no—
403C2×Dic3—32no—
404C2×Dic3—32no—
405C2×Dic3—61no—
406C2×Dic3—61no—
407C2×Dic3—61no—
408C2×Dic3—92no—
409C2×Dic3—92no—
410C2×Dic3—92no—
411C2×Dic3—92no—
412C4×D3—62no—
413C4×D3—62no—
414C4×D3—122no—
415Dic6—62no—
416Dic6—182no—
417Dic6—184no—
418Dic6—184no—
419Dic6—184no—
420C2×C32—21no—
421C2×C32—21no—
422C2×C32—21no—
423C2×C32—42no—
424C2×C32—41no—
425C2×C32—41no—
426C2×C32C2×#23511yescharacteristic
427C2×C32C2×#23711yescharacteristic
428C2×C32#100711yescharacteristic
429C2×C32D3×#4111yescharacteristic
430C2×C32D3×#4111yescharacteristic
431C2×C32#101011yescharacteristic
432C3×D3—122no—
433C3×D3—121no—
434C3×D3—121no—
435C2×Q8—91no—
436C2×Q8—272no—
437C2×D4—91no—
438C22×C4—91no—
439C22×C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
440C4⋊C4—91no—
441C4⋊C4—91no—
442C4⋊C4—91no—
443C4⋊C4—91no—
444C4⋊C4—92no—
445C4⋊C4—272no—
446C4⋊C4—272no—
447C4⋊C4—272no—
448C22⋊C4—92no—
449C22⋊C4—91no—
450C22⋊C4—92no—
451C22⋊C4—91no—
452C22⋊C4—272no—
453C22⋊C4—272no—
454C42—92no—
455C42—91no—
456C42—272no—
457C42—272no—
458C22×C3—21no—
459C22×C3—21no—
460C22×C3—21no—
461C22×C3—31no—
462C22×C3—31no—
463C22×C3—61no—
464C22×C3—61no—
465C22×C3—61no—
466C22×C3—61no—
467C22×C3—61no—
468C22×C3—61no—
469C22×C3—62no—
470C22×C3—61no—
471C22×C3—121no—
472C22×C3—121no—
473C22×C3—181no—
474C22×C3—181no—
475C22×C3—181no—
476C22×C3C22×D3211yescharacteristic
477C2×D3—61no—
478C2×D3—61no—
479C2×D3—61no—
480C2×D3—61no—
481C2×D3—61no—
482C2×D3—61no—
483C2×D3—62no—
484C2×D3—121no—
485C2×D3—121no—
486C3×C4—21no—
487C3×C4—32no—
488C3×C4—32no—
489C3×C4—61no—
490C3×C4—61no—
491C3×C4—62no—
492C3×C4—92no—
493C3×C4—121no—
494C3×C4—182no—
495C3×C4—182no—
496C3×C4—182no—
497C3×C4—182no—
498C3×C4—181no—
499C3×C4—181no—
500C3×C4#380412yes—
501Dic3—32no—
502Dic3—32no—
503Dic3—62no—
504Dic3—61no—
505Dic3—61no—
506Dic3—61no—
507Dic3—61no—
508Dic3—62no—
509Dic3—62no—
510Dic3—62no—
511Dic3—62no—
512Dic3—121no—
513Dic3—121no—
514Dic3—182no—
515Dic3—182no—
516Dic3—182no—
517Dic3—182no—
518C32—21no—
519C32#553311yescharacteristic, complemented
520C32#564111yescharacteristic, complemented
521C23—91no—
522C23C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
523Q8—182no—
524Q8—544no—
525D4—182no—
526C2×C4—22no—
527C2×C4—91no—
528C2×C4—92no—
529C2×C4—91no—
530C2×C4—92no—
531C2×C4—91no—
532C2×C4—92no—
533C2×C4—92no—
534C2×C4—182no—
535C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
536C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
537C2×C3—31no—
538C2×C3—31no—
539C2×C3—31no—
540C2×C3—31no—
541C2×C3—31no—
542C2×C3—31no—
543C2×C3—41no—
544C2×C3—61no—
545C2×C3—61no—
546C2×C3—61no—
547C2×C3—121no—
548C2×C3—121no—
549C2×C3—122no—
550C2×C3—361no—
551C2×C3#6429711yescharacteristic
552C2×C3#6430611yescharacteristic
553C2×C3C2×#380411yescharacteristic
554D3—121no—
555D3—121no—
556D3—122no—
557C22—21no—
558C22—21no—
559C22—21no—
560C22—181no—
561C22—181no—
562C22—181no—
563C22C22×((C3⋊D3)⋊C3)⋊C211yescharacteristic
564C4—21no—
565C4—92no—
566C4—181no—
567C4—182no—
568C4—181no—
569C4—182no—
570C4—182no—
571C4—182no—
572C4#8867112yes—
573C3—31no—
574C3—31no—
575C3—61no—
576C3#1059480911yescharacteristic
577C2—41no—
578C2—361no—
579C2#1172248211yescharacteristic
580C2#1172249111yescharacteristic
581C2C2×#8867111yescharacteristic
subgroupG/H
centerC22C22×((C3⋊D3)⋊C3)⋊C2
commutator, cosocleC22×H(3,𝔽3)C24
Frattini, socleC22×C3C22×D32
FittingC22×C4×H(3,𝔽3)C22
prime radicalC2×#6235C22
hypercenterC22×C4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#556
LCSG ▷ C22×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
2. #64135⋊C3C2semidirect
3. #64133⋊C3C2semidirect
4. #11721297C2semidirect
6. #11721117C2semidirect
7. #11721094C2semidirect
8. #11721092C2semidirect
9. #11721089C2semidirect
10. #11721084C2semidirect
11. #11721077C2semidirect
37. #64135D3semidirect
38. #64133D3semidirect
361. H(3,𝔽3)#556semidirect
519. C32#5533semidirect
520. C32#5641semidirect

Maximal subgroups

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

subgroupindex
C4×#62352
#64135⋊C32
#64133⋊C32
#117212972
#117212942
#117211172
#117210942
#117210922
#117210892
#117210842
#117210772
#105948413
#105948403