Finite Group Databasebeta

← #13150661

Subgroups — #13150661

Order 1728.

851 conjugacy classes of subgroups.

The subgroup lattice has 549 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11721824C211yesmaximal, characteristic
2#64133⋊C3C211yesmaximal, characteristic
3C2×C4×(C3⋊Dic3)⋊C3C211yesmaximal, characteristic
4#11721299C212yesmaximal
5#11721296C212yesmaximal
6#11721116C211yesmaximal, characteristic, complemented
7#11721091C212yesmaximal, complemented
8#11721090C211yesmaximal, characteristic, complemented
9#11721083C211yesmaximal, characteristic, complemented
10#11721082C211yesmaximal, characteristic, complemented
11#11721077C212yesmaximal, complemented
12#10594810—31nomaximal
13#10594808—31nomaximal
14C22×C4×H(3,𝔽3)C2211yescharacteristic
15C2×#6229C2211yescharacteristic
16C22×(C3⋊Dic3)⋊C3C2211yescharacteristic
17C22×(C3⋊Dic3)⋊C3C2211yescharacteristic
18C2×#634⋊C3C2211yescharacteristic
19C2×#6196C2212yes—
20#88552C2212yes—
21C4⋊#1084C2211yescharacteristic
22#88540C2212yes—
23#3757⋊C3C2212yes—
24C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
25#3750⋊C3C2212yes—
26C4×(C3⋊Dic3)⋊C3C2211yescharacteristic
27C4×(C3⋊Dic3)⋊C3C2211yescharacteristic
28C4×(C3⋊Dic3)⋊C3C2211yescharacteristic
29C4×(C3⋊Dic3)⋊C3C2211yescharacteristic
30#88449C2212yes—
31#88448C2212yes—
32#88448C2212yes—
33#88448C2212yes—
34#88448C2212yes—
35#88447C2212yes—
36#88447C2212yes—
37#64133D311yescharacteristic
38C2×C4×C3⋊Dic3D311yescharacteristic
39C3×#924—31no—
40C3×#924—31no—
41C3×#924—31no—
42C3×#924—61no—
43C2×C3×C4×Dic3—31no—
44#63982—32no—
45#63964—32no—
46#63957—32no—
47#63955—32no—
48#63919—31no—
49#63918—31no—
50#63873—31no—
51#63870—32no—
52#63867—32no—
53#63866—31no—
54#63844—31no—
55#63843—31no—
56#63843—31no—
57Dic3×Dic6—31no—
58#63838—32no—
59#63837—32no—
60C23×H(3,𝔽3)C2311yescharacteristic
61C2×C4×H(3,𝔽3)—22no—
62C2×C4×H(3,𝔽3)C2311yescharacteristic
63C2×C4×H(3,𝔽3)C2311yescharacteristic
64C2×#1084C2312yes—
65C2×#1084C2312yes—
66#6229—22no—
67C2×(C3⋊Dic3)⋊C3—22no—
68C2×(C3⋊Dic3)⋊C3Q814yescomplemented
69C2×(C3⋊Dic3)⋊C3C2312yes—
70C2×(C3⋊Dic3)⋊C3C2312yes—
71C2×(C3⋊Dic3)⋊C3C2311yescharacteristic
72C2×(C3⋊Dic3)⋊C3C2311yescharacteristic
73C2×(C3⋊Dic3)⋊C3C2311yescharacteristic
74C2×(C3⋊Dic3)⋊C3C2311yescharacteristic
75#634⋊C3—22no—
76#6196—24no—
77#5622—91no—
78#5622—91no—
79#5470—91no—
80C22×C32×C4—21no—
81C22×C32×C4C2×D311yescharacteristic
82C22×C32×C4C2×D311yescharacteristic
83C22×C3⋊Dic3C2×D311yescharacteristic
84C22×C3⋊Dic3C2×D311yescharacteristic
85C2×#634C2×D311yescharacteristic
86C22×C3×Dic3—31no—
87C22×C3×Dic3—31no—
88C2×C3×Dic6—31no—
89C2×C3×Dic6—31no—
90C2×C3×Dic6—31no—
91C2×C3×Dic6—61no—
92C2×#627—32no—
93C2×#627—32no—
94#3757C2×D312yes—
95C4⋊(C3⋊Dic3)C2×D311yescharacteristic
96#3750C2×D312yes—
97C4×C3⋊Dic3C2×D311yescharacteristic
98C4×C3⋊Dic3C2×D311yescharacteristic
99C4×C3⋊Dic3C2×D311yescharacteristic
100C4×C3⋊Dic3C2×D311yescharacteristic
101C3×#217—32no—
102C3×#217—32no—
103C3×#217—32no—
104C3×#217—62no—
105C3×C4⋊Dic3—31no—
106C3×C4⋊Dic3—31no—
107C3×C4⋊Dic3—31no—
108C3×C4⋊Dic3—61no—
109C3×Dic3⋊C4—32no—
110C3×Dic3⋊C4—32no—
111C3×Dic3⋊C4—32no—
112C3×Dic3⋊C4—62no—
113C3×C4×Dic3—31no—
114C3×C4×Dic3—31no—
115C3×C4×Dic3—31no—
116C3×C4×Dic3—31no—
117#3724—32no—
118#3724—32no—
119#3724—32no—
120#3724—32no—
121Dic3⋊Dic3—32no—
122Dic3⋊Dic3—32no—
123Dic3⋊Dic3—32no—
124Dic3⋊Dic3—32no—
125Dic3⋊Dic3—32no—
126Dic3⋊Dic3—32no—
127Dic32—32no—
128Dic32—32no—
129Dic32—32no—
130Dic32—32no—
131C22×H(3,𝔽3)—21no—
132C22×H(3,𝔽3)—21no—
133C22×H(3,𝔽3)C2×Q812yes—
134C22×H(3,𝔽3)C2411yescharacteristic
135C4×H(3,𝔽3)—21no—
136C4×H(3,𝔽3)#4112yes—
137#1084—22no—
138#1084—22no—
139(C3⋊Dic3)⋊C3—22no—
140(C3⋊Dic3)⋊C3—22no—
141(C3⋊Dic3)⋊C3—21no—
142(C3⋊Dic3)⋊C3C2×Q812yes—
143(C3⋊Dic3)⋊C3C2×Q812yes—
144(C3⋊Dic3)⋊C3#4112yes—
145C3×C22⋊Q8—91no—
146C3×C22⋊Q8—91no—
147C2×C3×C42—91no—
148Q8×Dic3—91no—
149Q8×Dic3—91no—
150#944—91no—
151#944—91no—
152#926—92no—
153#924—31no—
154#924—31no—
155#924—31no—
156#924—61no—
157#924—91no—
158#924—92no—
159C2×C4×Dic3—31no—
160C2×C4×Dic3—91no—
161#889—92no—
162#889—92no—
163#888—91no—
164#888—91no—
165#887—92no—
166#887—91no—
167#887—92no—
168#887—91no—
169#878—92no—
170#878—92no—
171#877—92no—
172#877—92no—
173#870—92no—
174#869—91no—
175#869—91no—
176C4×Dic6—92no—
177C4×Dic6—91no—
178C4×Dic6—91no—
179C23×C32—21no—
180C23×C32C22×D311yescharacteristic
181C23×C32C22×D311yescharacteristic
182C2×C32×C4—22no—
183C2×C32×C4—22no—
184C2×C32×C4—21no—
185C2×C32×C4—21no—
186C2×C32×C4—42no—
187C2×C32×C4C22×D311yescharacteristic
188C2×C32×C4C22×D311yescharacteristic
189C2×C32×C4C22×D311yescharacteristic
190C2×C32×C4C22×D311yescharacteristic
191C2×C3⋊Dic3—22no—
192C2×C3⋊Dic3Dic614yescomplemented
193C2×C3⋊Dic3C22×D312yes—
194C2×C3⋊Dic3C22×D312yes—
195C2×C3⋊Dic3C22×D311yescharacteristic
196C2×C3⋊Dic3C22×D311yescharacteristic
197C2×C3⋊Dic3C22×D311yescharacteristic
198C2×C3⋊Dic3C22×D311yescharacteristic
199#634—22no—
200C2×C3×Dic3—34no—
201C2×C3×Dic3—31no—
202C2×C3×Dic3—31no—
203C2×C3×Dic3—31no—
204C2×C3×Dic3—32no—
205C2×C3×Dic3—32no—
206C2×C3×Dic3—31no—
207C2×C3×Dic3—32no—
208C2×C3×Dic3—32no—
209C2×C3×Dic3—32no—
210C2×C3×Dic3—32no—
211C2×C3×Dic3—62no—
212C2×C3×Dic3—62no—
213C2×C3×Dic3—62no—
214C3×Dic6—62no—
215C3×Dic6—62no—
216C3×Dic6—62no—
217C3×Dic6—122no—
218#627—64no—
219#627—64no—
220#533—271noSylow
221C2×H(3,𝔽3)—22no—
222C2×H(3,𝔽3)C22×Q811yescharacteristic
223C2×H(3,𝔽3)C2×#4111yescharacteristic
224C2×H(3,𝔽3)C2×#4111yescharacteristic
225C2×C3×Q8—91no—
226C2×C3×Q8—91no—
227C22×C3×C4—31no—
228C22×C3×C4—31no—
229C22×C3×C4—61no—
230C22×C3×C4—91no—
231C22×C3×C4—91no—
232C22×C3×C4D3211yescharacteristic
233C22×Dic3—31no—
234C22×Dic3—31no—
235C22×Dic3—91no—
236C22×Dic3—91no—
237C2×Dic6—31no—
238C2×Dic6—31no—
239C2×Dic6—31no—
240C2×Dic6—61no—
241C2×Dic6—91no—
242C2×Dic6—92no—
243C2×Dic6—92no—
244C2×Dic6—92no—
245C3×C4⋊C4—91no—
246C3×C4⋊C4—92no—
247C3×C4⋊C4—91no—
248C3×C4⋊C4—92no—
249C3×C22⋊C4—92no—
250C3×C22⋊C4—92no—
251C3×C42—91no—
252C3×C42—91no—
253C3×C42—91no—
254C3×C42—91no—
255#217—32no—
256#217—32no—
257#217—32no—
258#217—62no—
259#217—92no—
260C4⋊Dic3—31no—
261C4⋊Dic3—31no—
262C4⋊Dic3—31no—
263C4⋊Dic3—61no—
264C4⋊Dic3—91no—
265C4⋊Dic3—92no—
266C4⋊Dic3—92no—
267C4⋊Dic3—92no—
268C4⋊Dic3—92no—
269C4⋊Dic3—92no—
270Dic3⋊C4—32no—
271Dic3⋊C4—32no—
272Dic3⋊C4—32no—
273Dic3⋊C4—62no—
274Dic3⋊C4—92no—
275Dic3⋊C4—92no—
276Dic3⋊C4—92no—
277Dic3⋊C4—92no—
278Dic3⋊C4—92no—
279Dic3⋊C4—92no—
280Dic3⋊C4—92no—
281Dic3⋊C4—92no—
282Dic3⋊C4—92no—
283Dic3⋊C4—92no—
284Dic3⋊C4—92no—
285C4×Dic3—31no—
286C4×Dic3—31no—
287C4×Dic3—31no—
288C4×Dic3—31no—
289C4×Dic3—91no—
290C4×Dic3—91no—
291C4×Dic3—91no—
292C4×Dic3—91no—
293C4×Dic3—92no—
294C4×Dic3—92no—
295C4×Dic3—92no—
296C4×Dic3—92no—
297C4×Dic3—92no—
298C4×Dic3—92no—
299C22×C32—21no—
300C22×C32—21no—
301C22×C32—21no—
302C22×C32—21no—
303C22×C32—21no—
304C22×C32—22no—
305C22×C32—41no—
306C22×C32—41no—
307C22×C32C2×Dic612yes—
308C22×C32D3×Q812yes—
309C22×C32C23×D311yescharacteristic
310C22×C32C23×D311yescharacteristic
311C32×C4—21no—
312C32×C4—21no—
313C32×C4—22no—
314C32×C4—41no—
315C32×C4#23512yes—
316C32×C4#23712yes—
317C3⋊Dic3—21no—
318C3⋊Dic3—22no—
319C3⋊Dic3—22no—
320C3⋊Dic3C2×Dic612yes—
321C3⋊Dic3C2×Dic612yes—
322C3⋊Dic3#23512yes—
323C3×Dic3—32no—
324C3×Dic3—32no—
325C3×Dic3—32no—
326C3×Dic3—62no—
327C3×Dic3—62no—
328C3×Dic3—62no—
329C3×Dic3—62no—
330C3×Dic3—62no—
331C3×Dic3—61no—
332C3×Dic3—62no—
333C3×Dic3—122no—
334C3×Dic3—122no—
335C4⋊Q8—271no—
336C4⋊Q8—271no—
337#125—272no—
338C22⋊Q8—91no—
339C22⋊Q8—91no—
340C22⋊Q8—272no—
341C4×Q8—272no—
342C4×Q8—271no—
343C4×Q8—271no—
344#117—272no—
345C2×C42—91no—
346H(3,𝔽3)#53311yescharacteristic, complemented, Sylow
347C23×C3—31no—
348C23×C3—31no—
349C23×C3—61no—
350C23×C3C2×D3211yescharacteristic
351C3×Q8—182no—
352C3×Q8—182no—
353C2×C3×C4—22no—
354C2×C3×C4—31no—
355C2×C3×C4—31no—
356C2×C3×C4—31no—
357C2×C3×C4—31no—
358C2×C3×C4—62no—
359C2×C3×C4—62no—
360C2×C3×C4—61no—
361C2×C3×C4—61no—
362C2×C3×C4—91no—
363C2×C3×C4—91no—
364C2×C3×C4—91no—
365C2×C3×C4—94no—
366C2×C3×C4—92no—
367C2×C3×C4—92no—
368C2×C3×C4—91no—
369C2×C3×C4—92no—
370C2×C3×C4—92no—
371C2×C3×C4—122no—
372C2×C3×C4—182no—
373C2×C3×C4C2×D3211yescharacteristic
374C2×C3×C4C2×D3211yescharacteristic
375C2×Dic3—31no—
376C2×Dic3—31no—
377C2×Dic3—31no—
378C2×Dic3—34no—
379C2×Dic3—32no—
380C2×Dic3—32no—
381C2×Dic3—32no—
382C2×Dic3—32no—
383C2×Dic3—31no—
384C2×Dic3—32no—
385C2×Dic3—32no—
386C2×Dic3—62no—
387C2×Dic3—62no—
388C2×Dic3—62no—
389C2×Dic3—94no—
390C2×Dic3—91no—
391C2×Dic3—91no—
392C2×Dic3—91no—
393C2×Dic3—91no—
394C2×Dic3—92no—
395C2×Dic3—92no—
396C2×Dic3—182no—
397Dic6—62no—
398Dic6—62no—
399Dic6—62no—
400Dic6—122no—
401Dic6—182no—
402Dic6—184no—
403Dic6—184no—
404Dic6—184no—
405C2×C32—22no—
406C2×C32—22no—
407C2×C32—21no—
408C2×C32—21no—
409C2×C32—21no—
410C2×C32—42no—
411C2×C32C22×Dic611yescharacteristic
412C2×C32C2×#23511yescharacteristic
413C2×C32C2×#23511yescharacteristic
414C2×C32C2×#23711yescharacteristic
415C2×C32C2×D3×Q811yescharacteristic
416C2×C32D3×#4111yescharacteristic
417C2×Q8—91no—
418C2×Q8—91no—
419C2×Q8—272no—
420C22×C4—91no—
421C22×C4—91no—
422C22×C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
423C4⋊C4—92no—
424C4⋊C4—92no—
425C4⋊C4—91no—
426C4⋊C4—91no—
427C4⋊C4—272no—
428C4⋊C4—272no—
429C4⋊C4—272no—
430C4⋊C4—272no—
431C4⋊C4—272no—
432C22⋊C4—92no—
433C22⋊C4—92no—
434C42—91no—
435C42—91no—
436C42—91no—
437C42—91no—
438C42—272no—
439C42—272no—
440C22×C3—21no—
441C22×C3—21no—
442C22×C3—31no—
443C22×C3—31no—
444C22×C3—32no—
445C22×C3—32no—
446C22×C3—61no—
447C22×C3—61no—
448C22×C3—61no—
449C22×C3—61no—
450C22×C3—61no—
451C22×C3—62no—
452C22×C3—121no—
453C22×C3—121no—
454C22×C3D3×Dic612yes—
455C22×C3C22×D3211yescharacteristic
456C3×C4—21no—
457C3×C4—32no—
458C3×C4—32no—
459C3×C4—61no—
460C3×C4—61no—
461C3×C4—62no—
462C3×C4—92no—
463C3×C4—92no—
464C3×C4—92no—
465C3×C4—121no—
466C3×C4—182no—
467C3×C4—181no—
468C3×C4—182no—
469C3×C4—182no—
470C3×C4—182no—
471C3×C4#380412yes—
472Dic3—32no—
473Dic3—32no—
474Dic3—32no—
475Dic3—62no—
476Dic3—62no—
477Dic3—62no—
478Dic3—62no—
479Dic3—62no—
480Dic3—61no—
481Dic3—62no—
482Dic3—92no—
483Dic3—92no—
484Dic3—92no—
485Dic3—122no—
486Dic3—122no—
487Dic3—182no—
488Dic3—182no—
489Dic3—181no—
490C32—21no—
491C32#547011yescharacteristic, complemented
492C32#562211yescharacteristic, complemented
493C23C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
494Q8—182no—
495Q8—182no—
496Q8—544no—
497C2×C4—22no—
498C2×C4—91no—
499C2×C4—91no—
500C2×C4—91no—
501C2×C4—94no—
502C2×C4—91no—
503C2×C4—92no—
504C2×C4—92no—
505C2×C4—92no—
506C2×C4—92no—
507C2×C4—182no—
508C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
509C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
510C2×C3—22no—
511C2×C3—31no—
512C2×C3—31no—
513C2×C3—31no—
514C2×C3—31no—
515C2×C3—31no—
516C2×C3—31no—
517C2×C3—62no—
518C2×C3—62no—
519C2×C3—61no—
520C2×C3—61no—
521C2×C3—61no—
522C2×C3—122no—
523C2×C3C2×D3×Dic611yescharacteristic
524C2×C3D3×#23511yescharacteristic
525C2×C3C2×#380411yescharacteristic
526C22—21no—
527C22—21no—
528C22((C3⋊D3)⋊C3)⋊Q812yes—
529C22C22×((C3⋊D3)⋊C3)⋊C211yescharacteristic
530C4—21no—
531C4—92no—
532C4—92no—
533C4—92no—
534C4—182no—
535C4—182no—
536C4—181no—
537C4—182no—
538C4—182no—
539C4#8867112yes—
540C3—31no—
541C3—31no—
542C3—61no—
543C3#1059480811yescharacteristic
544C2—22no—
545C2C2×((C3⋊D3)⋊C3)⋊Q811yescharacteristic
546C2#1172249011yescharacteristic
547C2C2×#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 radicalC23×H(3,𝔽3)C23
hypercenterC22×C4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#533
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
6. #11721116C2semidirect
7. #11721091C2semidirect
8. #11721090C2semidirect
9. #11721083C2semidirect
10. #11721082C2semidirect
11. #11721077C2semidirect
68. C2×(C3⋊Dic3)⋊C3Q8semidirect
192. C2×C3⋊Dic3Dic6semidirect
346. H(3,𝔽3)#533semidirect
491. C32#5470semidirect
492. C32#5622semidirect

Maximal subgroups

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

subgroupindex
#117218242
#64133⋊C32
C2×C4×(C3⋊Dic3)⋊C32
#117212992
#117212962
#117211162
#117210912
#117210902
#117210832
#117210822
#117210772
#105948103
#105948083