Finite Group Databasebeta

← #13180638

Subgroups — #13180638

Order 1728.

526 conjugacy classes of subgroups.

The subgroup lattice has 510 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11724368C211yesmaximal, characteristic, complemented
2#11724368C211yesmaximal, characteristic, complemented
3C2×#88945C211yesmaximal, characteristic, complemented
4C3×C22⋊PSU(3,𝔽2)C211yesmaximal, characteristic, complemented
5#67⋊#157C211yesmaximal, characteristic
6#67⋊#157C211yesmaximal, characteristic
7#67×#157C211yesmaximal, characteristic
8#11724153C211yesmaximal, characteristic, complemented
9#11724152C211yesmaximal, characteristic, complemented
10#11724152C211yesmaximal, characteristic, complemented
11#11724151C211yesmaximal, characteristic, complemented
12#11724151C211yesmaximal, characteristic, complemented
13#11724149C211yesmaximal, characteristic, complemented
14#11724148C211yesmaximal, characteristic, complemented
15D3×#3777C211yesmaximal, characteristic, complemented
16#10596545—31nomaximal
17C22×C3⋊#157C2211yescharacteristic
18C22×C3⋊#157C2211yescharacteristic
19C22×C3×#157C2211yescharacteristic
20C2×C3⋊PSU(3,𝔽2)C2211yescharacteristic
21C2×C3⋊PSU(3,𝔽2)C2211yescharacteristic
22C2×C3×PSU(3,𝔽2)C2211yescharacteristic
23C2×D3×#157C2211yescharacteristic
24C2×D3×#157C2211yescharacteristic
25C2×D3×#157C2211yescharacteristic
26C3⋊D3×#67C2211yescharacteristic
27#88994C2211yescharacteristic
28C3×C22⋊#157C2211yescharacteristic
29C3×C22⋊#157C2211yescharacteristic
30#88946C2211yescharacteristic
31#88946C2211yescharacteristic
32#88945C2211yescharacteristic
33#88945C2211yescharacteristic
34#88945C2211yescharacteristic
35#88945C2211yescharacteristic
36#88945C2211yescharacteristic
37#88945C2211yescharacteristic
38#88945C2211yescharacteristic
39#88945C2211yescharacteristic
40C3×#3777C2211yescharacteristic
41C3×#3777C2211yescharacteristic
42C3×#3777C2211yescharacteristic
43Dic3⋊#157C2211yescharacteristic
44Dic3⋊#157C2211yescharacteristic
45Dic3⋊#157C2211yescharacteristic
46(C2×D3)⋊#157C2211yescharacteristic
47(C2×D3)⋊#157C2211yescharacteristic
48(C2×D3)⋊#157C2211yescharacteristic
49Dic3×#157C2211yescharacteristic
50Dic3×#157C2211yescharacteristic
51Dic3×#157C2211yescharacteristic
52D4×#157—31no—
53D4×#157—31no—
54D4×#157—31no—
55C22⋊PSU(3,𝔽2)—31no—
56C22⋊PSU(3,𝔽2)—31no—
57C22⋊PSU(3,𝔽2)—31no—
58C22⋊PSU(3,𝔽2)—31no—
59C22⋊PSU(3,𝔽2)—31no—
60C22⋊PSU(3,𝔽2)D311yescharacteristic, complemented
61C2×#3777—31no—
62C2×#3777—31no—
63C4⋊PSU(3,𝔽2)—31no—
64C4×PSU(3,𝔽2)—31no—
65#64243—31no—
66#64243—31no—
67C22×C3×C3⋊D3C2311yescharacteristic
68C2×D3×C3⋊D3C2311yescharacteristic
69C2×C3⋊#157—22no—
70C2×C3⋊#157—41no—
71C2×C3⋊#157C2311yescharacteristic
72C2×C3⋊#157C2311yescharacteristic
73C2×C3⋊#157C2311yescharacteristic
74C2×C3⋊#157C2311yescharacteristic
75C2×C3⋊#157C2311yescharacteristic
76C2×C3⋊#157C2311yescharacteristic
77C2×C3×#157—41no—
78C2×C3×#157C2311yescharacteristic
79C2×C3×#157C2311yescharacteristic
80C2×C3×#157C2311yescharacteristic
81C2×C3×#157C2311yescharacteristic
82C2×C3×#157C2311yescharacteristic
83C2×C3×#157C2311yescharacteristic
84C3⋊PSU(3,𝔽2)—22no—
85C3⋊PSU(3,𝔽2)—41no—
86C3×PSU(3,𝔽2)—41no—
87D3×#157—22no—
88D3×#157—41no—
89D3×#157—41no—
90#6312Q811yescharacteristic, complemented
91C32×#67Q811yescharacteristic, complemented
92#6292Q811yescharacteristic, complemented
93#6290Q811yescharacteristic, complemented
94Dic3×C3⋊D3C2311yescharacteristic
95#5636—91nomaximal
96C22×#157—31no—
97C22×#157—31no—
98C22×#157—31no—
99C22×#157—31no—
100C22×#157—31no—
101C22×#157C2×D311yescharacteristic
102C2×PSU(3,𝔽2)—31no—
103C2×PSU(3,𝔽2)—31no—
104C2×PSU(3,𝔽2)C2×D311yescharacteristic
105D4×C3⋊D3—31no—
106D3×#67—121no—
107C22⋊#157—31no—
108C22⋊#157—31no—
109C22⋊#157—31no—
110C22⋊#157—31no—
111C22⋊#157C2×D311yescharacteristic
112C22⋊#157C2×D311yescharacteristic
113C4⋊#157—31no—
114C4⋊#157—31no—
115C4⋊#157—31no—
116C4×#157—31no—
117C4×#157—31no—
118C4×#157—31no—
119#3777—31no—
120#3777—31no—
121#3777—31no—
122#3777—31no—
123#3777—31no—
124#3777—31no—
125#3777—31no—
126#3777—31no—
127#3777—31no—
128#3777—31no—
129#3777C2×D311yescharacteristic
130#3777C2×D311yescharacteristic
131#3777C2×D311yescharacteristic
132C22×C33C2×Q811yescharacteristic
133C2×C3⋊(C3⋊D3)C2×Q811yescharacteristic
134C2×C3×C3⋊D3—21no—
135C2×C3×C3⋊D3—21no—
136C2×C3×C3⋊D3C2×Q811yescharacteristic
137C2×C3×C3⋊D3C2411yescharacteristic
138C2×C32×D3C2×Q811yescharacteristic
139D3×C3⋊D3—21no—
140D3×C3⋊D3—21no—
141C3⋊#157—21no—
142C3⋊#157—21no—
143C3⋊#157—21no—
144C3⋊#157—21no—
145C3⋊#157—21no—
146C3⋊#157#4112yes—
147C3×#157—21no—
148C3×#157—21no—
149C3×#157—21no—
150C3×#157—21no—
151C3×#157—21no—
152C3×#157—21no—
153C3⋊(C3⋊Dic3)C2×Q811yescharacteristic
154C32×Dic3C2×Q811yescharacteristic
155C3×C22⋊Q8—91no—
156#946—91no—
157#944—91no—
158C4×#67—91no—
159C2×Dic3⋊C4—91no—
160#896—91no—
161#896—91no—
162D3×C4⋊C4—91no—
163#889—91no—
164#889—91no—
165#887—91no—
166#67⋊C4—91no—
167#67⋊C4—91no—
168#878—91no—
169#878—91no—
170C22×C3⋊D3—31no—
171C22×C3⋊D3C22×D311yescharacteristic
172C22×C3×D3—121no—
173C2×D32—121no—
174C2×#157—31no—
175C2×#157—31no—
176C2×#157—31no—
177C2×#157—31no—
178C2×#157—31no—
179C2×#157—31no—
180C2×#157—41no—
181C2×#157—62no—
182C2×#157—62no—
183C2×#157—121no—
184C2×#157—121no—
185C2×#157—121no—
186C2×#157C22×D311yescharacteristic
187C2×#157C22×D311yescharacteristic
188C2×#157C22×D311yescharacteristic
189C2×#157C22×D311yescharacteristic
190C2×#157C22×D311yescharacteristic
191C2×#157C22×D311yescharacteristic
192PSU(3,𝔽2)—41no—
193PSU(3,𝔽2)—62no—
194PSU(3,𝔽2)—121no—
195C32×D4—31no—
196#638—31no—
197#638—31no—
198#638—121no—
199#638—241no—
200C4⋊(C3⋊D3)—31no—
201C4×C3⋊D3—31no—
202C3×#67—41no—
203#626—121no—
204#625—121no—
205D3×Dic3—121no—
206#554—271noSylow
207C2×C33—21no—
208C2×C33C22×Q811yescharacteristic
209C3⋊(C3⋊D3)—21no—
210C3×C3⋊D3—21no—
211C3×C3⋊D3C2×#4111yescharacteristic
212C3×C3⋊D3H(5,𝔽2)11yescharacteristic
213C32×D3—21no—
214C2×C3×Q8—91no—
215C22×C3×C4—91no—
216C2×#67—91no—
217C22×Dic3—91no—
218C22×Dic3—91no—
219D3×D4—361no—
220C2×C4×D3—91no—
221C2×C4×D3—91no—
222C2×C4×D3—91no—
223C2×Dic6—91no—
224C2×Dic6—91no—
225C3×C4⋊C4—91no—
226C3×C4⋊C4—91no—
227C3×C4⋊C4—91no—
228C3×C22⋊C4—91no—
229C3×C22⋊C4—91no—
230#217—91no—
231(C2×D3)⋊C4—91no—
232(C2×D3)⋊C4—91no—
233(C2×D3)⋊C4—91no—
234C4⋊Dic3—91no—
235C4⋊Dic3—91no—
236Dic3⋊C4—91no—
237Dic3⋊C4—91no—
238Dic3⋊C4—91no—
239Dic3⋊C4—91no—
240Dic3⋊C4—91no—
241Dic3⋊C4—91no—
242Dic3⋊C4—91no—
243Dic3⋊C4—91no—
244Dic3⋊C4—91no—
245Dic3⋊C4—91no—
246Dic3⋊C4—91no—
247C4×Dic3—91no—
248C4×Dic3—91no—
249C4×Dic3—91no—
250C22×C32—31no—
251C22×C32—41no—
252C22×C32—81no—
253C22×C32D3×Q811yescharacteristic
254C2×C3⋊D3—21no—
255C2×C3⋊D3—21no—
256C2×C3⋊D3—31no—
257C2×C3⋊D3—61no—
258C2×C3⋊D3—61no—
259C2×C3⋊D3—121no—
260C2×C3⋊D3—241no—
261C2×C3⋊D3D3×Q811yescharacteristic
262C2×C3⋊D3C23×D311yescharacteristic
263C2×C3×D3—41no—
264C2×C3×D3—121no—
265C2×C3×D3—121no—
266C2×C3×D3—241no—
267C2×C3×D3—241no—
268D32—241no—
269D32—241no—
270#157—21no—
271#157—21no—
272#157—21no—
273#157—21no—
274#157—21no—
275#157—21no—
276#157—32no—
277#157—61no—
278#157—61no—
279#157—61no—
280#157—61no—
281#157—61no—
282C32×C4—31no—
283C3⋊Dic3—31no—
284C3⋊Dic3—121no—
285C3⋊Dic3—241no—
286C3×Dic3—41no—
287C4⋊Q8—271no—
288#125—271no—
289#125—271no—
290C22⋊Q8—91no—
291C22⋊Q8—271no—
292C22⋊Q8—271no—
293C22⋊Q8—271no—
294C22⋊Q8—271no—
295C22⋊Q8—271no—
296C4×Q8—271no—
297C4×D4—271no—
298C4×D4—271no—
299C4×D4—271no—
300C2×C4⋊C4—271no—
301C2×C4⋊C4—271no—
302C33#55411yescharacteristic, complemented, Sylow
303C23×C3—91no—
304C22×D3—91no—
305C22×D3—121no—
306C22×D3—361no—
307C3×Q8—361no—
308C3×D4—121no—
309C2×C3×C4—91no—
310C2×C3×C4—91no—
311C2×C3×C4—91no—
312C2×C3×C4—91no—
313C2×C3×C4—91no—
314C2×C3×C4—91no—
315C2×C3×C4—361no—
316#67—91no—
317#67—91no—
318#67—91no—
319#67—361no—
320#67—361no—
321#67—721no—
322#67PSU(3,𝔽2)11yescharacteristic, complemented
323C2×Dic3—91no—
324C2×Dic3—91no—
325C2×Dic3—91no—
326C2×Dic3—91no—
327C2×Dic3—91no—
328C2×Dic3—91no—
329C2×Dic3—91no—
330C2×Dic3—182no—
331C2×Dic3—361no—
332D12—361no—
333C4×D3—182no—
334C4×D3—361no—
335C4×D3—361no—
336C4×D3—361no—
337Dic6—182no—
338Dic6—361no—
339C2×C32—21no—
340C2×C32—41no—
341C2×C32—61no—
342C2×C32—81no—
343C2×C32—81no—
344C2×C32—161no—
345C2×C32C2×D3×Q811yescharacteristic
346C3⋊D3—21no—
347C3⋊D3—61no—
348C3⋊D3—241no—
349C3⋊D3—481no—
350C3⋊D3#100411yescharacteristic
351C3⋊D3D3×#4111yescharacteristic
352C3×D3—81no—
353C3×D3—121no—
354C3×D3—121no—
355C3×D3—241no—
356C2×Q8—91no—
357C2×Q8—271no—
358C2×Q8—271no—
359C2×D4—271no—
360C22×C4—91no—
361C22×C4—271no—
362C22×C4—271no—
363C22×C4—271no—
364C22×C4—271no—
365C22×C4—271no—
366C4⋊C4—91no—
367C4⋊C4—91no—
368C4⋊C4—91no—
369C4⋊C4—271no—
370C4⋊C4—271no—
371C4⋊C4—271no—
372C4⋊C4—271no—
373C4⋊C4—271no—
374C4⋊C4—271no—
375C4⋊C4—271no—
376C4⋊C4—271no—
377C4⋊C4—271no—
378C4⋊C4—271no—
379C4⋊C4—271no—
380C4⋊C4—271no—
381C4⋊C4—271no—
382C22⋊C4—91no—
383C22⋊C4—91no—
384C22⋊C4—271no—
385C22⋊C4—271no—
386C22⋊C4—271no—
387C22⋊C4—271no—
388C42—271no—
389C42—271no—
390C42—271no—
391C22×C3—41no—
392C22×C3—81no—
393C22×C3—91no—
394C22×C3—91no—
395C22×C3—121no—
396C22×C3—181no—
397C22×C3—181no—
398C22×C3C2×PSU(3,𝔽2)11yescharacteristic
399C2×D3—91no—
400C2×D3—121no—
401C2×D3—121no—
402C2×D3—181no—
403C2×D3—181no—
404C2×D3—241no—
405C2×D3—241no—
406C2×D3—361no—
407C2×D3—721no—
408C2×D3—721no—
409C2×D3—721no—
410C2×D3C2×PSU(3,𝔽2)11yescharacteristic
411C3×C4—121no—
412C3×C4—181no—
413C3×C4—181no—
414C3×C4—181no—
415C3×C4—181no—
416C3×C4—181no—
417C3×C4—181no—
418Dic3—92no—
419Dic3—91no—
420Dic3—181no—
421Dic3—181no—
422Dic3—181no—
423Dic3—181no—
424Dic3—181no—
425Dic3—361no—
426Dic3—721no—
427Dic3C2×PSU(3,𝔽2)11yescharacteristic
428C32—41no—
429C32—81no—
430C32#563611yescharacteristic, complemented
431C23—91no—
432C23—271no—
433Q8—361no—
434Q8—542no—
435Q8—1081no—
436D4—31no—
437D4—271no—
438D4—271no—
439D4—271no—
440C2×C4—91no—
441C2×C4—91no—
442C2×C4—91no—
443C2×C4—91no—
444C2×C4—91no—
445C2×C4—91no—
446C2×C4—271no—
447C2×C4—271no—
448C2×C4—271no—
449C2×C4—271no—
450C2×C4—271no—
451C2×C4—271no—
452C2×C4—271no—
453C2×C4—361no—
454C2×C4—542no—
455C2×C4—542no—
456C2×C4—1081no—
457C2×C4—1081no—
458C2×C4—1081no—
459C2×C3—21no—
460C2×C3—41no—
461C2×C3—81no—
462C2×C3—81no—
463C2×C3—91no—
464C2×C3—91no—
465C2×C3—161no—
466C2×C3—181no—
467C2×C3—241no—
468C2×C3C22×PSU(3,𝔽2)11yescharacteristic
469D3—21no—
470D3—121no—
471D3—121no—
472D3—181no—
473D3—241no—
474D3—721no—
475D3—1441no—
476C22—31no—
477C22—91no—
478C22—91no—
479C22—181no—
480C22—181no—
481C22—271no—
482C22—541no—
483C22—541no—
484C22D3×PSU(3,𝔽2)11yescharacteristic
485C4—31no—
486C4—181no—
487C4—181no—
488C4—181no—
489C4—181no—
490C4—181no—
491C4—181no—
492C4—272no—
493C4—271no—
494C4—541no—
495C4—541no—
496C4—541no—
497C4—541no—
498C4—541no—
499C3—41no—
500C3—81no—
501C3#1059654511yescharacteristic, complemented
502C2—21no—
503C2—61no—
504C2—91no—
505C2—91no—
506C2—181no—
507C2—541no—
508C2C2×D3×PSU(3,𝔽2)11yescharacteristic
subgroupG/H
center, FrattiniC2C2×D3×PSU(3,𝔽2)
commutator, cosocleC2×C3×C3⋊D3C24
FittingC22×C33C2×Q8
socleC2×C33C22×Q8
prime radicalC3⋊D3×#67C22
hypercenterC22D3×PSU(3,𝔽2)
hypocenterC33#554
LCSG ▷ C2×C3×C3⋊D3 ▷ C33
UCSC1 ◁ C2 ◁ C22

Complements

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

NH ≅ G/Ntype
1. #11724368C2semidirect
2. #11724368C2semidirect
3. C2×#88945C2semidirect
4. C3×C22⋊PSU(3,𝔽2)C2semidirect
8. #11724153C2semidirect
9. #11724152C2semidirect
10. #11724152C2semidirect
11. #11724151C2semidirect
12. #11724151C2semidirect
13. #11724149C2semidirect
14. #11724148C2semidirect
15. D3×#3777C2semidirect
60. C22⋊PSU(3,𝔽2)D3semidirect
90. #6312Q8semidirect
91. C32×#67Q8semidirect
92. #6292Q8semidirect
93. #6290Q8semidirect
302. C33#554semidirect
322. #67PSU(3,𝔽2)semidirect
430. C32#5636semidirect
501. C3#10596545semidirect

Maximal subgroups

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

subgroupindex
#117243682
C2×#889452
C3×C22⋊PSU(3,𝔽2)2
#67⋊#1572
#67×#1572
#117241532
#117241522
#117241512
#117241492
#117241482
D3×#37772
#105965453
#56369