Finite Group Databasebeta

← #13156529

Subgroups — #13156529

Order 1728.

614 conjugacy classes of subgroups.

The subgroup lattice has 562 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11723441C211yesmaximal, characteristic
2C3×#63869C211yesmaximal, characteristic, complemented
3#11722076C211yesmaximal, characteristic, complemented
4#11722036C211yesmaximal, characteristic
5#11722035C211yesmaximal, characteristic
6C3×#63567C211yesmaximal, characteristic, complemented
7C3×Dic3×C3⋊C8C211yesmaximal, characteristic, complemented
8#10590872—31nomaximal
9#10590537—31nomaximal
10#10590293—31nomaximal
11C2×C4⋊(C3⋊(C3⋊D3))C2211yescharacteristic
12C2×C3×#634C2211yescharacteristic
13C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
14C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
15C32×C4⋊Dic3C2211yescharacteristic
16C32×C4×Dic3C2211yescharacteristic
17C2×C32×C3⋊C8C2211yescharacteristic
18#88804—21no—
19#88804—21no—
20C3×#3724—21no—
21C3×Dic3⋊Dic3—21no—
22#64084—31no—
23C3×#890—31no—
24C3×#870—31no—
25#63869D311yescharacteristic, complemented
26#63864—31no—
27#63864—31no—
28#63864—61no—
29#63648—31no—
30#63636—31no—
31C3×#816—31no—
32C3×C8×Dic3—31no—
33C3×#810—31no—
34C3×C4×C3⋊C8—31no—
35#63567—31no—
36#63567D311yescharacteristic, complemented
37#63566—31no—
38#63565—31no—
39#63564—31no—
40#63564—31no—
41#63564—31no—
42#63564—31no—
43#63564—61no—
44#63564—61no—
45Dic3×C3⋊C8D311yescharacteristic, complemented
46C22×C3⋊(C3⋊D3)—21no—
47C2×C33×C4C2311yescharacteristic
48C4⋊(C3⋊(C3⋊D3))—21no—
49C4⋊(C3⋊(C3⋊D3))—21no—
50C2×C3×C3⋊Dic3—21no—
51C2×C3×C3⋊Dic3—21no—
52C3×#634—21no—
53C3×#634—21no—
54C2×C32×Dic3—21no—
55C2×C32×Dic3D411yescharacteristic
56C2×C32×Dic3D411yescharacteristic
57C3×C3⋊(C3⋊C8)—21no—
58C32×C3⋊C8—21no—
59#5068—91no—
60#4800—91no—
61#4697—91no—
62C2×C4⋊(C3⋊D3)—31no—
63C2×C4⋊(C3⋊D3)—31no—
64C2×C4⋊(C3⋊D3)—31no—
65C2×C4⋊(C3⋊D3)—61no—
66C2×C4⋊(C3⋊D3)—61no—
67C2×C4⋊(C3⋊D3)—61no—
68C2×C4⋊(C3⋊D3)—121no—
69C2×#634C2×D311yescharacteristic
70C2×C3×Dic6—31no—
71C2×C3×Dic6—31no—
72C2×C3×Dic6—61no—
73C2×C32×C8—31no—
74C32×C4⋊C4—31no—
75C32×C42—31no—
76#3752—61no—
77#3752—61no—
78C4⋊(C3⋊Dic3)C2×D311yescharacteristic
79C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
80C3×C4⋊Dic3—21no—
81C3×C4⋊Dic3—31no—
82C3×C4⋊Dic3—31no—
83C3×C4⋊Dic3—61no—
84C3×C4⋊Dic3—61no—
85C3×C4⋊Dic3C2×D311yescharacteristic
86C3×C4⋊Dic3C2×D311yescharacteristic
87C3×Dic3⋊C4—61no—
88C3×Dic3⋊C4—61no—
89C3×Dic3⋊C4—61no—
90C3×C4×Dic3—21no—
91C3×C4×Dic3C2×D311yescharacteristic
92C3×C4×Dic3C2×D311yescharacteristic
93C2×C3×C3⋊C8—21no—
94C2×C3×C3⋊C8—31no—
95C2×C3×C3⋊C8—31no—
96C2×C3×C3⋊C8—61no—
97C2×C3×C3⋊C8C2×D311yescharacteristic
98C2×C3×C3⋊C8C2×D311yescharacteristic
99#3724—21no—
100Dic3⋊Dic3—21no—
101#3722—62no—
102#3722—62no—
103#3722—61no—
104#3722—61no—
105#3722—61no—
106#3722—61no—
107C22×C33C2×D411yescharacteristic
108C2×C3⋊(C3⋊D3)—41no—
109C2×C3⋊(C3⋊D3)—41no—
110C2×C3⋊(C3⋊D3)—41no—
111C33×C4#4111yescharacteristic
112C33×C4#4111yescharacteristic
113C3×C3⋊Dic3—41no—
114C3×C3⋊Dic3—41no—
115C32×Dic3—21no—
116C32×Dic3—21no—
117C32×Dic3—41no—
118C3×#125—91no—
119#947—91no—
120#890—31no—
121#875—91no—
122#875—91no—
123#875—181no—
124#870—31no—
125C3×#103—91no—
126C3×C4×C8—91no—
127#835—91no—
128#821—91no—
129#821—91no—
130#821—91no—
131#821—181no—
132#816—31no—
133#816—91no—
134C8×Dic3—31no—
135#810—31no—
136#810—91no—
137#809—91no—
138#809—91no—
139#809—91no—
140#809—181no—
141C4×C3⋊C8—31no—
142C22×C3⋊D3—61no—
143C22×C3⋊D3—62no—
144C22×C3⋊D3—61no—
145C22×C3⋊D3—61no—
146C22×C3⋊D3—121no—
147C22×C3⋊D3—121no—
148C22×C3⋊D3—122no—
149C2×C32×C4—21no—
150C2×C32×C4—21no—
151C2×C32×C4—21no—
152C2×C32×C4—31no—
153C2×C32×C4—31no—
154C2×C32×C4—41no—
155C2×C32×C4—61no—
156C2×C32×C4C22×D311yescharacteristic
157C2×C32×C4C22×D311yescharacteristic
158C2×C32×C4C22×D311yescharacteristic
159C2×C3⋊Dic3—21no—
160C2×C3⋊Dic3—21no—
161C4⋊(C3⋊D3)—61no—
162C4⋊(C3⋊D3)—61no—
163C4⋊(C3⋊D3)—62no—
164C4⋊(C3⋊D3)—62no—
165C4⋊(C3⋊D3)—61no—
166C4⋊(C3⋊D3)—61no—
167C4⋊(C3⋊D3)—61no—
168C4⋊(C3⋊D3)—61no—
169C4⋊(C3⋊D3)—121no—
170C4⋊(C3⋊D3)—121no—
171C4⋊(C3⋊D3)—121no—
172C4⋊(C3⋊D3)—121no—
173C4⋊(C3⋊D3)—122no—
174C4⋊(C3⋊D3)—122no—
175#634—21no—
176#634—21no—
177C2×C3×Dic3—21no—
178C2×C3×Dic3—21no—
179C2×C3×Dic3—21no—
180C2×C3×Dic3—21no—
181C2×C3×Dic3—41no—
182C2×C3×Dic3—62no—
183C2×C3×Dic3—61no—
184C2×C3×Dic3—61no—
185C2×C3×Dic3—61no—
186C2×C3×Dic3—61no—
187C2×C3×Dic3—121no—
188C2×C3×Dic3D1211yescharacteristic
189C2×C3×Dic3D1211yescharacteristic
190C2×C3×Dic3#6711yescharacteristic
191C2×C3×Dic3#6711yescharacteristic
192C3×Dic6—61no—
193C3×Dic6—61no—
194C3×Dic6—61no—
195C3×Dic6—61no—
196C3×Dic6—121no—
197C3×Dic6—121no—
198C32×C8—61no—
199C3⋊(C3⋊C8)—21no—
200C3×C3⋊C8—21no—
201C3×C3⋊C8—21no—
202C3×C3⋊C8—41no—
203C3×C3⋊C8—61no—
204C3×C3⋊C8—61no—
205C3×C3⋊C8—121no—
206#488—271noSylow
207C2×C33#12411yescharacteristic
208C2×C33#13511yescharacteristic
209C2×C33#13511yescharacteristic
210C3⋊(C3⋊D3)—81no—
211C2×C3×Q8—91no—
212C2×D12—91no—
213C2×D12—91no—
214C2×D12—91no—
215C2×D12—181no—
216C2×D12—181no—
217C2×D12—181no—
218C2×D12—361no—
219C2×Dic6—31no—
220C2×Dic6—31no—
221C2×Dic6—61no—
222C2×C3×C8—31no—
223C2×C3×C8—31no—
224C2×C3×C8—61no—
225C2×C3×C8—91no—
226C3×C4⋊C4—31no—
227C3×C4⋊C4—31no—
228C3×C4⋊C4—61no—
229C3×C4⋊C4—91no—
230C3×C4⋊C4—181no—
231C3×C4⋊C4—181no—
232C3×C42—31no—
233C3×C42—31no—
234C3×C42—61no—
235(C2×D3)⋊C4—181no—
236(C2×D3)⋊C4—181no—
237(C2×D3)⋊C4—181no—
238(C2×D3)⋊C4—181no—
239(C2×D3)⋊C4—182no—
240(C2×D3)⋊C4—182no—
241(C2×D3)⋊C4—181no—
242(C2×D3)⋊C4—181no—
243C4⋊Dic3—31no—
244C4⋊Dic3—31no—
245C4⋊Dic3—61no—
246C4⋊Dic3—61no—
247C4⋊Dic3D3211yescharacteristic
248Dic3⋊C4—61no—
249Dic3⋊C4—61no—
250Dic3⋊C4—61no—
251C4×Dic3D3211yescharacteristic
252C2×C3⋊C8—31no—
253C2×C3⋊C8—31no—
254C2×C3⋊C8—61no—
255C2×C3⋊C8D3211yescharacteristic
256C22×C32—21no—
257C22×C32—21no—
258C22×C32—21no—
259C22×C32—41no—
260C22×C32C2×D1211yescharacteristic
261C22×C32D3×D411yescharacteristic
262C22×C32C2×#6711yescharacteristic
263C2×C3⋊D3—122no—
264C2×C3⋊D3—122no—
265C2×C3⋊D3—122no—
266C2×C3⋊D3—124no—
267C2×C3⋊D3—121no—
268C2×C3⋊D3—121no—
269C2×C3⋊D3—121no—
270C2×C3⋊D3—121no—
271C2×C3⋊D3—121no—
272C2×C3⋊D3—122no—
273C2×C3⋊D3—122no—
274C2×C3⋊D3—121no—
275C2×C3⋊D3—121no—
276C2×C3⋊D3—121no—
277C2×C3⋊D3—121no—
278C2×C3⋊D3—241no—
279C2×C3⋊D3—241no—
280C2×C3⋊D3—241no—
281C2×C3⋊D3—241no—
282C2×C3⋊D3—242no—
283C2×C3⋊D3—242no—
284C32×C4—21no—
285C32×C4—21no—
286C32×C4—21no—
287C32×C4—21no—
288C32×C4—21no—
289C32×C4—21no—
290C32×C4—41no—
291C32×C4—41no—
292C32×C4—61no—
293C32×C4—61no—
294C32×C4—121no—
295C32×C4#23511yescharacteristic
296C32×C4#23511yescharacteristic
297C32×C4#23511yescharacteristic
298C32×C4#23711yescharacteristic
299C32×C4#23911yescharacteristic
300C32×C4#23911yescharacteristic
301C3⋊Dic3—41no—
302C3⋊Dic3—41no—
303C3×Dic3—21no—
304C3×Dic3—21no—
305C3×Dic3—21no—
306C3×Dic3—21no—
307C3×Dic3—41no—
308C3×Dic3—41no—
309C3×Dic3—41no—
310C3×Dic3—41no—
311C3×Dic3—81no—
312C3×Dic3—122no—
313C3×Dic3—121no—
314C3×Dic3—121no—
315C3×Dic3—121no—
316C3×Dic3—121no—
317C3×Dic3—241no—
318#125—91no—
319#124—271no—
320#103—91no—
321#103—271no—
322#102—271no—
323#102—271no—
324C4×C8—91no—
325C33#48811yescharacteristic, complemented, Sylow
326C22×D3—182no—
327C22×D3—181no—
328C22×D3—181no—
329C22×D3—181no—
330C22×D3—361no—
331C22×D3—361no—
332C22×D3—362no—
333C3×Q8—181no—
334C3×Q8—181no—
335C2×C3×C4—21no—
336C2×C3×C4—21no—
337C2×C3×C4—21no—
338C2×C3×C4—31no—
339C2×C3×C4—31no—
340C2×C3×C4—31no—
341C2×C3×C4—31no—
342C2×C3×C4—41no—
343C2×C3×C4—61no—
344C2×C3×C4—61no—
345C2×C3×C4—61no—
346C2×C3×C4—61no—
347C2×C3×C4—121no—
348C2×C3×C4—181no—
349C2×C3×C4—181no—
350C2×C3×C4C2×D3211yescharacteristic
351C2×C3×C4C2×D3211yescharacteristic
352C2×C3×C4C2×D3211yescharacteristic
353C2×Dic3—21no—
354C2×Dic3—62no—
355C2×Dic3—61no—
356C2×Dic3—61no—
357C2×Dic3—61no—
358C2×Dic3—61no—
359C2×Dic3—121no—
360C2×Dic3#62611yescharacteristic
361C2×Dic3#62611yescharacteristic
362D12—182no—
363D12—182no—
364D12—181no—
365D12—181no—
366D12—181no—
367D12—181no—
368D12—181no—
369D12—181no—
370D12—361no—
371D12—361no—
372D12—361no—
373D12—361no—
374D12—362no—
375D12—362no—
376Dic6—61no—
377Dic6—61no—
378Dic6—61no—
379Dic6—61no—
380Dic6—121no—
381Dic6—121no—
382C3×C8—61no—
383C3×C8—61no—
384C3×C8—121no—
385C3×C8—181no—
386C3⋊C8—21no—
387C3⋊C8—61no—
388C3⋊C8—61no—
389C3⋊C8—121no—
390C2×C32—21no—
391C2×C32—21no—
392C2×C32—21no—
393C2×C32—21no—
394C2×C32—21no—
395C2×C32—21no—
396C2×C32—21no—
397C2×C32—21no—
398C2×C32—21no—
399C2×C32—41no—
400C2×C32—41no—
401C2×C32—41no—
402C2×C32#87511yescharacteristic
403C2×C32#88511yescharacteristic
404C2×C32#90411yescharacteristic
405C2×C32#90411yescharacteristic
406C2×C32#91611yescharacteristic
407C2×C32#91911yescharacteristic
408C2×C32#94711yescharacteristic
409C2×C32#95011yescharacteristic
410C2×C32#95011yescharacteristic
411C3⋊D3—242no—
412C3⋊D3—242no—
413C3⋊D3—242no—
414C3⋊D3—244no—
415C3⋊D3—241no—
416C3⋊D3—241no—
417C3⋊D3—241no—
418C2×Q8—91no—
419C2×D4—271no—
420C2×C8—31no—
421C2×C8—91no—
422C4⋊C4—31no—
423C4⋊C4—91no—
424C4⋊C4—181no—
425C4⋊C4—181no—
426C22⋊C4—541no—
427C22⋊C4—541no—
428C42—31no—
429C22×C3—21no—
430C22×C3—21no—
431C22×C3—21no—
432C22×C3—41no—
433C22×C3D3×D1211yescharacteristic
434C22×C3C2×#62611yescharacteristic
435C22×C3D3×#6711yescharacteristic
436C2×D3—362no—
437C2×D3—362no—
438C2×D3—362no—
439C2×D3—362no—
440C2×D3—362no—
441C2×D3—364no—
442C2×D3—361no—
443C2×D3—361no—
444C2×D3—361no—
445C2×D3—361no—
446C2×D3—361no—
447C2×D3—361no—
448C2×D3—361no—
449C2×D3—361no—
450C2×D3—361no—
451C2×D3—721no—
452C2×D3—721no—
453C2×D3—721no—
454C2×D3—721no—
455C2×D3—722no—
456C2×D3—722no—
457C3×C4—21no—
458C3×C4—21no—
459C3×C4—21no—
460C3×C4—21no—
461C3×C4—21no—
462C3×C4—21no—
463C3×C4—41no—
464C3×C4—41no—
465C3×C4—61no—
466C3×C4—61no—
467C3×C4—61no—
468C3×C4—61no—
469C3×C4—121no—
470C3×C4—121no—
471C3×C4—121no—
472C3×C4—121no—
473C3×C4—241no—
474C3×C4—361no—
475C3×C4—361no—
476C3×C4#379611yescharacteristic
477C3×C4#379811yescharacteristic
478C3×C4#379911yescharacteristic
479C3×C4#379911yescharacteristic
480C3×C4#379911yescharacteristic
481C3×C4#380411yescharacteristic
482Dic3—21no—
483Dic3—21no—
484Dic3—41no—
485Dic3—122no—
486Dic3—121no—
487Dic3—121no—
488Dic3—121no—
489Dic3—121no—
490Dic3—241no—
491C32—21no—
492C32—21no—
493C32—21no—
494C32—41no—
495C32#469711yescharacteristic, complemented
496C32#480011yescharacteristic, complemented
497C32#506811yescharacteristic, complemented
498C23—541no—
499Q8—181no—
500Q8—181no—
501D4—541no—
502D4—541no—
503C2×C4—31no—
504C2×C4—31no—
505C2×C4—61no—
506C2×C4—181no—
507C2×C4—181no—
508C2×C4D3311yescharacteristic
509C8—61no—
510C8—181no—
511C2×C3—21no—
512C2×C3—21no—
513C2×C3—21no—
514C2×C3—21no—
515C2×C3—21no—
516C2×C3—21no—
517C2×C3—21no—
518C2×C3—21no—
519C2×C3—21no—
520C2×C3—41no—
521C2×C3—41no—
522C2×C3—41no—
523C2×C3#6380611yescharacteristic
524C2×C3#6380811yescharacteristic
525C2×C3#6382911yescharacteristic
526C2×C3#6382911yescharacteristic
527C2×C3#6385211yescharacteristic
528C2×C3#6386411yescharacteristic
529C2×C3#6390711yescharacteristic
530C2×C3#6393511yescharacteristic
531C2×C3#6394911yescharacteristic
532D3—722no—
533D3—722no—
534D3—722no—
535D3—724no—
536D3—721no—
537D3—721no—
538D3—721no—
539C22—1081no—
540C22—1081no—
541C22—1081no—
542C22D3×#62611yescharacteristic
543C4—61no—
544C4—61no—
545C4—121no—
546C4—361no—
547C4—361no—
548C4#8896211yescharacteristic
549C4#8896511yescharacteristic
550C3—21no—
551C3—21no—
552C3—21no—
553C3—41no—
554C3#1059029311yescharacteristic, complemented
555C3#1059053711yescharacteristic, complemented
556C3#1059087211yescharacteristic, complemented
557C2—2161no—
558C2#1172285311yescharacteristic
559C2#1172285611yescharacteristic
560C2#1172300311yescharacteristic
subgroupG/H
centerC22D3×#626
commutator, Fitting, cosocleC2×C33×C4C23
Frattini, hypercenterC2×C4D33
socleC22×C33C2×D4
prime radicalC2×C4⋊(C3⋊(C3⋊D3))C22
hypocenterC33#488
LCSG ▷ C2×C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
2. C3×#63869C2semidirect
3. #11722076C2semidirect
6. C3×#63567C2semidirect
7. C3×Dic3×C3⋊C8C2semidirect
25. #63869D3semidirect
36. #63567D3semidirect
45. Dic3×C3⋊C8D3semidirect
325. C33#488semidirect
495. C32#4697semidirect
496. C32#4800semidirect
497. C32#5068semidirect
554. C3#10590293semidirect
555. C3#10590537semidirect
556. C3#10590872semidirect

Maximal subgroups

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

subgroupindex
#117234412
C3×#638692
#117220762
#117220362
#117220352
C3×#635672
C3×Dic3×C3⋊C82
#105908723
#105905373
#105902933