Finite Group Databasebeta

← #13156338

Subgroups — #13156338

Order 1728.

656 conjugacy classes of subgroups.

The subgroup lattice has 552 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11723484C211yesmaximal, characteristic
2C3×#63869C211yesmaximal, characteristic, complemented
3#11722080C211yesmaximal, characteristic, complemented
4#11722036C211yesmaximal, characteristic
5#11722035C211yesmaximal, characteristic
6C3×#63573C211yesmaximal, characteristic, complemented
7C3×Dic3⋊(C3⋊C8)C211yesmaximal, characteristic, complemented
8#10590769—31nomaximal
9#10590470—31nomaximal
10#10590308—31nomaximal
11C2×C4⋊(C3⋊(C3⋊D3))C2211yescharacteristic
12C2×#6289—21no—
13C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
14C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
15C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
16C32×C4⋊Dic3C2211yescharacteristic
17C32×C4×Dic3C2211yescharacteristic
18C2×C32×C3⋊C8C2211yescharacteristic
19#88804—21no—
20C3×#3724—21no—
21C3×Dic3⋊Dic3—21no—
22C4×C4⋊(C3⋊D3)—31no—
23C3×#890—31no—
24C3×#870—31no—
25#63894—31no—
26#63894—31no—
27#63894—61no—
28#63869D311yescharacteristic, complemented
29#63648—31no—
30#63636—31no—
31C3×#817—31no—
32C3×Dic3⋊C8—31no—
33C3×#808—31no—
34C3×C4⋊(C3⋊C8)—31no—
35#63576—31no—
36#63575—31no—
37#63575—31no—
38#63573D311yescharacteristic, complemented
39Dic3⋊(C3⋊C8)D311yescharacteristic, complemented
40#63564—31no—
41#63564—31no—
42#63564—31no—
43#63564—31no—
44#63564—61no—
45#63564—61no—
46C22×C3⋊(C3⋊D3)—21no—
47C2×C33×C4C2311yescharacteristic
48C4⋊(C3⋊(C3⋊D3))—21no—
49C4⋊(C3⋊(C3⋊D3))Q812yes—
50C2×C3×C3⋊Dic3—21no—
51C2×C3×C3⋊Dic3—21no—
52C2×C32×Dic3—21no—
53C2×C32×Dic3D411yescharacteristic
54C2×C32×Dic3D411yescharacteristic
55#6289—42no—
56C3×C3⋊(C3⋊C8)—21no—
57C32×C3⋊C8—21no—
58#5066—91no—
59#4871—91no—
60#4847—91no—
61C2×C4⋊(C3⋊D3)—31no—
62C2×C4⋊(C3⋊D3)—31no—
63C2×C4⋊(C3⋊D3)—31no—
64C2×C4⋊(C3⋊D3)—61no—
65C2×C4⋊(C3⋊D3)—61no—
66C2×C4⋊(C3⋊D3)—61no—
67C2×C4⋊(C3⋊D3)—121no—
68C2×C4×C3⋊D3—61no—
69C2×#624—62no—
70C2×#624—61no—
71C2×#624—61no—
72C2×C32×C8—31no—
73C32×C4⋊C4—31no—
74C32×C42—31no—
75#3752—61no—
76C4⋊(C3⋊Dic3)C2×D311yescharacteristic
77C4⋊(C3⋊Dic3)C2×D311yescharacteristic
78C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
79C3×C4⋊Dic3—21no—
80C3×C4⋊Dic3—31no—
81C3×C4⋊Dic3—31no—
82C3×C4⋊Dic3—31no—
83C3×C4⋊Dic3—31no—
84C3×C4⋊Dic3—61no—
85C3×C4⋊Dic3—61no—
86C3×C4⋊Dic3—61no—
87C3×C4⋊Dic3C2×D311yescharacteristic
88C3×C4⋊Dic3C2×D311yescharacteristic
89C3×Dic3⋊C4—61no—
90C3×Dic3⋊C4—61no—
91C3×Dic3⋊C4—61no—
92C3×C4×Dic3—21no—
93C3×C4×Dic3C2×D311yescharacteristic
94C3×C4×Dic3C2×D311yescharacteristic
95C2×C3×C3⋊C8—21no—
96C2×C3×C3⋊C8—31no—
97C2×C3×C3⋊C8—31no—
98C2×C3×C3⋊C8—61no—
99C2×C3×C3⋊C8C2×D311yescharacteristic
100C2×C3×C3⋊C8C2×D311yescharacteristic
101#3724—21no—
102Dic3⋊Dic3—21no—
103#3722—62no—
104#3722—61no—
105#3722—61no—
106C22×C33C2×D411yescharacteristic
107C2×C3⋊(C3⋊D3)—22no—
108C2×C3⋊(C3⋊D3)—41no—
109C2×C3⋊(C3⋊D3)—41no—
110C33×C4C2×Q811yescharacteristic
111C33×C4#4111yescharacteristic
112C3×C3⋊Dic3—41no—
113C3×C3⋊Dic3—41no—
114C32×Dic3—21no—
115C32×Dic3—21no—
116C32×Dic3—41no—
117C3×#125—91no—
118#893—91no—
119#890—31no—
120C4×D12—91no—
121C4×D12—91no—
122C4×D12—181no—
123#870—31no—
124C3×#106—91no—
125C3×C4⋊C8—91no—
126#821—91no—
127#821—91no—
128#821—91no—
129#821—181no—
130#818—91no—
131#817—31no—
132Dic3⋊C8—31no—
133#809—91no—
134#809—91no—
135#809—91no—
136#809—181no—
137#808—31no—
138#807—91no—
139#807—91no—
140C4⋊(C3⋊C8)—31no—
141C22×C3⋊D3—61no—
142C22×C3⋊D3—62no—
143C22×C3⋊D3—61no—
144C22×C3⋊D3—61no—
145C22×C3⋊D3—121no—
146C22×C3⋊D3—121no—
147C22×C3⋊D3—122no—
148C2×C32×C4—21no—
149C2×C32×C4—21no—
150C2×C32×C4—21no—
151C2×C32×C4—31no—
152C2×C32×C4—31no—
153C2×C32×C4—41no—
154C2×C32×C4—61no—
155C2×C32×C4C22×D311yescharacteristic
156C2×C32×C4C22×D311yescharacteristic
157C2×C32×C4C22×D311yescharacteristic
158C2×C3⋊Dic3—21no—
159C2×C3⋊Dic3—21no—
160C4⋊(C3⋊D3)—32no—
161C4⋊(C3⋊D3)—32no—
162C4⋊(C3⋊D3)—32no—
163C4⋊(C3⋊D3)—62no—
164C4⋊(C3⋊D3)—62no—
165C4⋊(C3⋊D3)—62no—
166C4⋊(C3⋊D3)—61no—
167C4⋊(C3⋊D3)—62no—
168C4⋊(C3⋊D3)—61no—
169C4⋊(C3⋊D3)—61no—
170C4⋊(C3⋊D3)—122no—
171C4⋊(C3⋊D3)—121no—
172C4⋊(C3⋊D3)—121no—
173C4⋊(C3⋊D3)—122no—
174C4×C3⋊D3—122no—
175C2×C3×Dic3—21no—
176C2×C3×Dic3—21no—
177C2×C3×Dic3—21no—
178C2×C3×Dic3—21no—
179C2×C3×Dic3—41no—
180C2×C3×Dic3—62no—
181C2×C3×Dic3—61no—
182C2×C3×Dic3—61no—
183C2×C3×Dic3—61no—
184C2×C3×Dic3—61no—
185C2×C3×Dic3—121no—
186C2×C3×Dic3D1211yescharacteristic
187C2×C3×Dic3D1211yescharacteristic
188C2×C3×Dic3#6711yescharacteristic
189C2×C3×Dic3#6711yescharacteristic
190#624—124no—
191#624—122no—
192#624—122no—
193C32×C8—61no—
194C3⋊(C3⋊C8)—21no—
195C3×C3⋊C8—21no—
196C3×C3⋊C8—21no—
197C3×C3⋊C8—41no—
198C3×C3⋊C8—61no—
199C3×C3⋊C8—61no—
200C3×C3⋊C8—121no—
201#478—271noSylow
202C2×C33C22⋊Q811yescharacteristic
203C2×C33#13511yescharacteristic
204C2×C33AGL(1,ℤ/8)11yescharacteristic
205C3⋊(C3⋊D3)—42no—
206C2×D12—91no—
207C2×D12—91no—
208C2×D12—91no—
209C2×D12—181no—
210C2×D12—181no—
211C2×D12—181no—
212C2×D12—361no—
213C2×C4×D3—181no—
214C2×C4×D3—181no—
215C2×C4×D3—182no—
216C2×C4×D3—181no—
217C2×C3×C8—31no—
218C2×C3×C8—31no—
219C2×C3×C8—61no—
220C2×C3×C8—91no—
221C3×C4⋊C4—31no—
222C3×C4⋊C4—31no—
223C3×C4⋊C4—61no—
224C3×C4⋊C4—91no—
225C3×C4⋊C4—91no—
226C3×C4⋊C4—181no—
227C3×C4⋊C4—181no—
228C3×C42—31no—
229C3×C42—31no—
230C3×C42—61no—
231(C2×D3)⋊C4—181no—
232(C2×D3)⋊C4—181no—
233(C2×D3)⋊C4—182no—
234(C2×D3)⋊C4—181no—
235C4⋊Dic3—31no—
236C4⋊Dic3—31no—
237C4⋊Dic3—31no—
238C4⋊Dic3—31no—
239C4⋊Dic3—61no—
240C4⋊Dic3—61no—
241C4⋊Dic3—61no—
242C4⋊Dic3D3211yescharacteristic
243Dic3⋊C4—61no—
244Dic3⋊C4—61no—
245Dic3⋊C4—61no—
246C4×Dic3D3211yescharacteristic
247C2×C3⋊C8—31no—
248C2×C3⋊C8—31no—
249C2×C3⋊C8—61no—
250C2×C3⋊C8D3211yescharacteristic
251C22×C32—21no—
252C22×C32—21no—
253C22×C32—21no—
254C22×C32—41no—
255C22×C32C2×D1211yescharacteristic
256C22×C32D3×D411yescharacteristic
257C22×C32C2×#6711yescharacteristic
258C2×C3⋊D3—62no—
259C2×C3⋊D3—64no—
260C2×C3⋊D3—62no—
261C2×C3⋊D3—62no—
262C2×C3⋊D3—122no—
263C2×C3⋊D3—122no—
264C2×C3⋊D3—124no—
265C2×C3⋊D3—121no—
266C2×C3⋊D3—121no—
267C2×C3⋊D3—122no—
268C2×C3⋊D3—122no—
269C2×C3⋊D3—121no—
270C2×C3⋊D3—121no—
271C2×C3⋊D3—121no—
272C2×C3⋊D3—121no—
273C2×C3⋊D3—241no—
274C2×C3⋊D3—241no—
275C2×C3⋊D3—241no—
276C2×C3⋊D3—241no—
277C2×C3⋊D3—242no—
278C2×C3⋊D3—242no—
279C32×C4—21no—
280C32×C4—21no—
281C32×C4—21no—
282C32×C4—21no—
283C32×C4—21no—
284C32×C4—21no—
285C32×C4—41no—
286C32×C4—41no—
287C32×C4—61no—
288C32×C4—61no—
289C32×C4—121no—
290C32×C4#23511yescharacteristic
291C32×C4#23711yescharacteristic
292C32×C4D3×Q811yescharacteristic
293C32×C4D3×Q811yescharacteristic
294C32×C4D3×Q811yescharacteristic
295C32×C4#23911yescharacteristic
296C3⋊Dic3—41no—
297C3⋊Dic3—41no—
298C3×Dic3—22no—
299C3×Dic3—21no—
300C3×Dic3—21no—
301C3×Dic3—21no—
302C3×Dic3—21no—
303C3×Dic3—41no—
304C3×Dic3—41no—
305C3×Dic3—41no—
306C3×Dic3—81no—
307C3×Dic3—122no—
308C3×Dic3—121no—
309C3×Dic3—121no—
310C3×Dic3—121no—
311C3×Dic3—121no—
312C3×Dic3—241no—
313#125—91no—
314C4×D4—271no—
315#107—271no—
316#106—91no—
317C4⋊C8—91no—
318#102—271no—
319#102—271no—
320C33#47811yescharacteristic, complemented, Sylow
321C22×D3—182no—
322C22×D3—181no—
323C22×D3—181no—
324C22×D3—181no—
325C22×D3—361no—
326C22×D3—361no—
327C22×D3—362no—
328C2×C3×C4—21no—
329C2×C3×C4—21no—
330C2×C3×C4—21no—
331C2×C3×C4—31no—
332C2×C3×C4—31no—
333C2×C3×C4—31no—
334C2×C3×C4—31no—
335C2×C3×C4—41no—
336C2×C3×C4—61no—
337C2×C3×C4—61no—
338C2×C3×C4—61no—
339C2×C3×C4—61no—
340C2×C3×C4—121no—
341C2×C3×C4—181no—
342C2×C3×C4—181no—
343C2×C3×C4C2×D3211yescharacteristic
344C2×C3×C4C2×D3211yescharacteristic
345C2×C3×C4C2×D3211yescharacteristic
346C2×Dic3—21no—
347C2×Dic3—62no—
348C2×Dic3—61no—
349C2×Dic3—61no—
350C2×Dic3—61no—
351C2×Dic3—61no—
352C2×Dic3—121no—
353C2×Dic3#62611yescharacteristic
354C2×Dic3#62611yescharacteristic
355D12—92no—
356D12—92no—
357D12—92no—
358D12—182no—
359D12—182no—
360D12—182no—
361D12—182no—
362D12—181no—
363D12—181no—
364D12—181no—
365D12—362no—
366D12—361no—
367D12—361no—
368D12—362no—
369C4×D3—362no—
370C4×D3—364no—
371C4×D3—362no—
372C4×D3—362no—
373C3×C8—61no—
374C3×C8—61no—
375C3×C8—121no—
376C3×C8—181no—
377C3⋊C8—21no—
378C3⋊C8—61no—
379C3⋊C8—61no—
380C3⋊C8—121no—
381C2×C32—21no—
382C2×C32—21no—
383C2×C32—21no—
384C2×C32—21no—
385C2×C32—21no—
386C2×C32—21no—
387C2×C32—21no—
388C2×C32—21no—
389C2×C32—21no—
390C2×C32—41no—
391C2×C32—41no—
392C2×C32—41no—
393C2×C32#89611yescharacteristic
394C2×C32#89711yescharacteristic
395C2×C32#90411yescharacteristic
396C2×C32#90811yescharacteristic
397C2×C32#91411yescharacteristic
398C2×C32#91611yescharacteristic
399C2×C32#94611yescharacteristic
400C2×C32#94911yescharacteristic
401C2×C32#95011yescharacteristic
402C3⋊D3—122no—
403C3⋊D3—124no—
404C3⋊D3—122no—
405C3⋊D3—122no—
406C3⋊D3—242no—
407C3⋊D3—242no—
408C3⋊D3—244no—
409C2×D4—271no—
410C22×C4—541no—
411C2×C8—31no—
412C2×C8—91no—
413C4⋊C4—31no—
414C4⋊C4—91no—
415C4⋊C4—91no—
416C4⋊C4—181no—
417C4⋊C4—181no—
418C22⋊C4—541no—
419C42—31no—
420C22×C3—21no—
421C22×C3—21no—
422C22×C3—21no—
423C22×C3—41no—
424C22×C3D3×D1211yescharacteristic
425C22×C3C2×#62611yescharacteristic
426C22×C3D3×#6711yescharacteristic
427C2×D3—184no—
428C2×D3—182no—
429C2×D3—182no—
430C2×D3—182no—
431C2×D3—362no—
432C2×D3—362no—
433C2×D3—364no—
434C2×D3—362no—
435C2×D3—362no—
436C2×D3—361no—
437C2×D3—361no—
438C2×D3—361no—
439C2×D3—361no—
440C2×D3—361no—
441C2×D3—361no—
442C2×D3—721no—
443C2×D3—721no—
444C2×D3—721no—
445C2×D3—721no—
446C2×D3—722no—
447C2×D3—722no—
448C3×C4—21no—
449C3×C4—21no—
450C3×C4—21no—
451C3×C4—21no—
452C3×C4—21no—
453C3×C4—21no—
454C3×C4—41no—
455C3×C4—41no—
456C3×C4—62no—
457C3×C4—61no—
458C3×C4—61no—
459C3×C4—61no—
460C3×C4—61no—
461C3×C4—121no—
462C3×C4—121no—
463C3×C4—121no—
464C3×C4—241no—
465C3×C4—361no—
466C3×C4—361no—
467C3×C4#379611yescharacteristic
468C3×C4#379711yescharacteristic
469C3×C4#379711yescharacteristic
470C3×C4#379711yescharacteristic
471C3×C4#379911yescharacteristic
472C3×C4#380411yescharacteristic
473Dic3—21no—
474Dic3—21no—
475Dic3—41no—
476Dic3—122no—
477Dic3—121no—
478Dic3—121no—
479Dic3—121no—
480Dic3—121no—
481Dic3—241no—
482C32—21no—
483C32—21no—
484C32—21no—
485C32—41no—
486C32#484711yescharacteristic, complemented
487C32#487111yescharacteristic, complemented
488C32#506611yescharacteristic, complemented
489C23—541no—
490D4—272no—
491D4—541no—
492C2×C4—31no—
493C2×C4—31no—
494C2×C4—61no—
495C2×C4—181no—
496C2×C4—181no—
497C2×C4—1082no—
498C2×C4D3311yescharacteristic
499C8—61no—
500C8—181no—
501C2×C3—21no—
502C2×C3—21no—
503C2×C3—21no—
504C2×C3—21no—
505C2×C3—21no—
506C2×C3—21no—
507C2×C3—21no—
508C2×C3—21no—
509C2×C3—21no—
510C2×C3—41no—
511C2×C3—41no—
512C2×C3—41no—
513C2×C3#6379411yescharacteristic
514C2×C3#6380611yescharacteristic
515C2×C3#6382511yescharacteristic
516C2×C3#6382911yescharacteristic
517C2×C3#6387111yescharacteristic
518C2×C3#6388811yescharacteristic
519C2×C3#6389511yescharacteristic
520C2×C3#6393511yescharacteristic
521C2×C3#6393911yescharacteristic
522D3—364no—
523D3—362no—
524D3—362no—
525D3—362no—
526D3—722no—
527D3—722no—
528D3—724no—
529C22—542no—
530C22—1081no—
531C22—1081no—
532C22D3×#62611yescharacteristic
533C4—61no—
534C4—61no—
535C4—121no—
536C4—361no—
537C4—361no—
538C4#8895811yescharacteristic
539C4#8896211yescharacteristic
540C3—21no—
541C3—21no—
542C3—21no—
543C3—41no—
544C3#1059030811yescharacteristic, complemented
545C3#1059047011yescharacteristic, complemented
546C3#1059076911yescharacteristic, complemented
547C2—1082no—
548C2#1172285311yescharacteristic
549C2#1172287111yescharacteristic
550C2#1172297311yescharacteristic
subgroupG/H
centerC22D3×#626
commutator, Fitting, cosocleC2×C33×C4C23
Frattini, hypercenterC2×C4D33
socleC22×C33C2×D4
prime radicalC2×C4⋊(C3⋊(C3⋊D3))C22
hypocenterC33#478
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. #11722080C2semidirect
6. C3×#63573C2semidirect
7. C3×Dic3⋊(C3⋊C8)C2semidirect
28. #63869D3semidirect
38. #63573D3semidirect
39. Dic3⋊(C3⋊C8)D3semidirect
320. C33#478semidirect
486. C32#4847semidirect
487. C32#4871semidirect
488. C32#5066semidirect
544. C3#10590308semidirect
545. C3#10590470semidirect
546. C3#10590769semidirect

Maximal subgroups

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

subgroupindex
#117234842
C3×#638692
#117220802
#117220362
#117220352
C3×#635732
C3×Dic3⋊(C3⋊C8)2
#105907693
#105904703
#105903083