Finite Group Databasebeta

← #13156302

Subgroups — #13156302

Order 1728.

680 conjugacy classes of subgroups.

The subgroup lattice has 560 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11723484C211yesmaximal, characteristic
2C3×#63918C211yesmaximal, characteristic, complemented
3#11722079C211yesmaximal, characteristic, complemented
4#11722036C211yesmaximal, characteristic
5#11722035C211yesmaximal, characteristic
6C3×#63573C211yesmaximal, characteristic, complemented
7C3×Dic3⋊(C3⋊C8)C211yesmaximal, characteristic, complemented
8#10590759—31nomaximal
9#10590466—31nomaximal
10#10590300—31nomaximal
11C2×C4⋊(C3⋊(C3⋊D3))C2211yescharacteristic
12C2×#6289—21no—
13C2×C3×#627—21no—
14C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
15C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
16C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
17C32×C4⋊Dic3C2211yescharacteristic
18C32×C4×Dic3C2211yescharacteristic
19C2×C32×C3⋊C8C2211yescharacteristic
20#88804—21no—
21C3×Dic3⋊Dic3—21no—
22C4×C4⋊(C3⋊D3)—31no—
23C3×#888—31no—
24C3×#869—31no—
25#63918D311yescharacteristic, complemented
26#63894—31no—
27#63894—31no—
28#63894—61no—
29#63648—31no—
30#63636—31no—
31C3×#817—31no—
32C3×Dic3⋊C8—31no—
33C3×#808—31no—
34C3×C4⋊(C3⋊C8)—31no—
35#63574—31no—
36#63573—31no—
37#63573—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))Q812yescomplemented
50C2×C3×C3⋊Dic3—21no—
51C2×C3×C3⋊Dic3—21no—
52C2×C32×Dic3—21no—
53C2×C32×Dic3D411yescharacteristic
54C2×C32×Dic3D411yescharacteristic
55#6289—42no—
56C3×#627—42no—
57C3×C3⋊(C3⋊C8)—21no—
58C32×C3⋊C8—21no—
59#5084—91no—
60#4870—91no—
61#4842—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×C4×C3⋊D3—61no—
70C2×C3×Dic6—61no—
71C2×C3×Dic6—61no—
72C2×#627—21no—
73C2×#624—62no—
74C2×#624—61no—
75C2×#624—61no—
76C2×C32×C8—31no—
77C32×C4⋊C4—31no—
78C32×C42—31no—
79#3752—61no—
80C4⋊(C3⋊Dic3)C2×D311yescharacteristic
81C4⋊(C3⋊Dic3)C2×D311yescharacteristic
82C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
83C3×C4⋊Dic3—21no—
84C3×C4⋊Dic3—31no—
85C3×C4⋊Dic3—31no—
86C3×C4⋊Dic3—31no—
87C3×C4⋊Dic3—31no—
88C3×C4⋊Dic3—61no—
89C3×C4⋊Dic3—61no—
90C3×C4⋊Dic3—61no—
91C3×C4⋊Dic3C2×D311yescharacteristic
92C3×C4⋊Dic3C2×D311yescharacteristic
93C3×Dic3⋊C4—61no—
94C3×C4×Dic3—21no—
95C3×C4×Dic3C2×D311yescharacteristic
96C3×C4×Dic3C2×D311yescharacteristic
97C2×C3×C3⋊C8—21no—
98C2×C3×C3⋊C8—31no—
99C2×C3×C3⋊C8—31no—
100C2×C3×C3⋊C8—61no—
101C2×C3×C3⋊C8C2×D311yescharacteristic
102C2×C3×C3⋊C8C2×D311yescharacteristic
103Dic3⋊Dic3—21no—
104#3722—62no—
105#3722—61no—
106#3722—61no—
107C22×C33C2×D411yescharacteristic
108C2×C3⋊(C3⋊D3)—22no—
109C2×C3⋊(C3⋊D3)—41no—
110C2×C3⋊(C3⋊D3)—41no—
111C33×C4C2×Q811yescharacteristic
112C33×C4#4111yescharacteristic
113C3×C3⋊Dic3—41no—
114C3×C3⋊Dic3—41no—
115C32×Dic3—21no—
116C32×Dic3—41no—
117C32×Dic3SD1612yes—
118C3×C4⋊Q8—91no—
119#893—91no—
120#888—31no—
121C4×D12—91no—
122C4×D12—91no—
123C4×D12—181no—
124#869—31no—
125C3×#106—91no—
126C3×C4⋊C8—91no—
127#821—91no—
128#821—91no—
129#821—91no—
130#821—181no—
131#817—31no—
132#817—91no—
133Dic3⋊C8—31no—
134#809—91no—
135#809—91no—
136#809—91no—
137#809—181no—
138#808—31no—
139#808—91no—
140#808—91no—
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)—32no—
162C4⋊(C3⋊D3)—32no—
163C4⋊(C3⋊D3)—32no—
164C4⋊(C3⋊D3)—62no—
165C4⋊(C3⋊D3)—62no—
166C4⋊(C3⋊D3)—62no—
167C4⋊(C3⋊D3)—61no—
168C4⋊(C3⋊D3)—62no—
169C4⋊(C3⋊D3)—61no—
170C4⋊(C3⋊D3)—61no—
171C4⋊(C3⋊D3)—122no—
172C4⋊(C3⋊D3)—121no—
173C4⋊(C3⋊D3)—121no—
174C4⋊(C3⋊D3)—122no—
175C4×C3⋊D3—122no—
176C2×C3×Dic3—21no—
177C2×C3×Dic3—21no—
178C2×C3×Dic3—21no—
179C2×C3×Dic3—21no—
180C2×C3×Dic3—41no—
181C2×C3×Dic3—62no—
182C2×C3×Dic3—61no—
183C2×C3×Dic3—61no—
184C2×C3×Dic3—61no—
185C2×C3×Dic3—61no—
186C2×C3×Dic3—121no—
187C2×C3×Dic3D1211yescharacteristic
188C2×C3×Dic3D1211yescharacteristic
189C2×C3×Dic3#6711yescharacteristic
190C2×C3×Dic3#6711yescharacteristic
191C3×Dic6—122no—
192C3×Dic6—122no—
193#627—42no—
194#624—124no—
195#624—122no—
196#624—122no—
197C32×C8—61no—
198C3⋊(C3⋊C8)—21no—
199C3×C3⋊C8—21no—
200C3×C3⋊C8—21no—
201C3×C3⋊C8—41no—
202C3×C3⋊C8—61no—
203C3×C3⋊C8—61no—
204C3×C3⋊C8—121no—
205#476—271noSylow
206C2×C33C22⋊Q811yescharacteristic
207C2×C33C2×SD1611yescharacteristic
208C2×C33AGL(1,ℤ/8)11yescharacteristic
209C3⋊(C3⋊D3)—42no—
210C2×C3×Q8—181no—
211C2×D12—91no—
212C2×D12—91no—
213C2×D12—91no—
214C2×D12—181no—
215C2×D12—181no—
216C2×D12—181no—
217C2×D12—361no—
218C2×C4×D3—181no—
219C2×C4×D3—181no—
220C2×C4×D3—182no—
221C2×C4×D3—181no—
222C2×Dic6—61no—
223C2×Dic6—61no—
224C2×C3×C8—31no—
225C2×C3×C8—31no—
226C2×C3×C8—61no—
227C2×C3×C8—91no—
228C3×C4⋊C4—31no—
229C3×C4⋊C4—31no—
230C3×C4⋊C4—61no—
231C3×C4⋊C4—91no—
232C3×C4⋊C4—91no—
233C3×C4⋊C4—181no—
234C3×C42—31no—
235C3×C42—31no—
236C3×C42—61no—
237(C2×D3)⋊C4—181no—
238(C2×D3)⋊C4—181no—
239(C2×D3)⋊C4—182no—
240(C2×D3)⋊C4—181no—
241C4⋊Dic3—31no—
242C4⋊Dic3—31no—
243C4⋊Dic3—31no—
244C4⋊Dic3—31no—
245C4⋊Dic3—61no—
246C4⋊Dic3—61no—
247C4⋊Dic3—61no—
248C4⋊Dic3D3211yescharacteristic
249Dic3⋊C4—61no—
250C4×Dic3D3211yescharacteristic
251C2×C3⋊C8—31no—
252C2×C3⋊C8—31no—
253C2×C3⋊C8—61no—
254C2×C3⋊C8D3211yescharacteristic
255C22×C32—21no—
256C22×C32—21no—
257C22×C32—21no—
258C22×C32—41no—
259C22×C32C2×D1211yescharacteristic
260C22×C32D3×D411yescharacteristic
261C22×C32C2×#6711yescharacteristic
262C2×C3⋊D3—62no—
263C2×C3⋊D3—64no—
264C2×C3⋊D3—62no—
265C2×C3⋊D3—62no—
266C2×C3⋊D3—122no—
267C2×C3⋊D3—122no—
268C2×C3⋊D3—124no—
269C2×C3⋊D3—121no—
270C2×C3⋊D3—121no—
271C2×C3⋊D3—122no—
272C2×C3⋊D3—122no—
273C2×C3⋊D3—121no—
274C2×C3⋊D3—121no—
275C2×C3⋊D3—121no—
276C2×C3⋊D3—121no—
277C2×C3⋊D3—241no—
278C2×C3⋊D3—241no—
279C2×C3⋊D3—241no—
280C2×C3⋊D3—241no—
281C2×C3⋊D3—242no—
282C2×C3⋊D3—242no—
283C32×C4—21no—
284C32×C4—21no—
285C32×C4—21no—
286C32×C4—21no—
287C32×C4—21no—
288C32×C4—21no—
289C32×C4—32no—
290C32×C4—41no—
291C32×C4—41no—
292C32×C4—61no—
293C32×C4—121no—
294C32×C4#23511yescharacteristic
295C32×C4#23711yescharacteristic
296C32×C4D3×Q811yescharacteristic
297C32×C4D3×Q811yescharacteristic
298C32×C4D3×Q811yescharacteristic
299C32×C4#23911yescharacteristic
300C3⋊Dic3—41no—
301C3⋊Dic3—41no—
302C3×Dic3—21no—
303C3×Dic3—21no—
304C3×Dic3—22no—
305C3×Dic3—41no—
306C3×Dic3—41no—
307C3×Dic3—41no—
308C3×Dic3—81no—
309C3×Dic3—122no—
310C3×Dic3—121no—
311C3×Dic3—121no—
312C3×Dic3—121no—
313C3×Dic3—121no—
314C3×Dic3—241no—
315C3×Dic3#20412yes—
316C3×Dic3#21512yes—
317C4⋊Q8—91no—
318C4×D4—271no—
319#106—91no—
320#106—271no—
321C4⋊C8—91no—
322#102—271no—
323#102—271no—
324C33#47611yescharacteristic, complemented, Sylow
325C22×D3—182no—
326C22×D3—181no—
327C22×D3—181no—
328C22×D3—181no—
329C22×D3—361no—
330C22×D3—361no—
331C22×D3—362no—
332C3×Q8—362no—
333C2×C3×C4—21no—
334C2×C3×C4—21no—
335C2×C3×C4—21no—
336C2×C3×C4—31no—
337C2×C3×C4—31no—
338C2×C3×C4—31no—
339C2×C3×C4—31no—
340C2×C3×C4—41no—
341C2×C3×C4—61no—
342C2×C3×C4—61no—
343C2×C3×C4—61no—
344C2×C3×C4—61no—
345C2×C3×C4—121no—
346C2×C3×C4—181no—
347C2×C3×C4—181no—
348C2×C3×C4C2×D3211yescharacteristic
349C2×C3×C4C2×D3211yescharacteristic
350C2×C3×C4C2×D3211yescharacteristic
351C2×Dic3—21no—
352C2×Dic3—62no—
353C2×Dic3—61no—
354C2×Dic3—61no—
355C2×Dic3—61no—
356C2×Dic3—61no—
357C2×Dic3—121no—
358C2×Dic3#62611yescharacteristic
359C2×Dic3#62611yescharacteristic
360D12—92no—
361D12—92no—
362D12—92no—
363D12—182no—
364D12—182no—
365D12—182no—
366D12—182no—
367D12—181no—
368D12—181no—
369D12—181no—
370D12—362no—
371D12—361no—
372D12—361no—
373D12—362no—
374C4×D3—362no—
375C4×D3—362no—
376C4×D3—364no—
377C4×D3—362no—
378Dic6—122no—
379Dic6—122no—
380C3×C8—61no—
381C3×C8—61no—
382C3×C8—121no—
383C3×C8—181no—
384C3⋊C8—21no—
385C3⋊C8—61no—
386C3⋊C8—61no—
387C3⋊C8—121no—
388C2×C32—21no—
389C2×C32—21no—
390C2×C32—21no—
391C2×C32—21no—
392C2×C32—21no—
393C2×C32—21no—
394C2×C32—21no—
395C2×C32—21no—
396C2×C32—21no—
397C2×C32—41no—
398C2×C32—41no—
399C2×C32—41no—
400C2×C32#89611yescharacteristic
401C2×C32#89711yescharacteristic
402C2×C32C2×#20411yescharacteristic
403C2×C32#90811yescharacteristic
404C2×C32D3×SD1611yescharacteristic
405C2×C32#91411yescharacteristic
406C2×C32#93211yescharacteristic
407C2×C32C2×#21511yescharacteristic
408C2×C32#94611yescharacteristic
409C3⋊D3—122no—
410C3⋊D3—124no—
411C3⋊D3—122no—
412C3⋊D3—122no—
413C3⋊D3—242no—
414C3⋊D3—242no—
415C3⋊D3—244no—
416C2×Q8—181no—
417C2×D4—271no—
418C22×C4—541no—
419C2×C8—31no—
420C2×C8—91no—
421C4⋊C4—31no—
422C4⋊C4—91no—
423C4⋊C4—91no—
424C4⋊C4—181no—
425C22⋊C4—541no—
426C42—31no—
427C22×C3—21no—
428C22×C3—21no—
429C22×C3—21no—
430C22×C3—41no—
431C22×C3D3×D1211yescharacteristic
432C22×C3C2×#62611yescharacteristic
433C22×C3D3×#6711yescharacteristic
434C2×D3—184no—
435C2×D3—182no—
436C2×D3—182no—
437C2×D3—182no—
438C2×D3—362no—
439C2×D3—362no—
440C2×D3—364no—
441C2×D3—362no—
442C2×D3—362no—
443C2×D3—361no—
444C2×D3—361no—
445C2×D3—361no—
446C2×D3—361no—
447C2×D3—361no—
448C2×D3—361no—
449C2×D3—721no—
450C2×D3—721no—
451C2×D3—721no—
452C2×D3—721no—
453C2×D3—722no—
454C2×D3—722no—
455C3×C4—21no—
456C3×C4—21no—
457C3×C4—21no—
458C3×C4—21no—
459C3×C4—21no—
460C3×C4—21no—
461C3×C4—32no—
462C3×C4—32no—
463C3×C4—41no—
464C3×C4—41no—
465C3×C4—62no—
466C3×C4—61no—
467C3×C4—61no—
468C3×C4—121no—
469C3×C4—121no—
470C3×C4—121no—
471C3×C4—241no—
472C3×C4—361no—
473C3×C4—361no—
474C3×C4#379611yescharacteristic
475C3×C4#379711yescharacteristic
476C3×C4#379711yescharacteristic
477C3×C4#379711yescharacteristic
478C3×C4#379911yescharacteristic
479C3×C4#380411yescharacteristic
480Dic3—21no—
481Dic3—41no—
482Dic3—122no—
483Dic3—121no—
484Dic3—121no—
485Dic3—121no—
486Dic3—121no—
487Dic3—241no—
488Dic3#371712yes—
489C32—21no—
490C32—21no—
491C32—21no—
492C32—41no—
493C32#484211yescharacteristic, complemented
494C32#487011yescharacteristic, complemented
495C32#508411yescharacteristic, complemented
496C23—541no—
497Q8—362no—
498D4—272no—
499D4—541no—
500C2×C4—31no—
501C2×C4—31no—
502C2×C4—61no—
503C2×C4—181no—
504C2×C4—181no—
505C2×C4—1082no—
506C2×C4D3311yescharacteristic
507C8—61no—
508C8—181no—
509C2×C3—21no—
510C2×C3—21no—
511C2×C3—21no—
512C2×C3—21no—
513C2×C3—21no—
514C2×C3—21no—
515C2×C3—21no—
516C2×C3—21no—
517C2×C3—21no—
518C2×C3—41no—
519C2×C3—41no—
520C2×C3—41no—
521C2×C3D3×#20411yescharacteristic
522C2×C3#6379411yescharacteristic
523C2×C3#6383011yescharacteristic
524C2×C3C2×#371711yescharacteristic
525C2×C3#6387111yescharacteristic
526C2×C3#6388811yescharacteristic
527C2×C3#6389511yescharacteristic
528C2×C3#6393411yescharacteristic
529C2×C3D3×#21511yescharacteristic
530D3—364no—
531D3—362no—
532D3—362no—
533D3—362no—
534D3—722no—
535D3—722no—
536D3—724no—
537C22—542no—
538C22—1081no—
539C22—1081no—
540C22D3×#62611yescharacteristic
541C4—32no—
542C4—61no—
543C4—121no—
544C4—361no—
545C4—361no—
546C4#8895811yescharacteristic
547C4#8896211yescharacteristic
548C3—21no—
549C3—21no—
550C3—21no—
551C3—41no—
552C3#1059030011yescharacteristic, complemented
553C3#1059046611yescharacteristic, complemented
554C3#1059075911yescharacteristic, complemented
555C2—1082no—
556C2D3×#371711yescharacteristic
557C2#1172286811yescharacteristic
558C2#1172297311yescharacteristic
subgroupG/H
centerC22D3×#626
commutator, Fitting, cosocleC2×C33×C4C23
Frattini, hypercenterC2×C4D33
socleC22×C33C2×D4
prime radicalC2×C4⋊(C3⋊(C3⋊D3))C22
hypocenterC33#476
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×#63918C2semidirect
3. #11722079C2semidirect
6. C3×#63573C2semidirect
7. C3×Dic3⋊(C3⋊C8)C2semidirect
25. #63918D3semidirect
38. #63573D3semidirect
39. Dic3⋊(C3⋊C8)D3semidirect
49. C4⋊(C3⋊(C3⋊D3))Q8semidirect
324. C33#476semidirect
493. C32#4842semidirect
494. C32#4870semidirect
495. C32#5084semidirect
552. C3#10590300semidirect
553. C3#10590466semidirect
554. C3#10590759semidirect

Maximal subgroups

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

subgroupindex
#117234842
C3×#639182
#117220792
#117220362
#117220352
C3×#635732
C3×Dic3⋊(C3⋊C8)2
#105907593
#105904663
#105903003