Finite Group Databasebeta

← #13156545

Subgroups — #13156545

Order 1728.

598 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#11723439C211yesmaximal, characteristic
2C3×#63918C211yesmaximal, characteristic, complemented
3#11722075C211yesmaximal, characteristic
4#11722042C211yesmaximal, characteristic, complemented
5#11722041C211yesmaximal, characteristic, complemented
6C3×#63564C211yesmaximal, characteristic
7C3×#63554C211yesmaximal, characteristic, complemented
8#10590895—31nomaximal
9#10590531—31nomaximal
10#10590294—31nomaximal
11C2×#6308C2211yescharacteristic
12C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
13C2×C3×#627—21no—
14C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
15C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
16C32×C4⋊Dic3C2211yescharacteristic
17C32×C4×Dic3C2211yescharacteristic
18C2×C32×C3⋊C8C2211yescharacteristic
19#88803—21no—
20#88803—21no—
21C3×Dic3⋊Dic3—21no—
22#64084—31no—
23C3×#888—31no—
24C3×#869—31no—
25#63918D311yescharacteristic, complemented
26#63860—31no—
27#63860—31no—
28#63860—61no—
29#63643—31no—
30#63637—31no—
31C3×#821—31no—
32C3×#815—31no—
33C3×#809—31no—
34C3×#803—31no—
35#63567—31no—
36#63567—31no—
37#63567—31no—
38#63567—31no—
39#63567—61no—
40#63567—61no—
41#63564—31no—
42#63564D311yescharacteristic
43#63563—31no—
44#63562—31no—
45#63554D311yescharacteristic, complemented
46C22×C3×C3⋊D3—21no—
47C2×C33×C4C2311yescharacteristic
48C2×C3⋊(C3⋊Dic3)—21no—
49#6308—21no—
50#6308—21no—
51C2×C3×C3⋊Dic3—21no—
52C3×C4⋊(C3⋊D3)—21no—
53C3×C4⋊(C3⋊D3)—21no—
54C2×C32×Dic3—21no—
55C2×C32×Dic3D411yescharacteristic
56C2×C32×Dic3D411yescharacteristic
57C3×#627—42no—
58C3×C3⋊(C3⋊C8)—21no—
59C32×C3⋊C8—21no—
60#5089—91no—
61#4765—91no—
62#4714—91no—
63C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
64C2×#634—31no—
65C2×#634—31no—
66C2×#634—31no—
67C2×#634—61no—
68C2×#634—61no—
69C2×#634—61no—
70C2×#634—121no—
71C2×C3×D12—31no—
72C2×C3×D12—31no—
73C2×C3×D12—61no—
74C2×C3×Dic6—61no—
75C2×C3×Dic6—61no—
76C2×#627—21no—
77C2×C32×C8—31no—
78C32×C4⋊C4—31no—
79C32×C42—31no—
80#3752—61no—
81#3752—61no—
82C4⋊(C3⋊Dic3)C2×D311yescharacteristic
83C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
84C3×C4⋊Dic3—21no—
85C3×C4⋊Dic3—31no—
86C3×C4⋊Dic3—31no—
87C3×C4⋊Dic3—61no—
88C3×C4⋊Dic3—61no—
89C3×C4⋊Dic3C2×D311yescharacteristic
90C3×C4⋊Dic3C2×D311yescharacteristic
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
101Dic3⋊Dic3—21no—
102(C2×D3)⋊Dic3—62no—
103(C2×D3)⋊Dic3—62no—
104(C2×D3)⋊Dic3—61no—
105(C2×D3)⋊Dic3—61no—
106(C2×D3)⋊Dic3—61no—
107(C2×D3)⋊Dic3—61no—
108C22×C33C2×D411yescharacteristic
109C2×C3×C3⋊D3—41no—
110C2×C3×C3⋊D3—41no—
111C2×C3×C3⋊D3—41no—
112C33×C4#4111yescharacteristic
113C33×C4#4111yescharacteristic
114C3⋊(C3⋊Dic3)—41no—
115C3×C3⋊Dic3—41no—
116C32×Dic3—21no—
117C32×Dic3—21no—
118C32×Dic3—41no—
119C3×C4⋊Q8—91no—
120#936—91no—
121#888—31no—
122#875—91no—
123#875—91no—
124#875—181no—
125#869—31no—
126C3×#102—91no—
127C3×#97—91no—
128#832—91no—
129#821—31no—
130#821—91no—
131#816—91no—
132#816—91no—
133#816—91no—
134#816—181no—
135#815—31no—
136#810—91no—
137#810—91no—
138#810—91no—
139#810—181no—
140#809—31no—
141#809—91no—
142#803—31no—
143C22×C3⋊D3—21no—
144C22×C3×D3—62no—
145C22×C3×D3—61no—
146C22×C3×D3—61no—
147C2×C32×C4—21no—
148C2×C32×C4—21no—
149C2×C32×C4—21no—
150C2×C32×C4—31no—
151C2×C32×C4—31no—
152C2×C32×C4—41no—
153C2×C32×C4—61no—
154C2×C32×C4C22×D311yescharacteristic
155C2×C32×C4C22×D311yescharacteristic
156C2×C32×C4C22×D311yescharacteristic
157C2×C3⋊Dic3—21no—
158C2×C3⋊Dic3—61no—
159C2×C3⋊Dic3—62no—
160C2×C3⋊Dic3—61no—
161C2×C3⋊Dic3—61no—
162C2×C3⋊Dic3—121no—
163C2×C3⋊Dic3—121no—
164C2×C3⋊Dic3—122no—
165C4⋊(C3⋊D3)—21no—
166C4⋊(C3⋊D3)—21no—
167#634—61no—
168#634—61no—
169#634—62no—
170#634—62no—
171#634—61no—
172#634—61no—
173#634—61no—
174#634—61no—
175#634—121no—
176#634—121no—
177#634—121no—
178#634—121no—
179#634—122no—
180#634—122no—
181C2×C3×Dic3—21no—
182C2×C3×Dic3—21no—
183C2×C3×Dic3—21no—
184C2×C3×Dic3—21no—
185C2×C3×Dic3—41no—
186C2×C3×Dic3—61no—
187C2×C3×Dic3—61no—
188C2×C3×Dic3—121no—
189C2×C3×Dic3D1211yescharacteristic
190C2×C3×Dic3D1211yescharacteristic
191C2×C3×Dic3#6711yescharacteristic
192C2×C3×Dic3#6711yescharacteristic
193C3×D12—61no—
194C3×D12—61no—
195C3×D12—61no—
196C3×D12—61no—
197C3×D12—121no—
198C3×D12—121no—
199C3×Dic6—122no—
200C3×Dic6—122no—
201#627—42no—
202C32×C8—61no—
203C3⋊(C3⋊C8)—21no—
204C3×C3⋊C8—21no—
205C3×C3⋊C8—21no—
206C3×C3⋊C8—41no—
207C3×C3⋊C8—61no—
208C3×C3⋊C8—61no—
209C3×C3⋊C8—121no—
210#489—271noSylow
211C2×C33#12411yescharacteristic
212C2×C33AGL(1,ℤ/8)11yescharacteristic
213C2×C33#13711yescharacteristic
214C3×C3⋊D3—81no—
215C2×C3×Q8—181no—
216C2×C3×D4—91no—
217C2×D12—31no—
218C2×D12—31no—
219C2×D12—61no—
220C2×Dic6—61no—
221C2×Dic6—61no—
222C2×Dic6—91no—
223C2×Dic6—91no—
224C2×Dic6—91no—
225C2×Dic6—181no—
226C2×Dic6—181no—
227C2×Dic6—181no—
228C2×Dic6—361no—
229C2×C3×C8—31no—
230C2×C3×C8—31no—
231C2×C3×C8—61no—
232C2×C3×C8—91no—
233C3×C4⋊C4—31no—
234C3×C4⋊C4—31no—
235C3×C4⋊C4—61no—
236C3×C4⋊C4—91no—
237C3×C4⋊C4—181no—
238C3×C42—31no—
239C3×C42—31no—
240C3×C42—61no—
241#217—181no—
242#217—181no—
243(C2×D3)⋊C4—181no—
244(C2×D3)⋊C4—181no—
245(C2×D3)⋊C4—181no—
246(C2×D3)⋊C4—181no—
247(C2×D3)⋊C4—182no—
248(C2×D3)⋊C4—182no—
249C4⋊Dic3—31no—
250C4⋊Dic3—31no—
251C4⋊Dic3—61no—
252C4⋊Dic3—61no—
253C4⋊Dic3D3211yescharacteristic
254Dic3⋊C4—61no—
255C4×Dic3D3211yescharacteristic
256C2×C3⋊C8—31no—
257C2×C3⋊C8—31no—
258C2×C3⋊C8—61no—
259C2×C3⋊C8D3211yescharacteristic
260C22×C32—21no—
261C22×C32—21no—
262C22×C32—21no—
263C22×C32—41no—
264C22×C32C2×D1211yescharacteristic
265C22×C32D3×D411yescharacteristic
266C22×C32C2×#6711yescharacteristic
267C2×C3⋊D3—41no—
268C2×C3⋊D3—41no—
269C2×C3⋊D3—41no—
270C2×C3×D3—122no—
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—121no—
278C2×C3×D3—121no—
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#23511yescharacteristic
292C32×C4#23511yescharacteristic
293C32×C4#23711yescharacteristic
294C32×C4#23911yescharacteristic
295C32×C4#23911yescharacteristic
296C3⋊Dic3—41no—
297C3⋊Dic3—122no—
298C3⋊Dic3—122no—
299C3⋊Dic3—122no—
300C3⋊Dic3—124no—
301C3⋊Dic3—121no—
302C3⋊Dic3—121no—
303C3⋊Dic3—121no—
304C3×Dic3—21no—
305C3×Dic3—21no—
306C3×Dic3—21no—
307C3×Dic3—21no—
308C3×Dic3—41no—
309C3×Dic3—41no—
310C3×Dic3—41no—
311C3×Dic3—41no—
312C3×Dic3—81no—
313C3×Dic3—121no—
314C3×Dic3—121no—
315C3×Dic3—241no—
316C4⋊Q8—91no—
317#124—271no—
318#103—271no—
319#103—271no—
320#102—91no—
321#102—271no—
322#97—91no—
323C33#48911yescharacteristic, complemented, Sylow
324C23×C3—181no—
325C22×D3—62no—
326C22×D3—61no—
327C22×D3—61no—
328C3×Q8—362no—
329C3×D4—181no—
330C3×D4—181no—
331C2×C3×C4—21no—
332C2×C3×C4—21no—
333C2×C3×C4—21no—
334C2×C3×C4—31no—
335C2×C3×C4—31no—
336C2×C3×C4—31no—
337C2×C3×C4—31no—
338C2×C3×C4—41no—
339C2×C3×C4—61no—
340C2×C3×C4—61no—
341C2×C3×C4—61no—
342C2×C3×C4—61no—
343C2×C3×C4—121no—
344C2×C3×C4—181no—
345C2×C3×C4C2×D3211yescharacteristic
346C2×C3×C4C2×D3211yescharacteristic
347C2×C3×C4C2×D3211yescharacteristic
348C2×Dic3—21no—
349C2×Dic3—61no—
350C2×Dic3—61no—
351C2×Dic3—121no—
352C2×Dic3—182no—
353C2×Dic3—181no—
354C2×Dic3—181no—
355C2×Dic3—181no—
356C2×Dic3—361no—
357C2×Dic3—361no—
358C2×Dic3—362no—
359C2×Dic3#62611yescharacteristic
360C2×Dic3#62611yescharacteristic
361D12—61no—
362D12—61no—
363D12—61no—
364D12—61no—
365D12—121no—
366D12—121no—
367Dic6—122no—
368Dic6—122no—
369Dic6—182no—
370Dic6—182no—
371Dic6—181no—
372Dic6—181no—
373Dic6—181no—
374Dic6—181no—
375Dic6—181no—
376Dic6—181no—
377Dic6—361no—
378Dic6—361no—
379Dic6—361no—
380Dic6—361no—
381Dic6—362no—
382Dic6—362no—
383C3×C8—61no—
384C3×C8—61no—
385C3×C8—121no—
386C3×C8—181no—
387C3⋊C8—21no—
388C3⋊C8—61no—
389C3⋊C8—61no—
390C3⋊C8—121no—
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—21no—
400C2×C32—41no—
401C2×C32—41no—
402C2×C32—41no—
403C2×C32#87511yescharacteristic
404C2×C32#88511yescharacteristic
405C2×C32#90811yescharacteristic
406C2×C32#90911yescharacteristic
407C2×C32#91111yescharacteristic
408C2×C32#91511yescharacteristic
409C2×C32#93211yescharacteristic
410C2×C32#94211yescharacteristic
411C2×C32#94711yescharacteristic
412C3⋊D3—81no—
413C3×D3—242no—
414C3×D3—241no—
415C3×D3—241no—
416C2×Q8—181no—
417C2×Q8—271no—
418C2×D4—91no—
419C2×C8—31no—
420C2×C8—91no—
421C4⋊C4—31no—
422C4⋊C4—91no—
423C4⋊C4—181no—
424C22⋊C4—541no—
425C22⋊C4—541no—
426C42—31no—
427C22×C3—21no—
428C22×C3—21no—
429C22×C3—21no—
430C22×C3—41no—
431C22×C3—361no—
432C22×C3—361no—
433C22×C3—361no—
434C22×C3D3×D1211yescharacteristic
435C22×C3C2×#62611yescharacteristic
436C22×C3D3×#6711yescharacteristic
437C2×D3—122no—
438C2×D3—122no—
439C2×D3—122no—
440C2×D3—121no—
441C2×D3—121no—
442C2×D3—121no—
443C2×D3—121no—
444C2×D3—121no—
445C2×D3—121no—
446C3×C4—21no—
447C3×C4—21no—
448C3×C4—21no—
449C3×C4—21no—
450C3×C4—21no—
451C3×C4—21no—
452C3×C4—41no—
453C3×C4—41no—
454C3×C4—61no—
455C3×C4—61no—
456C3×C4—61no—
457C3×C4—61no—
458C3×C4—121no—
459C3×C4—121no—
460C3×C4—121no—
461C3×C4—121no—
462C3×C4—241no—
463C3×C4—361no—
464C3×C4#379611yescharacteristic
465C3×C4#379811yescharacteristic
466C3×C4#379911yescharacteristic
467C3×C4#379911yescharacteristic
468C3×C4#379911yescharacteristic
469C3×C4#380411yescharacteristic
470Dic3—21no—
471Dic3—21no—
472Dic3—41no—
473Dic3—121no—
474Dic3—121no—
475Dic3—241no—
476Dic3—362no—
477Dic3—362no—
478Dic3—362no—
479Dic3—364no—
480Dic3—361no—
481Dic3—361no—
482Dic3—361no—
483C32—21no—
484C32—21no—
485C32—21no—
486C32—41no—
487C32#471411yescharacteristic, complemented
488C32#476511yescharacteristic, complemented
489C32#508911yescharacteristic, complemented
490C23—181no—
491Q8—362no—
492Q8—541no—
493Q8—541no—
494D4—181no—
495D4—181no—
496C2×C4—31no—
497C2×C4—31no—
498C2×C4—61no—
499C2×C4—181no—
500C2×C4—541no—
501C2×C4D3311yescharacteristic
502C8—61no—
503C8—181no—
504C2×C3—21no—
505C2×C3—21no—
506C2×C3—21no—
507C2×C3—21no—
508C2×C3—21no—
509C2×C3—21no—
510C2×C3—21no—
511C2×C3—21no—
512C2×C3—21no—
513C2×C3—41no—
514C2×C3—41no—
515C2×C3—41no—
516C2×C3—721no—
517C2×C3#6379511yescharacteristic
518C2×C3#6380011yescharacteristic
519C2×C3#6383011yescharacteristic
520C2×C3#6383311yescharacteristic
521C2×C3#6385211yescharacteristic
522C2×C3#6386411yescharacteristic
523C2×C3#6390711yescharacteristic
524C2×C3#6393211yescharacteristic
525C2×C3#6394611yescharacteristic
526D3—242no—
527D3—241no—
528D3—241no—
529C22—361no—
530C22—361no—
531C22—361no—
532C22D3×#62611yescharacteristic
533C4—61no—
534C4—61no—
535C4—121no—
536C4—361no—
537C4—1081no—
538C4#8896211yescharacteristic
539C4#8896511yescharacteristic
540C3—21no—
541C3—21no—
542C3—21no—
543C3—41no—
544C3#1059029411yescharacteristic, complemented
545C3#1059053111yescharacteristic, complemented
546C3#1059089511yescharacteristic, complemented
547C2—721no—
548C2#1172286911yescharacteristic
549C2#1172288411yescharacteristic
550C2#1172300311yescharacteristic
subgroupG/H
centerC22D3×#626
commutator, Fitting, cosocleC2×C33×C4C23
Frattini, hypercenterC2×C4D33
socleC22×C33C2×D4
prime radicalC2×C3×C4⋊(C3⋊D3)C22
hypocenterC33#489
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
4. #11722042C2semidirect
5. #11722041C2semidirect
7. C3×#63554C2semidirect
25. #63918D3semidirect
45. #63554D3semidirect
323. C33#489semidirect
487. C32#4714semidirect
488. C32#4765semidirect
489. C32#5089semidirect
544. C3#10590294semidirect
545. C3#10590531semidirect
546. C3#10590895semidirect

Maximal subgroups

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

subgroupindex
#117234392
C3×#639182
#117220752
#117220422
#117220412
C3×#635642
C3×#635542
#105908953
#105905313
#105902943