Finite Group Databasebeta

← #13156312

Subgroups — #13156312

Order 1728.

662 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 662 conjugacy classes of subgroups — too many to draw. Only its 102 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11723536C211yesmaximal, characteristic, complemented
2#11723501C211yesmaximal, characteristic
3#11722075C211yesmaximal, characteristic
4#11722050C211yesmaximal, characteristic, complemented
5C3×#63574C211yesmaximal, characteristic, complemented
6C3×#63573C211yesmaximal, characteristic, complemented
7C3×#63564C211yesmaximal, characteristic
8#10590776—31nomaximal
9#10590472—31nomaximal
10#10590433—31nomaximal
11C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
12C2×#6293—21no—
13C2×Dic3×C3⋊D3—21no—
14C4×C3⋊(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#88806—21no—
21#88803—21no—
22#64102—31no—
23#64097—31no—
24#63919—31no—
25#63919—31no—
26#63919—61no—
27#63898—31no—
28#63898—31no—
29#63898—61no—
30#63641—31no—
31C3×#821—31no—
32C3×#817—31no—
33C3×#809—31no—
34C3×#808—31no—
35C3×#808—31no—
36C3×#808—31no—
37#63574D311yescharacteristic, complemented
38#63573D311yescharacteristic, complemented
39#63572—31no—
40#63572—31no—
41#63572—61no—
42#63564—31no—
43#63564D311yescharacteristic
44#63563—31no—
45#63562—31no—
46C22×C3×C3⋊D3—21no—
47C2×C33×C4C2311yescharacteristic
48C2×C3⋊(C3⋊Dic3)D411yescharacteristic
49C2×C3⋊(C3⋊Dic3)D411yescharacteristic
50C2×C3×C3⋊Dic3—21no—
51C3×C4⋊(C3⋊D3)—21no—
52C3×C4⋊(C3⋊D3)Q812yescomplemented
53C2×C32×Dic3—21no—
54C2×C32×Dic3—21no—
55#6293—42no—
56Dic3×C3⋊D3—42no—
57C3×C3⋊(C3⋊C8)—21no—
58C32×C3⋊C8—21no—
59#4870—91no—
60#4870—91no—
61#4761—91no—
62C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
63C2×C4×C3⋊D3—61no—
64C2×#634—61no—
65C2×C3×D12—31no—
66C2×C3×D12—31no—
67C2×C3×D12—61no—
68C2×#627—61no—
69C2×#627—61no—
70C2×#627—62no—
71C2×D3×Dic3—61no—
72C2×D3×Dic3—61no—
73C2×D3×Dic3—62no—
74C2×C32×C8—31no—
75C32×C4⋊C4—31no—
76C32×C4⋊C4—31no—
77#3752—61no—
78C4⋊(C3⋊Dic3)C2×D311yescharacteristic
79#3750—61no—
80C4×C3⋊Dic3—31no—
81C4×C3⋊Dic3—31no—
82C4×C3⋊Dic3—31no—
83C4×C3⋊Dic3—61no—
84C4×C3⋊Dic3—61no—
85C4×C3⋊Dic3—61no—
86C4×C3⋊Dic3—121no—
87C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
88C3×C4⋊Dic3—21no—
89C3×C4⋊Dic3—21no—
90C3×C4⋊Dic3—31no—
91C3×C4⋊Dic3—31no—
92C3×C4⋊Dic3—61no—
93C3×C4⋊Dic3C2×D311yescharacteristic
94C3×C4⋊Dic3C2×D311yescharacteristic
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
103#3724—61no—
104#3724—61no—
105#3724—62no—
106(C2×D3)⋊Dic3—61no—
107(C2×D3)⋊Dic3—61no—
108(C2×D3)⋊Dic3—62no—
109C22×C33C2×D411yescharacteristic
110C2×C3×C3⋊D3—22no—
111C2×C3×C3⋊D3—41no—
112C2×C3×C3⋊D3—41no—
113C33×C4C2×Q811yescharacteristic
114C33×C4#4111yescharacteristic
115C3⋊(C3⋊Dic3)—21no—
116C3⋊(C3⋊Dic3)SD1612yes—
117C3×C3⋊Dic3—41no—
118C32×Dic3—41no—
119C32×Dic3—41no—
120D4×Dic3—91no—
121#893—91no—
122#893—91no—
123#893—181no—
124#888—91no—
125#888—91no—
126#888—91no—
127#888—181no—
128C3×#106—91no—
129C3×#106—91no—
130C3×#102—91no—
131#832—91no—
132#821—31no—
133#821—91no—
134#817—31no—
135Dic3⋊C8—91no—
136Dic3⋊C8—91no—
137Dic3⋊C8—91no—
138Dic3⋊C8—181no—
139#809—31no—
140#809—91no—
141#808—31no—
142#808—31no—
143#808—31no—
144C22×C3⋊D3—21no—
145C22×C3×D3—62no—
146C22×C3×D3—61no—
147C22×C3×D3—61no—
148C2×C32×C4—21no—
149C2×C32×C4—21no—
150C2×C32×C4—21no—
151C2×C32×C4—41no—
152C2×C32×C4—61no—
153C2×C32×C4—61no—
154C2×C32×C4C22×D311yescharacteristic
155C2×C32×C4C22×D311yescharacteristic
156C2×C32×C4C22×D311yescharacteristic
157C2×C3⋊Dic3—21no—
158C2×C3⋊Dic3—31no—
159C2×C3⋊Dic3—31no—
160C2×C3⋊Dic3—31no—
161C2×C3⋊Dic3—31no—
162C2×C3⋊Dic3—31no—
163C2×C3⋊Dic3—31no—
164C2×C3⋊Dic3—61no—
165C2×C3⋊Dic3—61no—
166C2×C3⋊Dic3—61no—
167C2×C3⋊Dic3—61no—
168C2×C3⋊Dic3—61no—
169C2×C3⋊Dic3—61no—
170C2×C3⋊Dic3—121no—
171C2×C3⋊Dic3—121no—
172C4⋊(C3⋊D3)—21no—
173C4⋊(C3⋊D3)Dic612yescomplemented
174C4×C3⋊D3—122no—
175#634—122no—
176C2×C3×Dic3—22no—
177C2×C3×Dic3—22no—
178C2×C3×Dic3—21no—
179C2×C3×Dic3—21no—
180C2×C3×Dic3—21no—
181C2×C3×Dic3—21no—
182C2×C3×Dic3—62no—
183C2×C3×Dic3—61no—
184C2×C3×Dic3—61no—
185C3×D12—32no—
186C3×D12—32no—
187C3×D12—62no—
188C3×D12—62no—
189C3×D12—61no—
190C3×D12—61no—
191#627—124no—
192#627—122no—
193#627—122no—
194D3×Dic3—122no—
195D3×Dic3—122no—
196D3×Dic3—124no—
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×D4—91no—
211C22×Dic3—181no—
212C2×D12—31no—
213C2×D12—31no—
214C2×D12—61no—
215C2×C4×D3—181no—
216C2×C4×D3—181no—
217C2×C4×D3—182no—
218C2×Dic6—181no—
219C2×Dic6—181no—
220C2×Dic6—181no—
221C2×Dic6—182no—
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—31no—
229C3×C4⋊C4—31no—
230C3×C4⋊C4—61no—
231C3×C4⋊C4—61no—
232C3×C4⋊C4—91no—
233#217—181no—
234(C2×D3)⋊C4—181no—
235(C2×D3)⋊C4—181no—
236(C2×D3)⋊C4—182no—
237C4⋊Dic3—31no—
238C4⋊Dic3—31no—
239C4⋊Dic3—61no—
240C4⋊Dic3D3211yescharacteristic
241C4⋊Dic3D3211yescharacteristic
242Dic3⋊C4—181no—
243Dic3⋊C4—181no—
244Dic3⋊C4—181no—
245Dic3⋊C4—182no—
246C4×Dic3—91no—
247C4×Dic3—91no—
248C4×Dic3—91no—
249C4×Dic3—181no—
250C4×Dic3—181no—
251C4×Dic3—181no—
252C4×Dic3—361no—
253C2×C3⋊C8—31no—
254C2×C3⋊C8—31no—
255C2×C3⋊C8—61no—
256C2×C3⋊C8D3211yescharacteristic
257C22×C32—21no—
258C22×C32—21no—
259C22×C32—21no—
260C22×C32—41no—
261C22×C32D3×D411yescharacteristic
262C22×C32D3×D411yescharacteristic
263C22×C32D3×D411yescharacteristic
264C2×C3⋊D3—22no—
265C2×C3⋊D3—41no—
266C2×C3⋊D3—41no—
267C2×C3×D3—64no—
268C2×C3×D3—62no—
269C2×C3×D3—62no—
270C2×C3×D3—122no—
271C2×C3×D3—122no—
272C2×C3×D3—121no—
273C2×C3×D3—121no—
274C2×C3×D3—121no—
275C2×C3×D3—121no—
276C32×C4—21no—
277C32×C4—21no—
278C32×C4—21no—
279C32×C4—21no—
280C32×C4—21no—
281C32×C4—21no—
282C32×C4—41no—
283C32×C4—41no—
284C32×C4—121no—
285C32×C4—121no—
286C32×C4C2×Dic611yescharacteristic
287C32×C4#23511yescharacteristic
288C32×C4#23511yescharacteristic
289C32×C4#23711yescharacteristic
290C32×C4D3×Q811yescharacteristic
291C32×C4D3×Q811yescharacteristic
292C3⋊Dic3—32no—
293C3⋊Dic3—32no—
294C3⋊Dic3—32no—
295C3⋊Dic3—41no—
296C3⋊Dic3—62no—
297C3⋊Dic3—62no—
298C3⋊Dic3—62no—
299C3⋊Dic3—61no—
300C3⋊Dic3—62no—
301C3⋊Dic3—61no—
302C3⋊Dic3—61no—
303C3⋊Dic3—122no—
304C3⋊Dic3—121no—
305C3⋊Dic3—121no—
306C3⋊Dic3—122no—
307C3×Dic3—42no—
308C3×Dic3—42no—
309C3×Dic3—41no—
310C3×Dic3—41no—
311C3×Dic3—41no—
312C3×Dic3—41no—
313C3×Dic3—122no—
314C3×Dic3—121no—
315C3×Dic3—121no—
316C4⋊Q8—271no—
317C4×D4—271no—
318#106—91no—
319#106—91no—
320C4⋊C8—271no—
321#102—91no—
322#102—271no—
323C33#47611yescharacteristic, complemented, Sylow
324C23×C3—181no—
325C22×D3—62no—
326C22×D3—61no—
327C22×D3—61no—
328C3×D4—92no—
329C3×D4—181no—
330C2×C3×C4—21no—
331C2×C3×C4—21no—
332C2×C3×C4—21no—
333C2×C3×C4—41no—
334C2×C3×C4—62no—
335C2×C3×C4—62no—
336C2×C3×C4—61no—
337C2×C3×C4—61no—
338C2×C3×C4—61no—
339C2×C3×C4—61no—
340C2×C3×C4—181no—
341C2×C3×C4C2×D3211yescharacteristic
342C2×C3×C4C2×D3211yescharacteristic
343C2×C3×C4C2×D3211yescharacteristic
344C2×Dic3—21no—
345C2×Dic3—21no—
346C2×Dic3—62no—
347C2×Dic3—61no—
348C2×Dic3—61no—
349C2×Dic3—91no—
350C2×Dic3—91no—
351C2×Dic3—91no—
352C2×Dic3—91no—
353C2×Dic3—91no—
354C2×Dic3—91no—
355C2×Dic3—181no—
356C2×Dic3—181no—
357C2×Dic3—181no—
358C2×Dic3—181no—
359C2×Dic3—181no—
360C2×Dic3—181no—
361C2×Dic3—361no—
362C2×Dic3—361no—
363C2×Dic3—362no—
364D12—32no—
365D12—32no—
366D12—62no—
367D12—62no—
368D12—61no—
369D12—61no—
370C4×D3—362no—
371C4×D3—362no—
372C4×D3—364no—
373Dic6—362no—
374Dic6—364no—
375Dic6—362no—
376Dic6—362no—
377C3×C8—61no—
378C3×C8—61no—
379C3×C8—121no—
380C3×C8—181no—
381C3⋊C8—21no—
382C3⋊C8—61no—
383C3⋊C8—61no—
384C3⋊C8—121no—
385C2×C32—21no—
386C2×C32—21no—
387C2×C32—21no—
388C2×C32—21no—
389C2×C32—21no—
390C2×C32—21no—
391C2×C32—21no—
392C2×C32—21no—
393C2×C32—21no—
394C2×C32—41no—
395C2×C32—41no—
396C2×C32—41no—
397C2×C32#87811yescharacteristic
398C2×C32#89611yescharacteristic
399C2×C32#89611yescharacteristic
400C2×C32#91111yescharacteristic
401C2×C32D3×SD1611yescharacteristic
402C2×C32D3×SD1611yescharacteristic
403C2×C32D3×SD1611yescharacteristic
404C2×C32#91411yescharacteristic
405C2×C32#91411yescharacteristic
406C3⋊D3—42no—
407C3×D3—124no—
408C3×D3—122no—
409C3×D3—122no—
410C2×Q8—541no—
411C2×D4—91no—
412C22×C4—541no—
413C2×C8—31no—
414C2×C8—91no—
415C4⋊C4—31no—
416C4⋊C4—31no—
417C4⋊C4—91no—
418C4⋊C4—541no—
419C22⋊C4—541no—
420C42—271no—
421C22×C3—21no—
422C22×C3—21no—
423C22×C3—21no—
424C22×C3—41no—
425C22×C3—182no—
426C22×C3—361no—
427C22×C3—361no—
428C22×C3#380211yescharacteristic
429C22×C3#381111yescharacteristic
430C22×C3#381111yescharacteristic
431C2×D3—64no—
432C2×D3—62no—
433C2×D3—62no—
434C2×D3—122no—
435C2×D3—122no—
436C2×D3—121no—
437C2×D3—121no—
438C2×D3—121no—
439C2×D3—121no—
440C3×C4—21no—
441C3×C4—21no—
442C3×C4—21no—
443C3×C4—21no—
444C3×C4—21no—
445C3×C4—21no—
446C3×C4—41no—
447C3×C4—41no—
448C3×C4—122no—
449C3×C4—122no—
450C3×C4—121no—
451C3×C4—121no—
452C3×C4—121no—
453C3×C4—121no—
454C3×C4—361no—
455C3×C4D3×Dic611yescharacteristic
456C3×C4D3×Dic611yescharacteristic
457C3×C4#379511yescharacteristic
458C3×C4#379711yescharacteristic
459C3×C4#379811yescharacteristic
460C3×C4#380411yescharacteristic
461Dic3—41no—
462Dic3—41no—
463Dic3—92no—
464Dic3—92no—
465Dic3—92no—
466Dic3—122no—
467Dic3—121no—
468Dic3—121no—
469Dic3—182no—
470Dic3—182no—
471Dic3—182no—
472Dic3—182no—
473Dic3—181no—
474Dic3—181no—
475Dic3—181no—
476Dic3—362no—
477Dic3—361no—
478Dic3—361no—
479Dic3—362no—
480C32—21no—
481C32—21no—
482C32—21no—
483C32—41no—
484C32#476111yescharacteristic, complemented
485C32#487011yescharacteristic, complemented
486C32#487011yescharacteristic, complemented
487C23—181no—
488Q8—1082no—
489D4—92no—
490D4—181no—
491C2×C4—61no—
492C2×C4—61no—
493C2×C4—181no—
494C2×C4—271no—
495C2×C4—271no—
496C2×C4—1082no—
497C2×C4D3311yescharacteristic
498C8—61no—
499C8—181no—
500C2×C3—21no—
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—41no—
510C2×C3—41no—
511C2×C3—41no—
512C2×C3—362no—
513C2×C3#6379611yescharacteristic
514C2×C3#6379811yescharacteristic
515C2×C3#6386511yescharacteristic
516C2×C3#6389611yescharacteristic
517C2×C3#6389911yescharacteristic
518C2×C3#6393011yescharacteristic
519C2×C3#6393711yescharacteristic
520C2×C3#6394011yescharacteristic
521C2×C3#6394111yescharacteristic
522D3—124no—
523D3—122no—
524D3—122no—
525C22—182no—
526C22—361no—
527C22—361no—
528C22#8895411yescharacteristic
529C4—121no—
530C4—121no—
531C4—272no—
532C4—361no—
533C4—541no—
534C4#8895711yescharacteristic
535C4#8896011yescharacteristic
536C3—21no—
537C3—21no—
538C3—21no—
539C3—41no—
540C3#1059043311yescharacteristic, complemented
541C3#1059047211yescharacteristic, complemented
542C3#1059077611yescharacteristic, complemented
543C2—362no—
544C2#1172284111yescharacteristic
545C2#1172287811yescharacteristic
546C2#1172298311yescharacteristic
subgroupG/H
centerC22#88954
commutator, Fitting, cosocleC2×C33×C4C23
Frattini, hypercenterC2×C4D33
socleC22×C33C2×D4
prime radicalC2×C3×C4⋊(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
1. #11723536C2semidirect
4. #11722050C2semidirect
5. C3×#63574C2semidirect
6. C3×#63573C2semidirect
37. #63574D3semidirect
38. #63573D3semidirect
52. C3×C4⋊(C3⋊D3)Q8semidirect
173. C4⋊(C3⋊D3)Dic6semidirect
323. C33#476semidirect
484. C32#4761semidirect
485. C32#4870semidirect
486. C32#4870semidirect
540. C3#10590433semidirect
541. C3#10590472semidirect
542. C3#10590776semidirect

Maximal subgroups

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

subgroupindex
#117235362
#117235012
#117220752
#117220502
C3×#635742
C3×#635732
C3×#635642
#105907763
#105904723
#105904333