Finite Group Databasebeta

← #13156540

Subgroups — #13156540

Order 1728.

590 conjugacy classes of subgroups; only the 96 normal subgroups are shown.

normal subgroups only

This group has 590 conjugacy classes of subgroups — too many to draw. Only its 96 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#11723450C211yesmaximal, characteristic, complemented
2#11723436C211yesmaximal, characteristic
3#11722075C211yesmaximal, characteristic
4#11722018C211yesmaximal, characteristic, complemented
5C3×#63566C211yesmaximal, characteristic, complemented
6C3×#63565C211yesmaximal, characteristic, complemented
7C3×#63564C211yesmaximal, characteristic
8#10591539—31nomaximal
9#10590868—31nomaximal
10#10590539—31nomaximal
11C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
12C2×C32×Dic6C2211yescharacteristic
13C4×C3⋊(C3⋊Dic3)C2211yescharacteristic
14C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
15C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
16C32×C4⋊Dic3C2211yescharacteristic
17C2×C32×C3⋊C8C2211yescharacteristic
18#88807—21no—
19Dic3⋊(C3⋊Dic3)—21no—
20#88803—21no—
21#88803—21no—
22#64156—31no—
23#64099—31no—
24#63872—31no—
25#63872—31no—
26#63872—61no—
27#63863—31no—
28#63863—31no—
29#63863—61no—
30C8×C3⋊Dic3—31no—
31C3×#835—31no—
32C3×#835—31no—
33C3×#821—31no—
34C3×#816—31no—
35C3×#810—31no—
36C3×#809—31no—
37#63566D311yescharacteristic, complemented
38#63565D311yescharacteristic, complemented
39#63564—31no—
40#63564D311yescharacteristic
41#63563—31no—
42#63562—31no—
43#63553—31no—
44#63553—31no—
45#63553—61no—
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)—21no—
53C2×C32×Dic3—21no—
54C2×C32×Dic3—21no—
55C32×Dic6—21no—
56C32×Dic6—21no—
57C3×C3⋊(C3⋊C8)—21no—
58C32×C3⋊C8—21no—
59#4800—91no—
60#4800—91no—
61#4768—91no—
62C2×C32×Q8—31no—
63C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
64C2×C3×D12—31no—
65C2×C3×D12—31no—
66C2×C3×D12—61no—
67C2×C3×Dic6—21no—
68C2×C3×Dic6C2×D311yescharacteristic
69C2×C3×Dic6C2×D311yescharacteristic
70C2×C32×C8—31no—
71C32×C4⋊C4—31no—
72#3752—61no—
73#3752—61no—
74C4⋊(C3⋊Dic3)—61no—
75C4⋊(C3⋊Dic3)C2×D311yescharacteristic
76#3750—61no—
77C4×C3⋊Dic3—31no—
78C4×C3⋊Dic3—31no—
79C4×C3⋊Dic3—31no—
80C4×C3⋊Dic3—61no—
81C4×C3⋊Dic3—61no—
82C4×C3⋊Dic3—61no—
83C4×C3⋊Dic3—121no—
84C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
85C3×C4⋊Dic3—21no—
86C3×C4⋊Dic3—31no—
87C3×C4⋊Dic3—31no—
88C3×C4⋊Dic3—61no—
89C3×C4⋊Dic3C2×D311yescharacteristic
90C3×C4⋊Dic3C2×D311yescharacteristic
91C2×C3×C3⋊C8—21no—
92C2×C3×C3⋊C8—31no—
93C2×C3×C3⋊C8—31no—
94C2×C3×C3⋊C8—61no—
95C2×C3×C3⋊C8C2×D311yescharacteristic
96C2×C3×C3⋊C8C2×D311yescharacteristic
97Dic3⋊Dic3—61no—
98Dic3⋊Dic3—61no—
99Dic3⋊Dic3—61no—
100Dic3⋊Dic3—61no—
101Dic3⋊Dic3—62no—
102Dic3⋊Dic3—62no—
103(C2×D3)⋊Dic3—61no—
104(C2×D3)⋊Dic3—61no—
105(C2×D3)⋊Dic3—61no—
106(C2×D3)⋊Dic3—61no—
107(C2×D3)⋊Dic3—62no—
108(C2×D3)⋊Dic3—62no—
109C22×C33C2×D411yescharacteristic
110C2×C3×C3⋊D3—41no—
111C2×C3×C3⋊D3—41no—
112C2×C3×C3⋊D3—41no—
113C33×C4#4111yescharacteristic
114C33×C4#4111yescharacteristic
115C3⋊(C3⋊Dic3)—21no—
116C3⋊(C3⋊Dic3)—21no—
117C3×C3⋊Dic3—41no—
118C32×Dic3—41no—
119C32×Dic3—41no—
120#947—91no—
121#947—91no—
122#947—181no—
123#936—91no—
124#890—91no—
125#890—91no—
126#890—91no—
127#890—181no—
128C3×#103—91no—
129C3×#103—91no—
130C3×#102—91no—
131#835—31no—
132#835—31no—
133#832—91no—
134#821—31no—
135#821—91no—
136#816—31no—
137C8×Dic3—91no—
138C8×Dic3—91no—
139C8×Dic3—91no—
140C8×Dic3—181no—
141#810—31no—
142#809—31no—
143#809—91no—
144C22×C3⋊D3—21no—
145C22×C3×D3—62no—
146C22×C3×D3—61no—
147C22×C3×D3—61no—
148C32×Q8—61no—
149C32×Q8—61no—
150C2×C32×C4—21no—
151C2×C32×C4—21no—
152C2×C32×C4—21no—
153C2×C32×C4—41no—
154C2×C32×C4—61no—
155C2×C32×C4—61no—
156C2×C32×C4C22×D311yescharacteristic
157C2×C32×C4C22×D311yescharacteristic
158C2×C32×C4C22×D311yescharacteristic
159C2×C3⋊Dic3—21no—
160C2×C3⋊Dic3—31no—
161C2×C3⋊Dic3—31no—
162C2×C3⋊Dic3—31no—
163C2×C3⋊Dic3—31no—
164C2×C3⋊Dic3—31no—
165C2×C3⋊Dic3—31no—
166C2×C3⋊Dic3—61no—
167C2×C3⋊Dic3—61no—
168C2×C3⋊Dic3—61no—
169C2×C3⋊Dic3—61no—
170C2×C3⋊Dic3—61no—
171C2×C3⋊Dic3—61no—
172C2×C3⋊Dic3—121no—
173C2×C3⋊Dic3—121no—
174C4⋊(C3⋊D3)—21no—
175C4⋊(C3⋊D3)—21no—
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—61no—
186C3×D12—61no—
187C3×D12—61no—
188C3×D12—61no—
189C3×D12—121no—
190C3×D12—121no—
191C3×Dic6—21no—
192C3×Dic6—21no—
193C3×Dic6—21no—
194C3×Dic6—21no—
195C3×Dic6—41no—
196C3×Dic6—41no—
197C32×C8—61no—
198C3⋊(C3⋊C8)—21no—
199C3×C3⋊C8—22no—
200C3×C3⋊C8—21no—
201C3×C3⋊C8—21no—
202C3×C3⋊C8—62no—
203C3×C3⋊C8—61no—
204C3×C3⋊C8—61no—
205#488—271noSylow
206C2×C33#12411yescharacteristic
207C2×C33#13511yescharacteristic
208C2×C33#13511yescharacteristic
209C3×C3⋊D3—81no—
210C2×C3×Q8—31no—
211C2×C3×Q8—31no—
212C2×C3×Q8—61no—
213C2×C3×D4—91no—
214C2×D12—31no—
215C2×D12—31no—
216C2×D12—61no—
217C2×Dic6D3211yescharacteristic
218C2×C3×C8—31no—
219C2×C3×C8—31no—
220C2×C3×C8—61no—
221C2×C3×C8—91no—
222C3×C4⋊C4—31no—
223C3×C4⋊C4—31no—
224C3×C4⋊C4—61no—
225C3×C4⋊C4—91no—
226#217—181no—
227#217—181no—
228(C2×D3)⋊C4—181no—
229(C2×D3)⋊C4—181no—
230(C2×D3)⋊C4—181no—
231(C2×D3)⋊C4—181no—
232(C2×D3)⋊C4—182no—
233(C2×D3)⋊C4—182no—
234C4⋊Dic3—31no—
235C4⋊Dic3—31no—
236C4⋊Dic3—61no—
237C4⋊Dic3—181no—
238C4⋊Dic3—181no—
239C4⋊Dic3—181no—
240C4⋊Dic3—182no—
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—41no—
265C2×C3⋊D3—41no—
266C2×C3⋊D3—41no—
267C2×C3×D3—122no—
268C2×C3×D3—122no—
269C2×C3×D3—122no—
270C2×C3×D3—121no—
271C2×C3×D3—121no—
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×C4#23511yescharacteristic
287C32×C4#23511yescharacteristic
288C32×C4#23511yescharacteristic
289C32×C4#23711yescharacteristic
290C32×C4#23711yescharacteristic
291C32×C4#23711yescharacteristic
292C3⋊Dic3—41no—
293C3⋊Dic3—61no—
294C3⋊Dic3—61no—
295C3⋊Dic3—62no—
296C3⋊Dic3—62no—
297C3⋊Dic3—61no—
298C3⋊Dic3—61no—
299C3⋊Dic3—61no—
300C3⋊Dic3—61no—
301C3⋊Dic3—121no—
302C3⋊Dic3—121no—
303C3⋊Dic3—121no—
304C3⋊Dic3—121no—
305C3⋊Dic3—122no—
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—
316#125—271no—
317#124—271no—
318#103—91no—
319#103—91no—
320#102—91no—
321#102—271no—
322C4×C8—271no—
323C33#48811yescharacteristic, complemented, Sylow
324C23×C3—181no—
325C22×D3—62no—
326C22×D3—61no—
327C22×D3—61no—
328C3×Q8—61no—
329C3×Q8—61no—
330C3×Q8—61no—
331C3×Q8—61no—
332C3×Q8—121no—
333C3×Q8—121no—
334C3×D4—181no—
335C3×D4—181no—
336C2×C3×C4—21no—
337C2×C3×C4—21no—
338C2×C3×C4—21no—
339C2×C3×C4—41no—
340C2×C3×C4—62no—
341C2×C3×C4—62no—
342C2×C3×C4—61no—
343C2×C3×C4—61no—
344C2×C3×C4—61no—
345C2×C3×C4—61no—
346C2×C3×C4—181no—
347C2×C3×C4C2×D3211yescharacteristic
348C2×C3×C4C2×D3211yescharacteristic
349C2×C3×C4C2×D3211yescharacteristic
350C2×Dic3—21no—
351C2×Dic3—21no—
352C2×Dic3—62no—
353C2×Dic3—61no—
354C2×Dic3—61no—
355C2×Dic3—91no—
356C2×Dic3—91no—
357C2×Dic3—91no—
358C2×Dic3—91no—
359C2×Dic3—91no—
360C2×Dic3—91no—
361C2×Dic3—181no—
362C2×Dic3—181no—
363C2×Dic3—181no—
364C2×Dic3—181no—
365C2×Dic3—181no—
366C2×Dic3—181no—
367C2×Dic3—361no—
368C2×Dic3—361no—
369D12—61no—
370D12—61no—
371D12—61no—
372D12—61no—
373D12—121no—
374D12—121no—
375Dic6—21no—
376Dic6—21no—
377C3×C8—62no—
378C3×C8—61no—
379C3×C8—61no—
380C3×C8—181no—
381C3⋊C8—21no—
382C3⋊C8—62no—
383C3⋊C8—61no—
384C3⋊C8—61no—
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#88511yescharacteristic
398C2×C32#88511yescharacteristic
399C2×C32#88511yescharacteristic
400C2×C32#91211yescharacteristic
401C2×C32#91611yescharacteristic
402C2×C32#91611yescharacteristic
403C2×C32#91611yescharacteristic
404C2×C32#91911yescharacteristic
405C2×C32#91911yescharacteristic
406C3⋊D3—81no—
407C3×D3—242no—
408C3×D3—241no—
409C3×D3—241no—
410C2×Q8—31no—
411C2×D4—91no—
412C2×C8—31no—
413C2×C8—91no—
414C4⋊C4—31no—
415C4⋊C4—91no—
416C4⋊C4—541no—
417C4⋊C4—541no—
418C22⋊C4—541no—
419C22⋊C4—541no—
420C42—271no—
421C22×C3—21no—
422C22×C3—21no—
423C22×C3—21no—
424C22×C3—41no—
425C22×C3—361no—
426C22×C3—361no—
427C22×C3—361no—
428C22×C3#380211yescharacteristic
429C22×C3#381111yescharacteristic
430C22×C3#381111yescharacteristic
431C2×D3—122no—
432C2×D3—122no—
433C2×D3—122no—
434C2×D3—121no—
435C2×D3—121no—
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×C4#379511yescharacteristic
456C3×C4#379511yescharacteristic
457C3×C4#379811yescharacteristic
458C3×C4#380411yescharacteristic
459C3×C4#380411yescharacteristic
460C3×C4#380511yescharacteristic
461Dic3—41no—
462Dic3—41no—
463Dic3—122no—
464Dic3—121no—
465Dic3—121no—
466Dic3—182no—
467Dic3—182no—
468Dic3—181no—
469Dic3—181no—
470Dic3—181no—
471Dic3—181no—
472Dic3—181no—
473Dic3—181no—
474Dic3—361no—
475Dic3—361no—
476Dic3—361no—
477Dic3—361no—
478Dic3—362no—
479Dic3—362no—
480C32—21no—
481C32—21no—
482C32—21no—
483C32—41no—
484C32#476811yescharacteristic, complemented
485C32#480011yescharacteristic, complemented
486C32#480011yescharacteristic, complemented
487C23—181no—
488Q8—61no—
489Q8—61no—
490D4—181no—
491D4—181no—
492C2×C4—61no—
493C2×C4—61no—
494C2×C4—181no—
495C2×C4—271no—
496C2×C4—271no—
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—721no—
513C2×C3#6380911yescharacteristic
514C2×C3#6381011yescharacteristic
515C2×C3#6391311yescharacteristic
516C2×C3#6391511yescharacteristic
517C2×C3#6393611yescharacteristic
518C2×C3#6395011yescharacteristic
519C2×C3#6395011yescharacteristic
520C2×C3#6395011yescharacteristic
521C2×C3#6395311yescharacteristic
522D3—242no—
523D3—241no—
524D3—241no—
525C22—361no—
526C22—361no—
527C22—361no—
528C22#8895411yescharacteristic
529C4—121no—
530C4—121no—
531C4—361no—
532C4—541no—
533C4—541no—
534C4#8896011yescharacteristic
535C4#8896111yescharacteristic
536C3—21no—
537C3—21no—
538C3—21no—
539C3—41no—
540C3#1059053911yescharacteristic, complemented
541C3#1059086811yescharacteristic, complemented
542C3#1059153911yescharacteristic, complemented
543C2—721no—
544C2#1172286211yescharacteristic
545C2#1172286311yescharacteristic
546C2#1172300911yescharacteristic
subgroupG/H
centerC22#88954
commutator, Fitting, cosocleC2×C33×C4C23
Frattini, hypercenterC2×C4D33
socleC22×C33C2×D4
prime radicalC2×C3×C4⋊(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
1. #11723450C2semidirect
4. #11722018C2semidirect
5. C3×#63566C2semidirect
6. C3×#63565C2semidirect
37. #63566D3semidirect
38. #63565D3semidirect
323. C33#488semidirect
484. C32#4768semidirect
485. C32#4800semidirect
486. C32#4800semidirect
540. C3#10590539semidirect
541. C3#10590868semidirect
542. C3#10591539semidirect

Maximal subgroups

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

subgroupindex
#117234502
#117234362
#117220752
#117220182
C3×#635662
C3×#635652
C3×#635642
#105915393
#105908683
#105905393