Finite Group Databasebeta

← #13156432

Subgroups — #13156432

Order 1728.

616 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 616 conjugacy classes of subgroups — too many to draw. Only its 98 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#11723522C211yesmaximal, characteristic
2C3×#63849C211yesmaximal, characteristic, complemented
3#11722072C211yesmaximal, characteristic, complemented
4#11722057C211yesmaximal, characteristic, complemented
5#11722054C211yesmaximal, characteristic, complemented
6C3×#63562C211yesmaximal, characteristic, complemented
7C3×#63562C211yesmaximal, characteristic, complemented
8#10591425—31nomaximal
9#10590505—31nomaximal
10#10590502—31nomaximal
11C2×C3×C4×C3⋊D3C2211yescharacteristic
12C2×C32×D12C2211yescharacteristic
13C2×#6290—21no—
14C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
15C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
16C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
17C32×C4⋊Dic3C2211yescharacteristic
18C32×C4⋊Dic3C2211yescharacteristic
19(C2×D3)⋊(C3⋊Dic3)—21no—
20C3×#3722—21no—
21C3×Dic32—21no—
22#64147—31no—
23C3×#892—31no—
24C3×#892—31no—
25#63908—31no—
26#63908—31no—
27#63908—61no—
28#63849D311yescharacteristic, complemented
29#63635—31no—
30#63634—31no—
31C3×#832—31no—
32C3×#832—31no—
33C3×#809—31no—
34C3×#809—31no—
35#63575—31no—
36#63575—31no—
37#63575—61no—
38#63573—31no—
39#63573—31no—
40#63573—61no—
41#63562D311yescharacteristic, complemented
42#63562D311yescharacteristic, complemented
43#63557—31no—
44(C2×D3)⋊(C3⋊C8)—31no—
45(C2×D3)⋊(C3⋊C8)—31no—
46C22×C3×C3⋊D3D411yescharacteristic
47C22×C32×D3—21no—
48C2×C33×C4C2311yescharacteristic
49C2×C3⋊(C3⋊Dic3)—21no—
50C2×C3×C3⋊Dic3D411yescharacteristic
51C3×C4×C3⋊D3—21no—
52C3×C4×C3⋊D3—21no—
53C2×C32×Dic3—21no—
54C2×C32×Dic3—21no—
55C32×D12—21no—
56C32×D12—21no—
57#6290—42no—
58C3×C3⋊(C3⋊C8)—21no—
59C3×C3⋊(C3⋊C8)—21no—
60#5003—91no—
61#4784—91no—
62#4779—91no—
63C2×C32×D4—31no—
64C2×#638—61no—
65C2×C4×C3⋊D3C2×D311yescharacteristic
66C2×C3×D12—21no—
67C2×C3×D12C2×D311yescharacteristic
68C2×C3×D12C2×D311yescharacteristic
69C2×C3×C4×D3—31no—
70C2×C3×C4×D3—31no—
71C2×C3×C4×D3—61no—
72C2×#625—61no—
73C2×#625—61no—
74C2×#625—62no—
75C32×C4⋊C4—31no—
76C32×C4⋊C4—31no—
77#3757—61no—
78C4⋊(C3⋊Dic3)—31no—
79C4⋊(C3⋊Dic3)—31no—
80C4⋊(C3⋊Dic3)—31no—
81C4⋊(C3⋊Dic3)—61no—
82C4⋊(C3⋊Dic3)—61no—
83C4⋊(C3⋊Dic3)—61no—
84C4⋊(C3⋊Dic3)—121no—
85C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
86C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
87C3×(C2×D3)⋊C4—61no—
88C3×(C2×D3)⋊C4—61no—
89C3×C4⋊Dic3—21no—
90C3×C4⋊Dic3—21no—
91C3×C4⋊Dic3C2×D311yescharacteristic
92C3×C4⋊Dic3C2×D311yescharacteristic
93C3×C4⋊Dic3C2×D311yescharacteristic
94C3×C4⋊Dic3C2×D311yescharacteristic
95C3×C4×Dic3—61no—
96C3×C4×Dic3—61no—
97C2×C3×C3⋊C8—31no—
98C2×C3×C3⋊C8—31no—
99C2×C3×C3⋊C8—31no—
100C2×C3×C3⋊C8—31no—
101C2×C3×C3⋊C8—61no—
102C2×C3×C3⋊C8—61no—
103#3722—21no—
104(C2×D3)⋊Dic3—61no—
105(C2×D3)⋊Dic3—61no—
106(C2×D3)⋊Dic3—62no—
107Dic32—21no—
108C22×C33C2×D411yescharacteristic
109C2×C3×C3⋊D3—21no—
110C2×C3×C3⋊D3—21no—
111C2×C3×C3⋊D3—21no—
112C2×C32×D3—41no—
113C2×C32×D3—41no—
114C2×C32×D3—41no—
115C33×C4#4111yescharacteristic
116C33×C4#4111yescharacteristic
117C3⋊(C3⋊Dic3)—41no—
118C3×C3⋊Dic3—21no—
119C32×Dic3—41no—
120C32×Dic3—41no—
121C3×#117—91no—
122#938—91no—
123#938—91no—
124#938—181no—
125#930—91no—
126#892—31no—
127#892—31no—
128C3×#102—91no—
129C3×#102—91no—
130#832—31no—
131#832—31no—
132#830—91no—
133(C2×D3)⋊C8—91no—
134(C2×D3)⋊C8—91no—
135#818—91no—
136#817—91no—
137#809—31no—
138#809—31no—
139#808—91no—
140#808—91no—
141#808—181no—
142#807—91no—
143#807—91no—
144#807—181no—
145C23×C32—61no—
146C22×C3⋊D3#6711yescharacteristic
147C22×C3×D3—22no—
148C22×C3×D3—21no—
149C22×C3×D3—21no—
150C22×C3×D3—31no—
151C22×C3×D3—31no—
152C22×C3×D3—61no—
153C32×D4—61no—
154C32×D4—61no—
155C2×C32×C4—21no—
156C2×C32×C4—21no—
157C2×C32×C4—21no—
158C2×C32×C4—41no—
159C2×C32×C4—61no—
160C2×C32×C4—61no—
161C2×C32×C4C22×D311yescharacteristic
162C2×C32×C4C22×D311yescharacteristic
163C2×C32×C4C22×D311yescharacteristic
164#638—122no—
165C2×C3⋊Dic3—61no—
166C2×C3⋊Dic3—62no—
167C2×C3⋊Dic3—61no—
168C2×C3⋊Dic3—61no—
169C2×C3⋊Dic3—121no—
170C2×C3⋊Dic3—121no—
171C2×C3⋊Dic3—122no—
172C2×C3⋊Dic3#6711yescharacteristic
173C4×C3⋊D3—21no—
174C4×C3⋊D3—21no—
175C2×C3×Dic3—21no—
176C2×C3×Dic3—21no—
177C2×C3×Dic3—21no—
178C2×C3×Dic3—21no—
179C2×C3×Dic3—31no—
180C2×C3×Dic3—31no—
181C2×C3×Dic3—41no—
182C2×C3×Dic3—41no—
183C2×C3×Dic3—61no—
184C3×D12—21no—
185C3×D12—21no—
186C3×D12—21no—
187C3×D12—21no—
188C3×D12—41no—
189C3×D12—41no—
190C3×C4×D3—61no—
191C3×C4×D3—61no—
192C3×C4×D3—61no—
193C3×C4×D3—61no—
194C3×C4×D3—121no—
195C3×C4×D3—121no—
196#625—124no—
197#625—122no—
198#625—122no—
199C3⋊(C3⋊C8)—21no—
200C3⋊(C3⋊C8)—21no—
201C3×C3⋊C8—62no—
202C3×C3⋊C8—62no—
203C3×C3⋊C8—61no—
204C3×C3⋊C8—61no—
205C3×C3⋊C8—61no—
206C3×C3⋊C8—61no—
207#482—271noSylow
208C2×C33#12311yescharacteristic
209C2×C33#13511yescharacteristic
210C2×C33AGL(1,ℤ/8)11yescharacteristic
211C3×C3⋊D3—41no—
212C32×D3—81no—
213C2×C3×D4—31no—
214C2×C3×D4—31no—
215C2×C3×D4—61no—
216C22×C3×C4—91no—
217C2×#67—181no—
218C2×#67—181no—
219C2×#67—181no—
220C2×#67—182no—
221C2×D12D3211yescharacteristic
222C2×C4×D3—31no—
223C2×C4×D3—31no—
224C2×C4×D3—61no—
225C2×C3×C8—91no—
226C2×C3×C8—91no—
227C3×C4⋊C4—31no—
228C3×C4⋊C4—31no—
229C3×C4⋊C4—31no—
230C3×C4⋊C4—31no—
231C3×C4⋊C4—61no—
232C3×C4⋊C4—61no—
233C3×C22⋊C4—181no—
234C3×C42—181no—
235#217—181no—
236#217—181no—
237#217—182no—
238(C2×D3)⋊C4—61no—
239(C2×D3)⋊C4—61no—
240(C2×D3)⋊C4—181no—
241C4⋊Dic3—91no—
242C4⋊Dic3—91no—
243C4⋊Dic3—91no—
244C4⋊Dic3—181no—
245C4⋊Dic3—181no—
246C4⋊Dic3—181no—
247C4⋊Dic3—361no—
248C4⋊Dic3D3211yescharacteristic
249C4⋊Dic3D3211yescharacteristic
250C4×Dic3—61no—
251C4×Dic3—61no—
252C2×C3⋊C8—31no—
253C2×C3⋊C8—31no—
254C2×C3⋊C8—31no—
255C2×C3⋊C8—31no—
256C2×C3⋊C8—61no—
257C2×C3⋊C8—61no—
258C22×C32—21no—
259C22×C32—21no—
260C22×C32—21no—
261C22×C32—41no—
262C22×C32—121no—
263C22×C32—121no—
264C22×C32—121no—
265C22×C32D3×D411yescharacteristic
266C22×C32D3×D411yescharacteristic
267C22×C32C2×#6711yescharacteristic
268C2×C3⋊D3—21no—
269C2×C3⋊D3—21no—
270C2×C3⋊D3—21no—
271C2×C3×D3—42no—
272C2×C3×D3—42no—
273C2×C3×D3—42no—
274C2×C3×D3—41no—
275C2×C3×D3—41no—
276C2×C3×D3—41no—
277C2×C3×D3—41no—
278C2×C3×D3—41no—
279C2×C3×D3—41no—
280C2×C3×D3—62no—
281C2×C3×D3—61no—
282C2×C3×D3—61no—
283C2×C3×D3—61no—
284C2×C3×D3—61no—
285C2×C3×D3—61no—
286C2×C3×D3—61no—
287C2×C3×D3—121no—
288C2×C3×D3—121no—
289C32×C4—21no—
290C32×C4—21no—
291C32×C4—21no—
292C32×C4—21no—
293C32×C4—21no—
294C32×C4—21no—
295C32×C4—41no—
296C32×C4—41no—
297C32×C4—121no—
298C32×C4—121no—
299C32×C4#23511yescharacteristic
300C32×C4#23511yescharacteristic
301C32×C4#23511yescharacteristic
302C32×C4#23511yescharacteristic
303C32×C4#23711yescharacteristic
304C32×C4#23711yescharacteristic
305C3⋊Dic3—21no—
306C3⋊Dic3—121no—
307C3⋊Dic3—122no—
308C3⋊Dic3—121no—
309C3⋊Dic3—121no—
310C3⋊Dic3—241no—
311C3⋊Dic3—241no—
312C3⋊Dic3—242no—
313C3×Dic3—41no—
314C3×Dic3—41no—
315C3×Dic3—41no—
316C3×Dic3—41no—
317C3×Dic3—61no—
318C3×Dic3—61no—
319C3×Dic3—81no—
320C3×Dic3—81no—
321C3×Dic3—121no—
322#121—271no—
323#117—91no—
324#107—271no—
325#106—271no—
326#102—91no—
327#102—91no—
328C22⋊C8—271no—
329C33#48211yescharacteristic, complemented, Sylow
330C23×C3—62no—
331C23×C3—61no—
332C23×C3—61no—
333C23×C3—91no—
334C22×D3—21no—
335C22×D3—31no—
336C22×D3—31no—
337C22×D3—61no—
338C3×D4—61no—
339C3×D4—61no—
340C3×D4—61no—
341C3×D4—61no—
342C3×D4—121no—
343C3×D4—121no—
344C2×C3×C4—21no—
345C2×C3×C4—21no—
346C2×C3×C4—21no—
347C2×C3×C4—41no—
348C2×C3×C4—61no—
349C2×C3×C4—61no—
350C2×C3×C4—61no—
351C2×C3×C4—61no—
352C2×C3×C4—91no—
353C2×C3×C4—121no—
354C2×C3×C4—121no—
355C2×C3×C4—181no—
356C2×C3×C4—181no—
357C2×C3×C4C2×D3211yescharacteristic
358C2×C3×C4C2×D3211yescharacteristic
359C2×C3×C4C2×D3211yescharacteristic
360#67—362no—
361#67—364no—
362#67—362no—
363#67—362no—
364C2×Dic3—21no—
365C2×Dic3—21no—
366C2×Dic3—31no—
367C2×Dic3—31no—
368C2×Dic3—61no—
369C2×Dic3—182no—
370C2×Dic3—181no—
371C2×Dic3—181no—
372C2×Dic3—181no—
373C2×Dic3—361no—
374C2×Dic3—361no—
375C2×Dic3—362no—
376D12—21no—
377D12—21no—
378C4×D3—61no—
379C4×D3—61no—
380C4×D3—61no—
381C4×D3—61no—
382C4×D3—121no—
383C4×D3—121no—
384C3×C8—181no—
385C3×C8—181no—
386C3⋊C8—62no—
387C3⋊C8—62no—
388C3⋊C8—61no—
389C3⋊C8—61no—
390C3⋊C8—61no—
391C3⋊C8—61no—
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—21no—
401C2×C32—41no—
402C2×C32—41no—
403C2×C32—41no—
404C2×C32—241no—
405C2×C32#88311yescharacteristic
406C2×C32#88311yescharacteristic
407C2×C32#91111yescharacteristic
408C2×C32#91211yescharacteristic
409C2×C32#91411yescharacteristic
410C2×C32#91611yescharacteristic
411C2×C32#92911yescharacteristic
412C2×C32#94911yescharacteristic
413C2×C32#95011yescharacteristic
414C3⋊D3—41no—
415C3×D3—82no—
416C3×D3—81no—
417C3×D3—81no—
418C3×D3—122no—
419C3×D3—121no—
420C3×D3—121no—
421C2×D4—31no—
422C2×D4—541no—
423C22×C4—91no—
424C2×C8—91no—
425C2×C8—91no—
426C4⋊C4—31no—
427C4⋊C4—31no—
428C4⋊C4—271no—
429C22⋊C4—181no—
430C22⋊C4—541no—
431C42—181no—
432C22×C3—21no—
433C22×C3—21no—
434C22×C3—21no—
435C22×C3—41no—
436C22×C3—122no—
437C22×C3—122no—
438C22×C3—122no—
439C22×C3—121no—
440C22×C3—121no—
441C22×C3—121no—
442C22×C3—121no—
443C22×C3—121no—
444C22×C3—121no—
445C22×C3—181no—
446C22×C3—181no—
447C22×C3—181no—
448C22×C3D3×#6711yescharacteristic
449C22×C3D3×#6711yescharacteristic
450C22×C3#381111yescharacteristic
451C2×D3—41no—
452C2×D3—41no—
453C2×D3—41no—
454C2×D3—62no—
455C2×D3—61no—
456C2×D3—61no—
457C2×D3—61no—
458C2×D3—61no—
459C2×D3—61no—
460C2×D3—61no—
461C2×D3—121no—
462C2×D3—121no—
463C3×C4—21no—
464C3×C4—21no—
465C3×C4—21no—
466C3×C4—21no—
467C3×C4—21no—
468C3×C4—21no—
469C3×C4—41no—
470C3×C4—41no—
471C3×C4—121no—
472C3×C4—121no—
473C3×C4—121no—
474C3×C4—121no—
475C3×C4—181no—
476C3×C4—241no—
477C3×C4—241no—
478C3×C4#379511yescharacteristic
479C3×C4#379511yescharacteristic
480C3×C4#379811yescharacteristic
481C3×C4#379811yescharacteristic
482C3×C4#380411yescharacteristic
483C3×C4#380411yescharacteristic
484Dic3—41no—
485Dic3—41no—
486Dic3—61no—
487Dic3—61no—
488Dic3—121no—
489Dic3—362no—
490Dic3—361no—
491Dic3—361no—
492Dic3—361no—
493Dic3—721no—
494Dic3—721no—
495Dic3—722no—
496C32—21no—
497C32—21no—
498C32—21no—
499C32—41no—
500C32#477911yescharacteristic, complemented
501C32#478411yescharacteristic, complemented
502C32#500311yescharacteristic, complemented
503C23—61no—
504C23—91no—
505D4—61no—
506D4—61no—
507D4—1082no—
508C2×C4—61no—
509C2×C4—61no—
510C2×C4—91no—
511C2×C4—181no—
512C2×C4—181no—
513C2×C4—541no—
514C2×C4D3311yescharacteristic
515C8—181no—
516C8—181no—
517C2×C3—21no—
518C2×C3—21no—
519C2×C3—21no—
520C2×C3—21no—
521C2×C3—21no—
522C2×C3—21no—
523C2×C3—21no—
524C2×C3—21no—
525C2×C3—21no—
526C2×C3—41no—
527C2×C3—41no—
528C2×C3—41no—
529C2×C3—242no—
530C2×C3—241no—
531C2×C3—241no—
532C2×C3—361no—
533C2×C3#6390511yescharacteristic
534C2×C3#6390511yescharacteristic
535C2×C3#6392511yescharacteristic
536C2×C3#6393311yescharacteristic
537C2×C3#6393611yescharacteristic
538C2×C3#6393711yescharacteristic
539C2×C3#6393911yescharacteristic
540C2×C3#6394711yescharacteristic
541C2×C3#6397511yescharacteristic
542D3—81no—
543D3—122no—
544D3—121no—
545D3—121no—
546C22—121no—
547C22—121no—
548C22—121no—
549C22—181no—
550C22—181no—
551C22—181no—
552C22#8895211yescharacteristic
553C4—121no—
554C4—121no—
555C4—181no—
556C4—1081no—
557C4#8896011yescharacteristic
558C4#8896011yescharacteristic
559C3—21no—
560C3—21no—
561C3—21no—
562C3—41no—
563C3#1059050211yescharacteristic, complemented
564C3#1059050511yescharacteristic, complemented
565C3#1059142511yescharacteristic, complemented
566C2—241no—
567C2—361no—
568C2#1172285711yescharacteristic
569C2#1172287411yescharacteristic
570C2#1172299211yescharacteristic
subgroupG/H
centerC22#88952
commutator, Fitting, cosocleC2×C33×C4C23
Frattini, hypercenterC2×C4D33
socleC22×C33C2×D4
prime radical#11723522C2
hypocenterC33#482
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×#63849C2semidirect
3. #11722072C2semidirect
4. #11722057C2semidirect
5. #11722054C2semidirect
6. C3×#63562C2semidirect
7. C3×#63562C2semidirect
28. #63849D3semidirect
41. #63562D3semidirect
42. #63562D3semidirect
329. C33#482semidirect
500. C32#4779semidirect
501. C32#4784semidirect
502. C32#5003semidirect
563. C3#10590502semidirect
564. C3#10590505semidirect
565. C3#10591425semidirect

Maximal subgroups

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

subgroupindex
#117235222
C3×#638492
#117220722
#117220572
#117220542
C3×#635622
#105914253
#105905053
#105905023