Finite Group Databasebeta

← #13156433

Subgroups — #13156433

Order 1728.

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

normal subgroups only

This group has 672 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#11723526C211yesmaximal, characteristic
2C3×#63849C211yesmaximal, characteristic, complemented
3#11722072C211yesmaximal, characteristic, complemented
4#11722035C211yesmaximal, characteristic, complemented
5#11722035C211yesmaximal, characteristic, complemented
6C3×#63576C211yesmaximal, characteristic, complemented
7C3×#63574C211yesmaximal, characteristic, complemented
8#10590451—31nomaximal
9#10590425—31nomaximal
10#10590424—31nomaximal
11C2×C4⋊(C3⋊(C3⋊D3))C2211yescharacteristic
12C2×C3×C4×C3⋊D3C2211yescharacteristic
13C2×#6292—21no—
14C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
15C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
16C32×C4⋊Dic3C2211yescharacteristic
17C32×C4⋊Dic3C2211yescharacteristic
18C32×C4⋊Dic3C2211yescharacteristic
19#88804—21no—
20C3×#3722—21no—
21C3×Dic32—21no—
22#64104—31no—
23C3×#892—31no—
24C3×#892—31no—
25#63909—31no—
26#63909—31no—
27#63909—61no—
28#63849D311yescharacteristic, complemented
29#63636—31no—
30#63636—31no—
31C3×#808—31no—
32C3×#808—31no—
33C3×#807—31no—
34C3×#807—31no—
35#63576D311yescharacteristic, complemented
36#63574D311yescharacteristic, complemented
37#63564—31no—
38#63564—31no—
39#63564—31no—
40#63564—31no—
41#63564—61no—
42#63564—61no—
43#63557—31no—
44(C2×D3)⋊(C3⋊C8)—31no—
45(C2×D3)⋊(C3⋊C8)—31no—
46C22×C3⋊(C3⋊D3)—21no—
47C22×C3×C3⋊D3D411yescharacteristic
48C2×C33×C4C2311yescharacteristic
49C4⋊(C3⋊(C3⋊D3))—21no—
50C4⋊(C3⋊(C3⋊D3))—21no—
51C2×C3×C3⋊Dic3D411yescharacteristic
52C3×C4×C3⋊D3—21no—
53C3×C4×C3⋊D3—21no—
54C2×C32×Dic3—21no—
55C2×C32×Dic3—21no—
56C2×C32×Dic3—21no—
57#6292—42no—
58C3×C3⋊(C3⋊C8)—21no—
59C3×C3⋊(C3⋊C8)—21no—
60#5003—91no—
61#4885—91no—
62#4866—91no—
63C2×C4⋊(C3⋊D3)—31no—
64C2×C4⋊(C3⋊D3)—31no—
65C2×C4⋊(C3⋊D3)—31no—
66C2×C4⋊(C3⋊D3)—61no—
67C2×C4⋊(C3⋊D3)—61no—
68C2×C4⋊(C3⋊D3)—61no—
69C2×C4⋊(C3⋊D3)—61no—
70C2×C4⋊(C3⋊D3)—121no—
71C2×C4×C3⋊D3C2×D311yescharacteristic
72C2×C3×C4×D3—31no—
73C2×C3×C4×D3—31no—
74C2×C3×C4×D3—61no—
75C2×#626—61no—
76C2×#626—61no—
77C2×#626—62no—
78C32×C4⋊C4—31no—
79C32×C4⋊C4—31no—
80C32×C4⋊C4—31no—
81#3752—61no—
82C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
83C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
84C3×(C2×D3)⋊C4—61no—
85C3×(C2×D3)⋊C4—61no—
86C3×C4⋊Dic3—21no—
87C3×C4⋊Dic3—21no—
88C3×C4⋊Dic3—21no—
89C3×C4⋊Dic3C2×D311yescharacteristic
90C3×C4⋊Dic3C2×D311yescharacteristic
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#3722—61no—
105#3722—61no—
106#3722—62no—
107Dic32—21no—
108C22×C33C2×D411yescharacteristic
109C2×C3⋊(C3⋊D3)—41no—
110C2×C3⋊(C3⋊D3)—41no—
111C2×C3⋊(C3⋊D3)—41no—
112C2×C3×C3⋊D3—21no—
113C2×C3×C3⋊D3—21no—
114C2×C3×C3⋊D3—21no—
115C33×C4#4111yescharacteristic
116C33×C4#4111yescharacteristic
117C3×C3⋊Dic3—21no—
118C32×Dic3—41no—
119C32×Dic3—41no—
120C32×Dic3—41no—
121C3×#117—91no—
122#930—91no—
123#895—91no—
124#895—91no—
125#895—181no—
126#892—31no—
127#892—31no—
128C3×#107—91no—
129C3×#106—91no—
130#830—91no—
131#821—91no—
132#821—91no—
133(C2×D3)⋊C8—91no—
134(C2×D3)⋊C8—91no—
135#809—91no—
136#809—91no—
137#809—91no—
138#809—91no—
139#809—181no—
140#809—181no—
141#808—31no—
142#808—31no—
143#807—31no—
144#807—31no—
145C22×C3⋊D3—61no—
146C22×C3⋊D3—62no—
147C22×C3⋊D3—61no—
148C22×C3⋊D3—61no—
149C22×C3⋊D3—121no—
150C22×C3⋊D3—121no—
151C22×C3⋊D3—122no—
152C22×C3⋊D3#6711yescharacteristic
153C22×C3×D3—31no—
154C22×C3×D3—31no—
155C22×C3×D3—61no—
156C2×C32×C4—21no—
157C2×C32×C4—21no—
158C2×C32×C4—21no—
159C2×C32×C4—41no—
160C2×C32×C4—61no—
161C2×C32×C4—61no—
162C2×C32×C4—61no—
163C2×C32×C4C22×D311yescharacteristic
164C2×C32×C4C22×D311yescharacteristic
165C2×C32×C4C22×D311yescharacteristic
166C2×C3⋊Dic3#6711yescharacteristic
167C4⋊(C3⋊D3)—61no—
168C4⋊(C3⋊D3)—61no—
169C4⋊(C3⋊D3)—62no—
170C4⋊(C3⋊D3)—62no—
171C4⋊(C3⋊D3)—61no—
172C4⋊(C3⋊D3)—61no—
173C4⋊(C3⋊D3)—61no—
174C4⋊(C3⋊D3)—61no—
175C4⋊(C3⋊D3)—121no—
176C4⋊(C3⋊D3)—121no—
177C4⋊(C3⋊D3)—121no—
178C4⋊(C3⋊D3)—121no—
179C4⋊(C3⋊D3)—122no—
180C4⋊(C3⋊D3)—122no—
181C4⋊(C3⋊D3)—122no—
182C4×C3⋊D3—21no—
183C4×C3⋊D3—21no—
184C2×C3×Dic3—22no—
185C2×C3×Dic3—21no—
186C2×C3×Dic3—21no—
187C2×C3×Dic3—21no—
188C2×C3×Dic3—21no—
189C2×C3×Dic3—21no—
190C2×C3×Dic3—21no—
191C2×C3×Dic3—31no—
192C2×C3×Dic3—31no—
193C2×C3×Dic3—41no—
194C2×C3×Dic3—41no—
195C2×C3×Dic3—61no—
196C3×C4×D3—61no—
197C3×C4×D3—61no—
198C3×C4×D3—61no—
199C3×C4×D3—61no—
200C3×C4×D3—121no—
201C3×C4×D3—121no—
202#626—124no—
203#626—122no—
204#626—122no—
205C3⋊(C3⋊C8)—21no—
206C3⋊(C3⋊C8)—21no—
207C3×C3⋊C8—61no—
208C3×C3⋊C8—61no—
209C3×C3⋊C8—61no—
210C3×C3⋊C8—61no—
211C3×C3⋊C8—121no—
212C3×C3⋊C8—121no—
213#482—271noSylow
214C2×C33#12311yescharacteristic
215C2×C33#13511yescharacteristic
216C2×C33AGL(1,ℤ/8)11yescharacteristic
217C3⋊(C3⋊D3)—81no—
218C3×C3⋊D3—41no—
219C22×C3×C4—91no—
220C2×#67—181no—
221C2×D12—91no—
222C2×D12—91no—
223C2×D12—91no—
224C2×D12—181no—
225C2×D12—181no—
226C2×D12—181no—
227C2×D12—181no—
228C2×D12—181no—
229C2×D12—182no—
230C2×D12—361no—
231C2×C4×D3—31no—
232C2×C4×D3—31no—
233C2×C4×D3—61no—
234C2×C3×C8—91no—
235C2×C3×C8—91no—
236C3×C4⋊C4—31no—
237C3×C4⋊C4—31no—
238C3×C4⋊C4—31no—
239C3×C4⋊C4—31no—
240C3×C4⋊C4—31no—
241C3×C4⋊C4—31no—
242C3×C4⋊C4—61no—
243C3×C4⋊C4—61no—
244C3×C4⋊C4—61no—
245C3×C22⋊C4—181no—
246C3×C42—181no—
247(C2×D3)⋊C4—61no—
248(C2×D3)⋊C4—61no—
249(C2×D3)⋊C4—181no—
250(C2×D3)⋊C4—181no—
251(C2×D3)⋊C4—181no—
252(C2×D3)⋊C4—182no—
253C4⋊Dic3D3211yescharacteristic
254C4⋊Dic3D3211yescharacteristic
255C4⋊Dic3D3211yescharacteristic
256C4×Dic3—61no—
257C4×Dic3—61no—
258C2×C3⋊C8—31no—
259C2×C3⋊C8—31no—
260C2×C3⋊C8—31no—
261C2×C3⋊C8—31no—
262C2×C3⋊C8—61no—
263C2×C3⋊C8—61no—
264C22×C32—21no—
265C22×C32—21no—
266C22×C32—21no—
267C22×C32—41no—
268C22×C32D3×D411yescharacteristic
269C22×C32D3×D411yescharacteristic
270C22×C32C2×#6711yescharacteristic
271C2×C3⋊D3—21no—
272C2×C3⋊D3—21no—
273C2×C3⋊D3—21no—
274C2×C3⋊D3—122no—
275C2×C3⋊D3—122no—
276C2×C3⋊D3—122no—
277C2×C3⋊D3—124no—
278C2×C3⋊D3—121no—
279C2×C3⋊D3—121no—
280C2×C3⋊D3—121no—
281C2×C3⋊D3—121no—
282C2×C3⋊D3—121no—
283C2×C3⋊D3—122no—
284C2×C3⋊D3—122no—
285C2×C3⋊D3—121no—
286C2×C3⋊D3—121no—
287C2×C3⋊D3—121no—
288C2×C3⋊D3—121no—
289C2×C3⋊D3—241no—
290C2×C3⋊D3—241no—
291C2×C3⋊D3—241no—
292C2×C3⋊D3—241no—
293C2×C3⋊D3—242no—
294C2×C3⋊D3—242no—
295C2×C3×D3—62no—
296C2×C3×D3—61no—
297C2×C3×D3—61no—
298C2×C3×D3—61no—
299C2×C3×D3—61no—
300C2×C3×D3—61no—
301C2×C3×D3—61no—
302C2×C3×D3—121no—
303C2×C3×D3—121no—
304C32×C4—21no—
305C32×C4—21no—
306C32×C4—21no—
307C32×C4—21no—
308C32×C4—21no—
309C32×C4—21no—
310C32×C4—41no—
311C32×C4—41no—
312C32×C4—121no—
313C32×C4—121no—
314C32×C4—121no—
315C32×C4#23511yescharacteristic
316C32×C4#23511yescharacteristic
317C32×C4#23511yescharacteristic
318C32×C4#23511yescharacteristic
319C32×C4#23911yescharacteristic
320C32×C4#23911yescharacteristic
321C3⋊Dic3—21no—
322C3×Dic3—42no—
323C3×Dic3—41no—
324C3×Dic3—41no—
325C3×Dic3—41no—
326C3×Dic3—41no—
327C3×Dic3—41no—
328C3×Dic3—41no—
329C3×Dic3—61no—
330C3×Dic3—61no—
331C3×Dic3—81no—
332C3×Dic3—81no—
333C3×Dic3—121no—
334#121—271no—
335#117—91no—
336#107—91no—
337#106—91no—
338#102—271no—
339#102—271no—
340C22⋊C8—271no—
341C33#48211yescharacteristic, complemented, Sylow
342C23×C3—91no—
343C22×D3—31no—
344C22×D3—31no—
345C22×D3—61no—
346C22×D3—182no—
347C22×D3—181no—
348C22×D3—181no—
349C22×D3—181no—
350C22×D3—361no—
351C22×D3—361no—
352C22×D3—362no—
353C2×C3×C4—21no—
354C2×C3×C4—21no—
355C2×C3×C4—21no—
356C2×C3×C4—41no—
357C2×C3×C4—62no—
358C2×C3×C4—61no—
359C2×C3×C4—61no—
360C2×C3×C4—61no—
361C2×C3×C4—61no—
362C2×C3×C4—61no—
363C2×C3×C4—61no—
364C2×C3×C4—91no—
365C2×C3×C4—121no—
366C2×C3×C4—121no—
367C2×C3×C4—181no—
368C2×C3×C4—181no—
369C2×C3×C4C2×D3211yescharacteristic
370C2×C3×C4C2×D3211yescharacteristic
371C2×C3×C4C2×D3211yescharacteristic
372#67—362no—
373C2×Dic3—21no—
374C2×Dic3—21no—
375C2×Dic3—21no—
376C2×Dic3—31no—
377C2×Dic3—31no—
378C2×Dic3—61no—
379D12—182no—
380D12—182no—
381D12—181no—
382D12—181no—
383D12—181no—
384D12—181no—
385D12—181no—
386D12—181no—
387D12—361no—
388D12—361no—
389D12—361no—
390D12—361no—
391D12—362no—
392D12—362no—
393D12—364no—
394D12—362no—
395D12—362no—
396C4×D3—61no—
397C4×D3—61no—
398C4×D3—61no—
399C4×D3—61no—
400C4×D3—121no—
401C4×D3—121no—
402C3×C8—181no—
403C3×C8—181no—
404C3⋊C8—61no—
405C3⋊C8—61no—
406C3⋊C8—61no—
407C3⋊C8—61no—
408C3⋊C8—121no—
409C3⋊C8—121no—
410C2×C32—21no—
411C2×C32—21no—
412C2×C32—21no—
413C2×C32—21no—
414C2×C32—21no—
415C2×C32—21no—
416C2×C32—21no—
417C2×C32—21no—
418C2×C32—21no—
419C2×C32—41no—
420C2×C32—41no—
421C2×C32—41no—
422C2×C32#89411yescharacteristic
423C2×C32#89411yescharacteristic
424C2×C32#91111yescharacteristic
425C2×C32#91411yescharacteristic
426C2×C32#91611yescharacteristic
427C2×C32#91911yescharacteristic
428C2×C32#92911yescharacteristic
429C2×C32#94911yescharacteristic
430C2×C32#95011yescharacteristic
431C3⋊D3—41no—
432C3⋊D3—242no—
433C3⋊D3—242no—
434C3⋊D3—242no—
435C3⋊D3—244no—
436C3⋊D3—241no—
437C3⋊D3—241no—
438C3⋊D3—241no—
439C3×D3—122no—
440C3×D3—121no—
441C3×D3—121no—
442C2×D4—271no—
443C2×D4—541no—
444C22×C4—91no—
445C2×C8—91no—
446C2×C8—91no—
447C4⋊C4—31no—
448C4⋊C4—31no—
449C4⋊C4—31no—
450C22⋊C4—181no—
451C22⋊C4—541no—
452C42—181no—
453C22×C3—21no—
454C22×C3—21no—
455C22×C3—21no—
456C22×C3—41no—
457C22×C3—181no—
458C22×C3—181no—
459C22×C3—181no—
460C22×C3D3×#6711yescharacteristic
461C22×C3D3×#6711yescharacteristic
462C22×C3#381111yescharacteristic
463C2×D3—62no—
464C2×D3—61no—
465C2×D3—61no—
466C2×D3—61no—
467C2×D3—61no—
468C2×D3—61no—
469C2×D3—61no—
470C2×D3—121no—
471C2×D3—121no—
472C2×D3—362no—
473C2×D3—362no—
474C2×D3—362no—
475C2×D3—362no—
476C2×D3—362no—
477C2×D3—364no—
478C2×D3—361no—
479C2×D3—361no—
480C2×D3—361no—
481C2×D3—361no—
482C2×D3—361no—
483C2×D3—361no—
484C2×D3—361no—
485C2×D3—361no—
486C2×D3—361no—
487C2×D3—721no—
488C2×D3—721no—
489C2×D3—721no—
490C2×D3—721no—
491C2×D3—722no—
492C2×D3—722no—
493C3×C4—21no—
494C3×C4—21no—
495C3×C4—21no—
496C3×C4—21no—
497C3×C4—21no—
498C3×C4—21no—
499C3×C4—41no—
500C3×C4—41no—
501C3×C4—122no—
502C3×C4—121no—
503C3×C4—121no—
504C3×C4—121no—
505C3×C4—121no—
506C3×C4—121no—
507C3×C4—121no—
508C3×C4—181no—
509C3×C4—241no—
510C3×C4—241no—
511C3×C4#379811yescharacteristic
512C3×C4#379811yescharacteristic
513C3×C4#379911yescharacteristic
514C3×C4#379911yescharacteristic
515C3×C4#379911yescharacteristic
516C3×C4#379911yescharacteristic
517Dic3—41no—
518Dic3—41no—
519Dic3—41no—
520Dic3—61no—
521Dic3—61no—
522Dic3—121no—
523C32—21no—
524C32—21no—
525C32—21no—
526C32—41no—
527C32#486611yescharacteristic, complemented
528C32#488511yescharacteristic, complemented
529C32#500311yescharacteristic, complemented
530C23—91no—
531C23—541no—
532D4—541no—
533D4—541no—
534D4—1082no—
535C2×C4—61no—
536C2×C4—61no—
537C2×C4—61no—
538C2×C4—91no—
539C2×C4—181no—
540C2×C4—181no—
541C2×C4D3311yescharacteristic
542C8—181no—
543C8—181no—
544C2×C3—21no—
545C2×C3—21no—
546C2×C3—21no—
547C2×C3—21no—
548C2×C3—21no—
549C2×C3—21no—
550C2×C3—21no—
551C2×C3—21no—
552C2×C3—21no—
553C2×C3—41no—
554C2×C3—41no—
555C2×C3—41no—
556C2×C3—361no—
557C2×C3#6385011yescharacteristic
558C2×C3#6385011yescharacteristic
559C2×C3#6389711yescharacteristic
560C2×C3#6392511yescharacteristic
561C2×C3#6393511yescharacteristic
562C2×C3#6393711yescharacteristic
563C2×C3#6393911yescharacteristic
564C2×C3#6394911yescharacteristic
565C2×C3#6395011yescharacteristic
566D3—122no—
567D3—121no—
568D3—121no—
569D3—722no—
570D3—722no—
571D3—722no—
572D3—724no—
573D3—721no—
574D3—721no—
575D3—721no—
576C22—181no—
577C22—181no—
578C22—181no—
579C22—1081no—
580C22—1081no—
581C22—1081no—
582C22#8895211yescharacteristic
583C4—121no—
584C4—121no—
585C4—121no—
586C4—181no—
587C4#8896511yescharacteristic
588C4#8896511yescharacteristic
589C3—21no—
590C3—21no—
591C3—21no—
592C3—41no—
593C3#1059042411yescharacteristic, complemented
594C3#1059042511yescharacteristic, complemented
595C3#1059045111yescharacteristic, complemented
596C2—361no—
597C2—2161no—
598C2#1172285811yescharacteristic
599C2#1172287411yescharacteristic
600C2#1172299311yescharacteristic
subgroupG/H
centerC22#88952
commutator, Fitting, cosocleC2×C33×C4C23
Frattini, hypercenterC2×C4D33
socleC22×C33C2×D4
prime radical#11723526C2
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. #11722035C2semidirect
5. #11722035C2semidirect
6. C3×#63576C2semidirect
7. C3×#63574C2semidirect
28. #63849D3semidirect
35. #63576D3semidirect
36. #63574D3semidirect
341. C33#482semidirect
527. C32#4866semidirect
528. C32#4885semidirect
529. C32#5003semidirect
593. C3#10590424semidirect
594. C3#10590425semidirect
595. C3#10590451semidirect

Maximal subgroups

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

subgroupindex
#117235262
C3×#638492
#117220722
#117220352
C3×#635762
C3×#635742
#105904513
#105904253
#105904243