Finite Group Databasebeta

← #13155396

Subgroups — #13155396

Order 1728.

858 conjugacy classes of subgroups.

The subgroup lattice has 632 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#89041C211yesmaximal, characteristic
2C3⋊D3×C4⋊Dic3C211yesmaximal, characteristic, complemented
3C2×C3×#3712C211yesmaximal, characteristic, complemented
4#11722040C211yesmaximal, characteristic, complemented
5#11722034C211yesmaximal, characteristic, complemented
6C3×#63567C211yesmaximal, characteristic, complemented
7C3×#63564C211yesmaximal, characteristic, complemented
8#10590671—31nomaximal
9#10590670—31nomaximal
10#10590612—31nomaximal
11C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
12C2×C3×C4×C3⋊D3C2211yescharacteristic
13C2×C3×#634C2211yescharacteristic
14C2×#6295—21no—
15C2×#6294—21no—
16#89041—22no—
17#89041C414yescomplemented
18C2×Dic3×C3⋊D3—21no—
19C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
20C32×C4⋊Dic3C2211yescharacteristic
21C2×C32×C3⋊C8C2211yescharacteristic
22C2×C32×C3⋊C8C2211yescharacteristic
23#88807—21no—
24C3×#3712—22no—
25C2×#3799—31no—
26C2×#3796—31no—
27C2×#3795—31no—
28C4⋊C4×C3⋊D3—31no—
29C2×C3×ΓL(1,𝔽25)—31no—
30C2×C3×ΓL(1,𝔽25)—31no—
31D3×C4⋊Dic3—31no—
32D3×C4⋊Dic3—31no—
33D3×C4⋊Dic3—61no—
34C2×#3712D311yescharacteristic, complemented
35#63648—31no—
36#63643—31no—
37C3×#821—31no—
38C3×#816—31no—
39C3×#810—31no—
40C3×#809—31no—
41#63567D311yescharacteristic, complemented
42#63566—31no—
43#63566—31no—
44#63566—61no—
45#63564D311yescharacteristic, complemented
46#63563—31no—
47#63563—31no—
48#63563—61no—
49C22×C3×C3⋊D3—21no—
50C22×C3×C3⋊D3D411yescharacteristic
51C2×C33×C4C2311yescharacteristic
52C2×C3⋊(C3⋊Dic3)—21no—
53C2×C3×C3⋊Dic3—21no—
54C2×C3×C3⋊Dic3D411yescharacteristic
55C3×C4⋊(C3⋊D3)—21no—
56C3×C4⋊(C3⋊D3)C2×C412yescomplemented
57C3×C4×C3⋊D3C2×C412yes—
58C3×C4×C3⋊D3D412yes—
59C3×#634—21no—
60C3×#634C2×C412yescomplemented
61C2×C32×Dic3—21no—
62#6295—24no—
63#6294—24no—
64Dic3×C3⋊D3—42no—
65C32×C3⋊C8—21no—
66C32×C3⋊C8—21no—
67#5133—91no—
68#4803—91no—
69#4771—91no—
70C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
71C2×C4×C3⋊D3—61no—
72C2×C4×C3⋊D3C2×D311yescharacteristic
73C2×#634C2×D311yescharacteristic
74C2×C3×D12—31no—
75C2×C3×D12—31no—
76C2×C3×D12—61no—
77C2×C3×C4×D3—31no—
78C2×C3×C4×D3—31no—
79C2×C3×C4×D3—61no—
80C2×C3×Dic6—31no—
81C2×C3×Dic6—31no—
82C2×C3×Dic6—61no—
83C2×#626—61no—
84C2×#626—61no—
85C2×#625—61no—
86C2×#624—61no—
87C2×D3×Dic3—61no—
88C2×D3×Dic3—61no—
89C2×D3×Dic3—61no—
90C2×D3×Dic3—61no—
91C2×D3×Dic3—62no—
92#3799—34no—
93#3799—62no—
94#3796—34no—
95#3796—62no—
96#3795—34no—
97#3795—62no—
98C2×C32×C8—31no—
99C2×C32×C8—31no—
100C32×C4⋊C4—31no—
101C4⋊(C3⋊Dic3)—31no—
102C4⋊(C3⋊Dic3)—31no—
103C4⋊(C3⋊Dic3)—31no—
104C4⋊(C3⋊Dic3)—61no—
105C4⋊(C3⋊Dic3)—61no—
106C4⋊(C3⋊Dic3)—61no—
107C4⋊(C3⋊Dic3)—121no—
108#3750—61no—
109C3×C4⋊Dic3—21no—
110C3×C4⋊Dic3C2×D311yescharacteristic
111C3×C4⋊Dic3C2×D311yescharacteristic
112C2×C3×C3⋊C8—21no—
113C2×C3×C3⋊C8—21no—
114C2×C3×C3⋊C8C2×D311yescharacteristic
115C2×C3×C3⋊C8C2×D311yescharacteristic
116C2×C3×C3⋊C8C2×D311yescharacteristic
117C2×C3×C3⋊C8C2×D311yescharacteristic
118C3×ΓL(1,𝔽25)—62no—
119C3×ΓL(1,𝔽25)—62no—
120Dic3⋊Dic3—61no—
121Dic3⋊Dic3—61no—
122Dic3⋊Dic3—62no—
123#3712—22no—
124C22×C33C2×D411yescharacteristic
125C2×C3×C3⋊D3—21no—
126C2×C3×C3⋊D3—21no—
127C2×C3×C3⋊D3—22no—
128C2×C3×C3⋊D3—41no—
129C2×C3×C3⋊D3—41no—
130C2×C3×C3⋊D3C22⋊C412yes—
131C33×C4C22×C411yescharacteristic
132C33×C4C2×D411yescharacteristic
133C3⋊(C3⋊Dic3)—41no—
134C3×C3⋊Dic3—22no—
135C3×C3⋊Dic3C22⋊C412yes—
136C32×Dic3—41no—
137C2×#239—91no—
138C2×#237—91no—
139C2×#235—91no—
140C2×C3×M16—91no—
141C2×C4⋊Dic3—91no—
142C2×ΓL(1,𝔽25)—31no—
143C2×ΓL(1,𝔽25)—31no—
144D3×C4⋊C4—91no—
145D3×C4⋊C4—91no—
146D3×C4⋊C4—181no—
147C3×#103—91no—
148C3×#102—91no—
149#835—91no—
150#832—91no—
151#821—31no—
152#821—91no—
153#821—91no—
154#821—181no—
155#816—31no—
156#816—91no—
157#816—91no—
158#816—181no—
159#810—31no—
160#809—31no—
161C22×C3⋊D3—21no—
162C22×C3⋊D3D1211yescharacteristic
163C22×C3×D3—31no—
164C22×C3×D3—31no—
165C22×C3×D3—61no—
166C22×C3×D3—61no—
167C22×C3×D3—61no—
168C22×C3×D3—121no—
169C2×C32×C4—21no—
170C2×C32×C4—21no—
171C2×C32×C4—21no—
172C2×C32×C4—41no—
173C2×C32×C4—61no—
174C2×C32×C4C22×D311yescharacteristic
175C2×C32×C4C22×D311yescharacteristic
176C2×C32×C4C22×D311yescharacteristic
177C2×C3⋊Dic3—21no—
178C2×C3⋊Dic3—61no—
179C2×C3⋊Dic3—62no—
180C2×C3⋊Dic3—61no—
181C2×C3⋊Dic3—61no—
182C2×C3⋊Dic3—121no—
183C2×C3⋊Dic3—121no—
184C2×C3⋊Dic3—122no—
185C2×C3⋊Dic3D1211yescharacteristic
186C4⋊(C3⋊D3)—21no—
187C4⋊(C3⋊D3)C4×D312yescomplemented
188C4×C3⋊D3—122no—
189C4×C3⋊D3C4×D312yes—
190C4×C3⋊D3#6712yes—
191#634—21no—
192#634C4×D312yescomplemented
193C2×C3×Dic3—22no—
194C2×C3×Dic3—21no—
195C2×C3×Dic3—21no—
196C2×C3×Dic3—31no—
197C2×C3×Dic3—31no—
198C2×C3×Dic3—61no—
199C2×C3×Dic3—61no—
200C2×C3×Dic3—61no—
201C2×C3×Dic3—121no—
202C3×D12—32no—
203C3×D12—32no—
204C3×D12—62no—
205C3×D12—61no—
206C3×D12—61no—
207C3×D12—121no—
208C3×C4×D3—32no—
209C3×C4×D3—32no—
210C3×C4×D3—32no—
211C3×C4×D3—32no—
212C3×C4×D3—62no—
213C3×C4×D3—62no—
214C3×Dic6—32no—
215C3×Dic6—32no—
216C3×Dic6—62no—
217C3×Dic6—61no—
218C3×Dic6—61no—
219C3×Dic6—121no—
220#626—64no—
221#626—64no—
222#625—64no—
223#624—64no—
224D3×Dic3—64no—
225D3×Dic3—64no—
226D3×Dic3—124no—
227D3×Dic3—122no—
228D3×Dic3—122no—
229C32×C8—61no—
230C32×C8—61no—
231C3×C3⋊C8—21no—
232C3×C3⋊C8—21no—
233C3×C3⋊C8—21no—
234C3×C3⋊C8—21no—
235C3×C3⋊C8—41no—
236C3×C3⋊C8—41no—
237#417—271noSylow
238C2×C33C2×C22⋊C411yescharacteristic
239C2×C33AGL(1,ℤ/8)11yescharacteristic
240C2×C33#13711yescharacteristic
241C3×C3⋊D3—22no—
242C3×C3⋊D3—42no—
243C2×C3×Q8—91no—
244C2×C3×D4—91no—
245C22×C3×C4—91no—
246C2×#67—181no—
247C2×#67—181no—
248C22×Dic3—181no—
249C22×Dic3—181no—
250#239—94no—
251#239—182no—
252#237—94no—
253#237—182no—
254#235—94no—
255#235—182no—
256C2×D12—31no—
257C2×D12—31no—
258C2×D12—61no—
259C2×D12—181no—
260C2×C4×D3—31no—
261C2×C4×D3—31no—
262C2×C4×D3—61no—
263C2×C4×D3—181no—
264C2×C4×D3—181no—
265C2×C4×D3—181no—
266C2×C4×D3—181no—
267C2×C4×D3—182no—
268C2×Dic6—31no—
269C2×Dic6—31no—
270C2×Dic6—61no—
271C3×M16—182no—
272C2×C3×C8—31no—
273C2×C3×C8—31no—
274C2×C3×C8—31no—
275C2×C3×C8—31no—
276C2×C3×C8—61no—
277C2×C3×C8—61no—
278C3×C4⋊C4—31no—
279C3×C4⋊C4—31no—
280C3×C4⋊C4—61no—
281C4⋊Dic3—91no—
282C4⋊Dic3—91no—
283C4⋊Dic3—91no—
284C4⋊Dic3—181no—
285C4⋊Dic3—181no—
286C4⋊Dic3—181no—
287C4⋊Dic3—181no—
288C4⋊Dic3—361no—
289C4⋊Dic3D3211yescharacteristic
290Dic3⋊C4—181no—
291Dic3⋊C4—181no—
292Dic3⋊C4—182no—
293C2×C3⋊C8D3211yescharacteristic
294C2×C3⋊C8D3211yescharacteristic
295ΓL(1,𝔽25)—62no—
296ΓL(1,𝔽25)—62no—
297C22×C32—21no—
298C22×C32—21no—
299C22×C32—21no—
300C22×C32—41no—
301C22×C32C2×D1211yescharacteristic
302C22×C32D3×D411yescharacteristic
303C22×C32D3×D411yescharacteristic
304C2×C3⋊D3—21no—
305C2×C3⋊D3—21no—
306C2×C3⋊D3—22no—
307C2×C3⋊D3—41no—
308C2×C3⋊D3—41no—
309C2×C3⋊D3(C2×D3)⋊C412yes—
310C2×C3×D3—32no—
311C2×C3×D3—32no—
312C2×C3×D3—62no—
313C2×C3×D3—62no—
314C2×C3×D3—62no—
315C2×C3×D3—61no—
316C2×C3×D3—61no—
317C2×C3×D3—61no—
318C2×C3×D3—61no—
319C2×C3×D3—62no—
320C2×C3×D3—62no—
321C2×C3×D3—122no—
322C2×C3×D3—121no—
323C2×C3×D3—121no—
324C2×C3×D3—121no—
325C2×C3×D3—121no—
326C2×C3×D3—241no—
327C2×C3×D3—241no—
328C32×C4—21no—
329C32×C4—21no—
330C32×C4—21no—
331C32×C4—21no—
332C32×C4—21no—
333C32×C4—21no—
334C32×C4—41no—
335C32×C4—41no—
336C32×C4—121no—
337C32×C4C2×C4×D311yescharacteristic
338C32×C4C2×C4×D311yescharacteristic
339C32×C4C2×C4×D311yescharacteristic
340C32×C4D3×D411yescharacteristic
341C32×C4D3×D411yescharacteristic
342C32×C4C2×#6711yescharacteristic
343C3⋊Dic3—22no—
344C3⋊Dic3—121no—
345C3⋊Dic3—122no—
346C3⋊Dic3—121no—
347C3⋊Dic3—121no—
348C3⋊Dic3—241no—
349C3⋊Dic3—241no—
350C3⋊Dic3—242no—
351C3⋊Dic3(C2×D3)⋊C412yes—
352C3×Dic3—32no—
353C3×Dic3—32no—
354C3×Dic3—42no—
355C3×Dic3—41no—
356C3×Dic3—41no—
357C3×Dic3—62no—
358C3×Dic3—62no—
359C3×Dic3—62no—
360C3×Dic3—122no—
361C2×#41—271no—
362C2×M16—91no—
363C2×C4⋊C4—271no—
364#103—91no—
365#103—271no—
366#102—91no—
367#102—271no—
368C33#41711yescharacteristic, complemented, Sylow
369C23×C3—91no—
370C23×C3—181no—
371C22×D3—31no—
372C22×D3—31no—
373C22×D3—61no—
374C22×D3—61no—
375C22×D3—61no—
376C22×D3—121no—
377C3×Q8—92no—
378C3×Q8—181no—
379C3×D4—92no—
380C3×D4—181no—
381C2×C3×C4—21no—
382C2×C3×C4—21no—
383C2×C3×C4—21no—
384C2×C3×C4—41no—
385C2×C3×C4—62no—
386C2×C3×C4—61no—
387C2×C3×C4—61no—
388C2×C3×C4—91no—
389C2×C3×C4—92no—
390C2×C3×C4—92no—
391C2×C3×C4—181no—
392C2×C3×C4C2×D3211yescharacteristic
393C2×C3×C4C2×D3211yescharacteristic
394C2×C3×C4C2×D3211yescharacteristic
395#67—184no—
396#67—184no—
397C2×Dic3—21no—
398C2×Dic3—31no—
399C2×Dic3—31no—
400C2×Dic3—61no—
401C2×Dic3—61no—
402C2×Dic3—61no—
403C2×Dic3—121no—
404C2×Dic3—184no—
405C2×Dic3—182no—
406C2×Dic3—181no—
407C2×Dic3—181no—
408C2×Dic3—181no—
409C2×Dic3—361no—
410C2×Dic3—361no—
411C2×Dic3—362no—
412C2×Dic3—362no—
413D12—32no—
414D12—32no—
415D12—62no—
416D12—61no—
417D12—61no—
418D12—121no—
419D12—184no—
420C4×D3—32no—
421C4×D3—32no—
422C4×D3—32no—
423C4×D3—32no—
424C4×D3—62no—
425C4×D3—62no—
426C4×D3—184no—
427C4×D3—184no—
428C4×D3—364no—
429C4×D3—362no—
430C4×D3—362no—
431Dic6—32no—
432Dic6—32no—
433Dic6—62no—
434Dic6—61no—
435Dic6—61no—
436Dic6—121no—
437C3×C8—61no—
438C3×C8—61no—
439C3×C8—61no—
440C3×C8—61no—
441C3×C8—121no—
442C3×C8—121no—
443C3⋊C8—21no—
444C3⋊C8—21no—
445C2×C32—21no—
446C2×C32—21no—
447C2×C32—21no—
448C2×C32—21no—
449C2×C32—21no—
450C2×C32—21no—
451C2×C32—21no—
452C2×C32—21no—
453C2×C32—21no—
454C2×C32—41no—
455C2×C32—41no—
456C2×C32—41no—
457C2×C32D3×C22⋊C411yescharacteristic
458C2×C32D3×C22⋊C411yescharacteristic
459C2×C32#90811yescharacteristic
460C2×C32#90911yescharacteristic
461C2×C32#91111yescharacteristic
462C2×C32#91411yescharacteristic
463C2×C32#91511yescharacteristic
464C2×C32#91811yescharacteristic
465C2×C32C2×(C2×D3)⋊C411yescharacteristic
466C3⋊D3—22no—
467C3⋊D3—42no—
468C3×D3—64no—
469C3×D3—62no—
470C3×D3—62no—
471C3×D3—122no—
472C3×D3—122no—
473C3×D3—242no—
474#41—274no—
475#41—542no—
476C2×Q8—91no—
477C2×D4—91no—
478C2×D4—541no—
479C22×C4—91no—
480C22×C4—541no—
481C22×C4—541no—
482M16—182no—
483C2×C8—31no—
484C2×C8—31no—
485C4⋊C4—31no—
486C4⋊C4—271no—
487C4⋊C4—541no—
488C22×C3—21no—
489C22×C3—21no—
490C22×C3—21no—
491C22×C3—41no—
492C22×C3—92no—
493C22×C3—181no—
494C22×C3—181no—
495C22×C3—182no—
496C22×C3—361no—
497C22×C3—361no—
498C22×C3D3×D1211yescharacteristic
499C22×C3D3×D1211yescharacteristic
500C22×C3#381111yescharacteristic
501C2×D3—32no—
502C2×D3—32no—
503C2×D3—62no—
504C2×D3—62no—
505C2×D3—62no—
506C2×D3—61no—
507C2×D3—61no—
508C2×D3—61no—
509C2×D3—61no—
510C2×D3—62no—
511C2×D3—62no—
512C2×D3—122no—
513C2×D3—121no—
514C2×D3—121no—
515C2×D3—121no—
516C2×D3—121no—
517C2×D3—241no—
518C2×D3—241no—
519C3×C4—21no—
520C3×C4—21no—
521C3×C4—21no—
522C3×C4—21no—
523C3×C4—21no—
524C3×C4—21no—
525C3×C4—41no—
526C3×C4—41no—
527C3×C4—92no—
528C3×C4—122no—
529C3×C4—121no—
530C3×C4—121no—
531C3×C4—182no—
532C3×C4C4×D3211yescharacteristic
533C3×C4C4×D3211yescharacteristic
534C3×C4C4×D3211yescharacteristic
535C3×C4#380211yescharacteristic
536C3×C4D3×#6711yescharacteristic
537C3×C4D3×#6711yescharacteristic
538Dic3—32no—
539Dic3—32no—
540Dic3—41no—
541Dic3—62no—
542Dic3—62no—
543Dic3—62no—
544Dic3—122no—
545Dic3—362no—
546Dic3—361no—
547Dic3—361no—
548Dic3—361no—
549Dic3—721no—
550Dic3—721no—
551Dic3—722no—
552C32—21no—
553C32—21no—
554C32—21no—
555C32—41no—
556C32#477111yescharacteristic, complemented
557C32#480311yescharacteristic, complemented
558C32#513311yescharacteristic, complemented
559C23—91no—
560C23—181no—
561Q8—92no—
562Q8—181no—
563D4—92no—
564D4—181no—
565D4—544no—
566C2×C4—61no—
567C2×C4—91no—
568C2×C4—92no—
569C2×C4—92no—
570C2×C4—181no—
571C2×C4—544no—
572C2×C4—541no—
573C2×C4—1082no—
574C2×C4D3311yescharacteristic
575C8—61no—
576C8—61no—
577C2×C3—21no—
578C2×C3—21no—
579C2×C3—21no—
580C2×C3—21no—
581C2×C3—21no—
582C2×C3—21no—
583C2×C3—21no—
584C2×C3—21no—
585C2×C3—21no—
586C2×C3—41no—
587C2×C3—41no—
588C2×C3—41no—
589C2×C3—182no—
590C2×C3—362no—
591C2×C3#6379411yescharacteristic
592C2×C3#6379511yescharacteristic
593C2×C3#6380011yescharacteristic
594C2×C3#6380111yescharacteristic
595C2×C3D3×(C2×D3)⋊C411yescharacteristic
596C2×C3D3×(C2×D3)⋊C411yescharacteristic
597C2×C3#6392111yescharacteristic
598C2×C3#6393711yescharacteristic
599C2×C3#6395111yescharacteristic
600D3—64no—
601D3—62no—
602D3—62no—
603D3—122no—
604D3—122no—
605D3—242no—
606C22—92no—
607C22—181no—
608C22—181no—
609C22—182no—
610C22—361no—
611C22—361no—
612C22#8895511yescharacteristic
613C4—92no—
614C4—121no—
615C4—182no—
616C4—1081no—
617C4#8894911yescharacteristic
618C4#8895311yescharacteristic
619C3—21no—
620C3—21no—
621C3—21no—
622C3—41no—
623C3#1059061211yescharacteristic, complemented
624C3#1059067011yescharacteristic, complemented
625C3#1059067111yescharacteristic, complemented
626C2—182no—
627C2—362no—
628C2#1172287911yescharacteristic
629C2#1172289611yescharacteristic
630C2#1172290511yescharacteristic
subgroupG/H
centerC22#88955
commutatorC33×C4C22×C4
Frattini, hypercenterC2×C4D33
Fitting, cosocleC2×C33×C4C23
socleC22×C33C2×D4
prime radicalC2×#89041C2
hypocenterC33#417
LCSG ▷ 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⋊D3×C4⋊Dic3C2semidirect
3. C2×C3×#3712C2semidirect
4. #11722040C2semidirect
5. #11722034C2semidirect
6. C3×#63567C2semidirect
7. C3×#63564C2semidirect
17. #89041C4semidirect
34. C2×#3712D3semidirect
41. #63567D3semidirect
45. #63564D3semidirect
56. C3×C4⋊(C3⋊D3)C2×C4semidirect
60. C3×#634C2×C4semidirect
187. C4⋊(C3⋊D3)C4×D3semidirect
192. #634C4×D3semidirect
368. C33#417semidirect
556. C32#4771semidirect
557. C32#4803semidirect
558. C32#5133semidirect
623. C3#10590612semidirect
624. C3#10590670semidirect
625. C3#10590671semidirect

Maximal subgroups

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

subgroupindex
C2×#890412
C3⋊D3×C4⋊Dic32
C2×C3×#37122
#117220402
#117220342
C3×#635672
C3×#635642
#105906713
#105906703
#105906123