Finite Group Databasebeta

← #13155397

Subgroups — #13155397

Order 1728.

1026 conjugacy classes of subgroups.

The subgroup lattice has 688 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#89015C211yesmaximal, characteristic
2#11723503C211yesmaximal, characteristic, complemented
3C2×#88788C211yesmaximal, characteristic, complemented
4#11722076C211yesmaximal, characteristic, complemented
5C3×#63567C211yesmaximal, characteristic, complemented
6C3×#63563C211yesmaximal, characteristic, complemented
7C3×#63562C211yesmaximal, characteristic, complemented
8#10591418—31nomaximal
9#10590957—31nomaximal
10#10590612—31nomaximal
11C2×C4×C3⋊(C3⋊D3)C2211yescharacteristic
12C2×C3×#634C2211yescharacteristic
13C2×C32×D12C2211yescharacteristic
14C2×#6291—21no—
15C2×#6289—21no—
16C2×D3×C3⋊Dic3—21no—
17#89015—22no—
18#89015C414yescomplemented
19C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
20C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
21C32×C4⋊Dic3C2211yescharacteristic
22C2×C32×C3⋊C8C2211yescharacteristic
23#88806—21no—
24#88788—22no—
25C2×#3830—31no—
26C2×#3796—31no—
27C2×#3796—31no—
28C2×#3796—61no—
29C2×#3743—31no—
30C4⋊C4×C3⋊D3—31no—
31#63900—31no—
32#63900—31no—
33#63900—61no—
34C2×#3712—31no—
35C2×#3712—31no—
36C2×#3712—61no—
37C3×#832—31no—
38C3×#832—31no—
39C3×#821—31no—
40C3×#816—31no—
41C3×#810—31no—
42C3×#809—31no—
43#63567—31no—
44#63567D311yescharacteristic, complemented
45#63566—31no—
46#63565—31no—
47#63563D311yescharacteristic, complemented
48#63562D311yescharacteristic, complemented
49C22×C3⋊(C3⋊D3)D411yescharacteristic
50C22×C32×D3—21no—
51C2×C33×C4C2311yescharacteristic
52C2×C3⋊(C3⋊Dic3)D411yescharacteristic
53C4×C3⋊(C3⋊D3)C2×C412yes—
54C4×C3⋊(C3⋊D3)D412yes—
55C2×C3×C3⋊Dic3—21no—
56C2×C3×C3⋊Dic3—21no—
57C3×#634—21no—
58C3×#634C2×C412yescomplemented
59C2×C32×Dic3—21no—
60C32×D12—21no—
61C32×D12C2×C412yescomplemented
62#6291—24no—
63#6289—42no—
64D3×C3⋊Dic3—24no—
65C3×C3⋊(C3⋊C8)—21no—
66C32×C3⋊C8—21no—
67#4803—91no—
68#4771—91no—
69#4771—91no—
70C2×C32×D4—31no—
71C2×#638—61no—
72C22×C3⋊Dic3—61no—
73#3830—34no—
74#3830—62no—
75C2×C4×C3⋊D3—31no—
76C2×C4×C3⋊D3—31no—
77C2×C4×C3⋊D3—31no—
78C2×C4×C3⋊D3—61no—
79C2×C4×C3⋊D3—61no—
80C2×C4×C3⋊D3—61no—
81C2×C4×C3⋊D3—61no—
82C2×C4×C3⋊D3—121no—
83C2×#634C2×D311yescharacteristic
84C2×C3×D12—21no—
85C2×C3×D12C2×D311yescharacteristic
86C2×C3×D12C2×D311yescharacteristic
87C2×C3×Dic6—31no—
88C2×C3×Dic6—31no—
89C2×C3×Dic6—61no—
90C2×#626—61no—
91C2×#626—61no—
92C2×#626—62no—
93C2×#624—61no—
94C2×#624—61no—
95C2×#624—62no—
96C2×D3×Dic3—61no—
97C2×D3×Dic3—61no—
98C2×D3×Dic3—62no—
99#3796—34no—
100#3796—34no—
101#3796—62no—
102#3796—62no—
103#3796—64no—
104#3796—64no—
105C2×C32×C8—31no—
106C32×C4⋊C4—31no—
107C4⋊(C3⋊Dic3)C2×D311yescharacteristic
108#3750—61no—
109C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
110#3743—62no—
111C3×C4⋊Dic3—21no—
112C3×C4⋊Dic3—31no—
113C3×C4⋊Dic3—31no—
114C3×C4⋊Dic3—61no—
115C3×C4⋊Dic3C2×D311yescharacteristic
116C3×C4⋊Dic3C2×D311yescharacteristic
117C2×C3×C3⋊C8—21no—
118C2×C3×C3⋊C8—31no—
119C2×C3×C3⋊C8—31no—
120C2×C3×C3⋊C8—61no—
121C2×C3×C3⋊C8C2×D311yescharacteristic
122C2×C3×C3⋊C8C2×D311yescharacteristic
123#3724—61no—
124#3724—61no—
125#3724—62no—
126#3712—62no—
127#3712—62no—
128#3712—64no—
129C22×C33C2×D411yescharacteristic
130C2×C3⋊(C3⋊D3)—21no—
131C2×C3⋊(C3⋊D3)—21no—
132C2×C3⋊(C3⋊D3)C22⋊C412yes—
133C2×C32×D3—22no—
134C2×C32×D3—41no—
135C2×C32×D3—41no—
136C33×C4C22×C411yescharacteristic
137C33×C4C2×D411yescharacteristic
138C3⋊(C3⋊Dic3)C22⋊C412yes—
139C3×C3⋊Dic3—22no—
140C3×C3⋊Dic3—41no—
141C32×Dic3—41no—
142C2×#239—91no—
143C2×#237—91no—
144C2×#237—91no—
145C2×#237—181no—
146C2×ΓL(1,𝔽25)—91no—
147C2×ΓL(1,𝔽25)—91no—
148C2×ΓL(1,𝔽25)—91no—
149C2×ΓL(1,𝔽25)—181no—
150D3×C4⋊C4—91no—
151D3×C4⋊C4—91no—
152D3×C4⋊C4—91no—
153D3×C4⋊C4—181no—
154C3×#103—91no—
155C3×#102—91no—
156C3×#102—91no—
157#835—91no—
158#832—31no—
159#832—31no—
160#821—31no—
161#816—31no—
162#816—91no—
163#810—31no—
164#810—91no—
165#809—31no—
166C23×C32—61no—
167C22×C3⋊D3—31no—
168C22×C3⋊D3—31no—
169C22×C3⋊D3—31no—
170C22×C3⋊D3—61no—
171C22×C3⋊D3—61no—
172C22×C3⋊D3—61no—
173C22×C3⋊D3—121no—
174C22×C3×D3—22no—
175C22×C3×D3—21no—
176C22×C3×D3—21no—
177C32×D4—32no—
178C32×D4—61no—
179C2×C32×C4—21no—
180C2×C32×C4—21no—
181C2×C32×C4—21no—
182C2×C32×C4—41no—
183C2×C32×C4—61no—
184C2×C32×C4C22×D311yescharacteristic
185C2×C32×C4C22×D311yescharacteristic
186C2×C32×C4C22×D311yescharacteristic
187#638—64no—
188C2×C3⋊Dic3—21no—
189C2×C3⋊Dic3—21no—
190C2×C3⋊Dic3—31no—
191C2×C3⋊Dic3—31no—
192C2×C3⋊Dic3—31no—
193C2×C3⋊Dic3—61no—
194C2×C3⋊Dic3—61no—
195C2×C3⋊Dic3—61no—
196C2×C3⋊Dic3—64no—
197C2×C3⋊Dic3—121no—
198C4×C3⋊D3—32no—
199C4×C3⋊D3—32no—
200C4×C3⋊D3—32no—
201C4×C3⋊D3—32no—
202C4×C3⋊D3—32no—
203C4×C3⋊D3—32no—
204C4×C3⋊D3—62no—
205C4×C3⋊D3—62no—
206C4×C3⋊D3—62no—
207C4×C3⋊D3—62no—
208C4×C3⋊D3—62no—
209C4×C3⋊D3—62no—
210C4×C3⋊D3—122no—
211C4×C3⋊D3—122no—
212C4×C3⋊D3—122no—
213#634—21no—
214#634C4×D312yescomplemented
215C2×C3×Dic3—22no—
216C2×C3×Dic3—21no—
217C2×C3×Dic3—21no—
218C2×C3×Dic3—62no—
219C2×C3×Dic3—61no—
220C2×C3×Dic3—61no—
221C2×C3×Dic3—62no—
222C2×C3×Dic3—61no—
223C2×C3×Dic3—61no—
224C3×D12—22no—
225C3×D12—21no—
226C3×D12—21no—
227C3×D12—22no—
228C3×D12C4×D312yescomplemented
229C3×D12C4×D312yescomplemented
230C3×Dic6—32no—
231C3×Dic6—32no—
232C3×Dic6—62no—
233C3×Dic6—62no—
234C3×Dic6—61no—
235C3×Dic6—61no—
236#626—64no—
237#626—64no—
238#626—68no—
239#624—124no—
240#624—122no—
241#624—122no—
242D3×Dic3—64no—
243D3×Dic3—64no—
244D3×Dic3—68no—
245C32×C8—61no—
246C3⋊(C3⋊C8)—21no—
247C3×C3⋊C8—22no—
248C3×C3⋊C8—21no—
249C3×C3⋊C8—21no—
250C3×C3⋊C8—62no—
251C3×C3⋊C8—61no—
252C3×C3⋊C8—61no—
253#417—271noSylow
254C2×C33C2×C22⋊C411yescharacteristic
255C2×C33AGL(1,ℤ/8)11yescharacteristic
256C2×C33#13711yescharacteristic
257C3⋊(C3⋊D3)—22no—
258C32×D3—42no—
259C2×C3×Q8—91no—
260C2×C3×D4—31no—
261C2×C3×D4—31no—
262C2×C3×D4—61no—
263C2×#67—181no—
264C2×#67—181no—
265C2×#67—182no—
266C22×Dic3—181no—
267C22×Dic3—181no—
268C22×Dic3—182no—
269#239—94no—
270#239—182no—
271#237—94no—
272#237—94no—
273#237—184no—
274#237—182no—
275#237—182no—
276#237—184no—
277C2×D12—181no—
278C2×D12D3211yescharacteristic
279C2×C4×D3—91no—
280C2×C4×D3—91no—
281C2×C4×D3—91no—
282C2×C4×D3—181no—
283C2×C4×D3—181no—
284C2×C4×D3—181no—
285C2×C4×D3—181no—
286C2×C4×D3—181no—
287C2×C4×D3—181no—
288C2×C4×D3—181no—
289C2×C4×D3—182no—
290C2×C4×D3—361no—
291C2×Dic6—31no—
292C2×Dic6—31no—
293C2×Dic6—61no—
294C2×C3×C8—31no—
295C2×C3×C8—31no—
296C2×C3×C8—61no—
297C2×C3×C8—91no—
298C3×C4⋊C4—31no—
299C3×C4⋊C4—31no—
300C3×C4⋊C4—61no—
301C3×C4⋊C4—91no—
302C4⋊Dic3—31no—
303C4⋊Dic3—31no—
304C4⋊Dic3—61no—
305C4⋊Dic3D3211yescharacteristic
306Dic3⋊C4—181no—
307Dic3⋊C4—181no—
308Dic3⋊C4—181no—
309Dic3⋊C4—182no—
310C2×C3⋊C8—31no—
311C2×C3⋊C8—31no—
312C2×C3⋊C8—61no—
313C2×C3⋊C8D3211yescharacteristic
314ΓL(1,𝔽25)—182no—
315ΓL(1,𝔽25)—182no—
316ΓL(1,𝔽25)—182no—
317ΓL(1,𝔽25)—184no—
318C22×C32—21no—
319C22×C32—21no—
320C22×C32—21no—
321C22×C32—41no—
322C22×C32—62no—
323C22×C32—121no—
324C22×C32—121no—
325C22×C32D3×D411yescharacteristic
326C22×C32D3×D411yescharacteristic
327C22×C32D3×D411yescharacteristic
328C2×C3⋊D3—32no—
329C2×C3⋊D3—32no—
330C2×C3⋊D3—32no—
331C2×C3⋊D3—62no—
332C2×C3⋊D3—62no—
333C2×C3⋊D3—62no—
334C2×C3⋊D3—61no—
335C2×C3⋊D3—61no—
336C2×C3⋊D3—62no—
337C2×C3⋊D3—62no—
338C2×C3⋊D3—61no—
339C2×C3⋊D3—61no—
340C2×C3⋊D3—61no—
341C2×C3⋊D3—61no—
342C2×C3⋊D3—122no—
343C2×C3⋊D3—121no—
344C2×C3⋊D3—121no—
345C2×C3⋊D3—121no—
346C2×C3⋊D3—121no—
347C2×C3⋊D3—122no—
348C2×C3⋊D3—122no—
349C2×C3×D3—24no—
350C2×C3×D3—22no—
351C2×C3×D3—22no—
352C2×C3×D3—42no—
353C2×C3×D3—42no—
354C2×C3×D3—41no—
355C2×C3×D3—41no—
356C2×C3×D3—41no—
357C2×C3×D3—41no—
358C32×C4—21no—
359C32×C4—21no—
360C32×C4—21no—
361C32×C4—21no—
362C32×C4—21no—
363C32×C4—21no—
364C32×C4—41no—
365C32×C4—41no—
366C32×C4—121no—
367C32×C4C2×C4×D311yescharacteristic
368C32×C4C2×C4×D311yescharacteristic
369C32×C4C2×C4×D311yescharacteristic
370C32×C4D3×D411yescharacteristic
371C32×C4D3×D411yescharacteristic
372C32×C4D3×D411yescharacteristic
373C3⋊Dic3—22no—
374C3⋊Dic3—32no—
375C3⋊Dic3—32no—
376C3⋊Dic3—32no—
377C3⋊Dic3—41no—
378C3⋊Dic3—62no—
379C3⋊Dic3—62no—
380C3⋊Dic3—62no—
381C3⋊Dic3—122no—
382C3×Dic3—42no—
383C3×Dic3—41no—
384C3×Dic3—41no—
385C3×Dic3—64no—
386C3×Dic3—62no—
387C3×Dic3—62no—
388C3×Dic3—122no—
389C3×Dic3—121no—
390C3×Dic3—121no—
391C2×#41—271no—
392C2×M16—271no—
393C2×C4⋊C4—271no—
394#103—91no—
395#103—271no—
396#102—91no—
397#102—91no—
398C33#41711yescharacteristic, complemented, Sylow
399C23×C3—62no—
400C23×C3—61no—
401C23×C3—61no—
402C22×D3—21no—
403C22×D3—91no—
404C22×D3—91no—
405C22×D3—91no—
406C22×D3—181no—
407C22×D3—181no—
408C22×D3—181no—
409C22×D3—361no—
410C3×Q8—92no—
411C3×Q8—181no—
412C3×D4—32no—
413C3×D4—32no—
414C3×D4—62no—
415C3×D4—62no—
416C3×D4—61no—
417C3×D4—61no—
418C2×C3×C4—21no—
419C2×C3×C4—21no—
420C2×C3×C4—21no—
421C2×C3×C4—41no—
422C2×C3×C4—62no—
423C2×C3×C4—61no—
424C2×C3×C4—61no—
425C2×C3×C4—181no—
426C2×C3×C4—181no—
427C2×C3×C4C2×D3211yescharacteristic
428C2×C3×C4C2×D3211yescharacteristic
429C2×C3×C4C2×D3211yescharacteristic
430#67—184no—
431#67—184no—
432#67—188no—
433C2×Dic3—21no—
434C2×Dic3—62no—
435C2×Dic3—61no—
436C2×Dic3—61no—
437C2×Dic3—62no—
438C2×Dic3—61no—
439C2×Dic3—61no—
440C2×Dic3—91no—
441C2×Dic3—91no—
442C2×Dic3—91no—
443C2×Dic3—181no—
444C2×Dic3—181no—
445C2×Dic3—181no—
446C2×Dic3—184no—
447C2×Dic3—184no—
448C2×Dic3—188no—
449C2×Dic3—361no—
450D12—21no—
451D12—184no—
452D12#62412yescomplemented
453C4×D3—92no—
454C4×D3—92no—
455C4×D3—92no—
456C4×D3—92no—
457C4×D3—92no—
458C4×D3—92no—
459C4×D3—182no—
460C4×D3—182no—
461C4×D3—182no—
462C4×D3—182no—
463C4×D3—182no—
464C4×D3—182no—
465C4×D3—184no—
466C4×D3—362no—
467C4×D3—362no—
468C4×D3—362no—
469C4×D3—364no—
470C4×D3—362no—
471C4×D3—362no—
472Dic6—32no—
473Dic6—32no—
474Dic6—62no—
475Dic6—62no—
476Dic6—61no—
477Dic6—61no—
478C3×C8—62no—
479C3×C8—61no—
480C3×C8—61no—
481C3×C8—181no—
482C3⋊C8—21no—
483C3⋊C8—62no—
484C3⋊C8—61no—
485C3⋊C8—61no—
486C2×C32—21no—
487C2×C32—21no—
488C2×C32—21no—
489C2×C32—21no—
490C2×C32—21no—
491C2×C32—21no—
492C2×C32—21no—
493C2×C32—21no—
494C2×C32—21no—
495C2×C32—41no—
496C2×C32—41no—
497C2×C32—41no—
498C2×C32—122no—
499C2×C32D3×C22⋊C411yescharacteristic
500C2×C32D3×C22⋊C411yescharacteristic
501C2×C32D3×C22⋊C411yescharacteristic
502C2×C32#91111yescharacteristic
503C2×C32#91111yescharacteristic
504C2×C32#91411yescharacteristic
505C2×C32#91511yescharacteristic
506C2×C32#91511yescharacteristic
507C2×C32#91811yescharacteristic
508C3⋊D3—62no—
509C3⋊D3—64no—
510C3⋊D3—62no—
511C3⋊D3—62no—
512C3⋊D3—122no—
513C3⋊D3—122no—
514C3⋊D3—124no—
515C3×D3—44no—
516C3×D3—42no—
517C3×D3—42no—
518#41—274no—
519#41—542no—
520C2×Q8—91no—
521C2×D4—31no—
522C2×D4—541no—
523C22×C4—271no—
524C22×C4—541no—
525C22×C4—541no—
526M16—542no—
527C2×C8—31no—
528C2×C8—91no—
529C4⋊C4—31no—
530C4⋊C4—91no—
531C4⋊C4—541no—
532C22×C3—21no—
533C22×C3—21no—
534C22×C3—21no—
535C22×C3—41no—
536C22×C3—64no—
537C22×C3—62no—
538C22×C3—62no—
539C22×C3—122no—
540C22×C3—122no—
541C22×C3—121no—
542C22×C3—121no—
543C22×C3—121no—
544C22×C3—121no—
545C22×C3#380211yescharacteristic
546C22×C3#381111yescharacteristic
547C22×C3#381111yescharacteristic
548C2×D3—22no—
549C2×D3—41no—
550C2×D3—41no—
551C2×D3—92no—
552C2×D3—92no—
553C2×D3—92no—
554C2×D3—182no—
555C2×D3—182no—
556C2×D3—182no—
557C2×D3—182no—
558C2×D3—182no—
559C2×D3—181no—
560C2×D3—181no—
561C2×D3—181no—
562C2×D3—181no—
563C2×D3—181no—
564C2×D3—181no—
565C2×D3—362no—
566C2×D3—361no—
567C2×D3—361no—
568C2×D3—361no—
569C2×D3—361no—
570C2×D3—362no—
571C2×D3—362no—
572C3×C4—21no—
573C3×C4—21no—
574C3×C4—21no—
575C3×C4—21no—
576C3×C4—21no—
577C3×C4—21no—
578C3×C4—41no—
579C3×C4—41no—
580C3×C4—122no—
581C3×C4—121no—
582C3×C4—121no—
583C3×C4—182no—
584C3×C4—361no—
585C3×C4C4×D3211yescharacteristic
586C3×C4C4×D3211yescharacteristic
587C3×C4#380211yescharacteristic
588C3×C4C2×#62411yescharacteristic
589C3×C4#381111yescharacteristic
590C3×C4#381111yescharacteristic
591Dic3—41no—
592Dic3—64no—
593Dic3—62no—
594Dic3—62no—
595Dic3—92no—
596Dic3—92no—
597Dic3—92no—
598Dic3—122no—
599Dic3—121no—
600Dic3—121no—
601Dic3—182no—
602Dic3—182no—
603Dic3—182no—
604Dic3—362no—
605C32—21no—
606C32—21no—
607C32—21no—
608C32—41no—
609C32#477111yescharacteristic, complemented
610C32#477111yescharacteristic, complemented
611C32#480311yescharacteristic, complemented
612C23—61no—
613C23—271no—
614Q8—92no—
615Q8—181no—
616D4—32no—
617D4—61no—
618D4—544no—
619C2×C4—61no—
620C2×C4—181no—
621C2×C4—181no—
622C2×C4—271no—
623C2×C4—272no—
624C2×C4—272no—
625C2×C4—544no—
626C2×C4—1082no—
627C2×C4D3311yescharacteristic
628C8—61no—
629C8—181no—
630C2×C3—21no—
631C2×C3—21no—
632C2×C3—21no—
633C2×C3—21no—
634C2×C3—21no—
635C2×C3—21no—
636C2×C3—21no—
637C2×C3—21no—
638C2×C3—21no—
639C2×C3—41no—
640C2×C3—41no—
641C2×C3—41no—
642C2×C3—124no—
643C2×C3—122no—
644C2×C3—122no—
645C2×C3#6379811yescharacteristic
646C2×C3#6381111yescharacteristic
647C2×C3#6392111yescharacteristic
648C2×C3#6392111yescharacteristic
649C2×C3#6392711yescharacteristic
650C2×C3#6393111yescharacteristic
651C2×C3#6393711yescharacteristic
652C2×C3#6395111yescharacteristic
653C2×C3#6397411yescharacteristic
654D3—42no—
655D3—184no—
656D3—182no—
657D3—182no—
658D3—182no—
659D3—362no—
660D3—362no—
661D3—364no—
662C22—62no—
663C22—121no—
664C22—121no—
665C22—272no—
666C22—541no—
667C22—541no—
668C22#8895411yescharacteristic
669C4—121no—
670C4—182no—
671C4—272no—
672C4—361no—
673C4D3×#62411yescharacteristic
674C4#8895411yescharacteristic
675C3—21no—
676C3—21no—
677C3—21no—
678C3—41no—
679C3#1059061211yescharacteristic, complemented
680C3#1059095711yescharacteristic, complemented
681C3#1059141811yescharacteristic, complemented
682C2—122no—
683C2—542no—
684C2#1172287711yescharacteristic
685C2#1172289311yescharacteristic
686C2#1172290611yescharacteristic
subgroupG/H
centerC22#88954
commutatorC33×C4C22×C4
Frattini, hypercenterC2×C4D33
Fitting, cosocleC2×C33×C4C23
socleC22×C33C2×D4
prime radicalC2×#89015C2
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. #11723503C2semidirect
3. C2×#88788C2semidirect
4. #11722076C2semidirect
5. C3×#63567C2semidirect
6. C3×#63563C2semidirect
7. C3×#63562C2semidirect
18. #89015C4semidirect
44. #63567D3semidirect
47. #63563D3semidirect
48. #63562D3semidirect
58. C3×#634C2×C4semidirect
61. C32×D12C2×C4semidirect
214. #634C4×D3semidirect
228. C3×D12C4×D3semidirect
229. C3×D12C4×D3semidirect
398. C33#417semidirect
452. D12#624semidirect
609. C32#4771semidirect
610. C32#4771semidirect
611. C32#4803semidirect
679. C3#10590612semidirect
680. C3#10590957semidirect
681. C3#10591418semidirect

Maximal subgroups

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

subgroupindex
C2×#890152
#117235032
C2×#887882
#117220762
C3×#635672
C3×#635632
C3×#635622
#105914183
#105909573
#105906123