Finite Group Databasebeta

← C2×#11723657 #13178507

Subgroups — C2×#11723657 #13178507

Order 1728.

1426 conjugacy classes of subgroups.

The subgroup lattice has 584 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#3828C211yesmaximal, characteristic
2C2×C3×#3748C211yesmaximal, characteristic, complemented
3C22×C3×C3⋊(C3⋊C8)C211yesmaximal, characteristic, complemented
4C2×#88809C212yesmaximal, complemented
5#11723657C218yesmaximal, direct product factor
6C2×#88808C212yesmaximal, complemented
7C2×#63817—31nomaximal
8C2×#63815—31nomaximal
9C2×#63814—31nomaximal
10C22×C33×C4C2211yescharacteristic
11C2×C3×#638C412yes—
12C3×#3828C414yes—
13C3×#3828C2214yes—
14C2×C3×C4⋊(C3⋊D3)C411yescharacteristic
15C2×C3×C4×C3⋊D3C2212yes—
16C2×C3×#634C411yescharacteristic
17C3×#3748C2214yescomplemented
18C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
19C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
20C2×C3×C3⋊(C3⋊C8)C2212yes—
21C2×C3×C3⋊(C3⋊C8)C2214yescomplemented
22#88809C2218yescomplemented
23#88808C2218yescomplemented
24C2×#3828D311yescharacteristic
25C2×C3×#235—31no—
26C2×C3×#235—31no—
27C2×C3×#235—61no—
28C2×#3748D311yescharacteristic, complemented
29C22×C3⋊(C3⋊C8)D311yescharacteristic, complemented
30C2×C3×#208—31no—
31C2×C3×#208—31no—
32C2×C3×#208—61no—
33C22×C3×C3⋊C8—31no—
34C22×C3×C3⋊C8—31no—
35C22×C3×C3⋊C8—61no—
36C2×#3712—32no—
37C2×#3711—32no—
38C2×#3711—32no—
39#63817—38no—
40C2×#3710—32no—
41#63815—38no—
42#63814—38no—
43C2×D3×C3⋊C8—32no—
44C2×D3×C3⋊C8—32no—
45C23×C33C2×C411yescharacteristic
46C22×C3×C3⋊D3C2×C412yes—
47C2×C33×C4C2×C411yescharacteristic
48C2×C33×C4C2×C412yes—
49C2×C33×C4C2311yescharacteristic
50C2×C33×C4C2312yes—
51C3×#638C2×C418yes—
52C2×C3×C3⋊Dic3C2×C412yes—
53C3×C4⋊(C3⋊D3)C2×C414yes—
54C3×C4×C3⋊D3C2×C414yes—
55C3×C4×C3⋊D3C2314yes—
56C3×#634C2×C414yes—
57C3×C3⋊(C3⋊C8)C2312yes—
58C3×C3⋊(C3⋊C8)C2312yes—
59C3×C3⋊(C3⋊C8)C2314yes—
60C2×#948—91no—
61C2×#907—91no—
62C2×#901—91no—
63C22×C32×C4—21no—
64C22×C32×C4—21no—
65C22×C32×C4—21no—
66C22×C32×C4—41no—
67C22×C32×C4C2×D311yescharacteristic
68C22×C32×C4C2×D311yescharacteristic
69C22×C32×C4C2×D311yescharacteristic
70C2×#638Dic312yes—
71#3828Dic314yes—
72#3828C2×D314yes—
73C2×C4⋊(C3⋊D3)Dic311yescharacteristic
74C2×C4×C3⋊D3C2×D312yes—
75C2×#634Dic311yescharacteristic
76C2×C3×#67—32no—
77C2×C3×#67—32no—
78C2×C3×#67—62no—
79C3×#235—34no—
80C3×#235—34no—
81C3×#235—34no—
82C3×#235—34no—
83C3×#235—64no—
84C3×#235—64no—
85C2×C3×D12—31no—
86C2×C3×D12—31no—
87C2×C3×D12—61no—
88C2×C3×C4×D3—32no—
89C2×C3×C4×D3—32no—
90C2×C3×C4×D3—62no—
91C2×C3×Dic6—31no—
92C2×C3×Dic6—31no—
93C2×C3×Dic6—61no—
94#3748C2×D314yescomplemented
95C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
96C2×C3⋊(C3⋊C8)C2×D312yes—
97C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
98C2×C3⋊(C3⋊C8)C2×D314yescomplemented
99C3×#208—34no—
100C3×#208—34no—
101C3×#208—64no—
102C2×C3×C3⋊C8—31no—
103C2×C3×C3⋊C8—31no—
104C2×C3×C3⋊C8—31no—
105C2×C3×C3⋊C8—31no—
106C2×C3×C3⋊C8—32no—
107C2×C3×C3⋊C8—32no—
108C2×C3×C3⋊C8—34no—
109C2×C3×C3⋊C8—34no—
110C2×C3×C3⋊C8—61no—
111C2×C3×C3⋊C8—61no—
112C2×C3×C3⋊C8—62no—
113C2×C3×C3⋊C8—64no—
114#3712—38no—
115#3711—38no—
116#3711—38no—
117#3710—38no—
118D3×C3⋊C8—38no—
119D3×C3⋊C8—38no—
120C22×C33—22no—
121C22×C33C22×C412yes—
122C22×C33C22×C411yescharacteristic
123C2×C3×C3⋊D3—24no—
124C2×C3×C3⋊D3C22×C414yes—
125C33×C4C22×C412yes—
126C33×C4C22×C411yescharacteristic
127C33×C4C2411yescharacteristic
128C3×C3⋊Dic3C22×C414yes—
129C2×C3×#41—91no—
130C2×#235—31no—
131C2×#235—31no—
132C2×#235—61no—
133C2×C3×M16—91no—
134C22×C3×C8—91no—
135#948—98no—
136C2×#208—31no—
137C2×#208—31no—
138C2×#208—61no—
139C2×#208—92no—
140C22×C3⋊C8—31no—
141C22×C3⋊C8—31no—
142C22×C3⋊C8—61no—
143C22×C3⋊C8—92no—
144#907—98no—
145#901—98no—
146C2×ΓL(1,𝔽25)—92no—
147C2×ΓL(1,𝔽25)—92no—
148C2×D3×C8—92no—
149C2×D3×C8—92no—
150C23×C32—21no—
151C23×C32—21no—
152C23×C32—21no—
153C23×C32—41no—
154C23×C32C4×D311yescharacteristic
155C23×C32C4×D311yescharacteristic
156C23×C32C2×Dic311yescharacteristic
157C22×C3⋊D3C2×Dic312yes—
158C22×C3×D3—32no—
159C22×C3×D3—32no—
160C22×C3×D3—62no—
161C2×C32×C4—21no—
162C2×C32×C4—21no—
163C2×C32×C4—22no—
164C2×C32×C4—22no—
165C2×C32×C4—21no—
166C2×C32×C4—22no—
167C2×C32×C4—22no—
168C2×C32×C4—21no—
169C2×C32×C4—21no—
170C2×C32×C4—22no—
171C2×C32×C4—21no—
172C2×C32×C4—22no—
173C2×C32×C4—41no—
174C2×C32×C4—42no—
175C2×C32×C4—42no—
176C2×C32×C4—41no—
177C2×C32×C4C4×D311yescharacteristic
178C2×C32×C4C4×D311yescharacteristic
179C2×C32×C4C4×D312yes—
180C2×C32×C4C4×D312yes—
181C2×C32×C4C2×Dic311yescharacteristic
182C2×C32×C4C2×Dic312yes—
183C2×C32×C4C22×D311yescharacteristic
184C2×C32×C4C22×D311yescharacteristic
185C2×C32×C4C22×D312yes—
186C2×C32×C4C22×D312yes—
187C2×C32×C4C22×D311yescharacteristic
188C2×C32×C4C22×D312yes—
189#638C2×Dic318yes—
190C2×C3⋊Dic3C2×Dic312yes—
191C4⋊(C3⋊D3)C2×Dic314yes—
192C4×C3⋊D3C2×Dic314yes—
193C4×C3⋊D3C22×D314yes—
194#634C2×Dic314yes—
195C3×#67—38no—
196C3×#67—38no—
197C3×#67—68no—
198C2×C3×Dic3—32no—
199C2×C3×Dic3—32no—
200C2×C3×Dic3—62no—
201C3×D12—34no—
202C3×D12—34no—
203C3×D12—64no—
204C3×C4×D3—34no—
205C3×C4×D3—34no—
206C3×C4×D3—34no—
207C3×C4×D3—34no—
208C3×C4×D3—64no—
209C3×C4×D3—64no—
210C3×Dic6—34no—
211C3×Dic6—34no—
212C3×Dic6—64no—
213C3⋊(C3⋊C8)C22×D312yes—
214C3⋊(C3⋊C8)C22×D314yes—
215C3⋊(C3⋊C8)C22×D312yes—
216C3×C3⋊C8—32no—
217C3×C3⋊C8—34no—
218C3×C3⋊C8—32no—
219C3×C3⋊C8—34no—
220C3×C3⋊C8—32no—
221C3×C3⋊C8—32no—
222C3×C3⋊C8—62no—
223C3×C3⋊C8—62no—
224C3×C3⋊C8—64no—
225C2×#131—271noSylow
226C2×C33—22no—
227C2×C33#13112yescomplemented
228C2×C33C23×C411yescharacteristic
229C3×C3⋊D3—24no—
230C3×#41—94no—
231C3×#41—94no—
232C2×C3×Q8—91no—
233C2×C3×D4—92no—
234C2×C3×D4—91no—
235C22×C3×C4—21no—
236C22×C3×C4—21no—
237C22×C3×C4—21no—
238C22×C3×C4—41no—
239C22×C3×C4—92no—
240C22×C3×C4D3211yescharacteristic
241C22×C3×C4D3211yescharacteristic
242C22×C3×C4D3211yescharacteristic
243C2×#67—32no—
244C2×#67—32no—
245C2×#67—62no—
246#235—34no—
247#235—34no—
248#235—34no—
249#235—34no—
250#235—64no—
251#235—64no—
252C2×D12—31no—
253C2×D12—31no—
254C2×D12—61no—
255C2×C4×D3—32no—
256C2×C4×D3—32no—
257C2×C4×D3—62no—
258C2×Dic6—31no—
259C2×Dic6—31no—
260C2×Dic6—61no—
261C3×M16—94no—
262C2×C3×C8—91no—
263C2×C3×C8—92no—
264C2×C3×C8—91no—
265C2×C3×C8—94no—
266#208—34no—
267#208—34no—
268#208—64no—
269#208—98no—
270C2×C3⋊C8—31no—
271C2×C3⋊C8—32no—
272C2×C3⋊C8—31no—
273C2×C3⋊C8—34no—
274C2×C3⋊C8—31no—
275C2×C3⋊C8—31no—
276C2×C3⋊C8—34no—
277C2×C3⋊C8—32no—
278C2×C3⋊C8—61no—
279C2×C3⋊C8—61no—
280C2×C3⋊C8—64no—
281C2×C3⋊C8—62no—
282C2×C3⋊C8—98no—
283ΓL(1,𝔽25)—98no—
284ΓL(1,𝔽25)—98no—
285D3×C8—98no—
286D3×C8—98no—
287C22×C32—22no—
288C22×C32—24no—
289C22×C32—22no—
290C22×C32—22no—
291C22×C32—22no—
292C22×C32—21no—
293C22×C32—21no—
294C22×C32—22no—
295C22×C32—21no—
296C22×C32—22no—
297C22×C32—41no—
298C22×C32—42no—
299C22×C32—42no—
300C22×C32—42no—
301C22×C32—44no—
302C22×C32C2×C4×D312yes—
303C22×C32C2×C4×D312yes—
304C22×C32C2×C4×D311yescharacteristic
305C22×C32C2×C4×D311yescharacteristic
306C22×C32C22×Dic312yes—
307C22×C32C22×Dic311yescharacteristic
308C2×C3⋊D3—24no—
309C2×C3⋊D3C22×Dic314yes—
310C2×C3×D3—34no—
311C2×C3×D3—34no—
312C2×C3×D3—64no—
313C2×C3×D3—64no—
314C2×C3×D3—64no—
315C2×C3×D3—124no—
316C32×C4—21no—
317C32×C4—21no—
318C32×C4—21no—
319C32×C4—22no—
320C32×C4—22no—
321C32×C4—21no—
322C32×C4—21no—
323C32×C4—22no—
324C32×C4—21no—
325C32×C4—42no—
326C32×C4—41no—
327C32×C4—41no—
328C32×C4C2×C4×D312yes—
329C32×C4C2×C4×D312yes—
330C32×C4C2×C4×D311yescharacteristic
331C32×C4C2×C4×D311yescharacteristic
332C32×C4C22×Dic312yes—
333C32×C4C22×Dic311yescharacteristic
334C32×C4C23×D311yescharacteristic
335C32×C4C23×D311yescharacteristic
336C32×C4C23×D311yescharacteristic
337C3⋊Dic3C22×Dic314yes—
338C3×Dic3—34no—
339C3×Dic3—34no—
340C3×Dic3—64no—
341C2×#41—91no—
342#131—278no—
343C2×M16—91no—
344C2×M16—272no—
345C22×C8—91no—
346C22×C8—272no—
347C33C2×#13111yescharacteristic, complemented, Sylow
348C23×C3—21no—
349C23×C3—21no—
350C23×C3—21no—
351C23×C3—41no—
352C23×C3—92no—
353C23×C3D3×Dic311yescharacteristic
354C23×C3D3×Dic311yescharacteristic
355C23×C3#62411yescharacteristic
356C22×D3—32no—
357C22×D3—32no—
358C22×D3—62no—
359C3×Q8—94no—
360C3×D4—98no—
361C3×D4—94no—
362C2×C3×C4—21no—
363C2×C3×C4—21no—
364C2×C3×C4—22no—
365C2×C3×C4—22no—
366C2×C3×C4—21no—
367C2×C3×C4—22no—
368C2×C3×C4—22no—
369C2×C3×C4—22no—
370C2×C3×C4—22no—
371C2×C3×C4—21no—
372C2×C3×C4—21no—
373C2×C3×C4—21no—
374C2×C3×C4—41no—
375C2×C3×C4—42no—
376C2×C3×C4—42no—
377C2×C3×C4—41no—
378C2×C3×C4—92no—
379C2×C3×C4—94no—
380C2×C3×C4—94no—
381C2×C3×C4D3×Dic311yescharacteristic
382C2×C3×C4D3×Dic311yescharacteristic
383C2×C3×C4D3×Dic312yes—
384C2×C3×C4D3×Dic312yes—
385C2×C3×C4#62411yescharacteristic
386C2×C3×C4#62412yes—
387C2×C3×C4C2×D3212yes—
388C2×C3×C4C2×D3212yes—
389C2×C3×C4C2×D3212yes—
390C2×C3×C4C2×D3211yescharacteristic
391C2×C3×C4C2×D3211yescharacteristic
392C2×C3×C4C2×D3211yescharacteristic
393#67—38no—
394#67—38no—
395#67—68no—
396C2×Dic3—32no—
397C2×Dic3—32no—
398C2×Dic3—62no—
399D12—34no—
400D12—34no—
401D12—64no—
402C4×D3—34no—
403C4×D3—34no—
404C4×D3—34no—
405C4×D3—34no—
406C4×D3—64no—
407C4×D3—64no—
408Dic6—34no—
409Dic6—34no—
410Dic6—64no—
411C3×C8—92no—
412C3×C8—94no—
413C3×C8—92no—
414C3⋊C8—32no—
415C3⋊C8—34no—
416C3⋊C8—32no—
417C3⋊C8—32no—
418C3⋊C8—32no—
419C3⋊C8—34no—
420C3⋊C8—62no—
421C3⋊C8—62no—
422C3⋊C8—64no—
423C2×C32—22no—
424C2×C32—24no—
425C2×C32—22no—
426C2×C32—22no—
427C2×C32—21no—
428C2×C32—21no—
429C2×C32—21no—
430C2×C32—22no—
431C2×C32—22no—
432C2×C32—22no—
433C2×C32—42no—
434C2×C32—41no—
435C2×C32—42no—
436C2×C32—42no—
437C2×C32—44no—
438C2×C32#90112yescomplemented
439C2×C32#90712yescomplemented
440C2×C32#94812yescomplemented
441C2×C32C22×C4×D311yescharacteristic
442C2×C32C22×C4×D311yescharacteristic
443C2×C32C23×Dic311yescharacteristic
444C3⋊D3—24no—
445C3×D3—64no—
446C3×D3—64no—
447C3×D3—124no—
448#41—94no—
449#41—94no—
450C2×Q8—91no—
451C2×D4—92no—
452C2×D4—91no—
453C22×C4—92no—
454C22×C4#111311yescharacteristic
455M16—94no—
456M16—278no—
457C2×C8—94no—
458C2×C8—91no—
459C2×C8—91no—
460C2×C8—92no—
461C2×C8—278no—
462C22×C3—24no—
463C22×C3—22no—
464C22×C3—22no—
465C22×C3—22no—
466C22×C3—21no—
467C22×C3—22no—
468C22×C3—21no—
469C22×C3—21no—
470C22×C3—22no—
471C22×C3—22no—
472C22×C3—42no—
473C22×C3—42no—
474C22×C3—44no—
475C22×C3—42no—
476C22×C3—41no—
477C22×C3—94no—
478C22×C3—184no—
479C22×C3C2×D3×Dic312yes—
480C22×C3C2×D3×Dic312yes—
481C22×C3C2×D3×Dic311yescharacteristic
482C22×C3C2×D3×Dic311yescharacteristic
483C22×C3C2×#62412yes—
484C22×C3C2×#62411yescharacteristic
485C2×D3—34no—
486C2×D3—34no—
487C2×D3—64no—
488C2×D3—64no—
489C2×D3—64no—
490C2×D3—124no—
491C3×C4—21no—
492C3×C4—21no—
493C3×C4—21no—
494C3×C4—22no—
495C3×C4—21no—
496C3×C4—21no—
497C3×C4—21no—
498C3×C4—22no—
499C3×C4—22no—
500C3×C4—41no—
501C3×C4—41no—
502C3×C4—42no—
503C3×C4—94no—
504C3×C4C2×D3×Dic312yes—
505C3×C4C2×D3×Dic312yes—
506C3×C4C2×D3×Dic311yescharacteristic
507C3×C4C2×D3×Dic311yescharacteristic
508C3×C4C2×#62411yescharacteristic
509C3×C4C2×#62412yes—
510C3×C4C22×D3211yescharacteristic
511C3×C4C22×D3211yescharacteristic
512C3×C4C22×D3211yescharacteristic
513Dic3—34no—
514Dic3—34no—
515Dic3—64no—
516C32—21no—
517C32—21no—
518C32—21no—
519C32—41no—
520C32C2×#90111yescharacteristic, complemented
521C32C2×#90711yescharacteristic, complemented
522C32C2×#94811yescharacteristic, complemented
523C23—92no—
524C23#629411yescharacteristic
525Q8—94no—
526D4—98no—
527D4—94no—
528C2×C4—92no—
529C2×C4—94no—
530C2×C4—94no—
531C2×C4#629411yescharacteristic
532C2×C4#629412yes—
533C2×C4C2×#111312yes—
534C2×C4C2×#111311yescharacteristic
535C8—92no—
536C8—92no—
537C8—94no—
538C2×C3—24no—
539C2×C3—22no—
540C2×C3—22no—
541C2×C3—22no—
542C2×C3—21no—
543C2×C3—21no—
544C2×C3—21no—
545C2×C3—22no—
546C2×C3—22no—
547C2×C3—22no—
548C2×C3—42no—
549C2×C3—42no—
550C2×C3—44no—
551C2×C3—41no—
552C2×C3—42no—
553C2×C3—184no—
554C2×C3#6381412yescomplemented
555C2×C3#6381512yescomplemented
556C2×C3#6381712yescomplemented
557C2×C3C22×D3×Dic311yescharacteristic
558C2×C3C22×D3×Dic311yescharacteristic
559C2×C3C22×#62411yescharacteristic
560D3—64no—
561D3—64no—
562D3—124no—
563C22—22no—
564C22—94no—
565C22—184no—
566C22C2×#629412yes—
567C22C2×#629411yescharacteristic
568C4—94no—
569C4C2×#629411yescharacteristic
570C4C2×#629412yes—
571C4C22×#111311yescharacteristic
572C3—21no—
573C3—21no—
574C3—21no—
575C3—41no—
576C3C2×#6381411yescharacteristic, complemented
577C3C2×#6381511yescharacteristic, complemented
578C3C2×#6381711yescharacteristic, complemented
579C2—22no—
580C2—184no—
581C2#1172365712yesdirect product factor
582C2C22×#629411yescharacteristic
subgroupG/H
centerC2×C4C2×#1113
commutatorC2×C33C23×C4
FrattiniC4C22×#1113
FittingC22×C33×C4C22
socleC22×C33C22×C4
prime radicalC2×C3×#3828C2
cosocleC33×C4C24
hypercenterC22×C4#1113
hypocenterC33C2×#131
LCSG ▷ C2×C33 ▷ C33
UCSC1 ◁ C2×C4 ◁ C22×C4

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. C2×C3×#3748C2semidirect
3. C22×C3×C3⋊(C3⋊C8)C2semidirect
4. C2×#88809C2semidirect
5. #11723657C2direct (N × H)
6. C2×#88808C2semidirect
17. C3×#3748C22semidirect
21. C2×C3×C3⋊(C3⋊C8)C22semidirect
22. #88809C22semidirect
23. #88808C22semidirect
28. C2×#3748D3semidirect
29. C22×C3⋊(C3⋊C8)D3semidirect
94. #3748C2×D3semidirect
98. C2×C3⋊(C3⋊C8)C2×D3semidirect
227. C2×C33#131semidirect
347. C33C2×#131semidirect
438. C2×C32#901semidirect
439. C2×C32#907semidirect
440. C2×C32#948semidirect
520. C32C2×#901semidirect
521. C32C2×#907semidirect
522. C32C2×#948semidirect
554. C2×C3#63814semidirect
555. C2×C3#63815semidirect
556. C2×C3#63817semidirect
576. C3C2×#63814semidirect
577. C3C2×#63815semidirect
578. C3C2×#63817semidirect
581. C2#11723657direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#38282
C2×C3×#37482
C22×C3×C3⋊(C3⋊C8)2
C2×#888092
#117236572
C2×#888082
C2×#638173
C2×#638153
C2×#638143