Finite Group Databasebeta

← C22×#88960 #13181339

Subgroups — C22×#88960 #13181339

Order 1728.

4242 conjugacy classes of subgroups.

The subgroup lattice has 624 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#6295C211yesmaximal, characteristic, complemented
2C22×#6293C211yesmaximal, characteristic, complemented
3C22×#6290C211yesmaximal, characteristic, complemented
4C22×Dic3×C3⋊D3C211yesmaximal, characteristic, complemented
5C22×C3×#626C211yesmaximal, characteristic, complemented
6C22×C3×#624C211yesmaximal, characteristic, complemented
7C22×C3×D3×Dic3C211yesmaximal, characteristic, complemented
8C2×#88960C2124yesmaximal, direct product factor
9C22×#3804—31nomaximal
10C22×#3798—31nomaximal
11C22×#3795—31nomaximal
12C23×C3×C3⋊D3C2211yescharacteristic
13C23×C3×C3⋊D3C2211yescharacteristic
14C23×C32×D3C2211yescharacteristic
15C22×C3⋊(C3⋊Dic3)C2211yescharacteristic
16C22×C3×C3⋊Dic3C2211yescharacteristic
17C22×C32×Dic3C2211yescharacteristic
18C22×C32×Dic3C2211yescharacteristic
19C2×#6295C22112yescomplemented
20C2×#6293C22112yescomplemented
21C2×#6290C22112yescomplemented
22C2×Dic3×C3⋊D3C22112yescomplemented
23C2×C3×#626C22112yescomplemented
24C2×C3×#624C22112yescomplemented
25C2×C3×D3×Dic3C22112yescomplemented
26#88960C22164yesdirect product factor
27C22×#638—31no—
28C22×C4×C3⋊D3—31no—
29C22×#634—31no—
30C22×C3×#67—31no—
31C23×C3×Dic3—31no—
32C22×C3×D12—31no—
33C22×C3×C4×D3—31no—
34C22×C3×C4×D3—31no—
35C22×C3×C4×D3—31no—
36C22×#627—31no—
37C22×#627—31no—
38C22×#627—61no—
39C22×#626—31no—
40C22×#626—31no—
41C22×#626D311yescharacteristic, complemented
42C22×#625—31no—
43C22×#625—31no—
44C22×#625—31no—
45C22×#625—61no—
46C22×#624D311yescharacteristic, complemented
47C2×#3804—324no—
48C22×D3×Dic3—31no—
49C22×D3×Dic3—31no—
50C22×D3×Dic3—61no—
51C22×D3×Dic3D311yescharacteristic, complemented
52C2×#3798—324no—
53C2×#3795—324no—
54C23×C33C2311yescharacteristic
55C22×C3×C3⋊D3—24no—
56C22×C3×C3⋊D3—24no—
57C22×C3×C3⋊D3C2316yes—
58C22×C3×C3⋊D3C2316yes—
59C22×C32×D3—24no—
60C22×C32×D3C2316yes—
61C2×C3⋊(C3⋊Dic3)C2316yes—
62C2×C3×C3⋊Dic3C2316yes—
63C2×C32×Dic3C2316yes—
64C2×C32×Dic3C2316yes—
65#6295C23116yescomplemented
66#6293C23116yescomplemented
67#6290C23116yescomplemented
68Dic3×C3⋊D3C23116yescomplemented
69C3×#626C23116yescomplemented
70C3×#624C23116yescomplemented
71C3×D3×Dic3C23116yescomplemented
72C22×#237—91no—
73C22×#235—91no—
74C22×#235—91no—
75C24×C32—31no—
76C23×C3⋊D3C2×D311yescharacteristic
77C23×C3⋊D3C2×D311yescharacteristic
78C23×C3×D3—21no—
79C23×C3×D3—31no—
80C23×C3×D3—31no—
81C23×C3×D3—31no—
82C23×C3×D3—31no—
83C23×C3×D3—61no—
84C23×C3×D3—61no—
85C23×C3×D3C2×D311yescharacteristic
86C23×C3×D3C2×D311yescharacteristic
87C22×C32×C4—31no—
88C22×C32×C4—31no—
89C2×#638—312no—
90C22×C3⋊Dic3—31no—
91C22×C3⋊Dic3—31no—
92C22×C3⋊Dic3—31no—
93C22×C3⋊Dic3—61no—
94C22×C3⋊Dic3—61no—
95C22×C3⋊Dic3—61no—
96C22×C3⋊Dic3—121no—
97C22×C3⋊Dic3C2×D311yescharacteristic
98C2×C4×C3⋊D3—312no—
99C2×#634—312no—
100C2×C3×#67—312no—
101C22×C3×Dic3—21no—
102C22×C3×Dic3—21no—
103C22×C3×Dic3—31no—
104C22×C3×Dic3—31no—
105C22×C3×Dic3—312no—
106C22×C3×Dic3—61no—
107C22×C3×Dic3C2×D311yescharacteristic
108C22×C3×Dic3C2×D311yescharacteristic
109C22×C3×Dic3C2×D311yescharacteristic
110C22×C3×Dic3C2×D311yescharacteristic
111C2×C3×D12—312no—
112C2×C3×C4×D3—312no—
113C2×C3×C4×D3—312no—
114C2×C3×C4×D3—312no—
115C2×#627—312no—
116C2×#627—312no—
117C2×#627—612no—
118C2×#626—312no—
119C2×#626—312no—
120C2×#626C2×D3112yescomplemented
121C2×#625—312no—
122C2×#625—312no—
123C2×#625—312no—
124C2×#625—612no—
125C2×#624C2×D3112yescomplemented
126#3804—364no—
127C2×D3×Dic3—312no—
128C2×D3×Dic3—312no—
129C2×D3×Dic3—612no—
130C2×D3×Dic3C2×D3112yescomplemented
131#3798—364no—
132#3795—364no—
133C22×C33#4114yescomplemented
134C22×C33C2413yes—
135C2×C3×C3⋊D3—212no—
136C2×C3×C3⋊D3—212no—
137C2×C3×C3⋊D3C2414yes—
138C2×C3×C3⋊D3C2414yes—
139C2×C32×D3—212no—
140C2×C32×D3C2414yes—
141C3⋊(C3⋊Dic3)C2414yes—
142C3×C3⋊Dic3C2414yes—
143C32×Dic3C2414yes—
144C32×Dic3C2414yes—
145C22×C3×D4—91no—
146C23×C3×C4—91no—
147C23×C3×C4—91no—
148C22×#67—31no—
149C22×#67—91no—
150C22×#67—91no—
151C22×#67—91no—
152C22×#67—91no—
153C22×#67—91no—
154C22×#67—181no—
155C23×Dic3—31no—
156C23×Dic3—91no—
157C2×#237—924no—
158C2×#235—924no—
159C2×#235—924no—
160C22×D12—31no—
161C22×D12—91no—
162C22×C4×D3—31no—
163C22×C4×D3—31no—
164C22×C4×D3—31no—
165C22×C4×D3—91no—
166C22×C4×D3—91no—
167C22×C4×D3—181no—
168C22×Dic6—91no—
169C22×Dic6—91no—
170C22×Dic6—91no—
171C22×Dic6—181no—
172C23×C32—21no—
173C23×C32—21no—
174C23×C32—21no—
175C23×C32—36no—
176C23×C32—41no—
177C23×C32—64no—
178C23×C32C22×D311yescharacteristic
179C23×C32C22×D311yescharacteristic
180C23×C32C22×D311yescharacteristic
181C22×C3⋊D3—24no—
182C22×C3⋊D3—24no—
183C22×C3⋊D3C22×D316yes—
184C22×C3⋊D3C22×D316yes—
185C22×C3×D3—24no—
186C22×C3×D3—24no—
187C22×C3×D3—26no—
188C22×C3×D3—36no—
189C22×C3×D3—36no—
190C22×C3×D3—36no—
191C22×C3×D3—36no—
192C22×C3×D3—44no—
193C22×C3×D3—66no—
194C22×C3×D3—66no—
195C22×C3×D3—68no—
196C22×C3×D3—64no—
197C22×C3×D3—64no—
198C22×C3×D3—64no—
199C22×C3×D3—64no—
200C22×C3×D3—124no—
201C22×C3×D3C22×D316yes—
202C22×C3×D3C22×D316yes—
203C2×C32×C4—36no—
204C2×C32×C4—36no—
205#638—316no—
206C2×C3⋊Dic3—36no—
207C2×C3⋊Dic3—36no—
208C2×C3⋊Dic3—36no—
209C2×C3⋊Dic3—66no—
210C2×C3⋊Dic3—66no—
211C2×C3⋊Dic3—66no—
212C2×C3⋊Dic3—126no—
213C2×C3⋊Dic3C22×D316yes—
214C4×C3⋊D3—316no—
215#634—316no—
216C3×#67—316no—
217C2×C3×Dic3—26no—
218C2×C3×Dic3—26no—
219C2×C3×Dic3—36no—
220C2×C3×Dic3—36no—
221C2×C3×Dic3—316no—
222C2×C3×Dic3—66no—
223C2×C3×Dic3C22×D316yes—
224C2×C3×Dic3C22×D316yes—
225C2×C3×Dic3C22×D316yes—
226C2×C3×Dic3C22×D316yes—
227C3×D12—316no—
228C3×C4×D3—316no—
229C3×C4×D3—316no—
230C3×C4×D3—316no—
231#627—316no—
232#627—316no—
233#627—616no—
234#626—316no—
235#626—316no—
236#626C22×D3116yescomplemented
237#625—316no—
238#625—316no—
239#625—316no—
240#625—616no—
241#624C22×D3116yescomplemented
242D3×Dic3—316no—
243D3×Dic3—316no—
244D3×Dic3—616no—
245D3×Dic3C22×D3116yescomplemented
246C22×#41—271noSylow
247C2×C33C2×#4116yescomplemented
248C2×C33C2511yescharacteristic
249C3×C3⋊D3—24no—
250C3×C3⋊D3—24no—
251C32×D3—24no—
252C24×C3—31no—
253C24×C3—31no—
254C24×C3—61no—
255C24×C3—91no—
256C24×C3—91no—
257C23×D3—31no—
258C23×D3—31no—
259C23×D3—31no—
260C23×D3—31no—
261C23×D3—61no—
262C23×D3—61no—
263C23×D3D3211yescharacteristic
264C2×C3×D4—912no—
265C22×C3×C4—31no—
266C22×C3×C4—31no—
267C22×C3×C4—31no—
268C22×C3×C4—31no—
269C22×C3×C4—61no—
270C22×C3×C4—61no—
271C22×C3×C4—912no—
272C22×C3×C4—912no—
273C22×C3×C4—91no—
274C2×#67—312no—
275C2×#67—912no—
276C2×#67—912no—
277C2×#67—912no—
278C2×#67—912no—
279C2×#67—912no—
280C2×#67—1812no—
281C22×Dic3—31no—
282C22×Dic3—31no—
283C22×Dic3—312no—
284C22×Dic3—61no—
285C22×Dic3—91no—
286C22×Dic3—91no—
287C22×Dic3—91no—
288C22×Dic3—912no—
289C22×Dic3—181no—
290C22×Dic3—181no—
291C22×Dic3—181no—
292C22×Dic3—361no—
293C22×Dic3D3211yescharacteristic
294C22×Dic3D3211yescharacteristic
295#237—964no—
296#235—964no—
297#235—964no—
298C2×D12—312no—
299C2×D12—912no—
300C2×C4×D3—312no—
301C2×C4×D3—312no—
302C2×C4×D3—312no—
303C2×C4×D3—912no—
304C2×C4×D3—912no—
305C2×C4×D3—1812no—
306C2×Dic6—912no—
307C2×Dic6—912no—
308C2×Dic6—912no—
309C2×Dic6—1812no—
310C22×C32—23no—
311C22×C32—23no—
312C22×C32—23no—
313C22×C32—24no—
314C22×C32—24no—
315C22×C32—24no—
316C22×C32—34no—
317C22×C32—44no—
318C22×C32—43no—
319C22×C32—612no—
320C22×C32#23514yescomplemented
321C22×C32#23514yescomplemented
322C22×C32#23714yescomplemented
323C22×C32C23×D313yes—
324C22×C32C23×D313yes—
325C22×C32C23×D313yes—
326C2×C3⋊D3—212no—
327C2×C3⋊D3—212no—
328C2×C3⋊D3C23×D314yes—
329C2×C3⋊D3C23×D314yes—
330C2×C3×D3—212no—
331C2×C3×D3—212no—
332C2×C3×D3—24no—
333C2×C3×D3—34no—
334C2×C3×D3—34no—
335C2×C3×D3—34no—
336C2×C3×D3—34no—
337C2×C3×D3—412no—
338C2×C3×D3—64no—
339C2×C3×D3—64no—
340C2×C3×D3—624no—
341C2×C3×D3—612no—
342C2×C3×D3—612no—
343C2×C3×D3—612no—
344C2×C3×D3—612no—
345C2×C3×D3—1212no—
346C2×C3×D3C23×D314yes—
347C2×C3×D3C23×D314yes—
348C32×C4—34no—
349C32×C4—34no—
350C3⋊Dic3—34no—
351C3⋊Dic3—34no—
352C3⋊Dic3—34no—
353C3⋊Dic3—64no—
354C3⋊Dic3—64no—
355C3⋊Dic3—64no—
356C3⋊Dic3—124no—
357C3⋊Dic3C23×D314yes—
358C3×Dic3—24no—
359C3×Dic3—24no—
360C3×Dic3—34no—
361C3×Dic3—34no—
362C3×Dic3—64no—
363C3×Dic3C23×D314yes—
364C3×Dic3C23×D314yes—
365C3×Dic3C23×D314yes—
366C3×Dic3C23×D314yes—
367C2×#41—2724no—
368C22×Q8—271no—
369C22×D4—91no—
370C22×D4—271no—
371C22×D4—271no—
372C23×C4—91no—
373C23×C4—91no—
374C23×C4—271no—
375C33C22×#4111yescharacteristic, complemented, Sylow
376C23×C3—21no—
377C23×C3—21no—
378C23×C3—21no—
379C23×C3—36no—
380C23×C3—36no—
381C23×C3—41no—
382C23×C3—66no—
383C23×C3—64no—
384C23×C3—64no—
385C23×C3—96no—
386C23×C3—96no—
387C23×C3—124no—
388C23×C3—184no—
389C23×C3—184no—
390C23×C3C2×D3211yescharacteristic
391C23×C3C2×D3211yescharacteristic
392C23×C3C2×D3211yescharacteristic
393C22×D3—24no—
394C22×D3—36no—
395C22×D3—36no—
396C22×D3—36no—
397C22×D3—36no—
398C22×D3—66no—
399C22×D3—66no—
400C22×D3—68no—
401C22×D3—64no—
402C22×D3—64no—
403C22×D3—64no—
404C22×D3—64no—
405C22×D3—124no—
406C22×D3C2×D3216yes—
407C3×D4—916no—
408C2×C3×C4—36no—
409C2×C3×C4—36no—
410C2×C3×C4—36no—
411C2×C3×C4—36no—
412C2×C3×C4—66no—
413C2×C3×C4—66no—
414C2×C3×C4—916no—
415C2×C3×C4—916no—
416C2×C3×C4—96no—
417#67—316no—
418#67—916no—
419#67—916no—
420#67—916no—
421#67—916no—
422#67—916no—
423#67—1816no—
424C2×Dic3—36no—
425C2×Dic3—36no—
426C2×Dic3—316no—
427C2×Dic3—66no—
428C2×Dic3—96no—
429C2×Dic3—96no—
430C2×Dic3—96no—
431C2×Dic3—916no—
432C2×Dic3—186no—
433C2×Dic3—186no—
434C2×Dic3—186no—
435C2×Dic3—366no—
436C2×Dic3C2×D3216yes—
437C2×Dic3C2×D3216yes—
438D12—316no—
439D12—916no—
440C4×D3—316no—
441C4×D3—316no—
442C4×D3—316no—
443C4×D3—916no—
444C4×D3—916no—
445C4×D3—1816no—
446Dic6—916no—
447Dic6—916no—
448Dic6—916no—
449Dic6—1816no—
450C2×C32—26no—
451C2×C32—26no—
452C2×C32—26no—
453C2×C32—21no—
454C2×C32—21no—
455C2×C32—21no—
456C2×C32—46no—
457C2×C32—41no—
458C2×C32—64no—
459C2×C32C2×#23516yescomplemented
460C2×C32C2×#23516yescomplemented
461C2×C32C2×#23716yescomplemented
462C2×C32C24×D311yescharacteristic
463C2×C32C24×D311yescharacteristic
464C2×C32C24×D311yescharacteristic
465C3⋊D3—24no—
466C3⋊D3—24no—
467C3×D3—24no—
468C3×D3—24no—
469C3×D3—44no—
470C3×D3—68no—
471C3×D3—64no—
472C3×D3—64no—
473C3×D3—64no—
474C3×D3—64no—
475C3×D3—124no—
476C24—31no—
477C24—91no—
478C24—91no—
479#41—2764no—
480C2×Q8—2712no—
481C2×D4—912no—
482C2×D4—2712no—
483C2×D4—2712no—
484C22×C4—31no—
485C22×C4—31no—
486C22×C4—912no—
487C22×C4—912no—
488C22×C4—91no—
489C22×C4—2712no—
490C22×C4—271no—
491C22×C3—23no—
492C22×C3—23no—
493C22×C3—23no—
494C22×C3—24no—
495C22×C3—24no—
496C22×C3—24no—
497C22×C3—34no—
498C22×C3—34no—
499C22×C3—43no—
500C22×C3—44no—
501C22×C3—64no—
502C22×C3—612no—
503C22×C3—612no—
504C22×C3—94no—
505C22×C3—94no—
506C22×C3—1212no—
507C22×C3—1812no—
508C22×C3—1812no—
509C22×C3#379514yescomplemented
510C22×C3#379814yescomplemented
511C22×C3#380414yescomplemented
512C22×C3C22×D3213yes—
513C22×C3C22×D3213yes—
514C22×C3C22×D3213yes—
515C2×D3—212no—
516C2×D3—34no—
517C2×D3—34no—
518C2×D3—34no—
519C2×D3—34no—
520C2×D3—64no—
521C2×D3—64no—
522C2×D3—624no—
523C2×D3—612no—
524C2×D3—612no—
525C2×D3—612no—
526C2×D3—612no—
527C2×D3—1212no—
528C2×D3C22×D3214yes—
529C3×C4—34no—
530C3×C4—34no—
531C3×C4—34no—
532C3×C4—34no—
533C3×C4—64no—
534C3×C4—64no—
535C3×C4—94no—
536Dic3—34no—
537Dic3—34no—
538Dic3—64no—
539Dic3—94no—
540Dic3—94no—
541Dic3—94no—
542Dic3—184no—
543Dic3—184no—
544Dic3—184no—
545Dic3—364no—
546Dic3C22×D3214yes—
547Dic3C22×D3214yes—
548C32—21no—
549C32—21no—
550C32—21no—
551C32—41no—
552C32C22×#23511yescharacteristic, complemented
553C32C22×#23511yescharacteristic, complemented
554C32C22×#23711yescharacteristic, complemented
555C23—36no—
556C23—64no—
557C23—96no—
558C23—96no—
559C23—184no—
560C23—184no—
561C23D3311yescharacteristic
562Q8—2716no—
563D4—916no—
564D4—2716no—
565D4—2716no—
566C2×C4—36no—
567C2×C4—36no—
568C2×C4—916no—
569C2×C4—916no—
570C2×C4—96no—
571C2×C4—2716no—
572C2×C4—276no—
573C2×C3—26no—
574C2×C3—26no—
575C2×C3—26no—
576C2×C3—21no—
577C2×C3—21no—
578C2×C3—21no—
579C2×C3—46no—
580C2×C3—41no—
581C2×C3—64no—
582C2×C3—64no—
583C2×C3—124no—
584C2×C3—184no—
585C2×C3—184no—
586C2×C3C2×#379516yescomplemented
587C2×C3C2×#379816yescomplemented
588C2×C3C2×#380416yescomplemented
589C2×C3C23×D3211yescharacteristic
590C2×C3C23×D3211yescharacteristic
591C2×C3C23×D3211yescharacteristic
592D3—24no—
593D3—68no—
594D3—64no—
595D3—64no—
596D3—64no—
597D3—64no—
598D3—124no—
599C22—34no—
600C22—612no—
601C22—94no—
602C22—94no—
603C22—1812no—
604C22—1812no—
605C22#8896014yesdirect product factor
606C22C2×D3313yes—
607C4—34no—
608C4—34no—
609C4—94no—
610C4—274no—
611C3—21no—
612C3—21no—
613C3—21no—
614C3—41no—
615C3C22×#379511yescharacteristic, complemented
616C3C22×#379811yescharacteristic, complemented
617C3C22×#380411yescharacteristic, complemented
618C2—64no—
619C2—184no—
620C2—184no—
621C2C2×#8896016yesdirect product factor
622C2C22×D3311yescharacteristic
subgroupG/H
center, hypercenterC23D33
commutator, cosocleC2×C33C25
FrattiniC2C22×D33
Fitting, socleC23×C33C23
hypocenterC33C22×#41
LCSG ▷ C2×C33 ▷ C33
UCSC1 ◁ C23

Complements

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

NH ≅ G/Ntype
1. C22×#6295C2semidirect
2. C22×#6293C2semidirect
3. C22×#6290C2semidirect
4. C22×Dic3×C3⋊D3C2semidirect
5. C22×C3×#626C2semidirect
6. C22×C3×#624C2semidirect
7. C22×C3×D3×Dic3C2semidirect
8. C2×#88960C2direct (N × H)
19. C2×#6295C22semidirect
20. C2×#6293C22semidirect
21. C2×#6290C22semidirect
22. C2×Dic3×C3⋊D3C22semidirect
23. C2×C3×#626C22semidirect
24. C2×C3×#624C22semidirect
25. C2×C3×D3×Dic3C22semidirect
26. #88960C22direct (N × H)
41. C22×#626D3semidirect
46. C22×#624D3semidirect
51. C22×D3×Dic3D3semidirect
65. #6295C23semidirect
66. #6293C23semidirect
67. #6290C23semidirect
68. Dic3×C3⋊D3C23semidirect
69. C3×#626C23semidirect
70. C3×#624C23semidirect
71. C3×D3×Dic3C23semidirect
120. C2×#626C2×D3semidirect
125. C2×#624C2×D3semidirect
130. C2×D3×Dic3C2×D3semidirect
133. C22×C33#41semidirect
236. #626C22×D3semidirect
241. #624C22×D3semidirect
245. D3×Dic3C22×D3semidirect
247. C2×C33C2×#41semidirect
320. C22×C32#235semidirect
321. C22×C32#235semidirect
322. C22×C32#237semidirect
375. C33C22×#41semidirect
459. C2×C32C2×#235semidirect
460. C2×C32C2×#235semidirect
461. C2×C32C2×#237semidirect
509. C22×C3#3795semidirect
510. C22×C3#3798semidirect
511. C22×C3#3804semidirect
552. C32C22×#235semidirect
553. C32C22×#235semidirect
554. C32C22×#237semidirect
586. C2×C3C2×#3795semidirect
587. C2×C3C2×#3798semidirect
588. C2×C3C2×#3804semidirect
605. C22#88960direct (N × H)
615. C3C22×#3795semidirect
616. C3C22×#3798semidirect
617. C3C22×#3804semidirect
621. C2C2×#88960direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#62952
C22×#62932
C22×#62902
C22×Dic3×C3⋊D32
C22×C3×#6262
C22×C3×#6242
C22×C3×D3×Dic32
C2×#889602
C22×#38043
C22×#37983
C22×#37953