Finite Group Databasebeta

← C22×#88962 #13181341

Subgroups — C22×#88962 #13181341

Order 1728.

4402 conjugacy classes of subgroups.

The subgroup lattice has 642 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#6294C211yesmaximal, characteristic, complemented
2C22×#6292C211yesmaximal, characteristic, complemented
3C22×#6291C211yesmaximal, characteristic, complemented
4C22×#6289C211yesmaximal, characteristic, complemented
5C22×C3×#627C211yesmaximal, characteristic, complemented
6C22×C3×#626C211yesmaximal, characteristic, complemented
7C22×C3×D3×Dic3C211yesmaximal, characteristic, complemented
8C2×#88962C2124yesmaximal, direct product factor
9C22×#3804—31nomaximal
10C22×#3799—31nomaximal
11C22×#3796—31nomaximal
12C23×C3⋊(C3⋊D3)C2211yescharacteristic
13C23×C3×C3⋊D3C2211yescharacteristic
14C23×C32×D3C2211yescharacteristic
15C22×C3×C3⋊Dic3C2211yescharacteristic
16C22×C3×C3⋊Dic3C2211yescharacteristic
17C22×C32×Dic3C2211yescharacteristic
18C22×C32×Dic3C2211yescharacteristic
19C2×#6294C22112yescomplemented
20C2×#6292C22112yescomplemented
21C2×#6291C22112yescomplemented
22C2×#6289C22112yescomplemented
23C2×C3×#627C22112yescomplemented
24C2×C3×#626C22112yescomplemented
25C2×C3×D3×Dic3C22112yescomplemented
26#88962C22164yesdirect product factor
27C22×#638—31no—
28C22×C4⋊(C3⋊D3)—31no—
29C22×C4×C3⋊D3—31no—
30C22×C3×#67—31no—
31C23×C3×Dic3—31no—
32C22×C3×D12—31no—
33C22×C3×C4×D3—31no—
34C22×C3×Dic6—31no—
35C22×C3×Dic6—31no—
36C22×#627D311yescharacteristic, complemented
37C22×#626—31no—
38C22×#626—31no—
39C22×#626—31no—
40C22×#626—31no—
41C22×#626—61no—
42C22×#626—61no—
43C22×#626D311yescharacteristic, complemented
44C22×#624—31no—
45C22×#624—31no—
46C22×#624—31no—
47C22×#624—61no—
48C2×#3804—324no—
49C22×D3×Dic3—31no—
50C22×D3×Dic3—31no—
51C22×D3×Dic3D311yescharacteristic, complemented
52C2×#3799—324no—
53C2×#3796—324no—
54C23×C33C2311yescharacteristic
55C22×C3⋊(C3⋊D3)—24no—
56C22×C3⋊(C3⋊D3)C2316yes—
57C22×C3×C3⋊D3—24no—
58C22×C3×C3⋊D3C2316yes—
59C22×C32×D3—24no—
60C22×C32×D3C2316yes—
61C2×C3×C3⋊Dic3C2316yes—
62C2×C3×C3⋊Dic3C2316yes—
63C2×C32×Dic3C2316yes—
64C2×C32×Dic3C2316yes—
65#6294C23116yescomplemented
66#6292C23116yescomplemented
67#6291C23116yescomplemented
68#6289C23116yescomplemented
69C3×#627C23116yescomplemented
70C3×#626C23116yescomplemented
71C3×D3×Dic3C23116yescomplemented
72C22×#239—91no—
73C22×#237—91no—
74C22×#235—91no—
75C24×C32—31no—
76C23×C3⋊D3—31no—
77C23×C3⋊D3—31no—
78C23×C3⋊D3—31no—
79C23×C3⋊D3—61no—
80C23×C3⋊D3—61no—
81C23×C3⋊D3—61no—
82C23×C3⋊D3—121no—
83C23×C3⋊D3C2×D311yescharacteristic
84C23×C3×D3—21no—
85C23×C3×D3—31no—
86C23×C3×D3—31no—
87C23×C3×D3—61no—
88C23×C3×D3C2×D311yescharacteristic
89C23×C3×D3C2×D311yescharacteristic
90C22×C32×C4—31no—
91C22×C32×C4—31no—
92C2×#638—312no—
93C22×C3⋊Dic3C2×D311yescharacteristic
94C22×C3⋊Dic3C2×D311yescharacteristic
95C2×C4⋊(C3⋊D3)—312no—
96C2×C4×C3⋊D3—312no—
97C2×C3×#67—312no—
98C22×C3×Dic3—21no—
99C22×C3×Dic3—21no—
100C22×C3×Dic3—31no—
101C22×C3×Dic3—31no—
102C22×C3×Dic3—31no—
103C22×C3×Dic3—31no—
104C22×C3×Dic3—312no—
105C22×C3×Dic3—61no—
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×Dic6—312no—
114C2×C3×Dic6—312no—
115C2×#627C2×D3112yescomplemented
116C2×#626—312no—
117C2×#626—312no—
118C2×#626—312no—
119C2×#626—312no—
120C2×#626—612no—
121C2×#626—612no—
122C2×#626C2×D3112yescomplemented
123C2×#624—312no—
124C2×#624—312no—
125C2×#624—312no—
126C2×#624—612no—
127#3804—364no—
128C2×D3×Dic3—312no—
129C2×D3×Dic3—312no—
130C2×D3×Dic3C2×D3112yescomplemented
131#3799—364no—
132#3796—364no—
133C22×C33#4114yescomplemented
134C22×C33C2413yes—
135C2×C3⋊(C3⋊D3)—212no—
136C2×C3⋊(C3⋊D3)C2414yes—
137C2×C3×C3⋊D3—212no—
138C2×C3×C3⋊D3C2414yes—
139C2×C32×D3—212no—
140C2×C32×D3C2414yes—
141C3×C3⋊Dic3C2414yes—
142C3×C3⋊Dic3C2414yes—
143C32×Dic3C2414yes—
144C32×Dic3C2414yes—
145C22×C3×Q8—91no—
146C22×C3×D4—91no—
147C23×C3×C4—91no—
148C22×#67—31no—
149C22×#67—91no—
150C22×#67—91no—
151C22×#67—91no—
152C22×#67—181no—
153C23×Dic3—31no—
154C23×Dic3—91no—
155C2×#239—924no—
156C2×#237—924no—
157C2×#235—924no—
158C22×D12—31no—
159C22×D12—91no—
160C22×D12—91no—
161C22×D12—91no—
162C22×D12—181no—
163C22×C4×D3—31no—
164C22×C4×D3—91no—
165C22×C4×D3—91no—
166C22×C4×D3—91no—
167C22×C4×D3—91no—
168C22×C4×D3—91no—
169C22×C4×D3—181no—
170C22×Dic6—31no—
171C22×Dic6—31no—
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—36no—
183C22×C3⋊D3—36no—
184C22×C3⋊D3—36no—
185C22×C3⋊D3—66no—
186C22×C3⋊D3—66no—
187C22×C3⋊D3—66no—
188C22×C3⋊D3—64no—
189C22×C3⋊D3—68no—
190C22×C3⋊D3—64no—
191C22×C3⋊D3—64no—
192C22×C3⋊D3—126no—
193C22×C3⋊D3—124no—
194C22×C3⋊D3—124no—
195C22×C3⋊D3—128no—
196C22×C3⋊D3C22×D316yes—
197C22×C3×D3—24no—
198C22×C3×D3—24no—
199C22×C3×D3—26no—
200C22×C3×D3—36no—
201C22×C3×D3—36no—
202C22×C3×D3—44no—
203C22×C3×D3—66no—
204C22×C3×D3—64no—
205C22×C3×D3—64no—
206C22×C3×D3—124no—
207C22×C3×D3C22×D316yes—
208C22×C3×D3C22×D316yes—
209C2×C32×C4—36no—
210C2×C32×C4—36no—
211#638—316no—
212C2×C3⋊Dic3C22×D316yes—
213C2×C3⋊Dic3C22×D316yes—
214C4⋊(C3⋊D3)—316no—
215C4×C3⋊D3—316no—
216C3×#67—316no—
217C2×C3×Dic3—26no—
218C2×C3×Dic3—26no—
219C2×C3×Dic3—36no—
220C2×C3×Dic3—36no—
221C2×C3×Dic3—36no—
222C2×C3×Dic3—36no—
223C2×C3×Dic3—316no—
224C2×C3×Dic3—66no—
225C2×C3×Dic3—66no—
226C2×C3×Dic3C22×D316yes—
227C2×C3×Dic3C22×D316yes—
228C2×C3×Dic3C22×D316yes—
229C2×C3×Dic3C22×D316yes—
230C3×D12—316no—
231C3×C4×D3—316no—
232C3×Dic6—316no—
233C3×Dic6—316no—
234#627C22×D3116yescomplemented
235#626—316no—
236#626—316no—
237#626—316no—
238#626—316no—
239#626—616no—
240#626—616no—
241#626C22×D3116yescomplemented
242#624—316no—
243#624—316no—
244#624—316no—
245#624—616no—
246D3×Dic3—316no—
247D3×Dic3—316no—
248D3×Dic3C22×D3116yescomplemented
249C22×#41—271noSylow
250C2×C33C2×#4116yescomplemented
251C2×C33C2511yescharacteristic
252C3⋊(C3⋊D3)—24no—
253C3×C3⋊D3—24no—
254C32×D3—24no—
255C24×C3—31no—
256C24×C3—31no—
257C24×C3—61no—
258C24×C3—91no—
259C23×D3—31no—
260C23×D3—31no—
261C23×D3—61no—
262C23×D3—91no—
263C23×D3—91no—
264C23×D3—91no—
265C23×D3—181no—
266C23×D3—181no—
267C23×D3—181no—
268C23×D3—361no—
269C23×D3D3211yescharacteristic
270C2×C3×Q8—912no—
271C2×C3×D4—912no—
272C22×C3×C4—31no—
273C22×C3×C4—31no—
274C22×C3×C4—31no—
275C22×C3×C4—31no—
276C22×C3×C4—61no—
277C22×C3×C4—61no—
278C22×C3×C4—912no—
279C22×C3×C4—91no—
280C22×C3×C4—91no—
281C2×#67—312no—
282C2×#67—912no—
283C2×#67—912no—
284C2×#67—912no—
285C2×#67—1812no—
286C22×Dic3—31no—
287C22×Dic3—31no—
288C22×Dic3—31no—
289C22×Dic3—31no—
290C22×Dic3—312no—
291C22×Dic3—61no—
292C22×Dic3—61no—
293C22×Dic3—912no—
294C22×Dic3D3211yescharacteristic
295C22×Dic3D3211yescharacteristic
296#239—964no—
297#237—964no—
298#235—964no—
299C2×D12—312no—
300C2×D12—912no—
301C2×D12—912no—
302C2×D12—912no—
303C2×D12—1812no—
304C2×C4×D3—312no—
305C2×C4×D3—912no—
306C2×C4×D3—912no—
307C2×C4×D3—912no—
308C2×C4×D3—912no—
309C2×C4×D3—912no—
310C2×C4×D3—1812no—
311C2×Dic6—312no—
312C2×Dic6—312no—
313C22×C32—23no—
314C22×C32—23no—
315C22×C32—23no—
316C22×C32—24no—
317C22×C32—24no—
318C22×C32—24no—
319C22×C32—34no—
320C22×C32—44no—
321C22×C32—43no—
322C22×C32—612no—
323C22×C32#23514yescomplemented
324C22×C32#23714yescomplemented
325C22×C32#23914yescomplemented
326C22×C32C23×D313yes—
327C22×C32C23×D313yes—
328C22×C32C23×D313yes—
329C2×C3⋊D3—212no—
330C2×C3⋊D3—34no—
331C2×C3⋊D3—34no—
332C2×C3⋊D3—34no—
333C2×C3⋊D3—64no—
334C2×C3⋊D3—64no—
335C2×C3⋊D3—64no—
336C2×C3⋊D3—612no—
337C2×C3⋊D3—624no—
338C2×C3⋊D3—612no—
339C2×C3⋊D3—612no—
340C2×C3⋊D3—124no—
341C2×C3⋊D3—1212no—
342C2×C3⋊D3—1212no—
343C2×C3⋊D3—1224no—
344C2×C3⋊D3C23×D314yes—
345C2×C3×D3—212no—
346C2×C3×D3—212no—
347C2×C3×D3—24no—
348C2×C3×D3—34no—
349C2×C3×D3—34no—
350C2×C3×D3—412no—
351C2×C3×D3—64no—
352C2×C3×D3—612no—
353C2×C3×D3—612no—
354C2×C3×D3—1212no—
355C2×C3×D3C23×D314yes—
356C2×C3×D3C23×D314yes—
357C32×C4—34no—
358C32×C4—34no—
359C3⋊Dic3C23×D314yes—
360C3⋊Dic3C23×D314yes—
361C3×Dic3—24no—
362C3×Dic3—24no—
363C3×Dic3—34no—
364C3×Dic3—34no—
365C3×Dic3—34no—
366C3×Dic3—34no—
367C3×Dic3—64no—
368C3×Dic3—64no—
369C3×Dic3C23×D314yes—
370C3×Dic3C23×D314yes—
371C3×Dic3C23×D314yes—
372C3×Dic3C23×D314yes—
373C2×#41—2724no—
374C22×Q8—91no—
375C22×D4—91no—
376C22×D4—271no—
377C22×D4—271no—
378C23×C4—91no—
379C23×C4—271no—
380C23×C4—271no—
381C33C22×#4111yescharacteristic, complemented, Sylow
382C23×C3—21no—
383C23×C3—21no—
384C23×C3—21no—
385C23×C3—36no—
386C23×C3—36no—
387C23×C3—41no—
388C23×C3—66no—
389C23×C3—64no—
390C23×C3—64no—
391C23×C3—96no—
392C23×C3—124no—
393C23×C3—184no—
394C23×C3C2×D3211yescharacteristic
395C23×C3C2×D3211yescharacteristic
396C23×C3C2×D3211yescharacteristic
397C22×D3—24no—
398C22×D3—36no—
399C22×D3—36no—
400C22×D3—66no—
401C22×D3—64no—
402C22×D3—64no—
403C22×D3—96no—
404C22×D3—96no—
405C22×D3—96no—
406C22×D3—124no—
407C22×D3—186no—
408C22×D3—186no—
409C22×D3—186no—
410C22×D3—188no—
411C22×D3—184no—
412C22×D3—184no—
413C22×D3—184no—
414C22×D3—366no—
415C22×D3—364no—
416C22×D3—364no—
417C22×D3—368no—
418C22×D3C2×D3216yes—
419C3×Q8—916no—
420C3×D4—916no—
421C2×C3×C4—36no—
422C2×C3×C4—36no—
423C2×C3×C4—36no—
424C2×C3×C4—36no—
425C2×C3×C4—66no—
426C2×C3×C4—66no—
427C2×C3×C4—916no—
428C2×C3×C4—96no—
429C2×C3×C4—96no—
430#67—316no—
431#67—916no—
432#67—916no—
433#67—916no—
434#67—1816no—
435C2×Dic3—36no—
436C2×Dic3—36no—
437C2×Dic3—36no—
438C2×Dic3—36no—
439C2×Dic3—316no—
440C2×Dic3—66no—
441C2×Dic3—66no—
442C2×Dic3—916no—
443C2×Dic3C2×D3216yes—
444C2×Dic3C2×D3216yes—
445D12—316no—
446D12—916no—
447D12—916no—
448D12—916no—
449D12—1816no—
450C4×D3—316no—
451C4×D3—916no—
452C4×D3—916no—
453C4×D3—916no—
454C4×D3—916no—
455C4×D3—916no—
456C4×D3—1816no—
457Dic6—316no—
458Dic6—316no—
459C2×C32—26no—
460C2×C32—26no—
461C2×C32—26no—
462C2×C32—21no—
463C2×C32—21no—
464C2×C32—21no—
465C2×C32—46no—
466C2×C32—41no—
467C2×C32—64no—
468C2×C32C2×#23516yescomplemented
469C2×C32C2×#23716yescomplemented
470C2×C32C2×#23916yescomplemented
471C2×C32C24×D311yescharacteristic
472C2×C32C24×D311yescharacteristic
473C2×C32C24×D311yescharacteristic
474C3⋊D3—24no—
475C3⋊D3—64no—
476C3⋊D3—68no—
477C3⋊D3—64no—
478C3⋊D3—64no—
479C3⋊D3—124no—
480C3⋊D3—124no—
481C3⋊D3—128no—
482C3×D3—24no—
483C3×D3—24no—
484C3×D3—44no—
485C3×D3—64no—
486C3×D3—64no—
487C3×D3—124no—
488C24—31no—
489C24—91no—
490C24—271no—
491#41—2764no—
492C2×Q8—912no—
493C2×D4—912no—
494C2×D4—2712no—
495C2×D4—2712no—
496C22×C4—31no—
497C22×C4—31no—
498C22×C4—912no—
499C22×C4—91no—
500C22×C4—91no—
501C22×C4—2712no—
502C22×C4—2712no—
503C22×C3—23no—
504C22×C3—23no—
505C22×C3—23no—
506C22×C3—24no—
507C22×C3—24no—
508C22×C3—24no—
509C22×C3—34no—
510C22×C3—34no—
511C22×C3—43no—
512C22×C3—44no—
513C22×C3—64no—
514C22×C3—612no—
515C22×C3—612no—
516C22×C3—94no—
517C22×C3—1212no—
518C22×C3—1812no—
519C22×C3#379614yescomplemented
520C22×C3#379914yescomplemented
521C22×C3#380414yescomplemented
522C22×C3C22×D3213yes—
523C22×C3C22×D3213yes—
524C22×C3C22×D3213yes—
525C2×D3—212no—
526C2×D3—34no—
527C2×D3—34no—
528C2×D3—64no—
529C2×D3—612no—
530C2×D3—612no—
531C2×D3—94no—
532C2×D3—94no—
533C2×D3—94no—
534C2×D3—1212no—
535C2×D3—184no—
536C2×D3—184no—
537C2×D3—184no—
538C2×D3—1824no—
539C2×D3—1812no—
540C2×D3—1812no—
541C2×D3—1812no—
542C2×D3—364no—
543C2×D3—3612no—
544C2×D3—3612no—
545C2×D3—3624no—
546C2×D3C22×D3214yes—
547C3×C4—34no—
548C3×C4—34no—
549C3×C4—34no—
550C3×C4—34no—
551C3×C4—64no—
552C3×C4—64no—
553C3×C4—94no—
554C3×C4—94no—
555Dic3—34no—
556Dic3—34no—
557Dic3—34no—
558Dic3—34no—
559Dic3—64no—
560Dic3—64no—
561Dic3C22×D3214yes—
562Dic3C22×D3214yes—
563C32—21no—
564C32—21no—
565C32—21no—
566C32—41no—
567C32C22×#23511yescharacteristic, complemented
568C32C22×#23711yescharacteristic, complemented
569C32C22×#23911yescharacteristic, complemented
570C23—36no—
571C23—64no—
572C23—96no—
573C23—184no—
574C23—276no—
575C23—544no—
576C23D3311yescharacteristic
577Q8—916no—
578D4—916no—
579D4—2716no—
580D4—2716no—
581C2×C4—36no—
582C2×C4—36no—
583C2×C4—916no—
584C2×C4—96no—
585C2×C4—96no—
586C2×C4—2716no—
587C2×C4—2716no—
588C2×C3—26no—
589C2×C3—26no—
590C2×C3—26no—
591C2×C3—21no—
592C2×C3—21no—
593C2×C3—21no—
594C2×C3—46no—
595C2×C3—41no—
596C2×C3—64no—
597C2×C3—64no—
598C2×C3—124no—
599C2×C3—184no—
600C2×C3C2×#379616yescomplemented
601C2×C3C2×#379916yescomplemented
602C2×C3C2×#380416yescomplemented
603C2×C3C23×D3211yescharacteristic
604C2×C3C23×D3211yescharacteristic
605C2×C3C23×D3211yescharacteristic
606D3—24no—
607D3—64no—
608D3—64no—
609D3—124no—
610D3—188no—
611D3—184no—
612D3—184no—
613D3—184no—
614D3—364no—
615D3—364no—
616D3—368no—
617C22—34no—
618C22—612no—
619C22—94no—
620C22—1812no—
621C22—274no—
622C22—5412no—
623C22#8896214yesdirect product factor
624C22C2×D3313yes—
625C4—34no—
626C4—34no—
627C4—94no—
628C4—94no—
629C3—21no—
630C3—21no—
631C3—21no—
632C3—41no—
633C3C22×#379611yescharacteristic, complemented
634C3C22×#379911yescharacteristic, complemented
635C3C22×#380411yescharacteristic, complemented
636C2—64no—
637C2—184no—
638C2—544no—
639C2C2×#8896216yesdirect product factor
640C2C22×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×#6294C2semidirect
2. C22×#6292C2semidirect
3. C22×#6291C2semidirect
4. C22×#6289C2semidirect
5. C22×C3×#627C2semidirect
6. C22×C3×#626C2semidirect
7. C22×C3×D3×Dic3C2semidirect
8. C2×#88962C2direct (N × H)
19. C2×#6294C22semidirect
20. C2×#6292C22semidirect
21. C2×#6291C22semidirect
22. C2×#6289C22semidirect
23. C2×C3×#627C22semidirect
24. C2×C3×#626C22semidirect
25. C2×C3×D3×Dic3C22semidirect
26. #88962C22direct (N × H)
36. C22×#627D3semidirect
43. C22×#626D3semidirect
51. C22×D3×Dic3D3semidirect
65. #6294C23semidirect
66. #6292C23semidirect
67. #6291C23semidirect
68. #6289C23semidirect
69. C3×#627C23semidirect
70. C3×#626C23semidirect
71. C3×D3×Dic3C23semidirect
115. C2×#627C2×D3semidirect
122. C2×#626C2×D3semidirect
130. C2×D3×Dic3C2×D3semidirect
133. C22×C33#41semidirect
234. #627C22×D3semidirect
241. #626C22×D3semidirect
248. D3×Dic3C22×D3semidirect
250. C2×C33C2×#41semidirect
323. C22×C32#235semidirect
324. C22×C32#237semidirect
325. C22×C32#239semidirect
381. C33C22×#41semidirect
468. C2×C32C2×#235semidirect
469. C2×C32C2×#237semidirect
470. C2×C32C2×#239semidirect
519. C22×C3#3796semidirect
520. C22×C3#3799semidirect
521. C22×C3#3804semidirect
567. C32C22×#235semidirect
568. C32C22×#237semidirect
569. C32C22×#239semidirect
600. C2×C3C2×#3796semidirect
601. C2×C3C2×#3799semidirect
602. C2×C3C2×#3804semidirect
623. C22#88962direct (N × H)
633. C3C22×#3796semidirect
634. C3C22×#3799semidirect
635. C3C22×#3804semidirect
639. C2C2×#88962direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#62942
C22×#62922
C22×#62912
C22×#62892
C22×C3×#6272
C22×C3×#6262
C22×C3×D3×Dic32
C2×#889622
C22×#38043
C22×#37993
C22×#37963