Finite Group Databasebeta

← C2×#11721313 #13152078

Subgroups — C2×#11721313 #13152078

Order 1728.

1623 conjugacy classes of subgroups.

The subgroup lattice has 620 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#638⋊C3C211yesmaximal, characteristic, complemented
2C22×#638⋊C3C211yesmaximal, characteristic, complemented
3C22×#6198C211yesmaximal, characteristic, complemented
4C22×((C3⋊D3)⋊C3)⋊C4C211yesmaximal, characteristic, complemented
5C2×#88552C211yesmaximal, characteristic, complemented
6#11721313C218yesmaximal, direct product factor
7C2×#88450C211yesmaximal, characteristic, complemented
8C2×#88448C211yesmaximal, characteristic, complemented
9C2×#63971—31nomaximal
10C2×#63958—31nomaximal
11C24×H(3,𝔽3)C2211yescharacteristic
12C23×(C3⋊D3)⋊C3C2211yescharacteristic
13C23×(C3⋊D3)⋊C3C2211yescharacteristic
14C22×#1084C2211yescharacteristic
15C22×#1084C2211yescharacteristic
16C2×#638⋊C3—24no—
17C2×#638⋊C3—24no—
18C2×#638⋊C3C2214yescomplemented
19C2×#638⋊C3C2214yescomplemented
20C22×(C3⋊Dic3)⋊C3C2211yescharacteristic
21C22×(C3⋊Dic3)⋊C3C2211yescharacteristic
22C2×#6198—24no—
23C2×#6198C2214yescomplemented
24C2×((C3⋊D3)⋊C3)⋊C4—24no—
25C2×((C3⋊D3)⋊C3)⋊C4C2214yescomplemented
26#88552C2214yescomplemented
27#88450C2214yescomplemented
28#88448C2214yescomplemented
29C22×#638D311yescharacteristic, complemented
30C22×#638D311yescharacteristic, complemented
31C22×C3×#67—31no—
32C22×C3×#67—31no—
33C22×#626—31no—
34C22×#626—31no—
35C22×#624—31no—
36C22×D3×Dic3—31no—
37C2×C3×#217—31no—
38C2×C3×#217—31no—
39C2×C3×#217—61no—
40#63971—38no—
41C2×#3724—31no—
42C2×Dic3⋊Dic3—31no—
43C2×#3722—31no—
44C2×(C2×D3)⋊Dic3—31no—
45#63958—38no—
46C23×H(3,𝔽3)—22no—
47C23×H(3,𝔽3)—22no—
48C23×H(3,𝔽3)—22no—
49C23×H(3,𝔽3)C2312yes—
50C23×H(3,𝔽3)C2311yescharacteristic
51C22×(C3⋊D3)⋊C3—22no—
52C22×(C3⋊D3)⋊C3—22no—
53C22×(C3⋊D3)⋊C3—22no—
54C22×(C3⋊D3)⋊C3—22no—
55C22×(C3⋊D3)⋊C3—22no—
56C22×(C3⋊D3)⋊C3D414yescomplemented
57C22×(C3⋊D3)⋊C3C2312yes—
58C22×(C3⋊D3)⋊C3C2312yes—
59C2×#1084—22no—
60C2×#1084—22no—
61C2×#1084C2312yes—
62C2×#1084C2312yes—
63#638⋊C3—28no—
64#638⋊C3—28no—
65C2×(C3⋊Dic3)⋊C3—22no—
66C2×(C3⋊Dic3)⋊C3D414yescomplemented
67C2×(C3⋊Dic3)⋊C3C2312yes—
68C2×(C3⋊Dic3)⋊C3C2312yes—
69#6198—28no—
70((C3⋊D3)⋊C3)⋊C4—28no—
71C2×#939—91no—
72C2×#938—91no—
73C2×#884—91no—
74C24×C32—21no—
75C24×C32C2×D311yescharacteristic
76C24×C32C2×D311yescharacteristic
77C23×C3⋊D3C2×D311yescharacteristic
78C23×C3⋊D3C2×D311yescharacteristic
79C23×C3×D3—31no—
80C23×C3×D3—31no—
81C2×#638—24no—
82C2×#638—24no—
83C2×#638C2×D314yescomplemented
84C2×#638C2×D314yescomplemented
85C22×C3⋊Dic3C2×D311yescharacteristic
86C22×C3⋊Dic3C2×D311yescharacteristic
87C2×C3×#67—34no—
88C2×C3×#67—34no—
89C2×C3×#67—64no—
90C2×C3×#67—64no—
91C22×C3×Dic3—31no—
92C22×C3×Dic3—31no—
93C22×C3×Dic3—31no—
94C22×C3×Dic3—31no—
95C22×C3×Dic3—31no—
96C22×C3×Dic3—31no—
97C22×C3×Dic3—61no—
98C22×C3×Dic3—61no—
99C2×#626—34no—
100C2×#626—34no—
101C2×#626—64no—
102C2×#626—64no—
103C2×#624—34no—
104C2×#624—64no—
105C2×D3×Dic3—34no—
106C2×D3×Dic3—64no—
107C3×#217—34no—
108C3×#217—34no—
109C3×#217—64no—
110#3724—34no—
111Dic3⋊Dic3—34no—
112#3722—34no—
113(C2×D3)⋊Dic3—34no—
114C22×H(3,𝔽3)—22no—
115C22×H(3,𝔽3)—22no—
116C22×H(3,𝔽3)—22no—
117C22×H(3,𝔽3)—44no—
118C22×H(3,𝔽3)C2×D412yes—
119C22×H(3,𝔽3)C2×D412yes—
120C22×H(3,𝔽3)#4112yes—
121C22×H(3,𝔽3)C2411yescharacteristic
122C2×(C3⋊D3)⋊C3—28no—
123C2×(C3⋊D3)⋊C3—22no—
124C2×(C3⋊D3)⋊C3—22no—
125C2×(C3⋊D3)⋊C3—22no—
126C2×(C3⋊D3)⋊C3—22no—
127C2×(C3⋊D3)⋊C3—22no—
128C2×(C3⋊D3)⋊C3—44no—
129C2×(C3⋊D3)⋊C3C2×D414yescomplemented
130#1084—22no—
131#1084—22no—
132(C3⋊Dic3)⋊C3—22no—
133(C3⋊Dic3)⋊C3C2×D414yescomplemented
134C22×C3×D4—91no—
135C22×C3×D4—91no—
136C22×#67—31no—
137C22×#67—31no—
138C22×#67—91no—
139C22×#67—91no—
140C22×#67—91no—
141C22×#67—91no—
142C23×Dic3—91no—
143C22×D12—91no—
144C22×C4×D3—91no—
145C22×C4×D3—91no—
146C2×C3×C22⋊C4—91no—
147C2×#217—31no—
148C2×#217—31no—
149C2×#217—61no—
150C2×#217—91no—
151#939—98no—
152#938—98no—
153C2×(C2×D3)⋊C4—91no—
154C2×(C2×D3)⋊C4—91no—
155C2×C4⋊Dic3—91no—
156C2×Dic3⋊C4—91no—
157C2×Dic3⋊C4—91no—
158#884—98no—
159C23×C32—24no—
160C23×C32—22no—
161C23×C32—22no—
162C23×C32—22no—
163C23×C32—22no—
164C23×C32—22no—
165C23×C32—22no—
166C23×C32—21no—
167C23×C32—22no—
168C23×C32—42no—
169C23×C32—42no—
170C23×C32C22×D312yes—
171C23×C32C22×D312yes—
172C23×C32C22×D311yescharacteristic
173C23×C32C22×D311yescharacteristic
174C22×C3⋊D3—22no—
175C22×C3⋊D3—22no—
176C22×C3⋊D3—22no—
177C22×C3⋊D3—22no—
178C22×C3⋊D3—22no—
179C22×C3⋊D3#6714yescomplemented
180C22×C3⋊D3C22×D312yes—
181C22×C3⋊D3C22×D312yes—
182C22×C3×D3—34no—
183C22×C3×D3—32no—
184C22×C3×D3—32no—
185C22×C3×D3—62no—
186C22×C3×D3—62no—
187C22×C3×D3—62no—
188C22×C3×D3—62no—
189C22×C3×D3—62no—
190#638—28no—
191#638—28no—
192C2×C3⋊Dic3—22no—
193C2×C3⋊Dic3#6714yescomplemented
194C2×C3⋊Dic3C22×D312yes—
195C2×C3⋊Dic3C22×D312yes—
196C3×#67—68no—
197C3×#67—68no—
198C2×C3×Dic3—34no—
199C2×C3×Dic3—32no—
200C2×C3×Dic3—32no—
201C2×C3×Dic3—32no—
202C2×C3×Dic3—32no—
203C2×C3×Dic3—32no—
204C2×C3×Dic3—32no—
205C2×C3×Dic3—62no—
206C2×C3×Dic3—62no—
207C2×C3×Dic3—62no—
208C2×C3×Dic3—62no—
209C2×C3×Dic3—62no—
210C2×C3×Dic3—62no—
211C2×C3×Dic3—62no—
212C2×C3×Dic3—122no—
213C2×C3×Dic3—122no—
214#626—68no—
215#626—68no—
216#624—68no—
217D3×Dic3—68no—
218C2×#121—271noSylow
219C2×H(3,𝔽3)—42no—
220C2×H(3,𝔽3)#12114yescomplemented
221C2×H(3,𝔽3)C22×D411yescharacteristic
222C2×H(3,𝔽3)C22×D411yescharacteristic
223C2×H(3,𝔽3)C2×#4111yescharacteristic
224(C3⋊D3)⋊C3—24no—
225(C3⋊D3)⋊C3—42no—
226C24×C3—31no—
227C24×C3—31no—
228C24×C3—61no—
229C24×C3—91no—
230C24×C3—91no—
231C24×C3D3211yescharacteristic
232C23×D3—31no—
233C23×D3—31no—
234C23×D3—91no—
235C23×D3—91no—
236C2×C3×D4—94no—
237C2×C3×D4—94no—
238C2×C3×D4—184no—
239C2×C3×D4—184no—
240C22×C3×C4—91no—
241C22×C3×C4—91no—
242C22×C3×C4—91no—
243C22×C3×C4—91no—
244C2×#67—34no—
245C2×#67—34no—
246C2×#67—64no—
247C2×#67—64no—
248C2×#67—94no—
249C2×#67—94no—
250C2×#67—94no—
251C2×#67—94no—
252C2×#67—184no—
253C2×#67—184no—
254C2×#67—184no—
255C2×#67—184no—
256C22×Dic3—31no—
257C22×Dic3—31no—
258C22×Dic3—31no—
259C22×Dic3—31no—
260C22×Dic3—31no—
261C22×Dic3—31no—
262C22×Dic3—61no—
263C22×Dic3—61no—
264C22×Dic3—91no—
265C22×Dic3—91no—
266C22×Dic3—94no—
267C22×Dic3—184no—
268C2×D12—94no—
269C2×D12—184no—
270C2×C4×D3—94no—
271C2×C4×D3—94no—
272C2×C4×D3—184no—
273C2×C4×D3—184no—
274C3×C22⋊C4—94no—
275#217—34no—
276#217—34no—
277#217—64no—
278#217—94no—
279(C2×D3)⋊C4—94no—
280(C2×D3)⋊C4—94no—
281C4⋊Dic3—94no—
282Dic3⋊C4—94no—
283Dic3⋊C4—94no—
284C22×C32—24no—
285C22×C32—22no—
286C22×C32—22no—
287C22×C32—22no—
288C22×C32—22no—
289C22×C32—22no—
290C22×C32—22no—
291C22×C32—21no—
292C22×C32—22no—
293C22×C32—22no—
294C22×C32—22no—
295C22×C32—42no—
296C22×C32—42no—
297C22×C32—48no—
298C22×C32—44no—
299C22×C32—44no—
300C22×C32D3×D412yes—
301C22×C32D3×D412yes—
302C22×C32#23712yes—
303C22×C32#23712yes—
304C22×C32C2×#6712yes—
305C22×C32C2×#6712yes—
306C22×C32C23×D311yescharacteristic
307C22×C32C23×D311yescharacteristic
308C2×C3⋊D3—28no—
309C2×C3⋊D3—22no—
310C2×C3⋊D3—22no—
311C2×C3⋊D3—22no—
312C2×C3⋊D3—22no—
313C2×C3⋊D3—22no—
314C2×C3⋊D3—44no—
315C2×C3⋊D3C2×#6714yescomplemented
316C2×C3×D3—34no—
317C2×C3×D3—68no—
318C2×C3×D3—62no—
319C2×C3×D3—62no—
320C2×C3×D3—62no—
321C2×C3×D3—62no—
322C2×C3×D3—62no—
323C2×C3×D3—124no—
324C3⋊Dic3—22no—
325C3⋊Dic3C2×#6714yescomplemented
326C3×Dic3—34no—
327C3×Dic3—62no—
328C3×Dic3—62no—
329C3×Dic3—62no—
330C3×Dic3—62no—
331C3×Dic3—62no—
332C3×Dic3—122no—
333C3×Dic3—122no—
334C22×D4—91no—
335C22×D4—91no—
336C22×D4—271no—
337C23×C4—271no—
338#121—278no—
339C2×C4⋊C4—271no—
340C2×C22⋊C4—91no—
341C2×C22⋊C4—271no—
342H(3,𝔽3)C2×#12111yescharacteristic, complemented, Sylow
343C23×C3—22no—
344C23×C3—22no—
345C23×C3—22no—
346C23×C3—32no—
347C23×C3—32no—
348C23×C3—31no—
349C23×C3—31no—
350C23×C3—62no—
351C23×C3—62no—
352C23×C3—62no—
353C23×C3—62no—
354C23×C3—62no—
355C23×C3—62no—
356C23×C3—64no—
357C23×C3—62no—
358C23×C3—61no—
359C23×C3—92no—
360C23×C3—94no—
361C23×C3—92no—
362C23×C3—122no—
363C23×C3—122no—
364C23×C3—182no—
365C23×C3—182no—
366C23×C3—182no—
367C23×C3—182no—
368C23×C3—182no—
369C23×C3C2×D3212yes—
370C23×C3C2×D3211yescharacteristic
371C22×D3—32no—
372C22×D3—34no—
373C22×D3—32no—
374C22×D3—62no—
375C22×D3—62no—
376C22×D3—62no—
377C22×D3—62no—
378C22×D3—62no—
379C22×D3—94no—
380C22×D3—92no—
381C22×D3—92no—
382C22×D3—182no—
383C22×D3—182no—
384C22×D3—182no—
385C22×D3—182no—
386C22×D3—182no—
387C3×D4—188no—
388C3×D4—188no—
389C2×C3×C4—92no—
390C2×C3×C4—94no—
391C2×C3×C4—92no—
392C2×C3×C4—92no—
393C2×C3×C4—92no—
394C2×C3×C4—182no—
395C2×C3×C4—182no—
396C2×C3×C4—182no—
397#67—68no—
398#67—68no—
399#67—188no—
400#67—188no—
401#67—188no—
402#67—188no—
403C2×Dic3—32no—
404C2×Dic3—34no—
405C2×Dic3—32no—
406C2×Dic3—32no—
407C2×Dic3—32no—
408C2×Dic3—32no—
409C2×Dic3—32no—
410C2×Dic3—62no—
411C2×Dic3—62no—
412C2×Dic3—62no—
413C2×Dic3—62no—
414C2×Dic3—62no—
415C2×Dic3—62no—
416C2×Dic3—62no—
417C2×Dic3—94no—
418C2×Dic3—92no—
419C2×Dic3—92no—
420C2×Dic3—122no—
421C2×Dic3—122no—
422C2×Dic3—182no—
423C2×Dic3—188no—
424D12—188no—
425C4×D3—188no—
426C4×D3—188no—
427C2×C32—21no—
428C2×C32—21no—
429C2×C32—21no—
430C2×C32—24no—
431C2×C32—44no—
432C2×C32—42no—
433C2×C32—42no—
434C2×C32#93814yescomplemented
435C2×C32#93914yescomplemented
436C2×C32C2×D3×D411yescharacteristic
437C2×C32C2×D3×D411yescharacteristic
438C2×C32C2×#23711yescharacteristic
439C2×C32C2×#23711yescharacteristic
440C2×C32C22×#6711yescharacteristic
441C2×C32C22×#6711yescharacteristic
442C3⋊D3—24no—
443C3⋊D3—42no—
444C3×D3—64no—
445C3×D3—122no—
446C24—91no—
447C24—91no—
448C24((C3⋊D3)⋊C3)⋊C211yescharacteristic
449C2×D4—94no—
450C2×D4—94no—
451C2×D4—184no—
452C2×D4—184no—
453C2×D4—274no—
454C2×D4—544no—
455C22×C4—91no—
456C22×C4—91no—
457C22×C4—91no—
458C22×C4—91no—
459C22×C4—274no—
460C22×C4—544no—
461C4⋊C4—274no—
462C22⋊C4—94no—
463C22⋊C4—274no—
464C22×C3—22no—
465C22×C3—22no—
466C22×C3—22no—
467C22×C3—31no—
468C22×C3—31no—
469C22×C3—32no—
470C22×C3—32no—
471C22×C3—32no—
472C22×C3—32no—
473C22×C3—32no—
474C22×C3—32no—
475C22×C3—44no—
476C22×C3—62no—
477C22×C3—62no—
478C22×C3—62no—
479C22×C3—62no—
480C22×C3—62no—
481C22×C3—62no—
482C22×C3—64no—
483C22×C3—61no—
484C22×C3—62no—
485C22×C3—62no—
486C22×C3—62no—
487C22×C3—94no—
488C22×C3—122no—
489C22×C3—122no—
490C22×C3—124no—
491C22×C3—124no—
492C22×C3—128no—
493C22×C3—182no—
494C22×C3—182no—
495C22×C3—188no—
496C22×C3—182no—
497C22×C3—182no—
498C22×C3—182no—
499C22×C3—364no—
500C22×C3#380512yes—
501C22×C3D3×#6712yes—
502C22×C3D3×#6712yes—
503C22×C3C22×D3211yescharacteristic
504C2×D3—34no—
505C2×D3—62no—
506C2×D3—62no—
507C2×D3—68no—
508C2×D3—62no—
509C2×D3—62no—
510C2×D3—62no—
511C2×D3—94no—
512C2×D3—124no—
513C2×D3—182no—
514C2×D3—182no—
515C2×D3—188no—
516C2×D3—182no—
517C2×D3—182no—
518C2×D3—182no—
519C2×D3—364no—
520C3×C4—94no—
521C3×C4—182no—
522C3×C4—182no—
523C3×C4—182no—
524Dic3—34no—
525Dic3—62no—
526Dic3—62no—
527Dic3—62no—
528Dic3—62no—
529Dic3—62no—
530Dic3—94no—
531Dic3—122no—
532Dic3—122no—
533Dic3—182no—
534C32—21no—
535C32C2×#93811yescharacteristic, complemented
536C32C2×#93911yescharacteristic, complemented
537C23—22no—
538C23—22no—
539C23—22no—
540C23—92no—
541C23—94no—
542C23—92no—
543C23—182no—
544C23—182no—
545C23—182no—
546C23—182no—
547C23—182no—
548C23C2×((C3⋊D3)⋊C3)⋊C212yes—
549C23C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
550D4—188no—
551D4—188no—
552D4—548no—
553C2×C4—92no—
554C2×C4—94no—
555C2×C4—92no—
556C2×C4—92no—
557C2×C4—92no—
558C2×C4—182no—
559C2×C4—182no—
560C2×C4—182no—
561C2×C4—548no—
562C2×C3—31no—
563C2×C3—31no—
564C2×C3—31no—
565C2×C3—31no—
566C2×C3—31no—
567C2×C3—31no—
568C2×C3—34no—
569C2×C3—34no—
570C2×C3—42no—
571C2×C3—61no—
572C2×C3—61no—
573C2×C3—61no—
574C2×C3—64no—
575C2×C3—122no—
576C2×C3—122no—
577C2×C3—124no—
578C2×C3—184no—
579C2×C3—362no—
580C2×C3#6397014yes—
581C2×C3C2×#380511yescharacteristic
582C2×C3C2×D3×#6711yescharacteristic
583C2×C3C2×D3×#6711yescharacteristic
584D3—64no—
585D3—122no—
586D3—184no—
587D3—362no—
588C22—22no—
589C22—22no—
590C22—22no—
591C22—44no—
592C22—94no—
593C22—182no—
594C22—182no—
595C22—188no—
596C22—182no—
597C22—182no—
598C22—182no—
599C22—364no—
600C22#8867212yes—
601C22#8867612yes—
602C22#8867612yes—
603C22C22×((C3⋊D3)⋊C3)⋊C211yescharacteristic
604C4—94no—
605C4—182no—
606C4—182no—
607C4—182no—
608C3—31no—
609C3—31no—
610C3—61no—
611C3C2×#6397011yescharacteristic
612C2—42no—
613C2—184no—
614C2—362no—
615C2#1172131314yesdirect product factor
616C2C2×#8867211yescharacteristic
617C2C2×#8867611yescharacteristic
618C2C2×#8867611yescharacteristic
subgroupG/H
centerC23C2×((C3⋊D3)⋊C3)⋊C2
commutator, cosocleC22×H(3,𝔽3)C24
FrattiniC22×C3C22×D32
FittingC24×H(3,𝔽3)C22
socleC23×C3C2×D32
hypercenterC24((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)C2×#121
LCSG ▷ C22×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C23 ◁ C24

Complements

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

NH ≅ G/Ntype
1. C22×#638⋊C3C2semidirect
2. C22×#638⋊C3C2semidirect
3. C22×#6198C2semidirect
4. C22×((C3⋊D3)⋊C3)⋊C4C2semidirect
5. C2×#88552C2semidirect
6. #11721313C2direct (N × H)
7. C2×#88450C2semidirect
8. C2×#88448C2semidirect
18. C2×#638⋊C3C22semidirect
19. C2×#638⋊C3C22semidirect
23. C2×#6198C22semidirect
25. C2×((C3⋊D3)⋊C3)⋊C4C22semidirect
26. #88552C22semidirect
27. #88450C22semidirect
28. #88448C22semidirect
29. C22×#638D3semidirect
30. C22×#638D3semidirect
56. C22×(C3⋊D3)⋊C3D4semidirect
66. C2×(C3⋊Dic3)⋊C3D4semidirect
83. C2×#638C2×D3semidirect
84. C2×#638C2×D3semidirect
129. C2×(C3⋊D3)⋊C3C2×D4semidirect
133. (C3⋊Dic3)⋊C3C2×D4semidirect
179. C22×C3⋊D3#67semidirect
193. C2×C3⋊Dic3#67semidirect
220. C2×H(3,𝔽3)#121semidirect
315. C2×C3⋊D3C2×#67semidirect
325. C3⋊Dic3C2×#67semidirect
342. H(3,𝔽3)C2×#121semidirect
434. C2×C32#938semidirect
435. C2×C32#939semidirect
535. C32C2×#938semidirect
536. C32C2×#939semidirect
615. C2#11721313direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#638⋊C32
C22×#61982
C22×((C3⋊D3)⋊C3)⋊C42
C2×#885522
#117213132
C2×#884502
C2×#884482
C2×#639713
C2×#639583