Finite Group Databasebeta

← C2×(D32⋊D3)⋊C4 #13180540

Subgroups — C2×(D32⋊D3)⋊C4 #13180540

Order 1728.

1432 conjugacy classes of subgroups; only the 188 normal subgroups are shown.

normal subgroups only

This group has 1432 conjugacy classes of subgroups — too many to draw. Only its 188 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×D32⋊D3C211yesmaximal, characteristic
2C2×#88949C211yesmaximal, characteristic, complemented
3C2×D32×Dic3C211yesmaximal, characteristic, complemented
4C2×#88939C211yesmaximal, characteristic, complemented
5C2×#88934C211yesmaximal, characteristic, complemented
6C2×C3×D32⋊C4C211yesmaximal, characteristic, complemented
7C2×Dic3×#157C211yesmaximal, characteristic, complemented
8(D32⋊D3)⋊C4C218yesmaximal, direct product factor
9C2×C4×GO+(4,𝔽2)—31nomaximal
10C22×#1113C2211yescharacteristic
11C22×C3×D32C2211yescharacteristic
12C22×C3⋊#157C2211yescharacteristic
13C22×C3×#157C2211yescharacteristic
14C2×D32⋊D3C418yescomplemented
15C2×D32⋊D3C2214yes—
16C2×#6294C2211yescharacteristic
17C2×Dic3×C3⋊D3—21no—
18C2×Dic3×C3⋊D3C2211yescharacteristic
19C2×D3×C3⋊Dic3—21no—
20C2×C3×#624—21no—
21C2×C3×#624C2211yescharacteristic
22C2×C3×D3×Dic3—21no—
23#88949—22no—
24#88949C2214yescomplemented
25D32×Dic3—22no—
26D32×Dic3C2214yescomplemented
27#88939C2214yescomplemented
28#88934C2214yescomplemented
29C3×D32⋊C4C2214yescomplemented
30Dic3×#157C2214yescomplemented
31C22×GO+(4,𝔽2)—31no—
32C22×D3×Dic3—61no—
33C2×C4×D32—31no—
34C2×C4×D32—31no—
35C2×C4×D32—61no—
36C2×C4×#157—31no—
37C2×#3773—31no—
38C2×D32⋊C4—31no—
39C2×D32⋊C4D311yescharacteristic, complemented
40C4×GO+(4,𝔽2)—38no—
41C22×C3×C3⋊D3—21no—
42C22×C3×C3⋊D3C2311yescharacteristic
43C22×C32×D3—21no—
44C2×#1113—21no—
45C2×#1113—22no—
46C2×#1113C2×C414yescomplemented
47C2×#1113C2312yes—
48C2×C3×D32—21no—
49C2×C3×D32—22no—
50C2×C3×D32C2×C414yescomplemented
51C2×C3×D32C2312yes—
52C2×C3⋊#157C2×C414yescomplemented
53C2×C3⋊#157C2312yes—
54C2×C3×#157—22no—
55C2×C3×#157C2312yes—
56D32⋊D3C2×C4116yescomplemented
57C2×C3⋊(C3⋊Dic3)D411yescharacteristic, complemented
58C2×C3×C3⋊Dic3—21no—
59C2×C32×Dic3—21no—
60C2×C32×Dic3D411yescharacteristic, complemented
61#6294—21no—
62#6294C2312yes—
63Dic3×C3⋊D3—24no—
64Dic3×C3⋊D3D412yescomplemented
65Dic3×C3⋊D3C2312yes—
66D3×C3⋊Dic3—24no—
67C3×#624—21no—
68C3×#624—24no—
69C3×#624C2312yes—
70C3×D3×Dic3—24no—
71C2×#67⋊C4—91nomaximal
72C23×C3×D3—61no—
73C22×D32—31no—
74C22×D32—61no—
75C22×D32C2×D311yescharacteristic
76C22×#157—31no—
77C22×#157C2×D311yescharacteristic
78C2×GO+(4,𝔽2)—38no—
79C2×GO+(4,𝔽2)—34no—
80C22×C3⋊Dic3—61no—
81C2×C4×C3⋊D3—31no—
82C2×C4×C3⋊D3—61no—
83C22×C3×Dic3—61no—
84C2×C3×C4×D3—61no—
85C2×C3×C4×D3—61no—
86C2×C3×C4×D3—61no—
87C2×C3×C4×D3—61no—
88C2×#624—21no—
89C2×#624—31no—
90C2×#624C2×D311yescharacteristic
91C2×D3×Dic3—21no—
92C2×D3×Dic3—64no—
93C2×D3×Dic3—64no—
94C2×D3×Dic3—61no—
95C2×D3×Dic3—61no—
96C2×D3×Dic3—61no—
97C2×D3×Dic3—61no—
98C2×D3×Dic3—61no—
99C2×D3×Dic3—61no—
100C2×D3×Dic3—61no—
101C2×D3×Dic3—121no—
102C2×D3×Dic3—121no—
103C4×D32—34no—
104C4×D32—34no—
105C4×D32—62no—
106C4×D32—64no—
107C4×D32—64no—
108C4×D32—62no—
109C4×#157—34no—
110#3773—34no—
111D32⋊C4—34no—
112D32⋊C4C2×D314yescomplemented
113C22×C33C2×D411yescharacteristic
114C2×C3×C3⋊D3—24no—
115C2×C3×C3⋊D3—22no—
116C2×C3×C3⋊D3C22×C412yes—
117C2×C3×C3⋊D3C2×D411yescharacteristic
118C2×C3×C3⋊D3#4112yes—
119C2×C3×C3⋊D3C2411yescharacteristic
120C2×C32×D3—24no—
121C2×C32×D3—22no—
122#1113—24no—
123#1113—22no—
124#1113C22×C414yescomplemented
125C3×D32—24no—
126C3×D32—22no—
127C3×D32C22×C414yescomplemented
128C3⋊#157C22×C414yescomplemented
129C3×#157—22no—
130C3⋊(C3⋊Dic3)C2×D412yescomplemented
131C3×C3⋊Dic3—22no—
132C32×Dic3—22no—
133C32×Dic3C2×D412yescomplemented
134C22×#67—91no—
135C23×Dic3—91no—
136C22×C4×D3—91no—
137C22×C4×D3—181no—
138C22×C4×D3—181no—
139C2×C3×C22⋊C4—91no—
140C2×(C2×D3)⋊C4—91no—
141C2×Dic3⋊C4—91no—
142C2×C4×Dic3—91no—
143#67⋊C4—98no—
144C23×C32—61no—
145C22×C3⋊D3—21no—
146C22×C3⋊D3C22×D311yescharacteristic
147C22×C3×D3—21no—
148C22×C3×D3—21no—
149C22×C3×D3—41no—
150C22×C3×D3—64no—
151C22×C3×D3—62no—
152C22×C3×D3—61no—
153C22×C3×D3—61no—
154C22×C3×D3—64no—
155C22×C3×D3—62no—
156C22×C3×D3—61no—
157C22×C3×D3—61no—
158C22×C3×D3—121no—
159C2×D32—21no—
160C2×D32—22no—
161C2×D32—34no—
162C2×D32—32no—
163C2×D32—64no—
164C2×D32—62no—
165C2×D32—64no—
166C2×D32—62no—
167C2×D32—61no—
168C2×D32—62no—
169C2×D32C4×D314yescomplemented
170C2×D32C22×D312yes—
171C2×#157—22no—
172C2×#157—34no—
173C2×#157—32no—
174C2×#157C22×D312yes—
175GO+(4,𝔽2)—316no—
176C2×C32×C4—31no—
177C2×C32×C4—61no—
178C2×C3⋊Dic3—21no—
179C2×C3⋊Dic3—31no—
180C2×C3⋊Dic3—61no—
181C2×C3⋊Dic3—61no—
182C2×C3⋊Dic3—64no—
183C2×C3⋊Dic3—121no—
184C2×C3⋊Dic3—121no—
185C4×C3⋊D3—32no—
186C4×C3⋊D3—32no—
187C4×C3⋊D3—64no—
188C2×C3×Dic3—21no—
189C2×C3×Dic3—21no—
190C2×C3×Dic3—21no—
191C2×C3×Dic3—21no—
192C2×C3×Dic3—41no—
193C2×C3×Dic3—64no—
194C2×C3×Dic3—61no—
195C2×C3×Dic3—61no—
196C2×C3×Dic3—121no—
197C3×C4×D3—62no—
198C3×C4×D3—64no—
199C3×C4×D3—62no—
200C3×C4×D3—64no—
201C3×C4×D3—64no—
202#624—24no—
203#624—21no—
204#624—32no—
205#624—61no—
206#624C22×D312yes—
207D3×Dic3—24no—
208D3×Dic3—62no—
209D3×Dic3—62no—
210D3×Dic3—62no—
211D3×Dic3—62no—
212D3×Dic3—62no—
213D3×Dic3—64no—
214D3×Dic3—64no—
215D3×Dic3—62no—
216D3×Dic3—64no—
217D3×Dic3—64no—
218D3×Dic3—124no—
219D3×Dic3—124no—
220C2×C4×D4—271noSylow
221C2×C33C4×D412yescomplemented
222C2×C33C22×D411yescharacteristic
223C3×C3⋊D3—24no—
224C3×C3⋊D3C4×D412yescomplemented
225C3×C3⋊D3C23×C411yescharacteristic
226C3×C3⋊D3C2×#4111yescharacteristic
227C32×D3—24no—
228C24×C3—91no—
229C23×D3—61no—
230C23×D3—91no—
231C23×D3—181no—
232C22×C3×C4—91no—
233C22×C3×C4—91no—
234C22×C3×C4—181no—
235C22×C3×C4—181no—
236C2×#67—98no—
237C2×#67—94no—
238C22×Dic3—61no—
239C22×Dic3—94no—
240C22×Dic3—91no—
241C22×Dic3—91no—
242C22×Dic3—91no—
243C22×Dic3—181no—
244C22×Dic3—182no—
245C22×Dic3—181no—
246C22×Dic3—181no—
247C22×Dic3—361no—
248C2×C4×D3—61no—
249C2×C4×D3—61no—
250C2×C4×D3—61no—
251C2×C4×D3—61no—
252C2×C4×D3—94no—
253C2×C4×D3—184no—
254C2×C4×D3—184no—
255C2×C4×D3—181no—
256C2×C4×D3—181no—
257C2×C4×D3—181no—
258C2×C4×D3—182no—
259C2×C4×D3—184no—
260C2×C4×D3—184no—
261C2×C4×D3—181no—
262C2×C4×D3—181no—
263C2×C4×D3—181no—
264C2×C4×D3—361no—
265C3×C22⋊C4—94no—
266(C2×D3)⋊C4—94no—
267Dic3⋊C4—94no—
268C4×Dic3—94no—
269C22×C32—21no—
270C22×C32—21no—
271C22×C32—41no—
272C22×C32—41no—
273C22×C32—62no—
274C22×C32—64no—
275C22×C32D3×D411yescharacteristic
276C2×C3⋊D3—22no—
277C2×C3⋊D3—24no—
278C2×C3⋊D3C2×C4×D312yes—
279C2×C3⋊D3D3×D411yescharacteristic
280C2×C3⋊D3#23712yes—
281C2×C3⋊D3C23×D311yescharacteristic
282C2×C3×D3—22no—
283C2×C3×D3—24no—
284C2×C3×D3—24no—
285C2×C3×D3—22no—
286C2×C3×D3—42no—
287C2×C3×D3—44no—
288C2×C3×D3—64no—
289C2×C3×D3—62no—
290C2×C3×D3—62no—
291C2×C3×D3—61no—
292C2×C3×D3—61no—
293C2×C3×D3—62no—
294C2×C3×D3—62no—
295C2×C3×D3—61no—
296C2×C3×D3—61no—
297C2×C3×D3—68no—
298C2×C3×D3—64no—
299C2×C3×D3—62no—
300C2×C3×D3—64no—
301C2×C3×D3—64no—
302C2×C3×D3—62no—
303C2×C3×D3—122no—
304C2×C3×D3—124no—
305D32—24no—
306D32—22no—
307D32—34no—
308D32—64no—
309D32—68no—
310D32—64no—
311D32—64no—
312D32—62no—
313D32C2×C4×D314yescomplemented
314#157—22no—
315#157—34no—
316C32×C4—32no—
317C32×C4—62no—
318C3⋊Dic3—22no—
319C3⋊Dic3—32no—
320C3⋊Dic3—62no—
321C3⋊Dic3—62no—
322C3⋊Dic3—122no—
323C3⋊Dic3—122no—
324C3×Dic3—22no—
325C3×Dic3—22no—
326C3×Dic3—22no—
327C3×Dic3—22no—
328C3×Dic3—42no—
329C3×Dic3—62no—
330C3×Dic3—62no—
331C3×Dic3—122no—
332C22×D4—271no—
333C23×C4—271no—
334C23×C4—271no—
335C4×D4—278no—
336C2×C4⋊C4—271no—
337C2×C22⋊C4—91no—
338C2×C22⋊C4—271no—
339C2×C42—271no—
340C33C2×C4×D411yescharacteristic, complemented, Sylow
341C23×C3—61no—
342C23×C3—61no—
343C23×C3—92no—
344C23×C3—94no—
345C23×C3—91no—
346C23×C3—121no—
347C23×C3—181no—
348C23×C3—181no—
349C23×C3—182no—
350C22×D3—21no—
351C22×D3—64no—
352C22×D3—62no—
353C22×D3—61no—
354C22×D3—61no—
355C22×D3—62no—
356C22×D3—64no—
357C22×D3—61no—
358C22×D3—61no—
359C22×D3—92no—
360C22×D3—94no—
361C22×D3—121no—
362C22×D3—182no—
363C22×D3—184no—
364C22×D3—181no—
365C22×D3—182no—
366C22×D3—182no—
367C22×D3—184no—
368C2×C3×C4—61no—
369C2×C3×C4—61no—
370C2×C3×C4—61no—
371C2×C3×C4—61no—
372C2×C3×C4—92no—
373C2×C3×C4—92no—
374C2×C3×C4—121no—
375C2×C3×C4—181no—
376C2×C3×C4—184no—
377C2×C3×C4—184no—
378C2×C3×C4—181no—
379C2×C3×C4—182no—
380#67—916no—
381C2×Dic3—21no—
382C2×Dic3—61no—
383C2×Dic3—61no—
384C2×Dic3—64no—
385C2×Dic3—92no—
386C2×Dic3—94no—
387C2×Dic3—92no—
388C2×Dic3—91no—
389C2×Dic3—92no—
390C2×Dic3—92no—
391C2×Dic3—121no—
392C2×Dic3—181no—
393C2×Dic3—181no—
394C2×Dic3—184no—
395C2×Dic3—181no—
396C2×Dic3—184no—
397C2×Dic3—184no—
398C2×Dic3—361no—
399C2×Dic3—361no—
400C2×Dic3—364no—
401C2×Dic3GO+(4,𝔽2)11yescharacteristic, complemented
402C4×D3—62no—
403C4×D3—64no—
404C4×D3—62no—
405C4×D3—64no—
406C4×D3—64no—
407C4×D3—182no—
408C4×D3—182no—
409C4×D3—182no—
410C4×D3—182no—
411C4×D3—182no—
412C4×D3—182no—
413C4×D3—184no—
414C4×D3—184no—
415C4×D3—184no—
416C4×D3—364no—
417C2×C32—21no—
418C2×C32—21no—
419C2×C32—22no—
420C2×C32—22no—
421C2×C32—42no—
422C2×C32—42no—
423C2×C32—41no—
424C2×C32—41no—
425C2×C32—64no—
426C2×C32#67⋊C412yescomplemented
427C2×C32C2×D3×D411yescharacteristic
428C3⋊D3—24no—
429C3⋊D3#67⋊C412yescomplemented
430C3⋊D3C22×C4×D311yescharacteristic
431C3⋊D3C2×#23711yescharacteristic
432C3×D3—24no—
433C3×D3—24no—
434C3×D3—44no—
435C3×D3—62no—
436C3×D3—62no—
437C3×D3—61no—
438C3×D3—61no—
439C3×D3—61no—
440C3×D3—61no—
441C3×D3—64no—
442C3×D3—64no—
443C3×D3—124no—
444C24—91no—
445C24—271no—
446C2×D4—274no—
447C2×D4—278no—
448C22×C4—91no—
449C22×C4—91no—
450C22×C4—181no—
451C22×C4—181no—
452C22×C4—271no—
453C22×C4—274no—
454C22×C4—271no—
455C22×C4—274no—
456C22×C4—271no—
457C22×C4—541no—
458C22×C4—542no—
459C22×C4—542no—
460C22×C4—541no—
461C4⋊C4—274no—
462C22⋊C4—94no—
463C22⋊C4—274no—
464C42—274no—
465C22×C3—21no—
466C22×C3—21no—
467C22×C3—41no—
468C22×C3—41no—
469C22×C3—62no—
470C22×C3—64no—
471C22×C3—62no—
472C22×C3—64no—
473C22×C3—94no—
474C22×C3—91no—
475C22×C3—91no—
476C22×C3—92no—
477C22×C3—92no—
478C22×C3—122no—
479C22×C3—124no—
480C22×C3—182no—
481C22×C3—184no—
482C22×C3—182no—
483C22×C3—184no—
484C22×C3C2×GO+(4,𝔽2)11yescharacteristic
485C2×D3—22no—
486C2×D3—24no—
487C2×D3—64no—
488C2×D3—61no—
489C2×D3—61no—
490C2×D3—61no—
491C2×D3—61no—
492C2×D3—62no—
493C2×D3—62no—
494C2×D3—62no—
495C2×D3—62no—
496C2×D3—68no—
497C2×D3—64no—
498C2×D3—62no—
499C2×D3—64no—
500C2×D3—62no—
501C2×D3—64no—
502C2×D3—94no—
503C2×D3—122no—
504C2×D3—124no—
505C2×D3—184no—
506C2×D3—182no—
507C2×D3—184no—
508C2×D3—188no—
509C2×D3—184no—
510C3×C4—62no—
511C3×C4—62no—
512C3×C4—62no—
513C3×C4—62no—
514C3×C4—122no—
515C3×C4—182no—
516C3×C4—182no—
517Dic3—22no—
518Dic3—62no—
519Dic3—62no—
520Dic3—94no—
521Dic3—92no—
522Dic3—122no—
523Dic3—182no—
524Dic3—182no—
525Dic3—362no—
526Dic3—362no—
527Dic3C2×GO+(4,𝔽2)12yescomplemented
528C32—21no—
529C32—21no—
530C32—41no—
531C32—41no—
532C32C2×#67⋊C411yescharacteristic, complemented
533C23—61no—
534C23—92no—
535C23—94no—
536C23—91no—
537C23—181no—
538C23—182no—
539C23—181no—
540C23—272no—
541C23—274no—
542C23—541no—
543C23—542no—
544D4—2716no—
545C2×C4—31no—
546C2×C4—61no—
547C2×C4—92no—
548C2×C4—92no—
549C2×C4—184no—
550C2×C4—184no—
551C2×C4—181no—
552C2×C4—182no—
553C2×C4—181no—
554C2×C4—272no—
555C2×C4—272no—
556C2×C4—272no—
557C2×C4—272no—
558C2×C4—274no—
559C2×C4—271no—
560C2×C4—544no—
561C2×C4—541no—
562C2×C4—544no—
563C2×C3—22no—
564C2×C3—22no—
565C2×C3—21no—
566C2×C3—21no—
567C2×C3—42no—
568C2×C3—42no—
569C2×C3—41no—
570C2×C3—41no—
571C2×C3—64no—
572C2×C3—64no—
573C2×C3—91no—
574C2×C3—91no—
575C2×C3—92no—
576C2×C3—124no—
577C2×C3—184no—
578C2×C3C4×GO+(4,𝔽2)12yescomplemented
579C2×C3C22×GO+(4,𝔽2)11yescharacteristic
580D3—24no—
581D3—62no—
582D3—61no—
583D3—61no—
584D3—62no—
585D3—61no—
586D3—61no—
587D3—64no—
588D3—64no—
589D3—124no—
590C22—64no—
591C22—62no—
592C22—94no—
593C22—92no—
594C22—92no—
595C22—91no—
596C22—91no—
597C22—182no—
598C22—184no—
599C22—184no—
600C22—182no—
601C22—274no—
602C22—542no—
603C22—544no—
604C22D3×GO+(4,𝔽2)11yescharacteristic
605C4—32no—
606C4—62no—
607C4—182no—
608C4—182no—
609C4—274no—
610C4—272no—
611C3—21no—
612C3—21no—
613C3—41no—
614C3—41no—
615C3C2×C4×GO+(4,𝔽2)11yescharacteristic, complemented
616C2—64no—
617C2—91no—
618C2—91no—
619C2—92no—
620C2—184no—
621C2(D32⋊D3)⋊C412yesdirect product factor
622C2C2×D3×GO+(4,𝔽2)11yescharacteristic
subgroupG/H
center, hypercenterC22D3×GO+(4,𝔽2)
commutatorC3×C3⋊D3C23×C4
FrattiniC2C2×D3×GO+(4,𝔽2)
Fitting, socleC22×C33C2×D4
prime radicalC22×D32⋊D3C2
cosocleC2×C3×C3⋊D3C24
hypocenterC33C2×C4×D4
LCSG ▷ C3×C3⋊D3 ▷ C33
UCSC1 ◁ C22

Complements

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

NH ≅ G/Ntype
2. C2×#88949C2semidirect
3. C2×D32×Dic3C2semidirect
4. C2×#88939C2semidirect
5. C2×#88934C2semidirect
6. C2×C3×D32⋊C4C2semidirect
7. C2×Dic3×#157C2semidirect
8. (D32⋊D3)⋊C4C2direct (N × H)
14. C2×D32⋊D3C4semidirect
24. #88949C22semidirect
26. D32×Dic3C22semidirect
27. #88939C22semidirect
28. #88934C22semidirect
29. C3×D32⋊C4C22semidirect
30. Dic3×#157C22semidirect
39. C2×D32⋊C4D3semidirect
46. C2×#1113C2×C4semidirect
50. C2×C3×D32C2×C4semidirect
52. C2×C3⋊#157C2×C4semidirect
56. D32⋊D3C2×C4semidirect
57. C2×C3⋊(C3⋊Dic3)D4semidirect
60. C2×C32×Dic3D4semidirect
64. Dic3×C3⋊D3D4semidirect
112. D32⋊C4C2×D3semidirect
124. #1113C22×C4semidirect
127. C3×D32C22×C4semidirect
128. C3⋊#157C22×C4semidirect
130. C3⋊(C3⋊Dic3)C2×D4semidirect
133. C32×Dic3C2×D4semidirect
169. C2×D32C4×D3semidirect
221. C2×C33C4×D4semidirect
224. C3×C3⋊D3C4×D4semidirect
313. D32C2×C4×D3semidirect
340. C33C2×C4×D4semidirect
401. C2×Dic3GO+(4,𝔽2)semidirect
426. C2×C32#67⋊C4semidirect
429. C3⋊D3#67⋊C4semidirect
527. Dic3C2×GO+(4,𝔽2)semidirect
532. C32C2×#67⋊C4semidirect
578. C2×C3C4×GO+(4,𝔽2)semidirect
615. C3C2×C4×GO+(4,𝔽2)semidirect
621. C2(D32⋊D3)⋊C4direct (N × H)

Maximal subgroups

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

subgroupindex
C22×D32⋊D32
C2×#889492
C2×D32×Dic32
C2×#889392
C2×#889342
C2×C3×D32⋊C42
C2×Dic3×#1572
(D32⋊D3)⋊C42
C2×C4×GO+(4,𝔽2)3
C2×#67⋊C49