Finite Group Databasebeta

← C2×#11722398 #13164511

Subgroups — C2×#11722398 #13164511

Order 1728.

1360 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 1360 conjugacy classes of subgroups — too many to draw. Only its 91 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
1C23×C22⋊H(3,𝔽3)C211yesmaximal, characteristic, complemented
2C22×#6258C211yesmaximal, characteristic, complemented
3#11722398C214yesmaximal, direct product factor
4C2×#88613C211yesmaximal, characteristic, complemented
5C2×C3×GL(2,ℤ/4)—33nomaximal
6C2×C3×#953—31nomaximal
7C22×C22⋊H(3,𝔽3)—22no—
8C22×C22⋊H(3,𝔽3)C2211yescharacteristic
9C22×C22⋊H(3,𝔽3)C2212yescomplemented
10C2×#6258—22no—
11C2×#6258C2212yescomplemented
12C2×#6235—41nomaximal
13#88613C2212yescomplemented
14C25×C32D311yescharacteristic, complemented
15C23×C3×A4D313yescomplemented
16C22×C3×S4—33no—
17C22×C3×#67—31no—
18C22×C3×#67—31no—
19C22×C3×#67—31no—
20C3×GL(2,ℤ/4)—312no—
21C2×C3×SL(2,ℤ/4)—33no—
22C3×#953—34no—
23C3×#953—34no—
24C2×C3×#217—31no—
25C2×C3×#217—31no—
26C2×C3×#217—31no—
27C23×H(3,𝔽3)—41no—
28C22×#272—41no—
29C2×C22⋊H(3,𝔽3)—22no—
30C2×C22⋊H(3,𝔽3)D412yescomplemented
31C2×C22⋊H(3,𝔽3)C2311yescharacteristic
32#6258—22no—
33#6235—44no—
34C2×#1084—41no—
35C2×GL(2,ℤ/4)—33no—
36C2×C3×#120—91no—
37C2×#953—31no—
38C24×C32—22no—
39C24×C32—31no—
40C24×C32—31no—
41C24×C32—32no—
42C24×C32—62no—
43C24×C32C2×D311yescharacteristic
44C24×C32C2×D312yescomplemented
45C23×C3×D3—31no—
46C22×C3×A4—26no—
47C22×C3×A4C2×D313yes—
48C22×C3×A4C2×D316yescomplemented
49C2×C3×S4—36no—
50C2×C3×S4—66no—
51C2×C3×#67—32no—
52C2×C3×#67—34no—
53C2×C3×#67—34no—
54C2×C3×#67—32no—
55C2×C3×#67—62no—
56C2×C3×#67—64no—
57C2×C3×#67—62no—
58C2×C3×#67—61no—
59C2×C3×#67—62no—
60C2×C3×#67—61no—
61C2×C3×#67—123no—
62C22×C3×Dic3—31no—
63C22×C3×Dic3—31no—
64C22×C3×Dic3—31no—
65C3×SL(2,ℤ/4)—36no—
66C3×#217—32no—
67C3×#217—34no—
68C3×#217—34no—
69C3×#217—32no—
70C22×H(3,𝔽3)—41no—
71C22×H(3,𝔽3)—42no—
72C22×H(3,𝔽3)—82no—
73C2×#272—42no—
74C2×#272—82no—
75C22⋊H(3,𝔽3)C2×D411yescharacteristic, complemented
76#1084—42no—
77C25×C3—31no—
78C25×C3C3⋊D311yescharacteristic
79C23×A4—33no—
80C22×S4—33no—
81C22×C3×D4—91no—
82C22×C3×D4—91no—
83C22×C3×D4—91no—
84C22×#67—31no—
85C22×#67—31no—
86C22×#67—31no—
87GL(2,ℤ/4)—312no—
88C2×SL(2,ℤ/4)—33no—
89C3×#120—94no—
90C3×#120—94no—
91C2×C3×C22⋊C4—91no—
92C2×C3×C22⋊C4—91no—
93C2×C3×C22⋊C4—91no—
94#953—34no—
95#953—34no—
96C2×#217—31no—
97C2×#217—31no—
98C2×#217—31no—
99C23×C32—22no—
100C23×C32—31no—
101C23×C32—31no—
102C23×C32—32no—
103C23×C32—32no—
104C23×C32—34no—
105C23×C32—31no—
106C23×C32—32no—
107C23×C32—32no—
108C23×C32—43no—
109C23×C32—62no—
110C23×C32—62no—
111C23×C32—62no—
112C23×C32—61no—
113C23×C32—64no—
114C23×C32—62no—
115C23×C32—64no—
116C23×C32#6712yescomplemented
117C23×C32S411yescharacteristic, complemented
118C23×C32C22×D311yescharacteristic
119C22×C3×D3—32no—
120C22×C3×D3—62no—
121C22×C3×D3—62no—
122C22×C3×D3—61no—
123C22×C3×D3—61no—
124C22×C3×D3—123no—
125C2×C3×A4—26no—
126C2×C3×A4#6716yescomplemented
127C2×C3×A4C22×D313yes—
128C3×S4—66no—
129C3×#67—64no—
130C3×#67—62no—
131C3×#67—68no—
132C3×#67—64no—
133C3×#67—62no—
134C3×#67—64no—
135C3×#67—1212no—
136C2×C3×Dic3—32no—
137C2×C3×Dic3—32no—
138C2×C3×Dic3—32no—
139C2×C3×Dic3—62no—
140C2×C3×Dic3—62no—
141C2×C3×Dic3—61no—
142C2×C3×Dic3—61no—
143C2×C3×Dic3—123no—
144C2×#120—91noSylow
145C2×H(3,𝔽3)—42no—
146C2×H(3,𝔽3)—41no—
147C2×H(3,𝔽3)—82no—
148#272—82no—
149C24×C3—22no—
150C24×C3—31no—
151C24×C3—31no—
152C24×C3—31no—
153C24×C3—31no—
154C24×C3—32no—
155C24×C3—32no—
156C24×C3—31no—
157C24×C3—32no—
158C24×C3—62no—
159C24×C3—61no—
160C24×C3—61no—
161C24×C3—62no—
162C24×C3—62no—
163C24×C3—64no—
164C24×C3—62no—
165C24×C3—91no—
166C24×C3C2×C3⋊D311yescharacteristic
167C24×C3C2×C3⋊D312yes—
168C23×D3—31no—
169C22×A4—33no—
170C22×A4—36no—
171C22×A4—66no—
172C2×S4—36no—
173C2×S4—66no—
174C2×C3×D4—92no—
175C2×C3×D4—94no—
176C2×C3×D4—94no—
177C2×C3×D4—92no—
178C2×C3×D4—182no—
179C2×C3×D4—184no—
180C2×C3×D4—182no—
181C2×C3×D4—181no—
182C2×C3×D4—182no—
183C2×C3×D4—181no—
184C22×C3×C4—91no—
185C22×C3×C4—91no—
186C22×C3×C4—91no—
187C2×#67—32no—
188C2×#67—34no—
189C2×#67—34no—
190C2×#67—32no—
191C2×#67—62no—
192C2×#67—64no—
193C2×#67—62no—
194C2×#67—61no—
195C2×#67—62no—
196C2×#67—61no—
197C2×#67—123no—
198C22×Dic3—31no—
199C22×Dic3—31no—
200C22×Dic3—31no—
201SL(2,ℤ/4)—36no—
202C3×C22⋊C4—94no—
203C3×C22⋊C4—92no—
204C3×C22⋊C4—94no—
205C3×C22⋊C4—92no—
206#217—32no—
207#217—34no—
208#217—34no—
209#217—32no—
210C22×C32—22no—
211C22×C32—32no—
212C22×C32—31no—
213C22×C32—32no—
214C22×C32—32no—
215C22×C32—31no—
216C22×C32—31no—
217C22×C32—34no—
218C22×C32—32no—
219C22×C32—46no—
220C22×C32—43no—
221C22×C32—62no—
222C22×C32—62no—
223C22×C32—64no—
224C22×C32—61no—
225C22×C32—62no—
226C22×C32—64no—
227C22×C32—62no—
228C22×C32—86no—
229C22×C32C2×#6711yescharacteristic, complemented
230C22×C32C2×S411yescharacteristic
231C22×C32C2×S412yescomplemented
232C2×C3×D3—62no—
233C2×C3×D3—62no—
234C2×C3×D3—62no—
235C2×C3×D3—126no—
236C2×C3×D3—122no—
237C2×C3×D3—122no—
238C2×C3×D3—246no—
239C3×A4C2×#6713yescomplemented
240C3×Dic3—62no—
241C3×Dic3—62no—
242C3×Dic3—62no—
243C3×Dic3—126no—
244C25#27211yescharacteristic, complemented
245C22×D4—91no—
246C22×D4—91no—
247C22×D4—91no—
248#120—94no—
249#120—94no—
250C2×C22⋊C4—91no—
251C2×C22⋊C4—91no—
252C2×C22⋊C4—91no—
253H(3,𝔽3)—41noSylow
254C23×C3—22no—
255C23×C3—31no—
256C23×C3—31no—
257C23×C3—31no—
258C23×C3—32no—
259C23×C3—32no—
260C23×C3—31no—
261C23×C3—32no—
262C23×C3—32no—
263C23×C3—32no—
264C23×C3—34no—
265C23×C3—31no—
266C23×C3—32no—
267C23×C3—34no—
268C23×C3—32no—
269C23×C3—32no—
270C23×C3—31no—
271C23×C3—32no—
272C23×C3—31no—
273C23×C3—31no—
274C23×C3—62no—
275C23×C3—61no—
276C23×C3—61no—
277C23×C3—62no—
278C23×C3—62no—
279C23×C3—61no—
280C23×C3—62no—
281C23×C3—62no—
282C23×C3—64no—
283C23×C3—62no—
284C23×C3—62no—
285C23×C3—62no—
286C23×C3—61no—
287C23×C3—64no—
288C23×C3—62no—
289C23×C3—62no—
290C23×C3—62no—
291C23×C3—64no—
292C23×C3—62no—
293C23×C3—61no—
294C23×C3—62no—
295C23×C3—62no—
296C23×C3—64no—
297C23×C3—64no—
298C23×C3—64no—
299C23×C3—64no—
300C23×C3—64no—
301C23×C3—68no—
302C23×C3—64no—
303C23×C3—68no—
304C23×C3—92no—
305C23×C3—123no—
306C23×C3—182no—
307C23×C3—182no—
308C23×C3—181no—
309C23×C3—181no—
310C23×C3#63812yes—
311C23×C3C3⋊S411yescharacteristic
312C23×C3C22×C3⋊D311yescharacteristic
313C22×D3—32no—
314C22×D3—61no—
315C22×D3—62no—
316C22×D3—62no—
317C22×D3—61no—
318C22×D3—123no—
319C2×A4—36no—
320C2×A4—33no—
321C2×A4—66no—
322S4—66no—
323C3×D4—184no—
324C3×D4—182no—
325C3×D4—188no—
326C3×D4—184no—
327C3×D4—184no—
328C3×D4—182no—
329C2×C3×C4—92no—
330C2×C3×C4—92no—
331C2×C3×C4—92no—
332C2×C3×C4—182no—
333C2×C3×C4—182no—
334C2×C3×C4—181no—
335C2×C3×C4—181no—
336#67—64no—
337#67—62no—
338#67—68no—
339#67—64no—
340#67—64no—
341#67—62no—
342#67—1212no—
343C2×Dic3—32no—
344C2×Dic3—32no—
345C2×Dic3—32no—
346C2×Dic3—62no—
347C2×Dic3—62no—
348C2×Dic3—61no—
349C2×Dic3—61no—
350C2×Dic3—123no—
351C2×C32—22no—
352C2×C32—31no—
353C2×C32—31no—
354C2×C32—32no—
355C2×C32—46no—
356C2×C32—43no—
357C2×C32—62no—
358C2×C32—86no—
359C2×C32GL(2,ℤ/4)12yescomplemented
360C2×C32C22×S411yescharacteristic
361C3×D3—122no—
362C3×D3—246no—
363C24—22no—
364C24—31no—
365C24—31no—
366C24—32no—
367C24—62no—
368C24—91no—
369C24C2×#27211yescharacteristic
370C24C2×#27212yescomplemented
371C2×D4—92no—
372C2×D4—94no—
373C2×D4—94no—
374C2×D4—92no—
375C2×D4—182no—
376C2×D4—184no—
377C2×D4—182no—
378C2×D4—181no—
379C2×D4—182no—
380C2×D4—181no—
381C22×C4—91no—
382C22×C4—91no—
383C22×C4—91no—
384C22⋊C4—92no—
385C22⋊C4—94no—
386C22⋊C4—94no—
387C22⋊C4—92no—
388C22×C3—22no—
389C22×C3—32no—
390C22×C3—31no—
391C22×C3—31no—
392C22×C3—31no—
393C22×C3—32no—
394C22×C3—32no—
395C22×C3—32no—
396C22×C3—32no—
397C22×C3—31no—
398C22×C3—31no—
399C22×C3—32no—
400C22×C3—31no—
401C22×C3—34no—
402C22×C3—32no—
403C22×C3—34no—
404C22×C3—32no—
405C22×C3—31no—
406C22×C3—32no—
407C22×C3—31no—
408C22×C3—62no—
409C22×C3—62no—
410C22×C3—61no—
411C22×C3—61no—
412C22×C3—61no—
413C22×C3—62no—
414C22×C3—62no—
415C22×C3—62no—
416C22×C3—64no—
417C22×C3—62no—
418C22×C3—62no—
419C22×C3—64no—
420C22×C3—61no—
421C22×C3—62no—
422C22×C3—62no—
423C22×C3—62no—
424C22×C3—64no—
425C22×C3—61no—
426C22×C3—62no—
427C22×C3—62no—
428C22×C3—64no—
429C22×C3—64no—
430C22×C3—64no—
431C22×C3—62no—
432C22×C3—64no—
433C22×C3—64no—
434C22×C3—68no—
435C22×C3—64no—
436C22×C3—68no—
437C22×C3—62no—
438C22×C3—123no—
439C22×C3—126no—
440C22×C3—182no—
441C22×C3—182no—
442C22×C3—182no—
443C22×C3—246no—
444C22×C3—362no—
445C22×C3—362no—
446C22×C3C2×#63811yescharacteristic
447C22×C3C2×C3⋊S412yes—
448C22×C3C2×C3⋊S411yescharacteristic
449C2×D3—62no—
450C2×D3—62no—
451C2×D3—62no—
452C2×D3—126no—
453C2×D3—122no—
454C2×D3—122no—
455C2×D3—246no—
456A4—33no—
457C3×C4—182no—
458C3×C4—182no—
459C3×C4—182no—
460Dic3—62no—
461Dic3—62no—
462Dic3—62no—
463Dic3—126no—
464C32—43no—
465C32C2×GL(2,ℤ/4)11yescharacteristic, complemented
466C23—22no—
467C23—31no—
468C23—31no—
469C23—32no—
470C23—32no—
471C23—34no—
472C23—31no—
473C23—32no—
474C23—32no—
475C23—62no—
476C23—62no—
477C23—62no—
478C23—61no—
479C23—64no—
480C23—62no—
481C23—64no—
482C23—92no—
483C23—181no—
484C23—182no—
485C23—182no—
486C23—181no—
487C23#623512yescomplemented
488C23#625811yescharacteristic, complemented
489C23C22×#27211yescharacteristic
490D4—184no—
491D4—182no—
492D4—188no—
493D4—184no—
494D4—184no—
495D4—182no—
496C2×C4—92no—
497C2×C4—92no—
498C2×C4—92no—
499C2×C4—182no—
500C2×C4—182no—
501C2×C4—181no—
502C2×C4—181no—
503C2×C3—22no—
504C2×C3—31no—
505C2×C3—31no—
506C2×C3—32no—
507C2×C3—31no—
508C2×C3—31no—
509C2×C3—32no—
510C2×C3—31no—
511C2×C3—32no—
512C2×C3—62no—
513C2×C3—61no—
514C2×C3—61no—
515C2×C3—62no—
516C2×C3—64no—
517C2×C3—62no—
518C2×C3—62no—
519C2×C3—123no—
520C2×C3—126no—
521C2×C3—246no—
522C2×C3—362no—
523C2×C3#6426912yes—
524C2×C3C22×C3⋊S411yescharacteristic
525D3—122no—
526D3—246no—
527C22—22no—
528C22—32no—
529C22—31no—
530C22—31no—
531C22—31no—
532C22—32no—
533C22—32no—
534C22—34no—
535C22—32no—
536C22—62no—
537C22—62no—
538C22—64no—
539C22—61no—
540C22—62no—
541C22—64no—
542C22—62no—
543C22—182no—
544C22—182no—
545C22—182no—
546C22—362no—
547C22—362no—
548C22C2×#623511yescharacteristic, complemented
549C22C2×#625811yescharacteristic
550C22C2×#625812yescomplemented
551C4—182no—
552C4—182no—
553C4—182no—
554C3—31no—
555C3—123no—
556C3C2×#6426911yescharacteristic
557C2—22no—
558C2—31no—
559C2—31no—
560C2—32no—
561C2—62no—
562C2—362no—
563C2#1172239812yesdirect product factor
564C2C22×#625811yescharacteristic
subgroupG/H
centerC22×C3C2×C3⋊S4
commutator, cosocleC2×C22⋊H(3,𝔽3)C23
FrattiniC2×C3C22×C3⋊S4
FittingC25×C32D3
socleC24×C3C2×C3⋊D3
hypercenterC23×C3C3⋊S4
hypocenterC22⋊H(3,𝔽3)C2×D4
LCSG ▷ C2×C22⋊H(3,𝔽3) ▷ C22⋊H(3,𝔽3)
UCSC1 ◁ C22×C3 ◁ C23×C3

Complements

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

NH ≅ G/Ntype
1. C23×C22⋊H(3,𝔽3)C2semidirect
2. C22×#6258C2semidirect
3. #11722398C2direct (N × H)
4. C2×#88613C2semidirect
9. C22×C22⋊H(3,𝔽3)C22semidirect
11. C2×#6258C22semidirect
13. #88613C22semidirect
14. C25×C32D3semidirect
15. C23×C3×A4D3semidirect
30. C2×C22⋊H(3,𝔽3)D4semidirect
44. C24×C32C2×D3semidirect
48. C22×C3×A4C2×D3semidirect
75. C22⋊H(3,𝔽3)C2×D4semidirect
116. C23×C32#67semidirect
117. C23×C32S4semidirect
126. C2×C3×A4#67semidirect
229. C22×C32C2×#67semidirect
231. C22×C32C2×S4semidirect
239. C3×A4C2×#67semidirect
244. C25#272semidirect
359. C2×C32GL(2,ℤ/4)semidirect
370. C24C2×#272semidirect
465. C32C2×GL(2,ℤ/4)semidirect
487. C23#6235semidirect
488. C23#6258semidirect
548. C22C2×#6235semidirect
550. C22C2×#6258semidirect
563. C2#11722398direct (N × H)

Maximal subgroups

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

subgroupindex
C23×C22⋊H(3,𝔽3)2
C22×#62582
#117223982
C2×#886132
C2×C3×GL(2,ℤ/4)3
C2×C3×#9533
C2×#62354