Finite Group Databasebeta

← #10596575⋊C3 #13164397

Subgroups — #10596575⋊C3 #13164397

Order 1728.

710 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 710 conjugacy classes of subgroups — too many to draw. Only its 70 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
1C2×C4×C22⋊H(3,𝔽3)C211yesmaximal, characteristic, complemented
2#64269⋊C3C212yesmaximal, complemented
3C4⋊((C3⋊S4)⋊C3)C211yesmaximal, characteristic, complemented
4C4×(C3⋊S4)⋊C3C212yesmaximal, complemented
5#64259⋊C3C211yesmaximal, characteristic, complemented
6#10596575C311yesmaximal, characteristic, complemented
7C3×#5791—31nomaximal
8C22×C22⋊H(3,𝔽3)C2211yescharacteristic
9C2×(C3⋊S4)⋊C3C2212yes—
10#3828⋊C3—41nomaximal
11C4×C22⋊H(3,𝔽3)C2211yescharacteristic
12C4×C22⋊H(3,𝔽3)C2211yescharacteristic
13(C3⋊SL(2,ℤ/4))⋊C3C2212yes—
14C23×C32×C4D311yescharacteristic, complemented
15C2×C3×#235—31no—
16C2×C3×C4×A4—21no—
17C2×C3×C4×A4C2×C311yescharacteristic, complemented
18#64269C2×C312yescomplemented
19C4⋊(C3⋊S4)C2×C311yescharacteristic, complemented
20C4×C3⋊S4C2×C312yescomplemented
21#64259C2×C311yescharacteristic, complemented
22C3×#953—32no—
23C3×#930—31no—
24C3×#930—31no—
25C3×#929—32no—
26C3×C4×#67—32no—
27C3×C4×#67—32no—
28C3×#926—31no—
29C3×#924—31no—
30C3×#924—31no—
31C2×C22⋊H(3,𝔽3)—21no—
32C2×C22⋊H(3,𝔽3)C2311yescharacteristic
33(C3⋊S4)⋊C3—22no—
34C2×C4×H(3,𝔽3)—41no—
35#638⋊C3—42no—
36C4⋊((C3⋊D3)⋊C3)—41no—
37C4×(C3⋊D3)⋊C3—42no—
38#634⋊C3—41no—
39#5912—91no—
40C3×#525—91no—
41#5791—31no—
42C24×C32C2×D311yescharacteristic
43C22×C3×A4—21no—
44C22×C3×A4C22×C311yescharacteristic
45C2×C3⋊S4C22×C312yes—
46C22×C32×C4—31no—
47C22×C32×C4—31no—
48C22×C32×C4—31no—
49C22×C32×C4—31no—
50C22×C32×C4C2×D311yescharacteristic
51C22×C32×C4C2×D311yescharacteristic
52#3828—41no—
53C2×C3×#67—32no—
54C2×C3×#67—32no—
55C2×C3×#67—32no—
56C3×#235—61no—
57C3×#235—61no—
58C3×#235—61no—
59C3×#235—61no—
60C2×C3×D12—31no—
61C2×C3×C4×D3—32no—
62C2×C3×Dic6—31no—
63C3×C4×A4—21no—
64C3×C4×A4—21no—
65C3×C4×A4C22×C311yescharacteristic
66C3×C4×A4C22×C311yescharacteristic
67C3⋊SL(2,ℤ/4)C22×C312yes—
68C3×#217—32no—
69C3×#217—32no—
70C3×#217—32no—
71C3×(C2×D3)⋊C4—32no—
72C3×(C2×D3)⋊C4—32no—
73C3×C4⋊Dic3—31no—
74C3×C4⋊Dic3—31no—
75C3×Dic3⋊C4—32no—
76C3×Dic3⋊C4—32no—
77C3×C4×Dic3—32no—
78C22×H(3,𝔽3)—41no—
79C2×(C3⋊D3)⋊C3—42no—
80C22⋊H(3,𝔽3)#4111yescharacteristic, complemented
81C4×H(3,𝔽3)—41no—
82C4×H(3,𝔽3)—41no—
83(C3⋊Dic3)⋊C3—42no—
84C2×C3×#41—91no—
85C23×C3×C4—31no—
86C23×C3×C4C3×D311yescharacteristic
87C2×#235—31no—
88C2×C4×A4—31no—
89C2×C4×A4—61no—
90GL(2,ℤ/4)—92no—
91C4⋊S4—91no—
92C4×S4—92no—
93A4⋊Q8—91no—
94C3×#123—92no—
95C3×C22⋊Q8—91no—
96C3×C22⋊Q8—91no—
97C3×#121—91no—
98C3×#121—91no—
99C3×#120—92no—
100C3×C4×D4—92no—
101C3×C4×D4—92no—
102C3×#117—91no—
103#953—32no—
104#930—31no—
105#930—31no—
106#929—32no—
107C4×#67—32no—
108C4×#67—32no—
109#926—31no—
110#924—31no—
111#924—31no—
112C23×C32—21no—
113C23×C32—31no—
114C23×C32—31no—
115C23×C32—61no—
116C23×C32C22×D311yescharacteristic
117C22×C3×D3—32no—
118C2×C3×A4—22no—
119C2×C3×A4—21no—
120C2×C3×A4—21no—
121C2×C3×A4C23×C311yescharacteristic
122C3⋊S4—22no—
123C2×C32×C4—31no—
124C2×C32×C4—31no—
125C2×C32×C4—31no—
126C2×C32×C4—31no—
127C2×C32×C4—31no—
128C2×C32×C4—31no—
129C2×C32×C4—31no—
130C2×C32×C4—41no—
131C2×C32×C4—61no—
132C2×C32×C4—81no—
133C2×C32×C4S411yescharacteristic, complemented
134#638—42no—
135C4⋊(C3⋊D3)—41no—
136C4×C3⋊D3—42no—
137#634—41no—
138C3×#67—62no—
139C3×#67—62no—
140C3×#67—64no—
141C3×#67—62no—
142C3×#67—62no—
143C2×C3×Dic3—32no—
144C2×C3×Dic3—32no—
145C2×C3×Dic3—32no—
146C3×D12—61no—
147C3×D12—61no—
148C3×C4×D3—62no—
149C3×C4×D3—62no—
150C3×Dic6—61no—
151C3×Dic6—61no—
152#525—91noSylow
153C2×H(3,𝔽3)—41no—
154C2×H(3,𝔽3)—81no—
155(C3⋊D3)⋊C3—82no—
156C24×C3—31no—
157C24×C3C2×C3×D311yescharacteristic
158C22×A4—31no—
159C22×A4—61no—
160C2×S4—92no—
161C3×#41—181no—
162C3×#41—181no—
163C3×#41—181no—
164C3×#41—181no—
165C2×C3×Q8—91no—
166C2×C3×D4—91no—
167C2×C3×D4—92no—
168C2×C3×D4—92no—
169C2×C3×D4—92no—
170C22×C3×C4—31no—
171C22×C3×C4—31no—
172C22×C3×C4—31no—
173C22×C3×C4—31no—
174C22×C3×C4—31no—
175C22×C3×C4—31no—
176C22×C3×C4—31no—
177C22×C3×C4—31no—
178C22×C3×C4—31no—
179C22×C3×C4—31no—
180C22×C3×C4—61no—
181C22×C3×C4—61no—
182C22×C3×C4—61no—
183C22×C3×C4—61no—
184C22×C3×C4—92no—
185C22×C3×C4C2×C3×D311yescharacteristic
186C22×C3×C4C2×C3×D311yescharacteristic
187C2×#67—32no—
188C2×#67—32no—
189C2×#67—32no—
190#235—61no—
191#235—61no—
192#235—61no—
193#235—61no—
194#235—361no—
195C2×D12—31no—
196C2×C4×D3—32no—
197C2×Dic6—31no—
198C4×A4—31no—
199C4×A4—31no—
200C4×A4—61no—
201C4×A4—61no—
202SL(2,ℤ/4)—92no—
203C3×C4⋊C4—92no—
204C3×C4⋊C4—92no—
205C3×C4⋊C4—91no—
206C3×C4⋊C4—91no—
207C3×C22⋊C4—92no—
208C3×C22⋊C4—92no—
209C3×C22⋊C4—92no—
210C3×C22⋊C4—92no—
211C3×C22⋊C4—92no—
212C3×C42—92no—
213#217—32no—
214#217—32no—
215#217—32no—
216(C2×D3)⋊C4—32no—
217(C2×D3)⋊C4—32no—
218C4⋊Dic3—31no—
219C4⋊Dic3—31no—
220Dic3⋊C4—32no—
221Dic3⋊C4—32no—
222C4×Dic3—32no—
223C22×C32—31no—
224C22×C32—31no—
225C22×C32—32no—
226C22×C32—31no—
227C22×C32—41no—
228C22×C32—61no—
229C22×C32—61no—
230C22×C32—61no—
231C22×C32—81no—
232C22×C32#23511yescharacteristic, complemented
233C22×C32C2×S411yescharacteristic
234C2×C3⋊D3—42no—
235C2×C3×D3—62no—
236C2×C3×D3—62no—
237C2×C3×D3—62no—
238C3×A4—21no—
239C3×A4C3×#4111yescharacteristic, complemented
240C32×C4—31no—
241C32×C4—31no—
242C32×C4—41no—
243C32×C4—41no—
244C32×C4—81no—
245C32×C4—81no—
246C32×C4C2×S411yescharacteristic
247C32×C4C2×S411yescharacteristic
248C3⋊Dic3—42no—
249C3×Dic3—62no—
250C3×Dic3—62no—
251C3×Dic3—62no—
252C2×#41—91no—
253C23×C4(C3⋊D3)⋊C311yescharacteristic, complemented
254#123—92no—
255C22⋊Q8—91no—
256C22⋊Q8—91no—
257#121—91no—
258#121—91no—
259#120—92no—
260C4×D4—92no—
261C4×D4—92no—
262#117—91no—
263H(3,𝔽3)—41noSylow
264C23×C3—21no—
265C23×C3—31no—
266C23×C3—31no—
267C23×C3—31no—
268C23×C3—31no—
269C23×C3—31no—
270C23×C3—61no—
271C23×C3—61no—
272C23×C3—61no—
273C23×C3—61no—
274C23×C3—62no—
275C23×C3—61no—
276C23×C3—92no—
277C23×C3C22×C3×D311yescharacteristic
278C22×D3—32no—
279C2×A4—31no—
280C2×A4—61no—
281C2×A4—62no—
282C2×A4—61no—
283S4—182no—
284C3×Q8—181no—
285C3×Q8—181no—
286C3×D4—181no—
287C3×D4—181no—
288C3×D4—182no—
289C3×D4—184no—
290C3×D4—182no—
291C3×D4—182no—
292C3×D4—182no—
293C2×C3×C4—31no—
294C2×C3×C4—31no—
295C2×C3×C4—31no—
296C2×C3×C4—31no—
297C2×C3×C4—31no—
298C2×C3×C4—31no—
299C2×C3×C4—31no—
300C2×C3×C4—31no—
301C2×C3×C4—31no—
302C2×C3×C4—31no—
303C2×C3×C4—31no—
304C2×C3×C4—31no—
305C2×C3×C4—31no—
306C2×C3×C4—31no—
307C2×C3×C4—31no—
308C2×C3×C4—61no—
309C2×C3×C4—61no—
310C2×C3×C4—61no—
311C2×C3×C4—61no—
312C2×C3×C4—61no—
313C2×C3×C4—61no—
314C2×C3×C4—61no—
315C2×C3×C4—61no—
316C2×C3×C4—61no—
317C2×C3×C4—61no—
318C2×C3×C4—61no—
319C2×C3×C4—92no—
320C2×C3×C4—92no—
321C2×C3×C4—92no—
322C2×C3×C4—121no—
323C2×C3×C4—182no—
324C2×C3×C4—182no—
325C2×C3×C4—241no—
326C2×C3×C4C3×S411yescharacteristic
327#67—62no—
328#67—62no—
329#67—64no—
330#67—62no—
331#67—62no—
332#67—362no—
333C2×Dic3—32no—
334C2×Dic3—32no—
335C2×Dic3—32no—
336D12—61no—
337D12—61no—
338D12—361no—
339C4×D3—62no—
340C4×D3—62no—
341C4×D3—362no—
342Dic6—61no—
343Dic6—61no—
344Dic6—361no—
345C2×C32—21no—
346C2×C32—31no—
347C2×C32—31no—
348C2×C32—41no—
349C2×C32—61no—
350C2×C32—81no—
351C2×C32—82no—
352C2×C32—81no—
353C2×C32C22×S411yescharacteristic
354C3⋊D3—82no—
355C3×D3—122no—
356C24C2×(C3⋊D3)⋊C311yescharacteristic
357#41—181no—
358#41—181no—
359#41—181no—
360#41—181no—
361C2×Q8—91no—
362C2×D4—91no—
363C2×D4—92no—
364C2×D4—92no—
365C2×D4—92no—
366C22×C4—31no—
367C22×C4—31no—
368C22×C4—31no—
369C22×C4—31no—
370C22×C4—92no—
371C22×C4C2×(C3⋊D3)⋊C311yescharacteristic
372C22×C4C2×(C3⋊D3)⋊C311yescharacteristic
373C4⋊C4—91no—
374C4⋊C4—91no—
375C4⋊C4—92no—
376C4⋊C4—92no—
377C22⋊C4—92no—
378C22⋊C4—92no—
379C22⋊C4—92no—
380C22⋊C4—92no—
381C22⋊C4—92no—
382C42—92no—
383C22×C3—31no—
384C22×C3—31no—
385C22×C3—31no—
386C22×C3—32no—
387C22×C3—31no—
388C22×C3—31no—
389C22×C3—32no—
390C22×C3—31no—
391C22×C3—31no—
392C22×C3—31no—
393C22×C3—61no—
394C22×C3—61no—
395C22×C3—61no—
396C22×C3—62no—
397C22×C3—61no—
398C22×C3—61no—
399C22×C3—61no—
400C22×C3—61no—
401C22×C3—61no—
402C22×C3—61no—
403C22×C3—62no—
404C22×C3—62no—
405C22×C3—62no—
406C22×C3—121no—
407C22×C3—182no—
408C22×C3—182no—
409C22×C3—182no—
410C22×C3—241no—
411C22×C3C3×#23511yescharacteristic
412C22×C3C2×C3×S411yescharacteristic
413C2×D3—62no—
414C2×D3—62no—
415C2×D3—62no—
416C2×D3—362no—
417A4—31no—
418A4—61no—
419C3×C4—31no—
420C3×C4—31no—
421C3×C4—31no—
422C3×C4—31no—
423C3×C4—31no—
424C3×C4—31no—
425C3×C4—61no—
426C3×C4—61no—
427C3×C4—121no—
428C3×C4—121no—
429C3×C4—182no—
430C3×C4—182no—
431C3×C4—182no—
432C3×C4—241no—
433C3×C4—241no—
434C3×C4C2×C3×S411yescharacteristic
435C3×C4C2×C3×S411yescharacteristic
436Dic3—62no—
437Dic3—62no—
438Dic3—62no—
439Dic3—362no—
440C32—41no—
441C32—81no—
442C32#591211yescharacteristic, complemented
443C23—21no—
444C23—31no—
445C23—31no—
446C23—61no—
447C23—92no—
448C23C22×(C3⋊D3)⋊C311yescharacteristic
449Q8—181no—
450Q8—181no—
451D4—181no—
452D4—182no—
453D4—182no—
454D4—184no—
455D4—182no—
456D4—182no—
457D4—181no—
458C2×C4—31no—
459C2×C4—31no—
460C2×C4—31no—
461C2×C4—31no—
462C2×C4—31no—
463C2×C4—31no—
464C2×C4—31no—
465C2×C4—61no—
466C2×C4—92no—
467C2×C4—92no—
468C2×C4—92no—
469C2×C4—182no—
470C2×C4—182no—
471C2×C4(C3⋊S4)⋊C311yescharacteristic, complemented
472C2×C3—21no—
473C2×C3—31no—
474C2×C3—31no—
475C2×C3—31no—
476C2×C3—31no—
477C2×C3—31no—
478C2×C3—61no—
479C2×C3—61no—
480C2×C3—61no—
481C2×C3—61no—
482C2×C3—62no—
483C2×C3—61no—
484C2×C3—121no—
485C2×C3—241no—
486C2×C3—242no—
487C2×C3—241no—
488C2×C3—362no—
489C2×C3C22×C3×S411yescharacteristic
490D3—122no—
491D3—722no—
492C22—31no—
493C22—31no—
494C22—32no—
495C22—31no—
496C22—61no—
497C22—61no—
498C22—61no—
499C22—182no—
500C22—182no—
501C22—182no—
502C22#3828⋊C311yescharacteristic, complemented
503C22C2×(C3⋊S4)⋊C311yescharacteristic
504C4—31no—
505C4—31no—
506C4—182no—
507C4—182no—
508C4—182no—
509C4C2×(C3⋊S4)⋊C311yescharacteristic
510C4C2×(C3⋊S4)⋊C311yescharacteristic
511C3—31no—
512C3—121no—
513C3—241no—
514C3C3×#591211yescharacteristic
515C2—21no—
516C2—31no—
517C2—31no—
518C2—61no—
519C2—362no—
520C2C22×(C3⋊S4)⋊C311yescharacteristic
subgroupG/H
centerC4C2×(C3⋊S4)⋊C3
commutator, cosocleC2×C3×A4C23×C3
FrattiniC2×C3C22×C3×S4
FittingC23×C32×C4D3
socleC23×C3C22×C3×D3
hypercenterC2×C4(C3⋊S4)⋊C3
hypocenterC3×A4C3×#41
LCSG ▷ C2×C3×A4 ▷ C3×A4
UCSC1 ◁ C4 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
1. C2×C4×C22⋊H(3,𝔽3)C2semidirect
2. #64269⋊C3C2semidirect
3. C4⋊((C3⋊S4)⋊C3)C2semidirect
4. C4×(C3⋊S4)⋊C3C2semidirect
5. #64259⋊C3C2semidirect
6. #10596575C3semidirect
14. C23×C32×C4D3semidirect
17. C2×C3×C4×A4C2×C3semidirect
18. #64269C2×C3semidirect
19. C4⋊(C3⋊S4)C2×C3semidirect
20. C4×C3⋊S4C2×C3semidirect
21. #64259C2×C3semidirect
80. C22⋊H(3,𝔽3)#41semidirect
133. C2×C32×C4S4semidirect
232. C22×C32#235semidirect
239. C3×A4C3×#41semidirect
253. C23×C4(C3⋊D3)⋊C3semidirect
442. C32#5912semidirect
471. C2×C4(C3⋊S4)⋊C3semidirect
502. C22#3828⋊C3semidirect

Maximal subgroups

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

subgroupindex
C2×C4×C22⋊H(3,𝔽3)2
#64269⋊C32
C4⋊((C3⋊S4)⋊C3)2
C4×(C3⋊S4)⋊C32
#64259⋊C32
#105965753
C3×#57913
#3828⋊C34