Finite Group Databasebeta

← #10596577⋊C3 #13164403

Subgroups — #10596577⋊C3 #13164403

Order 1728.

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

normal subgroups only

This group has 718 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
1D4×C22⋊H(3,𝔽3)C211yesmaximal, characteristic, complemented
2#64269⋊C3C212yesmaximal, complemented
3C2×(C3⋊SL(2,ℤ/4))⋊C3C212yesmaximal, complemented
4C4×(C3⋊S4)⋊C3C211yesmaximal, characteristic, complemented
5#64259⋊C3C211yesmaximal, characteristic, complemented
6#10596577C311yesmaximal, characteristic, complemented
7C3×#5806—31nomaximal
8C22×C22⋊H(3,𝔽3)C2212yes—
9C2×(C3⋊S4)⋊C3C2211yescharacteristic
10#3830⋊C3—41nomaximal
11C4×C22⋊H(3,𝔽3)C2211yescharacteristic
12(C3⋊SL(2,ℤ/4))⋊C3C2212yes—
13(C3⋊SL(2,ℤ/4))⋊C3C2211yescharacteristic
14C22×C32×D4D311yescharacteristic, complemented
15C2×C3×#237—31no—
16C3×D4×A4—21no—
17C3×D4×A4C2×C311yescharacteristic, complemented
18#64269C2×C312yescomplemented
19C2×C3⋊SL(2,ℤ/4)C2×C312yescomplemented
20C4×C3⋊S4C2×C311yescharacteristic, complemented
21#64259C2×C311yescharacteristic, complemented
22C3×#953—32no—
23C2×C3×#217—32no—
24C3×#939—32no—
25C3×#938—31no—
26C3×#936—31no—
27C3×#935—32no—
28C3×D4×Dic3—31no—
29C3×C4×#67—31no—
30C3×#924—31no—
31C2×C22⋊H(3,𝔽3)—22no—
32C2×C22⋊H(3,𝔽3)C2311yescharacteristic
33(C3⋊S4)⋊C3—21no—
34D4×H(3,𝔽3)—41no—
35#638⋊C3—42no—
36C2×(C3⋊Dic3)⋊C3—42no—
37C4×(C3⋊D3)⋊C3—41no—
38#634⋊C3—41no—
39#5914—91no—
40C3×#546—91no—
41#5806—31no—
42C24×C32C2×D312yes—
43C22×C3×A4—22no—
44C22×C3×A4C22×C312yes—
45C2×C3⋊S4C22×C311yescharacteristic
46C2×C32×D4—31no—
47C2×C32×D4—32no—
48C2×C32×D4—31no—
49C22×C32×C4C2×D311yescharacteristic
50#3830—41no—
51C2×C3×#67—32no—
52C2×C3×#67—31no—
53C2×C3×#67—32no—
54C22×C3×Dic3—32no—
55C22×C3×Dic3—32no—
56C3×#237—62no—
57C3×#237—61no—
58C3×#237—61no—
59C2×C3×C4×D3—31no—
60C2×C3×Dic6—31no—
61C3×C4×A4—21no—
62C3×C4×A4C22×C311yescharacteristic
63C3⋊SL(2,ℤ/4)C22×C312yes—
64C3⋊SL(2,ℤ/4)C22×C311yescharacteristic
65C3×#217—32no—
66C3×#217—32no—
67C3×#217—32no—
68C3×#217—32no—
69C3×#217—32no—
70C3×#217—31no—
71C3×(C2×D3)⋊C4—31no—
72C3×C4⋊Dic3—31no—
73C3×Dic3⋊C4—32no—
74C3×Dic3⋊C4—31no—
75C3×C4×Dic3—31no—
76C22×H(3,𝔽3)—42no—
77C2×(C3⋊D3)⋊C3—41no—
78C22⋊H(3,𝔽3)#4111yescharacteristic, complemented
79C4×H(3,𝔽3)—41no—
80(C3⋊Dic3)⋊C3—42no—
81(C3⋊Dic3)⋊C3—41no—
82C2×C3×#41—91no—
83C22×C3×D4—31no—
84C22×C3×D4C3×D311yescharacteristic
85C2×#237—31no—
86D4×A4—31no—
87D4×A4—61no—
88GL(2,ℤ/4)—92no—
89C2×SL(2,ℤ/4)—92no—
90C4×S4—91no—
91A4⋊Q8—91no—
92C3×#124—91no—
93C3×#123—92no—
94C3×C22⋊Q8—91no—
95C3×#121—91no—
96C3×#121—92no—
97C3×#120—92no—
98C3×C4×D4—91no—
99C3×C4×D4—91no—
100C2×C3×C22⋊C4—92no—
101#953—32no—
102C2×#217—32no—
103#939—32no—
104#938—31no—
105#936—31no—
106#935—32no—
107D4×Dic3—31no—
108C4×#67—31no—
109#924—31no—
110C23×C32—22no—
111C23×C32—32no—
112C23×C32—32no—
113C23×C32—62no—
114C23×C32C22×D311yescharacteristic
115C22×C3×D3—31no—
116C2×C3×A4—22no—
117C2×C3×A4—21no—
118C2×C3×A4—42no—
119C2×C3×A4C23×C311yescharacteristic
120C3⋊S4—21no—
121C32×D4—32no—
122C32×D4—31no—
123C32×D4—41no—
124C32×D4—61no—
125C32×D4—81no—
126C32×D4S411yescharacteristic, complemented
127C2×C32×C4—31no—
128C2×C32×C4—31no—
129#638—42no—
130C2×C3⋊Dic3—42no—
131C4×C3⋊D3—41no—
132#634—41no—
133C3×#67—62no—
134C3×#67—61no—
135C3×#67—61no—
136C3×#67—62no—
137C3×#67—122no—
138C2×C3×Dic3—31no—
139C2×C3×Dic3—31no—
140C2×C3×Dic3—32no—
141C2×C3×Dic3—32no—
142C2×C3×Dic3—31no—
143C2×C3×Dic3—62no—
144C2×C3×Dic3—62no—
145C2×C3×Dic3—122no—
146C3×C4×D3—61no—
147C3×C4×D3—61no—
148C3×Dic6—61no—
149C3×Dic6—61no—
150#546—91noSylow
151C2×H(3,𝔽3)—41no—
152C2×H(3,𝔽3)—82no—
153(C3⋊D3)⋊C3—81no—
154C24×C3—32no—
155C24×C3C2×C3×D312yes—
156C22×A4—32no—
157C22×A4—62no—
158C2×S4—91no—
159C3×#41—182no—
160C3×#41—181no—
161C3×#41—181no—
162C2×C3×Q8—91no—
163C2×C3×D4—31no—
164C2×C3×D4—31no—
165C2×C3×D4—32no—
166C2×C3×D4—31no—
167C2×C3×D4—32no—
168C2×C3×D4—31no—
169C2×C3×D4—61no—
170C2×C3×D4—62no—
171C2×C3×D4—61no—
172C2×C3×D4—91no—
173C2×C3×D4—92no—
174C2×C3×D4—92no—
175C22×C3×C4—31no—
176C22×C3×C4—91no—
177C22×C3×C4—92no—
178C22×C3×C4—92no—
179C22×C3×C4C2×C3×D311yescharacteristic
180C2×#67—32no—
181C2×#67—31no—
182C2×#67—32no—
183C22×Dic3—32no—
184C22×Dic3—32no—
185#237—62no—
186#237—61no—
187#237—61no—
188#237—361no—
189C2×C4×D3—31no—
190C2×Dic6—31no—
191C4×A4—31no—
192C4×A4—61no—
193SL(2,ℤ/4)—92no—
194SL(2,ℤ/4)—91no—
195C3×C4⋊C4—91no—
196C3×C4⋊C4—92no—
197C3×C4⋊C4—91no—
198C3×C22⋊C4—91no—
199C3×C22⋊C4—91no—
200C3×C22⋊C4—92no—
201C3×C22⋊C4—92no—
202C3×C22⋊C4—92no—
203C3×C22⋊C4—92no—
204C3×C22⋊C4—92no—
205C3×C42—91no—
206#217—32no—
207#217—32no—
208#217—32no—
209#217—32no—
210#217—32no—
211#217—31no—
212(C2×D3)⋊C4—31no—
213C4⋊Dic3—31no—
214Dic3⋊C4—32no—
215Dic3⋊C4—31no—
216C4×Dic3—31no—
217C22×C32—31no—
218C22×C32—31no—
219C22×C32—32no—
220C22×C32—42no—
221C22×C32—62no—
222C22×C32—62no—
223C22×C32—62no—
224C22×C32—62no—
225C22×C32—82no—
226C22×C32#23711yescharacteristic, complemented
227C22×C32C2×S412yes—
228C2×C3⋊D3—41no—
229C2×C3×D3—61no—
230C2×C3×D3—61no—
231C2×C3×D3—61no—
232C3×A4—21no—
233C3×A4C3×#4111yescharacteristic, complemented
234C32×C4—31no—
235C32×C4—41no—
236C32×C4—81no—
237C32×C4C2×S411yescharacteristic
238C3⋊Dic3—42no—
239C3⋊Dic3—41no—
240C3×Dic3—61no—
241C3×Dic3—62no—
242C3×Dic3—61no—
243C3×Dic3—62no—
244C3×Dic3—61no—
245C2×#41—91no—
246C22×D4(C3⋊D3)⋊C311yescharacteristic, complemented
247#124—91no—
248#123—92no—
249C22⋊Q8—91no—
250#121—91no—
251#121—92no—
252#120—92no—
253C4×D4—91no—
254C4×D4—91no—
255C2×C22⋊C4—92no—
256H(3,𝔽3)—41noSylow
257C23×C3—22no—
258C23×C3—32no—
259C23×C3—32no—
260C23×C3—32no—
261C23×C3—32no—
262C23×C3—31no—
263C23×C3—62no—
264C23×C3—62no—
265C23×C3—62no—
266C23×C3—62no—
267C23×C3—62no—
268C23×C3—91no—
269C23×C3—122no—
270C23×C3C22×C3×D311yescharacteristic
271C22×D3—31no—
272C2×A4—31no—
273C2×A4—62no—
274C2×A4—61no—
275C2×A4—122no—
276S4—181no—
277C3×Q8—181no—
278C3×Q8—181no—
279C3×D4—32no—
280C3×D4—31no—
281C3×D4—32no—
282C3×D4—31no—
283C3×D4—31no—
284C3×D4—61no—
285C3×D4—62no—
286C3×D4—61no—
287C3×D4—61no—
288C3×D4—62no—
289C3×D4—121no—
290C3×D4—181no—
291C3×D4—181no—
292C3×D4—182no—
293C3×D4—182no—
294C3×D4—241no—
295C3×D4—362no—
296C3×D4C3×S411yescharacteristic
297C2×C3×C4—31no—
298C2×C3×C4—31no—
299C2×C3×C4—31no—
300C2×C3×C4—31no—
301C2×C3×C4—61no—
302C2×C3×C4—61no—
303C2×C3×C4—91no—
304C2×C3×C4—91no—
305C2×C3×C4—92no—
306C2×C3×C4—91no—
307C2×C3×C4—92no—
308C2×C3×C4—181no—
309C2×C3×C4—181no—
310C2×C3×C4—182no—
311C2×C3×C4—182no—
312C2×C3×C4—362no—
313#67—62no—
314#67—61no—
315#67—62no—
316#67—61no—
317#67—122no—
318#67—362no—
319C2×Dic3—31no—
320C2×Dic3—31no—
321C2×Dic3—32no—
322C2×Dic3—32no—
323C2×Dic3—31no—
324C2×Dic3—62no—
325C2×Dic3—62no—
326C2×Dic3—122no—
327C2×Dic3—362no—
328C4×D3—61no—
329C4×D3—61no—
330C4×D3—361no—
331Dic6—61no—
332Dic6—61no—
333Dic6—361no—
334C2×C32—22no—
335C2×C32—31no—
336C2×C32—31no—
337C2×C32—41no—
338C2×C32—62no—
339C2×C32—82no—
340C2×C32—81no—
341C2×C32—162no—
342C2×C32C22×S411yescharacteristic
343C3⋊D3—81no—
344C3×D3—121no—
345C24C2×(C3⋊D3)⋊C312yes—
346#41—182no—
347#41—181no—
348#41—181no—
349C2×Q8—91no—
350C2×D4—31no—
351C2×D4—32no—
352C2×D4—31no—
353C2×D4—92no—
354C2×D4—91no—
355C2×D4—92no—
356C22×C4—91no—
357C22×C4—92no—
358C22×C4—92no—
359C22×C4C2×(C3⋊D3)⋊C311yescharacteristic
360C4⋊C4—91no—
361C4⋊C4—91no—
362C4⋊C4—92no—
363C22⋊C4—91no—
364C22⋊C4—92no—
365C22⋊C4—92no—
366C22⋊C4—92no—
367C22⋊C4—92no—
368C22⋊C4—91no—
369C22⋊C4—92no—
370C42—91no—
371C22×C3—31no—
372C22×C3—31no—
373C22×C3—31no—
374C22×C3—32no—
375C22×C3—31no—
376C22×C3—32no—
377C22×C3—32no—
378C22×C3—31no—
379C22×C3—61no—
380C22×C3—61no—
381C22×C3—62no—
382C22×C3—62no—
383C22×C3—62no—
384C22×C3—62no—
385C22×C3—62no—
386C22×C3—62no—
387C22×C3—62no—
388C22×C3—62no—
389C22×C3—62no—
390C22×C3—122no—
391C22×C3—122no—
392C22×C3—122no—
393C22×C3—122no—
394C22×C3—122no—
395C22×C3—181no—
396C22×C3—181no—
397C22×C3—181no—
398C22×C3—242no—
399C22×C3C3×#23711yescharacteristic
400C22×C3C2×C3×S412yes—
401C2×D3—61no—
402C2×D3—61no—
403C2×D3—61no—
404C2×D3—361no—
405A4—31no—
406A4—61no—
407C3×C4—31no—
408C3×C4—31no—
409C3×C4—31no—
410C3×C4—61no—
411C3×C4—121no—
412C3×C4—181no—
413C3×C4—182no—
414C3×C4—181no—
415C3×C4—182no—
416C3×C4—181no—
417C3×C4—241no—
418C3×C4C2×C3×S411yescharacteristic
419Dic3—61no—
420Dic3—62no—
421Dic3—61no—
422Dic3—62no—
423Dic3—61no—
424Dic3—362no—
425Dic3—361no—
426C32—41no—
427C32—81no—
428C32#591411yescharacteristic, complemented
429C23—22no—
430C23—32no—
431C23—32no—
432C23—62no—
433C23—91no—
434C23C22×(C3⋊D3)⋊C311yescharacteristic
435Q8—181no—
436Q8—181no—
437D4—32no—
438D4—31no—
439D4—61no—
440D4—182no—
441D4—181no—
442D4—182no—
443D4—181no—
444D4—362no—
445D4(C3⋊S4)⋊C311yescharacteristic, complemented
446C2×C4—31no—
447C2×C4—31no—
448C2×C4—91no—
449C2×C4—91no—
450C2×C4—92no—
451C2×C4—91no—
452C2×C4—92no—
453C2×C4—181no—
454C2×C4—182no—
455C2×C4—182no—
456C2×C4—181no—
457C2×C4—362no—
458C2×C3—22no—
459C2×C3—31no—
460C2×C3—31no—
461C2×C3—31no—
462C2×C3—31no—
463C2×C3—31no—
464C2×C3—62no—
465C2×C3—61no—
466C2×C3—61no—
467C2×C3—62no—
468C2×C3—62no—
469C2×C3—122no—
470C2×C3—121no—
471C2×C3—242no—
472C2×C3—241no—
473C2×C3—361no—
474C2×C3—482no—
475C2×C3C22×C3×S411yescharacteristic
476D3—121no—
477D3—721no—
478C22—31no—
479C22—31no—
480C22—32no—
481C22—62no—
482C22—62no—
483C22—62no—
484C22—62no—
485C22—181no—
486C22—181no—
487C22—181no—
488C22#3830⋊C311yescharacteristic, complemented
489C22C2×(C3⋊S4)⋊C312yes—
490C4—31no—
491C4—181no—
492C4—182no—
493C4—181no—
494C4—181no—
495C4—182no—
496C4C2×(C3⋊S4)⋊C311yescharacteristic
497C3—31no—
498C3—121no—
499C3—241no—
500C3C3×#591411yescharacteristic
501C2—22no—
502C2—31no—
503C2—31no—
504C2—62no—
505C2—361no—
506C2C22×(C3⋊S4)⋊C311yescharacteristic
subgroupG/H
centerC2C22×(C3⋊S4)⋊C3
commutator, cosocleC2×C3×A4C23×C3
FrattiniC2×C3C22×C3×S4
FittingC22×C32×D4D3
socleC23×C3C22×C3×D3
hypercenterD4(C3⋊S4)⋊C3
hypocenterC3×A4C3×#41
LCSG ▷ C2×C3×A4 ▷ C3×A4
UCSC1 ◁ C2 ◁ D4

Complements

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

NH ≅ G/Ntype
1. D4×C22⋊H(3,𝔽3)C2semidirect
2. #64269⋊C3C2semidirect
3. C2×(C3⋊SL(2,ℤ/4))⋊C3C2semidirect
4. C4×(C3⋊S4)⋊C3C2semidirect
5. #64259⋊C3C2semidirect
6. #10596577C3semidirect
14. C22×C32×D4D3semidirect
17. C3×D4×A4C2×C3semidirect
18. #64269C2×C3semidirect
19. C2×C3⋊SL(2,ℤ/4)C2×C3semidirect
20. C4×C3⋊S4C2×C3semidirect
21. #64259C2×C3semidirect
78. C22⋊H(3,𝔽3)#41semidirect
126. C32×D4S4semidirect
226. C22×C32#237semidirect
233. C3×A4C3×#41semidirect
246. C22×D4(C3⋊D3)⋊C3semidirect
428. C32#5914semidirect
445. D4(C3⋊S4)⋊C3semidirect
488. C22#3830⋊C3semidirect

Maximal subgroups

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

subgroupindex
D4×C22⋊H(3,𝔽3)2
#64269⋊C32
C2×(C3⋊SL(2,ℤ/4))⋊C32
C4×(C3⋊S4)⋊C32
#64259⋊C32
#105965773
C3×#58063
#3830⋊C34