Finite Group Databasebeta

← #10592433⋊C3 #13141618

Subgroups — #10592433⋊C3 #13141618

Order 1728.

617 conjugacy classes of subgroups; only the 100 normal subgroups are shown.

normal subgroups only

This group has 617 conjugacy classes of subgroups — too many to draw. Only its 100 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×#3828⋊C3C211yesmaximal, characteristic, complemented
2#64139⋊C3C211yesmaximal, characteristic, complemented
3C2×#3745⋊C3C211yesmaximal, characteristic, complemented
4C2×#3744⋊C3C211yesmaximal, characteristic, complemented
5H(3,𝔽3)×C22⋊C8C211yesmaximal, characteristic, complemented
6#63648⋊C3C211yesmaximal, characteristic, complemented
7#63643⋊C3C211yesmaximal, characteristic, complemented
8#10592433C311yesmaximal, characteristic, complemented
9C3×#4734—31nomaximal
10C22×C4×H(3,𝔽3)C2211yescharacteristic
11C2×#638⋊C3—21no—
12C2×#638⋊C3—21no—
13#3828⋊C3—22no—
14#3828⋊C3—22no—
15C2×C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic
16C2×C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic
17C2×C4×(C3⋊D3)⋊C3—21no—
18C2×#634⋊C3C2211yescharacteristic
19C2×C8×H(3,𝔽3)C2211yescharacteristic
20C2×C8×H(3,𝔽3)C2211yescharacteristic
21#3752⋊C3—21no—
22C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
23#3745⋊C3—22no—
24#3744⋊C3—22no—
25C2×#3828C2×C311yescharacteristic, complemented
26C2×C3×#235—31no—
27#64139C2×C311yescharacteristic, complemented
28C2×#3745C2×C311yescharacteristic, complemented
29C2×#3744C2×C311yescharacteristic, complemented
30C3×#930—31no—
31C2×C3×D24—31no—
32C2×C3×#204—31no—
33C32×C22⋊C8—21no—
34C32×C22⋊C8D311yescharacteristic, complemented
35C32×C22⋊C8C2×C311yescharacteristic, complemented
36#63648C2×C311yescharacteristic, complemented
37#63643C2×C311yescharacteristic, complemented
38C3×#821—31no—
39C3×#816—31no—
40C23×H(3,𝔽3)D411yescharacteristic
41C22×(C3⋊D3)⋊C3—21no—
42C22×(C3⋊D3)⋊C3—21no—
43C2×C4×H(3,𝔽3)—21no—
44C2×C4×H(3,𝔽3)—21no—
45C2×C4×H(3,𝔽3)D411yescharacteristic
46C2×C4×H(3,𝔽3)C2311yescharacteristic
47#638⋊C3—24no—
48#638⋊C3—42no—
49C2×(C3⋊Dic3)⋊C3—21no—
50C2×(C3⋊Dic3)⋊C3—21no—
51C4⋊((C3⋊D3)⋊C3)—21no—
52C4⋊((C3⋊D3)⋊C3)—21no—
53C4⋊((C3⋊D3)⋊C3)—21no—
54C4⋊((C3⋊D3)⋊C3)D412yescomplemented
55C4×(C3⋊D3)⋊C3—24no—
56#634⋊C3—21no—
57#634⋊C3D412yescomplemented
58C8×H(3,𝔽3)—21no—
59C8×H(3,𝔽3)—21no—
60C3×#449—91no—
61#4734—31no—
62#4734—91no—
63C22×C32×C4—21no—
64C22×C32×C4C2×D311yescharacteristic
65C22×C32×C4C22×C311yescharacteristic
66C2×#638—21no—
67C2×#638—21no—
68#3828—22no—
69#3828—22no—
70C2×C4⋊(C3⋊D3)C22×C311yescharacteristic
71C2×C4⋊(C3⋊D3)C22×C311yescharacteristic
72C2×C4×C3⋊D3—21no—
73C2×#634C22×C311yescharacteristic
74C2×C3×#67—61no—
75C2×C3×#67—61no—
76C3×#235—62no—
77C3×#235—62no—
78C2×C3×D12—31no—
79C2×C3×D12—31no—
80C2×C3×C4×D3—61no—
81C2×C3×Dic6—31no—
82C2×C32×C8—21no—
83C2×C32×C8—21no—
84C2×C32×C8C2×D311yescharacteristic
85C2×C32×C8C2×D311yescharacteristic
86C2×C32×C8C22×C311yescharacteristic
87C2×C32×C8C22×C311yescharacteristic
88#3752—21no—
89C4⋊(C3⋊Dic3)C22×C311yescharacteristic
90#3745—22no—
91#3744—22no—
92C3×(C2×D3)⋊C4—61no—
93C3×C4⋊Dic3—31no—
94C3×D24—62no—
95C3×#204—62no—
96C22×H(3,𝔽3)—21no—
97C22×H(3,𝔽3)—21no—
98C22×H(3,𝔽3)—21no—
99C22×H(3,𝔽3)C2×D411yescharacteristic
100C2×(C3⋊D3)⋊C3—22no—
101C2×(C3⋊D3)⋊C3—41no—
102C2×(C3⋊D3)⋊C3—41no—
103C2×(C3⋊D3)⋊C3—41no—
104C2×(C3⋊D3)⋊C3—41no—
105C2×(C3⋊D3)⋊C3—41no—
106C4×H(3,𝔽3)—21no—
107C4×H(3,𝔽3)C2×D411yescharacteristic
108C4×H(3,𝔽3)C2×D411yescharacteristic
109(C3⋊Dic3)⋊C3—22no—
110(C3⋊Dic3)⋊C3—41no—
111C2×C3×#41—91no—
112C2×#235—31no—
113C2×#235—91no—
114C2×C3×SD16—91no—
115C2×C3×D8—91no—
116C3×#121—91no—
117#930—31no—
118#930—91no—
119C2×D24—31no—
120C2×D24—91no—
121C2×#204—31no—
122C2×#204—91no—
123C3×#103—91no—
124C3×#102—91no—
125C3×C22⋊C8—31no—
126C3×C22⋊C8—31no—
127C3×C22⋊C8—61no—
128C3×C22⋊C8C3×D311yescharacteristic
129#821—31no—
130#821—91no—
131#816—31no—
132#816—91no—
133C23×C32—21no—
134C23×C32D1211yescharacteristic
135C23×C32C3×D411yescharacteristic
136C22×C3⋊D3—21no—
137C22×C3⋊D3—21no—
138C22×C3×D3—61no—
139C22×C3×D3—61no—
140C2×C32×C4—21no—
141C2×C32×C4—21no—
142C2×C32×C4—21no—
143C2×C32×C4—21no—
144C2×C32×C4—21no—
145C2×C32×C4—21no—
146C2×C32×C4—41no—
147C2×C32×C4—41no—
148C2×C32×C4D1211yescharacteristic
149C2×C32×C4C3×D411yescharacteristic
150C2×C32×C4C22×D311yescharacteristic
151C2×C32×C4C23×C311yescharacteristic
152#638—24no—
153#638—42no—
154C2×C3⋊Dic3—21no—
155C2×C3⋊Dic3—21no—
156C4⋊(C3⋊D3)—21no—
157C4⋊(C3⋊D3)—21no—
158C4⋊(C3⋊D3)—21no—
159C4⋊(C3⋊D3)C3×D412yescomplemented
160C4×C3⋊D3—24no—
161#634—21no—
162#634C3×D412yescomplemented
163C3×#67—64no—
164C3×#67—122no—
165C2×C3×Dic3—61no—
166C2×C3×Dic3—61no—
167C3×D12—32no—
168C3×D12—61no—
169C3×D12—61no—
170C3×D12—61no—
171C3×C4×D3—64no—
172C3×Dic6—32no—
173C3×Dic6—61no—
174C32×C8—21no—
175C32×C8—21no—
176C32×C8—21no—
177C32×C8—21no—
178C32×C8—41no—
179C32×C8—41no—
180#449—91noSylow
181C2×H(3,𝔽3)—41no—
182C2×H(3,𝔽3)#12011yescharacteristic
183C2×H(3,𝔽3)#13511yescharacteristic
184C2×H(3,𝔽3)AGL(1,ℤ/8)11yescharacteristic
185(C3⋊D3)⋊C3—42no—
186(C3⋊D3)⋊C3—81no—
187C3×#41—182no—
188C3×#41—182no—
189C2×C3×Q8—91no—
190C2×C3×D4—91no—
191C2×C3×D4—91no—
192C2×C3×D4—181no—
193C2×C3×D4—181no—
194C22×C3×C4—31no—
195C22×C3×C4—31no—
196C22×C3×C4—61no—
197C22×C3×C4—181no—
198C22×C3×C4C2×C3×D311yescharacteristic
199C2×#67—61no—
200C2×#67—61no—
201C2×#67—181no—
202C2×#67—181no—
203#235—62no—
204#235—62no—
205#235—182no—
206#235—182no—
207C2×D12—31no—
208C2×D12—31no—
209C2×D12—91no—
210C2×D12—91no—
211C2×C4×D3—61no—
212C2×C4×D3—181no—
213C2×Dic6—31no—
214C2×Dic6—91no—
215C3×SD16—182no—
216C3×D8—182no—
217C2×C3×C8—31no—
218C2×C3×C8—31no—
219C2×C3×C8—31no—
220C2×C3×C8—31no—
221C2×C3×C8—61no—
222C2×C3×C8—61no—
223C2×C3×C8C2×C3×D311yescharacteristic
224C2×C3×C8C2×C3×D311yescharacteristic
225C3×C4⋊C4—91no—
226C3×C22⋊C4—181no—
227(C2×D3)⋊C4—61no—
228(C2×D3)⋊C4—181no—
229C4⋊Dic3—31no—
230C4⋊Dic3—91no—
231D24—62no—
232D24—182no—
233#204—62no—
234#204—182no—
235C22×C32—22no—
236C22×C32—21no—
237C22×C32—21no—
238C22×C32—21no—
239C22×C32—21no—
240C22×C32—21no—
241C22×C32—21no—
242C22×C32—21no—
243C22×C32—41no—
244C22×C32—41no—
245C22×C32C2×D1211yescharacteristic
246C22×C32C2×C3×D411yescharacteristic
247C2×C3⋊D3—22no—
248C2×C3⋊D3—41no—
249C2×C3⋊D3—41no—
250C2×C3⋊D3—41no—
251C2×C3⋊D3—41no—
252C2×C3⋊D3—41no—
253C2×C3×D3—62no—
254C2×C3×D3—121no—
255C2×C3×D3—121no—
256C2×C3×D3—121no—
257C2×C3×D3—121no—
258C2×C3×D3—121no—
259C32×C4—21no—
260C32×C4—21no—
261C32×C4—21no—
262C32×C4—21no—
263C32×C4—41no—
264C32×C4D3×D411yescharacteristic
265C32×C4D3×D411yescharacteristic
266C32×C4C2×C3×D411yescharacteristic
267C32×C4C2×C3×D411yescharacteristic
268C3⋊Dic3—22no—
269C3⋊Dic3—41no—
270C3×Dic3—62no—
271C3×Dic3—121no—
272C2×#41—91no—
273C2×SD16—91no—
274C2×D8—91no—
275#121—91no—
276#103—91no—
277#102—91no—
278C22⋊C8(C3⋊D3)⋊C311yescharacteristic, complemented
279H(3,𝔽3)#44911yescharacteristic, complemented, Sylow
280C23×C3—31no—
281C23×C3—31no—
282C23×C3—61no—
283C23×C3—181no—
284C23×C3—181no—
285C23×C3C3×D1211yescharacteristic
286C22×D3—61no—
287C22×D3—61no—
288C22×D3—181no—
289C22×D3—181no—
290C3×Q8—92no—
291C3×Q8—181no—
292C3×D4—92no—
293C3×D4—181no—
294C3×D4—181no—
295C3×D4—181no—
296C3×D4—184no—
297C3×D4—362no—
298C2×C3×C4—21no—
299C2×C3×C4—21no—
300C2×C3×C4—31no—
301C2×C3×C4—31no—
302C2×C3×C4—31no—
303C2×C3×C4—31no—
304C2×C3×C4—61no—
305C2×C3×C4—61no—
306C2×C3×C4—61no—
307C2×C3×C4—61no—
308C2×C3×C4—61no—
309C2×C3×C4—61no—
310C2×C3×C4—121no—
311C2×C3×C4—121no—
312C2×C3×C4—184no—
313C2×C3×C4—181no—
314C2×C3×C4—181no—
315C2×C3×C4C3×D1211yescharacteristic
316C2×C3×C4C22×C3×D311yescharacteristic
317#67—64no—
318#67—122no—
319#67—184no—
320#67—362no—
321C2×Dic3—61no—
322C2×Dic3—61no—
323C2×Dic3—181no—
324C2×Dic3—181no—
325D12—32no—
326D12—61no—
327D12—61no—
328D12—61no—
329D12—92no—
330D12—181no—
331D12—181no—
332D12—181no—
333C4×D3—64no—
334C4×D3—184no—
335Dic6—32no—
336Dic6—61no—
337Dic6—92no—
338Dic6—181no—
339C3×C8—21no—
340C3×C8—21no—
341C3×C8—61no—
342C3×C8—61no—
343C3×C8—61no—
344C3×C8—61no—
345C3×C8—121no—
346C3×C8—121no—
347C2×C32—21no—
348C2×C32—21no—
349C2×C32—21no—
350C2×C32—42no—
351C2×C32—41no—
352C2×C32—41no—
353C2×C32#88211yescharacteristic
354C2×C32#90411yescharacteristic
355C2×C32#90811yescharacteristic
356C2×C32C3×#12011yescharacteristic
357C2×C32C3×#13511yescharacteristic
358C2×C32C3×AGL(1,ℤ/8)11yescharacteristic
359C3⋊D3—42no—
360C3⋊D3—81no—
361C3×D3—122no—
362C3×D3—241no—
363#41—182no—
364#41—182no—
365C2×Q8—91no—
366C2×D4—91no—
367C2×D4—91no—
368C2×D4—181no—
369C2×D4—181no—
370C22×C4—181no—
371C22×C4C2×(C3⋊D3)⋊C311yescharacteristic
372SD16—182no—
373D8—182no—
374C2×C8C2×(C3⋊D3)⋊C311yescharacteristic
375C2×C8C2×(C3⋊D3)⋊C311yescharacteristic
376C4⋊C4—91no—
377C22⋊C4—181no—
378C22×C3—21no—
379C22×C3—21no—
380C22×C3—21no—
381C22×C3—31no—
382C22×C3—31no—
383C22×C3—61no—
384C22×C3—61no—
385C22×C3—61no—
386C22×C3—61no—
387C22×C3—61no—
388C22×C3—61no—
389C22×C3—62no—
390C22×C3—61no—
391C22×C3—121no—
392C22×C3—121no—
393C22×C3—182no—
394C22×C3—361no—
395C22×C3—361no—
396C22×C3—361no—
397C22×C3—361no—
398C22×C3—361no—
399C22×C3C2×C3×D1211yescharacteristic
400C2×D3—62no—
401C2×D3—121no—
402C2×D3—121no—
403C2×D3—121no—
404C2×D3—121no—
405C2×D3—121no—
406C2×D3—182no—
407C2×D3—361no—
408C2×D3—361no—
409C2×D3—361no—
410C2×D3—361no—
411C2×D3—361no—
412C3×C4—21no—
413C3×C4—31no—
414C3×C4—31no—
415C3×C4—31no—
416C3×C4—31no—
417C3×C4—61no—
418C3×C4—61no—
419C3×C4—61no—
420C3×C4—61no—
421C3×C4—121no—
422C3×C4—182no—
423C3×C4—361no—
424C3×C4C3×D3×D411yescharacteristic
425C3×C4C3×D3×D411yescharacteristic
426Dic3—62no—
427Dic3—121no—
428Dic3—182no—
429Dic3—361no—
430C32—21no—
431C32#473411yescharacteristic, complemented
432C32C3×#44911yescharacteristic, complemented
433C23—181no—
434C23—181no—
435C23C4⋊((C3⋊D3)⋊C3)11yescharacteristic
436Q8—92no—
437Q8—181no—
438D4—92no—
439D4—184no—
440D4—181no—
441D4—181no—
442D4—181no—
443D4—362no—
444C2×C4—21no—
445C2×C4—21no—
446C2×C4—184no—
447C2×C4—181no—
448C2×C4—181no—
449C2×C4C4⋊((C3⋊D3)⋊C3)11yescharacteristic
450C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
451C8—21no—
452C8—21no—
453C2×C3—31no—
454C2×C3—31no—
455C2×C3—31no—
456C2×C3—31no—
457C2×C3—31no—
458C2×C3—31no—
459C2×C3—41no—
460C2×C3—61no—
461C2×C3—61no—
462C2×C3—61no—
463C2×C3—121no—
464C2×C3—121no—
465C2×C3—122no—
466C2×C3—362no—
467C2×C3—721no—
468C2×C3C3×#88211yescharacteristic
469C2×C3C3×#90411yescharacteristic
470C2×C3C3×#90811yescharacteristic
471D3—122no—
472D3—241no—
473D3—362no—
474D3—721no—
475C22—21no—
476C22—21no—
477C22—21no—
478C22—182no—
479C22—361no—
480C22—361no—
481C22—361no—
482C22—361no—
483C22—361no—
484C22C2×C4⋊((C3⋊D3)⋊C3)11yescharacteristic
485C4—21no—
486C4—182no—
487C4—361no—
488C4D4×(C3⋊D3)⋊C311yescharacteristic
489C4D4×(C3⋊D3)⋊C311yescharacteristic
490C3—31no—
491C3—31no—
492C3—61no—
493C3C3×#473411yescharacteristic
494C2—41no—
495C2—362no—
496C2—721no—
497C2#64091⋊C311yescharacteristic
498C2#64113⋊C311yescharacteristic
499C2#64117⋊C311yescharacteristic
subgroupG/H
centerC22C2×C4⋊((C3⋊D3)⋊C3)
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C3×C4C22×C3×D3
FittingH(3,𝔽3)×C22⋊C8C2
socleC22×C3C2×C3×D12
hypercenterC22⋊C8(C3⋊D3)⋊C3
hypocenterC32C3×#449
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22⋊C8

Complements

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

NH ≅ G/Ntype
1. C2×#3828⋊C3C2semidirect
2. #64139⋊C3C2semidirect
3. C2×#3745⋊C3C2semidirect
4. C2×#3744⋊C3C2semidirect
5. H(3,𝔽3)×C22⋊C8C2semidirect
6. #63648⋊C3C2semidirect
7. #63643⋊C3C2semidirect
8. #10592433C3semidirect
25. C2×#3828C2×C3semidirect
27. #64139C2×C3semidirect
28. C2×#3745C2×C3semidirect
29. C2×#3744C2×C3semidirect
34. C32×C22⋊C8D3semidirect
35. C32×C22⋊C8C2×C3semidirect
36. #63648C2×C3semidirect
37. #63643C2×C3semidirect
54. C4⋊((C3⋊D3)⋊C3)D4semidirect
57. #634⋊C3D4semidirect
159. C4⋊(C3⋊D3)C3×D4semidirect
162. #634C3×D4semidirect
278. C22⋊C8(C3⋊D3)⋊C3semidirect
279. H(3,𝔽3)#449semidirect
431. C32#4734semidirect
432. C32C3×#449semidirect

Maximal subgroups

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

subgroupindex
C2×#3828⋊C32
#64139⋊C32
C2×#3745⋊C32
C2×#3744⋊C32
H(3,𝔽3)×C22⋊C82
#63648⋊C32
#63643⋊C32
#105924333
C3×#47343