Finite Group Databasebeta

← #10592480⋊C3 #13141798

Subgroups — #10592480⋊C3 #13141798

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×#3830⋊C3C211yesmaximal, characteristic, complemented
2C2×#3753⋊C3C211yesmaximal, characteristic, complemented
3C2×#3744⋊C3C211yesmaximal, characteristic, complemented
4#64104⋊C3C211yesmaximal, characteristic, complemented
5H(3,𝔽3)×#102C211yesmaximal, characteristic, complemented
6#63647⋊C3C211yesmaximal, characteristic, complemented
7#63637⋊C3C211yesmaximal, characteristic, complemented
8#10592480C311yesmaximal, characteristic, complemented
9C3×#4781—31nomaximal
10C2×D4×H(3,𝔽3)C2211yescharacteristic
11C2×#638⋊C3—21no—
12C22×(C3⋊Dic3)⋊C3—21no—
13#3830⋊C3—22no—
14#3830⋊C3—22no—
15C2×C4⋊((C3⋊D3)⋊C3)—21no—
16C2×C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic
17C2×C4×(C3⋊D3)⋊C3C2211yescharacteristic
18C2×#634⋊C3C2211yescharacteristic
19C2×C8×H(3,𝔽3)C2211yescharacteristic
20C4⋊C4×H(3,𝔽3)C2211yescharacteristic
21#3753⋊C3—22no—
22#3752⋊C3—21no—
23C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
24#3744⋊C3—22no—
25C2×#3830C2×C311yescharacteristic, complemented
26C2×C3×#237—31no—
27C2×#3753C2×C311yescharacteristic, complemented
28C2×#3744C2×C311yescharacteristic, complemented
29#64104C2×C311yescharacteristic, complemented
30C2×C3×#213—31no—
31C2×C3×#204—31no—
32C3×#895—31no—
33C32×#102—21no—
34C32×#102D311yescharacteristic, complemented
35C32×#102C2×C311yescharacteristic, complemented
36#63647C2×C311yescharacteristic, complemented
37#63637C2×C311yescharacteristic, complemented
38C3×(C2×D3)⋊C8—31no—
39C3×#810—31no—
40C23×H(3,𝔽3)—21no—
41C22×(C3⋊D3)⋊C3—21no—
42C22×(C3⋊D3)⋊C3D411yescharacteristic
43D4×H(3,𝔽3)—21no—
44D4×H(3,𝔽3)D412yescomplemented
45C2×C4×H(3,𝔽3)—21no—
46C2×C4×H(3,𝔽3)C2311yescharacteristic
47#638⋊C3—24no—
48C2×(C3⋊Dic3)⋊C3—24no—
49C2×(C3⋊Dic3)⋊C3—21no—
50C2×(C3⋊Dic3)⋊C3D411yescharacteristic
51C4⋊((C3⋊D3)⋊C3)—21no—
52C4⋊((C3⋊D3)⋊C3)—21no—
53C4⋊((C3⋊D3)⋊C3)—42no—
54C4×(C3⋊D3)⋊C3—21no—
55C4×(C3⋊D3)⋊C3—21no—
56#634⋊C3—21no—
57#634⋊C3D412yescomplemented
58C8×H(3,𝔽3)—21no—
59(C3⋊(C3⋊C8))⋊C3—21no—
60C3×#449—91no—
61#4781—31no—
62#4781—91no—
63C2×C32×D4—21no—
64C2×C32×D4C2×D311yescharacteristic
65C2×C32×D4C22×C311yescharacteristic
66C2×#638—21no—
67C22×C3⋊Dic3—21no—
68#3830—22no—
69#3830—22no—
70C2×C4⋊(C3⋊D3)—21no—
71C2×C4⋊(C3⋊D3)C22×C311yescharacteristic
72C2×C4×C3⋊D3C22×C311yescharacteristic
73C2×#634C22×C311yescharacteristic
74C2×C3×#67—61no—
75C22×C3×Dic3—61no—
76C3×#237—62no—
77C3×#237—62no—
78C2×C3×D12—31no—
79C2×C3×D12—61no—
80C2×C3×C4×D3—31no—
81C2×C3×Dic6—31no—
82C2×C32×C8—21no—
83C2×C32×C8C2×D311yescharacteristic
84C2×C32×C8C22×C311yescharacteristic
85C32×C4⋊C4—21no—
86C32×C4⋊C4C2×D311yescharacteristic
87C32×C4⋊C4C22×C311yescharacteristic
88#3753—22no—
89#3752—21no—
90C2×C3⋊(C3⋊C8)C22×C311yescharacteristic
91#3744—22no—
92C3×#213—62no—
93C3×(C2×D3)⋊C4—61no—
94C2×C3×C3⋊C8—31no—
95C3×#204—62no—
96C22×H(3,𝔽3)—22no—
97C22×H(3,𝔽3)—41no—
98C22×H(3,𝔽3)—41no—
99C22×H(3,𝔽3)C2×D411yescharacteristic
100C2×(C3⋊D3)⋊C3—21no—
101C2×(C3⋊D3)⋊C3—21no—
102C2×(C3⋊D3)⋊C3—21no—
103C2×(C3⋊D3)⋊C3—41no—
104C2×(C3⋊D3)⋊C3—41no—
105C2×(C3⋊D3)⋊C3—41no—
106C4×H(3,𝔽3)—41no—
107C4×H(3,𝔽3)C2×D411yescharacteristic
108C4×H(3,𝔽3)C2×D411yescharacteristic
109(C3⋊Dic3)⋊C3—22no—
110(C3⋊Dic3)⋊C3—21no—
111C2×C3×#41—91no—
112C2×#237—31no—
113C2×#237—91no—
114C2×C3×SD16—91no—
115C2×C3×D8—91no—
116C3×#121—91no—
117C2×#213—31no—
118C2×#213—91no—
119C2×#204—31no—
120C2×#204—91no—
121#895—31no—
122#895—91no—
123C3×#103—91no—
124C3×#102—31no—
125C3×#102—31no—
126C3×#102—61no—
127C3×#102C3×D311yescharacteristic
128C3×C22⋊C8—91no—
129(C2×D3)⋊C8—31no—
130(C2×D3)⋊C8—91no—
131#810—31no—
132#810—91no—
133C23×C32—21no—
134C23×C32—21no—
135C23×C32—41no—
136C22×C3⋊D3—21no—
137C22×C3⋊D3C3×D411yescharacteristic
138C22×C3×D3—31no—
139C22×C3×D3—61no—
140C32×D4—21no—
141C32×D4—21no—
142C32×D4—22no—
143C32×D4—41no—
144C32×D4D1212yescomplemented
145C32×D4C3×D412yescomplemented
146C2×C32×C4—21no—
147C2×C32×C4—21no—
148C2×C32×C4—21no—
149C2×C32×C4—41no—
150C2×C32×C4C22×D311yescharacteristic
151C2×C32×C4C23×C311yescharacteristic
152#638—24no—
153C2×C3⋊Dic3—24no—
154C2×C3⋊Dic3—21no—
155C2×C3⋊Dic3C3×D411yescharacteristic
156C4⋊(C3⋊D3)—21no—
157C4⋊(C3⋊D3)—21no—
158C4⋊(C3⋊D3)—42no—
159C4×C3⋊D3—21no—
160C4×C3⋊D3—21no—
161#634—21no—
162#634C3×D412yescomplemented
163C3×#67—64no—
164C2×C3×Dic3—31no—
165C2×C3×Dic3—64no—
166C2×C3×Dic3—61no—
167C3×D12—61no—
168C3×D12—61no—
169C3×D12—122no—
170C3×C4×D3—61no—
171C3×C4×D3—61no—
172C3×Dic6—32no—
173C3×Dic6—61no—
174C32×C8—21no—
175C32×C8—21no—
176C32×C8—41no—
177C3⋊(C3⋊C8)—21no—
178C3×C3⋊C8—61no—
179#449—91noSylow
180C2×H(3,𝔽3)—42no—
181C2×H(3,𝔽3)#12011yescharacteristic
182C2×H(3,𝔽3)#13511yescharacteristic
183C2×H(3,𝔽3)AGL(1,ℤ/8)11yescharacteristic
184(C3⋊D3)⋊C3—41no—
185(C3⋊D3)⋊C3—81no—
186C3×#41—182no—
187C3×#41—182no—
188C2×C3×Q8—91no—
189C2×C3×D4—31no—
190C2×C3×D4—31no—
191C2×C3×D4—61no—
192C2×C3×D4—91no—
193C2×C3×D4—181no—
194C2×C3×D4—181no—
195C2×C3×D4C2×C3×D311yescharacteristic
196C22×C3×C4—91no—
197C22×C3×C4—181no—
198C2×#67—61no—
199C2×#67—181no—
200C22×Dic3—61no—
201C22×Dic3—181no—
202#237—62no—
203#237—62no—
204#237—182no—
205#237—182no—
206C2×D12—31no—
207C2×D12—61no—
208C2×D12—91no—
209C2×D12—181no—
210C2×C4×D3—31no—
211C2×C4×D3—91no—
212C2×Dic6—31no—
213C2×Dic6—91no—
214C3×SD16—182no—
215C3×D8—182no—
216C2×C3×C8—31no—
217C2×C3×C8—31no—
218C2×C3×C8—61no—
219C2×C3×C8—91no—
220C2×C3×C8C2×C3×D311yescharacteristic
221C3×C4⋊C4—31no—
222C3×C4⋊C4—31no—
223C3×C4⋊C4—61no—
224C3×C4⋊C4C2×C3×D311yescharacteristic
225C3×C22⋊C4—181no—
226#213—62no—
227#213—182no—
228(C2×D3)⋊C4—61no—
229(C2×D3)⋊C4—181no—
230C2×C3⋊C8—31no—
231C2×C3⋊C8—91no—
232#204—62no—
233#204—182no—
234C22×C32—22no—
235C22×C32—22no—
236C22×C32—21no—
237C22×C32—42no—
238C22×C32—41no—
239C22×C32—41no—
240C22×C32—41no—
241C22×C32—41no—
242C22×C32—81no—
243C22×C32—81no—
244C22×C32D3×D411yescharacteristic
245C22×C32C2×C3×D411yescharacteristic
246C2×C3⋊D3—21no—
247C2×C3⋊D3—21no—
248C2×C3⋊D3—21no—
249C2×C3⋊D3—41no—
250C2×C3⋊D3—41no—
251C2×C3⋊D3—41no—
252C2×C3×D3—61no—
253C2×C3×D3—61no—
254C2×C3×D3—61no—
255C2×C3×D3—121no—
256C2×C3×D3—121no—
257C2×C3×D3—121no—
258C32×C4—21no—
259C32×C4—21no—
260C32×C4—41no—
261C32×C4—41no—
262C32×C4—81no—
263C32×C4C2×D1211yescharacteristic
264C32×C4D3×D411yescharacteristic
265C32×C4C2×C3×D411yescharacteristic
266C32×C4C2×C3×D411yescharacteristic
267C3⋊Dic3—22no—
268C3⋊Dic3—21no—
269C3×Dic3—62no—
270C3×Dic3—61no—
271C2×#41—91no—
272C2×SD16—91no—
273C2×D8—91no—
274#121—91no—
275#103—91no—
276#102(C3⋊D3)⋊C311yescharacteristic, complemented
277C22⋊C8—91no—
278H(3,𝔽3)#44911yescharacteristic, complemented, Sylow
279C23×C3—21no—
280C23×C3—61no—
281C23×C3—61no—
282C23×C3—91no—
283C23×C3—121no—
284C23×C3—181no—
285C22×D3—31no—
286C22×D3—61no—
287C22×D3—91no—
288C22×D3—181no—
289C3×Q8—92no—
290C3×Q8—181no—
291C3×D4—21no—
292C3×D4—32no—
293C3×D4—32no—
294C3×D4—61no—
295C3×D4—61no—
296C3×D4—62no—
297C3×D4—121no—
298C3×D4—181no—
299C3×D4—181no—
300C3×D4—184no—
301C3×D4—362no—
302C3×D4C3×D1212yes—
303C2×C3×C4—21no—
304C2×C3×C4—31no—
305C2×C3×C4—31no—
306C2×C3×C4—61no—
307C2×C3×C4—61no—
308C2×C3×C4—61no—
309C2×C3×C4—91no—
310C2×C3×C4—121no—
311C2×C3×C4—181no—
312C2×C3×C4—181no—
313C2×C3×C4—181no—
314C2×C3×C4—184no—
315C2×C3×C4C22×C3×D311yescharacteristic
316#67—64no—
317#67—184no—
318C2×Dic3—31no—
319C2×Dic3—64no—
320C2×Dic3—61no—
321C2×Dic3—91no—
322C2×Dic3—181no—
323C2×Dic3—184no—
324D12—61no—
325D12—61no—
326D12—122no—
327D12—181no—
328D12—181no—
329D12—362no—
330C4×D3—61no—
331C4×D3—61no—
332C4×D3—181no—
333C4×D3—181no—
334Dic6—32no—
335Dic6—61no—
336Dic6—92no—
337Dic6—181no—
338C3×C8—21no—
339C3×C8—61no—
340C3×C8—61no—
341C3×C8—121no—
342C3×C8—181no—
343C3⋊C8—61no—
344C3⋊C8—181no—
345C2×C32—21no—
346C2×C32—21no—
347C2×C32—21no—
348C2×C32—42no—
349C2×C32—42no—
350C2×C32—82no—
351C2×C32#88211yescharacteristic
352C2×C32#91111yescharacteristic
353C2×C32#91611yescharacteristic
354C2×C32C3×#12011yescharacteristic
355C2×C32C3×#13511yescharacteristic
356C2×C32C3×AGL(1,ℤ/8)11yescharacteristic
357C3⋊D3—41no—
358C3⋊D3—81no—
359C3×D3—121no—
360C3×D3—241no—
361#41—182no—
362#41—182no—
363C2×Q8—91no—
364C2×D4—91no—
365C2×D4—181no—
366C2×D4—181no—
367C2×D4C2×(C3⋊D3)⋊C311yescharacteristic
368C22×C4—91no—
369C22×C4—181no—
370SD16—182no—
371D8—182no—
372C2×C8—91no—
373C2×C8C2×(C3⋊D3)⋊C311yescharacteristic
374C4⋊C4C2×(C3⋊D3)⋊C311yescharacteristic
375C22⋊C4—181no—
376C22×C3—22no—
377C22×C3—31no—
378C22×C3—31no—
379C22×C3—41no—
380C22×C3—41no—
381C22×C3—62no—
382C22×C3—62no—
383C22×C3—61no—
384C22×C3—122no—
385C22×C3—121no—
386C22×C3—121no—
387C22×C3—121no—
388C22×C3—121no—
389C22×C3—181no—
390C22×C3—181no—
391C22×C3—181no—
392C22×C3—241no—
393C22×C3—241no—
394C22×C3—361no—
395C22×C3—361no—
396C22×C3—361no—
397C22×C3C3×D3×D411yescharacteristic
398C2×D3—61no—
399C2×D3—61no—
400C2×D3—61no—
401C2×D3—121no—
402C2×D3—121no—
403C2×D3—121no—
404C2×D3—181no—
405C2×D3—181no—
406C2×D3—181no—
407C2×D3—361no—
408C2×D3—361no—
409C2×D3—361no—
410C3×C4—31no—
411C3×C4—31no—
412C3×C4—31no—
413C3×C4—31no—
414C3×C4—41no—
415C3×C4—61no—
416C3×C4—61no—
417C3×C4—121no—
418C3×C4—121no—
419C3×C4—181no—
420C3×C4—182no—
421C3×C4—241no—
422C3×C4C2×C3×D1211yescharacteristic
423C3×C4C3×D3×D411yescharacteristic
424Dic3—62no—
425Dic3—61no—
426Dic3—182no—
427Dic3—181no—
428C32—21no—
429C32#478111yescharacteristic, complemented
430C32C3×#44911yescharacteristic, complemented
431C23—21no—
432C23—91no—
433C23—181no—
434Q8—92no—
435Q8—181no—
436D4—21no—
437D4—184no—
438D4—181no—
439D4—181no—
440D4—362no—
441D4C4⋊((C3⋊D3)⋊C3)12yescomplemented
442C2×C4—21no—
443C2×C4—91no—
444C2×C4—184no—
445C2×C4—181no—
446C2×C4—181no—
447C2×C4—181no—
448C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
449C8—21no—
450C8—181no—
451C2×C3—31no—
452C2×C3—31no—
453C2×C3—31no—
454C2×C3—31no—
455C2×C3—31no—
456C2×C3—31no—
457C2×C3—42no—
458C2×C3—61no—
459C2×C3—61no—
460C2×C3—61no—
461C2×C3—122no—
462C2×C3—122no—
463C2×C3—242no—
464C2×C3—361no—
465C2×C3—721no—
466C2×C3C3×#88211yescharacteristic
467C2×C3C3×#91111yescharacteristic
468C2×C3C3×#91611yescharacteristic
469D3—121no—
470D3—241no—
471D3—361no—
472D3—721no—
473C22—22no—
474C22—41no—
475C22—41no—
476C22—181no—
477C22—181no—
478C22—181no—
479C22—361no—
480C22—361no—
481C22—361no—
482C22D4×(C3⋊D3)⋊C311yescharacteristic
483C4—41no—
484C4—181no—
485C4—182no—
486C4C2×C4⋊((C3⋊D3)⋊C3)11yescharacteristic
487C4D4×(C3⋊D3)⋊C311yescharacteristic
488C3—31no—
489C3—31no—
490C3—61no—
491C3C3×#478111yescharacteristic
492C2—42no—
493C2—361no—
494C2—721no—
495C2#64091⋊C311yescharacteristic
496C2#64120⋊C311yescharacteristic
497C2#64125⋊C311yescharacteristic
subgroupG/H
centerC22D4×(C3⋊D3)⋊C3
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C3×C4C22×C3×D3
FittingH(3,𝔽3)×#102C2
socleC22×C3C3×D3×D4
hypercenter#102(C3⋊D3)⋊C3
hypocenterC32C3×#449
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C2×C4 ◁ #102

Complements

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

NH ≅ G/Ntype
1. C2×#3830⋊C3C2semidirect
2. C2×#3753⋊C3C2semidirect
3. C2×#3744⋊C3C2semidirect
4. #64104⋊C3C2semidirect
5. H(3,𝔽3)×#102C2semidirect
6. #63647⋊C3C2semidirect
7. #63637⋊C3C2semidirect
8. #10592480C3semidirect
25. C2×#3830C2×C3semidirect
27. C2×#3753C2×C3semidirect
28. C2×#3744C2×C3semidirect
29. #64104C2×C3semidirect
34. C32×#102D3semidirect
35. C32×#102C2×C3semidirect
36. #63647C2×C3semidirect
37. #63637C2×C3semidirect
44. D4×H(3,𝔽3)D4semidirect
57. #634⋊C3D4semidirect
144. C32×D4D12semidirect
145. C32×D4C3×D4semidirect
162. #634C3×D4semidirect
276. #102(C3⋊D3)⋊C3semidirect
278. H(3,𝔽3)#449semidirect
429. C32#4781semidirect
430. C32C3×#449semidirect
441. D4C4⋊((C3⋊D3)⋊C3)semidirect

Maximal subgroups

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

subgroupindex
C2×#3830⋊C32
C2×#3753⋊C32
C2×#3744⋊C32
#64104⋊C32
H(3,𝔽3)×#1022
#63647⋊C32
#63637⋊C32
#105924803
C3×#47813