Finite Group Databasebeta

← #10592871⋊C3 #13143377

Subgroups — #10592871⋊C3 #13143377

Order 1728.

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

normal subgroups only

This group has 635 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×#88722C211yesmaximal, characteristic, complemented
2C2×SD16×H(3,𝔽3)C211yesmaximal, characteristic, complemented
3#64147⋊C3C211yesmaximal, characteristic, complemented
4C2×#3753⋊C3C211yesmaximal, characteristic, complemented
5#63662⋊C3C211yesmaximal, characteristic, complemented
6#63648⋊C3C211yesmaximal, characteristic, complemented
7#63647⋊C3C211yesmaximal, characteristic, complemented
8#10592871C311yesmaximal, characteristic, complemented
9C3×#5172—31nomaximal
10C2×Q8×H(3,𝔽3)C2211yescharacteristic
11C2×D4×H(3,𝔽3)C2211yescharacteristic
12C2×#638⋊C3—21no—
13#88722—22no—
14#88722—22no—
15C2×C4⋊((C3⋊D3)⋊C3)—21no—
16C2×C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic
17C2×C4×(C3⋊D3)⋊C3—21no—
18C2×C4×(C3⋊D3)⋊C3C2211yescharacteristic
19SD16×H(3,𝔽3)—22no—
20C2×C8×H(3,𝔽3)C2211yescharacteristic
21#3757⋊C3—21no—
22#3753⋊C3—22no—
23C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
24C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
25C2×#3832C2×C311yescharacteristic, complemented
26C2×C3×#239—31no—
27C2×C32×SD16—21no—
28C2×C32×SD16D311yescharacteristic, complemented
29C2×C32×SD16C2×C311yescharacteristic, complemented
30#64147C2×C311yescharacteristic, complemented
31C2×#3753C2×C311yescharacteristic, complemented
32C3×#938—31no—
33C2×C3×#213—31no—
34#63662C2×C311yescharacteristic, complemented
35#63648C2×C311yescharacteristic, complemented
36#63647C2×C311yescharacteristic, complemented
37C3×#835—31no—
38C3×#821—31no—
39C3×(C2×D3)⋊C8—31no—
40C23×H(3,𝔽3)—21no—
41C22×(C3⋊D3)⋊C3—21no—
42C22×(C3⋊D3)⋊C3D411yescharacteristic
43Q8×H(3,𝔽3)—21no—
44Q8×H(3,𝔽3)D412yescomplemented
45D4×H(3,𝔽3)—21no—
46D4×H(3,𝔽3)—21no—
47C2×C4×H(3,𝔽3)—21no—
48C2×C4×H(3,𝔽3)C2311yescharacteristic
49#638⋊C3—42no—
50C2×(C3⋊Dic3)⋊C3—21no—
51C2×(C3⋊Dic3)⋊C3D411yescharacteristic
52C4⋊((C3⋊D3)⋊C3)—24no—
53C4⋊((C3⋊D3)⋊C3)—21no—
54C4⋊((C3⋊D3)⋊C3)D412yescomplemented
55C4×(C3⋊D3)⋊C3—24no—
56C4×(C3⋊D3)⋊C3—21no—
57C4×(C3⋊D3)⋊C3—21no—
58C8×H(3,𝔽3)—21no—
59(C3⋊(C3⋊C8))⋊C3—21no—
60C3×#449—91no—
61#5172—31no—
62#5172—91no—
63C2×C32×Q8—21no—
64C2×C32×Q8C2×D311yescharacteristic
65C2×C32×Q8C22×C311yescharacteristic
66C2×C32×D4—21no—
67C2×C32×D4C2×D311yescharacteristic
68C2×C32×D4C22×C311yescharacteristic
69C2×#638—21no—
70#3832—22no—
71#3832—22no—
72C2×C4⋊(C3⋊D3)—21no—
73C2×C4⋊(C3⋊D3)C22×C311yescharacteristic
74C2×C4×C3⋊D3—21no—
75C2×C4×C3⋊D3C22×C311yescharacteristic
76C2×C3×#67—61no—
77C3×#239—62no—
78C3×#239—62no—
79C2×C3×D12—31no—
80C2×C3×D12—61no—
81C2×C3×C4×D3—31no—
82C2×C3×C4×D3—61no—
83C32×SD16—24no—
84C32×SD16—22no—
85C32×SD16—22no—
86C2×C32×C8—21no—
87C2×C32×C8C2×D311yescharacteristic
88C2×C32×C8C22×C311yescharacteristic
89#3757—21no—
90#3753—22no—
91C4⋊(C3⋊Dic3)C22×C311yescharacteristic
92C2×C3⋊(C3⋊C8)C22×C311yescharacteristic
93C3×#217—61no—
94C3×#213—62no—
95C3×C4⋊Dic3—31no—
96C2×C3×C3⋊C8—31no—
97C22×H(3,𝔽3)—41no—
98C22×H(3,𝔽3)—41no—
99C22×H(3,𝔽3)—41no—
100C22×H(3,𝔽3)C2×D411yescharacteristic
101C2×(C3⋊D3)⋊C3—21no—
102C2×(C3⋊D3)⋊C3—22no—
103C2×(C3⋊D3)⋊C3—21no—
104C2×(C3⋊D3)⋊C3—21no—
105C2×(C3⋊D3)⋊C3—41no—
106C2×(C3⋊D3)⋊C3—41no—
107C4×H(3,𝔽3)—22no—
108C4×H(3,𝔽3)C2×D411yescharacteristic
109C4×H(3,𝔽3)C2×D411yescharacteristic
110(C3⋊Dic3)⋊C3—21no—
111(C3⋊Dic3)⋊C3—41no—
112C2×C3×#41—91no—
113C2×#239—31no—
114C2×#239—91no—
115C2×C3×SD16—31no—
116C2×C3×SD16—31no—
117C2×C3×SD16—61no—
118C2×C3×SD16C3×D311yescharacteristic
119C2×C3×D8—91no—
120C3×#121—91no—
121#938—31no—
122#938—91no—
123C2×#213—31no—
124C2×#213—91no—
125C3×#103—91no—
126C3×#102—91no—
127C3×C22⋊C8—91no—
128#835—31no—
129#835—91no—
130#821—31no—
131#821—91no—
132(C2×D3)⋊C8—31no—
133(C2×D3)⋊C8—91no—
134C23×C32—21no—
135C23×C32—21no—
136C23×C32—41no—
137C22×C3⋊D3—21no—
138C22×C3⋊D3C3×D411yescharacteristic
139C22×C3×D3—31no—
140C22×C3×D3—61no—
141C32×Q8—21no—
142C32×Q8—21no—
143C32×Q8—22no—
144C32×Q8—41no—
145C32×Q8#6712yescomplemented
146C32×Q8C3×D412yescomplemented
147C32×D4—22no—
148C32×D4—21no—
149C32×D4—21no—
150C32×D4—21no—
151C32×D4—21no—
152C32×D4—41no—
153C2×C32×C4—21no—
154C2×C32×C4—21no—
155C2×C32×C4—21no—
156C2×C32×C4—41no—
157C2×C32×C4C22×D311yescharacteristic
158C2×C32×C4C23×C311yescharacteristic
159#638—42no—
160C2×C3⋊Dic3—21no—
161C2×C3⋊Dic3C3×D411yescharacteristic
162C4⋊(C3⋊D3)—24no—
163C4⋊(C3⋊D3)—21no—
164C4⋊(C3⋊D3)C3×D412yescomplemented
165C4×C3⋊D3—24no—
166C4×C3⋊D3—21no—
167C4×C3⋊D3—21no—
168C3×#67—122no—
169C2×C3×Dic3—31no—
170C2×C3×Dic3—61no—
171C3×D12—32no—
172C3×D12—64no—
173C3×D12—61no—
174C3×C4×D3—64no—
175C3×C4×D3—61no—
176C3×C4×D3—61no—
177C32×C8—22no—
178C32×C8—21no—
179C32×C8—21no—
180C3⋊(C3⋊C8)—21no—
181C3×C3⋊C8—61no—
182#449—91noSylow
183C2×H(3,𝔽3)—81no—
184C2×H(3,𝔽3)#12011yescharacteristic
185C2×H(3,𝔽3)#13511yescharacteristic
186C2×H(3,𝔽3)AGL(1,ℤ/8)11yescharacteristic
187(C3⋊D3)⋊C3—41no—
188(C3⋊D3)⋊C3—42no—
189C3×#41—182no—
190C3×#41—182no—
191C2×C3×Q8—31no—
192C2×C3×Q8—31no—
193C2×C3×Q8—61no—
194C2×C3×Q8C2×C3×D311yescharacteristic
195C2×C3×D4—31no—
196C2×C3×D4—31no—
197C2×C3×D4—61no—
198C2×C3×D4—91no—
199C2×C3×D4—181no—
200C2×C3×D4—181no—
201C2×C3×D4C2×C3×D311yescharacteristic
202C22×C3×C4—91no—
203C22×C3×C4—181no—
204C2×#67—61no—
205C2×#67—181no—
206#239—62no—
207#239—62no—
208#239—182no—
209#239—182no—
210C2×D12—31no—
211C2×D12—61no—
212C2×D12—91no—
213C2×D12—181no—
214C2×C4×D3—31no—
215C2×C4×D3—61no—
216C2×C4×D3—91no—
217C2×C4×D3—181no—
218C3×SD16—22no—
219C3×SD16—62no—
220C3×SD16—62no—
221C3×SD16—64no—
222C3×D8—182no—
223C2×C3×C8—31no—
224C2×C3×C8—31no—
225C2×C3×C8—61no—
226C2×C3×C8—91no—
227C2×C3×C8C2×C3×D311yescharacteristic
228C3×C4⋊C4—91no—
229C3×C22⋊C4—181no—
230#217—61no—
231#217—181no—
232#213—62no—
233#213—182no—
234C4⋊Dic3—31no—
235C4⋊Dic3—91no—
236C2×C3⋊C8—31no—
237C2×C3⋊C8—91no—
238C22×C32—21no—
239C22×C32—42no—
240C22×C32—41no—
241C22×C32—41no—
242C22×C32—41no—
243C22×C32—41no—
244C22×C32—41no—
245C22×C32—41no—
246C22×C32—81no—
247C22×C32—81no—
248C22×C32D3×D411yescharacteristic
249C22×C32C2×C3×D411yescharacteristic
250C2×C3⋊D3—21no—
251C2×C3⋊D3—22no—
252C2×C3⋊D3—21no—
253C2×C3⋊D3—21no—
254C2×C3⋊D3—41no—
255C2×C3⋊D3—41no—
256C2×C3×D3—61no—
257C2×C3×D3—61no—
258C2×C3×D3—62no—
259C2×C3×D3—61no—
260C2×C3×D3—121no—
261C2×C3×D3—121no—
262C32×C4—22no—
263C32×C4—22no—
264C32×C4—21no—
265C32×C4—21no—
266C32×C4—42no—
267C32×C4D3×D411yescharacteristic
268C32×C4C2×#6711yescharacteristic
269C32×C4C2×C3×D411yescharacteristic
270C32×C4C2×C3×D411yescharacteristic
271C3⋊Dic3—21no—
272C3⋊Dic3—41no—
273C3×Dic3—61no—
274C3×Dic3—121no—
275C2×#41—91no—
276C2×SD16(C3⋊D3)⋊C311yescharacteristic, complemented
277C2×D8—91no—
278#121—91no—
279#103—91no—
280#102—91no—
281C22⋊C8—91no—
282H(3,𝔽3)#44911yescharacteristic, complemented, Sylow
283C23×C3—21no—
284C23×C3—61no—
285C23×C3—61no—
286C23×C3—91no—
287C23×C3—121no—
288C23×C3—181no—
289C22×D3—31no—
290C22×D3—61no—
291C22×D3—91no—
292C22×D3—181no—
293C3×Q8—21no—
294C3×Q8—32no—
295C3×Q8—32no—
296C3×Q8—61no—
297C3×Q8—61no—
298C3×Q8—62no—
299C3×Q8—121no—
300C3×Q8C3×#6712yes—
301C3×D4—21no—
302C3×D4—21no—
303C3×D4—61no—
304C3×D4—61no—
305C3×D4—61no—
306C3×D4—61no—
307C3×D4—62no—
308C3×D4—92no—
309C3×D4—121no—
310C3×D4—181no—
311C3×D4—184no—
312C3×D4—362no—
313C2×C3×C4—21no—
314C2×C3×C4—31no—
315C2×C3×C4—31no—
316C2×C3×C4—61no—
317C2×C3×C4—61no—
318C2×C3×C4—61no—
319C2×C3×C4—91no—
320C2×C3×C4—121no—
321C2×C3×C4—181no—
322C2×C3×C4—181no—
323C2×C3×C4—184no—
324C2×C3×C4—181no—
325C2×C3×C4C22×C3×D311yescharacteristic
326#67—122no—
327#67—362no—
328C2×Dic3—31no—
329C2×Dic3—61no—
330C2×Dic3—91no—
331C2×Dic3—181no—
332D12—32no—
333D12—64no—
334D12—61no—
335D12—92no—
336D12—181no—
337D12—184no—
338C4×D3—64no—
339C4×D3—61no—
340C4×D3—61no—
341C4×D3—181no—
342C4×D3—181no—
343C4×D3—184no—
344C3×C8—21no—
345C3×C8—61no—
346C3×C8—61no—
347C3×C8—62no—
348C3×C8—181no—
349C3⋊C8—61no—
350C3⋊C8—181no—
351C2×C32—21no—
352C2×C32—21no—
353C2×C32—21no—
354C2×C32—82no—
355C2×C32—81no—
356C2×C32—81no—
357C2×C32#91411yescharacteristic
358C2×C32#91611yescharacteristic
359C2×C32#93711yescharacteristic
360C2×C32C3×#12011yescharacteristic
361C2×C32C3×#13511yescharacteristic
362C2×C32C3×AGL(1,ℤ/8)11yescharacteristic
363C3⋊D3—41no—
364C3⋊D3—42no—
365C3×D3—121no—
366C3×D3—122no—
367#41—182no—
368#41—182no—
369C2×Q8C2×(C3⋊D3)⋊C311yescharacteristic
370C2×D4—91no—
371C2×D4—181no—
372C2×D4—181no—
373C2×D4C2×(C3⋊D3)⋊C311yescharacteristic
374C22×C4—91no—
375C22×C4—181no—
376SD16—22no—
377D8—182no—
378C2×C8—91no—
379C2×C8C2×(C3⋊D3)⋊C311yescharacteristic
380C4⋊C4—91no—
381C22⋊C4—181no—
382C22×C3—31no—
383C22×C3—31no—
384C22×C3—41no—
385C22×C3—41no—
386C22×C3—41no—
387C22×C3—61no—
388C22×C3—121no—
389C22×C3—121no—
390C22×C3—121no—
391C22×C3—121no—
392C22×C3—121no—
393C22×C3—121no—
394C22×C3—122no—
395C22×C3—182no—
396C22×C3—181no—
397C22×C3—181no—
398C22×C3—181no—
399C22×C3—241no—
400C22×C3—241no—
401C22×C3—361no—
402C22×C3—361no—
403C22×C3C3×D3×D411yescharacteristic
404C2×D3—61no—
405C2×D3—61no—
406C2×D3—62no—
407C2×D3—61no—
408C2×D3—121no—
409C2×D3—121no—
410C2×D3—182no—
411C2×D3—181no—
412C2×D3—181no—
413C2×D3—181no—
414C2×D3—361no—
415C2×D3—361no—
416C3×C4—22no—
417C3×C4—31no—
418C3×C4—31no—
419C3×C4—31no—
420C3×C4—31no—
421C3×C4—62no—
422C3×C4—62no—
423C3×C4—61no—
424C3×C4—61no—
425C3×C4—122no—
426C3×C4—181no—
427C3×C4—361no—
428C3×C4C3×D3×D411yescharacteristic
429C3×C4C2×C3×#6711yescharacteristic
430Dic3—61no—
431Dic3—121no—
432Dic3—181no—
433Dic3—361no—
434C32—21no—
435C32#517211yescharacteristic, complemented
436C32C3×#44911yescharacteristic, complemented
437C23—21no—
438C23—91no—
439C23—181no—
440Q8—21no—
441Q8#638⋊C312yescomplemented
442D4—21no—
443D4—21no—
444D4—92no—
445D4—184no—
446D4—181no—
447D4—362no—
448C2×C4—21no—
449C2×C4—91no—
450C2×C4—184no—
451C2×C4—181no—
452C2×C4—181no—
453C2×C4—181no—
454C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
455C8—21no—
456C8—181no—
457C2×C3—31no—
458C2×C3—31no—
459C2×C3—31no—
460C2×C3—31no—
461C2×C3—31no—
462C2×C3—31no—
463C2×C3—61no—
464C2×C3—61no—
465C2×C3—61no—
466C2×C3—81no—
467C2×C3—241no—
468C2×C3—241no—
469C2×C3—242no—
470C2×C3—361no—
471C2×C3—362no—
472C2×C3C3×#91411yescharacteristic
473C2×C3C3×#91611yescharacteristic
474C2×C3C3×#93711yescharacteristic
475D3—121no—
476D3—122no—
477D3—361no—
478D3—362no—
479C22—41no—
480C22—41no—
481C22—41no—
482C22—181no—
483C22—181no—
484C22—182no—
485C22—181no—
486C22—361no—
487C22—361no—
488C22D4×(C3⋊D3)⋊C311yescharacteristic
489C4—22no—
490C4—181no—
491C4—361no—
492C4D4×(C3⋊D3)⋊C311yescharacteristic
493C4C2×#638⋊C311yescharacteristic
494C3—31no—
495C3—31no—
496C3—61no—
497C3C3×#517211yescharacteristic
498C2—81no—
499C2—361no—
500C2—362no—
501C2#64123⋊C311yescharacteristic
502C2#64125⋊C311yescharacteristic
503C2#64146⋊C311yescharacteristic
subgroupG/H
centerC22D4×(C3⋊D3)⋊C3
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C3×C4C22×C3×D3
FittingC2×SD16×H(3,𝔽3)C2
socleC22×C3C3×D3×D4
hypercenterC2×SD16(C3⋊D3)⋊C3
hypocenterC32C3×#449
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×SD16

Complements

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

NH ≅ G/Ntype
1. C2×#88722C2semidirect
2. C2×SD16×H(3,𝔽3)C2semidirect
3. #64147⋊C3C2semidirect
4. C2×#3753⋊C3C2semidirect
5. #63662⋊C3C2semidirect
6. #63648⋊C3C2semidirect
7. #63647⋊C3C2semidirect
8. #10592871C3semidirect
25. C2×#3832C2×C3semidirect
28. C2×C32×SD16D3semidirect
29. C2×C32×SD16C2×C3semidirect
30. #64147C2×C3semidirect
31. C2×#3753C2×C3semidirect
34. #63662C2×C3semidirect
35. #63648C2×C3semidirect
36. #63647C2×C3semidirect
44. Q8×H(3,𝔽3)D4semidirect
54. C4⋊((C3⋊D3)⋊C3)D4semidirect
145. C32×Q8#67semidirect
146. C32×Q8C3×D4semidirect
164. C4⋊(C3⋊D3)C3×D4semidirect
276. C2×SD16(C3⋊D3)⋊C3semidirect
282. H(3,𝔽3)#449semidirect
435. C32#5172semidirect
436. C32C3×#449semidirect
441. Q8#638⋊C3semidirect

Maximal subgroups

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

subgroupindex
C2×#887222
C2×SD16×H(3,𝔽3)2
#64147⋊C32
C2×#3753⋊C32
#63662⋊C32
#63648⋊C32
#63647⋊C32
#105928713
C3×#51723