Finite Group Databasebeta

← #13167053

Subgroups — #13167053

Order 1728.

682 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 682 conjugacy classes of subgroups — too many to draw. Only its 83 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
1C3×#64306C211yesmaximal, characteristic
2#11724511C211yesmaximal, characteristic, complemented
3C4×#6294C211yesmaximal, characteristic, complemented
4#11722777C214yesmaximal, complemented
5#10593380—31nomaximal
6C2×C3×C4×C3⋊D3C2211yescharacteristic
7C32×#235—21no—
8C2×#6294—21no—
9C3×#3811—21no—
10C3×#3805C2212yes—
11C3×#3802C411yescharacteristic, complemented
12C3×C4×D32—21no—
13C3×#3798C412yescomplemented
14C3×#3797C411yescharacteristic, complemented
15C3×#3796—21no—
16#88993C2212yescomplemented
17#88989—21no—
18#88988—21no—
19C3×C4×C3⋊Dic3—21no—
20#88813C2212yescomplemented
21C3×D3×#41—61no—
22#64306D311yescharacteristic
23#64283—31no—
24C4×#624—31no—
25C4×D3×Dic3—61no—
26#63741—34no—
27C22×C3×C3⋊D3D411yescharacteristic
28C2×C3×D32—21no—
29C2×C33×C4D411yescharacteristic
30C2×C3×C3⋊Dic3—22no—
31C2×C3×C3⋊Dic3C2311yescharacteristic
32C3×C4×C3⋊D3C2×C411yescharacteristic
33C3×C4×C3⋊D3C2×C411yescharacteristic
34C3×C4×C3⋊D3D411yescharacteristic
35C3×C4×C3⋊D3D411yescharacteristic
36C32×#67—22no—
37C32×D12—21no—
38C32×C4×D3—22no—
39C32×Dic6—21no—
40#6294—24no—
41C3×#627C2×C412yescomplemented
42C3×#626—22no—
43C3×#625C2×C412yescomplemented
44C3×#624—21no—
45C3×D3×Dic3—22no—
46C3⋊#622—22no—
47C2×#837—91nomaximal
48C32×#41—61no—
49C2×C4×C3⋊D3C2×D311yescharacteristic
50C3×#239—61no—
51C3×D3×Q8—61no—
52C3×#237—61no—
53C3×#237—62no—
54C3×D3×D4—61no—
55C3×D3×D4—62no—
56C3×#235—21no—
57C3×#235—21no—
58C3×#235—41no—
59C3×#235—62no—
60C2×C3×C4×D3—62no—
61C2×C3×C4×D3—61no—
62C2×C3×C4×D3—61no—
63#3811—21no—
64C2×#624—61no—
65#3805C2×D312yes—
66C2×D3×Dic3—62no—
67#3802Dic311yescharacteristic, complemented
68C4×D32—21no—
69#3798Dic312yescomplemented
70#3797Dic311yescharacteristic, complemented
71#3796—21no—
72#3792—32no—
73#3788—61no—
74#3787—61no—
75C4×C3⋊Dic3—21no—
76C3×C4×Dic3—61no—
77C3×C4×Dic3—61no—
78C3×C4×Dic3—121no—
79Dic32—32no—
80Dic32—62no—
81C22×C33C2×D411yescharacteristic
82C2×C3×C3⋊D3—22no—
83C2×C3×C3⋊D3C22⋊C411yescharacteristic
84C2×C3×C3⋊D3C22⋊C411yescharacteristic
85C2×C32×D3—22no—
86C3×D32—42no—
87C33×C4C22⋊C411yescharacteristic
88C33×C4C22⋊C411yescharacteristic
89C3×C3⋊Dic3—24no—
90C3×C3⋊Dic3C22×C411yescharacteristic
91C3×C3⋊Dic3C2×D411yescharacteristic
92C32×Dic3—22no—
93C2×C3×#41—91no—
94D3×#41—61no—
95C2×C4×Dic3—91no—
96C2×#208—91no—
97C42×D3—181no—
98#837—94no—
99C22×C3⋊D3#6711yescharacteristic
100C22×C3×D3—61no—
101C22×C3×D3—61no—
102C22×C3×D3—62no—
103C2×D32—21no—
104C32×Q8—61no—
105C32×D4—61no—
106C32×D4—62no—
107C2×C32×C4—21no—
108C2×C32×C4—21no—
109C2×C32×C4—41no—
110C2×C32×C4—41no—
111C2×C32×C4—62no—
112C2×C32×C4#6711yescharacteristic
113C2×C3⋊Dic3—22no—
114C2×C3⋊Dic3C22×D311yescharacteristic
115C4×C3⋊D3C2×Dic311yescharacteristic
116C4×C3⋊D3C2×Dic311yescharacteristic
117C4×C3⋊D3#6711yescharacteristic
118C4×C3⋊D3#6711yescharacteristic
119C3×#67—22no—
120C3×#67—22no—
121C3×#67—42no—
122C3×#67—62no—
123C3×#67—62no—
124C2×C3×Dic3—61no—
125C2×C3×Dic3—61no—
126C2×C3×Dic3—62no—
127C2×C3×Dic3—62no—
128C2×C3×Dic3—62no—
129C2×C3×Dic3—122no—
130C3×D12—21no—
131C3×D12—21no—
132C3×D12—41no—
133C3×D12—62no—
134C3×C4×D3—22no—
135C3×C4×D3—22no—
136C3×C4×D3—42no—
137C3×C4×D3—61no—
138C3×C4×D3—61no—
139C3×C4×D3—61no—
140C3×C4×D3—61no—
141C3×C4×D3—61no—
142C3×C4×D3—61no—
143C3×C4×D3—61no—
144C3×C4×D3—61no—
145C3×C4×D3—62no—
146C3×C4×D3—62no—
147C3×Dic6—21no—
148C3×Dic6—21no—
149C3×Dic6—41no—
150C3×Dic6—62no—
151#627C2×Dic312yescomplemented
152#626—22no—
153#625C2×Dic312yescomplemented
154#624—21no—
155#624—64no—
156D3×Dic3—22no—
157D3×Dic3—68no—
158#622—62no—
159C2×#104—271noSylow
160C2×C33—21no—
161C2×C33C2×C22⋊C411yescharacteristic
162C3×C3⋊D3—21no—
163C3×C3⋊D3#10412yescomplemented
164C32×D3—42no—
165C3×#41—61no—
166C3×#41—61no—
167C3×#41—92no—
168C3×#41—92no—
169C3×#41—121no—
170C3×#41—181no—
171C2×C3×Q8—91no—
172C2×C3×D4—91no—
173C2×C3×D4—181no—
174C22×C3×C4—91no—
175C22×C3×C4—181no—
176C22×Dic3—181no—
177#239—61no—
178D3×Q8—61no—
179#237—62no—
180#237—61no—
181D3×D4—61no—
182D3×D4—62no—
183#235—21no—
184#235—62no—
185C2×C4×D3—61no—
186C2×C4×D3—61no—
187C2×C4×D3—62no—
188C2×C4×D3—182no—
189C3×C42—181no—
190C4×Dic3—61no—
191C4×Dic3—61no—
192C4×Dic3—92no—
193C4×Dic3—121no—
194C4×Dic3—182no—
195#208—92no—
196#208—181no—
197C2×C3⋊C8—181no—
198C22×C32—21no—
199C22×C32—21no—
200C22×C32—41no—
201C22×C32—41no—
202C22×C32—62no—
203C22×C32C2×#6711yescharacteristic
204C2×C3⋊D3—22no—
205C2×C3⋊D3#21711yescharacteristic
206C2×C3⋊D3#21711yescharacteristic
207C2×C3×D3—22no—
208C2×C3×D3—22no—
209C2×C3×D3—42no—
210C2×C3×D3—61no—
211C2×C3×D3—61no—
212C2×C3×D3—61no—
213C2×C3×D3—61no—
214C2×C3×D3—64no—
215C2×C3×D3—122no—
216C2×C3×D3—124no—
217D32—42no—
218C32×C4—21no—
219C32×C4—21no—
220C32×C4—21no—
221C32×C4—21no—
222C32×C4—41no—
223C32×C4—41no—
224C32×C4—41no—
225C32×C4—41no—
226C32×C4—62no—
227C32×C4#21711yescharacteristic
228C32×C4#21711yescharacteristic
229C3⋊Dic3—24no—
230C3⋊Dic3C22×Dic311yescharacteristic
231C3⋊Dic3C2×#6711yescharacteristic
232C3×Dic3—22no—
233C3×Dic3—22no—
234C3×Dic3—42no—
235C3×Dic3—61no—
236C3×Dic3—61no—
237C3×Dic3—61no—
238C3×Dic3—61no—
239C3×Dic3—64no—
240C3×Dic3—64no—
241C3×Dic3—124no—
242C2×#41—91no—
243C2×M16—271no—
244C2×C42—271no—
245#104—274no—
246C33C2×#10411yescharacteristic, complemented, Sylow
247C23×C3—91no—
248C23×C3—181no—
249C22×D3—61no—
250C22×D3—61no—
251C22×D3—62no—
252C3×Q8—61no—
253C3×Q8—61no—
254C3×Q8—92no—
255C3×Q8—121no—
256C3×D4—61no—
257C3×D4—62no—
258C3×D4—62no—
259C3×D4—61no—
260C3×D4—92no—
261C3×D4—121no—
262C3×D4—122no—
263C3×D4—182no—
264C2×C3×C4—21no—
265C2×C3×C4—21no—
266C2×C3×C4—41no—
267C2×C3×C4—41no—
268C2×C3×C4—62no—
269C2×C3×C4—62no—
270C2×C3×C4—91no—
271C2×C3×C4—91no—
272C2×C3×C4—91no—
273C2×C3×C4—91no—
274C2×C3×C4—91no—
275C2×C3×C4—122no—
276C2×C3×C4—181no—
277C2×C3×C4—182no—
278C2×C3×C4—182no—
279C2×C3×C4GO+(4,𝔽2)11yescharacteristic
280#67—22no—
281#67—62no—
282#67—62no—
283C2×Dic3—61no—
284C2×Dic3—61no—
285C2×Dic3—62no—
286C2×Dic3—62no—
287C2×Dic3—62no—
288C2×Dic3—122no—
289C2×Dic3—184no—
290D12—21no—
291D12—62no—
292C4×D3—22no—
293C4×D3—61no—
294C4×D3—61no—
295C4×D3—61no—
296C4×D3—61no—
297C4×D3—61no—
298C4×D3—61no—
299C4×D3—61no—
300C4×D3—61no—
301C4×D3—62no—
302C4×D3—62no—
303C4×D3—188no—
304Dic6—21no—
305Dic6—62no—
306C3⋊C8—182no—
307C2×C32—22no—
308C2×C32—21no—
309C2×C32—21no—
310C2×C32—21no—
311C2×C32—41no—
312C2×C32—41no—
313C2×C32—41no—
314C2×C32—42no—
315C2×C32—81no—
316C2×C32—122no—
317C2×C32C2×#21711yescharacteristic
318C3⋊D3—21no—
319C3⋊D3#83712yescomplemented
320C3×D3—42no—
321C3×D3—42no—
322C3×D3—62no—
323C3×D3—62no—
324C3×D3—62no—
325C3×D3—82no—
326C3×D3—121no—
327#41—61no—
328#41—92no—
329#41—92no—
330#41—181no—
331C2×Q8—91no—
332C2×D4—91no—
333C2×D4—181no—
334C22×C4—91no—
335C22×C4—181no—
336C22×C4—541no—
337M16—272no—
338M16—541no—
339C2×C8—541no—
340C42—181no—
341C42—272no—
342C22×C3—21no—
343C22×C3—21no—
344C22×C3—41no—
345C22×C3—41no—
346C22×C3—62no—
347C22×C3—62no—
348C22×C3—91no—
349C22×C3—91no—
350C22×C3—122no—
351C22×C3—182no—
352C22×C3—362no—
353C22×C3C2×GO+(4,𝔽2)11yescharacteristic
354C2×D3—22no—
355C2×D3—61no—
356C2×D3—61no—
357C2×D3—61no—
358C2×D3—61no—
359C2×D3—64no—
360C2×D3—122no—
361C2×D3—124no—
362C3×C4—21no—
363C3×C4—21no—
364C3×C4—21no—
365C3×C4—21no—
366C3×C4—41no—
367C3×C4—41no—
368C3×C4—41no—
369C3×C4—41no—
370C3×C4—62no—
371C3×C4—62no—
372C3×C4—91no—
373C3×C4—91no—
374C3×C4—122no—
375C3×C4—184no—
376C3×C4D32⋊C411yescharacteristic
377C3×C4D32⋊C411yescharacteristic
378Dic3—22no—
379Dic3—61no—
380Dic3—61no—
381Dic3—61no—
382Dic3—61no—
383Dic3—64no—
384Dic3—64no—
385Dic3—124no—
386C32—21no—
387C32—21no—
388C32—41no—
389C32—41no—
390C32C2×#83711yescharacteristic, complemented
391C23—91no—
392C23—181no—
393Q8—61no—
394Q8—92no—
395D4—62no—
396D4—61no—
397D4—92no—
398D4—182no—
399C2×C4—62no—
400C2×C4—91no—
401C2×C4—91no—
402C2×C4—91no—
403C2×C4—91no—
404C2×C4—91no—
405C2×C4—181no—
406C2×C4—182no—
407C2×C4—182no—
408C2×C4—544no—
409C2×C4D32⋊D311yescharacteristic
410C8—542no—
411C2×C3—22no—
412C2×C3—21no—
413C2×C3—21no—
414C2×C3—21no—
415C2×C3—41no—
416C2×C3—42no—
417C2×C3—41no—
418C2×C3—41no—
419C2×C3—81no—
420C2×C3—92no—
421C2×C3—122no—
422C2×C3—122no—
423C2×C3—181no—
424C2×C3—242no—
425C2×C3C2×D32⋊C411yescharacteristic
426D3—42no—
427D3—62no—
428D3—62no—
429D3—62no—
430D3—121no—
431C22—62no—
432C22—91no—
433C22—91no—
434C22—182no—
435C22—362no—
436C22C2×D32⋊D311yescharacteristic
437C4—62no—
438C4—91no—
439C4—91no—
440C4—184no—
441C4D32⋊Dic311yescharacteristic
442C4D32⋊Dic311yescharacteristic
443C3—21no—
444C3—21no—
445C3—41no—
446C3—41no—
447C3#1059338011yescharacteristic, complemented
448C2—21no—
449C2—92no—
450C2—122no—
451C2—181no—
452C2C2×D32⋊Dic311yescharacteristic
subgroupG/H
centerC4D32⋊Dic3
commutatorC3×C3⋊Dic3C22×C4
FrattiniC22C2×D32⋊D3
FittingC2×C33×C4D4
socleC2×C33C2×C22⋊C4
prime radicalC3×#64306C2
cosocleC2×C3×C3⋊Dic3C23
hypercenterC2×C4D32⋊D3
hypocenterC33C2×#104
LCSG ▷ C3×C3⋊Dic3 ▷ C2×C33 ▷ C33
UCSC1 ◁ C4 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
2. #11724511C2semidirect
3. C4×#6294C2semidirect
4. #11722777C2semidirect
11. C3×#3802C4semidirect
13. C3×#3798C4semidirect
14. C3×#3797C4semidirect
16. #88993C22semidirect
20. #88813C22semidirect
41. C3×#627C2×C4semidirect
43. C3×#625C2×C4semidirect
67. #3802Dic3semidirect
69. #3798Dic3semidirect
70. #3797Dic3semidirect
151. #627C2×Dic3semidirect
153. #625C2×Dic3semidirect
163. C3×C3⋊D3#104semidirect
246. C33C2×#104semidirect
319. C3⋊D3#837semidirect
390. C32C2×#837semidirect
447. C3#10593380semidirect

Maximal subgroups

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

subgroupindex
C3×#643062
#117245112
C4×#62942
#117227772
#105933803
C2×#8379