Finite Group Databasebeta

← #13167054

Subgroups — #13167054

Order 1728.

614 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 614 conjugacy classes of subgroups — too many to draw. Only its 79 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
3#11723667C211yesmaximal, characteristic, complemented
4#11722777C214yesmaximal, complemented
5#10593381—31nomaximal
6C2×C3×C4×C3⋊D3C2211yescharacteristic
7C32×#235—21no—
8C3×#3811—21no—
9C3×#3805C2212yes—
10C3×#3802C411yescharacteristic, complemented
11C3×C4×D32—21no—
12C3×#3798C412yescomplemented
13C3×#3797C411yescharacteristic, complemented
14C3×#3796—21no—
15#88993C2212yescomplemented
16#88989—21no—
17#88988—21no—
18C3×C4⋊(C3⋊Dic3)—21no—
19#88814—21no—
20#88813C2212yescomplemented
21C3×D3×#41—61no—
22#64306D311yescharacteristic
23#64283—31no—
24#63849—31no—
25#63841—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—
40C3×#627C2×C412yes—
41C3×#626—22no—
42C3×#625C2×C412yes—
43C3×#624—21no—
44C3×D3×Dic3—22no—
45C3⋊#622—22no—
46#5236—91nomaximal
47C32×#41—61no—
48C2×C4×C3⋊D3C2×D311yescharacteristic
49C3×#239—61no—
50C3×D3×Q8—61no—
51C3×#237—61no—
52C3×#237—62no—
53C3×D3×D4—61no—
54C3×D3×D4—62no—
55C3×#235—21no—
56C3×#235—21no—
57C3×#235—41no—
58C3×#235—62no—
59C2×C3×C4×D3—62no—
60C2×C3×C4×D3—61no—
61C2×C3×C4×D3—61no—
62#3811—21no—
63#3805C2×D312yes—
64#3802Dic311yescharacteristic, complemented
65C4×D32—21no—
66#3798Dic312yescomplemented
67#3797Dic311yescharacteristic, complemented
68#3796—21no—
69#3792—32no—
70#3788—61no—
71#3787—61no—
72C4⋊(C3⋊Dic3)—21no—
73C3×C4⋊Dic3—61no—
74C3×C4⋊Dic3—61no—
75C3×C4⋊Dic3—121no—
76#3722—61no—
77(C2×D3)⋊Dic3—62no—
78Dic32—32no—
79Dic32—62no—
80C22×C33C2×D411yescharacteristic
81C2×C3×C3⋊D3—41no—
82C2×C3×C3⋊D3C22⋊C411yescharacteristic
83C2×C3×C3⋊D3C22⋊C411yescharacteristic
84C2×C32×D3—22no—
85C3×D32—42no—
86C33×C4C22⋊C411yescharacteristic
87C33×C4C22⋊C411yescharacteristic
88C3×C3⋊Dic3—42no—
89C3×C3⋊Dic3C22×C411yescharacteristic
90C3×C3⋊Dic3C2×D411yescharacteristic
91C32×Dic3—22no—
92C2×C3×#41—91no—
93D3×#41—61no—
94#926—91no—
95C2×#208—91no—
96#892—181no—
97#837—94no—
98C22×C3⋊D3#6711yescharacteristic
99C22×C3×D3—61no—
100C22×C3×D3—61no—
101C22×C3×D3—62no—
102C2×D32—21no—
103C32×Q8—61no—
104C32×D4—61no—
105C32×D4—62no—
106C2×C32×C4—21no—
107C2×C32×C4—21no—
108C2×C32×C4—41no—
109C2×C32×C4—41no—
110C2×C32×C4—62no—
111C2×C32×C4#6711yescharacteristic
112C2×C3⋊Dic3—22no—
113C2×C3⋊Dic3C22×D311yescharacteristic
114C4×C3⋊D3C2×Dic311yescharacteristic
115C4×C3⋊D3C2×Dic311yescharacteristic
116C4×C3⋊D3#6711yescharacteristic
117C4×C3⋊D3#6711yescharacteristic
118C3×#67—22no—
119C3×#67—22no—
120C3×#67—42no—
121C3×#67—62no—
122C3×#67—62no—
123C2×C3×Dic3—61no—
124C2×C3×Dic3—61no—
125C2×C3×Dic3—62no—
126C2×C3×Dic3—62no—
127C2×C3×Dic3—62no—
128C2×C3×Dic3—122no—
129C3×D12—21no—
130C3×D12—21no—
131C3×D12—41no—
132C3×D12—62no—
133C3×C4×D3—22no—
134C3×C4×D3—22no—
135C3×C4×D3—42no—
136C3×C4×D3—61no—
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—62no—
145C3×C4×D3—62no—
146C3×Dic6—21no—
147C3×Dic6—21no—
148C3×Dic6—41no—
149C3×Dic6—62no—
150#627C2×Dic312yes—
151#626—22no—
152#625C2×Dic312yes—
153#624—21no—
154D3×Dic3—22no—
155#622—62no—
156#421—271noSylow
157C2×C33—21no—
158C2×C33C2×C22⋊C411yescharacteristic
159C3×C3⋊D3—21no—
160C3×C3⋊D3—21no—
161C32×D3—42no—
162C3×#41—61no—
163C3×#41—61no—
164C3×#41—92no—
165C3×#41—92no—
166C3×#41—121no—
167C3×#41—181no—
168C2×C3×Q8—91no—
169C2×C3×D4—91no—
170C2×C3×D4—181no—
171C22×C3×C4—91no—
172C22×C3×C4—181no—
173#239—61no—
174D3×Q8—61no—
175#237—62no—
176#237—61no—
177D3×D4—61no—
178D3×D4—62no—
179#235—21no—
180#235—62no—
181C2×C4×D3—61no—
182C2×C4×D3—61no—
183C2×C4×D3—62no—
184C3×C4⋊C4—181no—
185#217—181no—
186(C2×D3)⋊C4—182no—
187C4⋊Dic3—61no—
188C4⋊Dic3—61no—
189C4⋊Dic3—121no—
190C4×Dic3—92no—
191C4×Dic3—182no—
192#208—92no—
193#208—181no—
194C2×C3⋊C8—181no—
195C22×C32—21no—
196C22×C32—21no—
197C22×C32—41no—
198C22×C32—41no—
199C22×C32—62no—
200C22×C32C2×#6711yescharacteristic
201C2×C3⋊D3—41no—
202C2×C3⋊D3#21711yescharacteristic
203C2×C3⋊D3#21711yescharacteristic
204C2×C3×D3—22no—
205C2×C3×D3—22no—
206C2×C3×D3—42no—
207C2×C3×D3—61no—
208C2×C3×D3—61no—
209C2×C3×D3—61no—
210C2×C3×D3—61no—
211C2×C3×D3—124no—
212C2×C3×D3—122no—
213C2×C3×D3—121no—
214C2×C3×D3—121no—
215D32—42no—
216C32×C4—21no—
217C32×C4—21no—
218C32×C4—21no—
219C32×C4—21no—
220C32×C4—41no—
221C32×C4—41no—
222C32×C4—41no—
223C32×C4—41no—
224C32×C4—62no—
225C32×C4#21711yescharacteristic
226C32×C4#21711yescharacteristic
227C3⋊Dic3—42no—
228C3⋊Dic3C22×Dic311yescharacteristic
229C3⋊Dic3C2×#6711yescharacteristic
230C3×Dic3—22no—
231C3×Dic3—22no—
232C3×Dic3—42no—
233C3×Dic3—61no—
234C3×Dic3—61no—
235C3×Dic3—61no—
236C3×Dic3—61no—
237C3×Dic3—122no—
238C3×Dic3—122no—
239C3×Dic3—242no—
240C2×#41—91no—
241C2×M16—271no—
242#117—271no—
243#104—274no—
244C33#42111yescharacteristic, complemented, Sylow
245C23×C3—91no—
246C23×C3—181no—
247C22×D3—61no—
248C22×D3—61no—
249C22×D3—62no—
250C3×Q8—61no—
251C3×Q8—61no—
252C3×Q8—92no—
253C3×Q8—121no—
254C3×D4—61no—
255C3×D4—62no—
256C3×D4—62no—
257C3×D4—61no—
258C3×D4—92no—
259C3×D4—121no—
260C3×D4—122no—
261C3×D4—182no—
262C2×C3×C4—21no—
263C2×C3×C4—21no—
264C2×C3×C4—41no—
265C2×C3×C4—41no—
266C2×C3×C4—62no—
267C2×C3×C4—62no—
268C2×C3×C4—91no—
269C2×C3×C4—91no—
270C2×C3×C4—91no—
271C2×C3×C4—91no—
272C2×C3×C4—91no—
273C2×C3×C4—122no—
274C2×C3×C4—181no—
275C2×C3×C4—182no—
276C2×C3×C4—182no—
277C2×C3×C4GO+(4,𝔽2)11yescharacteristic
278#67—22no—
279#67—62no—
280#67—62no—
281C2×Dic3—61no—
282C2×Dic3—61no—
283C2×Dic3—62no—
284C2×Dic3—62no—
285C2×Dic3—62no—
286C2×Dic3—122no—
287D12—21no—
288D12—62no—
289C4×D3—22no—
290C4×D3—61no—
291C4×D3—61no—
292C4×D3—61no—
293C4×D3—61no—
294C4×D3—61no—
295C4×D3—61no—
296C4×D3—61no—
297C4×D3—61no—
298C4×D3—62no—
299C4×D3—62no—
300Dic6—21no—
301Dic6—62no—
302C3⋊C8—182no—
303C2×C32—21no—
304C2×C32—21no—
305C2×C32—21no—
306C2×C32—21no—
307C2×C32—21no—
308C2×C32—41no—
309C2×C32—41no—
310C2×C32—41no—
311C2×C32—41no—
312C2×C32—41no—
313C2×C32—81no—
314C2×C32—122no—
315C2×C32C2×#21711yescharacteristic
316C3⋊D3—21no—
317C3⋊D3—21no—
318C3×D3—42no—
319C3×D3—42no—
320C3×D3—62no—
321C3×D3—82no—
322C3×D3—121no—
323C3×D3—121no—
324C3×D3—121no—
325#41—61no—
326#41—92no—
327#41—92no—
328#41—181no—
329C2×Q8—91no—
330C2×D4—91no—
331C2×D4—181no—
332C22×C4—91no—
333C22×C4—181no—
334M16—272no—
335M16—541no—
336C2×C8—541no—
337C4⋊C4—181no—
338C22⋊C4—541no—
339C42—272no—
340C22×C3—21no—
341C22×C3—21no—
342C22×C3—41no—
343C22×C3—41no—
344C22×C3—62no—
345C22×C3—62no—
346C22×C3—91no—
347C22×C3—91no—
348C22×C3—122no—
349C22×C3—361no—
350C22×C3—362no—
351C22×C3C2×GO+(4,𝔽2)11yescharacteristic
352C2×D3—22no—
353C2×D3—61no—
354C2×D3—61no—
355C2×D3—61no—
356C2×D3—61no—
357C2×D3—124no—
358C2×D3—122no—
359C2×D3—121no—
360C2×D3—121no—
361C3×C4—21no—
362C3×C4—21no—
363C3×C4—21no—
364C3×C4—21no—
365C3×C4—41no—
366C3×C4—41no—
367C3×C4—41no—
368C3×C4—41no—
369C3×C4—62no—
370C3×C4—62no—
371C3×C4—91no—
372C3×C4—91no—
373C3×C4—122no—
374C3×C4—362no—
375C3×C4D32⋊C411yescharacteristic
376C3×C4D32⋊C411yescharacteristic
377Dic3—22no—
378Dic3—61no—
379Dic3—61no—
380Dic3—61no—
381Dic3—61no—
382Dic3—122no—
383Dic3—122no—
384Dic3—242no—
385C32—21no—
386C32—21no—
387C32—41no—
388C32—41no—
389C32#523611yescharacteristic, complemented
390C23—91no—
391C23—181no—
392Q8—61no—
393Q8—92no—
394D4—62no—
395D4—61no—
396D4—92no—
397D4—182no—
398C2×C4—62no—
399C2×C4—91no—
400C2×C4—91no—
401C2×C4—91no—
402C2×C4—91no—
403C2×C4—91no—
404C2×C4—181no—
405C2×C4—182no—
406C2×C4—182no—
407C2×C4D32⋊D311yescharacteristic
408C8—542no—
409C2×C3—21no—
410C2×C3—21no—
411C2×C3—21no—
412C2×C3—21no—
413C2×C3—21no—
414C2×C3—41no—
415C2×C3—41no—
416C2×C3—41no—
417C2×C3—41no—
418C2×C3—41no—
419C2×C3—81no—
420C2×C3—122no—
421C2×C3—122no—
422C2×C3—181no—
423C2×C3—181no—
424C2×C3—242no—
425C2×C3C2×D32⋊C411yescharacteristic
426D3—42no—
427D3—62no—
428D3—121no—
429D3—121no—
430D3—121no—
431C22—62no—
432C22—91no—
433C22—91no—
434C22—361no—
435C22—362no—
436C22C2×D32⋊D311yescharacteristic
437C4—62no—
438C4—91no—
439C4—91no—
440C4—362no—
441C4D32⋊Dic311yescharacteristic
442C4D32⋊Dic311yescharacteristic
443C3—21no—
444C3—21no—
445C3—41no—
446C3—41no—
447C3#1059338111yescharacteristic, complemented
448C2—21no—
449C2—122no—
450C2—181no—
451C2—181no—
452C2C2×D32⋊Dic311yescharacteristic
subgroupG/H
centerC2C2×D32⋊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
hypocenterC33#421
LCSG ▷ C3×C3⋊Dic3 ▷ C2×C33 ▷ C33
UCSC1 ◁ C2 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
2. #11724511C2semidirect
3. #11723667C2semidirect
4. #11722777C2semidirect
10. C3×#3802C4semidirect
12. C3×#3798C4semidirect
13. C3×#3797C4semidirect
15. #88993C22semidirect
20. #88813C22semidirect
64. #3802Dic3semidirect
66. #3798Dic3semidirect
67. #3797Dic3semidirect
244. C33#421semidirect
389. C32#5236semidirect
447. C3#10593381semidirect

Maximal subgroups

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

subgroupindex
C3×#643062
#117245112
#117236672
#117227772
#105933813
#52369