Finite Group Databasebeta

← #13167047

Subgroups — #13167047

Order 1728.

682 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 682 conjugacy classes of subgroups — too many to draw. Only its 75 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
1#11724618C211yesmaximal, characteristic
2#11724511C211yesmaximal, characteristic, complemented
3C3×C4×#624C211yesmaximal, characteristic, complemented
4#11722776C214yesmaximal, complemented
5#10593380—31nomaximal
6C3×#3828—21no—
7C2×C3×C4×C3⋊D3C2211yescharacteristic
8#89049—21no—
9#89046C2212yes—
10C4⋊#1113C411yescharacteristic, complemented
11C4×#1113—21no—
12#89042C412yescomplemented
13#89041—21no—
14#89040C411yescharacteristic, complemented
15C2×C3×#624—21no—
16#88993C2212yescomplemented
17#88989—21no—
18#88988—21no—
19C32×C4×Dic3—21no—
20C3×Dic32C2212yescomplemented
21#64306—31no—
22D3×#235—61no—
23#64283—31no—
24C3×C42×D3—61no—
25C4×#624D311yescharacteristic, complemented
26#63741—34no—
27C22×C3×C3⋊D3D411yescharacteristic
28C2×#1113—21no—
29C2×C33×C4D411yescharacteristic
30C3×#638—22no—
31C2×C3×C3⋊Dic3C2311yescharacteristic
32C3×C4⋊(C3⋊D3)—21no—
33C3×C4×C3⋊D3—22no—
34C3×C4×C3⋊D3C2×C411yescharacteristic
35C3×C4×C3⋊D3C2×C411yescharacteristic
36C3×C4×C3⋊D3D411yescharacteristic
37C3×C4×C3⋊D3D411yescharacteristic
38C3×#634—21no—
39C2×C32×Dic3—22no—
40#6296C2×C412yescomplemented
41#6295—22no—
42#6295C2×C412yescomplemented
43#6294—21no—
44#6294—22no—
45C3×#624—24no—
46C3⋊#622—22no—
47C2×#805—91nomaximal
48#3828—21no—
49C2×C4×C3⋊D3C2×D311yescharacteristic
50C3×#235—61no—
51C3×#235—61no—
52C3×#235—121no—
53C2×C3×C4×D3—62no—
54C2×C3×C4×D3—61no—
55C2×C3×C4×D3—61no—
56#3811—61no—
57D3×#67—62no—
58C2×#624—21no—
59#3805—32no—
60#3804—62no—
61#3802—31no—
62D3×D12—61no—
63C4×D32—62no—
64C4×D32—61no—
65#3799—61no—
66#3798—32no—
67#3798—62no—
68#3797—31no—
69#3796—61no—
70#3795—61no—
71D3×Dic6—61no—
72#3792—32no—
73#3788—61no—
74#3787—61no—
75C32×C42—61no—
76C3×C4×Dic3—21no—
77C3×C4×Dic3—21no—
78C3×C4×Dic3—41no—
79C3×C4×Dic3—62no—
80Dic32C2×D312yescomplemented
81C22×C33C2×D411yescharacteristic
82C2×C3×C3⋊D3—22no—
83C2×C3×C3⋊D3—22no—
84C2×C3×C3⋊D3C22⋊C411yescharacteristic
85C2×C3×C3⋊D3C22⋊C411yescharacteristic
86#1113—42no—
87C33×C4C22⋊C411yescharacteristic
88C33×C4C22⋊C411yescharacteristic
89C3×C3⋊Dic3—22no—
90C3×C3⋊Dic3C22×C411yescharacteristic
91C3×C3⋊Dic3C2×D411yescharacteristic
92C32×Dic3—24no—
93D3×#41—181no—
94C2×#235—91no—
95C2×C3×C42—91no—
96C2×#208—91no—
97C42×D3—61no—
98#805—94no—
99C22×C3⋊D3#6711yescharacteristic
100C22×C3×D3—61no—
101C22×C3×D3—61no—
102C2×D32—62no—
103C2×D32—61no—
104C2×C32×C4—21no—
105C2×C32×C4—21no—
106C2×C32×C4—41no—
107C2×C32×C4—41no—
108C2×C32×C4—62no—
109C2×C32×C4#6711yescharacteristic
110#638—22no—
111C2×C3⋊Dic3C22×D311yescharacteristic
112C4⋊(C3⋊D3)—21no—
113C4×C3⋊D3—22no—
114C4×C3⋊D3C4×D311yescharacteristic
115C4×C3⋊D3C4×D311yescharacteristic
116C4×C3⋊D3D1211yescharacteristic
117C4×C3⋊D3D1211yescharacteristic
118#634—21no—
119C3×#67—62no—
120C3×#67—62no—
121C3×#67—122no—
122C2×C3×Dic3—22no—
123C2×C3×Dic3—22no—
124C2×C3×Dic3—42no—
125C2×C3×Dic3—61no—
126C2×C3×Dic3—61no—
127C3×D12—61no—
128C3×D12—61no—
129C3×D12—121no—
130C3×C4×D3—61no—
131C3×C4×D3—61no—
132C3×C4×D3—61no—
133C3×C4×D3—61no—
134C3×C4×D3—61no—
135C3×C4×D3—61no—
136C3×C4×D3—61no—
137C3×C4×D3—61no—
138C3×C4×D3—68no—
139C3×C4×D3—62no—
140C3×C4×D3—62no—
141C3×C4×D3—122no—
142C3×Dic6—61no—
143C3×Dic6—61no—
144C3×Dic6—121no—
145#627—32no—
146#627—62no—
147#626—62no—
148#626—62no—
149#626—62no—
150#625—32no—
151#625—62no—
152#624—24no—
153#624—62no—
154#624—61no—
155D3×Dic3—62no—
156D3×Dic3—62no—
157D3×Dic3—62no—
158#622—62no—
159C2×#104—271noSylow
160C2×C33—21no—
161C2×C33C2×C22⋊C411yescharacteristic
162C3×C3⋊D3—21no—
163C3×C3⋊D3—42no—
164C3×C3⋊D3#10412yescomplemented
165C3×#41—181no—
166C22×C3×C4—91no—
167C22×C3×C4—181no—
168C2×#67—181no—
169#239—181no—
170D3×Q8—181no—
171#237—182no—
172#237—181no—
173D3×D4—181no—
174D3×D4—182no—
175#235—61no—
176#235—61no—
177#235—92no—
178#235—92no—
179#235—121no—
180#235—182no—
181#235—181no—
182C2×D12—91no—
183C2×C4×D3—61no—
184C2×C4×D3—61no—
185C2×C4×D3—62no—
186C2×C4×D3—181no—
187C2×C4×D3—182no—
188C2×Dic6—91no—
189C3×C42—61no—
190C3×C42—61no—
191C3×C42—92no—
192C3×C42—121no—
193C4×Dic3—21no—
194C4×Dic3—62no—
195#208—92no—
196#208—181no—
197C2×C3⋊C8—181no—
198C22×C32—21no—
199C22×C32—21no—
200C22×C32—41no—
201C22×C32—41no—
202C22×C32C2×#6711yescharacteristic
203C2×C3⋊D3—22no—
204C2×C3⋊D3—22no—
205C2×C3⋊D3(C2×D3)⋊C411yescharacteristic
206C2×C3⋊D3(C2×D3)⋊C411yescharacteristic
207C2×C3×D3—61no—
208C2×C3×D3—61no—
209C2×C3×D3—61no—
210C2×C3×D3—61no—
211C2×C3×D3—64no—
212C2×C3×D3—62no—
213C2×C3×D3—62no—
214C2×C3×D3—122no—
215C2×C3×D3—122no—
216D32—124no—
217D32—122no—
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—64no—
227C32×C4(C2×D3)⋊C411yescharacteristic
228C32×C4(C2×D3)⋊C411yescharacteristic
229C3⋊Dic3—22no—
230C3⋊Dic3C2×C4×D311yescharacteristic
231C3⋊Dic3C2×D1211yescharacteristic
232C3×Dic3—24no—
233C3×Dic3—24no—
234C3×Dic3—44no—
235C3×Dic3—61no—
236C3×Dic3—61no—
237C3×Dic3—61no—
238C3×Dic3—61no—
239C3×Dic3—62no—
240C3×Dic3—62no—
241C3×Dic3—122no—
242C2×#41—271no—
243C2×M16—271no—
244C2×C42—91no—
245#104—274no—
246C33C2×#10411yescharacteristic, complemented, Sylow
247C23×C3—91no—
248C22×D3—61no—
249C22×D3—61no—
250C22×D3—181no—
251C22×D3—182no—
252C3×Q8—181no—
253C3×D4—181no—
254C3×D4—182no—
255C2×C3×C4—21no—
256C2×C3×C4—21no—
257C2×C3×C4—41no—
258C2×C3×C4—41no—
259C2×C3×C4—62no—
260C2×C3×C4—62no—
261C2×C3×C4—91no—
262C2×C3×C4—91no—
263C2×C3×C4—91no—
264C2×C3×C4—91no—
265C2×C3×C4—91no—
266C2×C3×C4—122no—
267C2×C3×C4—182no—
268C2×C3×C4—184no—
269C2×C3×C4GO+(4,𝔽2)11yescharacteristic
270#67—62no—
271#67—62no—
272#67—122no—
273#67—182no—
274#67—182no—
275#67—182no—
276C2×Dic3—22no—
277C2×Dic3—61no—
278C2×Dic3—61no—
279C2×Dic3—181no—
280C2×Dic3—182no—
281D12—61no—
282D12—61no—
283D12—92no—
284D12—121no—
285D12—182no—
286C4×D3—61no—
287C4×D3—61no—
288C4×D3—61no—
289C4×D3—61no—
290C4×D3—61no—
291C4×D3—61no—
292C4×D3—61no—
293C4×D3—61no—
294C4×D3—68no—
295C4×D3—62no—
296C4×D3—62no—
297C4×D3—122no—
298C4×D3—182no—
299C4×D3—182no—
300C4×D3—182no—
301Dic6—61no—
302Dic6—61no—
303Dic6—92no—
304Dic6—121no—
305Dic6—182no—
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×C32C2×(C2×D3)⋊C411yescharacteristic
317C3⋊D3—21no—
318C3⋊D3—42no—
319C3⋊D3#80512yescomplemented
320C3×D3—62no—
321C3×D3—62no—
322C3×D3—62no—
323C3×D3—121no—
324C3×D3—122no—
325C3×D3—122no—
326C3×D3—242no—
327#41—181no—
328#41—272no—
329#41—272no—
330#41—541no—
331C2×Q8—271no—
332C2×D4—271no—
333C2×D4—541no—
334C22×C4—91no—
335C22×C4—181no—
336C22×C4—541no—
337M16—272no—
338M16—541no—
339C2×C8—541no—
340C42—61no—
341C42—92no—
342C22×C3—21no—
343C22×C3—21no—
344C22×C3—41no—
345C22×C3—41no—
346C22×C3—91no—
347C22×C3—91no—
348C22×C3—182no—
349C22×C3—182no—
350C22×C3C2×GO+(4,𝔽2)11yescharacteristic
351C2×D3—61no—
352C2×D3—61no—
353C2×D3—61no—
354C2×D3—61no—
355C2×D3—64no—
356C2×D3—62no—
357C2×D3—62no—
358C2×D3—122no—
359C2×D3—122no—
360C2×D3—364no—
361C2×D3—362no—
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—64no—
371C3×C4—64no—
372C3×C4—91no—
373C3×C4—91no—
374C3×C4—124no—
375C3×C4—182no—
376C3×C4D32⋊C411yescharacteristic
377C3×C4D32⋊C411yescharacteristic
378Dic3—24no—
379Dic3—61no—
380Dic3—61no—
381Dic3—61no—
382Dic3—61no—
383Dic3—62no—
384Dic3—62no—
385Dic3—122no—
386C32—21no—
387C32—21no—
388C32—41no—
389C32—41no—
390C32C2×#80511yescharacteristic, complemented
391C23—91no—
392C23—541no—
393Q8—181no—
394Q8—272no—
395D4—182no—
396D4—181no—
397D4—272no—
398D4—542no—
399C2×C4—62no—
400C2×C4—91no—
401C2×C4—91no—
402C2×C4—91no—
403C2×C4—91no—
404C2×C4—91no—
405C2×C4—184no—
406C2×C4—182no—
407C2×C4—542no—
408C2×C4—541no—
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—181no—
422C2×C3—362no—
423C2×C3C2×D32⋊C411yescharacteristic
424D3—62no—
425D3—62no—
426D3—62no—
427D3—121no—
428D3—122no—
429D3—122no—
430D3—242no—
431C22—91no—
432C22—91no—
433C22—182no—
434C22—182no—
435C22—1082no—
436C22C2×D32⋊D311yescharacteristic
437C4—64no—
438C4—91no—
439C4—91no—
440C4—182no—
441C4#8893211yescharacteristic
442C4#8893211yescharacteristic
443C3—21no—
444C3—21no—
445C3—41no—
446C3—41no—
447C3#1059338011yescharacteristic, complemented
448C2—21no—
449C2—92no—
450C2—181no—
451C2—362no—
452C2C2×#8893211yescharacteristic
subgroupG/H
centerC4#88932
commutatorC3×C3⋊Dic3C22×C4
FrattiniC22C2×D32⋊D3
FittingC2×C33×C4D4
socleC2×C33C2×C22⋊C4
prime radical#11724618C2
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. C3×C4×#624C2semidirect
4. #11722776C2semidirect
10. C4⋊#1113C4semidirect
12. #89042C4semidirect
14. #89040C4semidirect
16. #88993C22semidirect
20. C3×Dic32C22semidirect
25. C4×#624D3semidirect
40. #6296C2×C4semidirect
42. #6295C2×C4semidirect
80. Dic32C2×D3semidirect
164. C3×C3⋊D3#104semidirect
246. C33C2×#104semidirect
319. C3⋊D3#805semidirect
390. C32C2×#805semidirect
447. C3#10593380semidirect

Maximal subgroups

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

subgroupindex
#117246182
#117245112
C3×C4×#6242
#117227762
#105933803
C2×#8059