Finite Group Databasebeta

← #13159360

Subgroups — #13159360

Order 1728.

1004 conjugacy classes of subgroups; only the 170 normal subgroups are shown.

normal subgroups only

This group has 1004 conjugacy classes of subgroups — too many to draw. Only its 170 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×#64102C211yesmaximal, characteristic, complemented
2#11723494C211yesmaximal, characteristic, complemented
3C2×#88791C211yesmaximal, characteristic
4#11722179C211yesmaximal, characteristic, complemented
5C32×#807C211yesmaximal, characteristic, complemented
6#11722033C211yesmaximal, characteristic, complemented
7C3⋊C8×C3⋊Dic3C211yesmaximal, characteristic, complemented
8#10592571—31nomaximal
9#10590456—34nomaximal
10C2×C4⋊(C3⋊(C3⋊D3))C2211yescharacteristic
11C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
12C2×C3×C4×C3⋊D3—21no—
13C2×#6289—21no—
14C33×C4⋊C4C2211yescharacteristic
15C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
16C3×#3752—21no—
17C3×C4×C3⋊Dic3C2211yescharacteristic
18C32×C4⋊Dic3C2211yescharacteristic
19C2×C32×C3⋊C8C2211yescharacteristic
20#88804—21no—
21#88791C414yescomplemented
22C2×#3745—31no—
23#64102—31no—
24#64102D311yescharacteristic, complemented
25C3×#893—34no—
26#63894—34no—
27C2×#3714—34no—
28C32×#107—31no—
29C8×C3⋊Dic3—31no—
30#63636—31no—
31#63636—31no—
32#63636—34no—
33#63636—64no—
34C3×#807D314yescomplemented
35#63562—34no—
36Dic3×C3⋊C8—34no—
37C22×C3⋊(C3⋊D3)—21no—
38C22×C3×C3⋊D3—21no—
39C2×C33×C4—21no—
40C2×C33×C4C2311yescharacteristic
41C4⋊(C3⋊(C3⋊D3))—21no—
42C4⋊(C3⋊(C3⋊D3))C2×C412yescomplemented
43C2×C3×C3⋊Dic3D411yescharacteristic
44C2×C3×C3⋊Dic3D411yescharacteristic
45C3×C4⋊(C3⋊D3)—21no—
46C3×C4⋊(C3⋊D3)C2×C412yescomplemented
47C3×C4×C3⋊D3—42no—
48C2×C32×Dic3—21no—
49#6289—42no—
50C3⋊(C3⋊(C3⋊C8))—21no—
51C32×C3⋊C8C2×C412yescomplemented
52C4×#213—91no—
53#4872—94no—
54C2×C4⋊(C3⋊D3)—31no—
55C2×C4⋊(C3⋊D3)—34no—
56C2×C4⋊(C3⋊D3)—64no—
57C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
58C2×C4×C3⋊D3—21no—
59C2×C4×C3⋊D3—61no—
60C2×C3×D12—34no—
61C2×C3×C4×D3—64no—
62C2×#624—64no—
63C2×C32×C8—31no—
64C32×C4⋊C4—24no—
65C32×C4⋊C4—31no—
66C32×C4⋊C4C2×D314yes—
67C32×C4⋊C4C2×D311yescharacteristic
68#3752—21no—
69#3752—61no—
70C4×C3⋊Dic3C2×D311yescharacteristic
71C2×C3⋊(C3⋊C8)—31no—
72C2×C3⋊(C3⋊C8)—34no—
73C2×C3⋊(C3⋊C8)—64no—
74#3745—34no—
75C3×(C2×D3)⋊C4—64no—
76C3×C4⋊Dic3C2×D314yes—
77C3×C4×Dic3—34no—
78C2×C3×C3⋊C8C2×D314yes—
79#3722—64no—
80#3714—316no—
81C22×C33C2×D411yescharacteristic
82C2×C3⋊(C3⋊D3)—22no—
83C2×C3⋊(C3⋊D3)—41no—
84C2×C3⋊(C3⋊D3)—41no—
85C2×C3×C3⋊D3—22no—
86C2×C3×C3⋊D3—41no—
87C2×C3×C3⋊D3—41no—
88C33×C4—41no—
89C33×C4C22×C411yescharacteristic
90C33×C4#4111yescharacteristic
91C3×C3⋊Dic3—21no—
92C3×C3⋊Dic3D812yescomplemented
93C32×Dic3—41no—
94C3×C4×D4—91no—
95C2×#213—91no—
96C2×D24—94no—
97#893—34no—
98#893—94no—
99C4×D12—91no—
100C3×#107—34no—
101#832—91no—
102C8×Dic3—94no—
103#809—94no—
104#809—94no—
105#809—91no—
106#809—184no—
107#807C3⋊D311yescharacteristic, complemented
108C4×C3⋊C8—91no—
109C22×C3⋊D3—21no—
110C22×C3⋊D3—61no—
111C22×C3⋊D3—64no—
112C22×C3⋊D3—124no—
113C22×C3×D3—64no—
114C2×C32×C4—21no—
115C2×C32×C4—24no—
116C2×C32×C4—24no—
117C2×C32×C4—44no—
118C2×C32×C4—61no—
119C2×C32×C4C22×D314yes—
120C2×C32×C4C22×D311yescharacteristic
121C2×C3⋊Dic3#6711yescharacteristic
122C2×C3⋊Dic3#6711yescharacteristic
123C4⋊(C3⋊D3)—21no—
124C4⋊(C3⋊D3)—32no—
125C4⋊(C3⋊D3)—38no—
126C4⋊(C3⋊D3)—68no—
127C4⋊(C3⋊D3)—61no—
128C4⋊(C3⋊D3)—64no—
129C4⋊(C3⋊D3)—124no—
130C4⋊(C3⋊D3)C4×D312yescomplemented
131C4×C3⋊D3—42no—
132C4×C3⋊D3—122no—
133C2×C3×Dic3—24no—
134C2×C3×Dic3—34no—
135C2×C3×Dic3—34no—
136C3×D12—38no—
137C3×D12—64no—
138C3×C4×D3—128no—
139#624—128no—
140C32×C8—32no—
141C3⋊(C3⋊C8)—61no—
142C3⋊(C3⋊C8)—64no—
143C3⋊(C3⋊C8)—124no—
144C3×C3⋊C8C4×D318yescomplemented
145C4×D8—271noSylow
146C2×C33C4×D411yescharacteristic
147C2×C33C2×D811yescharacteristic
148C2×C33#13511yescharacteristic
149C3⋊(C3⋊D3)—42no—
150C3×C3⋊D3—42no—
151C2×C3×D4—91no—
152C22×C3×C4—181no—
153C2×D12—34no—
154C2×D12—94no—
155C2×D12—91no—
156C2×D12—184no—
157C2×C4×D3—64no—
158C2×C4×D3—181no—
159C2×C4×D3—184no—
160C2×C3×C8—34no—
161C3×C4⋊C4—24no—
162C3×C4⋊C4—34no—
163C3×C4⋊C4D3214yes—
164C3×C4⋊C4C2×C3⋊D311yescharacteristic
165C3×C22⋊C4—181no—
166C3×C42—91no—
167#213—94no—
168(C2×D3)⋊C4—64no—
169(C2×D3)⋊C4—181no—
170(C2×D3)⋊C4—184no—
171C4⋊Dic3C2×C3⋊D311yescharacteristic
172C4×Dic3—34no—
173C2×C3⋊C8—94no—
174C2×C3⋊C8—91no—
175C2×C3⋊C8—184no—
176C2×C3⋊C8C2×C3⋊D311yescharacteristic
177D24—916no—
178C22×C32—24no—
179C22×C32D3×D414yes—
180C22×C32C2×#6711yescharacteristic
181C2×C3⋊D3—22no—
182C2×C3⋊D3—41no—
183C2×C3⋊D3—41no—
184C2×C3⋊D3—62no—
185C2×C3⋊D3—68no—
186C2×C3⋊D3—128no—
187C2×C3⋊D3—121no—
188C2×C3⋊D3—121no—
189C2×C3⋊D3—124no—
190C2×C3⋊D3—124no—
191C2×C3⋊D3—244no—
192C2×C3⋊D3—244no—
193C2×C3×D3—68no—
194C2×C3×D3—124no—
195C2×C3×D3—124no—
196C32×C4—24no—
197C32×C4—24no—
198C32×C4—41no—
199C32×C4—44no—
200C32×C4—84no—
201C32×C4—121no—
202C32×C4C2×C4×D314yes—
203C32×C4C2×C4×D311yescharacteristic
204C32×C4#23511yescharacteristic
205C32×C4#23914yes—
206C3⋊Dic3—21no—
207C3⋊Dic3#21312yescomplemented
208C3×Dic3—38no—
209C3×Dic3—44no—
210C3×Dic3—64no—
211C2×D8—271no—
212C4×D4—91no—
213C4×D4—271no—
214#107—31no—
215#102—271no—
216#102—271no—
217C4×C8—271no—
218C33C4×D811yescharacteristic, complemented, Sylow
219C23×C3—181no—
220C22×D3—64no—
221C22×D3—184no—
222C22×D3—181no—
223C22×D3—364no—
224C3×D4—92no—
225C3×D4—181no—
226C2×C3×C4—24no—
227C2×C3×C4—21no—
228C2×C3×C4—24no—
229C2×C3×C4—44no—
230C2×C3×C4—64no—
231C2×C3×C4—91no—
232C2×C3×C4—91no—
233C2×C3×C4—362no—
234C2×C3×C4C2×D3214yes—
235C2×C3×C4C22×C3⋊D311yescharacteristic
236C2×Dic3—21no—
237C2×Dic3—34no—
238C2×Dic3—34no—
239D12—38no—
240D12—64no—
241D12—98no—
242D12—92no—
243D12—188no—
244D12—184no—
245D12—181no—
246D12—364no—
247C4×D3—128no—
248C4×D3—368no—
249C4×D3—362no—
250C3×C8—38no—
251C3⋊C8—184no—
252C3⋊C8—181no—
253C3⋊C8—364no—
254C3⋊C8C4×C3⋊D312yescomplemented
255C2×C32—24no—
256C2×C32—24no—
257C2×C32—24no—
258C2×C32#89314yes—
259C2×C32D3×D814yes—
260C2×C32#91914yes—
261C2×C32C4×#6711yescharacteristic
262C2×C32C2×#21311yescharacteristic
263C2×C32#95011yescharacteristic
264C3⋊D3—42no—
265C3⋊D3—122no—
266C3⋊D3—128no—
267C3⋊D3—248no—
268C3×D3—128no—
269C2×D4—91no—
270C2×D4—271no—
271C22×C4—181no—
272C22×C4—541no—
273D8—274no—
274C2×C8—31no—
275C2×C8—271no—
276C4⋊C4—31no—
277C4⋊C4D3×C3⋊D311yescharacteristic
278C22⋊C4—181no—
279C22⋊C4—541no—
280C42—91no—
281C22×C3—24no—
282C22×C3—182no—
283C22×C3—361no—
284C22×C3—361no—
285C22×C3D3×#6714yes—
286C22×C3D4×C3⋊D311yescharacteristic
287C2×D3—68no—
288C2×D3—124no—
289C2×D3—124no—
290C2×D3—182no—
291C2×D3—188no—
292C2×D3—368no—
293C2×D3—361no—
294C2×D3—361no—
295C2×D3—364no—
296C2×D3—364no—
297C2×D3—724no—
298C2×D3—724no—
299C3×C4—24no—
300C3×C4—24no—
301C3×C4—41no—
302C3×C4—44no—
303C3×C4—84no—
304C3×C4—92no—
305C3×C4—124no—
306C3×C4—181no—
307C3×C4#379914yes—
308C3×C4C4×D3214yes—
309C3×C4C2×C4×C3⋊D311yescharacteristic
310C3×C4#383211yescharacteristic
311Dic3—38no—
312Dic3—41no—
313Dic3—64no—
314C32—24no—
315C32#487214yescomplemented
316C32C4×#21311yescharacteristic, complemented
317C23—181no—
318C23—541no—
319D4—92no—
320D4—181no—
321D4—272no—
322D4—541no—
323C2×C4—21no—
324C2×C4—61no—
325C2×C4—91no—
326C2×C4—91no—
327C2×C4—362no—
328C2×C4—1082no—
329C2×C4C2×D3×C3⋊D311yescharacteristic
330C8—32no—
331C8—541no—
332C2×C3—24no—
333C2×C3—24no—
334C2×C3—24no—
335C2×C3—362no—
336C2×C3#6390414yes—
337C2×C3D3×#21314yes—
338C2×C3#6394914yes—
339C2×C3#6410211yescharacteristic
340C2×C3D8×C3⋊D311yescharacteristic
341C2×C3#6412811yescharacteristic
342D3—128no—
343D3—362no—
344D3—368no—
345D3—728no—
346C22—182no—
347C22—361no—
348C22—361no—
349C22—542no—
350C22—1081no—
351C22—1081no—
352C22C3⋊D3×#6711yescharacteristic
353C4—41no—
354C4—92no—
355C4—121no—
356C4—181no—
357C4#8901811yescharacteristic
358C4C4×D3×C3⋊D311yescharacteristic
359C3—24no—
360C3#1059045614yescomplemented
361C3#1059257111yescharacteristic, complemented
362C2—362no—
363C2—1082no—
364C2#1172351211yescharacteristic
365C2C3⋊D3×#21311yescharacteristic
366C2#1172358511yescharacteristic
subgroupG/H
centerC22C3⋊D3×#67
commutatorC33×C4C22×C4
FrattiniC2×C4C2×D3×C3⋊D3
FittingC33×C4⋊C4C22
socleC22×C33C2×D4
prime radicalC2×#88791C2
cosocleC2×C33×C4C23
hypercenterC4⋊C4D3×C3⋊D3
hypocenterC33C4×D8
LCSG ▷ C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
1. C3×#64102C2semidirect
2. #11723494C2semidirect
4. #11722179C2semidirect
5. C32×#807C2semidirect
6. #11722033C2semidirect
7. C3⋊C8×C3⋊Dic3C2semidirect
21. #88791C4semidirect
24. #64102D3semidirect
34. C3×#807D3semidirect
42. C4⋊(C3⋊(C3⋊D3))C2×C4semidirect
46. C3×C4⋊(C3⋊D3)C2×C4semidirect
51. C32×C3⋊C8C2×C4semidirect
92. C3×C3⋊Dic3D8semidirect
107. #807C3⋊D3semidirect
130. C4⋊(C3⋊D3)C4×D3semidirect
144. C3×C3⋊C8C4×D3semidirect
207. C3⋊Dic3#213semidirect
218. C33C4×D8semidirect
254. C3⋊C8C4×C3⋊D3semidirect
315. C32#4872semidirect
316. C32C4×#213semidirect
360. C3#10590456semidirect
361. C3#10592571semidirect

Maximal subgroups

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

subgroupindex
C3×#641022
#117234942
C2×#887912
#117221792
C32×#8072
#117220332
C3⋊C8×C3⋊Dic32
#105925713
#105904563