Finite Group Databasebeta

← #13159372

Subgroups — #13159372

Order 1728.

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

normal subgroups only

This group has 876 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×#64096C211yesmaximal, characteristic, complemented
2D12×C3⋊Dic3C211yesmaximal, characteristic
3C2×#88792C211yesmaximal, characteristic
4#11722178C211yesmaximal, characteristic, complemented
5C32×#809C211yesmaximal, characteristic
6#11722040C211yesmaximal, characteristic, complemented
7C3⋊C8×C3⋊Dic3C211yesmaximal, characteristic, complemented
8#10592457—31nomaximal
9#10590453—34nomaximal
10C2×C3×#634C2211yescharacteristic
11C2×C32×D12C2211yescharacteristic
12C2×D3×C3⋊Dic3—21no—
13C33×C4⋊C4C2211yescharacteristic
14C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
15C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
16C3×#3750—21no—
17C3×C4×C3⋊Dic3—21no—
18C3×C4×C3⋊Dic3C2211yescharacteristic
19C2×C32×C3⋊C8C2211yescharacteristic
20(C2×D3)⋊(C3⋊Dic3)—21no—
21#88792C414yescomplemented
22D4×C3⋊Dic3—31no—
23C2×#3754—31no—
24#64096D311yescharacteristic, complemented
25C3×#887—34no—
26Dic3×D12—34no—
27C2×#3715—34no—
28C32×#102—31no—
29#63643—31no—
30C8×C3⋊Dic3—31no—
31#63635—31no—
32#63635—34no—
33#63635—64no—
34C3×#809D314yes—
35#63566—34no—
36Dic3×C3⋊C8—34no—
37C22×C32×D3—21no—
38C2×C33×C4—21no—
39C2×C33×C4C2311yescharacteristic
40C2×C3⋊(C3⋊Dic3)—21no—
41C2×C3×C3⋊Dic3—21no—
42C2×C3×C3⋊Dic3D411yescharacteristic
43C2×C3×C3⋊Dic3D411yescharacteristic
44C3×#634—21no—
45C3×#634C2×C412yescomplemented
46C32×D12—21no—
47C32×D12C2×C412yes—
48D3×C3⋊Dic3—42no—
49C3⋊(C3⋊(C3⋊C8))C2×C412yescomplemented
50C32×C3⋊C8—21no—
51C4×#215—91no—
52#4758—94no—
53C2×C32×D4—31no—
54C22×C3⋊Dic3—61no—
55C2×#634C2×D311yescharacteristic
56C2×C3×D12C2×D314yes—
57C2×C3×Dic6—34no—
58C2×D3×Dic3—64no—
59C2×C32×C8—31no—
60C32×C4⋊C4—24no—
61C32×C4⋊C4C2×D314yes—
62C32×C4⋊C4C2×D311yescharacteristic
63#3757—61no—
64#3754—34no—
65C4⋊(C3⋊Dic3)—31no—
66C4⋊(C3⋊Dic3)—34no—
67C4⋊(C3⋊Dic3)—64no—
68#3750—21no—
69C4×C3⋊Dic3—21no—
70C4×C3⋊Dic3C2×D311yescharacteristic
71C2×C3⋊(C3⋊C8)—31no—
72C2×C3⋊(C3⋊C8)—34no—
73C2×C3⋊(C3⋊C8)—64no—
74C3×Dic3⋊C4—64no—
75C3×C4×Dic3—34no—
76C3×C4×Dic3—64no—
77C2×C3×C3⋊C8C2×D314yes—
78(C2×D3)⋊Dic3—64no—
79#3715—316no—
80C22×C33C2×D411yescharacteristic
81C2×C32×D3—22no—
82C2×C32×D3—41no—
83C2×C32×D3—41no—
84C33×C4—41no—
85C33×C4C22×C411yescharacteristic
86C33×C4#4111yescharacteristic
87C3⋊(C3⋊Dic3)—41no—
88C3×C3⋊Dic3—22no—
89C3×C3⋊Dic3—21no—
90C3×C3⋊Dic3SD1612yescomplemented
91C3×C4×Q8—91no—
92C2×#215—91no—
93D4×Dic3—94no—
94C2×#214—94no—
95#887—34no—
96C4×D12—91no—
97C3×#102—34no—
98#835—91no—
99#816—94no—
100C8×Dic3—94no—
101#809C3⋊D311yescharacteristic
102#808—94no—
103#808—91no—
104#808—184no—
105C4×C3⋊C8—91no—
106C23×C32—61no—
107C22×C3×D3—24no—
108C32×D4—32no—
109C32×D4—61no—
110C2×C32×C4—21no—
111C2×C32×C4—24no—
112C2×C32×C4—24no—
113C2×C32×C4—44no—
114C2×C32×C4C22×D314yes—
115C2×C32×C4C22×D311yescharacteristic
116C2×C3⋊Dic3—21no—
117C2×C3⋊Dic3—61no—
118C2×C3⋊Dic3—64no—
119C2×C3⋊Dic3—124no—
120C2×C3⋊Dic3—122no—
121C2×C3⋊Dic3#6711yescharacteristic
122C2×C3⋊Dic3#6711yescharacteristic
123#634—21no—
124#634C4×D312yescomplemented
125C2×C3×Dic3—34no—
126C2×C3×Dic3—34no—
127C2×C3×Dic3—64no—
128C3×D12—24no—
129C3×D12C4×D318yes—
130C3×Dic6—38no—
131C3×Dic6—64no—
132D3×Dic3—128no—
133C32×C8—61no—
134C3⋊(C3⋊C8)—32no—
135C3⋊(C3⋊C8)—38no—
136C3⋊(C3⋊C8)—68no—
137C3×C3⋊C8—24no—
138C4×SD16—271noSylow
139C2×C33C4×D411yescharacteristic
140C2×C33C2×SD1611yescharacteristic
141C2×C33#13511yescharacteristic
142C32×D3—42no—
143C2×C3×Q8—91no—
144C2×C3×D4—34no—
145C22×Dic3—184no—
146C2×D12C2×C3⋊D311yescharacteristic
147C2×C4×D3—181no—
148C2×Dic6—34no—
149C2×C3×C8—34no—
150C3×C4⋊C4—24no—
151C3×C4⋊C4—181no—
152C3×C4⋊C4D3214yes—
153C3×C4⋊C4C2×C3⋊D311yescharacteristic
154C3×C42—91no—
155C3×C42—181no—
156#217—184no—
157#215—94no—
158#214—916no—
159(C2×D3)⋊C4—181no—
160C4⋊Dic3—94no—
161C4⋊Dic3—91no—
162C4⋊Dic3—184no—
163Dic3⋊C4—64no—
164C4×Dic3—34no—
165C4×Dic3—64no—
166C2×C3⋊C8—94no—
167C2×C3⋊C8—91no—
168C2×C3⋊C8—184no—
169C2×C3⋊C8C2×C3⋊D311yescharacteristic
170C22×C32—24no—
171C22×C32—62no—
172C22×C32—121no—
173C22×C32—121no—
174C22×C32D3×D414yes—
175C22×C32C2×#6711yescharacteristic
176C2×C3×D3—28no—
177C2×C3×D3—44no—
178C2×C3×D3—44no—
179C32×C4—24no—
180C32×C4—24no—
181C32×C4—41no—
182C32×C4—44no—
183C32×C4—84no—
184C32×C4C2×C4×D314yes—
185C32×C4C2×C4×D311yescharacteristic
186C32×C4#23511yescharacteristic
187C32×C4#23714yes—
188C3⋊Dic3—22no—
189C3⋊Dic3—21no—
190C3⋊Dic3—121no—
191C3⋊Dic3—124no—
192C3⋊Dic3—244no—
193C3⋊Dic3#21512yescomplemented
194C3×Dic3—38no—
195C3×Dic3—68no—
196C3×Dic3—64no—
197C2×SD16—271no—
198C4×Q8—91no—
199C4×D4—271no—
200#106—271no—
201#103—271no—
202#102—31no—
203C4×C8—271no—
204C33C4×SD1611yescharacteristic, complemented, Sylow
205C23×C3—64no—
206C22×D3—21no—
207C3×Q8—92no—
208C3×Q8—181no—
209C3×D4—38no—
210C3×D4—64no—
211C2×C3×C4—24no—
212C2×C3×C4—21no—
213C2×C3×C4—24no—
214C2×C3×C4—44no—
215C2×C3×C4—91no—
216C2×C3×C4—91no—
217C2×C3×C4—181no—
218C2×C3×C4C2×D3214yes—
219C2×C3×C4C22×C3⋊D311yescharacteristic
220C2×Dic3—34no—
221C2×Dic3—34no—
222C2×Dic3—64no—
223C2×Dic3—184no—
224C2×Dic3—181no—
225C2×Dic3—364no—
226C2×Dic3—368no—
227D12—21no—
228D12C4×C3⋊D312yes—
229C4×D3—362no—
230Dic6—38no—
231Dic6—64no—
232C3×C8—64no—
233C3⋊C8—21no—
234C3⋊C8—98no—
235C3⋊C8—92no—
236C3⋊C8—188no—
237C2×C32—24no—
238C2×C32—24no—
239C2×C32—24no—
240C2×C32—122no—
241C2×C32#67⋊C414yes—
242C2×C32#91214yes—
243C2×C32D3×SD1614yes—
244C2×C32C4×#6711yescharacteristic
245C2×C32C2×#21511yescharacteristic
246C2×C32#95011yescharacteristic
247C3×D3—48no—
248C2×Q8—91no—
249C2×D4—31no—
250C22×C4—541no—
251SD16—274no—
252C2×C8—31no—
253C2×C8—271no—
254C4⋊C4—181no—
255C4⋊C4—271no—
256C4⋊C4D3×C3⋊D311yescharacteristic
257C22⋊C4—541no—
258C42—91no—
259C42—181no—
260C22×C3—24no—
261C22×C3—68no—
262C22×C3—124no—
263C22×C3—124no—
264C22×C3D3×#6714yes—
265C22×C3D4×C3⋊D311yescharacteristic
266C2×D3—22no—
267C2×D3—41no—
268C2×D3—41no—
269C3×C4—24no—
270C3×C4—24no—
271C3×C4—41no—
272C3×C4—44no—
273C3×C4—84no—
274C3×C4—92no—
275C3×C4—182no—
276C3×C4—181no—
277C3×C4#379514yes—
278C3×C4C4×D3214yes—
279C3×C4C2×C4×C3⋊D311yescharacteristic
280C3×C4#383011yescharacteristic
281Dic3—38no—
282Dic3—68no—
283Dic3—64no—
284Dic3—361no—
285Dic3—364no—
286Dic3—724no—
287C32—24no—
288C32#475814yescomplemented
289C32C4×#21511yescharacteristic, complemented
290C23—61no—
291Q8—92no—
292Q8—181no—
293D4—32no—
294D4—61no—
295C2×C4—21no—
296C2×C4—91no—
297C2×C4—91no—
298C2×C4—181no—
299C2×C4—541no—
300C2×C4—1082no—
301C2×C4C2×D3×C3⋊D311yescharacteristic
302C8—61no—
303C8—272no—
304C2×C3—24no—
305C2×C3—24no—
306C2×C3—24no—
307C2×C3—128no—
308C2×C3#6387914yes—
309C2×C3#6393314yes—
310C2×C3D3×#21514yes—
311C2×C3#6409011yescharacteristic
312C2×C3#6412111yescharacteristic
313C2×C3SD16×C3⋊D311yescharacteristic
314D3—42no—
315C22—62no—
316C22—121no—
317C22—121no—
318C22C3⋊D3×#6711yescharacteristic
319C4—41no—
320C4—92no—
321C4—181no—
322C4—182no—
323C4—1081no—
324C4#8901411yescharacteristic
325C4C4×D3×C3⋊D311yescharacteristic
326C3—24no—
327C3#1059045314yescomplemented
328C3#1059245711yescharacteristic, complemented
329C2—122no—
330C2#1172350711yescharacteristic
331C2#1172355911yescharacteristic
332C2C3⋊D3×#21511yescharacteristic
subgroupG/H
centerC22C3⋊D3×#67
commutatorC33×C4C22×C4
FrattiniC2×C4C2×D3×C3⋊D3
FittingC33×C4⋊C4C22
socleC22×C33C2×D4
prime radicalC2×C32×D12C22
cosocleC2×C33×C4C23
hypercenterC4⋊C4D3×C3⋊D3
hypocenterC33C4×SD16
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×#64096C2semidirect
4. #11722178C2semidirect
6. #11722040C2semidirect
7. C3⋊C8×C3⋊Dic3C2semidirect
21. #88792C4semidirect
24. #64096D3semidirect
45. C3×#634C2×C4semidirect
49. C3⋊(C3⋊(C3⋊C8))C2×C4semidirect
90. C3×C3⋊Dic3SD16semidirect
124. #634C4×D3semidirect
193. C3⋊Dic3#215semidirect
204. C33C4×SD16semidirect
288. C32#4758semidirect
289. C32C4×#215semidirect
327. C3#10590453semidirect
328. C3#10592457semidirect

Maximal subgroups

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

subgroupindex
C3×#640962
D12×C3⋊Dic32
C2×#887922
#117221782
C32×#8092
#117220402
C3⋊C8×C3⋊Dic32
#105924573
#105904533