Finite Group Databasebeta

← #13159379

Subgroups — #13159379

Order 1728.

920 conjugacy classes of subgroups; only the 166 normal subgroups are shown.

normal subgroups only

This group has 920 conjugacy classes of subgroups — too many to draw. Only its 166 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
2#11723494C211yesmaximal, characteristic
3C2×#88797C211yesmaximal, characteristic
4#11722179C211yesmaximal, characteristic
5C32×#807C211yesmaximal, characteristic, complemented
6#11722039C211yesmaximal, characteristic, complemented
7#11722020C211yesmaximal, characteristic, complemented
8#10592588—31nomaximal
9#10590457—34nomaximal
10C2×C4⋊(C3⋊(C3⋊D3))C2211yescharacteristic
11C2×C3×#634C2211yescharacteristic
12C2×#6289—21no—
13C33×C4⋊C4C2211yescharacteristic
14C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
15C3×#3750—21no—
16C3×C4×C3⋊Dic3—21no—
17C3×C4×C3⋊Dic3C2211yescharacteristic
18C32×C4⋊Dic3C2211yescharacteristic
19C2×C32×C3⋊C8C2211yescharacteristic
20#88804—21no—
21#88797C414yescomplemented
22C2×#3744—31no—
23#64102—31no—
24#64096D311yescharacteristic, complemented
25C3×#887—34no—
26#63894—34no—
27C2×#3717—34no—
28C32×#107—31no—
29#63642—31no—
30#63637—31no—
31#63636—31no—
32#63636—34no—
33#63636—64no—
34C3×#807D314yescomplemented
35#63565—34no—
36#63554—34no—
37C22×C3⋊(C3⋊D3)—21no—
38C2×C33×C4—21no—
39C2×C33×C4C2311yescharacteristic
40C4⋊(C3⋊(C3⋊D3))—21no—
41C4⋊(C3⋊(C3⋊D3))C2×C412yes—
42C2×C3×C3⋊Dic3—21no—
43C2×C3×C3⋊Dic3D411yescharacteristic
44C2×C3×C3⋊Dic3D411yescharacteristic
45C3×#634—21no—
46C3×#634C2×C412yescomplemented
47C2×C32×Dic3—21no—
48#6289—42no—
49C3⋊(C3⋊(C3⋊C8))—21no—
50C32×C3⋊C8C2×C412yescomplemented
51#5026—91no—
52#4889—94no—
53C2×C4⋊(C3⋊D3)—31no—
54C2×C4⋊(C3⋊D3)—34no—
55C2×C4⋊(C3⋊D3)—64no—
56C2×C4×C3⋊D3—61no—
57C2×#634C2×D311yescharacteristic
58C2×C3×Dic6—34no—
59C2×#624—64no—
60C2×C32×C8—31no—
61C32×C4⋊C4—24no—
62C32×C4⋊C4—31no—
63C32×C4⋊C4C2×D314yes—
64C32×C4⋊C4C2×D311yescharacteristic
65#3752—61no—
66#3750—21no—
67C4×C3⋊Dic3—21no—
68C4×C3⋊Dic3C2×D311yescharacteristic
69C2×C3⋊(C3⋊C8)—31no—
70C2×C3⋊(C3⋊C8)—34no—
71C2×C3⋊(C3⋊C8)—64no—
72#3744—34no—
73C3×C4⋊Dic3C2×D314yes—
74C3×Dic3⋊C4—64no—
75C3×C4×Dic3—34no—
76C3×C4×Dic3—64no—
77C2×C3×C3⋊C8C2×D314yes—
78#3722—64no—
79#3717—316no—
80C22×C33C2×D411yescharacteristic
81C2×C3⋊(C3⋊D3)—22no—
82C2×C3⋊(C3⋊D3)—41no—
83C2×C3⋊(C3⋊D3)—41no—
84C33×C4—41no—
85C33×C4C22×C411yescharacteristic
86C33×C4#4111yescharacteristic
87C3×C3⋊Dic3—22no—
88C3×C3⋊Dic3—21no—
89C3×C3⋊Dic3—21no—
90C32×Dic3—41no—
91C3×C4×Q8—91no—
92C2×#215—91no—
93C2×#204—94no—
94#893—94no—
95#887—34no—
96C4×D12—91no—
97C3×#107—34no—
98#835—91no—
99#815—94no—
100#810—94no—
101#809—94no—
102#809—91no—
103#809—184no—
104#807C3⋊D311yescharacteristic, complemented
105#803—91no—
106C22×C3⋊D3—61no—
107C22×C3⋊D3—64no—
108C22×C3⋊D3—124no—
109C2×C32×C4—21no—
110C2×C32×C4—24no—
111C2×C32×C4—24no—
112C2×C32×C4—44no—
113C2×C32×C4—61no—
114C2×C32×C4C22×D314yes—
115C2×C32×C4C22×D311yescharacteristic
116C2×C3⋊Dic3—21no—
117C2×C3⋊Dic3#6711yescharacteristic
118C2×C3⋊Dic3#6711yescharacteristic
119C4⋊(C3⋊D3)—32no—
120C4⋊(C3⋊D3)—38no—
121C4⋊(C3⋊D3)—68no—
122C4⋊(C3⋊D3)—61no—
123C4⋊(C3⋊D3)—64no—
124C4⋊(C3⋊D3)—124no—
125C4×C3⋊D3—122no—
126#634—21no—
127#634C4×D312yescomplemented
128C2×C3×Dic3—24no—
129C2×C3×Dic3—34no—
130C2×C3×Dic3—34no—
131C2×C3×Dic3—64no—
132C3×Dic6—38no—
133C3×Dic6—64no—
134#624—128no—
135C32×C8—32no—
136C3⋊(C3⋊C8)—61no—
137C3⋊(C3⋊C8)—64no—
138C3⋊(C3⋊C8)—124no—
139C3×C3⋊C8C4×D318yescomplemented
140#440—271noSylow
141C2×C33C4×D411yescharacteristic
142C2×C33AGL(1,ℤ/8)11yescharacteristic
143C2×C33#13711yescharacteristic
144C3⋊(C3⋊D3)—42no—
145C2×C3×Q8—91no—
146C2×D12—94no—
147C2×D12—91no—
148C2×D12—184no—
149C2×C4×D3—181no—
150C2×C4×D3—184no—
151C2×Dic6—34no—
152C2×C3×C8—34no—
153C3×C4⋊C4—24no—
154C3×C4⋊C4—34no—
155C3×C4⋊C4—181no—
156C3×C4⋊C4D3214yes—
157C3×C4⋊C4C2×C3⋊D311yescharacteristic
158C3×C42—91no—
159C3×C42—181no—
160#215—94no—
161(C2×D3)⋊C4—181no—
162(C2×D3)⋊C4—184no—
163C4⋊Dic3C2×C3⋊D311yescharacteristic
164Dic3⋊C4—64no—
165C4×Dic3—34no—
166C4×Dic3—64no—
167C2×C3⋊C8—94no—
168C2×C3⋊C8—91no—
169C2×C3⋊C8—184no—
170C2×C3⋊C8C2×C3⋊D311yescharacteristic
171#204—916no—
172C22×C32—24no—
173C22×C32D3×D414yes—
174C22×C32C2×#6711yescharacteristic
175C2×C3⋊D3—62no—
176C2×C3⋊D3—68no—
177C2×C3⋊D3—128no—
178C2×C3⋊D3—121no—
179C2×C3⋊D3—121no—
180C2×C3⋊D3—124no—
181C2×C3⋊D3—124no—
182C2×C3⋊D3—244no—
183C2×C3⋊D3—244no—
184C32×C4—24no—
185C32×C4—24no—
186C32×C4—41no—
187C32×C4—44no—
188C32×C4—84no—
189C32×C4—121no—
190C32×C4C2×C4×D314yes—
191C32×C4C2×C4×D311yescharacteristic
192C32×C4#23511yescharacteristic
193C32×C4#23914yes—
194C3⋊Dic3—22no—
195C3⋊Dic3—21no—
196C3⋊Dic3—21no—
197C3×Dic3—44no—
198C3×Dic3—68no—
199C3×Dic3—64no—
200C3×Dic3—64no—
201C2×SD16—271no—
202C4×Q8—91no—
203C4×D4—271no—
204#107—31no—
205#103—271no—
206#102—271no—
207#97—271no—
208C33#44011yescharacteristic, complemented, Sylow
209C22×D3—184no—
210C22×D3—181no—
211C22×D3—364no—
212C3×Q8—92no—
213C3×Q8—181no—
214C2×C3×C4—24no—
215C2×C3×C4—21no—
216C2×C3×C4—24no—
217C2×C3×C4—44no—
218C2×C3×C4—64no—
219C2×C3×C4—91no—
220C2×C3×C4—91no—
221C2×C3×C4—181no—
222C2×C3×C4C2×D3214yes—
223C2×C3×C4C22×C3⋊D311yescharacteristic
224C2×Dic3—21no—
225C2×Dic3—34no—
226C2×Dic3—34no—
227C2×Dic3—64no—
228D12—98no—
229D12—92no—
230D12—188no—
231D12—184no—
232D12—181no—
233D12—364no—
234C4×D3—362no—
235C4×D3—368no—
236Dic6—38no—
237Dic6—64no—
238C3×C8—38no—
239C3⋊C8—184no—
240C3⋊C8—181no—
241C3⋊C8—364no—
242C3⋊C8C4×C3⋊D312yescomplemented
243C2×C32—24no—
244C2×C32—24no—
245C2×C32—24no—
246C2×C32#89314yes—
247C2×C32#91114yes—
248C2×C32#91814yes—
249C2×C32C4×#6711yescharacteristic
250C2×C32#94211yescharacteristic
251C2×C32#94911yescharacteristic
252C3⋊D3—122no—
253C3⋊D3—128no—
254C3⋊D3—248no—
255C2×Q8—91no—
256C2×D4—271no—
257C22×C4—541no—
258SD16—274no—
259C2×C8—31no—
260C2×C8—271no—
261C4⋊C4—31no—
262C4⋊C4—181no—
263C4⋊C4D3×C3⋊D311yescharacteristic
264C22⋊C4—541no—
265C42—91no—
266C42—181no—
267C22×C3—24no—
268C22×C3D3×#6714yes—
269C22×C3D4×C3⋊D311yescharacteristic
270C2×D3—182no—
271C2×D3—188no—
272C2×D3—368no—
273C2×D3—361no—
274C2×D3—361no—
275C2×D3—364no—
276C2×D3—364no—
277C2×D3—724no—
278C2×D3—724no—
279C3×C4—24no—
280C3×C4—24no—
281C3×C4—41no—
282C3×C4—44no—
283C3×C4—84no—
284C3×C4—124no—
285C3×C4—182no—
286C3×C4—181no—
287C3×C4—181no—
288C3×C4#379914yes—
289C3×C4C4×D3214yes—
290C3×C4C2×C4×C3⋊D311yescharacteristic
291C3×C4#383211yescharacteristic
292Dic3—41no—
293Dic3—68no—
294Dic3—64no—
295Dic3—64no—
296C32—24no—
297C32#488914yescomplemented
298C32#502611yescharacteristic, complemented
299C23—541no—
300Q8—92no—
301Q8—181no—
302D4—272no—
303D4—541no—
304C2×C4—21no—
305C2×C4—61no—
306C2×C4—91no—
307C2×C4—91no—
308C2×C4—181no—
309C2×C4—1082no—
310C2×C4C2×D3×C3⋊D311yescharacteristic
311C8—32no—
312C8—541no—
313C2×C3—24no—
314C2×C3—24no—
315C2×C3—24no—
316C2×C3#6390414yes—
317C2×C3#6392514yes—
318C2×C3#6394814yes—
319C2×C3#6410211yescharacteristic
320C2×C3#6412011yescharacteristic
321C2×C3#6412711yescharacteristic
322D3—362no—
323D3—368no—
324D3—728no—
325C22—542no—
326C22—1081no—
327C22—1081no—
328C22C3⋊D3×#6711yescharacteristic
329C4—41no—
330C4—121no—
331C4—181no—
332C4—181no—
333C4—182no—
334C4#8901811yescharacteristic
335C4C4×D3×C3⋊D311yescharacteristic
336C3—24no—
337C3#1059045714yescomplemented
338C3#1059258811yescharacteristic, complemented
339C2—1082no—
340C2#1172351211yescharacteristic
341C2#1172354611yescharacteristic
342C2#1172358611yescharacteristic
subgroupG/H
centerC22C3⋊D3×#67
commutatorC33×C4C22×C4
FrattiniC2×C4C2×D3×C3⋊D3
FittingC33×C4⋊C4C22
socleC22×C33C2×D4
prime radicalC2×C4⋊(C3⋊(C3⋊D3))C22
cosocleC2×C33×C4C23
hypercenterC4⋊C4D3×C3⋊D3
hypocenterC33#440
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
5. C32×#807C2semidirect
6. #11722039C2semidirect
7. #11722020C2semidirect
21. #88797C4semidirect
24. #64096D3semidirect
34. C3×#807D3semidirect
46. C3×#634C2×C4semidirect
50. C32×C3⋊C8C2×C4semidirect
104. #807C3⋊D3semidirect
127. #634C4×D3semidirect
139. C3×C3⋊C8C4×D3semidirect
208. C33#440semidirect
242. C3⋊C8C4×C3⋊D3semidirect
297. C32#4889semidirect
298. C32#5026semidirect
337. C3#10590457semidirect
338. C3#10592588semidirect

Maximal subgroups

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

subgroupindex
C3×#640962
#117234942
C2×#887972
#117221792
C32×#8072
#117220392
#117220202
#105925883
#105904573