Finite Group Databasebeta

← #13158986

Subgroups — #13158986

Order 1728.

1096 conjugacy classes of subgroups; only the 186 normal subgroups are shown.

normal subgroups only

This group has 1096 conjugacy classes of subgroups — too many to draw. Only its 186 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
1C4×C4⋊(C3⋊(C3⋊D3))C211yesmaximal, characteristic
2C3×C4×#634C211yesmaximal, characteristic, complemented
3C2×#88797C211yesmaximal, characteristic
4C32×#803C211yesmaximal, characteristic, complemented
5#11722055C211yesmaximal, characteristic, complemented
6#11722040C211yesmaximal, characteristic, complemented
7#11722036C211yesmaximal, characteristic
8#10592409—31nomaximal
9#10590362—34nomaximal
10C2×C4⋊(C3⋊(C3⋊D3))C2211yescharacteristic
11C2×C4×C3⋊(C3⋊D3)—21no—
12C2×C3×#634C2211yescharacteristic
13C33×C42C2211yescharacteristic
14#88859—21no—
15C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
16C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
17C3×#3750—21no—
18C3×C4×C3⋊Dic3—21no—
19C2×C32×C3⋊C8C2211yescharacteristic
20C2×C32×C3⋊C8C2211yescharacteristic
21#88797C414yescomplemented
22C2×#3744—31no—
23C4×C4⋊(C3⋊D3)—31no—
24C4×C4⋊(C3⋊D3)—34no—
25C4×C4⋊(C3⋊D3)—64no—
26C4×#634D311yescharacteristic, complemented
27C3×C4×Dic6—34no—
28C2×#3717—34no—
29C32×#97—31no—
30#63648—31no—
31#63645—31no—
32#63643—31no—
33C3×#803D314yescomplemented
34#63575—34no—
35#63566—34no—
36#63564—34no—
37C22×C3⋊(C3⋊D3)—21no—
38C2×C33×C4D411yescharacteristic
39C2×C33×C4D411yescharacteristic
40C2×C33×C4C2311yescharacteristic
41C2×C3⋊(C3⋊Dic3)—21no—
42C4⋊(C3⋊(C3⋊D3))—21no—
43C4⋊(C3⋊(C3⋊D3))C2×C412yes—
44C4×C3⋊(C3⋊D3)—42no—
45C2×C3×C3⋊Dic3—21no—
46C2×C3×C3⋊Dic3—21no—
47C3×#634—21no—
48C3×#634C2×C412yescomplemented
49C32×C3⋊C8—21no—
50C32×C3⋊C8C2×C412yescomplemented
51#5026—91no—
52#4710—94no—
53C2×C4⋊(C3⋊D3)—31no—
54C2×C4⋊(C3⋊D3)—34no—
55C2×C4⋊(C3⋊D3)—64no—
56C2×C4×C3⋊D3—68no—
57C2×C4×C3⋊D3—64no—
58C2×C4×C3⋊D3—61no—
59C2×#634C2×D311yescharacteristic
60C2×C3×Dic6—34no—
61C2×C32×C8—31no—
62C2×C32×C8—31no—
63C32×C42—24no—
64C32×C42C2×D314yes—
65C32×C42C2×D311yescharacteristic
66#3752—68no—
67#3752—64no—
68#3752—61no—
69C4⋊(C3⋊Dic3)—31no—
70C4⋊(C3⋊Dic3)—34no—
71C4⋊(C3⋊Dic3)—64no—
72C4⋊(C3⋊Dic3)C2×D311yescharacteristic
73#3750—21no—
74C4×C3⋊Dic3—21no—
75#3744—34no—
76C3×C4⋊Dic3—34no—
77C3×Dic3⋊C4—64no—
78C3×C4×Dic3—64no—
79C2×C3×C3⋊C8C2×D314yes—
80C2×C3×C3⋊C8C2×D314yes—
81#3717—316no—
82C22×C33C2×D411yescharacteristic
83C2×C3⋊(C3⋊D3)—22no—
84C2×C3⋊(C3⋊D3)—41no—
85C2×C3⋊(C3⋊D3)—41no—
86C33×C4—21no—
87C33×C4—21no—
88C33×C4C22×C411yescharacteristic
89C33×C4#4111yescharacteristic
90C3⋊(C3⋊Dic3)—41no—
91C3×C3⋊Dic3—22no—
92C3×C3⋊Dic3—41no—
93C3×C4×Q8—91no—
94C2×#215—91no—
95C2×#204—94no—
96C4×D12—94no—
97C4×D12—91no—
98C4×D12—184no—
99C4×Dic6—34no—
100C3×#97—34no—
101#835—91no—
102#821—94no—
103#818—94no—
104#816—94no—
105#809—91no—
106#807—91no—
107#803C3⋊D311yescharacteristic, complemented
108C22×C3⋊D3—68no—
109C22×C3⋊D3—64no—
110C22×C3⋊D3—61no—
111C2×C32×C4—24no—
112C2×C32×C4—24no—
113C2×C32×C4—24no—
114C2×C32×C4D1214yes—
115C2×C32×C4D1214yes—
116C2×C32×C4#6711yescharacteristic
117C2×C32×C4#6711yescharacteristic
118C2×C32×C4C22×D314yes—
119C2×C32×C4C22×D311yescharacteristic
120C2×C3⋊Dic3—21no—
121C2×C3⋊Dic3—21no—
122C2×C3⋊Dic3—68no—
123C2×C3⋊Dic3—64no—
124C2×C3⋊Dic3—61no—
125C4⋊(C3⋊D3)—32no—
126C4⋊(C3⋊D3)—38no—
127C4⋊(C3⋊D3)—68no—
128C4⋊(C3⋊D3)—68no—
129C4⋊(C3⋊D3)—64no—
130C4⋊(C3⋊D3)—61no—
131C4×C3⋊D3—1216no—
132C4×C3⋊D3—128no—
133C4×C3⋊D3—122no—
134#634—21no—
135#634C4×D312yescomplemented
136C2×C3×Dic3—64no—
137C2×C3×Dic3—64no—
138C3×Dic6—38no—
139C3×Dic6—64no—
140C32×C8—32no—
141C32×C8—61no—
142C3×C3⋊C8—24no—
143C3×C3⋊C8C4×D318yescomplemented
144#440—271noSylow
145C2×C33C4×D411yescharacteristic
146C2×C33AGL(1,ℤ/8)11yescharacteristic
147C2×C33#13711yescharacteristic
148C3⋊(C3⋊D3)—42no—
149C2×C3×Q8—91no—
150C2×D12—94no—
151C2×D12—91no—
152C2×D12—184no—
153C2×C4×D3—188no—
154C2×C4×D3—184no—
155C2×C4×D3—181no—
156C2×Dic6—34no—
157C2×C3×C8—34no—
158C2×C3×C8—34no—
159C3×C4⋊C4—91no—
160C3×C4⋊C4—181no—
161C3×C42—24no—
162C3×C42—181no—
163C3×C42D3214yes—
164C3×C42C2×C3⋊D311yescharacteristic
165#215—94no—
166(C2×D3)⋊C4—188no—
167(C2×D3)⋊C4—184no—
168(C2×D3)⋊C4—181no—
169C4⋊Dic3—34no—
170C4⋊Dic3—94no—
171C4⋊Dic3—91no—
172C4⋊Dic3—184no—
173Dic3⋊C4—64no—
174C4×Dic3—64no—
175C2×C3⋊C8C2×C3⋊D311yescharacteristic
176C2×C3⋊C8C2×C3⋊D311yescharacteristic
177#204—916no—
178C22×C32—24no—
179C22×C32C2×D1214yes—
180C22×C32C2×#6711yescharacteristic
181C2×C3⋊D3—616no—
182C2×C3⋊D3—68no—
183C2×C3⋊D3—62no—
184C2×C3⋊D3—128no—
185C2×C3⋊D3—128no—
186C2×C3⋊D3—124no—
187C2×C3⋊D3—124no—
188C2×C3⋊D3—121no—
189C2×C3⋊D3—121no—
190C32×C4—28no—
191C32×C4—24no—
192C32×C4—21no—
193C32×C4—21no—
194C32×C4—24no—
195C32×C4—24no—
196C32×C4—24no—
197C32×C4—44no—
198C32×C4C2×C4×D314yes—
199C32×C4C2×C4×D311yescharacteristic
200C32×C4#23514yes—
201C32×C4#23511yescharacteristic
202C3⋊Dic3—22no—
203C3⋊Dic3—41no—
204C3⋊Dic3—128no—
205C3⋊Dic3—124no—
206C3⋊Dic3—121no—
207C3×Dic3—68no—
208C3×Dic3—124no—
209C2×SD16—271no—
210C4×Q8—91no—
211C4×D4—271no—
212#107—271no—
213#103—271no—
214#102—271no—
215#97—31no—
216C33#44011yescharacteristic, complemented, Sylow
217C22×D3—188no—
218C22×D3—184no—
219C22×D3—181no—
220C3×Q8—92no—
221C3×Q8—181no—
222C2×C3×C4—24no—
223C2×C3×C4—24no—
224C2×C3×C4—24no—
225C2×C3×C4—181no—
226C2×C3×C4—181no—
227C2×C3×C4#62614yes—
228C2×C3×C4#62614yes—
229C2×C3×C4C4⋊(C3⋊D3)11yescharacteristic
230C2×C3×C4C4⋊(C3⋊D3)11yescharacteristic
231C2×C3×C4C2×D3214yes—
232C2×C3×C4C22×C3⋊D311yescharacteristic
233C2×Dic3—64no—
234C2×Dic3—64no—
235C2×Dic3—188no—
236C2×Dic3—184no—
237C2×Dic3—181no—
238D12—98no—
239D12—92no—
240D12—188no—
241D12—188no—
242D12—184no—
243D12—181no—
244C4×D3—3616no—
245C4×D3—368no—
246C4×D3—362no—
247Dic6—38no—
248Dic6—64no—
249C3×C8—38no—
250C3×C8—64no—
251C3⋊C8—21no—
252C3⋊C8C4×C3⋊D312yescomplemented
253C2×C32—24no—
254C2×C32—24no—
255C2×C32—24no—
256C2×C32C4×D1214yes—
257C2×C32#90814yes—
258C2×C32#90914yes—
259C2×C32C4×#6711yescharacteristic
260C2×C32#94211yescharacteristic
261C2×C32#94911yescharacteristic
262C3⋊D3—1216no—
263C3⋊D3—128no—
264C3⋊D3—122no—
265C2×Q8—91no—
266C2×D4—271no—
267C22×C4—541no—
268SD16—274no—
269C2×C8—31no—
270C2×C8—31no—
271C4⋊C4—91no—
272C4⋊C4—181no—
273C4⋊C4—271no—
274C22⋊C4—541no—
275C42—181no—
276C42D3×C3⋊D311yescharacteristic
277C22×C3—24no—
278C22×C3C2×#62614yes—
279C22×C3C2×C4⋊(C3⋊D3)11yescharacteristic
280C2×D3—1816no—
281C2×D3—182no—
282C2×D3—188no—
283C2×D3—368no—
284C2×D3—368no—
285C2×D3—364no—
286C2×D3—364no—
287C2×D3—361no—
288C2×D3—361no—
289C3×C4—28no—
290C3×C4—24no—
291C3×C4—24no—
292C3×C4—21no—
293C3×C4—21no—
294C3×C4—24no—
295C3×C4—24no—
296C3×C4—44no—
297C3×C4—182no—
298C3×C4—361no—
299C3×C4#379814yes—
300C3×C4C4×D3214yes—
301C3×C4C2×C4×C3⋊D311yescharacteristic
302C3×C4#382811yescharacteristic
303Dic3—68no—
304Dic3—124no—
305Dic3—368no—
306Dic3—364no—
307Dic3—361no—
308C32—24no—
309C32#471014yescomplemented
310C32#502611yescharacteristic, complemented
311C23—541no—
312Q8—92no—
313Q8—181no—
314D4—272no—
315D4—541no—
316C2×C4—181no—
317C2×C4—181no—
318C2×C4—541no—
319C2×C4—1082no—
320C2×C4#629211yescharacteristic
321C2×C4#629211yescharacteristic
322C2×C4C2×D3×C3⋊D311yescharacteristic
323C8—32no—
324C8—61no—
325C2×C3—24no—
326C2×C3—24no—
327C2×C3—24no—
328C2×C3#6382514yes—
329C2×C3#6383314yes—
330C2×C3C4×#62614yes—
331C2×C3C4×C4⋊(C3⋊D3)11yescharacteristic
332C2×C3#6411711yescharacteristic
333C2×C3#6411811yescharacteristic
334D3—3616no—
335D3—362no—
336D3—368no—
337C22—542no—
338C22—1081no—
339C22—1081no—
340C22C2×#629211yescharacteristic
341C4—21no—
342C4—21no—
343C4—182no—
344C4—361no—
345C4—1081no—
346C4#8902011yescharacteristic
347C4C4×D3×C3⋊D311yescharacteristic
348C3—24no—
349C3#1059036214yescomplemented
350C3#1059240911yescharacteristic, complemented
351C2—1082no—
352C2#1172337111yescharacteristic
353C2#1172338711yescharacteristic
354C2C4×#629211yescharacteristic
subgroupG/H
centerC22C2×#6292
commutatorC33×C4C22×C4
FrattiniC2×C4C2×D3×C3⋊D3
FittingC33×C42C22
socleC22×C33C2×D4
prime radicalC2×C4⋊(C3⋊(C3⋊D3))C22
cosocleC2×C33×C4C23
hypercenterC42D3×C3⋊D3
hypocenterC33#440
LCSG ▷ C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C42

Complements

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

NH ≅ G/Ntype
2. C3×C4×#634C2semidirect
4. C32×#803C2semidirect
5. #11722055C2semidirect
6. #11722040C2semidirect
21. #88797C4semidirect
26. C4×#634D3semidirect
33. C3×#803D3semidirect
48. C3×#634C2×C4semidirect
50. C32×C3⋊C8C2×C4semidirect
107. #803C3⋊D3semidirect
135. #634C4×D3semidirect
143. C3×C3⋊C8C4×D3semidirect
216. C33#440semidirect
252. C3⋊C8C4×C3⋊D3semidirect
309. C32#4710semidirect
310. C32#5026semidirect
349. C3#10590362semidirect
350. C3#10592409semidirect

Maximal subgroups

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

subgroupindex
C4×C4⋊(C3⋊(C3⋊D3))2
C3×C4×#6342
C2×#887972
C32×#8032
#117220552
#117220402
#117220362
#105924093
#105903623