Finite Group Databasebeta

← #13158953

Subgroups — #13158953

Order 1728.

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

normal subgroups only

This group has 952 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
1C3×C4×C4⋊(C3⋊D3)C211yesmaximal, characteristic, complemented
2C32×C4×D12C211yesmaximal, characteristic, complemented
3C2×#88789C211yesmaximal, characteristic
4#11722173C211yesmaximal, characteristic, complemented
5#11722053C211yesmaximal, characteristic, complemented
6#11722033C211yesmaximal, characteristic, complemented
7#11722031C211yesmaximal, characteristic, complemented
8#10592713—31nomaximal
9#10590344—34nomaximal
10C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
11C2×C3×C4×C3⋊D3—21no—
12C2×C32×D12C2211yescharacteristic
13C2×C32×C4×D3—21no—
14C33×C42C2211yescharacteristic
15C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
16C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
17C3×#3752—21no—
18C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
19C32×(C2×D3)⋊C4—21no—
20C32×C4⋊Dic3C2211yescharacteristic
21#88789C414yescomplemented
22C32×C4×D4—31no—
23C2×#3753—31no—
24C4×C4⋊(C3⋊D3)D311yescharacteristic, complemented
25C3×C4×D12—34no—
26C3×C4×D12D314yescomplemented
27C2×#3713—34no—
28#63659—31no—
29#63636—31no—
30#63635—31no—
31#63630—31no—
32#63630—34no—
33#63630—64no—
34#63574—34no—
35#63562—34no—
36#63562—34no—
37C22×C3×C3⋊D3—21no—
38C22×C32×D3—21no—
39C2×C33×C4D411yescharacteristic
40C2×C33×C4D411yescharacteristic
41C2×C33×C4C2311yescharacteristic
42C2×C3×C3⋊Dic3—21no—
43C3×C4⋊(C3⋊D3)—21no—
44C3×C4⋊(C3⋊D3)C2×C412yescomplemented
45C3×C4×C3⋊D3—42no—
46C2×C32×Dic3—21no—
47C32×D12—21no—
48C32×D12C2×C412yescomplemented
49C32×C4×D3—42no—
50C3⋊(C3⋊(C3⋊C8))—21no—
51C3⋊(C3⋊(C3⋊C8))C2×C412yescomplemented
52#5014—94no—
53#5014—91no—
54C2×C32×D4—31no—
55C22×C32×C4—61no—
56C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
57C2×C4×C3⋊D3—21no—
58C2×C3×D12—34no—
59C2×C3×D12C2×D314yes—
60C2×C3×C4×D3—24no—
61C2×C3×C4×D3—64no—
62C32×C4⋊C4—31no—
63C32×C22⋊C4—61no—
64C32×C42—24no—
65C32×C42C2×D314yes—
66C32×C42C2×D311yescharacteristic
67#3753—34no—
68#3752—21no—
69C4⋊(C3⋊Dic3)C2×D311yescharacteristic
70C2×C3⋊(C3⋊C8)—31no—
71C2×C3⋊(C3⋊C8)—34no—
72C2×C3⋊(C3⋊C8)—31no—
73C2×C3⋊(C3⋊C8)—34no—
74C2×C3⋊(C3⋊C8)—64no—
75C2×C3⋊(C3⋊C8)—64no—
76C3×(C2×D3)⋊C4—24no—
77C3×(C2×D3)⋊C4—64no—
78C3×C4⋊Dic3—34no—
79C3×C4⋊Dic3C2×D314yes—
80#3713—316no—
81C22×C33C2×D411yescharacteristic
82C2×C3×C3⋊D3—22no—
83C2×C3×C3⋊D3—41no—
84C2×C3×C3⋊D3—41no—
85C2×C32×D3—22no—
86C2×C32×D3—41no—
87C2×C32×D3—41no—
88C33×C4—21no—
89C33×C4—21no—
90C33×C4C22×C411yescharacteristic
91C33×C4#4111yescharacteristic
92C3×C3⋊Dic3—41no—
93C32×Dic3—41no—
94C3×C4×D4—34no—
95C3×C4×D4—91no—
96C2×#213—94no—
97C2×#213—91no—
98C4×D12—34no—
99C4×D12C3⋊D311yescharacteristic, complemented
100#832—94no—
101#832—91no—
102#809—94no—
103#809—91no—
104#808—94no—
105#808—91no—
106#803—94no—
107#803—91no—
108#803—184no—
109C23×C32—61no—
110C22×C3⋊D3—21no—
111C22×C3×D3—24no—
112C22×C3×D3—64no—
113C32×D4—32no—
114C32×D4—61no—
115C2×C32×C4—24no—
116C2×C32×C4—24no—
117C2×C32×C4—24no—
118C2×C32×C4—61no—
119C2×C32×C4—122no—
120C2×C32×C4#6714yes—
121C2×C32×C4#6714yes—
122C2×C32×C4#6711yescharacteristic
123C2×C32×C4#6711yescharacteristic
124C2×C32×C4C22×D314yes—
125C2×C32×C4C22×D311yescharacteristic
126C2×C3⋊Dic3—21no—
127C4⋊(C3⋊D3)—21no—
128C4⋊(C3⋊D3)C4×D312yescomplemented
129C4×C3⋊D3—42no—
130C2×C3×Dic3—24no—
131C2×C3×Dic3—64no—
132C3×D12—24no—
133C3×D12—38no—
134C3×D12—64no—
135C3×D12C4×D318yescomplemented
136C3×C4×D3—48no—
137C3×C4×D3—128no—
138C3⋊(C3⋊C8)—32no—
139C3⋊(C3⋊C8)—38no—
140C3⋊(C3⋊C8)—68no—
141C3⋊(C3⋊C8)—68no—
142C3⋊(C3⋊C8)—64no—
143C3⋊(C3⋊C8)—61no—
144#442—271noSylow
145C2×C33C4×D411yescharacteristic
146C2×C33AGL(1,ℤ/8)11yescharacteristic
147C2×C33AGL(1,ℤ/8)11yescharacteristic
148C3×C3⋊D3—42no—
149C32×D3—42no—
150C2×C3×D4—34no—
151C2×C3×D4—91no—
152C22×C3×C4—64no—
153C22×C3×C4—181no—
154C2×D12—34no—
155C2×D12C2×C3⋊D311yescharacteristic
156C2×C4×D3—21no—
157C2×C4×D3—64no—
158C3×C4⋊C4—34no—
159C3×C4⋊C4—91no—
160C3×C22⋊C4—64no—
161C3×C22⋊C4—181no—
162C3×C42—24no—
163C3×C42D3214yes—
164C3×C42C2×C3⋊D311yescharacteristic
165#213—916no—
166#213—94no—
167(C2×D3)⋊C4—21no—
168(C2×D3)⋊C4—64no—
169C4⋊Dic3—34no—
170C4⋊Dic3C2×C3⋊D311yescharacteristic
171C2×C3⋊C8—94no—
172C2×C3⋊C8—91no—
173C2×C3⋊C8—94no—
174C2×C3⋊C8—91no—
175C2×C3⋊C8—184no—
176C2×C3⋊C8—184no—
177C22×C32—24no—
178C22×C32—62no—
179C22×C32—121no—
180C22×C32—121no—
181C22×C32C2×#6714yes—
182C22×C32C2×#6711yescharacteristic
183C2×C3⋊D3—22no—
184C2×C3⋊D3—41no—
185C2×C3⋊D3—41no—
186C2×C3×D3—28no—
187C2×C3×D3—44no—
188C2×C3×D3—44no—
189C2×C3×D3—68no—
190C2×C3×D3—124no—
191C2×C3×D3—124no—
192C32×C4—21no—
193C32×C4—21no—
194C32×C4—24no—
195C32×C4—24no—
196C32×C4—24no—
197C32×C4—24no—
198C32×C4—44no—
199C32×C4—44no—
200C32×C4—121no—
201C32×C4C2×C4×D314yes—
202C32×C4C2×C4×D311yescharacteristic
203C32×C4#23514yes—
204C32×C4#23511yescharacteristic
205C3⋊Dic3—41no—
206C3×Dic3—44no—
207C3×Dic3—124no—
208C2×D8—271no—
209C4×D4—31no—
210C4×D4—91no—
211#106—271no—
212#102—271no—
213#102—271no—
214#97—271no—
215C33#44211yescharacteristic, complemented, Sylow
216C23×C3—64no—
217C23×C3—181no—
218C22×D3—21no—
219C22×D3—64no—
220C3×D4—38no—
221C3×D4—64no—
222C3×D4—92no—
223C3×D4—181no—
224C2×C3×C4—24no—
225C2×C3×C4—24no—
226C2×C3×C4—24no—
227C2×C3×C4—64no—
228C2×C3×C4—128no—
229C2×C3×C4—181no—
230C2×C3×C4—362no—
231C2×C3×C4#62514yes—
232C2×C3×C4#62514yes—
233C2×C3×C4#63811yescharacteristic
234C2×C3×C4#63811yescharacteristic
235C2×C3×C4C2×D3214yes—
236C2×C3×C4C22×C3⋊D311yescharacteristic
237C2×Dic3—21no—
238C2×Dic3—64no—
239D12—21no—
240D12—38no—
241D12—64no—
242D12C4×C3⋊D312yescomplemented
243C4×D3—42no—
244C4×D3—128no—
245C3⋊C8—98no—
246C3⋊C8—92no—
247C3⋊C8—188no—
248C3⋊C8—188no—
249C3⋊C8—184no—
250C3⋊C8—181no—
251C2×C32—24no—
252C2×C32—24no—
253C2×C32—24no—
254C2×C32—122no—
255C2×C32C4×#6714yes—
256C2×C32C4×#6711yescharacteristic
257C2×C32#93214yes—
258C2×C32#93211yescharacteristic
259C2×C32#94914yes—
260C2×C32#94911yescharacteristic
261C3⋊D3—42no—
262C3×D3—48no—
263C3×D3—128no—
264C2×D4—31no—
265C2×D4—91no—
266C22×C4—61no—
267C22×C4—181no—
268D8—274no—
269C2×C8—271no—
270C2×C8—271no—
271C4⋊C4—31no—
272C4⋊C4—91no—
273C22⋊C4—61no—
274C22⋊C4—181no—
275C42D3×C3⋊D311yescharacteristic
276C22×C3—24no—
277C22×C3—68no—
278C22×C3—124no—
279C22×C3—124no—
280C22×C3—182no—
281C22×C3—361no—
282C22×C3—361no—
283C22×C3C2×#62514yes—
284C22×C3C2×#63811yescharacteristic
285C2×D3—22no—
286C2×D3—41no—
287C2×D3—41no—
288C2×D3—68no—
289C2×D3—124no—
290C2×D3—124no—
291C3×C4—24no—
292C3×C4—24no—
293C3×C4—21no—
294C3×C4—21no—
295C3×C4—24no—
296C3×C4—24no—
297C3×C4—44no—
298C3×C4—44no—
299C3×C4—124no—
300C3×C4—361no—
301C3×C4#379814yes—
302C3×C4C4×D3214yes—
303C3×C4C2×C4×C3⋊D311yescharacteristic
304C3×C4#382811yescharacteristic
305Dic3—41no—
306Dic3—124no—
307C32—24no—
308C32#501414yescomplemented
309C32#501411yescharacteristic, complemented
310C23—61no—
311C23—181no—
312D4—32no—
313D4—61no—
314D4—92no—
315D4—181no—
316C2×C4—61no—
317C2×C4—122no—
318C2×C4—181no—
319C2×C4—362no—
320C2×C4#629011yescharacteristic
321C2×C4#629011yescharacteristic
322C2×C4C2×D3×C3⋊D311yescharacteristic
323C8—272no—
324C8—541no—
325C2×C3—24no—
326C2×C3—24no—
327C2×C3—24no—
328C2×C3—128no—
329C2×C3—362no—
330C2×C3#6382314yes—
331C2×C3#6382314yes—
332C2×C3C4×#62514yes—
333C2×C3C4×#63811yescharacteristic
334C2×C3#6414111yescharacteristic
335C2×C3#6415811yescharacteristic
336D3—42no—
337D3—128no—
338C22—62no—
339C22—121no—
340C22—121no—
341C22—182no—
342C22—361no—
343C22—361no—
344C22C2×#629011yescharacteristic
345C4—21no—
346C4—21no—
347C4—121no—
348C4—361no—
349C4#8902011yescharacteristic
350C4C4×D3×C3⋊D311yescharacteristic
351C3—24no—
352C3#1059034414yescomplemented
353C3#1059271311yescharacteristic, complemented
354C2—122no—
355C2—362no—
356C2#1172336411yescharacteristic
357C2#1172336511yescharacteristic
358C2C4×#629011yescharacteristic
subgroupG/H
centerC22C2×#6290
commutatorC33×C4C22×C4
FrattiniC2×C4C2×D3×C3⋊D3
FittingC33×C42C22
socleC22×C33C2×D4
prime radicalC2×#88789C2
cosocleC2×C33×C4C23
hypercenterC42D3×C3⋊D3
hypocenterC33#442
LCSG ▷ C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C42

Complements

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

NH ≅ G/Ntype
1. C3×C4×C4⋊(C3⋊D3)C2semidirect
2. C32×C4×D12C2semidirect
4. #11722173C2semidirect
5. #11722053C2semidirect
6. #11722033C2semidirect
7. #11722031C2semidirect
21. #88789C4semidirect
24. C4×C4⋊(C3⋊D3)D3semidirect
26. C3×C4×D12D3semidirect
44. C3×C4⋊(C3⋊D3)C2×C4semidirect
48. C32×D12C2×C4semidirect
51. C3⋊(C3⋊(C3⋊C8))C2×C4semidirect
99. C4×D12C3⋊D3semidirect
128. C4⋊(C3⋊D3)C4×D3semidirect
135. C3×D12C4×D3semidirect
215. C33#442semidirect
242. D12C4×C3⋊D3semidirect
308. C32#5014semidirect
309. C32#5014semidirect
352. C3#10590344semidirect
353. C3#10592713semidirect

Maximal subgroups

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

subgroupindex
C3×C4×C4⋊(C3⋊D3)2
C32×C4×D122
C2×#887892
#117221732
#117220532
#117220332
#117220312
#105927133
#105903443