Finite Group Databasebeta

← #13180976

Subgroups — #13180976

Order 1728.

1256 conjugacy classes of subgroups; only the 174 normal subgroups are shown.

normal subgroups only

This group has 1256 conjugacy classes of subgroups — too many to draw. Only its 174 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
1C2×C3⋊D3×D12C211yesmaximal, characteristic
2#11724514C211yesmaximal, characteristic, complemented
3C3×#64286C211yesmaximal, characteristic, complemented
4D3×C22⋊#157C214yesmaximal, complemented
5#11724259C214yesmaximal, complemented
6D12×#157C214yesmaximal, complemented
7#10596642—31nomaximal
8C22×D3×C3⋊D3C2212yes—
9C2×D3×#157C2218yescomplemented
10C2×C4⋊(C3⋊(C3⋊D3))C411yescharacteristic, complemented
11C2×C3×C4×C3⋊D3C2211yescharacteristic
12C2×C32×D12C411yescharacteristic, complemented
13C2×#6291C412yescomplemented
14C3⋊D3×D12C412yescomplemented
15C3⋊D3×D12C412yescomplemented
16C3⋊D3×D12C2212yes—
17C3⋊D3×D12C2212yes—
18#88994C2212yes—
19C4⋊(C3⋊#157)C2212yescomplemented
20C4×C3⋊#157C2212yescomplemented
21C3×C22⋊#157C2212yes—
22C3×C4⋊#157C2212yescomplemented
23C3×C4×#157C2212yescomplemented
24(C2×D3)⋊#157C2218yescomplemented
25C2×D4×C3⋊D3—31no—
26C2×D3×D12—62no—
27C2×C22⋊#157—34no—
28D4×#157—34no—
29D4×#157—34no—
30#64286—31no—
31#64286D311yescharacteristic, complemented
32C22×C3⋊(C3⋊D3)C2×C412yescomplemented
33C22×C3×C3⋊D3C2311yescharacteristic
34C22×C32×D3C2×C412yescomplemented
35C2×D3×C3⋊D3—44no—
36C2×D3×C3⋊D3C2×C414yescomplemented
37C2×D3×C3⋊D3C2314yes—
38C2×C3⋊#157C2314yes—
39C2×C3×#157C2314yes—
40D3×#157—48no—
41C2×C33×C4C2×C411yescharacteristic, complemented
42C4⋊(C3⋊(C3⋊D3))C2×C412yescomplemented
43C4⋊(C3⋊(C3⋊D3))C2×C412yescomplemented
44C2×C3×C3⋊Dic3C2×C411yescharacteristic, complemented
45C3×C4×C3⋊D3C2×C411yescharacteristic, complemented
46C3×C4×C3⋊D3C2×C411yescharacteristic, complemented
47C3×C4×C3⋊D3C2311yescharacteristic
48C3×C4×C3⋊D3C2311yescharacteristic
49C32×D12C2×C412yescomplemented
50C32×D12C2×C412yescomplemented
51#6291C2×C414yescomplemented
52#6291C2×C414yescomplemented
53#5521—91nomaximal
54C23×C3⋊D3—32no—
55C22×D32—64no—
56C22×#157—38no—
57C2×C32×D4—31no—
58C2×#638—32no—
59D4×C3⋊D3—32no—
60D4×C3⋊D3—32no—
61D4×C3⋊D3—32no—
62D4×C3⋊D3—32no—
63C2×C4⋊(C3⋊D3)—31no—
64C2×C4⋊(C3⋊D3)—62no—
65C2×C4⋊(C3⋊D3)—122no—
66C2×C4×C3⋊D3C2×D311yescharacteristic
67C2×C3×D12—22no—
68C2×C3×C4×D3—62no—
69C2×#626—64no—
70D3×D12—64no—
71D3×D12—64no—
72D3×D12—64no—
73D3×D12—64no—
74C22⋊#157—32no—
75C22⋊#157—38no—
76C22⋊#157C2×D312yes—
77C4⋊#157—32no—
78C4⋊#157C2×D312yescomplemented
79C4×#157—32no—
80C4×#157C2×D312yescomplemented
81C22×C33C22×C411yescharacteristic
82C2×C3⋊(C3⋊D3)—42no—
83C2×C3⋊(C3⋊D3)C22×C414yes—
84C2×C3×C3⋊D3—22no—
85C2×C3×C3⋊D3C22×C411yescharacteristic
86C2×C3×C3⋊D3C2411yescharacteristic
87C2×C32×D3—42no—
88C2×C32×D3C22×C414yes—
89D3×C3⋊D3—28no—
90D3×C3⋊D3—44no—
91C3⋊#157—24no—
92C3×#157—24no—
93C33×C4C22×C411yescharacteristic
94C33×C4C22×C411yescharacteristic
95C3×C3⋊Dic3C22×C411yescharacteristic
96C3×C3⋊Dic3C22×C411yescharacteristic
97C2×D3×D4—182no—
98C22×D12—91no—
99C3×#117—91no—
100#926—91no—
101#893—94no—
102D3×C22⋊C4—94no—
103C4×D12—94no—
104C23×C32—32no—
105C22×C3⋊D3—32no—
106C22×C3⋊D3—34no—
107C22×C3⋊D3—34no—
108C22×C3⋊D3—64no—
109C22×C3⋊D3—124no—
110C22×C3⋊D3—124no—
111C22×C3⋊D3C22×D311yescharacteristic
112C22×C3×D3—24no—
113C22×C3×D3—62no—
114C2×D32—68no—
115C2×D32—68no—
116C2×D32—1216no—
117C2×#157—34no—
118C2×#157—128no—
119C2×#157C22×D314yes—
120C32×D4—32no—
121C32×D4—32no—
122C2×C32×C4—22no—
123C2×C32×C4—42no—
124C2×C32×C4C4×D311yescharacteristic, complemented
125#638—34no—
126#638—34no—
127C2×C3⋊Dic3C4×D311yescharacteristic, complemented
128C4⋊(C3⋊D3)—32no—
129C4⋊(C3⋊D3)—32no—
130C4⋊(C3⋊D3)—64no—
131C4⋊(C3⋊D3)—64no—
132C4⋊(C3⋊D3)—124no—
133C4⋊(C3⋊D3)—124no—
134C4×C3⋊D3C4×D311yescharacteristic, complemented
135C4×C3⋊D3C4×D311yescharacteristic, complemented
136C4×C3⋊D3C22×D311yescharacteristic
137C4×C3⋊D3C22×D311yescharacteristic
138C2×C3×Dic3—62no—
139C3×D12—24no—
140C3×D12—24no—
141C3×C4×D3—62no—
142C3×C4×D3—62no—
143C3×C4×D3—62no—
144C3×C4×D3—62no—
145#626—68no—
146#626—68no—
147#518—271noSylow
148C2×C33—21no—
149C2×C33C23×C411yescharacteristic
150C3⋊(C3⋊D3)—24no—
151C3×C3⋊D3—21no—
152C3×C3⋊D3H(5,𝔽2)12yes—
153C32×D3—24no—
154C23×D3—92no—
155C23×D3—184no—
156C2×C3×D4—62no—
157C22×C3×C4—91no—
158C2×#67—184no—
159D3×D4—184no—
160D3×D4—184no—
161D3×D4—184no—
162D3×D4—184no—
163C2×D12—92no—
164C2×D12—91no—
165C2×D12—92no—
166C2×D12—92no—
167C2×D12—92no—
168C2×D12—92no—
169C2×D12—182no—
170C2×D12—362no—
171C2×D12#15711yescharacteristic, complemented
172C2×C4×D3—62no—
173C2×C4×D3—98no—
174C3×C4⋊C4—92no—
175C3×C22⋊C4—92no—
176C3×C42—92no—
177#217—92no—
178(C2×D3)⋊C4—98no—
179C4⋊Dic3—92no—
180C4×Dic3—92no—
181C22×C32—22no—
182C22×C32—34no—
183C22×C32—42no—
184C22×C32—122no—
185C22×C32C2×C4×D311yescharacteristic
186C2×C3⋊D3—22no—
187C2×C3⋊D3—34no—
188C2×C3⋊D3—68no—
189C2×C3⋊D3—68no—
190C2×C3⋊D3—128no—
191C2×C3⋊D3—124no—
192C2×C3⋊D3—122no—
193C2×C3⋊D3—128no—
194C2×C3⋊D3—248no—
195C2×C3⋊D3C2×C4×D311yescharacteristic
196C2×C3⋊D3C23×D311yescharacteristic
197C2×C3×D3—28no—
198C2×C3×D3—48no—
199C2×C3×D3—62no—
200C2×C3×D3—62no—
201C2×C3×D3—68no—
202D32—1216no—
203D32—1216no—
204#157—24no—
205#157—64no—
206C32×C4—22no—
207C32×C4—22no—
208C32×C4—42no—
209C32×C4—42no—
210C32×C4C2×C4×D311yescharacteristic
211C32×C4C2×C4×D311yescharacteristic
212C3⋊Dic3C2×C4×D311yescharacteristic
213C3⋊Dic3C2×C4×D311yescharacteristic
214C3×Dic3—62no—
215C3×Dic3—62no—
216C22×D4—271no—
217C4×D4—274no—
218C4×D4—274no—
219#117—91no—
220#117—271no—
221C2×C22⋊C4—274no—
222C33#51811yescharacteristic, complemented, Sylow
223C23×C3—64no—
224C23×C3—91no—
225C22×D3—62no—
226C22×D3—92no—
227C22×D3—94no—
228C22×D3—94no—
229C22×D3—188no—
230C22×D3—188no—
231C22×D3—184no—
232C22×D3—364no—
233C22×D3—364no—
234C22×D3—3616no—
235C22×D3C2×#15712yescomplemented
236C3×D4—64no—
237C3×D4—64no—
238C2×C3×C4—22no—
239C2×C3×C4—42no—
240C2×C3×C4—91no—
241C2×C3×C4—91no—
242C2×C3×C4—91no—
243C2×C3×C4—91no—
244C2×C3×C4—91no—
245C2×C3×C4—94no—
246C2×C3×C4C2×#15711yescharacteristic, complemented
247#67—188no—
248#67—188no—
249C2×Dic3—62no—
250C2×Dic3—94no—
251D12—94no—
252D12—94no—
253D12—92no—
254D12—92no—
255D12—184no—
256D12—184no—
257D12—364no—
258D12—364no—
259D12C2×#15712yescomplemented
260D12C2×#15712yescomplemented
261C4×D3—62no—
262C4×D3—62no—
263C4×D3—62no—
264C4×D3—62no—
265C4×D3—368no—
266C2×C32—21no—
267C2×C32—24no—
268C2×C32—22no—
269C2×C32—44no—
270C2×C32—42no—
271C2×C32—64no—
272C2×C32C22×C4×D311yescharacteristic
273C3⋊D3—21no—
274C3⋊D3—64no—
275C3⋊D3—128no—
276C3⋊D3—248no—
277C3⋊D3#100912yes—
278C3×D3—48no—
279C3×D3—64no—
280C3×D3—64no—
281C24—272no—
282C2×D4—31no—
283C2×D4—272no—
284C2×D4—272no—
285C2×D4—272no—
286C2×D4—272no—
287C2×D4—272no—
288C2×D4—271no—
289C22×C4—91no—
290C22×C4—278no—
291C4⋊C4—92no—
292C4⋊C4—272no—
293C22⋊C4—92no—
294C22⋊C4—278no—
295C22⋊C4—272no—
296C42—92no—
297C42—272no—
298C22×C3—22no—
299C22×C3—42no—
300C22×C3—68no—
301C22×C3—91no—
302C22×C3—91no—
303C22×C3—128no—
304C22×C3—182no—
305C22×C3C22×#15711yescharacteristic
306C2×D3—42no—
307C2×D3—62no—
308C2×D3—62no—
309C2×D3—68no—
310C2×D3—94no—
311C2×D3—188no—
312C2×D3—188no—
313C2×D3—368no—
314C2×D3—3616no—
315C2×D3—364no—
316C2×D3—3616no—
317C2×D3—362no—
318C2×D3—368no—
319C2×D3—728no—
320C2×D3C22×#15714yes—
321C3×C4—22no—
322C3×C4—22no—
323C3×C4—42no—
324C3×C4—42no—
325C3×C4—91no—
326C3×C4—91no—
327C3×C4—184no—
328C3×C4C22×#15711yescharacteristic
329C3×C4C22×#15711yescharacteristic
330Dic3—62no—
331Dic3—62no—
332Dic3—184no—
333C32—22no—
334C32—42no—
335C32#552111yescharacteristic, complemented
336C23—32no—
337C23—91no—
338C23—274no—
339C23—274no—
340C23—272no—
341C23—1084no—
342D4—32no—
343D4—32no—
344D4—274no—
345D4—274no—
346D4—272no—
347D4—272no—
348C2×C4—91no—
349C2×C4—91no—
350C2×C4—91no—
351C2×C4—91no—
352C2×C4—91no—
353C2×C4—94no—
354C2×C4—274no—
355C2×C4—1088no—
356C2×C4D3×#15711yescharacteristic, complemented
357C2×C3—24no—
358C2×C3—21no—
359C2×C3—22no—
360C2×C3—44no—
361C2×C3—42no—
362C2×C3—92no—
363C2×C3—128no—
364C2×C3—181no—
365C2×C3C23×#15711yescharacteristic
366D3—24no—
367D3—64no—
368D3—64no—
369D3—184no—
370D3—368no—
371D3—728no—
372C22—34no—
373C22—91no—
374C22—91no—
375C22—122no—
376C22—182no—
377C22—274no—
378C22—548no—
379C22—1082no—
380C22—1084no—
381C22C2×D3×#15711yescharacteristic
382C4—91no—
383C4—91no—
384C4—184no—
385C4—544no—
386C4C2×D3×#15711yescharacteristic
387C4C2×D3×#15711yescharacteristic
388C3—22no—
389C3—42no—
390C3#1059664211yescharacteristic, complemented
391C2—21no—
392C2—64no—
393C2—92no—
394C2—181no—
395C2—544no—
396C2C22×D3×#15711yescharacteristic
subgroupG/H
center, FrattiniC2C22×D3×#157
commutator, socleC2×C33C23×C4
FittingC2×C33×C4C2×C4
prime radicalC2×C3⋊D3×D12C2
cosocleC2×C3×C3⋊D3C24
hypercenterC2×C4D3×#157
hypocenterC33#518
LCSG ▷ C2×C33 ▷ C33
UCSC1 ◁ C2 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
2. #11724514C2semidirect
3. C3×#64286C2semidirect
4. D3×C22⋊#157C2semidirect
5. #11724259C2semidirect
6. D12×#157C2semidirect
9. C2×D3×#157C22semidirect
10. C2×C4⋊(C3⋊(C3⋊D3))C4semidirect
12. C2×C32×D12C4semidirect
13. C2×#6291C4semidirect
14. C3⋊D3×D12C4semidirect
15. C3⋊D3×D12C4semidirect
19. C4⋊(C3⋊#157)C22semidirect
20. C4×C3⋊#157C22semidirect
22. C3×C4⋊#157C22semidirect
23. C3×C4×#157C22semidirect
24. (C2×D3)⋊#157C22semidirect
31. #64286D3semidirect
32. C22×C3⋊(C3⋊D3)C2×C4semidirect
34. C22×C32×D3C2×C4semidirect
36. C2×D3×C3⋊D3C2×C4semidirect
41. C2×C33×C4C2×C4semidirect
42. C4⋊(C3⋊(C3⋊D3))C2×C4semidirect
43. C4⋊(C3⋊(C3⋊D3))C2×C4semidirect
44. C2×C3×C3⋊Dic3C2×C4semidirect
45. C3×C4×C3⋊D3C2×C4semidirect
46. C3×C4×C3⋊D3C2×C4semidirect
49. C32×D12C2×C4semidirect
50. C32×D12C2×C4semidirect
51. #6291C2×C4semidirect
52. #6291C2×C4semidirect
78. C4⋊#157C2×D3semidirect
80. C4×#157C2×D3semidirect
124. C2×C32×C4C4×D3semidirect
127. C2×C3⋊Dic3C4×D3semidirect
134. C4×C3⋊D3C4×D3semidirect
135. C4×C3⋊D3C4×D3semidirect
171. C2×D12#157semidirect
222. C33#518semidirect
235. C22×D3C2×#157semidirect
246. C2×C3×C4C2×#157semidirect
259. D12C2×#157semidirect
260. D12C2×#157semidirect
335. C32#5521semidirect
356. C2×C4D3×#157semidirect
390. C3#10596642semidirect

Maximal subgroups

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

subgroupindex
C2×C3⋊D3×D122
#117245142
C3×#642862
D3×C22⋊#1572
#117242592
D12×#1572
#105966423
#55219