Finite Group Databasebeta

← #13151452

Subgroups — #13151452

Order 1728.

1249 conjugacy classes of subgroups; only the 227 normal subgroups are shown.

normal subgroups only

This group has 1249 conjugacy classes of subgroups — too many to draw. Only its 227 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×#88744C211yesmaximal, characteristic
2D4×(C3⋊Dic3)⋊C3C212yesmaximal, complemented
3#11721298C212yesmaximal, complemented
4C2×#88447C212yesmaximal, complemented
5#11721293C214yesmaximal, complemented
6C4×#6199C211yesmaximal, characteristic, complemented
7#11721089C211yesmaximal, characteristic, complemented
8#11721083C212yesmaximal, complemented
9Dic3×#237—32nomaximal
10C2×D4×H(3,𝔽3)C2211yescharacteristic
11C2×#6235C2212yes—
12C22×#1084C2212yes—
13#88744C418yescomplemented
14C2×C4×#272C2211yescharacteristic
15C2×#6229C2211yescharacteristic
16C22×(C3⋊Dic3)⋊C3C2214yes—
17C2×#6199C2212yes—
18#3757⋊C3C2214yes—
19C4⋊((C3⋊Dic3)⋊C3)C2212yes—
20C4×(C3⋊Dic3)⋊C3C2212yes—
21#88451C2212yes—
22#88449C2214yes—
23#88447C2214yes—
24#88447C2211yescharacteristic
25#88447C2212yes—
26#88447C2211yescharacteristic
27C2×C3×#237—32no—
28C2×C3×#237—61no—
29D4×C3⋊Dic3D312yescomplemented
30C3×D4×Dic3—32no—
31#67⋊Dic3—34no—
32Dic3×#67—34no—
33#63955—34no—
34C2×Dic32—34no—
35C4×D3×Dic3—32no—
36#63843—32no—
37Dic3×Dic6—32no—
38#63841—32no—
39C23×H(3,𝔽3)C2312yes—
40C22×#272C2311yescharacteristic
41D4×H(3,𝔽3)C2×C414yescomplemented
42C2×C4×H(3,𝔽3)C2311yescharacteristic
43#6235C2×C418yescomplemented
44C2×#1084C2×C418yescomplemented
45C2×#1084C2311yescharacteristic
46C2×#1084C2312yes—
47C4×#272C2×C414yescomplemented
48#6229C2×C414yescomplemented
49C2×(C3⋊Dic3)⋊C3—28no—
50C2×(C3⋊Dic3)⋊C3C2312yes—
51C2×(C3⋊Dic3)⋊C3C2314yes—
52C2×(C3⋊Dic3)⋊C3C2312yes—
53#6199—24no—
54Dic3×#41—91no—
55C4×#237—92no—
56C2×C32×D4—21no—
57C2×C32×D4C2×D312yes—
58C22×C3⋊Dic3C2×D314yes—
59C2×C3×#67—34no—
60C2×C3×#67—62no—
61C22×C3×Dic3—34no—
62C22×C3×Dic3—34no—
63C22×C3×Dic3—62no—
64C3×#237—316no—
65C3×#237—68no—
66C2×C3×C4×D3—32no—
67C2×C3×C4×D3—61no—
68C2×C3×Dic6—32no—
69C2×C3×Dic6—61no—
70C2×D3×Dic3—32no—
71C2×D3×Dic3—32no—
72#3757C2×D314yes—
73C4⋊(C3⋊Dic3)C2×D312yes—
74C4×C3⋊Dic3C2×D312yes—
75C3×#217—34no—
76C3×C4⋊Dic3—32no—
77C3×C4×Dic3—32no—
78Dic3⋊Dic3—34no—
79Dic3⋊Dic3—34no—
80(C2×D3)⋊Dic3—34no—
81Dic32—34no—
82Dic32—34no—
83Dic32—34no—
84Dic32—32no—
85Dic32—32no—
86C22×H(3,𝔽3)—24no—
87C22×H(3,𝔽3)C22×C414yes—
88C22×H(3,𝔽3)C2411yescharacteristic
89C2×#272—22no—
90C2×#272C22×C412yes—
91C4×H(3,𝔽3)C22×C412yes—
92#1084C22×C414yes—
93#1084C22×C412yes—
94(C3⋊Dic3)⋊C3—22no—
95(C3⋊Dic3)⋊C3—24no—
96(C3⋊Dic3)⋊C3#4114yescomplemented
97C2×C3×#41—91no—
98C2×#237—32no—
99C2×#237—61no—
100C3×C4×D4—92no—
101Q8×Dic3—92no—
102D4×Dic3—32no—
103D4×Dic3—92no—
104D4×Dic3—94no—
105C4×#67—94no—
106#926—91no—
107#926—92no—
108C2×C4×Dic3—91no—
109C2×C4×Dic3—92no—
110C2×C4×Dic3—94no—
111#892—92no—
112#887—92no—
113#67⋊C4—94no—
114#877—94no—
115C42×D3—92no—
116C4×Dic6—92no—
117C23×C32—22no—
118C23×C32C22×D314yes—
119C22×C3×D3—32no—
120C22×C3×D3—61no—
121C32×D4—24no—
122C32×D4C4×D318yescomplemented
123C2×C32×C4—21no—
124C2×C32×C4C22×D312yes—
125C2×C3⋊Dic3—28no—
126C2×C3⋊Dic3C22×D314yes—
127C2×C3⋊Dic3C22×D312yes—
128C2×C3⋊Dic3C22×D312yes—
129C3×#67—316no—
130C3×#67—68no—
131C2×C3×Dic3—316no—
132C2×C3×Dic3—32no—
133C2×C3×Dic3—32no—
134C2×C3×Dic3—34no—
135C2×C3×Dic3—34no—
136C2×C3×Dic3—32no—
137C2×C3×Dic3—68no—
138C2×C3×Dic3—62no—
139C2×C3×Dic3—61no—
140C2×C3×Dic3—68no—
141C3×C4×D3—38no—
142C3×C4×D3—64no—
143C3×Dic6—38no—
144C3×Dic6—64no—
145D3×Dic3—64no—
146D3×Dic3—64no—
147C4×#41—271noSylow
148C2×H(3,𝔽3)—24no—
149C2×H(3,𝔽3)C23×C411yescharacteristic
150C2×H(3,𝔽3)C2×#4112yes—
151#272—22no—
152C3×#41—98no—
153C2×C3×Q8—91no—
154C2×C3×D4—32no—
155C2×C3×D4—61no—
156C2×C3×D4—92no—
157C2×C3×D4D3211yescharacteristic
158C22×C3×C4—92no—
159C22×C3×C4—91no—
160C22×C3×C4—94no—
161C2×#67—34no—
162C2×#67—62no—
163C22×Dic3—34no—
164C22×Dic3—34no—
165C22×Dic3—62no—
166C22×Dic3—94no—
167C22×Dic3—92no—
168#237—316no—
169#237—68no—
170C2×C4×D3—32no—
171C2×C4×D3—61no—
172C2×C4×D3—92no—
173C2×C4×D3—92no—
174C2×Dic6—32no—
175C2×Dic6—61no—
176C3×C4⋊C4—92no—
177C3×C22⋊C4—94no—
178C3×C42—92no—
179#217—34no—
180#217—94no—
181#217—92no—
182(C2×D3)⋊C4—94no—
183C4⋊Dic3—32no—
184C4⋊Dic3—92no—
185C4⋊Dic3—94no—
186Dic3⋊C4—94no—
187Dic3⋊C4—94no—
188C4×Dic3—32no—
189C4×Dic3—92no—
190C4×Dic3—91no—
191C4×Dic3—91no—
192C4×Dic3—92no—
193C4×Dic3—94no—
194C4×Dic3—94no—
195C4×Dic3—92no—
196C4×Dic3—92no—
197C4×Dic3—94no—
198C4×Dic3—94no—
199C22×C32—24no—
200C22×C32—24no—
201C22×C32—21no—
202C22×C32—24no—
203C22×C32—44no—
204C22×C32C2×C4×D318yes—
205C22×C32C23×D312yes—
206C2×C3×D3—34no—
207C2×C3×D3—62no—
208C2×C3×D3—62no—
209C2×C3×D3—62no—
210C2×C3×D3—122no—
211C32×C4—22no—
212C32×C4C2×C4×D314yes—
213C3⋊Dic3—24no—
214C3⋊Dic3—22no—
215C3⋊Dic3#23714yescomplemented
216C3×Dic3—38no—
217C3×Dic3—34no—
218C3×Dic3—34no—
219C3×Dic3—64no—
220C3×Dic3—62no—
221C3×Dic3—62no—
222C3×Dic3—64no—
223C2×#41—91no—
224C4×Q8—272no—
225C4×D4—92no—
226C4×D4—274no—
227#117—272no—
228#117—271no—
229C2×C42—271no—
230C2×C42—272no—
231H(3,𝔽3)C4×#4111yescharacteristic, complemented, Sylow
232C23×C3—34no—
233C23×C3—62no—
234C23×C3—91no—
235C23×C3C2×D3212yes—
236C22×D3—32no—
237C22×D3—61no—
238C3×Q8—94no—
239C3×D4—38no—
240C3×D4—64no—
241C3×D4—98no—
242C3×D4#62414yes—
243C2×C3×C4—32no—
244C2×C3×C4—61no—
245C2×C3×C4—92no—
246C2×C3×C4—91no—
247C2×C3×C4—92no—
248C2×C3×C4—98no—
249C2×C3×C4—94no—
250C2×C3×C4—94no—
251C2×C3×C4—92no—
252C2×C3×C4—188no—
253C2×C3×C4C2×D3211yescharacteristic
254#67—316no—
255#67—68no—
256C2×Dic3—32no—
257C2×Dic3—34no—
258C2×Dic3—316no—
259C2×Dic3—32no—
260C2×Dic3—32no—
261C2×Dic3—34no—
262C2×Dic3—62no—
263C2×Dic3—61no—
264C2×Dic3—68no—
265C2×Dic3—68no—
266C2×Dic3—92no—
267C2×Dic3—92no—
268C2×Dic3—94no—
269C2×Dic3—188no—
270C2×Dic3—184no—
271C4×D3—38no—
272C4×D3—64no—
273C4×D3—184no—
274C4×D3—184no—
275Dic6—38no—
276Dic6—64no—
277C2×C32—28no—
278C2×C32—22no—
279C2×C32—21no—
280C2×C32—44no—
281C2×C32C22×C4×D312yes—
282C2×C32C2×#23712yes—
283C2×C32D3×#4112yes—
284C3×D3—64no—
285C3×D3—122no—
286#41—98no—
287C2×Q8—91no—
288C2×D4—92no—
289C2×D4((C3⋊D3)⋊C3)⋊C211yescharacteristic
290C22×C4—92no—
291C22×C4—91no—
292C22×C4—94no—
293C22×C4—272no—
294C4⋊C4—92no—
295C4⋊C4—274no—
296C22⋊C4—94no—
297C22⋊C4—272no—
298C42—92no—
299C42—271no—
300C42—271no—
301C42—272no—
302C42—274no—
303C22×C3—24no—
304C22×C3—32no—
305C22×C3—38no—
306C22×C3—64no—
307C22×C3—64no—
308C22×C3—61no—
309C22×C3—64no—
310C22×C3—92no—
311C22×C3—124no—
312C22×C3—182no—
313C22×C3C2×#62414yes—
314C22×C3C22×D3211yescharacteristic
315C2×D3—34no—
316C2×D3—62no—
317C2×D3—62no—
318C2×D3—62no—
319C2×D3—122no—
320C3×C4—34no—
321C3×C4—62no—
322C3×C4—94no—
323C3×C4—92no—
324C3×C4—94no—
325C3×C4—182no—
326C3×C4—184no—
327C3×C4C2×#62412yes—
328Dic3—38no—
329Dic3—34no—
330Dic3—34no—
331Dic3—64no—
332Dic3—62no—
333Dic3—64no—
334Dic3—62no—
335Dic3—94no—
336Dic3—182no—
337Dic3—184no—
338C32—21no—
339C32C4×#23712yescomplemented
340C23—91no—
341C23C2×((C3⋊D3)⋊C3)⋊C212yes—
342Q8—94no—
343D4—98no—
344D4#619914yescomplemented
345C2×C4—92no—
346C2×C4—91no—
347C2×C4—92no—
348C2×C4—98no—
349C2×C4—94no—
350C2×C4—92no—
351C2×C4—94no—
352C2×C4—188no—
353C2×C4—544no—
354C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
355C2×C3—24no—
356C2×C3—32no—
357C2×C3—32no—
358C2×C3—32no—
359C2×C3—68no—
360C2×C3—62no—
361C2×C3—61no—
362C2×C3—124no—
363C2×C3—182no—
364C2×C3D3×#23712yes—
365C2×C3C22×#62411yescharacteristic
366D3—64no—
367D3—122no—
368C22—24no—
369C22—92no—
370C22—182no—
371C22C2×#619914yes—
372C22C22×((C3⋊D3)⋊C3)⋊C211yescharacteristic
373C4—94no—
374C4—92no—
375C4—94no—
376C4—184no—
377C4—182no—
378C4C2×#619912yes—
379C3—32no—
380C3—61no—
381C3#1059519011yescharacteristic
382C2—24no—
383C2—182no—
384C2#1172250412yes—
385C2C22×#619911yescharacteristic
subgroupG/H
centerC22C22×((C3⋊D3)⋊C3)⋊C2
commutatorC2×H(3,𝔽3)C23×C4
Frattini, socleC22×C3C22×D32
FittingC2×D4×H(3,𝔽3)C22
prime radicalC2×#88744C2
cosocleC22×H(3,𝔽3)C24
hypercenterC2×D4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)C4×#41
LCSG ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
2. D4×(C3⋊Dic3)⋊C3C2semidirect
3. #11721298C2semidirect
4. C2×#88447C2semidirect
5. #11721293C2semidirect
6. C4×#6199C2semidirect
7. #11721089C2semidirect
8. #11721083C2semidirect
13. #88744C4semidirect
29. D4×C3⋊Dic3D3semidirect
41. D4×H(3,𝔽3)C2×C4semidirect
43. #6235C2×C4semidirect
44. C2×#1084C2×C4semidirect
47. C4×#272C2×C4semidirect
48. #6229C2×C4semidirect
96. (C3⋊Dic3)⋊C3#41semidirect
122. C32×D4C4×D3semidirect
215. C3⋊Dic3#237semidirect
231. H(3,𝔽3)C4×#41semidirect
339. C32C4×#237semidirect
344. D4#6199semidirect

Maximal subgroups

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

subgroupindex
C2×#887442
D4×(C3⋊Dic3)⋊C32
#117212982
C2×#884472
#117212932
C4×#61992
#117210892
#117210832
Dic3×#2373