Finite Group Databasebeta

← #13151458

Subgroups — #13151458

Order 1728.

1419 conjugacy classes of subgroups; only the 219 normal subgroups are shown.

normal subgroups only

This group has 1419 conjugacy classes of subgroups — too many to draw. Only its 219 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×D4×#272C211yesmaximal, characteristic
2D4×(C3⋊Dic3)⋊C3C212yesmaximal, complemented
3#11721320C212yesmaximal, complemented
4C2×#88451C212yesmaximal, complemented
5#11721293C214yesmaximal, complemented
6#11721149C212yesmaximal, complemented
7#11721097C211yesmaximal, characteristic, complemented
8#11721089C211yesmaximal, characteristic, complemented
9#10595215—32nomaximal
10C23×#272C2212yes—
11C2×D4×H(3,𝔽3)C2211yescharacteristic
12C2×#6235C2212yes—
13D4×#272C418yescomplemented
14C2×C4⋊#272C2211yescharacteristic
15C2×C4×#272C2211yescharacteristic
16C22×(C3⋊Dic3)⋊C3C2214yescomplemented
17C2×#6199C2214yescomplemented
18#3757⋊C3C2214yescomplemented
19C4⋊((C3⋊Dic3)⋊C3)C2212yescomplemented
20C4×(C3⋊Dic3)⋊C3C2212yescomplemented
21#88451C2211yescharacteristic
22#88451C2212yes—
23#88451C2211yescharacteristic
24#88451C2214yescomplemented
25#88449C2212yescomplemented
26#88447C2212yescomplemented
27C2×C3×D3×D4—32no—
28C2×C3×D3×D4—61no—
29D4×C3⋊Dic3D312yescomplemented
30C3×D4×Dic3—32no—
31#67⋊Dic3—34no—
32Dic3×#67—34no—
33D3×#217—34no—
34C2×(C2×D3)⋊Dic3—34no—
35#63898—32no—
36Dic3×D12—32no—
37#63855—32no—
38#63841—32no—
39C23×H(3,𝔽3)C2312yes—
40C22×#272—44no—
41C22×#272C2×C418yescomplemented
42C22×#272C2311yescharacteristic
43C22×#272C2312yes—
44D4×H(3,𝔽3)C2×C414yescomplemented
45C2×C4×H(3,𝔽3)C2311yescharacteristic
46#6235C2×C418yescomplemented
47C2×#1084C2311yescharacteristic
48C4⋊#272C2×C414yescomplemented
49C4×#272C2×C414yescomplemented
50C2×(C3⋊Dic3)⋊C3—44no—
51C2×(C3⋊Dic3)⋊C3C2312yes—
52C2×(C3⋊Dic3)⋊C3C2314yes—
53C2×(C3⋊Dic3)⋊C3C2312yes—
54#6199—44no—
55#5798—91no—
56#5545—92no—
57C23×C3×D3—34no—
58C23×C3×D3—62no—
59C2×C32×D4—21no—
60C2×C32×D4C2×D312yes—
61C22×C3⋊Dic3C2×D314yescomplemented
62C2×C3×#67—34no—
63C2×C3×#67—62no—
64C22×C3×Dic3—34no—
65C3×D3×D4—316no—
66C3×D3×D4—68no—
67C2×C3×D12—32no—
68C2×C3×D12—61no—
69C2×C3×C4×D3—32no—
70C2×C3×C4×D3—61no—
71C2×D3×Dic3—34no—
72C2×D3×Dic3—34no—
73#3757C2×D314yescomplemented
74C4⋊(C3⋊Dic3)C2×D312yescomplemented
75C4×C3⋊Dic3C2×D312yescomplemented
76C3×#217—34no—
77C3×C4⋊Dic3—32no—
78C3×C4×Dic3—32no—
79Dic3⋊Dic3—32no—
80Dic3⋊Dic3—32no—
81(C2×D3)⋊Dic3—34no—
82(C2×D3)⋊Dic3—32no—
83(C2×D3)⋊Dic3—32no—
84(C2×D3)⋊Dic3—34no—
85(C2×D3)⋊Dic3—34no—
86Dic32—34no—
87C22×H(3,𝔽3)—24no—
88C22×H(3,𝔽3)C22×C414yes—
89C22×H(3,𝔽3)C2411yescharacteristic
90C2×#272—22no—
91C2×#272—24no—
92C2×#272—48no—
93C2×#272C22×C412yes—
94C2×#272C22×C414yes—
95C4×H(3,𝔽3)C22×C412yes—
96#1084C22×C412yes—
97(C3⋊Dic3)⋊C3—22no—
98(C3⋊Dic3)⋊C3—24no—
99(C3⋊Dic3)⋊C3—22no—
100C22×C3×D4—91no—
101C2×D3×D4—32no—
102C2×D3×D4—61no—
103C3×C4×D4—92no—
104C2×#217—92no—
105C2×#217—92no—
106D4×Dic3—32no—
107D4×Dic3—92no—
108D4×Dic3—92no—
109D4×Dic3—94no—
110C4×#67—94no—
111C2×(C2×D3)⋊C4—94no—
112#926—91no—
113#926—91no—
114#893—92no—
115#892—92no—
116#67⋊C4—94no—
117D3×C22⋊C4—94no—
118C4×D12—92no—
119#872—92no—
120C23×C32—22no—
121C23×C32C22×D314yes—
122C22×C3×D3—32no—
123C22×C3×D3—34no—
124C22×C3×D3—316no—
125C22×C3×D3—61no—
126C22×C3×D3—68no—
127C22×C3×D3—62no—
128C22×C3×D3—124no—
129C22×C3×D3—124no—
130C22×C3×D3—128no—
131C32×D4—24no—
132C32×D4C4×D318yescomplemented
133C2×C32×C4—21no—
134C2×C32×C4C22×D312yes—
135C2×C3⋊Dic3—44no—
136C2×C3⋊Dic3C22×D314yes—
137C2×C3⋊Dic3C22×D312yes—
138C2×C3⋊Dic3C22×D312yes—
139C3×#67—316no—
140C3×#67—68no—
141C2×C3×Dic3—32no—
142C2×C3×Dic3—34no—
143C2×C3×Dic3—32no—
144C2×C3×Dic3—32no—
145C2×C3×Dic3—61no—
146C2×C3×Dic3—124no—
147C3×D12—38no—
148C3×D12—64no—
149C3×C4×D3—38no—
150C3×C4×D3—64no—
151D3×Dic3—124no—
152D3×Dic3—124no—
153#518—271noSylow
154C2×H(3,𝔽3)—24no—
155C2×H(3,𝔽3)C23×C411yescharacteristic
156C2×H(3,𝔽3)H(5,𝔽2)12yes—
157#272—22no—
158#272—24no—
159C24×C3—92no—
160C23×D3—34no—
161C23×D3—62no—
162C2×C3×D4—32no—
163C2×C3×D4—61no—
164C2×C3×D4—92no—
165C2×C3×D4—98no—
166C2×C3×D4—91no—
167C2×C3×D4D3211yescharacteristic
168C22×C3×C4—91no—
169C22×C3×C4—94no—
170C2×#67—34no—
171C2×#67—62no—
172C22×Dic3—34no—
173C22×Dic3—94no—
174C22×Dic3—94no—
175D3×D4—316no—
176D3×D4—68no—
177C2×D12—32no—
178C2×D12—61no—
179C2×C4×D3—32no—
180C2×C4×D3—61no—
181C2×C4×D3—94no—
182C2×C4×D3—94no—
183C3×C4⋊C4—92no—
184C3×C22⋊C4—94no—
185C3×C42—92no—
186#217—34no—
187#217—94no—
188#217—91no—
189#217—91no—
190#217—92no—
191#217—94no—
192(C2×D3)⋊C4—92no—
193(C2×D3)⋊C4—92no—
194(C2×D3)⋊C4—94no—
195(C2×D3)⋊C4—94no—
196(C2×D3)⋊C4—94no—
197C4⋊Dic3—32no—
198C4⋊Dic3—92no—
199C4⋊Dic3—92no—
200Dic3⋊C4—92no—
201Dic3⋊C4—92no—
202C4×Dic3—32no—
203C4×Dic3—92no—
204C4×Dic3—92no—
205C4×Dic3—94no—
206C22×C32—24no—
207C22×C32—24no—
208C22×C32—21no—
209C22×C32—24no—
210C22×C32—44no—
211C22×C32C2×C4×D318yes—
212C22×C32C23×D312yes—
213C2×C3×D3—34no—
214C2×C3×D3—38no—
215C2×C3×D3—62no—
216C2×C3×D3—64no—
217C2×C3×D3—64no—
218C2×C3×D3—64no—
219C2×C3×D3—62no—
220C2×C3×D3—64no—
221C2×C3×D3—62no—
222C2×C3×D3—124no—
223C2×C3×D3—1216no—
224C2×C3×D3—1216no—
225C32×C4—22no—
226C32×C4C2×C4×D314yes—
227C3⋊Dic3—24no—
228C3⋊Dic3—22no—
229C3⋊Dic3—22no—
230C3×Dic3—34no—
231C3×Dic3—62no—
232C3×Dic3—62no—
233C3×Dic3—64no—
234C3×Dic3—62no—
235C22×D4—91no—
236C4×D4—92no—
237C4×D4—272no—
238C4×D4—274no—
239#117—271no—
240#117—271no—
241C2×C22⋊C4—272no—
242C2×C22⋊C4—272no—
243H(3,𝔽3)#51811yescharacteristic, complemented, Sylow
244C23×C3—34no—
245C23×C3—62no—
246C23×C3—91no—
247C23×C3—92no—
248C23×C3—98no—
249C23×C3—364no—
250C23×C3C2×D3212yes—
251C22×D3—32no—
252C22×D3—34no—
253C22×D3—316no—
254C22×D3—61no—
255C22×D3—62no—
256C22×D3—68no—
257C22×D3—124no—
258C22×D3—124no—
259C22×D3—128no—
260C3×D4—38no—
261C3×D4—64no—
262C3×D4—98no—
263C3×D4—94no—
264C3×D4#62414yes—
265C2×C3×C4—32no—
266C2×C3×C4—61no—
267C2×C3×C4—91no—
268C2×C3×C4—94no—
269C2×C3×C4—94no—
270C2×C3×C4—92no—
271C2×C3×C4—92no—
272C2×C3×C4—364no—
273C2×C3×C4C2×D3211yescharacteristic
274#67—316no—
275#67—68no—
276C2×Dic3—32no—
277C2×Dic3—32no—
278C2×Dic3—32no—
279C2×Dic3—34no—
280C2×Dic3—61no—
281C2×Dic3—92no—
282C2×Dic3—92no—
283C2×Dic3—94no—
284C2×Dic3—124no—
285C2×Dic3—364no—
286C2×Dic3—364no—
287D12—38no—
288D12—64no—
289C4×D3—38no—
290C4×D3—64no—
291C4×D3—364no—
292C4×D3—364no—
293C2×C32—28no—
294C2×C32—22no—
295C2×C32—21no—
296C2×C32—44no—
297C2×C32C22×C4×D312yes—
298C2×C32#100412yes—
299C2×C32#100912yes—
300C3×D3—64no—
301C3×D3—68no—
302C3×D3—64no—
303C3×D3—124no—
304C24—92no—
305C2×D4—92no—
306C2×D4—98no—
307C2×D4—91no—
308C2×D4((C3⋊D3)⋊C3)⋊C211yescharacteristic
309C22×C4—91no—
310C22×C4—94no—
311C22×C4—274no—
312C4⋊C4—92no—
313C4⋊C4—272no—
314C22⋊C4—94no—
315C22⋊C4—271no—
316C22⋊C4—271no—
317C22⋊C4—272no—
318C22⋊C4—274no—
319C42—92no—
320C42—272no—
321C22×C3—24no—
322C22×C3—32no—
323C22×C3—38no—
324C22×C3—64no—
325C22×C3—64no—
326C22×C3—61no—
327C22×C3—64no—
328C22×C3—92no—
329C22×C3—94no—
330C22×C3—124no—
331C22×C3—184no—
332C22×C3—182no—
333C22×C3—368no—
334C22×C3C2×#62414yes—
335C22×C3C22×D3211yescharacteristic
336C2×D3—34no—
337C2×D3—38no—
338C2×D3—62no—
339C2×D3—64no—
340C2×D3—64no—
341C2×D3—64no—
342C2×D3—62no—
343C2×D3—62no—
344C2×D3—64no—
345C2×D3—124no—
346C2×D3—1216no—
347C2×D3—1216no—
348C3×C4—34no—
349C3×C4—62no—
350C3×C4—92no—
351C3×C4—182no—
352C3×C4—184no—
353C3×C4—182no—
354C3×C4C2×#62412yes—
355Dic3—34no—
356Dic3—62no—
357Dic3—62no—
358Dic3—64no—
359Dic3—62no—
360Dic3—182no—
361Dic3—184no—
362Dic3—182no—
363C32—21no—
364C32#554512yescomplemented
365C23—91no—
366C23—92no—
367C23—98no—
368C23—364no—
369C23C2×((C3⋊D3)⋊C3)⋊C212yes—
370D4—98no—
371D4—94no—
372D4#619914yescomplemented
373C2×C4—91no—
374C2×C4—94no—
375C2×C4—92no—
376C2×C4—92no—
377C2×C4—94no—
378C2×C4—364no—
379C2×C4—1084no—
380C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
381C2×C3—24no—
382C2×C3—32no—
383C2×C3—32no—
384C2×C3—32no—
385C2×C3—68no—
386C2×C3—62no—
387C2×C3—61no—
388C2×C3—124no—
389C2×C3—182no—
390C2×C3—184no—
391C2×C3#6431412yes—
392C2×C3C22×#62411yescharacteristic
393D3—64no—
394D3—68no—
395D3—64no—
396D3—124no—
397C22—24no—
398C22—92no—
399C22—94no—
400C22—184no—
401C22—182no—
402C22—368no—
403C22C2×#619914yes—
404C22C22×((C3⋊D3)⋊C3)⋊C211yescharacteristic
405C4—92no—
406C4—182no—
407C4—184no—
408C4—182no—
409C4C2×#619912yes—
410C3—32no—
411C3—61no—
412C3#1059522311yescharacteristic
413C2—24no—
414C2—182no—
415C2—184no—
416C2#1172250712yes—
417C2C22×#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×D4×#272C2
cosocleC22×H(3,𝔽3)C24
hypercenterC2×D4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#518
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. #11721320C2semidirect
4. C2×#88451C2semidirect
5. #11721293C2semidirect
6. #11721149C2semidirect
7. #11721097C2semidirect
8. #11721089C2semidirect
13. D4×#272C4semidirect
16. C22×(C3⋊Dic3)⋊C3C22semidirect
17. C2×#6199C22semidirect
18. #3757⋊C3C22semidirect
19. C4⋊((C3⋊Dic3)⋊C3)C22semidirect
20. C4×(C3⋊Dic3)⋊C3C22semidirect
24. #88451C22semidirect
25. #88449C22semidirect
26. #88447C22semidirect
29. D4×C3⋊Dic3D3semidirect
41. C22×#272C2×C4semidirect
44. D4×H(3,𝔽3)C2×C4semidirect
46. #6235C2×C4semidirect
48. C4⋊#272C2×C4semidirect
49. C4×#272C2×C4semidirect
61. C22×C3⋊Dic3C2×D3semidirect
73. #3757C2×D3semidirect
74. C4⋊(C3⋊Dic3)C2×D3semidirect
75. C4×C3⋊Dic3C2×D3semidirect
132. C32×D4C4×D3semidirect
243. H(3,𝔽3)#518semidirect
364. C32#5545semidirect
372. D4#6199semidirect

Maximal subgroups

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

subgroupindex
C2×D4×#2722
D4×(C3⋊Dic3)⋊C32
#117213202
C2×#884512
#117212932
#117211492
#117210972
#117210892
#105952153