Finite Group Databasebeta

← #13147898

Subgroups — #13147898

Order 1728.

1259 conjugacy classes of subgroups; only the 203 normal subgroups are shown.

normal subgroups only

This group has 1259 conjugacy classes of subgroups — too many to draw. Only its 203 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×#88655C211yesmaximal, characteristic
2#11721718C211yesmaximal, characteristic, complemented
3C4×C4⋊((C3⋊D3)⋊C3)C212yesmaximal, complemented
4#11721160C214yesmaximal, complemented
5#11721136C212yesmaximal, complemented
6#11721122C214yesmaximal, complemented
7#11721089C211yesmaximal, characteristic, complemented
8#10593994—32nomaximal
9C22×((C3⋊D3)⋊C3)⋊C2C2212yes—
10C2×C4×#272C2211yescharacteristic
11C2×C4⋊((C3⋊D3)⋊C3)C2212yes—
12C2×C4×(C3⋊D3)⋊C3C2214yescomplemented
13C2×#6198C2212yes—
14C2×((C3⋊D3)⋊C3)⋊C4C2214yescomplemented
15#88655C418yescomplemented
16C42×H(3,𝔽3)C2211yescharacteristic, complemented
17#88542C2212yes—
18#88540C2212yescomplemented
19C4×#1084C2211yescharacteristic, complemented
20#3752⋊C3C2214yescomplemented
21C4⋊((C3⋊Dic3)⋊C3)C2212yescomplemented
22#88451C2212yes—
23#88450C2214yescomplemented
24#88447C2212yescomplemented
25C2×D3×D12—32no—
26C4×C4⋊(C3⋊D3)D312yescomplemented
27C3×C4×D12—32no—
28C3×#872—32no—
29C3×#872—61no—
30#63921—34no—
31D3×(C2×D3)⋊C4—34no—
32#63904—34no—
33#63894—32no—
34Dic3×D12—32no—
35#63881—34no—
36#63841—32no—
37C22×#272C2311yescharacteristic
38C22×(C3⋊D3)⋊C3C2314yes—
39C2×((C3⋊D3)⋊C3)⋊C2—44no—
40C2×((C3⋊D3)⋊C3)⋊C2C2×C418yescomplemented
41C2×C4×H(3,𝔽3)C2312yes—
42C2×C4×H(3,𝔽3)C2311yescharacteristic
43C2×#1084C2312yes—
44C2×#1084C2311yescharacteristic
45C4×#272C2×C414yescomplemented
46C2×(C3⋊Dic3)⋊C3C2314yes—
47C4⋊((C3⋊D3)⋊C3)C2×C418yescomplemented
48C4×(C3⋊D3)⋊C3—44no—
49#6198C2×C418yescomplemented
50((C3⋊D3)⋊C3)⋊C4—44no—
51#5545—92no—
52#5521—91no—
53C22×D32—34no—
54C2×C4⋊(C3⋊D3)C2×D312yes—
55C2×C4×C3⋊D3C2×D314yescomplemented
56C2×C3×D12—32no—
57C2×C3×C4×D3—32no—
58C2×C3×C4×D3—34no—
59C2×C3×C4×D3—61no—
60C2×#626—34no—
61C2×#624—34no—
62C2×D3×Dic3—34no—
63D3×D12—316no—
64C32×C42—21no—
65C32×C42C2×D312yescomplemented
66#3752C2×D314yescomplemented
67C4⋊(C3⋊Dic3)C2×D312yescomplemented
68C3×(C2×D3)⋊C4—34no—
69C3×(C2×D3)⋊C4—32no—
70C3×(C2×D3)⋊C4—32no—
71C3×(C2×D3)⋊C4—62no—
72C3×C4⋊Dic3—32no—
73C3×Dic3⋊C4—32no—
74C3×Dic3⋊C4—32no—
75C3×Dic3⋊C4—62no—
76C3×C4×Dic3—32no—
77C3×C4×Dic3—61no—
78#3722—34no—
79(C2×D3)⋊Dic3—34no—
80(C2×D3)⋊Dic3—34no—
81Dic32—34no—
82C22×H(3,𝔽3)C2411yescharacteristic
83C2×#272—22no—
84C2×#272C22×C412yes—
85C2×(C3⋊D3)⋊C3—24no—
86C2×(C3⋊D3)⋊C3—24no—
87C2×(C3⋊D3)⋊C3C22×C418yes—
88((C3⋊D3)⋊C3)⋊C2—48no—
89C4×H(3,𝔽3)—22no—
90C4×H(3,𝔽3)C22×C412yes—
91#1084—22no—
92#1084C22×C412yes—
93(C3⋊Dic3)⋊C3—24no—
94C2×D3×D4—92no—
95C22×D12—91no—
96C3×C4×D4—92no—
97C3×#117—91no—
98D4×Dic3—92no—
99C4×#67—94no—
100C2×(C2×D3)⋊C4—94no—
101#926—91no—
102#893—92no—
103#893—94no—
104#892—92no—
105#67⋊C4—94no—
106D3×C22⋊C4—94no—
107D3×C22⋊C4—94no—
108C4×D12—32no—
109C4×D12—92no—
110C4×D12—92no—
111#872—32no—
112#872—61no—
113C22×C3⋊D3C22×D314yes—
114C22×C3×D3—32no—
115C22×C3×D3—34no—
116C22×C3×D3—61no—
117C2×D32—316no—
118C2×D32—124no—
119C2×D32—124no—
120C2×C32×C4—21no—
121C2×C32×C4—22no—
122C2×C32×C4C22×D312yes—
123C2×C32×C4C22×D312yes—
124C2×C32×C4C22×D312yes—
125C2×C3⋊Dic3C22×D314yes—
126C4⋊(C3⋊D3)C4×D318yescomplemented
127C4×C3⋊D3—44no—
128C2×C3×Dic3—32no—
129C2×C3×Dic3—34no—
130C2×C3×Dic3—32no—
131C2×C3×Dic3—32no—
132C2×C3×Dic3—61no—
133C2×C3×Dic3—62no—
134C3×D12—38no—
135C3×C4×D3—38no—
136C3×C4×D3—64no—
137C3×C4×D3—124no—
138#626—316no—
139#624—124no—
140D3×Dic3—124no—
141#518—271noSylow
142C2×H(3,𝔽3)C23×C411yescharacteristic
143C2×H(3,𝔽3)H(5,𝔽2)12yes—
144#272—22no—
145(C3⋊D3)⋊C3—28no—
146C23×D3—92no—
147C23×D3—94no—
148C2×C3×D4—92no—
149C22×C3×C4—94no—
150C22×C3×C4—91no—
151C2×#67—94no—
152C22×Dic3—94no—
153D3×D4—916no—
154C2×D12—32no—
155C2×D12—92no—
156C2×D12—98no—
157C2×D12—92no—
158C2×C4×D3—32no—
159C2×C4×D3—34no—
160C2×C4×D3—61no—
161C2×C4×D3—94no—
162C2×C4×D3—94no—
163C2×C4×D3—94no—
164C3×C4⋊C4—92no—
165C3×C4⋊C4—92no—
166C3×C22⋊C4—94no—
167C3×C22⋊C4—92no—
168C3×C42—32no—
169C3×C42—61no—
170C3×C42—91no—
171C3×C42D3211yescharacteristic
172#217—92no—
173#217—94no—
174(C2×D3)⋊C4—32no—
175(C2×D3)⋊C4—32no—
176(C2×D3)⋊C4—34no—
177(C2×D3)⋊C4—62no—
178(C2×D3)⋊C4—94no—
179(C2×D3)⋊C4—94no—
180(C2×D3)⋊C4—94no—
181(C2×D3)⋊C4—94no—
182C4⋊Dic3—32no—
183C4⋊Dic3—92no—
184Dic3⋊C4—32no—
185Dic3⋊C4—32no—
186Dic3⋊C4—62no—
187C4×Dic3—32no—
188C4×Dic3—61no—
189C4×Dic3—92no—
190C4×Dic3—94no—
191C22×C32—21no—
192C22×C32C23×D312yes—
193C2×C3⋊D3—24no—
194C2×C3⋊D3—24no—
195C2×C3⋊D3C2×C4×D318yes—
196C2×C3×D3—38no—
197C2×C3×D3—34no—
198C2×C3×D3—62no—
199C2×C3×D3—64no—
200C2×C3×D3—62no—
201C2×C3×D3—64no—
202C2×C3×D3—62no—
203C2×C3×D3—122no—
204D32—1216no—
205C32×C4—22no—
206C32×C4—22no—
207C32×C4—22no—
208C32×C4—42no—
209C32×C4C2×C4×D314yes—
210C3⋊Dic3—24no—
211C3×Dic3—34no—
212C3×Dic3—62no—
213C3×Dic3—62no—
214C3×Dic3—64no—
215C3×Dic3—62no—
216C3×Dic3—122no—
217C22×D4—271no—
218C4×D4—92no—
219C4×D4—272no—
220C4×D4—274no—
221#117—91no—
222#117—271no—
223C2×C22⋊C4—274no—
224H(3,𝔽3)#51811yescharacteristic, complemented, Sylow
225C23×C3—94no—
226C23×C3—91no—
227C22×D3—32no—
228C22×D3—34no—
229C22×D3—61no—
230C22×D3—94no—
231C22×D3—98no—
232C22×D3—916no—
233C22×D3—364no—
234C22×D3—364no—
235C22×D3—364no—
236C3×D4—98no—
237C2×C3×C4—32no—
238C2×C3×C4—32no—
239C2×C3×C4—32no—
240C2×C3×C4—61no—
241C2×C3×C4—62no—
242C2×C3×C4—91no—
243C2×C3×C4—94no—
244C2×C3×C4—94no—
245C2×C3×C4—92no—
246C2×C3×C4—364no—
247C2×C3×C4C2×D3212yes—
248C2×C3×C4C2×D3211yescharacteristic
249#67—916no—
250C2×Dic3—32no—
251C2×Dic3—32no—
252C2×Dic3—32no—
253C2×Dic3—34no—
254C2×Dic3—61no—
255C2×Dic3—62no—
256C2×Dic3—94no—
257C2×Dic3—364no—
258D12—38no—
259D12—98no—
260D12—98no—
261C4×D3—38no—
262C4×D3—64no—
263C4×D3—124no—
264C4×D3—364no—
265C4×D3—364no—
266C4×D3—364no—
267C2×C32—21no—
268C2×C32—22no—
269C2×C32C22×C4×D312yes—
270C2×C32#100412yes—
271C2×C32#100912yes—
272C3⋊D3—28no—
273C3×D3—68no—
274C3×D3—64no—
275C3×D3—122no—
276C24—272no—
277C2×D4—92no—
278C2×D4—278no—
279C2×D4—272no—
280C22×C4—91no—
281C22×C4—94no—
282C22×C4—274no—
283C4⋊C4—92no—
284C4⋊C4—92no—
285C22⋊C4—92no—
286C22⋊C4—94no—
287C22⋊C4—272no—
288C22⋊C4—274no—
289C42—91no—
290C42—272no—
291C42((C3⋊D3)⋊C3)⋊C211yescharacteristic, complemented
292C22×C3—32no—
293C22×C3—61no—
294C22×C3—98no—
295C22×C3—92no—
296C22×C3—184no—
297C22×C3—184no—
298C22×C3—182no—
299C22×C3C22×D3211yescharacteristic
300C2×D3—34no—
301C2×D3—38no—
302C2×D3—62no—
303C2×D3—62no—
304C2×D3—62no—
305C2×D3—64no—
306C2×D3—64no—
307C2×D3—98no—
308C2×D3—122no—
309C2×D3—184no—
310C2×D3—184no—
311C2×D3—368no—
312C2×D3—3616no—
313C3×C4—22no—
314C3×C4—34no—
315C3×C4—62no—
316C3×C4—62no—
317C3×C4—62no—
318C3×C4—92no—
319C3×C4—122no—
320C3×C4—184no—
321C3×C4—182no—
322C3×C4C4×D3212yes—
323Dic3—34no—
324Dic3—62no—
325Dic3—62no—
326Dic3—62no—
327Dic3—64no—
328Dic3—122no—
329Dic3—184no—
330C32—21no—
331C32#554512yescomplemented
332C23—91no—
333C23—94no—
334C23—278no—
335C23—1084no—
336D4—98no—
337D4—278no—
338C2×C4—91no—
339C2×C4—94no—
340C2×C4—92no—
341C2×C4—94no—
342C2×C4—364no—
343C2×C4—1084no—
344C2×C4C2×((C3⋊D3)⋊C3)⋊C212yes—
345C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
346C2×C3—32no—
347C2×C3—32no—
348C2×C3—32no—
349C2×C3—61no—
350C2×C3—62no—
351C2×C3—188no—
352C2×C3—182no—
353C2×C3C2×C4×D3211yescharacteristic
354C2×C3#6430712yes—
355D3—64no—
356D3—68no—
357D3—122no—
358D3—188no—
359C22—92no—
360C22—98no—
361C22—184no—
362C22—184no—
363C22—182no—
364C22—1088no—
365C22C22×((C3⋊D3)⋊C3)⋊C211yescharacteristic
366C4—22no—
367C4—92no—
368C4—184no—
369C4—182no—
370C4C4×((C3⋊D3)⋊C3)⋊C212yes—
371C3—32no—
372C3—61no—
373C3#1059399611yescharacteristic
374C2—182no—
375C2—188no—
376C2C2×C4×((C3⋊D3)⋊C3)⋊C211yescharacteristic
377C2#1172249212yes—
subgroupG/H
centerC22C22×((C3⋊D3)⋊C3)⋊C2
commutatorC2×H(3,𝔽3)C23×C4
Frattini, socleC22×C3C22×D32
FittingC42×H(3,𝔽3)C22
prime radicalC2×#88655C2
cosocleC22×H(3,𝔽3)C24
hypercenterC42((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#518
LCSG ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C42

Complements

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

NH ≅ G/Ntype
2. #11721718C2semidirect
3. C4×C4⋊((C3⋊D3)⋊C3)C2semidirect
4. #11721160C2semidirect
5. #11721136C2semidirect
6. #11721122C2semidirect
7. #11721089C2semidirect
12. C2×C4×(C3⋊D3)⋊C3C22semidirect
14. C2×((C3⋊D3)⋊C3)⋊C4C22semidirect
15. #88655C4semidirect
16. C42×H(3,𝔽3)C22semidirect
18. #88540C22semidirect
19. C4×#1084C22semidirect
20. #3752⋊C3C22semidirect
21. C4⋊((C3⋊Dic3)⋊C3)C22semidirect
23. #88450C22semidirect
24. #88447C22semidirect
26. C4×C4⋊(C3⋊D3)D3semidirect
40. C2×((C3⋊D3)⋊C3)⋊C2C2×C4semidirect
45. C4×#272C2×C4semidirect
47. C4⋊((C3⋊D3)⋊C3)C2×C4semidirect
49. #6198C2×C4semidirect
55. C2×C4×C3⋊D3C2×D3semidirect
65. C32×C42C2×D3semidirect
66. #3752C2×D3semidirect
67. C4⋊(C3⋊Dic3)C2×D3semidirect
126. C4⋊(C3⋊D3)C4×D3semidirect
224. H(3,𝔽3)#518semidirect
291. C42((C3⋊D3)⋊C3)⋊C2semidirect
331. C32#5545semidirect

Maximal subgroups

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

subgroupindex
C2×#886552
#117217182
C4×C4⋊((C3⋊D3)⋊C3)2
#117211602
#117211362
#117211222
#117210892
#105939943