Finite Group Databasebeta

← #13152079

Subgroups — #13152079

Order 1728.

1343 conjugacy classes of subgroups; only the 179 normal subgroups are shown.

normal subgroups only

This group has 1343 conjugacy classes of subgroups — too many to draw. Only its 179 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
1C22×#638⋊C3C212yesmaximal, complemented
2C2×#88552C211yesmaximal, characteristic, complemented
3#11721318C214yesmaximal, complemented
4#11721316C214yesmaximal, complemented
5#11721306C214yesmaximal, complemented
6#10595519—32nomaximal
7C24×H(3,𝔽3)C2211yescharacteristic
8C23×(C3⋊D3)⋊C3C2212yescomplemented
9C22×#1084C2212yescomplemented
10C2×#638⋊C3—28no—
11C2×#638⋊C3C2214yes—
12C2×#638⋊C3C2214yescomplemented
13C22×(C3⋊Dic3)⋊C3C2212yescomplemented
14C2×#6198C2214yescomplemented
15#88552C2214yescomplemented
16#88450C2218yescomplemented
17#88449C2214yescomplemented
18C22×#638D312yescomplemented
19C22×C3×#67—32no—
20C2×C3×#217—32no—
21C2×C3×#217—61no—
22#63981—34no—
23#63978—38no—
24#63977—34no—
25#63966—34no—
26#63962—34no—
27C23×H(3,𝔽3)—42no—
28C23×H(3,𝔽3)D414yescomplemented
29C23×H(3,𝔽3)C2312yes—
30C23×H(3,𝔽3)C2311yescharacteristic
31C22×(C3⋊D3)⋊C3—24no—
32C22×(C3⋊D3)⋊C3—44no—
33C22×(C3⋊D3)⋊C3C2314yes—
34C2×#1084—42no—
35C2×#1084C2314yes—
36#638⋊C3—28no—
37#638⋊C3—28no—
38C2×(C3⋊Dic3)⋊C3—24no—
39C2×(C3⋊Dic3)⋊C3C2314yes—
40#6198—44no—
41#5804—92no—
42#5493—91no—
43C24×C32—21no—
44C24×C32C2×D312yes—
45C23×C3⋊D3C2×D312yescomplemented
46C23×C3×D3—32no—
47C2×#638—28no—
48C2×#638C2×D314yes—
49C2×#638C2×D314yescomplemented
50C22×C3⋊Dic3C2×D312yescomplemented
51C2×C3×#67—34no—
52C2×C3×#67—34no—
53C2×C3×#67—68no—
54C22×C3×Dic3—32no—
55C22×C3×Dic3—34no—
56C22×C3×Dic3—62no—
57C2×#626—38no—
58C3×#217—34no—
59C3×#217—34no—
60C3×#217—64no—
61Dic3⋊Dic3—38no—
62#3722—38no—
63(C2×D3)⋊Dic3—38no—
64C22×H(3,𝔽3)—22no—
65C22×H(3,𝔽3)—24no—
66C22×H(3,𝔽3)—44no—
67C22×H(3,𝔽3)C2×D414yes—
68C22×H(3,𝔽3)C2×D412yes—
69C22×H(3,𝔽3)C2411yescharacteristic
70C2×(C3⋊D3)⋊C3—24no—
71C2×(C3⋊D3)⋊C3—28no—
72C2×(C3⋊D3)⋊C3—48no—
73#1084—24no—
74(C3⋊Dic3)⋊C3—24no—
75C22×C3×D4—92no—
76C22×#67—32no—
77C22×#67—92no—
78C2×C3×C22⋊C4—91no—
79#953—94no—
80C2×#217—32no—
81C2×#217—61no—
82#939—98no—
83#937—94no—
84#935—94no—
85#930—94no—
86#929—94no—
87#886—94no—
88#882—94no—
89C23×C32—24no—
90C23×C32—21no—
91C23×C32—22no—
92C23×C32—44no—
93C23×C32—44no—
94C23×C32#6718yescomplemented
95C23×C32C22×D312yes—
96C23×C32C22×D312yes—
97C23×C32C22×D312yes—
98C22×C3⋊D3—24no—
99C22×C3⋊D3—44no—
100C22×C3⋊D3C22×D314yes—
101C22×C3×D3—34no—
102C22×C3×D3—64no—
103C22×C3×D3—124no—
104#638—28no—
105#638—28no—
106C2×C3⋊Dic3—24no—
107C2×C3⋊Dic3C22×D314yes—
108C3×#67—68no—
109C3×#67—68no—
110C2×C3×Dic3—34no—
111C2×C3×Dic3—38no—
112C2×C3×Dic3—64no—
113C2×C3×Dic3—64no—
114C2×C3×Dic3—124no—
115C2×C3×Dic3—124no—
116#626—128no—
117#534—271noSylow
118C2×H(3,𝔽3)—22no—
119C2×H(3,𝔽3)—24no—
120C2×H(3,𝔽3)C22×D411yescharacteristic
121C2×H(3,𝔽3)H(5,𝔽2)12yes—
122(C3⋊D3)⋊C3—44no—
123C24×C3—32no—
124C24×C3—61no—
125C24×C3—92no—
126C24×C3D3211yescharacteristic
127C23×D3—32no—
128C23×D3—92no—
129C2×C3×D4—94no—
130C2×C3×D4—94no—
131C2×C3×D4—188no—
132C22×C3×C4—92no—
133C22×C3×C4—92no—
134C2×#67—34no—
135C2×#67—34no—
136C2×#67—68no—
137C2×#67—94no—
138C2×#67—94no—
139C2×#67—98no—
140C2×#67—188no—
141C22×Dic3—32no—
142C22×Dic3—34no—
143C22×Dic3—62no—
144C22×Dic3—92no—
145C2×D12—94no—
146C3×C22⋊C4—94no—
147#217—34no—
148#217—34no—
149#217—64no—
150#217—98no—
151(C2×D3)⋊C4—98no—
152(C2×D3)⋊C4—98no—
153C4⋊Dic3—94no—
154Dic3⋊C4—98no—
155C22×C32—24no—
156C22×C32—24no—
157C22×C32—24no—
158C22×C32—24no—
159C22×C32—21no—
160C22×C32—22no—
161C22×C32—24no—
162C22×C32—44no—
163C22×C32—48no—
164C22×C32—48no—
165C22×C32C2×#6714yes—
166C22×C32C2×#6714yes—
167C22×C32C2×#6714yes—
168C22×C32C23×D312yes—
169C2×C3⋊D3—28no—
170C2×C3⋊D3—24no—
171C2×C3⋊D3—48no—
172C2×C3×D3—68no—
173C2×C3×D3—64no—
174C2×C3×D3—128no—
175C3⋊Dic3—24no—
176C3×Dic3—68no—
177C3×Dic3—64no—
178C3×Dic3—124no—
179C22×D4—92no—
180#123—274no—
181#121—274no—
182#120—274no—
183C2×C22⋊C4—91no—
184H(3,𝔽3)#53411yescharacteristic, complemented, Sylow
185C23×C3—38no—
186C23×C3—32no—
187C23×C3—32no—
188C23×C3—32no—
189C23×C3—42no—
190C23×C3—64no—
191C23×C3—61no—
192C23×C3—62no—
193C23×C3—94no—
194C23×C3—124no—
195C23×C3—124no—
196C23×C3—184no—
197C23×C3—364no—
198C23×C3#62514yes—
199C23×C3C2×D3212yes—
200C23×C3C2×D3211yescharacteristic
201C22×D3—34no—
202C22×D3—64no—
203C22×D3—94no—
204C22×D3—124no—
205C22×D3—184no—
206C22×D3—364no—
207C3×D4—188no—
208C3×D4—188no—
209C2×C3×C4—94no—
210C2×C3×C4—94no—
211C2×C3×C4—184no—
212C2×C3×C4—362no—
213#67—68no—
214#67—68no—
215#67—188no—
216#67—188no—
217#67—368no—
218C2×Dic3—38no—
219C2×Dic3—34no—
220C2×Dic3—64no—
221C2×Dic3—64no—
222C2×Dic3—94no—
223C2×Dic3—124no—
224C2×Dic3—124no—
225C2×Dic3—184no—
226D12—364no—
227C2×C32—24no—
228C2×C32—24no—
229C2×C32—24no—
230C2×C32—24no—
231C2×C32—21no—
232C2×C32—22no—
233C2×C32—44no—
234C2×C32#100414yes—
235C2×C32C22×#6712yes—
236C3⋊D3—44no—
237C3×D3—124no—
238C24—92no—
239C24((C3⋊D3)⋊C3)⋊C211yescharacteristic
240C2×D4—94no—
241C2×D4—94no—
242C2×D4—188no—
243C2×D4—274no—
244C22×C4—92no—
245C22×C4—92no—
246C4⋊C4—274no—
247C22⋊C4—94no—
248C22⋊C4—278no—
249C22×C3—22no—
250C22×C3—24no—
251C22×C3—32no—
252C22×C3—34no—
253C22×C3—34no—
254C22×C3—34no—
255C22×C3—44no—
256C22×C3—64no—
257C22×C3—64no—
258C22×C3—64no—
259C22×C3—64no—
260C22×C3—61no—
261C22×C3—62no—
262C22×C3—64no—
263C22×C3—124no—
264C22×C3—128no—
265C22×C3—128no—
266C22×C3—188no—
267C22×C3—184no—
268C22×C3—368no—
269C22×C3C2×#62514yes—
270C22×C3C2×#62512yes—
271C22×C3C22×D3211yescharacteristic
272C2×D3—64no—
273C2×D3—68no—
274C2×D3—128no—
275C2×D3—184no—
276C2×D3—188no—
277C2×D3—368no—
278C3×C4—184no—
279C3×C4—184no—
280Dic3—68no—
281Dic3—64no—
282Dic3—124no—
283Dic3—184no—
284C32—21no—
285C32#580412yescomplemented
286C23—42no—
287C23—94no—
288C23—184no—
289C23—364no—
290C23#619814yescomplemented
291C23C2×((C3⋊D3)⋊C3)⋊C212yes—
292C23C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
293D4—188no—
294D4—188no—
295D4—1084no—
296C2×C4—94no—
297C2×C4—94no—
298C2×C4—184no—
299C2×C4—362no—
300C2×C3—24no—
301C2×C3—22no—
302C2×C3—32no—
303C2×C3—34no—
304C2×C3—64no—
305C2×C3—64no—
306C2×C3—64no—
307C2×C3—64no—
308C2×C3—61no—
309C2×C3—62no—
310C2×C3—124no—
311C2×C3—364no—
312C2×C3C22×#62511yescharacteristic
313C2×C3#6433012yes—
314D3—124no—
315D3—364no—
316C22—22no—
317C22—24no—
318C22—44no—
319C22—184no—
320C22—188no—
321C22—368no—
322C22C2×#619814yes—
323C22C2×#619812yes—
324C22C22×((C3⋊D3)⋊C3)⋊C211yescharacteristic
325C4—184no—
326C4—184no—
327C3—32no—
328C3—61no—
329C3#1059551811yescharacteristic
330C2—24no—
331C2—22no—
332C2—364no—
333C2C22×#619811yescharacteristic
334C2#1172254512yes—
subgroupG/H
centerC22C22×((C3⋊D3)⋊C3)⋊C2
commutator, cosocleC22×H(3,𝔽3)C24
Frattini, socleC22×C3C22×D32
FittingC24×H(3,𝔽3)C22
hypercenterC24((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#534
LCSG ▷ C22×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C24

Complements

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

NH ≅ G/Ntype
1. C22×#638⋊C3C2semidirect
2. C2×#88552C2semidirect
3. #11721318C2semidirect
4. #11721316C2semidirect
5. #11721306C2semidirect
8. C23×(C3⋊D3)⋊C3C22semidirect
9. C22×#1084C22semidirect
12. C2×#638⋊C3C22semidirect
13. C22×(C3⋊Dic3)⋊C3C22semidirect
14. C2×#6198C22semidirect
15. #88552C22semidirect
16. #88450C22semidirect
17. #88449C22semidirect
18. C22×#638D3semidirect
28. C23×H(3,𝔽3)D4semidirect
45. C23×C3⋊D3C2×D3semidirect
49. C2×#638C2×D3semidirect
50. C22×C3⋊Dic3C2×D3semidirect
94. C23×C32#67semidirect
184. H(3,𝔽3)#534semidirect
285. C32#5804semidirect
290. C23#6198semidirect

Maximal subgroups

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

subgroupindex
C22×#638⋊C32
C2×#885522
#117213182
#117213162
#117213062
#105955193