Finite Group Databasebeta

← #13144831

Subgroups — #13144831

Order 1728.

1771 conjugacy classes of subgroups; only the 199 normal subgroups are shown.

normal subgroups only

This group has 1771 conjugacy classes of subgroups — too many to draw. Only its 199 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×D4×H(3,𝔽3)C211yesmaximal, characteristic, complemented
2C22×#6235C212yesmaximal, complemented
3C2×#88552C212yesmaximal, complemented
4C2×#88542C211yesmaximal, characteristic, complemented
5#11720574C211yesmaximal, characteristic, complemented
6C3×#5223—34nomaximal
7C24×H(3,𝔽3)C2212yes—
8C23×#272C2211yescharacteristic
9C2×D4×H(3,𝔽3)—24no—
10C2×D4×H(3,𝔽3)—22no—
11C22×C4×H(3,𝔽3)C2211yescharacteristic
12C2×#6235—24no—
13C2×#6235—24no—
14C2×#6235—24no—
15C22×#1084C2211yescharacteristic
16C22×#1084C2212yes—
17#88552—24no—
18#88542—22no—
19C22×C32×D4D314yescomplemented
20C22×C3×#67—38no—
21C2×C3×#217—38no—
22C2×C3×(C2×D3)⋊C4—34no—
23C3×C3⋊#95—34no—
24C23×H(3,𝔽3)—22no—
25C23×H(3,𝔽3)—22no—
26C23×H(3,𝔽3)—22no—
27C23×H(3,𝔽3)—22no—
28C23×H(3,𝔽3)—22no—
29C23×H(3,𝔽3)—22no—
30C23×H(3,𝔽3)D414yescomplemented
31C23×H(3,𝔽3)C2311yescharacteristic
32C22×#272—22no—
33C22×#272—22no—
34C22×#272—21no—
35C22×#272—21no—
36C22×#272D412yescomplemented
37D4×H(3,𝔽3)—28no—
38C2×C4×H(3,𝔽3)—22no—
39C2×C4×H(3,𝔽3)D412yescomplemented
40#6235—216no—
41C2×#1084—22no—
42C2×#1084—22no—
43C2×#1084—21no—
44C2×#1084—22no—
45C2×#1084D414yescomplemented
46C3×#392—91no—
47#5223—34no—
48C24×C32C2×D318yes—
49C23×C3×D3—34no—
50C2×C32×D4—216no—
51C2×C32×D4—28no—
52C22×C32×C4C2×D314yes—
53C2×C3×#67—616no—
54C2×C3×#67—616no—
55C2×C3×#67—616no—
56C22×C3×Dic3—38no—
57C22×C3×Dic3—34no—
58C3×#217—616no—
59C3×(C2×D3)⋊C4—68no—
60C22×H(3,𝔽3)—24no—
61C22×H(3,𝔽3)—24no—
62C22×H(3,𝔽3)—24no—
63C22×H(3,𝔽3)—42no—
64C22×H(3,𝔽3)—42no—
65C22×H(3,𝔽3)—42no—
66C22×H(3,𝔽3)—42no—
67C22×H(3,𝔽3)C2×D411yescharacteristic
68C22×H(3,𝔽3)C2×D412yes—
69C22×H(3,𝔽3)C2×D412yes—
70C22×H(3,𝔽3)C2×D411yescharacteristic
71C22×H(3,𝔽3)#4111yescharacteristic
72C2×#272—24no—
73C2×#272—22no—
74C2×#272—41no—
75C2×#272—42no—
76C2×#272—41no—
77C4×H(3,𝔽3)—22no—
78#1084—24no—
79#1084—41no—
80C22×C3×D4—34no—
81C22×C3×D4—92no—
82C22×C3×D4C3⋊D311yescharacteristic
83C22×#67—38no—
84C2×C3×C22⋊C4—92no—
85C2×C3×C22⋊C4—91no—
86C2×#217—38no—
87C2×(C2×D3)⋊C4—34no—
88C3×#95—91no—
89C3⋊#95—34no—
90C23×C32—28no—
91C23×C32—28no—
92C23×C32—28no—
93C23×C32—28no—
94C23×C32—28no—
95C23×C32—28no—
96C23×C32#67116yescomplemented
97C23×C32C22×D314yes—
98C22×C3×D3—38no—
99C22×C3×D3—68no—
100C22×C3×D3—64no—
101C22×C3×D3—64no—
102C22×C3×D3—68no—
103C32×D4—232no—
104C2×C32×C4—28no—
105C2×C32×C4#6718yescomplemented
106C3×#67—664no—
107C2×C3×Dic3—316no—
108C2×C3×Dic3—68no—
109C2×C3×Dic3—68no—
110C2×C3×Dic3—68no—
111C2×C3×Dic3—64no—
112#392—91noSylow
113C2×H(3,𝔽3)—44no—
114C2×H(3,𝔽3)#12012yes—
115C2×H(3,𝔽3)#12011yescharacteristic
116C2×H(3,𝔽3)#12111yescharacteristic
117C2×H(3,𝔽3)#12112yes—
118C2×H(3,𝔽3)#12711yescharacteristic
119#272—42no—
120C24×C3—38no—
121C24×C3—91no—
122C24×C3C2×C3⋊D312yes—
123C23×D3—34no—
124C2×C3×D4—24no—
125C2×C3×D4—22no—
126C2×C3×D4—68no—
127C2×C3×D4—616no—
128C2×C3×D4—184no—
129C2×C3×D4—184no—
130C2×C3×D4—184no—
131C22×C3×C4—34no—
132C22×C3×C4—92no—
133C22×C3×C4—91no—
134C22×C3×C4C2×C3⋊D311yescharacteristic
135C2×#67—616no—
136C2×#67—616no—
137C2×#67—616no—
138C22×Dic3—38no—
139C22×Dic3—34no—
140C3×C22⋊C4—184no—
141C3×C22⋊C4—182no—
142#217—616no—
143(C2×D3)⋊C4—68no—
144C22×C32—216no—
145C22×C32—216no—
146C22×C32—216no—
147C22×C32—48no—
148C22×C32—48no—
149C22×C32—48no—
150C22×C32—48no—
151C22×C32D3×D414yes—
152C22×C32D3×D418yes—
153C22×C32#23714yes—
154C22×C32C2×#6718yes—
155C22×C32C2×#6714yes—
156C2×C3×D3—616no—
157C2×C3×D3—68no—
158C2×C3×D3—128no—
159C2×C3×D3—124no—
160C2×C3×D3—124no—
161C32×C4—28no—
162C3×Dic3—616no—
163C3×Dic3—124no—
164C22×D4—92no—
165C22×D4#27211yescharacteristic, complemented
166C2×C22⋊C4—92no—
167C2×C22⋊C4—91no—
168#95—91no—
169H(3,𝔽3)#39211yescharacteristic, complemented, Sylow
170C23×C3—22no—
171C23×C3—22no—
172C23×C3—22no—
173C23×C3—22no—
174C23×C3—22no—
175C23×C3—22no—
176C23×C3—316no—
177C23×C3—34no—
178C23×C3—68no—
179C23×C3—68no—
180C23×C3—68no—
181C23×C3—68no—
182C23×C3—68no—
183C23×C3—68no—
184C23×C3—92no—
185C23×C3—182no—
186C23×C3—181no—
187C23×C3—181no—
188C23×C3—182no—
189C23×C3#63814yes—
190C23×C3C22×C3⋊D311yescharacteristic
191C22×D3—38no—
192C22×D3—64no—
193C22×D3—68no—
194C22×D3—68no—
195C22×D3—64no—
196C3×D4—28no—
197C3×D4—632no—
198C3×D4—1816no—
199C2×C3×C4—22no—
200C2×C3×C4—38no—
201C2×C3×C4—68no—
202C2×C3×C4—94no—
203C2×C3×C4—182no—
204C2×C3×C4—182no—
205C2×C3×C4—182no—
206C2×C3×C4—181no—
207C2×C3×C4#63812yes—
208#67—664no—
209C2×Dic3—316no—
210C2×Dic3—68no—
211C2×Dic3—68no—
212C2×Dic3—64no—
213C2×Dic3—68no—
214C2×C32—416no—
215C2×C32#93718yes—
216C2×C32#93814yes—
217C2×C32#93918yes—
218C2×C32#94014yes—
219C2×C32#95314yes—
220C3×D3—128no—
221C24—91no—
222C24C2×#27212yes—
223C2×D4—24no—
224C2×D4—22no—
225C2×D4—184no—
226C2×D4—184no—
227C2×D4—184no—
228C22×C4—92no—
229C22×C4—91no—
230C22×C4C2×#27211yescharacteristic
231C22⋊C4—184no—
232C22⋊C4—182no—
233C22×C3—24no—
234C22×C3—24no—
235C22×C3—24no—
236C22×C3—38no—
237C22×C3—34no—
238C22×C3—34no—
239C22×C3—34no—
240C22×C3—38no—
241C22×C3—42no—
242C22×C3—42no—
243C22×C3—42no—
244C22×C3—42no—
245C22×C3—616no—
246C22×C3—616no—
247C22×C3—616no—
248C22×C3—128no—
249C22×C3—128no—
250C22×C3—128no—
251C22×C3—128no—
252C22×C3—184no—
253C22×C3—182no—
254C22×C3—362no—
255C22×C3—361no—
256C22×C3—361no—
257C22×C3D4×C3⋊D311yescharacteristic
258C22×C3D4×C3⋊D312yes—
259C22×C3#383011yescharacteristic
260C22×C3C2×#63812yes—
261C22×C3C2×#63811yescharacteristic
262C2×D3—616no—
263C2×D3—68no—
264C2×D3—128no—
265C2×D3—124no—
266C2×D3—124no—
267C3×C4—22no—
268C3×C4—68no—
269C3×C4—184no—
270C3×C4—361no—
271Dic3—616no—
272Dic3—124no—
273C32#522314yescomplemented
274C23—22no—
275C23—22no—
276C23—22no—
277C23—22no—
278C23—22no—
279C23—22no—
280C23—92no—
281C23—182no—
282C23—181no—
283C23—181no—
284C23—182no—
285C23#623514yescomplemented
286C23C22×#27211yescharacteristic
287D4—28no—
288D4—1816no—
289C2×C4—22no—
290C2×C4—94no—
291C2×C4—182no—
292C2×C4—182no—
293C2×C4—182no—
294C2×C4—181no—
295C2×C4#623512yescomplemented
296C2×C3—38no—
297C2×C3—34no—
298C2×C3—38no—
299C2×C3—34no—
300C2×C3—34no—
301C2×C3—44no—
302C2×C3—1216no—
303C2×C3—362no—
304C2×C3#6414612yes—
305C2×C3#6414711yescharacteristic
306C2×C3#6414812yes—
307C2×C3#6414911yescharacteristic
308C2×C3#6416211yescharacteristic
309D3—128no—
310C22—24no—
311C22—24no—
312C22—24no—
313C22—42no—
314C22—42no—
315C22—42no—
316C22—42no—
317C22—184no—
318C22—182no—
319C22—362no—
320C22—361no—
321C22—361no—
322C22D4×#27211yescharacteristic
323C22D4×#27212yes—
324C22#8874411yescharacteristic
325C22C2×#623512yes—
326C22C2×#623511yescharacteristic
327C4—22no—
328C4—184no—
329C4—361no—
330C3—34no—
331C3#1059292211yescharacteristic
332C2—44no—
333C2—362no—
334C2#1172184712yes—
335C2#1172184811yescharacteristic
336C2#1172184912yes—
337C2#1172185011yescharacteristic
338C2#1172187611yescharacteristic
subgroupG/H
center, Frattini, socleC23×C3C22×C3⋊D3
commutator, cosocleC23×H(3,𝔽3)C23
FittingC22×D4×H(3,𝔽3)C2
hypercenterC22×C3×D4C3⋊D3
hypocenterH(3,𝔽3)#392
LCSG ▷ C23×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C23×C3 ◁ C22×C3×D4

Complements

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

NH ≅ G/Ntype
1. C22×D4×H(3,𝔽3)C2semidirect
2. C22×#6235C2semidirect
3. C2×#88552C2semidirect
4. C2×#88542C2semidirect
5. #11720574C2semidirect
19. C22×C32×D4D3semidirect
30. C23×H(3,𝔽3)D4semidirect
36. C22×#272D4semidirect
39. C2×C4×H(3,𝔽3)D4semidirect
45. C2×#1084D4semidirect
96. C23×C32#67semidirect
105. C2×C32×C4#67semidirect
165. C22×D4#272semidirect
169. H(3,𝔽3)#392semidirect
273. C32#5223semidirect
285. C23#6235semidirect
295. C2×C4#6235semidirect

Maximal subgroups

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

subgroupindex
C22×D4×H(3,𝔽3)2
C22×#62352
C2×#885522
C2×#885422
#117205742
C3×#52233