Finite Group Databasebeta

← #13144304

Subgroups — #13144304

Order 1728.

1573 conjugacy classes of subgroups; only the 233 normal subgroups are shown.

normal subgroups only

This group has 1573 conjugacy classes of subgroups — too many to draw. Only its 233 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×#6235C211yesmaximal, characteristic
2C23×#1084C211yesmaximal, characteristic, complemented
3C2×C22⋊C4×H(3,𝔽3)C211yesmaximal, characteristic, complemented
4C2×#88542C212yesmaximal, complemented
5#11720574C212yesmaximal, complemented
6C3×#4958—34nomaximal
7C24×H(3,𝔽3)C2211yescharacteristic
8C23×#272C2211yescharacteristic
9C22×C4×H(3,𝔽3)C2212yes—
10C2×#6235—22no—
11C2×#6235—22no—
12C2×#6235C414yescomplemented
13C22×#1084—22no—
14C22×#1084—24no—
15C22×#1084C2211yescharacteristic
16C22×#1084C2212yes—
17C22⋊C4×H(3,𝔽3)—22no—
18#88542—24no—
19C22×C3×#67—34no—
20C23×C3×Dic3—34no—
21C2×C32×C22⋊C4D314yescomplemented
22C2×C3×(C2×D3)⋊C4—38no—
23C3×C3⋊#95—38no—
24C23×H(3,𝔽3)—22no—
25C23×H(3,𝔽3)—22no—
26C23×H(3,𝔽3)C2×C412yes—
27C23×H(3,𝔽3)D412yes—
28C23×H(3,𝔽3)D412yes—
29C23×H(3,𝔽3)C2311yescharacteristic
30C22×#272—21no—
31C22×#272—21no—
32C22×#272—22no—
33C22×#272—22no—
34C22×#272C2×C412yescomplemented
35C2×C4×H(3,𝔽3)—22no—
36C2×C4×H(3,𝔽3)—22no—
37C2×C4×H(3,𝔽3)—22no—
38#6235—28no—
39C2×#1084—28no—
40C2×#1084—22no—
41C2×#1084—22no—
42C2×#1084—21no—
43C2×#1084—21no—
44C2×#1084C2×C412yescomplemented
45C2×#1084D414yescomplemented
46C3×#386—91no—
47#4958—34no—
48C24×C32C2×D314yes—
49C23×C3×D3—34no—
50C22×C32×C4C2×D318yes—
51C2×C3×#67—316no—
52C2×C3×#67—68no—
53C2×C3×#67—68no—
54C22×C3×Dic3—38no—
55C22×C3×Dic3—34no—
56C22×C3×Dic3—616no—
57C22×C3×Dic3—68no—
58C32×C22⋊C4—28no—
59C3×(C2×D3)⋊C4—616no—
60C22×H(3,𝔽3)—24no—
61C22×H(3,𝔽3)—22no—
62C22×H(3,𝔽3)—24no—
63C22×H(3,𝔽3)—22no—
64C22×H(3,𝔽3)C22⋊C414yescomplemented
65C22×H(3,𝔽3)C22×C411yescharacteristic
66C22×H(3,𝔽3)C2×D411yescharacteristic
67C22×H(3,𝔽3)C2×D411yescharacteristic
68C22×H(3,𝔽3)C2×D412yes—
69C22×H(3,𝔽3)#4112yes—
70C2×#272—22no—
71C2×#272—24no—
72C2×#272—41no—
73C2×#272—42no—
74C2×#272—41no—
75C4×H(3,𝔽3)—42no—
76#1084—22no—
77#1084—24no—
78C22×C3×D4—91no—
79C23×C3×C4—91no—
80C22×#67—34no—
81C23×Dic3—34no—
82C2×C3×C22⋊C4—34no—
83C2×C3×C22⋊C4—92no—
84C2×C3×C22⋊C4C3⋊D311yescharacteristic
85C2×(C2×D3)⋊C4—38no—
86C3×#95—92no—
87C3⋊#95—38no—
88C23×C32—28no—
89C23×C32—28no—
90C23×C32C4×D318yes—
91C23×C32D1218yes—
92C23×C32#6718yes—
93C23×C32C22×D314yes—
94C22×C3×D3—38no—
95C22×C3×D3—64no—
96C22×C3×D3—68no—
97C22×C3×D3—68no—
98C22×C3×D3—64no—
99C2×C32×C4—28no—
100C2×C32×C4—28no—
101C2×C32×C4—28no—
102C3×#67—632no—
103C2×C3×Dic3—38no—
104C2×C3×Dic3—316no—
105C2×C3×Dic3—632no—
106C2×C3×Dic3—68no—
107C2×C3×Dic3—64no—
108C2×C3×Dic3—68no—
109C2×C3×Dic3—64no—
110#386—91noSylow
111C2×H(3,𝔽3)—24no—
112C2×H(3,𝔽3)C2×C22⋊C411yescharacteristic
113C2×H(3,𝔽3)C4×D412yes—
114C2×H(3,𝔽3)#12011yescharacteristic
115C2×H(3,𝔽3)#12112yes—
116C2×H(3,𝔽3)#12311yescharacteristic
117#272—42no—
118C24×C3—34no—
119C24×C3—91no—
120C24×C3C2×C3⋊D311yescharacteristic
121C23×D3—34no—
122C2×C3×D4—94no—
123C2×C3×D4—182no—
124C2×C3×D4—182no—
125C22×C3×C4—38no—
126C22×C3×C4—91no—
127C22×C3×C4—92no—
128C22×C3×C4—184no—
129C22×C3×C4—182no—
130C22×C3×C4C2×C3⋊D312yes—
131C2×#67—316no—
132C2×#67—68no—
133C2×#67—68no—
134C22×Dic3—34no—
135C22×Dic3—38no—
136C22×Dic3—616no—
137C22×Dic3—68no—
138C3×C22⋊C4—22no—
139C3×C22⋊C4—68no—
140C3×C22⋊C4—184no—
141(C2×D3)⋊C4—616no—
142C22×C32—216no—
143C22×C32—28no—
144C22×C32—28no—
145C22×C32—216no—
146C22×C32(C2×D3)⋊C4116yescomplemented
147C22×C32C2×C4×D314yes—
148C22×C32C2×D1214yes—
149C22×C32D3×D418yes—
150C22×C32#23718yes—
151C22×C32C2×#6714yes—
152C2×C3×D3—616no—
153C2×C3×D3—68no—
154C2×C3×D3—124no—
155C2×C3×D3—124no—
156C2×C3×D3—128no—
157C32×C4—48no—
158C3×Dic3—616no—
159C3×Dic3—68no—
160C22×D4—91no—
161C23×C4—91no—
162C2×C22⋊C4—92no—
163C2×C22⋊C4#27211yescharacteristic, complemented
164#95—92no—
165H(3,𝔽3)#38611yescharacteristic, complemented, Sylow
166C23×C3—22no—
167C23×C3—22no—
168C23×C3—38no—
169C23×C3—38no—
170C23×C3—38no—
171C23×C3—34no—
172C23×C3—68no—
173C23×C3—68no—
174C23×C3—92no—
175C23×C3—181no—
176C23×C3—182no—
177C23×C3—182no—
178C23×C3—181no—
179C23×C3C4×C3⋊D312yes—
180C23×C3C4⋊(C3⋊D3)12yes—
181C23×C3#63812yes—
182C23×C3C22×C3⋊D311yescharacteristic
183C22×D3—38no—
184C22×D3—68no—
185C22×D3—64no—
186C22×D3—64no—
187C22×D3—68no—
188C3×D4—188no—
189C2×C3×C4—22no—
190C2×C3×C4—22no—
191C2×C3×C4—22no—
192C2×C3×C4—68no—
193C2×C3×C4—68no—
194C2×C3×C4—68no—
195C2×C3×C4—92no—
196C2×C3×C4—94no—
197C2×C3×C4—188no—
198C2×C3×C4—182no—
199C2×C3×C4—181no—
200C2×C3×C4—182no—
201C2×C3×C4—181no—
202#67—632no—
203C2×Dic3—316no—
204C2×Dic3—38no—
205C2×Dic3—68no—
206C2×Dic3—68no—
207C2×Dic3—64no—
208C2×Dic3—64no—
209C2×Dic3—632no—
210C2×C32—216no—
211C2×C32#67⋊C418yes—
212C2×C32#88214yes—
213C2×C32#88614yes—
214C2×C32C2×(C2×D3)⋊C414yes—
215C2×C32#93918yes—
216C3×D3—128no—
217C24—91no—
218C24C2×#27211yescharacteristic
219C2×D4—94no—
220C2×D4—182no—
221C2×D4—182no—
222C22×C4—91no—
223C22×C4—92no—
224C22×C4—184no—
225C22×C4—182no—
226C22×C4C2×#27212yes—
227C22⋊C4—22no—
228C22⋊C4—184no—
229C22×C3—24no—
230C22×C3—22no—
231C22×C3—22no—
232C22×C3—24no—
233C22×C3—34no—
234C22×C3—34no—
235C22×C3—38no—
236C22×C3—38no—
237C22×C3—34no—
238C22×C3—316no—
239C22×C3—68no—
240C22×C3—616no—
241C22×C3—68no—
242C22×C3—616no—
243C22×C3—184no—
244C22×C3—182no—
245C22×C3—361no—
246C22×C3—361no—
247C22×C3—362no—
248C22×C3#375214yes—
249C22×C3C2×C4×C3⋊D311yescharacteristic
250C22×C3C2×C4⋊(C3⋊D3)11yescharacteristic
251C22×C3D4×C3⋊D312yes—
252C22×C3#383012yes—
253C22×C3C2×#63811yescharacteristic
254C2×D3—68no—
255C2×D3—616no—
256C2×D3—124no—
257C2×D3—124no—
258C2×D3—128no—
259C3×C4—42no—
260C3×C4—128no—
261C3×C4—184no—
262C3×C4—182no—
263Dic3—616no—
264Dic3—68no—
265C32#495814yescomplemented
266C23—22no—
267C23—22no—
268C23—92no—
269C23—181no—
270C23—182no—
271C23—182no—
272C23—181no—
273C23C4×#27212yes—
274C23C4⋊#27212yes—
275C23#623512yes—
276C23C22×#27211yescharacteristic
277D4—188no—
278C2×C4—22no—
279C2×C4—22no—
280C2×C4—22no—
281C2×C4—92no—
282C2×C4—94no—
283C2×C4—188no—
284C2×C4—182no—
285C2×C4—181no—
286C2×C4—182no—
287C2×C4—181no—
288C2×C3—24no—
289C2×C3—38no—
290C2×C3—34no—
291C2×C3—34no—
292C2×C3—34no—
293C2×C3—38no—
294C2×C3—616no—
295C2×C3—362no—
296C2×C3#6409012yes—
297C2×C3#6409111yescharacteristic
298C2×C3#6409511yescharacteristic
299C2×C3C2×#375211yescharacteristic
300C2×C3#6414812yes—
301D3—128no—
302C22—24no—
303C22—22no—
304C22—22no—
305C22—24no—
306C22—184no—
307C22—182no—
308C22—361no—
309C22—361no—
310C22—362no—
311C22#8854214yescomplemented
312C22C2×C4×#27211yescharacteristic
313C22C2×C4⋊#27211yescharacteristic
314C22D4×#27212yes—
315C22#8874412yes—
316C22C2×#623511yescharacteristic
317C4—42no—
318C4—184no—
319C4—182no—
320C3—34no—
321C3#1059265711yescharacteristic
322C2—24no—
323C2—362no—
324C2#1172173712yes—
325C2#1172173811yescharacteristic
326C2#1172174211yescharacteristic
327C2C2×#8854211yescharacteristic
328C2#1172184912yes—
subgroupG/H
center, Frattini, socleC23×C3C22×C3⋊D3
commutatorC22×H(3,𝔽3)C22×C4
FittingC2×C22⋊C4×H(3,𝔽3)C2
prime radicalC22×#6235C2
cosocleC23×H(3,𝔽3)C23
hypercenterC2×C3×C22⋊C4C3⋊D3
hypocenterH(3,𝔽3)#386
LCSG ▷ C22×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C23×C3 ◁ C2×C3×C22⋊C4

Complements

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

NH ≅ G/Ntype
2. C23×#1084C2semidirect
3. C2×C22⋊C4×H(3,𝔽3)C2semidirect
4. C2×#88542C2semidirect
5. #11720574C2semidirect
12. C2×#6235C4semidirect
21. C2×C32×C22⋊C4D3semidirect
34. C22×#272C2×C4semidirect
44. C2×#1084C2×C4semidirect
45. C2×#1084D4semidirect
64. C22×H(3,𝔽3)C22⋊C4semidirect
146. C22×C32(C2×D3)⋊C4semidirect
163. C2×C22⋊C4#272semidirect
165. H(3,𝔽3)#386semidirect
265. C32#4958semidirect
311. C22#88542semidirect

Maximal subgroups

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

subgroupindex
C22×#62352
C23×#10842
C2×C22⋊C4×H(3,𝔽3)2
C2×#885422
#117205742
C3×#49583