Finite Group Databasebeta

← #13141164

Subgroups — #13141164

Order 1728.

1055 conjugacy classes of subgroups; only the 171 normal subgroups are shown.

normal subgroups only

This group has 1055 conjugacy classes of subgroups — too many to draw. Only its 171 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
1C23×#1084C211yesmaximal, characteristic
2C2×#3757⋊C3C212yesmaximal
3C2×#88449C212yesmaximal, complemented
4#11720334C212yesmaximal, complemented
5#10591716—32nomaximal
6C24×H(3,𝔽3)C2211yescharacteristic
7C22×#1084—24no—
8C22×#1084C414yescomplemented
9C22×#1084C2211yescharacteristic
10C22×#1084C2211yescharacteristic
11C22×(C3⋊Dic3)⋊C3C2214yes—
12#3757⋊C3—24no—
13#88449—24no—
14C23×C3×Dic3—32no—
15C23×C3×Dic3—61no—
16C2×#3757D312yes—
17C2×C3×#217—32no—
18C2×Dic3⋊Dic3—34no—
19#63579—34no—
20C23×H(3,𝔽3)—22no—
21C23×H(3,𝔽3)—22no—
22C23×H(3,𝔽3)C2×C412yes—
23C23×H(3,𝔽3)D412yes—
24C23×H(3,𝔽3)Q812yes—
25C23×H(3,𝔽3)C2311yescharacteristic
26C2×#1084—22no—
27C2×#1084—28no—
28C2×#1084C2×C412yes—
29C2×#1084C2×C412yes—
30C2×#1084D414yes—
31C2×(C3⋊Dic3)⋊C3—24no—
32C2×(C3⋊Dic3)⋊C3—24no—
33C2×(C3⋊Dic3)⋊C3—24no—
34#5212—91no—
35#4942—92no—
36C24×C32—21no—
37C24×C32C2×D312yes—
38C22×C3⋊Dic3C2×D314yes—
39C22×C3×Dic3—32no—
40C22×C3×Dic3—38no—
41C22×C3×Dic3—32no—
42C22×C3×Dic3—34no—
43C22×C3×Dic3—68no—
44C22×C3×Dic3—64no—
45C22×C3×Dic3—61no—
46C22×C3×Dic3—61no—
47C22×C3×Dic3—68no—
48#3757—24no—
49C3×#217—64no—
50Dic3⋊Dic3—68no—
51C22×H(3,𝔽3)—22no—
52C22×H(3,𝔽3)—22no—
53C22×H(3,𝔽3)—24no—
54C22×H(3,𝔽3)—24no—
55C22×H(3,𝔽3)C4⋊C414yescomplemented
56C22×H(3,𝔽3)C22×C411yescharacteristic
57C22×H(3,𝔽3)C2×D412yes—
58C22×H(3,𝔽3)C2×D411yescharacteristic
59C22×H(3,𝔽3)C2×Q811yescharacteristic
60C22×H(3,𝔽3)#4112yes—
61#1084—22no—
62#1084C22⋊C414yescomplemented
63(C3⋊Dic3)⋊C3—44no—
64C23×C3×C4—91no—
65C23×Dic3—32no—
66C23×Dic3—61no—
67C2×C3×C22⋊C4—92no—
68C2×#217—32no—
69C2×#217—92no—
70C2×C4⋊Dic3—92no—
71C2×Dic3⋊C4—94no—
72C3⋊#95—92no—
73C3⋊#95—94no—
74C23×C32—24no—
75C23×C32—24no—
76C23×C32—24no—
77C23×C32—24no—
78C23×C32—22no—
79C23×C32—22no—
80C23×C32—22no—
81C23×C32—21no—
82C23×C32Dic614yes—
83C23×C32C4×D314yes—
84C23×C32#6714yes—
85C23×C32C22×D312yes—
86C2×C3⋊Dic3—24no—
87C2×C3⋊Dic3—24no—
88C2×C3⋊Dic3—24no—
89C2×C3×Dic3—34no—
90C2×C3×Dic3—38no—
91C2×C3×Dic3—34no—
92C2×C3×Dic3—62no—
93C2×C3×Dic3—62no—
94C2×C3×Dic3—64no—
95C2×C3×Dic3—616no—
96C2×C3×Dic3—616no—
97C2×C3×Dic3—64no—
98C2×C3×Dic3—64no—
99C2×C3×Dic3—64no—
100C2×C3×Dic3—64no—
101C2×C3×Dic3—64no—
102#380—271noSylow
103C2×H(3,𝔽3)—24no—
104C2×H(3,𝔽3)C2×C22⋊C411yescharacteristic
105C2×H(3,𝔽3)C2×C4⋊C411yescharacteristic
106C2×H(3,𝔽3)#11711yescharacteristic
107C2×H(3,𝔽3)#12112yes—
108C2×H(3,𝔽3)C22⋊Q812yes—
109C24×C3—32no—
110C24×C3—61no—
111C24×C3D3211yescharacteristic
112C22×C3×C4—91no—
113C22×C3×C4—94no—
114C22×C3×C4—91no—
115C22×C3×C4—94no—
116C22×C3×C4—184no—
117C22×Dic3—32no—
118C22×Dic3—32no—
119C22×Dic3—38no—
120C22×Dic3—34no—
121C22×Dic3—61no—
122C22×Dic3—64no—
123C22×Dic3—61no—
124C22×Dic3—68no—
125C22×Dic3—68no—
126C22×Dic3—94no—
127C3×C22⋊C4—184no—
128#217—64no—
129#217—184no—
130C4⋊Dic3—184no—
131Dic3⋊C4—188no—
132C22×C32—24no—
133C22×C32—24no—
134C22×C32—28no—
135C22×C32—28no—
136C22×C32—24no—
137C22×C32—28no—
138C22×C32—24no—
139C22×C32—28no—
140C22×C32—24no—
141C22×C32—22no—
142C22×C32—21no—
143C22×C32—21no—
144C22×C32—21no—
145C22×C32—22no—
146C22×C32Dic3⋊C418yescomplemented
147C22×C32C2×Dic612yes—
148C22×C32C2×C4×D312yes—
149C22×C32D3×D414yes—
150C22×C32#23714yes—
151C22×C32C2×#6712yes—
152C3⋊Dic3—44no—
153C3×Dic3—38no—
154C3×Dic3—64no—
155C3×Dic3—64no—
156C3×Dic3—64no—
157C3×Dic3—124no—
158C23×C4—91no—
159C2×C4⋊C4—272no—
160C2×C22⋊C4—92no—
161#95—272no—
162H(3,𝔽3)#38011yescharacteristic, complemented, Sylow
163C23×C3—22no—
164C23×C3—22no—
165C23×C3—34no—
166C23×C3—34no—
167C23×C3—34no—
168C23×C3—32no—
169C23×C3—64no—
170C23×C3—64no—
171C23×C3—64no—
172C23×C3—64no—
173C23×C3—62no—
174C23×C3—62no—
175C23×C3—62no—
176C23×C3—61no—
177C23×C3#62412yes—
178C23×C3#62512yes—
179C23×C3#62712yes—
180C23×C3C2×D3211yescharacteristic
181C2×C3×C4—92no—
182C2×C3×C4—94no—
183C2×C3×C4—92no—
184C2×C3×C4—188no—
185C2×C3×C4—182no—
186C2×C3×C4—184no—
187C2×C3×C4—184no—
188C2×C3×C4—184no—
189C2×Dic3—38no—
190C2×Dic3—34no—
191C2×Dic3—34no—
192C2×Dic3—62no—
193C2×Dic3—64no—
194C2×Dic3—62no—
195C2×Dic3—616no—
196C2×Dic3—64no—
197C2×Dic3—616no—
198C2×Dic3—64no—
199C2×Dic3—64no—
200C2×Dic3—64no—
201C2×Dic3—64no—
202C2×Dic3—184no—
203C2×Dic3—184no—
204C2×Dic3—184no—
205C2×C32—28no—
206C2×C32—28no—
207C2×C32—21no—
208C2×C32—21no—
209C2×C32—21no—
210C2×C32—22no—
211C2×C32—22no—
212C2×C32#87712yes—
213C2×C32#87814yes—
214C2×C32D3×C22⋊C412yes—
215C2×C32C2×Dic3⋊C412yes—
216C2×C32#93914yes—
217C24((C3⋊D3)⋊C3)⋊C211yescharacteristic
218C22×C4—91no—
219C22×C4—94no—
220C22×C4—91no—
221C22×C4—94no—
222C22×C4—184no—
223C4⋊C4—544no—
224C22⋊C4—184no—
225C22×C3—22no—
226C22×C3—22no—
227C22×C3—24no—
228C22×C3—24no—
229C22×C3—32no—
230C22×C3—34no—
231C22×C3—32no—
232C22×C3—32no—
233C22×C3—34no—
234C22×C3—38no—
235C22×C3—64no—
236C22×C3—64no—
237C22×C3—68no—
238C22×C3—68no—
239C22×C3—64no—
240C22×C3—64no—
241C22×C3—68no—
242C22×C3—68no—
243C22×C3—61no—
244C22×C3—62no—
245C22×C3—61no—
246C22×C3—61no—
247C22×C3—62no—
248C22×C3—64no—
249C22×C3#372414yes—
250C22×C3#380512yes—
251C22×C3C2×#62411yescharacteristic
252C22×C3C2×#62511yescharacteristic
253C22×C3C2×#62711yescharacteristic
254C22×C3#381112yes—
255C3×C4—94no—
256C3×C4—182no—
257C3×C4—364no—
258Dic3—38no—
259Dic3—64no—
260Dic3—64no—
261Dic3—64no—
262Dic3—124no—
263Dic3—364no—
264C32—21no—
265C32#494212yescomplemented
266C23—22no—
267C23—22no—
268C23#619612yes—
269C23#619812yes—
270C23#619912yes—
271C23C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
272C2×C4—92no—
273C2×C4—94no—
274C2×C4—92no—
275C2×C4—188no—
276C2×C4—182no—
277C2×C4—184no—
278C2×C4—184no—
279C2×C4—184no—
280C2×C3—24no—
281C2×C3—32no—
282C2×C3—32no—
283C2×C3—34no—
284C2×C3—34no—
285C2×C3—32no—
286C2×C3—68no—
287C2×C3—68no—
288C2×C3—61no—
289C2×C3—61no—
290C2×C3—62no—
291C2×C3—62no—
292C2×C3—61no—
293C2×C3#6395711yescharacteristic
294C2×C3C2×#372411yescharacteristic
295C2×C3#6397411yescharacteristic
296C2×C3#6398012yes—
297C2×C3#6398212yes—
298C22—22no—
299C22—24no—
300C22—24no—
301C22—22no—
302C22#8844914yescomplemented
303C22C2×#619611yescharacteristic
304C22#8867212yes—
305C22C2×#619811yescharacteristic
306C22C2×#619911yescharacteristic
307C22#8867712yes—
308C4—94no—
309C4—182no—
310C4—364no—
311C3—32no—
312C3—61no—
313C3#1059172011yescharacteristic
314C2—24no—
315C2#1172129811yescharacteristic
316C2C2×#8844911yescharacteristic
317C2#1172130912yes—
318C2#1172131812yes—
319C2#1172132011yescharacteristic
subgroupG/H
centerC23C2×((C3⋊D3)⋊C3)⋊C2
commutatorC22×H(3,𝔽3)C22×C4
Frattini, socleC23×C3C2×D32
Fitting, prime radicalC24×H(3,𝔽3)C22
cosocleC23×H(3,𝔽3)C23
hypercenterC24((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#380
LCSG ▷ C22×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C23 ◁ C24

Complements

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

NH ≅ G/Ntype
3. C2×#88449C2semidirect
4. #11720334C2semidirect
8. C22×#1084C4semidirect
55. C22×H(3,𝔽3)C4⋊C4semidirect
62. #1084C22⋊C4semidirect
146. C22×C32Dic3⋊C4semidirect
162. H(3,𝔽3)#380semidirect
265. C32#4942semidirect
302. C22#88449semidirect

Maximal subgroups

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

subgroupindex
C23×#10842
C2×#3757⋊C32
C2×#884492
#117203342
#105917163