Finite Group Databasebeta

← C4×#88451 #13140201

Subgroups — C4×#88451 #13140201

Order 1728.

1139 conjugacy classes of subgroups; only the 195 normal subgroups are shown.

normal subgroups only

This group has 1139 conjugacy classes of subgroups — too many to draw. Only its 195 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×C4×#272C211yesmaximal, characteristic
2C2×C4×(C3⋊Dic3)⋊C3C212yesmaximal, complemented
3C2×#88451C212yesmaximal
4#11720334C212yesmaximal, complemented
5C4×(C2×D3)⋊Dic3—32nomaximal
6C23×#272C2211yescharacteristic
7C22×C4×H(3,𝔽3)C2211yescharacteristic
8C22×#1084C2211yescharacteristic
9C2×C4×#272—24no—
10C2×C4×#272C414yescomplemented
11C22×(C3⋊Dic3)⋊C3C2214yes—
12C4×(C3⋊Dic3)⋊C3—24no—
13#88451C418yesdirect product factor
14C22×C3×C4×D3—32no—
15C22×C3×C4×D3—61no—
16C2×C4×C3⋊Dic3D312yescomplemented
17C2×C3×C4×Dic3—32no—
18C2×(C2×D3)⋊Dic3—34no—
19#63579—34no—
20C23×H(3,𝔽3)C2311yescharacteristic
21C22×#272—22no—
22C22×#272—22no—
23C22×#272C2×C414yes—
24C22×#272C2×C412yes—
25C2×C4×H(3,𝔽3)C2×C412yes—
26C2×C4×H(3,𝔽3)D414yes—
27C2×#1084—22no—
28C2×#1084C2×C412yes—
29C4×#272—28no—
30C2×(C3⋊Dic3)⋊C3—24no—
31C2×(C3⋊Dic3)⋊C3—24no—
32C2×(C3⋊Dic3)⋊C3C2×C418yescomplemented
33C4×#217—91no—
34C4×(C2×D3)⋊C4—92no—
35C23×C3×D3—32no—
36C23×C3×D3—61no—
37C22×C32×C4—21no—
38C22×C32×C4C2×D312yes—
39C22×C3⋊Dic3C2×D314yes—
40C22×C3×Dic3—34no—
41C22×C3×Dic3—32no—
42C22×C3×Dic3—61no—
43C2×C3×C4×D3—38no—
44C2×C3×C4×D3—64no—
45C2×C3×C4×D3—68no—
46C2×C3×C4×D3—64no—
47C2×C3×C4×D3—64no—
48C4×C3⋊Dic3—24no—
49C3×C4×Dic3—64no—
50(C2×D3)⋊Dic3—316no—
51C22×H(3,𝔽3)C22×C411yescharacteristic
52C22×H(3,𝔽3)C22×C412yes—
53C22×H(3,𝔽3)C2×D412yes—
54C22×H(3,𝔽3)#4112yes—
55C2×#272—22no—
56C2×#272—22no—
57C2×#272—28no—
58C2×#272C4214yescomplemented
59C4×H(3,𝔽3)C22⋊C414yescomplemented
60#1084—22no—
61(C3⋊Dic3)⋊C3—28no—
62C23×C3×C4—91no—
63C22×C4×D3—32no—
64C22×C4×D3—61no—
65C2×C3×C42—92no—
66C2×#217—92no—
67C2×(C2×D3)⋊C4—94no—
68C2×C4×Dic3—32no—
69C2×C4×Dic3—92no—
70C3⋊#95—92no—
71C3⋊#95—94no—
72C23×C32—21no—
73C23×C32C22×D312yes—
74C22×C3×D3—38no—
75C22×C3×D3—34no—
76C22×C3×D3—64no—
77C22×C3×D3—62no—
78C22×C3×D3—64no—
79C22×C3×D3—64no—
80C22×C3×D3—62no—
81C22×C3×D3—64no—
82C22×C3×D3—62no—
83C2×C32×C4—22no—
84C2×C32×C4—24no—
85C2×C32×C4C4×D314yes—
86C2×C32×C4D1214yes—
87C2×C32×C4#6714yes—
88C2×C3⋊Dic3—24no—
89C2×C3⋊Dic3—24no—
90C2×C3⋊Dic3C4×D318yescomplemented
91C2×C3×Dic3—38no—
92C2×C3×Dic3—34no—
93C2×C3×Dic3—62no—
94C2×C3×Dic3—64no—
95C2×C3×Dic3—64no—
96C2×C3×Dic3—62no—
97C2×C3×Dic3—64no—
98C2×C3×Dic3—62no—
99C3×C4×D3—616no—
100C3×C4×D3—616no—
101C4×C22⋊C4—271noSylow
102C2×H(3,𝔽3)C2×C4211yescharacteristic
103C2×H(3,𝔽3)C2×C22⋊C411yescharacteristic
104C2×H(3,𝔽3)#11711yescharacteristic
105C2×H(3,𝔽3)C4×D414yes—
106#272—24no—
107C24×C3—91no—
108C23×D3—32no—
109C23×D3—61no—
110C22×C3×C4—32no—
111C22×C3×C4—61no—
112C22×C3×C4—94no—
113C22×C3×C4—91no—
114C22×C3×C4—94no—
115C22×C3×C4—184no—
116C22×C3×C4D3211yescharacteristic
117C22×Dic3—32no—
118C22×Dic3—34no—
119C22×Dic3—61no—
120C22×Dic3—94no—
121C2×C4×D3—38no—
122C2×C4×D3—64no—
123C2×C4×D3—64no—
124C2×C4×D3—64no—
125C2×C4×D3—68no—
126C3×C42—184no—
127#217—98no—
128(C2×D3)⋊C4—916no—
129C4×Dic3—64no—
130C4×Dic3—184no—
131C22×C32—22no—
132C22×C32—22no—
133C22×C32—21no—
134C22×C32—22no—
135C22×C32C2×C4×D314yes—
136C22×C32C2×C4×D312yes—
137C22×C32C2×D1212yes—
138C22×C32#23514yes—
139C22×C32C2×#6712yes—
140C2×C3×D3—38no—
141C2×C3×D3—64no—
142C2×C3×D3—64no—
143C2×C3×D3—64no—
144C2×C3×D3—616no—
145C2×C3×D3—64no—
146C2×C3×D3—68no—
147C2×C3×D3—64no—
148C2×C3×D3—68no—
149C32×C4—24no—
150C32×C4(C2×D3)⋊C418yescomplemented
151C3⋊Dic3—28no—
152C3×Dic3—64no—
153C3×Dic3—68no—
154C3×Dic3—64no—
155C23×C4—91no—
156C2×C22⋊C4—272no—
157C2×C42—92no—
158#95—272no—
159H(3,𝔽3)C4×C22⋊C411yescharacteristic, complemented, Sylow
160C23×C3—32no—
161C23×C3—61no—
162C23×C3—94no—
163C23×C3—92no—
164C23×C3—182no—
165C23×C3—182no—
166C23×C3C2×D3211yescharacteristic
167C22×D3—38no—
168C22×D3—34no—
169C22×D3—62no—
170C22×D3—64no—
171C22×D3—62no—
172C22×D3—62no—
173C22×D3—64no—
174C22×D3—64no—
175C22×D3—64no—
176C2×C3×C4—34no—
177C2×C3×C4—34no—
178C2×C3×C4—34no—
179C2×C3×C4—62no—
180C2×C3×C4—64no—
181C2×C3×C4—98no—
182C2×C3×C4—92no—
183C2×C3×C4—184no—
184C2×C3×C4—184no—
185C2×C3×C4—182no—
186C2×C3×C4—188no—
187C2×C3×C4#62412yes—
188C2×C3×C4#62614yes—
189C2×Dic3—34no—
190C2×Dic3—38no—
191C2×Dic3—62no—
192C2×Dic3—62no—
193C2×Dic3—62no—
194C2×Dic3—64no—
195C2×Dic3—64no—
196C2×Dic3—64no—
197C2×Dic3—98no—
198C2×Dic3—184no—
199C2×Dic3—184no—
200C4×D3—616no—
201C4×D3—616no—
202C2×C32—21no—
203C2×C32—21no—
204C2×C32—24no—
205C2×C32—21no—
206C2×C32C42×D312yes—
207C2×C32#87212yes—
208C2×C32C4×D1214yes—
209C2×C32C2×(C2×D3)⋊C412yes—
210C2×C32C4×#6714yes—
211C3×D3—68no—
212C3×D3—68no—
213C24—91no—
214C22×C4—91no—
215C22×C4—94no—
216C22×C4—94no—
217C22×C4—184no—
218C22×C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
219C22⋊C4—278no—
220C42—184no—
221C22×C3—34no—
222C22×C3—34no—
223C22×C3—32no—
224C22×C3—32no—
225C22×C3—32no—
226C22×C3—62no—
227C22×C3—62no—
228C22×C3—61no—
229C22×C3—62no—
230C22×C3—94no—
231C22×C3—182no—
232C22×C3—188no—
233C22×C3—182no—
234C22×C3#379812yes—
235C22×C3C4×D3212yes—
236C22×C3C2×#62411yescharacteristic
237C22×C3C2×#62612yes—
238C2×D3—38no—
239C2×D3—64no—
240C2×D3—64no—
241C2×D3—64no—
242C2×D3—68no—
243C2×D3—68no—
244C2×D3—64no—
245C2×D3—64no—
246C2×D3—616no—
247C3×C4—38no—
248C3×C4—64no—
249C3×C4—188no—
250C3×C4—182no—
251C3×C4#372214yes—
252Dic3—64no—
253Dic3—64no—
254Dic3—68no—
255Dic3—188no—
256C32—21no—
257C32C4×(C2×D3)⋊C412yescomplemented
258C23—94no—
259C23—92no—
260C23—182no—
261C23—182no—
262C23C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
263C2×C4—98no—
264C2×C4—92no—
265C2×C4—184no—
266C2×C4—184no—
267C2×C4—182no—
268C2×C4—188no—
269C2×C4#619912yes—
270C2×C4#620014yes—
271C2×C3—32no—
272C2×C3—32no—
273C2×C3—32no—
274C2×C3—34no—
275C2×C3—34no—
276C2×C3—61no—
277C2×C3—61no—
278C2×C3—61no—
279C2×C3—64no—
280C2×C3—184no—
281C2×C3#6387411yescharacteristic
282C2×C3C4×#62411yescharacteristic
283C2×C3C4×#62614yes—
284C2×C3C2×#372211yescharacteristic
285D3—68no—
286D3—68no—
287C22—94no—
288C22—182no—
289C22—182no—
290C22—188no—
291C22#8865012yes—
292C22C4×((C3⋊D3)⋊C3)⋊C212yes—
293C22C2×#619911yescharacteristic
294C22C2×#620012yes—
295C4—182no—
296C4—188no—
297C4#8845114yesdirect product factor
298C3—32no—
299C3—61no—
300C3C4×#372211yescharacteristic
301C2—184no—
302C2#1172109711yescharacteristic
303C2C4×#619911yescharacteristic
304C2C4×#620014yes—
305C2C2×#8845111yescharacteristic
subgroupG/H
center, hypercenterC22×C4((C3⋊D3)⋊C3)⋊C2
commutatorC2×H(3,𝔽3)C2×C42
Frattini, socleC23×C3C2×D32
FittingC22×C4×H(3,𝔽3)C22
prime radicalC23×#272C22
cosocleC23×H(3,𝔽3)C23
hypocenterH(3,𝔽3)C4×C22⋊C4
LCSG ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
2. C2×C4×(C3⋊Dic3)⋊C3C2semidirect
4. #11720334C2semidirect
10. C2×C4×#272C4semidirect
13. #88451C4direct (N × H)
16. C2×C4×C3⋊Dic3D3semidirect
32. C2×(C3⋊Dic3)⋊C3C2×C4semidirect
58. C2×#272C42semidirect
59. C4×H(3,𝔽3)C22⋊C4semidirect
90. C2×C3⋊Dic3C4×D3semidirect
150. C32×C4(C2×D3)⋊C4semidirect
159. H(3,𝔽3)C4×C22⋊C4semidirect
257. C32C4×(C2×D3)⋊C4semidirect
297. C4#88451direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C4×#2722
C2×C4×(C3⋊Dic3)⋊C32
C2×#884512
#117203342
C4×(C2×D3)⋊Dic33