Finite Group Databasebeta

← C2×#11721080 #13150658

Subgroups — C2×#11721080 #13150658

Order 1728.

1171 conjugacy classes of subgroups; only the 219 normal subgroups are shown.

normal subgroups only

This group has 1171 conjugacy classes of subgroups — too many to draw. Only its 219 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×#6229C211yesmaximal, characteristic
2C22×#6196C211yesmaximal, characteristic
3C2×#3750⋊C3C212yesmaximal
4C2×#88448C212yesmaximal
5C2×#88447C211yesmaximal, characteristic
6#11721080C218yesmaximal, direct product factor
7C2×#63839—32nomaximal
8C22×C4×H(3,𝔽3)C2211yescharacteristic
9C22×#1084C2211yescharacteristic
10C22×#1084C2211yescharacteristic
11C2×#6229—24no—
12C2×#6229C2214yes—
13C22×(C3⋊Dic3)⋊C3C2212yes—
14C22×(C3⋊Dic3)⋊C3C2212yes—
15C2×#6196—24no—
16C2×#6196C2214yes—
17#3750⋊C3C2218yes—
18#88448C2218yes—
19#88447C2214yes—
20C22×C3×Dic6—32no—
21C22×C3×Dic6—61no—
22C22×#627—32no—
23C2×#3750D312yes—
24C2×C3×Dic3⋊C4—32no—
25C2×#3724—32no—
26C2×Dic3⋊Dic3—32no—
27C2×Dic32—32no—
28#63839—316no—
29C23×H(3,𝔽3)C2311yescharacteristic
30C2×C4×H(3,𝔽3)—22no—
31C2×C4×H(3,𝔽3)C2312yes—
32C2×#1084—22no—
33C2×#1084D414yes—
34C2×#1084C2312yes—
35C2×#1084C2312yes—
36#6229—28no—
37C2×(C3⋊Dic3)⋊C3—24no—
38C2×(C3⋊Dic3)⋊C3Q818yescomplemented
39C2×(C3⋊Dic3)⋊C3C2314yes—
40C2×(C3⋊Dic3)⋊C3C2314yes—
41#6196—28no—
42C2×#944—91no—
43C2×#888—92no—
44C22×C32×C4—21no—
45C22×C32×C4C2×D312yes—
46C22×C3⋊Dic3C2×D312yes—
47C22×C3⋊Dic3C2×D312yes—
48C22×C3×Dic3—32no—
49C22×C3×Dic3—32no—
50C22×C3×Dic3—32no—
51C22×C3×Dic3—32no—
52C22×C3×Dic3—61no—
53C22×C3×Dic3—61no—
54C2×C3×Dic6—38no—
55C2×C3×Dic6—68no—
56C2×C3×Dic6—64no—
57C2×C3×Dic6—68no—
58C2×#627—38no—
59C2×#627—68no—
60#3750C2×D318yes—
61C3×Dic3⋊C4—38no—
62#3724—38no—
63Dic3⋊Dic3—38no—
64Dic32—38no—
65C22×H(3,𝔽3)C2×D412yes—
66C22×H(3,𝔽3)C2×Q814yes—
67C22×H(3,𝔽3)C2411yescharacteristic
68C4×H(3,𝔽3)—22no—
69#1084—22no—
70#1084C2×D414yes—
71(C3⋊Dic3)⋊C3—24no—
72(C3⋊Dic3)⋊C3C2×Q818yescomplemented
73C22×C3×Q8—91no—
74C22×Dic6—32no—
75C22×Dic6—61no—
76C22×Dic6—91no—
77C22×Dic6—92no—
78C2×C3×C4⋊C4—92no—
79#944—98no—
80C2×C4⋊Dic3—92no—
81C2×Dic3⋊C4—32no—
82C2×Dic3⋊C4—92no—
83C2×Dic3⋊C4—92no—
84C2×Dic3⋊C4—92no—
85C2×C4×Dic3—91no—
86C2×C4×Dic3—92no—
87#888—916no—
88C23×C32—21no—
89C23×C32C22×D312yes—
90C2×C32×C4—24no—
91C2×C32×C4—24no—
92C2×C32×C4—22no—
93C2×C32×C4C22×D314yes—
94C2×C3⋊Dic3—24no—
95C2×C3⋊Dic3Dic618yescomplemented
96C2×C3⋊Dic3C22×D314yes—
97C2×C3⋊Dic3C22×D314yes—
98C2×C3×Dic3—38no—
99C2×C3×Dic3—38no—
100C2×C3×Dic3—34no—
101C2×C3×Dic3—34no—
102C2×C3×Dic3—34no—
103C2×C3×Dic3—34no—
104C2×C3×Dic3—64no—
105C2×C3×Dic3—62no—
106C2×C3×Dic3—62no—
107C2×C3×Dic3—64no—
108C2×C3×Dic3—64no—
109C2×C3×Dic3—64no—
110C3×Dic6—616no—
111C3×Dic6—616no—
112#627—616no—
113C2×C4⋊Q8—271noSylow
114C2×H(3,𝔽3)C4⋊Q814yescomplemented
115C2×H(3,𝔽3)C22×D411yescharacteristic
116C2×H(3,𝔽3)C22×Q812yes—
117C2×C3×Q8—94no—
118C2×C3×Q8—184no—
119C22×C3×C4—32no—
120C22×C3×C4—61no—
121C22×C3×C4—92no—
122C22×C3×C4—91no—
123C22×C3×C4—92no—
124C22×C3×C4—91no—
125C22×C3×C4D3211yescharacteristic
126C22×Dic3—32no—
127C22×Dic3—32no—
128C22×Dic3—32no—
129C22×Dic3—32no—
130C22×Dic3—61no—
131C22×Dic3—61no—
132C22×Dic3—92no—
133C22×Dic3—92no—
134C2×Dic6—38no—
135C2×Dic6—64no—
136C2×Dic6—68no—
137C2×Dic6—68no—
138C2×Dic6—94no—
139C2×Dic6—98no—
140C2×Dic6—188no—
141C2×Dic6—184no—
142C3×C4⋊C4—98no—
143C4⋊Dic3—98no—
144Dic3⋊C4—38no—
145Dic3⋊C4—98no—
146Dic3⋊C4—98no—
147Dic3⋊C4—98no—
148C4×Dic3—94no—
149C4×Dic3—98no—
150C22×C32—21no—
151C22×C32—24no—
152C22×C32—22no—
153C22×C32C2×Dic614yes—
154C22×C32D3×D414yes—
155C22×C32D3×Q814yes—
156C22×C32C23×D312yes—
157C32×C4—24no—
158C32×C4—24no—
159C3⋊Dic3—24no—
160C3⋊Dic3C2×Dic618yescomplemented
161C3×Dic3—38no—
162C3×Dic3—38no—
163C3×Dic3—64no—
164C3×Dic3—64no—
165C3×Dic3—64no—
166C3×Dic3—64no—
167C22×Q8—91no—
168C22×Q8—271no—
169C4⋊Q8—278no—
170C2×C4⋊C4—92no—
171C2×C4⋊C4—272no—
172C2×C42—271no—
173H(3,𝔽3)C2×C4⋊Q811yescharacteristic, complemented, Sylow
174C23×C3—32no—
175C23×C3—61no—
176C23×C3C2×D3211yescharacteristic
177C3×Q8—188no—
178C2×C3×C4—22no—
179C2×C3×C4—34no—
180C2×C3×C4—64no—
181C2×C3×C4—64no—
182C2×C3×C4—62no—
183C2×C3×C4—94no—
184C2×C3×C4—98no—
185C2×C3×C4—92no—
186C2×C3×C4—94no—
187C2×C3×C4—94no—
188C2×C3×C4—92no—
189C2×C3×C4—184no—
190C2×C3×C4—182no—
191C2×C3×C4C2×D3212yes—
192C2×Dic3—34no—
193C2×Dic3—38no—
194C2×Dic3—34no—
195C2×Dic3—38no—
196C2×Dic3—34no—
197C2×Dic3—34no—
198C2×Dic3—62no—
199C2×Dic3—64no—
200C2×Dic3—62no—
201C2×Dic3—64no—
202C2×Dic3—64no—
203C2×Dic3—64no—
204C2×Dic3—98no—
205C2×Dic3—94no—
206C2×Dic3—94no—
207C2×Dic3—184no—
208Dic6—616no—
209Dic6—616no—
210Dic6—1816no—
211Dic6—188no—
212C2×C32—22no—
213C2×C32—21no—
214C2×C32—24no—
215C2×C32#88818yescomplemented
216C2×C32C22×Dic612yes—
217C2×C32C2×D3×D412yes—
218C2×C32C2×D3×Q812yes—
219C2×Q8—94no—
220C2×Q8—184no—
221C2×Q8—274no—
222C2×Q8—544no—
223C22×C4—92no—
224C22×C4—91no—
225C22×C4—92no—
226C22×C4—91no—
227C22×C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
228C4⋊C4—98no—
229C4⋊C4—278no—
230C42—274no—
231C22×C3—32no—
232C22×C3—34no—
233C22×C3—34no—
234C22×C3—34no—
235C22×C3—61no—
236C22×C3—64no—
237C22×C3—62no—
238C22×C3D3×Dic614yes—
239C22×C3#381112yes—
240C22×C3C22×D3211yescharacteristic
241C3×C4—22no—
242C3×C4—64no—
243C3×C4—64no—
244C3×C4—98no—
245C3×C4—94no—
246C3×C4—184no—
247C3×C4—182no—
248Dic3—38no—
249Dic3—38no—
250Dic3—64no—
251Dic3—64no—
252Dic3—64no—
253Dic3—64no—
254Dic3—98no—
255Dic3—184no—
256C32—21no—
257C32C2×#88812yescomplemented
258C23C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
259Q8—188no—
260Q8—548no—
261C2×C4—22no—
262C2×C4—94no—
263C2×C4—92no—
264C2×C4—98no—
265C2×C4—94no—
266C2×C4—94no—
267C2×C4—92no—
268C2×C4—182no—
269C2×C4—184no—
270C2×C4C2×((C3⋊D3)⋊C3)⋊C212yes—
271C2×C3—32no—
272C2×C3—38no—
273C2×C3—32no—
274C2×C3—32no—
275C2×C3—61no—
276C2×C3—64no—
277C2×C3—62no—
278C2×C3#6384014yes—
279C2×C3C2×D3×Dic612yes—
280C2×C3C2×#381111yescharacteristic
281C22((C3⋊D3)⋊C3)⋊Q814yes—
282C22#8867712yes—
283C22C22×((C3⋊D3)⋊C3)⋊C211yescharacteristic
284C4—22no—
285C4—94no—
286C4—98no—
287C4—182no—
288C4—184no—
289C3—32no—
290C3—61no—
291C3C2×#6384011yescharacteristic
292C2#1172108014yesdirect product factor
293C2C2×((C3⋊D3)⋊C3)⋊Q812yes—
294C2C2×#8867711yescharacteristic
subgroupG/H
centerC23C2×((C3⋊D3)⋊C3)⋊C2
commutator, cosocleC22×H(3,𝔽3)C24
FrattiniC22×C3C22×D32
FittingC22×C4×H(3,𝔽3)C22
socleC23×C3C2×D32
prime radicalC23×H(3,𝔽3)C23
hypercenterC22×C4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)C2×C4⋊Q8
LCSG ▷ C22×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C23 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
6. #11721080C2direct (N × H)
38. C2×(C3⋊Dic3)⋊C3Q8semidirect
72. (C3⋊Dic3)⋊C3C2×Q8semidirect
95. C2×C3⋊Dic3Dic6semidirect
114. C2×H(3,𝔽3)C4⋊Q8semidirect
160. C3⋊Dic3C2×Dic6semidirect
173. H(3,𝔽3)C2×C4⋊Q8semidirect
215. C2×C32#888semidirect
257. C32C2×#888semidirect
292. C2#11721080direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#62292
C22×#61962
C2×#3750⋊C32
C2×#884482
C2×#884472
#117210802
C2×#638393