Finite Group Databasebeta

← #13139024

Subgroups — #13139024

Order 1728.

593 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 593 conjugacy classes of subgroups — too many to draw. Only its 87 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
1C2×#88651C211yesmaximal, characteristic
2C2×#88527C211yesmaximal, characteristic
3#11721089C211yesmaximal, characteristic
4#63643⋊C3C212yesmaximal, complemented
5#11720313C212yesmaximal, complemented
6#10590948—32nomaximal
7C2×C4×#272C2211yescharacteristic
8C2×#634⋊C3C2212yes—
9C2×#6199—21no—
10C2×#6196—21no—
11#88651—22no—
12#88651C414yescomplemented
13C2×C8×H(3,𝔽3)C2211yescharacteristic
14C2×#6180C2211yescharacteristic
15#88527—22no—
16C4⋊((C3⋊Dic3)⋊C3)C2212yes—
17#88451—21no—
18#88447—21no—
19C2×D3×Dic6—32no—
20C2×C3×ΓL(1,𝔽25)—32no—
21C2×C3×ΓL(1,𝔽25)—61no—
22#63841—32no—
23#63643D312yescomplemented
24C3×#816—32no—
25#63567—32no—
26#63566—32no—
27C22×#272D411yescharacteristic
28C2×C4×H(3,𝔽3)C2311yescharacteristic
29C2×#1084D411yescharacteristic
30C4×#272C2×C412yes—
31C4×#272D412yes—
32C2×(C3⋊Dic3)⋊C3—22no—
33C2×(C3⋊Dic3)⋊C3—22no—
34#634⋊C3—22no—
35#634⋊C3C2×C414yes—
36#6199—24no—
37#6196—24no—
38C8×H(3,𝔽3)—21no—
39#6180—21no—
40#5120—91no—
41#4802—92no—
42C2×#634C2×D312yes—
43C2×C3×C4×D3—32no—
44C2×C3×C4×D3—61no—
45C2×C3×Dic6—32no—
46C2×#627—62no—
47C2×D3×Dic3—62no—
48D3×Dic6—38no—
49D3×Dic6—64no—
50C2×C32×C8—21no—
51C2×C32×C8C2×D312yes—
52C4⋊(C3⋊Dic3)C2×D312yes—
53C3×C4⋊Dic3—32no—
54C2×C3×C3⋊C8—32no—
55C2×C3×C3⋊C8—61no—
56C3×ΓL(1,𝔽25)—64no—
57C3×ΓL(1,𝔽25)—64no—
58(C2×D3)⋊Dic3—62no—
59Dic32—62no—
60C22×H(3,𝔽3)C2×D411yescharacteristic
61C2×#272—22no—
62C2×#272C22⋊C412yes—
63C4×H(3,𝔽3)C22×C411yescharacteristic
64C4×H(3,𝔽3)C2×D411yescharacteristic
65#1084C22⋊C412yes—
66(C3⋊Dic3)⋊C3—24no—
67(C3⋊Dic3)⋊C3—42no—
68C2×D3×Q8—92no—
69C22×Dic6—91no—
70C2×C3×M16—91no—
71#926—91no—
72C2×ΓL(1,𝔽25)—32no—
73C2×ΓL(1,𝔽25)—61no—
74#892—92no—
75C3×#103—92no—
76#835—92no—
77#816—32no—
78#816—92no—
79#816—92no—
80#810—92no—
81C22×C3×D3—32no—
82C22×C3×D3—61no—
83C2×C32×C4—21no—
84C2×C32×C4C22×D312yes—
85C2×C3⋊Dic3—22no—
86C2×C3⋊Dic3—22no—
87#634—22no—
88#634C4×D314yes—
89C2×C3×Dic3—32no—
90C2×C3×Dic3—61no—
91C2×C3×Dic3—62no—
92C2×C3×Dic3—62no—
93C3×C4×D3—34no—
94C3×C4×D3—34no—
95C3×C4×D3—62no—
96C3×C4×D3—62no—
97C3×Dic6—34no—
98C3×Dic6—62no—
99#627—68no—
100D3×Dic3—68no—
101C32×C8—22no—
102C32×C8—22no—
103C3×C3⋊C8—62no—
104C3×C3⋊C8—62no—
105#419—271noSylow
106C2×H(3,𝔽3)C2×C22⋊C411yescharacteristic
107C2×H(3,𝔽3)#13712yes—
108#272—22no—
109C2×C3×Q8—92no—
110C22×C3×C4—91no—
111C22×Dic3—181no—
112D3×Q8—98no—
113D3×Q8—184no—
114C2×C4×D3—32no—
115C2×C4×D3—61no—
116C2×C4×D3—182no—
117C2×Dic6—32no—
118C2×Dic6—92no—
119C2×Dic6—94no—
120C2×Dic6—182no—
121C2×Dic6—182no—
122C2×Dic6—181no—
123C3×M16—182no—
124C2×C3×C8—32no—
125C2×C3×C8—61no—
126C2×C3×C8—91no—
127C2×C3×C8D3211yescharacteristic
128C3×C4⋊C4—92no—
129#217—181no—
130(C2×D3)⋊C4—182no—
131C4⋊Dic3—32no—
132C4⋊Dic3—92no—
133C4×Dic3—182no—
134C4×Dic3—181no—
135C2×C3⋊C8—32no—
136C2×C3⋊C8—61no—
137ΓL(1,𝔽25)—64no—
138ΓL(1,𝔽25)—64no—
139C22×C32—21no—
140C22×C32D3×D412yes—
141C2×C3×D3—34no—
142C2×C3×D3—62no—
143C2×C3×D3—62no—
144C2×C3×D3—62no—
145C2×C3×D3—122no—
146C32×C4—21no—
147C32×C4—21no—
148C32×C4C2×C4×D312yes—
149C32×C4D3×D412yes—
150C3⋊Dic3—24no—
151C3⋊Dic3—42no—
152C3×Dic3—34no—
153C3×Dic3—62no—
154C3×Dic3—64no—
155C3×Dic3—122no—
156C22×Q8—271no—
157C2×M16—91no—
158#117—271no—
159#103—92no—
160#103—272no—
161H(3,𝔽3)#41911yescharacteristic, complemented, Sylow
162C23×C3—91no—
163C22×D3—32no—
164C22×D3—61no—
165C3×Q8—94no—
166C3×Q8—182no—
167C2×C3×C4—32no—
168C2×C3×C4—61no—
169C2×C3×C4—91no—
170C2×C3×C4—92no—
171C2×C3×C4—92no—
172C2×C3×C4—182no—
173C2×C3×C4—182no—
174C2×C3×C4C2×D3211yescharacteristic
175C2×Dic3—32no—
176C2×Dic3—61no—
177C2×Dic3—62no—
178C2×Dic3—62no—
179C2×Dic3—182no—
180C2×Dic3—182no—
181C2×Dic3—184no—
182C4×D3—34no—
183C4×D3—34no—
184C4×D3—62no—
185C4×D3—62no—
186C4×D3—188no—
187Dic6—34no—
188Dic6—62no—
189Dic6—94no—
190Dic6—182no—
191Dic6—188no—
192Dic6—184no—
193C3×C8—21no—
194C3×C8—62no—
195C3×C8—62no—
196C3×C8—181no—
197C3⋊C8—62no—
198C3⋊C8—62no—
199C2×C32—22no—
200C2×C32—21no—
201C2×C32D3×C22⋊C412yes—
202C2×C32#91512yes—
203C2×C32#91812yes—
204C3×D3—64no—
205C3×D3—122no—
206C2×Q8—92no—
207C2×Q8—274no—
208C2×Q8—541no—
209C2×Q8—542no—
210C22×C4—91no—
211C22×C4—541no—
212M16—182no—
213C2×C8—91no—
214C2×C8((C3⋊D3)⋊C3)⋊C211yescharacteristic
215C4⋊C4—92no—
216C22⋊C4—541no—
217C42—541no—
218C22×C3—32no—
219C22×C3—61no—
220C22×C3—92no—
221C22×C3—182no—
222C22×C3#380211yescharacteristic
223C2×D3—34no—
224C2×D3—62no—
225C2×D3—62no—
226C2×D3—62no—
227C2×D3—122no—
228C3×C4—32no—
229C3×C4—32no—
230C3×C4—61no—
231C3×C4—61no—
232C3×C4—92no—
233C3×C4—184no—
234C3×C4—362no—
235C3×C4C4×D3211yescharacteristic
236C3×C4#381111yescharacteristic
237Dic3—34no—
238Dic3—62no—
239Dic3—64no—
240Dic3—122no—
241Dic3—184no—
242Dic3—362no—
243C32—21no—
244C32#480212yescomplemented
245C23—91no—
246Q8—94no—
247Q8—182no—
248Q8—544no—
249C2×C4—91no—
250C2×C4—92no—
251C2×C4—92no—
252C2×C4—182no—
253C2×C4—182no—
254C2×C4—544no—
255C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
256C8—21no—
257C8—181no—
258C2×C3—32no—
259C2×C3—32no—
260C2×C3—32no—
261C2×C3—62no—
262C2×C3—61no—
263C2×C3—182no—
264C2×C3#6381112yes—
265C2×C3#6392111yescharacteristic
266D3—64no—
267D3—122no—
268C22—92no—
269C22—182no—
270C22#8865511yescharacteristic
271C4—92no—
272C4—184no—
273C4—362no—
274C4C4×((C3⋊D3)⋊C3)⋊C211yescharacteristic
275C4#8867711yescharacteristic
276C3—32no—
277C3—61no—
278C3#1059069111yescharacteristic
279C2—182no—
280C2#1172084512yes—
281C2#1172116111yescharacteristic
subgroupG/H
centerC22#88655
commutatorC4×H(3,𝔽3)C22×C4
FrattiniC2×C3×C4C2×D32
FittingC2×C8×H(3,𝔽3)C22
socleC22×C3#3802
prime radicalC22×#272D4
cosocleC2×C4×H(3,𝔽3)C23
hypercenterC2×C8((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#419
LCSG ▷ C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×C8

Complements

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

NH ≅ G/Ntype
4. #63643⋊C3C2semidirect
5. #11720313C2semidirect
12. #88651C4semidirect
23. #63643D3semidirect
161. H(3,𝔽3)#419semidirect
244. C32#4802semidirect

Maximal subgroups

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

subgroupindex
C2×#886512
C2×#885272
#117210892
#63643⋊C32
#117203132
#105909483