Finite Group Databasebeta

← #13138042

Subgroups — #13138042

Order 1728.

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

normal subgroups only

This group has 601 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
2#11721760C211yesmaximal, characteristic
3C2×#88435C211yesmaximal, characteristic
4#63637⋊C3C212yesmaximal, complemented
5#11720309C212yesmaximal, complemented
6#10590600—32nomaximal
7C2×C4×#272C2211yescharacteristic
8C2×#634⋊C3C2212yes—
9C2×#6199—21no—
10C2×#6196—21no—
11#88651—22no—
12#88651C414yescomplemented
13C4⋊C4×H(3,𝔽3)C2211yescharacteristic
14#88542—21no—
15C4⋊#1084C2211yescharacteristic
16C4×#1084—21no—
17C2×(C3⋊(C3⋊C8))⋊C3C2212yes—
18#88435—22no—
19C2×D3×Dic6—32no—
20C3×#892—32no—
21C3×#892—61no—
22C2×#3711—32no—
23#63637D312yescomplemented
24C3×#810—32no—
25#63567—32no—
26#63565—32no—
27C22×#272D411yescharacteristic
28C2×C4×H(3,𝔽3)—21no—
29C2×C4×H(3,𝔽3)C2311yescharacteristic
30C2×#1084—21no—
31C2×#1084D411yescharacteristic
32C4×#272C2×C412yes—
33C4×#272D412yes—
34C2×(C3⋊Dic3)⋊C3—22no—
35#634⋊C3—22no—
36#634⋊C3C2×C414yes—
37#6199—24no—
38#6196—24no—
39(C3⋊(C3⋊C8))⋊C3—22no—
40#5004—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—
50C32×C4⋊C4—21no—
51C32×C4⋊C4C2×D312yes—
52C2×C3⋊(C3⋊C8)C2×D312yes—
53C3×(C2×D3)⋊C4—62no—
54C3×(C2×D3)⋊C4—62no—
55C3×C4⋊Dic3—32no—
56C3×C4⋊Dic3—61no—
57C3×C4×Dic3—62no—
58C3×C4×Dic3—62no—
59C2×C3×C3⋊C8—32no—
60#3711—64no—
61C22×H(3,𝔽3)C2×D411yescharacteristic
62C2×#272—21no—
63C2×#272—21no—
64C2×#272C22⋊C412yes—
65C4×H(3,𝔽3)—41no—
66C4×H(3,𝔽3)C22×C411yescharacteristic
67C4×H(3,𝔽3)C2×D411yescharacteristic
68#1084—41no—
69#1084C22⋊C412yes—
70(C3⋊Dic3)⋊C3—24no—
71C2×D3×Q8—92no—
72C22×Dic6—91no—
73C3×#117—91no—
74C2×#208—91no—
75C2×ΓL(1,𝔽25)—92no—
76#892—32no—
77#892—61no—
78C3×#103—92no—
79#835—92no—
80#816—92no—
81#810—32no—
82#810—92no—
83#810—92no—
84C22×C3×D3—32no—
85C22×C3×D3—61no—
86C2×C32×C4—22no—
87C2×C32×C4—22no—
88C2×C32×C4—21no—
89C2×C32×C4C22×D312yes—
90C2×C3⋊Dic3—22no—
91#634—22no—
92#634C4×D314yes—
93C2×C3×Dic3—32no—
94C2×C3×Dic3—61no—
95C2×C3×Dic3—62no—
96C2×C3×Dic3—62no—
97C2×C3×Dic3—62no—
98C3×C4×D3—34no—
99C3×C4×D3—34no—
100C3×C4×D3—62no—
101C3×C4×D3—62no—
102C3×Dic6—34no—
103C3×Dic6—62no—
104#627—68no—
105D3×Dic3—68no—
106C3⋊(C3⋊C8)—22no—
107C3×C3⋊C8—62no—
108#419—271noSylow
109C2×H(3,𝔽3)C2×C22⋊C411yescharacteristic
110C2×H(3,𝔽3)#13711yescharacteristic
111C2×H(3,𝔽3)#13711yescharacteristic
112#272—22no—
113C2×C3×Q8—92no—
114C22×C3×C4—91no—
115C22×Dic3—181no—
116D3×Q8—98no—
117D3×Q8—184no—
118C2×C4×D3—32no—
119C2×C4×D3—61no—
120C2×C4×D3—182no—
121C2×Dic6—32no—
122C2×Dic6—92no—
123C2×Dic6—94no—
124C2×Dic6—182no—
125C2×Dic6—182no—
126C2×Dic6—181no—
127C2×C3×C8—92no—
128C3×C4⋊C4—32no—
129C3×C4⋊C4—61no—
130C3×C4⋊C4—91no—
131C3×C4⋊C4D3211yescharacteristic
132C3×C22⋊C4—181no—
133C3×C42—181no—
134(C2×D3)⋊C4—62no—
135(C2×D3)⋊C4—62no—
136C4⋊Dic3—32no—
137C4⋊Dic3—61no—
138C4×Dic3—62no—
139C4×Dic3—62no—
140#208—182no—
141C2×C3⋊C8—32no—
142C2×C3⋊C8—92no—
143ΓL(1,𝔽25)—184no—
144C22×C32—21no—
145C22×C32D3×D412yes—
146C2×C3×D3—34no—
147C2×C3×D3—62no—
148C2×C3×D3—62no—
149C2×C3×D3—62no—
150C2×C3×D3—121no—
151C2×C3×D3—121no—
152C32×C4—21no—
153C32×C4—21no—
154C32×C4—42no—
155C32×C4—42no—
156C32×C4C2×C4×D312yes—
157C32×C4D3×D412yes—
158C3⋊Dic3—24no—
159C3×Dic3—34no—
160C3×Dic3—62no—
161C3×Dic3—64no—
162C3×Dic3—122no—
163C3×Dic3—122no—
164C22×Q8—271no—
165C2×M16—271no—
166#117—91no—
167#103—92no—
168#103—272no—
169H(3,𝔽3)#41911yescharacteristic, complemented, Sylow
170C23×C3—91no—
171C22×D3—32no—
172C22×D3—61no—
173C3×Q8—94no—
174C3×Q8—182no—
175C2×C3×C4—21no—
176C2×C3×C4—32no—
177C2×C3×C4—62no—
178C2×C3×C4—62no—
179C2×C3×C4—61no—
180C2×C3×C4—91no—
181C2×C3×C4—92no—
182C2×C3×C4—92no—
183C2×C3×C4—182no—
184C2×C3×C4—181no—
185C2×C3×C4C2×D3211yescharacteristic
186C2×Dic3—32no—
187C2×Dic3—61no—
188C2×Dic3—62no—
189C2×Dic3—62no—
190C2×Dic3—62no—
191C2×Dic3—182no—
192C2×Dic3—184no—
193C4×D3—34no—
194C4×D3—34no—
195C4×D3—62no—
196C4×D3—62no—
197C4×D3—188no—
198Dic6—34no—
199Dic6—62no—
200Dic6—94no—
201Dic6—182no—
202Dic6—188no—
203Dic6—184no—
204C3×C8—182no—
205C3⋊C8—62no—
206C3⋊C8—182no—
207C2×C32—21no—
208C2×C32—21no—
209C2×C32—21no—
210C2×C32D3×C22⋊C412yes—
211C2×C32#91512yes—
212C2×C32#91812yes—
213C3×D3—64no—
214C3×D3—122no—
215C2×Q8—92no—
216C2×Q8—274no—
217C2×Q8—541no—
218C2×Q8—542no—
219C22×C4—91no—
220C22×C4—541no—
221M16—542no—
222C2×C8—92no—
223C4⋊C4—91no—
224C4⋊C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
225C22⋊C4—181no—
226C42—181no—
227C22×C3—32no—
228C22×C3—61no—
229C22×C3—92no—
230C22×C3—181no—
231C22×C3—181no—
232C22×C3#381111yescharacteristic
233C2×D3—34no—
234C2×D3—62no—
235C2×D3—62no—
236C2×D3—62no—
237C2×D3—121no—
238C2×D3—121no—
239C3×C4—32no—
240C3×C4—32no—
241C3×C4—41no—
242C3×C4—61no—
243C3×C4—61no—
244C3×C4—92no—
245C3×C4—122no—
246C3×C4—122no—
247C3×C4—184no—
248C3×C4—361no—
249C3×C4C4×D3211yescharacteristic
250C3×C4#380211yescharacteristic
251Dic3—34no—
252Dic3—62no—
253Dic3—64no—
254Dic3—122no—
255Dic3—122no—
256Dic3—184no—
257C32—21no—
258C32#480212yescomplemented
259C23—91no—
260Q8—94no—
261Q8—182no—
262Q8—544no—
263C2×C4—21no—
264C2×C4—91no—
265C2×C4—92no—
266C2×C4—92no—
267C2×C4—182no—
268C2×C4—181no—
269C2×C4—544no—
270C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
271C8—182no—
272C2×C3—32no—
273C2×C3—32no—
274C2×C3—32no—
275C2×C3—61no—
276C2×C3—61no—
277C2×C3—61no—
278C2×C3—182no—
279C2×C3#6392111yescharacteristic
280C2×C3#6393111yescharacteristic
281C2×C3#6394511yescharacteristic
282D3—64no—
283D3—122no—
284C22—92no—
285C22—181no—
286C22—181no—
287C22#8867711yescharacteristic
288C4—41no—
289C4—92no—
290C4—184no—
291C4—361no—
292C4C4×((C3⋊D3)⋊C3)⋊C211yescharacteristic
293C4#8865511yescharacteristic
294C3—32no—
295C3—61no—
296C3#1059060211yescharacteristic
297C2—182no—
298C2#1172116111yescharacteristic
299C2#1172118911yescharacteristic
300C2#1172122711yescharacteristic
subgroupG/H
centerC22#88677
commutatorC4×H(3,𝔽3)C22×C4
FrattiniC2×C3×C4C2×D32
FittingC4⋊C4×H(3,𝔽3)C22
socleC22×C3#3811
prime radicalC22×#272D4
cosocleC2×C4×H(3,𝔽3)C23
hypercenterC4⋊C4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#419
LCSG ▷ C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C2×C4 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
4. #63637⋊C3C2semidirect
5. #11720309C2semidirect
12. #88651C4semidirect
23. #63637D3semidirect
169. H(3,𝔽3)#419semidirect
258. C32#4802semidirect

Maximal subgroups

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

subgroupindex
C2×#886512
#117217602
C2×#884352
#63637⋊C32
#117203092
#105906003