Finite Group Databasebeta

← #13140994

Subgroups — #13140994

Order 1728.

592 conjugacy classes of subgroups; only the 77 normal subgroups are shown.

normal subgroups only

This group has 592 conjugacy classes of subgroups — too many to draw. Only its 77 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#11721863C211yesmaximal, characteristic
3C2×#3756⋊C3C212yesmaximal, complemented
4#11720323C211yesmaximal, characteristic
5#11720309C212yesmaximal, complemented
6#10591653—32nomaximal
7C2×Q8×H(3,𝔽3)C2211yescharacteristic
8C2×C4×#272C2211yescharacteristic
9C2×#634⋊C3C2212yes—
10C2×#6199—21no—
11C2×#6196—21no—
12#88651—24no—
13#88542—21no—
14C4⋊#1084C2211yescharacteristic
15#88540—21no—
16#3756⋊C3—24no—
17C2×(C3⋊(C3⋊C8))⋊C3C2212yes—
18C2×D3×Dic6—32no—
19C2×#3756D312yescomplemented
20C3×#946—32no—
21C3×#946—61no—
22C2×C3×#216—32no—
23#63567—32no—
24#63565—32no—
25(C2×D3)⋊(C3⋊C8)—32no—
26C22×#272D411yescharacteristic
27Q8×H(3,𝔽3)—22no—
28C2×C4×H(3,𝔽3)—21no—
29C2×C4×H(3,𝔽3)C2311yescharacteristic
30C2×#1084—21no—
31C2×#1084D411yescharacteristic
32C4×#272—22no—
33C2×(C3⋊Dic3)⋊C3—22no—
34#634⋊C3—22no—
35#634⋊C3D414yes—
36#6199—24no—
37#6196—24no—
38(C3⋊(C3⋊C8))⋊C3—22no—
39#5186—92no—
40#5050—91no—
41C2×C32×Q8—21no—
42C2×C32×Q8C2×D312yes—
43C2×#634C2×D312yes—
44C2×C3×C4×D3—32no—
45C2×C3×C4×D3—61no—
46C2×C3×Dic6—32no—
47C2×#627—62no—
48C2×D3×Dic3—62no—
49D3×Dic6—64no—
50D3×Dic6—64no—
51#3756—24no—
52C2×C3⋊(C3⋊C8)C2×D312yes—
53C3×#216—64no—
54C3×(C2×D3)⋊C4—62no—
55C3×(C2×D3)⋊C4—62no—
56C3×C4⋊Dic3—32no—
57C3×C4⋊Dic3—61no—
58C3×Dic3⋊C4—62no—
59C3×Dic3⋊C4—62no—
60C2×C3×C3⋊C8—32no—
61C22×H(3,𝔽3)C2×D411yescharacteristic
62C2×#272—21no—
63C2×#272—21no—
64C2×#272Q1612yescomplemented
65C4×H(3,𝔽3)—41no—
66C4×H(3,𝔽3)C2×D412yes—
67#1084—21no—
68#1084—41no—
69(C3⋊Dic3)⋊C3—24no—
70C2×D3×Q8—92no—
71C22×Dic6—91no—
72C2×C3×Q16—92no—
73C3×C22⋊Q8—91no—
74#946—32no—
75#946—61no—
76C2×#216—32no—
77C2×#216—92no—
78#835—92no—
79#830—91no—
80(C2×D3)⋊C8—92no—
81#816—92no—
82#810—92no—
83C22×C3×D3—32no—
84C22×C3×D3—61no—
85C32×Q8—22no—
86C32×Q8—22no—
87C32×Q8—42no—
88C2×C32×C4—22no—
89C2×C32×C4—22no—
90C2×C32×C4—21no—
91C2×C32×C4C22×D312yes—
92C2×C3⋊Dic3—22no—
93#634—22no—
94#634#6714yes—
95C2×C3×Dic3—32no—
96C2×C3×Dic3—61no—
97C2×C3×Dic3—62no—
98C2×C3×Dic3—62no—
99C2×C3×Dic3—62no—
100C3×C4×D3—62no—
101C3×C4×D3—62no—
102C3×C4×D3—122no—
103C3×Dic6—34no—
104C3×Dic6—62no—
105#627—68no—
106D3×Dic3—68no—
107C3⋊(C3⋊C8)—22no—
108C3×C3⋊C8—62no—
109#451—271noSylow
110C2×H(3,𝔽3)#12011yescharacteristic
111C2×H(3,𝔽3)C2×Q1611yescharacteristic
112C2×H(3,𝔽3)#13711yescharacteristic
113#272—22no—
114C2×C3×Q8—32no—
115C2×C3×Q8—61no—
116C2×C3×Q8—92no—
117C2×C3×Q8D3211yescharacteristic
118C22×C3×C4—91no—
119C22×Dic3—181no—
120D3×Q8—184no—
121D3×Q8—184no—
122C2×C4×D3—32no—
123C2×C4×D3—61no—
124C2×C4×D3—182no—
125C2×Dic6—32no—
126C2×Dic6—92no—
127C2×Dic6—184no—
128C2×Dic6—182no—
129C2×Dic6—181no—
130C3×Q16—184no—
131C2×C3×C8—92no—
132C3×C4⋊C4—91no—
133C3×C4⋊C4—181no—
134C3×C22⋊C4—181no—
135#216—64no—
136#216—184no—
137(C2×D3)⋊C4—62no—
138(C2×D3)⋊C4—62no—
139C4⋊Dic3—32no—
140C4⋊Dic3—61no—
141Dic3⋊C4—62no—
142Dic3⋊C4—62no—
143C2×C3⋊C8—32no—
144C2×C3⋊C8—92no—
145C22×C32—21no—
146C22×C32D3×D412yes—
147C2×C3×D3—34no—
148C2×C3×D3—62no—
149C2×C3×D3—62no—
150C2×C3×D3—62no—
151C2×C3×D3—121no—
152C2×C3×D3—121no—
153C32×C4—22no—
154C32×C4—42no—
155C32×C4—42no—
156C32×C4D3×D412yes—
157C32×C4C2×#6712yes—
158C3⋊Dic3—24no—
159C3×Dic3—64no—
160C3×Dic3—62no—
161C3×Dic3—121no—
162C3×Dic3—122no—
163C3×Dic3—122no—
164C22×Q8—271no—
165C2×Q16—92no—
166C22⋊Q8—91no—
167#103—272no—
168C22⋊C8—271no—
169H(3,𝔽3)#45111yescharacteristic, complemented, Sylow
170C23×C3—91no—
171C22×D3—32no—
172C22×D3—61no—
173C3×Q8—22no—
174C3×Q8—62no—
175C3×Q8—62no—
176C3×Q8—94no—
177C3×Q8—122no—
178C3×Q8—182no—
179C2×C3×C4—21no—
180C2×C3×C4—32no—
181C2×C3×C4—62no—
182C2×C3×C4—62no—
183C2×C3×C4—61no—
184C2×C3×C4—91no—
185C2×C3×C4—182no—
186C2×C3×C4—182no—
187C2×C3×C4—181no—
188C2×C3×C4C2×D3211yescharacteristic
189C2×Dic3—32no—
190C2×Dic3—61no—
191C2×Dic3—62no—
192C2×Dic3—62no—
193C2×Dic3—62no—
194C2×Dic3—182no—
195C2×Dic3—184no—
196C4×D3—62no—
197C4×D3—62no—
198C4×D3—122no—
199C4×D3—188no—
200Dic6—34no—
201Dic6—62no—
202Dic6—94no—
203Dic6—182no—
204Dic6—184no—
205Dic6—188no—
206C3×C8—182no—
207C3⋊C8—62no—
208C3⋊C8—182no—
209C2×C32—21no—
210C2×C32—21no—
211C2×C32—21no—
212C2×C32D3×Q1612yes—
213C2×C32#91812yes—
214C2×C32#93712yes—
215C3×D3—64no—
216C3×D3—122no—
217C2×Q8—92no—
218C2×Q8—541no—
219C2×Q8—544no—
220C2×Q8((C3⋊D3)⋊C3)⋊C211yescharacteristic
221C22×C4—91no—
222C22×C4—541no—
223Q16—184no—
224C2×C8—92no—
225C4⋊C4—91no—
226C4⋊C4—181no—
227C22⋊C4—181no—
228C22×C3—32no—
229C22×C3—61no—
230C22×C3—92no—
231C22×C3—181no—
232C22×C3—181no—
233C22×C3#381111yescharacteristic
234C2×D3—34no—
235C2×D3—62no—
236C2×D3—62no—
237C2×D3—62no—
238C2×D3—121no—
239C2×D3—121no—
240C3×C4—32no—
241C3×C4—32no—
242C3×C4—41no—
243C3×C4—62no—
244C3×C4—122no—
245C3×C4—122no—
246C3×C4—184no—
247C3×C4—181no—
248C3×C4—361no—
249C3×C4D3×#6712yes—
250Dic3—62no—
251Dic3—64no—
252Dic3—121no—
253Dic3—122no—
254Dic3—122no—
255Dic3—184no—
256C32—21no—
257C32#518612yescomplemented
258C23—91no—
259Q8—22no—
260Q8—94no—
261Q8—182no—
262Q8—544no—
263C2×C4—21no—
264C2×C4—91no—
265C2×C4—182no—
266C2×C4—182no—
267C2×C4—181no—
268C2×C4—544no—
269C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
270C8—182no—
271C2×C3—32no—
272C2×C3—32no—
273C2×C3—32no—
274C2×C3—61no—
275C2×C3—61no—
276C2×C3—61no—
277C2×C3—182no—
278C2×C3#6394411yescharacteristic
279C2×C3#6394511yescharacteristic
280C2×C3#6398311yescharacteristic
281D3—64no—
282D3—122no—
283C22—92no—
284C22—181no—
285C22—181no—
286C22#8867711yescharacteristic
287C4—41no—
288C4—181no—
289C4—184no—
290C4—361no—
291C4#8867612yes—
292C3—32no—
293C3—61no—
294C3#1059165411yescharacteristic
295C2—182no—
296C2#1172122611yescharacteristic
297C2#1172122711yescharacteristic
298C2#1172132411yescharacteristic
subgroupG/H
centerC22#88677
commutator, cosocleC2×C4×H(3,𝔽3)C23
FrattiniC2×C3×C4C2×D32
FittingC2×Q8×H(3,𝔽3)C22
socleC22×C3#3811
prime radicalC22×#272D4
hypercenterC2×Q8((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#451
LCSG ▷ C2×C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×Q8

Complements

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

NH ≅ G/Ntype
3. C2×#3756⋊C3C2semidirect
5. #11720309C2semidirect
19. C2×#3756D3semidirect
64. C2×#272Q16semidirect
169. H(3,𝔽3)#451semidirect
257. C32#5186semidirect

Maximal subgroups

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

subgroupindex
C2×#886512
#117218632
C2×#3756⋊C32
#117203232
#117203092
#105916533