Finite Group Databasebeta

← #13137873

Subgroups — #13137873

Order 1728.

514 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 514 conjugacy classes of subgroups — too many to draw. Only its 71 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
1#11721761C211yesmaximal, characteristic
2#11721099C211yesmaximal, characteristic, complemented
3C2×#88437C212yesmaximal
4#63637⋊C3C212yesmaximal, complemented
5#11720304C211yesmaximal, characteristic, complemented
6#10590559—32nomaximal
7C2×C4⋊#272C2211yescharacteristic
8C2×C4×#272—21no—
9C2×#634⋊C3C2212yes—
10C4⋊C4×H(3,𝔽3)C2211yescharacteristic
11#88542—21no—
12C4×#1084C2211yescharacteristic
13C2×(C3⋊(C3⋊C8))⋊C3C2212yes—
14#88449—21no—
15#88449—21no—
16#88437—24no—
17C3×#893—32no—
18C3×#893—61no—
19#63866—32no—
20C2×#3716—32no—
21C2×#3715—32no—
22#63637D312yescomplemented
23C3×#810—32no—
24Dic3⋊(C3⋊C8)—32no—
25C22×#272—21no—
26C2×C4×H(3,𝔽3)—21no—
27C2×C4×H(3,𝔽3)C2311yescharacteristic
28C2×#1084D411yescharacteristic
29C2×#1084D411yescharacteristic
30C4⋊#272—21no—
31C4⋊#272D412yes—
32C4×#272—42no—
33C2×(C3⋊Dic3)⋊C3—22no—
34#634⋊C3—22no—
35#634⋊C3—22no—
36(C3⋊(C3⋊C8))⋊C3—22no—
37#5019—91no—
38#4818—92no—
39C2×#634C2×D312yes—
40C2×C3×D12—32no—
41C2×C3×D12—61no—
42C2×C3×C4×D3—62no—
43C2×C3×C4×D3—62no—
44C2×C3×Dic6—32no—
45C32×C4⋊C4—21no—
46C32×C4⋊C4C2×D312yes—
47C2×C3⋊(C3⋊C8)C2×D312yes—
48C3×(C2×D3)⋊C4—62no—
49C3×(C2×D3)⋊C4—62no—
50C3×C4×Dic3—32no—
51C3×C4×Dic3—61no—
52C2×C3×C3⋊C8—32no—
53Dic3⋊Dic3—62no—
54Dic3⋊Dic3—62no—
55#3716—64no—
56#3715—64no—
57C22×H(3,𝔽3)C2×D411yescharacteristic
58C2×#272—22no—
59C2×#272—41no—
60C2×#272—41no—
61C4×H(3,𝔽3)—41no—
62C4×H(3,𝔽3)C2×D411yescharacteristic
63C4×H(3,𝔽3)#4111yescharacteristic
64#1084—21no—
65#1084SD1612yescomplemented
66(C3⋊Dic3)⋊C3—42no—
67C3×C4×D4—91no—
68#944—92no—
69C2×#215—92no—
70C2×#214—92no—
71C2×#204—92no—
72#893—32no—
73#893—61no—
74#869—91no—
75C3×#103—92no—
76Dic3⋊C8—92no—
77#810—32no—
78#810—92no—
79C4⋊(C3⋊C8)—91no—
80C22×C3×D3—62no—
81C22×C3×D3—62no—
82C2×C32×C4—22no—
83C2×C32×C4—22no—
84C2×C32×C4—21no—
85C2×C32×C4C22×D312yes—
86C2×C3⋊Dic3—22no—
87#634—22no—
88#634—22no—
89C2×C3×Dic3—32no—
90C2×C3×Dic3—32no—
91C2×C3×Dic3—61no—
92C2×C3×Dic3—61no—
93C2×C3×Dic3—62no—
94C3×D12—34no—
95C3×D12—62no—
96C3×D12—62no—
97C3×D12—62no—
98C3×C4×D3—124no—
99C3×C4×D3—124no—
100C3×Dic6—62no—
101C3×Dic6—62no—
102C3⋊(C3⋊C8)—22no—
103C3×C3⋊C8—62no—
104#461—271noSylow
105C2×H(3,𝔽3)#12111yescharacteristic
106C2×H(3,𝔽3)C2×SD1611yescharacteristic
107C2×H(3,𝔽3)#13711yescharacteristic
108#272—42no—
109C2×C3×Q8—92no—
110C2×C3×D4—91no—
111C22×C3×C4—181no—
112C2×D12—32no—
113C2×D12—61no—
114C2×C4×D3—62no—
115C2×C4×D3—62no—
116C2×Dic6—32no—
117C2×Dic6—92no—
118C2×C3×C8—92no—
119C3×C4⋊C4—32no—
120C3×C4⋊C4—61no—
121C3×C4⋊C4D3211yescharacteristic
122C3×C22⋊C4—181no—
123C3×C42—91no—
124#215—184no—
125#214—184no—
126(C2×D3)⋊C4—62no—
127(C2×D3)⋊C4—62no—
128C4⋊Dic3—181no—
129C4⋊Dic3—181no—
130Dic3⋊C4—182no—
131Dic3⋊C4—182no—
132C4×Dic3—32no—
133C4×Dic3—61no—
134C2×C3⋊C8—32no—
135C2×C3⋊C8—92no—
136#204—184no—
137C22×C32—21no—
138C22×C32D3×D412yes—
139C2×C3×D3—64no—
140C2×C3×D3—64no—
141C2×C3×D3—122no—
142C2×C3×D3—122no—
143C2×C3×D3—122no—
144C2×C3×D3—122no—
145C32×C4—21no—
146C32×C4—21no—
147C32×C4—42no—
148C32×C4—42no—
149C32×C4#23512yes—
150C32×C4D3×D412yes—
151C3⋊Dic3—42no—
152C3×Dic3—34no—
153C3×Dic3—62no—
154C3×Dic3—62no—
155C3×Dic3—121no—
156C3×Dic3—122no—
157C2×SD16—272no—
158C4⋊Q8—271no—
159C4×D4—91no—
160C4⋊C8—271no—
161#103—92no—
162H(3,𝔽3)#46111yescharacteristic, complemented, Sylow
163C23×C3—181no—
164C22×D3—62no—
165C22×D3—62no—
166C3×Q8—182no—
167C3×Q8—182no—
168C3×D4—92no—
169C3×D4—181no—
170C2×C3×C4—21no—
171C2×C3×C4—32no—
172C2×C3×C4—62no—
173C2×C3×C4—62no—
174C2×C3×C4—61no—
175C2×C3×C4—91no—
176C2×C3×C4—91no—
177C2×C3×C4—182no—
178C2×C3×C4—362no—
179C2×C3×C4C2×D3211yescharacteristic
180C2×Dic3—32no—
181C2×Dic3—32no—
182C2×Dic3—61no—
183C2×Dic3—61no—
184C2×Dic3—62no—
185C2×Dic3—182no—
186D12—34no—
187D12—62no—
188D12—62no—
189D12—62no—
190C4×D3—124no—
191C4×D3—124no—
192Dic6—62no—
193Dic6—62no—
194Dic6—182no—
195Dic6—182no—
196C3×C8—182no—
197C3⋊C8—62no—
198C3⋊C8—182no—
199C2×C32—21no—
200C2×C32—21no—
201C2×C32—21no—
202C2×C32#88412yes—
203C2×C32D3×SD1612yes—
204C2×C32#91812yes—
205C3×D3—124no—
206C3×D3—124no—
207C2×Q8—92no—
208C2×D4—91no—
209C22×C4—181no—
210SD16—544no—
211C2×C8—92no—
212C4⋊C4—541no—
213C4⋊C4—541no—
214C4⋊C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
215C22⋊C4—181no—
216C42—91no—
217C22×C3—32no—
218C22×C3—61no—
219C22×C3—182no—
220C22×C3—361no—
221C22×C3—361no—
222C22×C3#381111yescharacteristic
223C2×D3—64no—
224C2×D3—64no—
225C2×D3—122no—
226C2×D3—122no—
227C2×D3—122no—
228C2×D3—122no—
229C3×C4—32no—
230C3×C4—32no—
231C3×C4—41no—
232C3×C4—61no—
233C3×C4—61no—
234C3×C4—92no—
235C3×C4—122no—
236C3×C4—122no—
237C3×C4—181no—
238C3×C4—362no—
239C3×C4#379811yescharacteristic
240C3×C4#380211yescharacteristic
241Dic3—34no—
242Dic3—62no—
243Dic3—62no—
244Dic3—121no—
245Dic3—122no—
246Dic3—362no—
247C32—21no—
248C32#481812yescomplemented
249C23—181no—
250Q8—182no—
251Q8—182no—
252D4—92no—
253D4—181no—
254C2×C4—21no—
255C2×C4—91no—
256C2×C4—91no—
257C2×C4—182no—
258C2×C4—362no—
259C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
260C8—182no—
261C2×C3—32no—
262C2×C3—32no—
263C2×C3—32no—
264C2×C3—61no—
265C2×C3—61no—
266C2×C3—61no—
267C2×C3—362no—
268C2×C3#6391211yescharacteristic
269C2×C3#6393011yescharacteristic
270C2×C3#6394511yescharacteristic
271D3—124no—
272D3—124no—
273C22—182no—
274C22—361no—
275C22—361no—
276C22#8867711yescharacteristic
277C4—41no—
278C4—92no—
279C4—181no—
280C4—362no—
281C4#8865011yescharacteristic
282C4#8865511yescharacteristic
283C3—32no—
284C3—61no—
285C3#1059056511yescharacteristic
286C2—362no—
287C2#1172114211yescharacteristic
288C2#1172118811yescharacteristic
289C2#1172122711yescharacteristic
subgroupG/H
centerC22#88677
commutator, cosocleC2×C4×H(3,𝔽3)C23
FrattiniC2×C3×C4C2×D32
FittingC4⋊C4×H(3,𝔽3)C22
socleC22×C3#3811
prime radicalC2×C4⋊#272C22
hypercenterC4⋊C4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#461
LCSG ▷ C2×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
2. #11721099C2semidirect
4. #63637⋊C3C2semidirect
5. #11720304C2semidirect
22. #63637D3semidirect
65. #1084SD16semidirect
162. H(3,𝔽3)#461semidirect
248. C32#4818semidirect

Maximal subgroups

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

subgroupindex
#117217612
#117210992
C2×#884372
#63637⋊C32
#117203042
#105905593