Finite Group Databasebeta

← #13151321

Subgroups — #13151321

Order 1728.

934 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 934 conjugacy classes of subgroups — too many to draw. Only its 133 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#11721850C211yesmaximal, characteristic
2#64144⋊C3C212yesmaximal, complemented
3#11721294C214yesmaximal, complemented
4#11721293C214yesmaximal, complemented
5#11721151C212yesmaximal, complemented
6#11721078C212yesmaximal, complemented
7#10595200—32nomaximal
8C2×D4×H(3,𝔽3)C2211yescharacteristic
9C2×#6235C2214yes—
10C2×C4⋊#272C2211yescharacteristic
11C22×(C3⋊Dic3)⋊C3C2212yes—
12C2×#6199C2212yes—
13C2×#6196C2212yes—
14C4×#1084C2211yescharacteristic
15#3757⋊C3C2214yes—
16#3757⋊C3C2212yes—
17#3750⋊C3C2214yes—
18#88451C2214yes—
19#88451C2212yes—
20#88449C2212yes—
21#88447C2214yes—
22#64144D312yescomplemented
23C3×#940—32no—
24C3×#940—61no—
25C3×#935—32no—
26#67⋊Dic3—34no—
27Dic3×#67—34no—
28#63959—34no—
29#63959—34no—
30#63884—34no—
31#63837—34no—
32C23×H(3,𝔽3)C2312yes—
33C22×#272C2312yes—
34D4×H(3,𝔽3)—41no—
35C2×C4×H(3,𝔽3)C2311yescharacteristic
36#6235—28no—
37C2×#1084C2312yes—
38C4⋊#272—41no—
39C2×(C3⋊Dic3)⋊C3—42no—
40C2×(C3⋊Dic3)⋊C3C2314yes—
41C2×(C3⋊Dic3)⋊C3C2314yes—
42#6199—42no—
43#6196—24no—
44#5713—91no—
45#5664—92no—
46C2×C32×D4—21no—
47C2×C32×D4C2×D312yes—
48C22×C3⋊Dic3C2×D312yes—
49C2×C3×#67—34no—
50C2×C3×#67—34no—
51C2×C3×#67—64no—
52C22×C3×Dic3—32no—
53C2×C3×D12—32no—
54C2×C3×D12—61no—
55C2×#627—34no—
56C2×D3×Dic3—34no—
57#3757C2×D312yes—
58#3757C2×D314yes—
59#3750C2×D314yes—
60C3×#217—32no—
61C3×#217—34no—
62C3×Dic3⋊C4—34no—
63C3×C4×Dic3—32no—
64C3×C4×Dic3—61no—
65Dic3⋊Dic3—34no—
66(C2×D3)⋊Dic3—34no—
67(C2×D3)⋊Dic3—34no—
68(C2×D3)⋊Dic3—34no—
69Dic32—34no—
70Dic32—34no—
71C22×H(3,𝔽3)—24no—
72C22×H(3,𝔽3)—22no—
73C22×H(3,𝔽3)C2411yescharacteristic
74C2×#272—24no—
75C2×#272—22no—
76C4×H(3,𝔽3)—21no—
77#1084#4114yes—
78(C3⋊Dic3)⋊C3—24no—
79(C3⋊Dic3)⋊C3—24no—
80C3×#127—91no—
81C3×#123—92no—
82#940—32no—
83#940—61no—
84#936—94no—
85#935—32no—
86#935—92no—
87#935—92no—
88D4×Dic3—94no—
89C4×#67—94no—
90#894—94no—
91#887—94no—
92#885—94no—
93#885—94no—
94#67⋊C4—94no—
95C4×Dic6—92no—
96C23×C32—22no—
97C23×C32C22×D312yes—
98C23×C32C22×D312yes—
99C22×C3×D3—34no—
100C22×C3×D3—62no—
101C32×D4—42no—
102C32×D4—42no—
103C2×C32×C4—21no—
104C2×C32×C4C22×D312yes—
105C2×C3⋊Dic3—42no—
106C2×C3⋊Dic3C22×D314yes—
107C2×C3⋊Dic3C22×D314yes—
108C3×#67—68no—
109C3×#67—68no—
110C3×#67—128no—
111C2×C3×Dic3—34no—
112C2×C3×Dic3—34no—
113C2×C3×Dic3—34no—
114C2×C3×Dic3—62no—
115C2×C3×Dic3—122no—
116C3×D12—122no—
117C3×D12—122no—
118#627—68no—
119D3×Dic3—124no—
120#559—271noSylow
121C2×H(3,𝔽3)—42no—
122C2×H(3,𝔽3)C2×#4112yes—
123C2×H(3,𝔽3)H(5,𝔽2)11yescharacteristic
124#272—42no—
125C2×C3×D4—32no—
126C2×C3×D4—61no—
127C2×C3×D4—94no—
128C2×C3×D4—91no—
129C2×C3×D4D3211yescharacteristic
130C22×C3×C4—92no—
131C2×#67—34no—
132C2×#67—34no—
133C2×#67—64no—
134C22×Dic3—32no—
135C22×Dic3—92no—
136C22×Dic3—92no—
137C2×D12—32no—
138C2×D12—61no—
139C2×C4×D3—94no—
140C2×Dic6—92no—
141C2×Dic6—94no—
142C3×C4⋊C4—94no—
143C3×C22⋊C4—94no—
144C3×C22⋊C4—92no—
145C3×C42—91no—
146#217—34no—
147#217—32no—
148#217—92no—
149#217—94no—
150#217—94no—
151#217—92no—
152(C2×D3)⋊C4—94no—
153(C2×D3)⋊C4—94no—
154(C2×D3)⋊C4—94no—
155C4⋊Dic3—92no—
156Dic3⋊C4—34no—
157Dic3⋊C4—94no—
158Dic3⋊C4—94no—
159C4×Dic3—32no—
160C4×Dic3—61no—
161C4×Dic3—94no—
162C4×Dic3—94no—
163C4×Dic3—94no—
164C22×C32—22no—
165C22×C32—22no—
166C22×C32—24no—
167C22×C32—24no—
168C22×C32—21no—
169C22×C32—44no—
170C22×C32—42no—
171C22×C32C23×D312yes—
172C2×C3×D3—64no—
173C2×C3×D3—64no—
174C2×C3×D3—64no—
175C2×C3×D3—122no—
176C2×C3×D3—124no—
177C32×C4—22no—
178C32×C4—22no—
179C3⋊Dic3—24no—
180C3⋊Dic3—24no—
181C3×Dic3—38no—
182C3×Dic3—64no—
183C3×Dic3—64no—
184C3×Dic3—64no—
185#127—91no—
186#124—274no—
187#123—92no—
188#123—272no—
189C4×Q8—272no—
190C4×D4—274no—
191H(3,𝔽3)#55911yescharacteristic, complemented, Sylow
192C23×C3—32no—
193C23×C3—32no—
194C23×C3—62no—
195C23×C3—92no—
196C23×C3C2×D3212yes—
197C22×D3—34no—
198C22×D3—62no—
199C3×D4—41no—
200C3×D4—122no—
201C3×D4—122no—
202C3×D4—188no—
203C3×D4—361no—
204C2×C3×C4—32no—
205C2×C3×C4—61no—
206C2×C3×C4—92no—
207C2×C3×C4—94no—
208C2×C3×C4—94no—
209C2×C3×C4—362no—
210C2×C3×C4C2×D3211yescharacteristic
211#67—68no—
212#67—68no—
213#67—128no—
214C2×Dic3—34no—
215C2×Dic3—34no—
216C2×Dic3—34no—
217C2×Dic3—62no—
218C2×Dic3—94no—
219C2×Dic3—94no—
220C2×Dic3—122no—
221C2×Dic3—362no—
222C2×Dic3—362no—
223D12—122no—
224D12—122no—
225C4×D3—364no—
226Dic6—188no—
227Dic6—184no—
228C2×C32—22no—
229C2×C32—21no—
230C2×C32—42no—
231C2×C32—42no—
232C2×C32—82no—
233C2×C32#100412yes—
234C2×C32D3×#4114yes—
235C3×D3—124no—
236C3×D3—242no—
237C2×Q8—272no—
238C2×D4—94no—
239C2×D4—91no—
240C2×D4((C3⋊D3)⋊C3)⋊C211yescharacteristic
241C22×C4—92no—
242C22×C4—272no—
243C4⋊C4—94no—
244C4⋊C4—272no—
245C22⋊C4—94no—
246C22⋊C4—92no—
247C22⋊C4—274no—
248C22⋊C4—272no—
249C42—91no—
250C42—274no—
251C22×C3—24no—
252C22×C3—22no—
253C22×C3—32no—
254C22×C3—62no—
255C22×C3—64no—
256C22×C3—64no—
257C22×C3—62no—
258C22×C3—61no—
259C22×C3—124no—
260C22×C3—122no—
261C22×C3—184no—
262C22×C3—182no—
263C22×C3C22×D3211yescharacteristic
264C2×D3—64no—
265C2×D3—64no—
266C2×D3—64no—
267C2×D3—122no—
268C2×D3—124no—
269C3×C4—21no—
270C3×C4—62no—
271C3×C4—62no—
272C3×C4—94no—
273C3×C4—184no—
274C3×C4—184no—
275Dic3—38no—
276Dic3—64no—
277Dic3—64no—
278Dic3—64no—
279Dic3—184no—
280Dic3—184no—
281C32—21no—
282C32#566412yescomplemented
283C23—92no—
284C23C2×((C3⋊D3)⋊C3)⋊C212yes—
285Q8—544no—
286D4—41no—
287D4—188no—
288D4—361no—
289C2×C4—92no—
290C2×C4—94no—
291C2×C4—94no—
292C2×C4—362no—
293C2×C4—1082no—
294C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
295C2×C3—34no—
296C2×C3—32no—
297C2×C3—42no—
298C2×C3—62no—
299C2×C3—61no—
300C2×C3—122no—
301C2×C3—122no—
302C2×C3—242no—
303C2×C3—362no—
304C2×C3#6431212yes—
305C2×C3#6433011yescharacteristic
306D3—124no—
307D3—242no—
308C22—24no—
309C22—22no—
310C22—184no—
311C22—182no—
312C22C22×((C3⋊D3)⋊C3)⋊C211yescharacteristic
313C4—21no—
314C4—94no—
315C4—184no—
316C4—184no—
317C3—32no—
318C3—61no—
319C3#1059521211yescharacteristic
320C2—42no—
321C2—362no—
322C2#1172250512yes—
323C2#1172254511yescharacteristic
subgroupG/H
centerC22C22×((C3⋊D3)⋊C3)⋊C2
commutator, cosocleC22×H(3,𝔽3)C24
Frattini, socleC22×C3C22×D32
FittingC2×D4×H(3,𝔽3)C22
prime radical#11721850C2
hypercenterC2×D4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#559
LCSG ▷ C22×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
2. #64144⋊C3C2semidirect
3. #11721294C2semidirect
4. #11721293C2semidirect
5. #11721151C2semidirect
6. #11721078C2semidirect
22. #64144D3semidirect
191. H(3,𝔽3)#559semidirect
282. C32#5664semidirect

Maximal subgroups

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

subgroupindex
#117218502
#64144⋊C32
#117212942
#117212932
#117211512
#117210782
#105952003