Finite Group Databasebeta

← #13150611

Subgroups — #13150611

Order 1728.

748 conjugacy classes of subgroups; only the 139 normal subgroups are shown.

normal subgroups only

This group has 748 conjugacy classes of subgroups — too many to draw. Only its 139 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#11721826C211yesmaximal, characteristic
2#64133⋊C3C212yesmaximal
3#11721297C214yesmaximal
4#11721099C211yesmaximal, characteristic, complemented
5#11721095C211yesmaximal, characteristic, complemented
6#11721085C214yesmaximal, complemented
7#11721084C212yesmaximal, complemented
8#10594824—32nomaximal
9C22×C4×H(3,𝔽3)C2211yescharacteristic
10C2×#634⋊C3C2212yes—
11#88552C2212yes—
12C4⋊#1084C2212yes—
13C4×#1084C2211yescharacteristic
14C4×#1084C2211yescharacteristic
15#3757⋊C3C2214yes—
16C4⋊((C3⋊Dic3)⋊C3)C2212yes—
17#3750⋊C3C2214yes—
18#88449C2214yes—
19#88448C2214yes—
20#88448C2214yes—
21#88447C2214yes—
22#64133D312yes—
23C3×#926—32no—
24C3×#926—61no—
25C3×#924—32no—
26#63956—34no—
27#63956—34no—
28#63869—32no—
29#63866—32no—
30#63847—34no—
31#63846—34no—
32#63844—32no—
33Dic3×Dic6—32no—
34C23×H(3,𝔽3)C2311yescharacteristic
35C2×C4×H(3,𝔽3)—22no—
36C2×C4×H(3,𝔽3)C2311yescharacteristic
37C2×C4×H(3,𝔽3)C2311yescharacteristic
38C2×#1084C2312yes—
39C2×#1084C2312yes—
40C2×(C3⋊Dic3)⋊C3C2314yes—
41C2×(C3⋊Dic3)⋊C3C2314yes—
42#634⋊C3—24no—
43#5625—92no—
44#5518—91no—
45C22×C32×C4—21no—
46C22×C32×C4C2×D312yes—
47C2×#634C2×D312yes—
48C2×C3×Dic6—32no—
49#3757C2×D314yes—
50C4⋊(C3⋊Dic3)C2×D312yes—
51#3750C2×D314yes—
52C3×#217—34no—
53C3×#217—32no—
54C3×#217—32no—
55C3×#217—62no—
56C3×C4⋊Dic3—32no—
57C3×C4⋊Dic3—32no—
58C3×C4⋊Dic3—32no—
59C3×C4⋊Dic3—62no—
60C3×Dic3⋊C4—34no—
61C3×C4×Dic3—32no—
62C3×C4×Dic3—32no—
63C3×C4×Dic3—61no—
64C3×C4×Dic3—61no—
65#3724—34no—
66#3724—34no—
67Dic3⋊Dic3—34no—
68Dic3⋊Dic3—34no—
69Dic3⋊Dic3—34no—
70Dic3⋊Dic3—34no—
71Dic32—34no—
72Dic32—34no—
73C22×H(3,𝔽3)—21no—
74C22×H(3,𝔽3)—22no—
75C22×H(3,𝔽3)C2411yescharacteristic
76C4×H(3,𝔽3)—21no—
77C4×H(3,𝔽3)#4112yes—
78#1084—22no—
79#1084—22no—
80(C3⋊Dic3)⋊C3—24no—
81(C3⋊Dic3)⋊C3—24no—
82C3×C22⋊Q8—92no—
83C3×#117—91no—
84Q8×Dic3—92no—
85#944—92no—
86#926—32no—
87#926—61no—
88#924—32no—
89#924—92no—
90#890—92no—
91#890—94no—
92#889—94no—
93#889—94no—
94#887—92no—
95#879—94no—
96#879—94no—
97#879—94no—
98#870—91no—
99#869—91no—
100C4×Dic6—92no—
101C23×C32—21no—
102C23×C32C22×D312yes—
103C2×C32×C4—24no—
104C2×C32×C4—24no—
105C2×C32×C4—21no—
106C2×C32×C4—21no—
107C2×C32×C4C22×D312yes—
108C2×C32×C4C22×D312yes—
109C2×C3⋊Dic3C22×D314yes—
110C2×C3⋊Dic3C22×D314yes—
111#634—24no—
112C2×C3×Dic3—34no—
113C2×C3×Dic3—34no—
114C2×C3×Dic3—32no—
115C2×C3×Dic3—32no—
116C2×C3×Dic3—32no—
117C2×C3×Dic3—32no—
118C2×C3×Dic3—62no—
119C2×C3×Dic3—62no—
120C3×Dic6—64no—
121#541—271noSylow
122C2×H(3,𝔽3)—41no—
123C2×H(3,𝔽3)C2×#4111yescharacteristic
124C2×H(3,𝔽3)2−1+412yes—
125C2×C3×Q8—92no—
126C22×C3×C4—32no—
127C22×C3×C4—61no—
128C22×C3×C4D3211yescharacteristic
129C2×Dic6—32no—
130C2×Dic6—92no—
131C3×C4⋊C4—92no—
132C3×C4⋊C4—94no—
133C3×C4⋊C4—92no—
134C3×C22⋊C4—94no—
135C3×C22⋊C4—92no—
136C3×C42—91no—
137C3×C42—91no—
138#217—32no—
139#217—32no—
140#217—34no—
141#217—62no—
142#217—94no—
143C4⋊Dic3—32no—
144C4⋊Dic3—32no—
145C4⋊Dic3—32no—
146C4⋊Dic3—62no—
147C4⋊Dic3—92no—
148C4⋊Dic3—94no—
149C4⋊Dic3—94no—
150C4⋊Dic3—94no—
151Dic3⋊C4—34no—
152Dic3⋊C4—94no—
153Dic3⋊C4—94no—
154Dic3⋊C4—94no—
155Dic3⋊C4—94no—
156Dic3⋊C4—94no—
157Dic3⋊C4—94no—
158Dic3⋊C4—94no—
159C4×Dic3—32no—
160C4×Dic3—32no—
161C4×Dic3—61no—
162C4×Dic3—61no—
163C4×Dic3—94no—
164C4×Dic3—94no—
165C4×Dic3—94no—
166C22×C32—22no—
167C22×C32—22no—
168C22×C32—22no—
169C22×C32—22no—
170C22×C32—21no—
171C22×C32—42no—
172C22×C32C23×D312yes—
173C32×C4—22no—
174C32×C4—22no—
175C32×C4—22no—
176C32×C4#23714yes—
177C3⋊Dic3—24no—
178C3⋊Dic3—24no—
179C3×Dic3—64no—
180C3×Dic3—64no—
181C3×Dic3—62no—
182C3×Dic3—62no—
183C3×Dic3—62no—
184C3×Dic3—62no—
185C3×Dic3—122no—
186C3×Dic3—122no—
187C4⋊Q8—271no—
188#126—274no—
189#125—271no—
190#125—274no—
191C22⋊Q8—92no—
192C4×Q8—272no—
193#117—91no—
194H(3,𝔽3)#54111yescharacteristic, complemented, Sylow
195C23×C3—32no—
196C23×C3—61no—
197C23×C3C2×D3211yescharacteristic
198C3×Q8—184no—
199C2×C3×C4—22no—
200C2×C3×C4—32no—
201C2×C3×C4—32no—
202C2×C3×C4—64no—
203C2×C3×C4—64no—
204C2×C3×C4—61no—
205C2×C3×C4—61no—
206C2×C3×C4—94no—
207C2×C3×C4—94no—
208C2×C3×C4—92no—
209C2×C3×C4—92no—
210C2×C3×C4C2×D3211yescharacteristic
211C2×C3×C4C2×D3211yescharacteristic
212C2×Dic3—32no—
213C2×Dic3—32no—
214C2×Dic3—32no—
215C2×Dic3—32no—
216C2×Dic3—34no—
217C2×Dic3—34no—
218C2×Dic3—62no—
219C2×Dic3—62no—
220C2×Dic3—94no—
221C2×Dic3—94no—
222Dic6—64no—
223Dic6—184no—
224C2×C32—22no—
225C2×C32—21no—
226C2×C32—42no—
227C2×C32—42no—
228C2×C32C2×#23712yes—
229C2×C32#100712yes—
230C2×C32#101012yes—
231C2×Q8—92no—
232C22×C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
233C4⋊C4—92no—
234C4⋊C4—92no—
235C4⋊C4—94no—
236C4⋊C4—274no—
237C4⋊C4—274no—
238C4⋊C4—274no—
239C22⋊C4—92no—
240C22⋊C4—94no—
241C42—91no—
242C42—91no—
243C42—274no—
244C22×C3—22no—
245C22×C3—21no—
246C22×C3—32no—
247C22×C3—62no—
248C22×C3—62no—
249C22×C3—62no—
250C22×C3—62no—
251C22×C3—61no—
252C22×C3—122no—
253C22×C3C22×D3211yescharacteristic
254C3×C4—21no—
255C3×C4—34no—
256C3×C4—62no—
257C3×C4—62no—
258C3×C4—62no—
259C3×C4—184no—
260C3×C4—184no—
261C3×C4—182no—
262C3×C4—182no—
263C3×C4#380512yes—
264Dic3—62no—
265Dic3—62no—
266Dic3—62no—
267Dic3—62no—
268Dic3—64no—
269Dic3—64no—
270Dic3—122no—
271Dic3—122no—
272Dic3—184no—
273Dic3—184no—
274C32—21no—
275C32#562512yescomplemented
276C23C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
277Q8—184no—
278C2×C4—22no—
279C2×C4—92no—
280C2×C4—92no—
281C2×C4—94no—
282C2×C4—94no—
283C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
284C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
285C2×C3—32no—
286C2×C3—32no—
287C2×C3—32no—
288C2×C3—41no—
289C2×C3—61no—
290C2×C3—62no—
291C2×C3—122no—
292C2×C3—122no—
293C2×C3#6429712yes—
294C2×C3C2×#380511yescharacteristic
295C22—21no—
296C22—22no—
297C22C22×((C3⋊D3)⋊C3)⋊C211yescharacteristic
298C4—21no—
299C4—184no—
300C4—182no—
301C4—182no—
302C4—184no—
303C4#8867212yes—
304C3—32no—
305C3—61no—
306C3#1059482611yescharacteristic
307C2—41no—
308C2#1172248212yes—
309C2C2×#8867211yescharacteristic
subgroupG/H
centerC22C22×((C3⋊D3)⋊C3)⋊C2
commutator, cosocleC22×H(3,𝔽3)C24
Frattini, socleC22×C3C22×D32
FittingC22×C4×H(3,𝔽3)C22
prime radicalC23×H(3,𝔽3)C23
hypercenterC22×C4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#541
LCSG ▷ C22×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
4. #11721099C2semidirect
5. #11721095C2semidirect
6. #11721085C2semidirect
7. #11721084C2semidirect
194. H(3,𝔽3)#541semidirect
275. C32#5625semidirect

Maximal subgroups

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

subgroupindex
#117218262
#64133⋊C32
#117212972
#117210992
#117210952
#117210852
#117210842
#105948243