Finite Group Databasebeta

← #13147759

Subgroups — #13147759

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#11721718C211yesmaximal, characteristic
2C4×#634⋊C3C212yesmaximal, complemented
3#11721099C211yesmaximal, characteristic, complemented
4#11721095C211yesmaximal, characteristic, complemented
5#11721094C212yesmaximal
6#11721092C214yesmaximal
7#11721091C214yesmaximal, complemented
8#10593880—32nomaximal
9C2×C4×#272C2211yescharacteristic
10C2×#634⋊C3C2212yes—
11C42×H(3,𝔽3)C2211yescharacteristic
12#88542C2212yes—
13#88540C2212yes—
14C4×#1084C2211yescharacteristic
15C4⋊((C3⋊Dic3)⋊C3)C2212yes—
16#3750⋊C3C2214yes—
17C4×(C3⋊Dic3)⋊C3C2214yes—
18#88451C2214yes—
19#88449C2214yes—
20#88448C2214yes—
21#88448C2214yes—
22C4×#634D312yescomplemented
23C3×#872—32no—
24C3×#872—61no—
25C3×C4×Dic6—32no—
26#63870—34no—
27#63869—32no—
28#63867—34no—
29#63866—32no—
30#63862—34no—
31#63859—34no—
32#63857—32no—
33#63856—32no—
34C22×#272C2311yescharacteristic
35C2×C4×H(3,𝔽3)C2312yes—
36C2×C4×H(3,𝔽3)C2311yescharacteristic
37C2×#1084C2311yescharacteristic
38C2×#1084C2312yes—
39C4×#272—22no—
40C2×(C3⋊Dic3)⋊C3C2314yes—
41C2×(C3⋊Dic3)⋊C3C2314yes—
42#634⋊C3—24no—
43#5583—92no—
44#5518—91no—
45C2×#634C2×D312yes—
46C2×C3×C4×D3—32no—
47C2×C3×C4×D3—61no—
48C2×C3×Dic6—32no—
49C32×C42—21no—
50C32×C42C2×D312yes—
51C4⋊(C3⋊Dic3)C2×D312yes—
52#3750C2×D314yes—
53C4×C3⋊Dic3C2×D314yes—
54C3×(C2×D3)⋊C4—32no—
55C3×(C2×D3)⋊C4—32no—
56C3×(C2×D3)⋊C4—62no—
57C3×C4⋊Dic3—32no—
58C3×Dic3⋊C4—34no—
59C3×Dic3⋊C4—32no—
60C3×Dic3⋊C4—32no—
61C3×Dic3⋊C4—62no—
62C3×C4×Dic3—34no—
63C3×C4×Dic3—32no—
64C3×C4×Dic3—61no—
65#3724—34no—
66#3724—34no—
67Dic3⋊Dic3—34no—
68Dic3⋊Dic3—34no—
69Dic3⋊Dic3—34no—
70Dic3⋊Dic3—34no—
71(C2×D3)⋊Dic3—34no—
72(C2×D3)⋊Dic3—34no—
73C22×H(3,𝔽3)C2411yescharacteristic
74C2×#272—21no—
75C2×#272—22no—
76C4×H(3,𝔽3)—22no—
77C4×H(3,𝔽3)#4112yes—
78#1084—21no—
79#1084—22no—
80(C3⋊Dic3)⋊C3—24no—
81(C3⋊Dic3)⋊C3—24no—
82C3×C4×Q8—92no—
83C3×#117—91no—
84#946—92no—
85#944—92no—
86#924—92no—
87#898—94no—
88#897—92no—
89#890—92no—
90#889—94no—
91#889—94no—
92#879—94no—
93#876—94no—
94#872—32no—
95#872—61no—
96#870—94no—
97#870—91no—
98#869—91no—
99C4×Dic6—32no—
100C4×Dic6—92no—
101C22×C3×D3—32no—
102C22×C3×D3—61no—
103C2×C32×C4—21no—
104C2×C32×C4—22no—
105C2×C32×C4C22×D312yes—
106C2×C32×C4C22×D312yes—
107C2×C32×C4C22×D312yes—
108C2×C3⋊Dic3C22×D314yes—
109C2×C3⋊Dic3C22×D314yes—
110#634—24no—
111C2×C3×Dic3—34no—
112C2×C3×Dic3—34no—
113C2×C3×Dic3—32no—
114C2×C3×Dic3—32no—
115C2×C3×Dic3—32no—
116C2×C3×Dic3—61no—
117C2×C3×Dic3—62no—
118C3×C4×D3—64no—
119C3×C4×D3—64no—
120C3×Dic6—64no—
121#541—271noSylow
122C2×H(3,𝔽3)C2×#4111yescharacteristic
123C2×H(3,𝔽3)2−1+412yes—
124#272—41no—
125C2×C3×Q8—92no—
126C22×C3×C4—91no—
127C2×C4×D3—32no—
128C2×C4×D3—61no—
129C2×Dic6—32no—
130C2×Dic6—92no—
131C3×C4⋊C4—94no—
132C3×C4⋊C4—92no—
133C3×C4⋊C4—92no—
134C3×C22⋊C4—92no—
135C3×C42—32no—
136C3×C42—61no—
137C3×C42—94no—
138C3×C42—91no—
139C3×C42D3211yescharacteristic
140#217—94no—
141(C2×D3)⋊C4—32no—
142(C2×D3)⋊C4—32no—
143(C2×D3)⋊C4—62no—
144(C2×D3)⋊C4—94no—
145(C2×D3)⋊C4—94no—
146C4⋊Dic3—32no—
147C4⋊Dic3—92no—
148C4⋊Dic3—94no—
149C4⋊Dic3—94no—
150C4⋊Dic3—94no—
151Dic3⋊C4—32no—
152Dic3⋊C4—32no—
153Dic3⋊C4—34no—
154Dic3⋊C4—62no—
155Dic3⋊C4—94no—
156Dic3⋊C4—94no—
157Dic3⋊C4—94no—
158Dic3⋊C4—94no—
159Dic3⋊C4—94no—
160Dic3⋊C4—94no—
161Dic3⋊C4—94no—
162C4×Dic3—32no—
163C4×Dic3—34no—
164C4×Dic3—61no—
165C4×Dic3—94no—
166C22×C32—21no—
167C22×C32C23×D312yes—
168C2×C3×D3—62no—
169C2×C3×D3—62no—
170C2×C3×D3—62no—
171C2×C3×D3—62no—
172C2×C3×D3—122no—
173C32×C4—22no—
174C32×C4—22no—
175C32×C4—22no—
176C32×C4—42no—
177C32×C4#23514yes—
178C3⋊Dic3—24no—
179C3⋊Dic3—24no—
180C3×Dic3—62no—
181C3×Dic3—64no—
182C3×Dic3—62no—
183C3×Dic3—64no—
184C3×Dic3—62no—
185C3×Dic3—62no—
186C3×Dic3—122no—
187C4⋊Q8—271no—
188#126—274no—
189#125—274no—
190#125—271no—
191C22⋊Q8—272no—
192C4×Q8—92no—
193#117—91no—
194H(3,𝔽3)#54111yescharacteristic, complemented, Sylow
195C23×C3—91no—
196C22×D3—32no—
197C22×D3—61no—
198C3×Q8—184no—
199C2×C3×C4—32no—
200C2×C3×C4—32no—
201C2×C3×C4—32no—
202C2×C3×C4—61no—
203C2×C3×C4—62no—
204C2×C3×C4—94no—
205C2×C3×C4—94no—
206C2×C3×C4—91no—
207C2×C3×C4—92no—
208C2×C3×C4—182no—
209C2×C3×C4C2×D3212yes—
210C2×C3×C4C2×D3211yescharacteristic
211C2×Dic3—32no—
212C2×Dic3—32no—
213C2×Dic3—32no—
214C2×Dic3—34no—
215C2×Dic3—34no—
216C2×Dic3—61no—
217C2×Dic3—62no—
218C2×Dic3—94no—
219C2×Dic3—94no—
220C4×D3—64no—
221C4×D3—64no—
222Dic6—64no—
223Dic6—184no—
224C2×C32—22no—
225C2×C32—21no—
226C2×C32C2×#23512yes—
227C2×C32#100712yes—
228C2×C32#101012yes—
229C3×D3—122no—
230C3×D3—122no—
231C2×Q8—92no—
232C22×C4—91no—
233C4⋊C4—92no—
234C4⋊C4—92no—
235C4⋊C4—94no—
236C4⋊C4—274no—
237C4⋊C4—274no—
238C4⋊C4—274no—
239C22⋊C4—92no—
240C22⋊C4—274no—
241C42—91no—
242C42—94no—
243C42((C3⋊D3)⋊C3)⋊C211yescharacteristic
244C22×C3—32no—
245C22×C3—61no—
246C22×C3—182no—
247C22×C3—181no—
248C22×C3C22×D3211yescharacteristic
249C2×D3—62no—
250C2×D3—62no—
251C2×D3—62no—
252C2×D3—62no—
253C2×D3—122no—
254C3×C4—22no—
255C3×C4—34no—
256C3×C4—62no—
257C3×C4—62no—
258C3×C4—62no—
259C3×C4—122no—
260C3×C4—184no—
261C3×C4—184no—
262C3×C4—182no—
263C3×C4—181no—
264C3×C4#379812yes—
265Dic3—62no—
266Dic3—62no—
267Dic3—62no—
268Dic3—62no—
269Dic3—64no—
270Dic3—64no—
271Dic3—122no—
272Dic3—184no—
273Dic3—184no—
274C32—21no—
275C32#558312yescomplemented
276C23—91no—
277Q8—184no—
278C2×C4—92no—
279C2×C4—94no—
280C2×C4—94no—
281C2×C4—91no—
282C2×C4—182no—
283C2×C4C2×((C3⋊D3)⋊C3)⋊C212yes—
284C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
285C2×C3—32no—
286C2×C3—32no—
287C2×C3—32no—
288C2×C3—62no—
289C2×C3—61no—
290C2×C3—361no—
291C2×C3#6429712yes—
292C2×C3C2×#379811yescharacteristic
293D3—122no—
294D3—122no—
295C22—182no—
296C22—181no—
297C22C22×((C3⋊D3)⋊C3)⋊C211yescharacteristic
298C4—22no—
299C4—182no—
300C4—184no—
301C4—181no—
302C4—184no—
303C4#8865012yes—
304C3—32no—
305C3—61no—
306C3#1059388311yescharacteristic
307C2—361no—
308C2#1172248212yes—
309C2C2×#8865011yescharacteristic
subgroupG/H
centerC22C22×((C3⋊D3)⋊C3)⋊C2
commutator, cosocleC22×H(3,𝔽3)C24
Frattini, socleC22×C3C22×D32
FittingC42×H(3,𝔽3)C22
prime radicalC22×#272C23
hypercenterC42((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#541
LCSG ▷ C22×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C42

Complements

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

NH ≅ G/Ntype
2. C4×#634⋊C3C2semidirect
3. #11721099C2semidirect
4. #11721095C2semidirect
7. #11721091C2semidirect
22. C4×#634D3semidirect
194. H(3,𝔽3)#541semidirect
275. C32#5583semidirect

Maximal subgroups

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

subgroupindex
#117217182
C4×#634⋊C32
#117210992
#117210952
#117210942
#117210922
#117210912
#105938803