Finite Group Databasebeta

← #13149394

Subgroups — #13149394

Order 1728.

770 conjugacy classes of subgroups; only the 153 normal subgroups are shown.

normal subgroups only

This group has 770 conjugacy classes of subgroups — too many to draw. Only its 153 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#11721755C211yesmaximal, characteristic
2#64096⋊C3C212yesmaximal
3C4×#6196C212yesmaximal
4C4×#6196C211yesmaximal, characteristic
5#11721099C211yesmaximal, characteristic
6#11721098C212yesmaximal
7#11721091C214yesmaximal
8#11721090C212yesmaximal
9#10594595—32nomaximal
10C2×#6229C2211yescharacteristic
11C2×#634⋊C3C2212yes—
12C2×#6196C2211yescharacteristic
13C4⋊C4×H(3,𝔽3)C2211yescharacteristic
14#88540C2212yes—
15C4×#1084C2212yes—
16C4×#1084C2211yescharacteristic
17#3750⋊C3C2214yes—
18C4×(C3⋊Dic3)⋊C3C2212yes—
19C4×(C3⋊Dic3)⋊C3C2214yes—
20#88449C2212yes—
21#88449C2211yescharacteristic
22#88449C2212yes—
23#88448C2214yes—
24#88448C2214yes—
25#88448C2212yes—
26#64096D312yes—
27C3×#887—32no—
28C3×#887—32no—
29C3×#887—61no—
30C4×#627—34no—
31C4×#627—32no—
32#63873—32no—
33#63870—34no—
34#63867—34no—
35#63867—34no—
36#63866—32no—
37#63866—32no—
38C2×C4×H(3,𝔽3)C2312yes—
39C2×C4×H(3,𝔽3)C2311yescharacteristic
40C2×#1084C2312yes—
41C2×#1084C2311yescharacteristic
42C2×#1084C2311yescharacteristic
43#6229—22no—
44C2×(C3⋊Dic3)⋊C3C2314yes—
45C2×(C3⋊Dic3)⋊C3C2312yes—
46C2×(C3⋊Dic3)⋊C3C2312yes—
47#634⋊C3—24no—
48#6196Q814yescomplemented
49#5567—91no—
50#5567—92no—
51C2×#634C2×D312yes—
52C2×C3×Dic6—32no—
53C2×C3×Dic6—32no—
54C2×C3×Dic6—61no—
55C2×#627—32no—
56C32×C4⋊C4—21no—
57C32×C4⋊C4C2×D312yes—
58#3750C2×D314yes—
59C4×C3⋊Dic3C2×D314yes—
60C4×C3⋊Dic3C2×D312yes—
61C3×Dic3⋊C4—34no—
62C3×Dic3⋊C4—34no—
63C3×Dic3⋊C4—62no—
64C3×C4×Dic3—32no—
65C3×C4×Dic3—32no—
66C3×C4×Dic3—34no—
67C3×C4×Dic3—34no—
68C3×C4×Dic3—62no—
69C3×C4×Dic3—61no—
70#3724—34no—
71#3724—32no—
72#3724—34no—
73Dic3⋊Dic3—34no—
74Dic3⋊Dic3—34no—
75Dic3⋊Dic3—34no—
76Dic3⋊Dic3—34no—
77Dic3⋊Dic3—32no—
78Dic3⋊Dic3—32no—
79C22×H(3,𝔽3)C2411yescharacteristic
80C4×H(3,𝔽3)—22no—
81C4×H(3,𝔽3)#4112yes—
82#1084—22no—
83#1084—21no—
84#1084C2×Q812yes—
85(C3⋊Dic3)⋊C3—24no—
86(C3⋊Dic3)⋊C3—22no—
87(C3⋊Dic3)⋊C3C2×Q814yes—
88C3×C4×Q8—92no—
89C3×C4×Q8—91no—
90#944—92no—
91#944—92no—
92#944—92no—
93#889—94no—
94#889—94no—
95#889—94no—
96#887—32no—
97#887—32no—
98#887—61no—
99#887—92no—
100#870—92no—
101#870—94no—
102#869—91no—
103#869—92no—
104C4×Dic6—92no—
105C4×Dic6—91no—
106C4×Dic6—94no—
107C4×Dic6—92no—
108C2×C32×C4—21no—
109C2×C32×C4—22no—
110C2×C32×C4C22×D314yes—
111C2×C32×C4C22×D312yes—
112C2×C3⋊Dic3C22×D314yes—
113C2×C3⋊Dic3C22×D312yes—
114C2×C3⋊Dic3C22×D312yes—
115#634—24no—
116C2×C3×Dic3—32no—
117C2×C3×Dic3—32no—
118C2×C3×Dic3—32no—
119C2×C3×Dic3—34no—
120C2×C3×Dic3—32no—
121C2×C3×Dic3—34no—
122C2×C3×Dic3—61no—
123C2×C3×Dic3—62no—
124C2×C3×Dic3—61no—
125C3×Dic6—64no—
126C3×Dic6—64no—
127C3×Dic6—64no—
128#627—38no—
129#557—271noSylow
130C2×H(3,𝔽3)C22×Q811yescharacteristic
131C2×H(3,𝔽3)C2×#4111yescharacteristic
132C2×H(3,𝔽3)2−1+411yescharacteristic
133C2×C3×Q8—92no—
134C2×C3×Q8—91no—
135C2×Dic6—32no—
136C2×Dic6—32no—
137C2×Dic6—61no—
138C2×Dic6—92no—
139C2×Dic6—91no—
140C2×Dic6—92no—
141C3×C4⋊C4—32no—
142C3×C4⋊C4—61no—
143C3×C4⋊C4—94no—
144C3×C4⋊C4—92no—
145C3×C4⋊C4D3211yescharacteristic
146C3×C42—92no—
147C3×C42—94no—
148C3×C42—91no—
149C3×C42—92no—
150C4⋊Dic3—92no—
151C4⋊Dic3—91no—
152C4⋊Dic3—92no—
153C4⋊Dic3—94no—
154C4⋊Dic3—92no—
155C4⋊Dic3—94no—
156Dic3⋊C4—34no—
157Dic3⋊C4—34no—
158Dic3⋊C4—62no—
159Dic3⋊C4—94no—
160Dic3⋊C4—92no—
161Dic3⋊C4—94no—
162Dic3⋊C4—94no—
163Dic3⋊C4—92no—
164Dic3⋊C4—92no—
165Dic3⋊C4—94no—
166Dic3⋊C4—94no—
167Dic3⋊C4—94no—
168Dic3⋊C4—94no—
169C4×Dic3—34no—
170C4×Dic3—32no—
171C4×Dic3—34no—
172C4×Dic3—32no—
173C4×Dic3—62no—
174C4×Dic3—61no—
175C4×Dic3—94no—
176C4×Dic3—92no—
177C22×C32—21no—
178C22×C32C23×D312yes—
179C32×C4—24no—
180C32×C4—22no—
181C32×C4—42no—
182C32×C4#23514yes—
183C3⋊Dic3—24no—
184C3⋊Dic3—22no—
185C3⋊Dic3D3×Q814yes—
186C3×Dic3—34no—
187C3×Dic3—34no—
188C3×Dic3—62no—
189C3×Dic3—64no—
190C3×Dic3—64no—
191C3×Dic3—64no—
192C3×Dic3—62no—
193C3×Dic3—62no—
194C3×Dic3—121no—
195C4⋊Q8—271no—
196C4⋊Q8—272no—
197#125—272no—
198#125—274no—
199C4×Q8—91no—
200C4×Q8—92no—
201C4×Q8—272no—
202C4×Q8—271no—
203H(3,𝔽3)#55711yescharacteristic, complemented, Sylow
204C3×Q8—184no—
205C3×Q8—182no—
206C2×C3×C4—32no—
207C2×C3×C4—34no—
208C2×C3×C4—61no—
209C2×C3×C4—62no—
210C2×C3×C4—92no—
211C2×C3×C4—91no—
212C2×C3×C4—94no—
213C2×C3×C4—92no—
214C2×C3×C4—92no—
215C2×C3×C4—91no—
216C2×C3×C4C2×D3212yes—
217C2×C3×C4C2×D3211yescharacteristic
218C2×Dic3—32no—
219C2×Dic3—32no—
220C2×Dic3—32no—
221C2×Dic3—34no—
222C2×Dic3—32no—
223C2×Dic3—34no—
224C2×Dic3—61no—
225C2×Dic3—62no—
226C2×Dic3—61no—
227C2×Dic3—92no—
228C2×Dic3—92no—
229C2×Dic3—94no—
230Dic6—64no—
231Dic6—64no—
232Dic6—64no—
233Dic6—94no—
234Dic6—98no—
235Dic6—184no—
236C2×C32—21no—
237C2×C32—21no—
238C2×C32—21no—
239C2×C32C2×#23512yes—
240C2×C32C2×D3×Q812yes—
241C2×C32#101012yes—
242C2×Q8—91no—
243C2×Q8—92no—
244C2×Q8—271no—
245C4⋊C4—92no—
246C4⋊C4—94no—
247C4⋊C4—271no—
248C4⋊C4—272no—
249C4⋊C4—272no—
250C4⋊C4—274no—
251C4⋊C4—272no—
252C4⋊C4—274no—
253C4⋊C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
254C42—92no—
255C42—94no—
256C42—91no—
257C42—92no—
258C22×C3—32no—
259C22×C3—61no—
260C22×C3C22×D3211yescharacteristic
261C3×C4—22no—
262C3×C4—34no—
263C3×C4—64no—
264C3×C4—62no—
265C3×C4—94no—
266C3×C4—92no—
267C3×C4—122no—
268C3×C4—184no—
269C3×C4—182no—
270C3×C4—182no—
271C3×C4—181no—
272C3×C4#379812yes—
273Dic3—34no—
274Dic3—34no—
275Dic3—62no—
276Dic3—64no—
277Dic3—62no—
278Dic3—64no—
279Dic3—64no—
280Dic3—62no—
281Dic3—94no—
282Dic3—121no—
283Dic3—184no—
284Dic3—182no—
285C32—21no—
286C32#556712yescomplemented
287Q8—182no—
288Q8—184no—
289Q8—274no—
290C2×C4—91no—
291C2×C4—92no—
292C2×C4—91no—
293C2×C4—92no—
294C2×C4—92no—
295C2×C4—94no—
296C2×C4C2×((C3⋊D3)⋊C3)⋊C212yes—
297C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
298C2×C3—32no—
299C2×C3—32no—
300C2×C3—32no—
301C2×C3—61no—
302C2×C3—61no—
303C2×C3—61no—
304C2×C3C2×#379811yescharacteristic
305C2×C3#6430911yescharacteristic
306C2×C3D32×Q811yescharacteristic
307C22C22×((C3⋊D3)⋊C3)⋊C211yescharacteristic
308C4—22no—
309C4—92no—
310C4—94no—
311C4—181no—
312C4—182no—
313C4—182no—
314C4—184no—
315C4#8865012yes—
316C3—32no—
317C3—61no—
318C3#1059459511yescharacteristic
319C2C2×#8865011yescharacteristic
320C2#1172250211yescharacteristic
321C2Q8×((C3⋊D3)⋊C3)⋊C211yescharacteristic
subgroupG/H
centerC22C22×((C3⋊D3)⋊C3)⋊C2
commutator, prime radical, cosocleC22×H(3,𝔽3)C24
Frattini, socleC22×C3C22×D32
FittingC4⋊C4×H(3,𝔽3)C22
hypercenterC4⋊C4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#557
LCSG ▷ C22×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
48. #6196Q8semidirect
203. H(3,𝔽3)#557semidirect
286. C32#5567semidirect

Maximal subgroups

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

subgroupindex
#117217552
#64096⋊C32
C4×#61962
#117210992
#117210982
#117210912
#117210902
#105945953