Finite Group Databasebeta

← #13149411

Subgroups — #13149411

Order 1728.

770 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 770 conjugacy classes of subgroups — too many to draw. Only its 149 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#64099⋊C3C212yesmaximal
3#11721117C211yesmaximal, characteristic
4#11721117C212yesmaximal
5C4×#6196C211yesmaximal, characteristic
6#11721086C214yesmaximal
7#11721083C212yesmaximal
8#11721078C212yesmaximal
9#10594321—32nomaximal
10C2×#6229C2211yescharacteristic
11C2×#6196C2212yes—
12C2×#6196C2211yescharacteristic
13C4⋊C4×H(3,𝔽3)C2211yescharacteristic
14#88540C2212yes—
15C4×#1084C2212yes—
16C4×#1084C2211yescharacteristic
17C4⋊((C3⋊Dic3)⋊C3)C2212yes—
18C4⋊((C3⋊Dic3)⋊C3)C2214yes—
19#3750⋊C3C2214yes—
20C4×(C3⋊Dic3)⋊C3C2212yes—
21#88449C2211yescharacteristic
22#88449C2214yes—
23#88449C2212yes—
24#88448C2212yes—
25#88447C2214yes—
26#64099D312yes—
27C3×#890—32no—
28C3×#887—32no—
29C3×#887—61no—
30#63918—34no—
31#63918—32no—
32C4×#627—32no—
33#63846—34no—
34#63845—34no—
35#63843—32no—
36Dic3×Dic6—32no—
37#63837—34no—
38C2×C4×H(3,𝔽3)C2312yes—
39C2×C4×H(3,𝔽3)C2311yescharacteristic
40C2×#1084C2311yescharacteristic
41C2×#1084C2312yes—
42C2×#1084C2311yescharacteristic
43#6229Q814yescomplemented
44C2×(C3⋊Dic3)⋊C3C2314yes—
45C2×(C3⋊Dic3)⋊C3C2312yes—
46C2×(C3⋊Dic3)⋊C3C2312yes—
47#6196—24no—
48#6196—22no—
49#5685—92no—
50#5570—91no—
51C2×C3×Dic6—32no—
52C2×C3×Dic6—61no—
53C2×#627—34no—
54C2×#627—32no—
55C32×C4⋊C4—21no—
56C32×C4⋊C4C2×D312yes—
57C4⋊(C3⋊Dic3)C2×D312yes—
58C4⋊(C3⋊Dic3)C2×D314yes—
59#3750C2×D314yes—
60C4×C3⋊Dic3C2×D312yes—
61C3×C4⋊Dic3—32no—
62C3×C4⋊Dic3—34no—
63C3×Dic3⋊C4—34no—
64C3×Dic3⋊C4—34no—
65C3×Dic3⋊C4—62no—
66C3×C4×Dic3—34no—
67C3×C4×Dic3—32no—
68C3×C4×Dic3—32no—
69C3×C4×Dic3—62no—
70C3×C4×Dic3—61no—
71#3724—32no—
72Dic3⋊Dic3—32no—
73Dic3⋊Dic3—34no—
74Dic3⋊Dic3—34no—
75Dic3⋊Dic3—34no—
76Dic3⋊Dic3—32no—
77Dic32—34no—
78Dic32—34no—
79C22×H(3,𝔽3)C2411yescharacteristic
80C4×H(3,𝔽3)—22no—
81C4×H(3,𝔽3)C2×Q812yes—
82#1084—21no—
83#1084C2×Q814yes—
84#1084#4112yes—
85(C3⋊Dic3)⋊C3—24no—
86(C3⋊Dic3)⋊C3—22no—
87(C3⋊Dic3)⋊C3—22no—
88C3×#125—92no—
89C3×C4×Q8—91no—
90Q8×Dic3—92no—
91#890—32no—
92#890—92no—
93#890—94no—
94#889—94no—
95#889—94no—
96#888—92no—
97#888—94no—
98#887—32no—
99#887—61no—
100#887—94no—
101#887—92no—
102#869—92no—
103#869—91no—
104C4×Dic6—91no—
105C4×Dic6—92no—
106C4×Dic6—92no—
107C4×Dic6—92no—
108C2×C32×C4—21no—
109C2×C32×C4—22no—
110C2×C32×C4C22×D314yes—
111C2×C32×C4C22×D312yes—
112C2×C3⋊Dic3C22×D312yes—
113C2×C3⋊Dic3C22×D314yes—
114C2×C3⋊Dic3C22×D312yes—
115C2×C3×Dic3—34no—
116C2×C3×Dic3—32no—
117C2×C3×Dic3—32no—
118C2×C3×Dic3—34no—
119C2×C3×Dic3—32no—
120C2×C3×Dic3—32no—
121C2×C3×Dic3—61no—
122C2×C3×Dic3—62no—
123C2×C3×Dic3—61no—
124C3×Dic6—38no—
125C3×Dic6—64no—
126#627—68no—
127#627—64no—
128#557—271noSylow
129C2×H(3,𝔽3)C22×Q811yescharacteristic
130C2×H(3,𝔽3)C2×#4111yescharacteristic
131C2×H(3,𝔽3)2−1+411yescharacteristic
132C2×C3×Q8—91no—
133C2×Dic6—32no—
134C2×Dic6—61no—
135C2×Dic6—91no—
136C2×Dic6—92no—
137C2×Dic6—92no—
138C2×Dic6—94no—
139C3×C4⋊C4—32no—
140C3×C4⋊C4—61no—
141C3×C4⋊C4—92no—
142C3×C4⋊C4—94no—
143C3×C4⋊C4—94no—
144C3×C4⋊C4—92no—
145C3×C4⋊C4D3211yescharacteristic
146C3×C42—92no—
147C3×C42—92no—
148C3×C42—91no—
149C4⋊Dic3—34no—
150C4⋊Dic3—32no—
151C4⋊Dic3—92no—
152C4⋊Dic3—94no—
153C4⋊Dic3—91no—
154C4⋊Dic3—92no—
155C4⋊Dic3—94no—
156C4⋊Dic3—92no—
157Dic3⋊C4—34no—
158Dic3⋊C4—34no—
159Dic3⋊C4—62no—
160Dic3⋊C4—94no—
161Dic3⋊C4—92no—
162Dic3⋊C4—92no—
163Dic3⋊C4—94no—
164Dic3⋊C4—92no—
165Dic3⋊C4—94no—
166Dic3⋊C4—94no—
167C4×Dic3—32no—
168C4×Dic3—34no—
169C4×Dic3—32no—
170C4×Dic3—62no—
171C4×Dic3—61no—
172C4×Dic3—92no—
173C4×Dic3—94no—
174C4×Dic3—94no—
175C4×Dic3—94no—
176C22×C32—21no—
177C22×C32C23×D312yes—
178C32×C4—24no—
179C32×C4—22no—
180C32×C4—42no—
181C32×C4D3×Q814yes—
182C3⋊Dic3—24no—
183C3⋊Dic3—22no—
184C3⋊Dic3—22no—
185C3×Dic3—38no—
186C3×Dic3—34no—
187C3×Dic3—64no—
188C3×Dic3—62no—
189C3×Dic3—64no—
190C3×Dic3—62no—
191C3×Dic3—62no—
192C3×Dic3—62no—
193C3×Dic3—121no—
194C4⋊Q8—271no—
195C4⋊Q8—272no—
196#125—92no—
197#125—274no—
198C4×Q8—91no—
199C4×Q8—271no—
200C4×Q8—272no—
201C4×Q8—272no—
202H(3,𝔽3)#55711yescharacteristic, complemented, Sylow
203C3×Q8—94no—
204C2×C3×C4—32no—
205C2×C3×C4—34no—
206C2×C3×C4—61no—
207C2×C3×C4—62no—
208C2×C3×C4—92no—
209C2×C3×C4—91no—
210C2×C3×C4—92no—
211C2×C3×C4—94no—
212C2×C3×C4—92no—
213C2×C3×C4—91no—
214C2×C3×C4C2×D3212yes—
215C2×C3×C4C2×D3211yescharacteristic
216C2×Dic3—32no—
217C2×Dic3—34no—
218C2×Dic3—32no—
219C2×Dic3—32no—
220C2×Dic3—32no—
221C2×Dic3—34no—
222C2×Dic3—61no—
223C2×Dic3—62no—
224C2×Dic3—61no—
225C2×Dic3—92no—
226C2×Dic3—92no—
227C2×Dic3—94no—
228Dic6—38no—
229Dic6—64no—
230Dic6—184no—
231Dic6—188no—
232Dic6—182no—
233Dic6—184no—
234C2×C32—21no—
235C2×C32—21no—
236C2×C32—21no—
237C2×C32C2×D3×Q812yes—
238C2×C32D3×#4112yes—
239C2×C32#101012yes—
240C2×Q8—91no—
241C2×Q8—271no—
242C2×Q8—272no—
243C4⋊C4—92no—
244C4⋊C4—94no—
245C4⋊C4—92no—
246C4⋊C4—94no—
247C4⋊C4—271no—
248C4⋊C4—272no—
249C4⋊C4—274no—
250C4⋊C4—272no—
251C4⋊C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
252C42—92no—
253C42—91no—
254C42—92no—
255C42—274no—
256C22×C3—32no—
257C22×C3—61no—
258C22×C3C22×D3211yescharacteristic
259C3×C4—22no—
260C3×C4—34no—
261C3×C4—64no—
262C3×C4—62no—
263C3×C4—94no—
264C3×C4—92no—
265C3×C4—122no—
266C3×C4—184no—
267C3×C4—182no—
268C3×C4—182no—
269C3×C4—181no—
270C3×C4#379712yes—
271Dic3—38no—
272Dic3—34no—
273Dic3—64no—
274Dic3—62no—
275Dic3—62no—
276Dic3—62no—
277Dic3—62no—
278Dic3—64no—
279Dic3—121no—
280Dic3—182no—
281Dic3—184no—
282Dic3—182no—
283C32—21no—
284C32#568512yescomplemented
285Q8—94no—
286Q8—542no—
287Q8—544no—
288C2×C4—91no—
289C2×C4—92no—
290C2×C4—92no—
291C2×C4—91no—
292C2×C4—92no—
293C2×C4—94no—
294C2×C4C2×((C3⋊D3)⋊C3)⋊C212yes—
295C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
296C2×C3—32no—
297C2×C3—32no—
298C2×C3—32no—
299C2×C3—61no—
300C2×C3—61no—
301C2×C3—61no—
302C2×C3C2×#379711yescharacteristic
303C2×C3#6430911yescharacteristic
304C2×C3#6431911yescharacteristic
305C22C22×((C3⋊D3)⋊C3)⋊C211yescharacteristic
306C4—22no—
307C4—92no—
308C4—94no—
309C4—181no—
310C4—182no—
311C4—182no—
312C4—184no—
313C4#8865112yes—
314C3—32no—
315C3—61no—
316C3#1059432411yescharacteristic
317C2C2×#8865111yescharacteristic
318C2#1172250211yescharacteristic
319C2#1172252011yescharacteristic
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
43. #6229Q8semidirect
202. H(3,𝔽3)#557semidirect
284. C32#5685semidirect

Maximal subgroups

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

subgroupindex
#117217552
#64099⋊C32
#117211172
C4×#61962
#117210862
#117210832
#117210782
#105943213