Finite Group Databasebeta

← #13155507

Subgroups — #13155507

Order 1728.

558 conjugacy classes of subgroups; only the 128 normal subgroups are shown.

normal subgroups only

This group has 558 conjugacy classes of subgroups — too many to draw. Only its 128 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
1C2×#88786C211yesmaximal, characteristic
2C3×D3×#208C212yesmaximal
3#11722081C211yesmaximal, characteristic, complemented
4#11722059C212yesmaximal, complemented
5C3×#63578C211yesmaximal, characteristic, complemented
6#10591143—31nomaximal
7#10590973—32nomaximal
8C2×C32×C4×D3C2211yescharacteristic
9C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
10C3×#3748C2212yes—
11C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
12C32×#208C2212yes—
13#88786C414yes—
14C3×#3711—22no—
15C3×D3×C3⋊C8—22no—
16C2×#3748—31no—
17C2×C3×#208—32no—
18C3×D3×M16—32no—
19C2×#3711—32no—
20C2×#3711—61no—
21D3×#208D312yes—
22#63649—32no—
23C3×#822—32no—
24#63578—31no—
25#63578—32no—
26#63578—32no—
27#63578—62no—
28#63578D311yescharacteristic, complemented
29#63577—32no—
30C22×C32×D3Q811yescharacteristic
31C2×C33×C4C2311yescharacteristic
32C2×C32×Dic3Q811yescharacteristic
33C32×C4×D3C2×C412yes—
34C32×C4×D3D412yes—
35C3⋊(C3⋊(C3⋊C8))C2×C412yes—
36C3×C3⋊(C3⋊C8)—22no—
37C3×C3⋊(C3⋊C8)C2×C412yes—
38C32×C3⋊C8—22no—
39#5124—92no—
40#4893—91no—
41C22×C32×C4—31no—
42C2×C3×C4×D3—21no—
43C2×C3×C4×D3C2×D312yes—
44C32×M16—32no—
45#3748—34no—
46#3748C2×D312yes—
47C2×C3⋊(C3⋊C8)—32no—
48C2×C3⋊(C3⋊C8)—31no—
49C2×C3⋊(C3⋊C8)—61no—
50C2×C3⋊(C3⋊C8)—62no—
51C2×C3⋊(C3⋊C8)—121no—
52C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
53C3×#208—22no—
54C3×#208—32no—
55C3×#208—32no—
56C3×#208—62no—
57C3×#208—62no—
58C3×#208C2×D312yes—
59C3×#208C2×D312yes—
60C2×C3×C3⋊C8—32no—
61C2×C3×C3⋊C8—62no—
62C2×C3×C3⋊C8—61no—
63C3×ΓL(1,𝔽25)—62no—
64C3×D3×C8—62no—
65#3711—22no—
66#3711—34no—
67#3711—34no—
68#3711—64no—
69D3×C3⋊C8—22no—
70C22×C33C2×Q811yescharacteristic
71C2×C32×D3—41no—
72C2×C32×D3C4⋊C412yes—
73C33×C4C22×C411yescharacteristic
74C33×C4C2×D411yescharacteristic
75C32×Dic3C4⋊C412yes—
76C2×C3×M16—92no—
77C2×#208—32no—
78C2×#208—92no—
79C2×#208—181no—
80D3×M16—32no—
81C2×ΓL(1,𝔽25)—91no—
82C3×#108—91no—
83#822—32no—
84#822—92no—
85#822—92no—
86#822—92no—
87#822—92no—
88#822—182no—
89#819—91no—
90C23×C32—31no—
91C22×C3×D3—21no—
92C22×C3×D3Dic612yes—
93C2×C32×C4—21no—
94C2×C32×C4—22no—
95C2×C32×C4—31no—
96C2×C32×C4—32no—
97C2×C32×C4—32no—
98C2×C32×C4—41no—
99C2×C32×C4C22×D311yescharacteristic
100C2×C32×C4C22×D312yes—
101C2×C3×Dic3—21no—
102C2×C3×Dic3Dic612yes—
103C3×C4×D3—22no—
104C3×C4×D3—22no—
105C3×C4×D3C4×D312yes—
106C3×C4×D3C4×D312yes—
107C3×C4×D3#6712yes—
108C3×C4×D3#6712yes—
109C32×C8—62no—
110C3⋊(C3⋊C8)—22no—
111C3⋊(C3⋊C8)—34no—
112C3⋊(C3⋊C8)—32no—
113C3⋊(C3⋊C8)—62no—
114C3⋊(C3⋊C8)—64no—
115C3⋊(C3⋊C8)—122no—
116C3⋊(C3⋊C8)C4×D312yes—
117C3×C3⋊C8—22no—
118C3×C3⋊C8—22no—
119C3×C3⋊C8—34no—
120C3×C3⋊C8—42no—
121C3×C3⋊C8—62no—
122C3×C3⋊C8—62no—
123C3×C3⋊C8—62no—
124C3×C3⋊C8—122no—
125#430—271noSylow
126C2×C33—21no—
127C2×C33C2×C4⋊C411yescharacteristic
128C32×D3—22no—
129C22×C3×C4—32no—
130C22×C3×C4—61no—
131C2×C4×D3D3211yescharacteristic
132C3×M16—32no—
133C3×M16—32no—
134C3×M16—62no—
135C3×M16—92no—
136C3×M16—182no—
137C2×C3×C8—91no—
138C2×C3×C8—182no—
139#208—32no—
140#208—32no—
141#208—62no—
142#208—62no—
143#208—94no—
144#208—94no—
145#208—184no—
146#208D3212yes—
147C2×C3⋊C8—32no—
148C2×C3⋊C8—61no—
149C2×C3⋊C8—62no—
150C2×C3⋊C8—91no—
151C2×C3⋊C8—92no—
152C2×C3⋊C8—182no—
153C2×C3⋊C8—181no—
154C2×C3⋊C8—361no—
155ΓL(1,𝔽25)—62no—
156ΓL(1,𝔽25)—94no—
157D3×C8—62no—
158C22×C32—22no—
159C22×C32—21no—
160C22×C32—32no—
161C22×C32—41no—
162C22×C32—121no—
163C22×C32C2×Dic612yes—
164C22×C32D3×Q811yescharacteristic
165C2×C3×D3—22no—
166C2×C3×D3—42no—
167C2×C3×D3—42no—
168C2×C3×D3Dic3⋊C412yes—
169C2×C3×D3Dic3⋊C412yes—
170C32×C4—22no—
171C32×C4—21no—
172C32×C4—21no—
173C32×C4—22no—
174C32×C4—32no—
175C32×C4—41no—
176C32×C4—41no—
177C32×C4C2×C4×D311yescharacteristic
178C32×C4C2×C4×D312yes—
179C32×C4D3×D411yescharacteristic
180C32×C4C2×#6712yes—
181C3×Dic3—22no—
182C3×Dic3Dic3⋊C412yes—
183C3×Dic3Dic3⋊C412yes—
184C2×M16—92no—
185C2×M16—271no—
186#108—91no—
187#108—271no—
188#108—272no—
189C33#43011yescharacteristic, complemented, Sylow
190C23×C3—32no—
191C23×C3—61no—
192C22×D3#62711yescharacteristic
193C2×C3×C4—21no—
194C2×C3×C4—22no—
195C2×C3×C4—32no—
196C2×C3×C4—32no—
197C2×C3×C4—32no—
198C2×C3×C4—32no—
199C2×C3×C4—32no—
200C2×C3×C4—41no—
201C2×C3×C4—62no—
202C2×C3×C4—62no—
203C2×C3×C4—61no—
204C2×C3×C4C2×D3212yes—
205C2×C3×C4C2×D3211yescharacteristic
206C2×Dic3#62711yescharacteristic
207C4×D3#62412yes—
208C4×D3#62512yes—
209C3×C8—62no—
210C3×C8—62no—
211C3×C8—92no—
212C3×C8—122no—
213C3×C8—182no—
214C3⋊C8—22no—
215C3⋊C8—34no—
216C3⋊C8—62no—
217C3⋊C8—62no—
218C3⋊C8—62no—
219C3⋊C8—94no—
220C3⋊C8—92no—
221C3⋊C8—122no—
222C3⋊C8—184no—
223C3⋊C8—182no—
224C3⋊C8—362no—
225C2×C32—21no—
226C2×C32—22no—
227C2×C32—22no—
228C2×C32—22no—
229C2×C32—21no—
230C2×C32—41no—
231C2×C32—42no—
232C2×C32—42no—
233C2×C32—62no—
234C2×C32D3×C4⋊C411yescharacteristic
235C2×C32C2×Dic3⋊C412yes—
236C3×D3—22no—
237C3×D3—22no—
238C3×D3—42no—
239C22×C4—31no—
240M16—32no—
241M16—92no—
242M16—182no—
243M16—274no—
244C2×C8—91no—
245C2×C8—182no—
246C2×C8—271no—
247C22×C3—21no—
248C22×C3—22no—
249C22×C3—32no—
250C22×C3—32no—
251C22×C3—41no—
252C22×C3—62no—
253C22×C3—122no—
254C22×C3—122no—
255C22×C3D3×Dic612yes—
256C22×C3C2×#62711yescharacteristic
257C2×D3—41no—
258C2×D3#372412yes—
259C3×C4—21no—
260C3×C4—22no—
261C3×C4—22no—
262C3×C4—21no—
263C3×C4—32no—
264C3×C4—32no—
265C3×C4—41no—
266C3×C4—41no—
267C3×C4—62no—
268C3×C4C4×D3212yes—
269C3×C4C2×#62411yescharacteristic
270C3×C4C2×#62511yescharacteristic
271C3×C4D3×#6712yes—
272Dic3#372412yes—
273C32—21no—
274C32—22no—
275C32—41no—
276C32#489311yescharacteristic, complemented
277C32#512412yescomplemented
278C23—31no—
279C2×C4—31no—
280C2×C4—32no—
281C2×C4—32no—
282C2×C4D3311yescharacteristic
283C8—62no—
284C8—92no—
285C8—182no—
286C8—272no—
287C2×C3—22no—
288C2×C3—21no—
289C2×C3—22no—
290C2×C3—21no—
291C2×C3—22no—
292C2×C3—42no—
293C2×C3—42no—
294C2×C3—41no—
295C2×C3—62no—
296C2×C3—62no—
297C2×C3—122no—
298C2×C3D3×Dic3⋊C412yes—
299C2×C3C2×#372411yescharacteristic
300D3—22no—
301C22—32no—
302C22—121no—
303C22D3×#62711yescharacteristic
304C4—32no—
305C4D3×#62411yescharacteristic
306C4D3×#62511yescharacteristic
307C3—21no—
308C3—22no—
309C3—41no—
310C3#1059097312yescomplemented
311C3#1059114311yescharacteristic, complemented
312C2—21no—
313C2—62no—
314C2D3×#372411yescharacteristic
subgroupG/H
centerC4D3×#625
commutatorC33×C4C22×C4
Frattini, hypercenterC2×C4D33
Fitting, cosocleC2×C33×C4C23
socleC2×C33C2×C4⋊C4
prime radicalC22×C32×D3Q8
hypocenterC33#430
LCSG ▷ C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C4 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
3. #11722081C2semidirect
4. #11722059C2semidirect
5. C3×#63578C2semidirect
28. #63578D3semidirect
189. C33#430semidirect
276. C32#4893semidirect
277. C32#5124semidirect
310. C3#10590973semidirect
311. C3#10591143semidirect

Maximal subgroups

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

subgroupindex
C2×#887862
C3×D3×#2082
#117220812
#117220592
C3×#635782
#105911433
#105909733