Finite Group Databasebeta

← #13161048

Subgroups — #13161048

Order 1728.

1084 conjugacy classes of subgroups; only the 236 normal subgroups are shown.

normal subgroups only

This group has 1084 conjugacy classes of subgroups — too many to draw. Only its 236 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×C32×#237C211yesmaximal, characteristic
2D3×#3748C212yesmaximal, complemented
3#11722203C211yesmaximal, characteristic, complemented
4C3×#63660C211yesmaximal, characteristic, complemented
5#11722028C212yesmaximal, complemented
6#10592936—31nomaximal
7#10591501—34nomaximal
8C2×C33×D4C2211yescharacteristic
9C2×C32×#67C412yes—
10C22×C32×Dic3C412yes—
11C32×#237—22no—
12C32×#237—22no—
13C2×C32×C4×D3C2211yescharacteristic
14C2×C32×Dic6C2211yescharacteristic
15#88855C2212yes—
16C3×#3748C2212yes—
17#88786—22no—
18D3×C3⋊(C3⋊C8)—22no—
19C2×C32×#41—31no—
20C2×C3×#237D314yes—
21C2×#3748—32no—
22D3×#208—38no—
23#63663—32no—
24#63660—31no—
25#63660—34no—
26#63660—64no—
27#63660D311yescharacteristic, complemented
28C3×#833—34no—
29#63560—38no—
30C23×C33C2×C412yes—
31C22×C32×D3C2×C411yescharacteristic
32C33×D4—22no—
33C2×C33×C4C2311yescharacteristic
34C32×#67—24no—
35C2×C32×Dic3—22no—
36C2×C32×Dic3—22no—
37C2×C32×Dic3C2×C412yes—
38C2×C32×Dic3C2×C411yescharacteristic
39C32×C4×D3D412yes—
40C32×C4×D3D412yes—
41C32×Dic6—22no—
42C3⋊(C3⋊(C3⋊C8))—22no—
43C3×C3⋊(C3⋊C8)—22no—
44#5237—94no—
45#4745—91no—
46C32×#41—62no—
47C32×#41—62no—
48C2×C32×Q8—31no—
49C2×C32×D4—24no—
50C2×C32×D4—32no—
51C2×C32×D4C2×D311yescharacteristic
52C2×C32×D4C2×D314yes—
53C22×C32×C4—31no—
54C22×C32×C4—32no—
55C2×C3×#67Dic318yes—
56C22×C3×Dic3Dic318yes—
57C3×#237—28no—
58C3×#237—28no—
59C2×C3×C4×D3C2×D314yes—
60C2×C3×Dic6C2×D314yes—
61#3748—32no—
62#3748—38no—
63#3748—62no—
64#3748—68no—
65#3748C2×D312yes—
66C2×C3⋊(C3⋊C8)—62no—
67C3×#208—38no—
68#3711—68no—
69D3×C3⋊C8—68no—
70C22×C33—22no—
71C22×C33—42no—
72C22×C33C22×C411yescharacteristic
73C2×C32×D3—41no—
74C2×C32×D3C22⋊C412yes—
75C33×C4C2×D412yes—
76C32×Dic3—22no—
77C32×Dic3C22⋊C411yescharacteristic
78C32×Dic3C22⋊C411yescharacteristic
79C2×C3×#41—34no—
80C2×#237C3⋊D311yescharacteristic
81C2×#208—98no—
82D3×M16—92no—
83C3×#100—91no—
84#836—98no—
85#833—34no—
86#833—94no—
87#833—91no—
88#833—184no—
89#824—92no—
90C23×C32—28no—
91C23×C32—31no—
92C23×C32C4×D312yes—
93C23×C32C2×Dic318yes—
94C22×C3×D3C2×Dic314yes—
95C32×Q8—62no—
96C32×D4—22no—
97C32×D4—28no—
98C32×D4—48no—
99C32×D4—64no—
100C2×C32×C4—24no—
101C2×C32×C4—31no—
102C2×C32×C4—32no—
103C2×C32×C4—32no—
104C2×C32×C4—32no—
105C2×C32×C4—62no—
106C2×C32×C4—62no—
107C2×C32×C4C22×D314yes—
108C2×C32×C4C22×D311yescharacteristic
109C3×#67—216no—
110C2×C3×Dic3—28no—
111C2×C3×Dic3—28no—
112C2×C3×Dic3C2×Dic318yes—
113C2×C3×Dic3C2×Dic314yes—
114C3×C4×D3#6718yes—
115C3×C4×D3#6718yes—
116C3×Dic6—28no—
117C3⋊(C3⋊C8)—22no—
118C3⋊(C3⋊C8)—62no—
119C3⋊(C3⋊C8)—68no—
120C3⋊(C3⋊C8)—128no—
121C3×C3⋊C8—68no—
122#413—271noSylow
123C2×C33—21no—
124C2×C33—42no—
125C2×C33C2×C22⋊C411yescharacteristic
126C32×D3—22no—
127C3×#41—68no—
128C3×#41—68no—
129C2×C3×Q8—34no—
130C2×C3×D4—24no—
131C2×C3×D4—38no—
132C2×C3×D4D3214yes—
133C2×C3×D4C2×C3⋊D311yescharacteristic
134C22×C3×C4—34no—
135C22×C3×C4—38no—
136C2×#67C3⋊Dic312yes—
137C22×Dic3C3⋊Dic312yes—
138#237—22no—
139#237—22no—
140C2×C4×D3C2×C3⋊D311yescharacteristic
141C2×Dic6C2×C3⋊D311yescharacteristic
142C3×M16—92no—
143#208—38no—
144#208—98no—
145#208—92no—
146#208—188no—
147#208—188no—
148C2×C3⋊C8—188no—
149ΓL(1,𝔽25)—182no—
150D3×C8—182no—
151C22×C32—22no—
152C22×C32—24no—
153C22×C32—28no—
154C22×C32—32no—
155C22×C32—48no—
156C22×C32—42no—
157C22×C32—48no—
158C22×C32—88no—
159C22×C32—121no—
160C22×C32C2×C4×D311yescharacteristic
161C22×C32C22×Dic314yes—
162C2×C3×D3—44no—
163C2×C3×D3#21718yes—
164C32×C4—28no—
165C32×C4—31no—
166C32×C4—31no—
167C32×C4—62no—
168C32×C4D3×D412yes—
169C32×C4C2×#6718yes—
170C3×Dic3—28no—
171C3×Dic3#21714yes—
172C3×Dic3#21714yes—
173C2×#41—31no—
174C2×M16—272no—
175#101—272no—
176#100—91no—
177#100—271no—
178C33#41311yescharacteristic, complemented, Sylow
179C23×C3—28no—
180C23×C3—34no—
181C23×C3D3×Dic318yes—
182C23×C3C2×C3⋊Dic312yes—
183C22×D3C2×C3⋊Dic311yescharacteristic
184C3×Q8—68no—
185C3×D4—28no—
186C3×D4—22no—
187C3×D4—48no—
188C3×D4—616no—
189C2×C3×C4—24no—
190C2×C3×C4—34no—
191C2×C3×C4—38no—
192C2×C3×C4—38no—
193C2×C3×C4—38no—
194C2×C3×C4—68no—
195C2×C3×C4—68no—
196C2×C3×C4C2×D3214yes—
197C2×C3×C4C22×C3⋊D311yescharacteristic
198#67—24no—
199C2×Dic3—22no—
200C2×Dic3—22no—
201C2×Dic3C2×C3⋊Dic312yes—
202C2×Dic3C2×C3⋊Dic311yescharacteristic
203C4×D3#63812yes—
204C4×D3#63812yes—
205Dic6—22no—
206C3×C8—182no—
207C3⋊C8—68no—
208C3⋊C8—188no—
209C3⋊C8—182no—
210C3⋊C8—368no—
211C2×C32—21no—
212C2×C32—24no—
213C2×C32—24no—
214C2×C32—44no—
215C2×C32—42no—
216C2×C32—48no—
217C2×C32—62no—
218C2×C32—88no—
219C2×C32D3×C22⋊C411yescharacteristic
220C2×C32C2×#21714yes—
221C3×D3—28no—
222#41—62no—
223#41—62no—
224C2×Q8—31no—
225C2×D4—32no—
226C2×D4D3×C3⋊D311yescharacteristic
227C22×C4—31no—
228C22×C4—32no—
229M16—92no—
230M16—272no—
231M16—542no—
232C2×C8—542no—
233C22×C3—22no—
234C22×C3—28no—
235C22×C3—24no—
236C22×C3—38no—
237C22×C3—48no—
238C22×C3—48no—
239C22×C3—42no—
240C22×C3—88no—
241C22×C3—124no—
242C22×C3C2×D3×Dic314yes—
243C22×C3C22×C3⋊Dic311yescharacteristic
244C2×D3—41no—
245C2×D3#375712yes—
246C3×C4—28no—
247C3×C4—34no—
248C3×C4—34no—
249C3×C4—68no—
250C3×C4D3×#6718yes—
251C3×C4C2×#63812yes—
252Dic3—22no—
253Dic3#375711yescharacteristic
254Dic3#375711yescharacteristic
255C32—24no—
256C32#474511yescharacteristic, complemented
257C32#523714yescomplemented
258C23—31no—
259C23D3×C3⋊Dic312yes—
260Q8—62no—
261D4—22no—
262D4—64no—
263C2×C4—31no—
264C2×C4—32no—
265C2×C4—32no—
266C2×C4—32no—
267C2×C4—62no—
268C2×C4—62no—
269C2×C4C2×D3×C3⋊D311yescharacteristic
270C8—182no—
271C8—542no—
272C2×C3—24no—
273C2×C3—21no—
274C2×C3—24no—
275C2×C3—44no—
276C2×C3—48no—
277C2×C3—42no—
278C2×C3—68no—
279C2×C3—88no—
280C2×C3D3×#21714yes—
281C2×C3C2×#375711yescharacteristic
282D3—22no—
283C22—22no—
284C22—32no—
285C22—42no—
286C22—121no—
287C22C2×D3×C3⋊Dic311yescharacteristic
288C4—31no—
289C4—31no—
290C4—62no—
291C4D3×#63812yes—
292C3—24no—
293C3#1059150114yescomplemented
294C3#1059293611yescharacteristic, complemented
295C2—21no—
296C2—42no—
297C2—62no—
298C2D3×#375711yescharacteristic
subgroupG/H
centerC2D3×#3757
commutatorC22×C33C22×C4
FrattiniC2×C4C2×D3×C3⋊D3
FittingC2×C33×D4C22
socleC2×C33C2×C22⋊C4
prime radicalC2×C32×#237C2
cosocleC2×C33×C4C23
hypercenterC2×D4D3×C3⋊D3
hypocenterC33#413
LCSG ▷ C22×C33 ▷ C2×C33 ▷ C33
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
2. D3×#3748C2semidirect
3. #11722203C2semidirect
4. C3×#63660C2semidirect
5. #11722028C2semidirect
27. #63660D3semidirect
178. C33#413semidirect
256. C32#4745semidirect
257. C32#5237semidirect
293. C3#10591501semidirect
294. C3#10592936semidirect

Maximal subgroups

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

subgroupindex
C2×C32×#2372
D3×#37482
#117222032
C3×#636602
#117220282
#105929363
#105915013