Finite Group Databasebeta

← #13140486

Subgroups — #13140486

Order 1728.

550 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 550 conjugacy classes of subgroups — too many to draw. Only its 89 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
1C9×#1004C211yesmaximal, characteristic, complemented
2C3×#63502C211yesmaximal, characteristic, complemented
3#11720882C212yesmaximal, complemented
4#11720575C211yesmaximal, characteristic, complemented
5#11720301C212yesmaximal, complemented
6#10591509—31nomaximal
7#10588922—31nomaximal
8C2×C3×D4×C9C2211yescharacteristic
9C2×C9×#67—22no—
10C2×C3×#611—22no—
11C9×#237—22no—
12D3×D4×C9—22no—
13C9×#235C2212yes—
14C2×C3×D36C2211yescharacteristic, complemented
15#88532C2212yescomplemented
16C3×C4×Dic9C2211yescharacteristic, complemented
17#88425—22no—
18#88421—22no—
19C3×#1004D311yescharacteristic
20C3×#940—31no—
21#63823—32no—
22C9×H(5,𝔽2)—31no—
23#63660—31no—
24#63569—32no—
25#63512—32no—
26#63502D311yescharacteristic, complemented
27#63396—32no—
28#63391—31no—
29#63391—61no—
30C23×C3×C9D412yes—
31C22×D3×C9—22no—
32C22×C3×D9—21no—
33C3×D4×C9—22no—
34C2×C3×C4×C9C2311yescharacteristic
35C9×#67—24no—
36C9×#67—22no—
37C3×#611—44no—
38C2×C9×Dic3—22no—
39C2×C3×Dic9—21no—
40C9×D12D412yescomplemented
41C4×D3×C9—22no—
42C3×D36—22no—
43C9×Dic6D412yescomplemented
44C3⋊(C9⋊C8)—22no—
45#5241—91no—
46#5077—91no—
47C2×C32×D4C2×D311yescharacteristic
48C2×C3×#67—22no—
49C2×C3×#67—62no—
50C3×#237—22no—
51C3×D3×D4—22no—
52C3×#235C2×D312yes—
53C2×C3×D12—31no—
54#3748—32no—
55C3×C4×Dic3—31no—
56#3715—62no—
57#3713—62no—
58C9×#41—32no—
59C9×#41—62no—
60C2×D4×C9—21no—
61C2×D4×C9—62no—
62C2×D4×C9—62no—
63C2×D4×C9C2×D311yescharacteristic
64C2×#611—22no—
65C2×D36C2×D311yescharacteristic, complemented
66#3675—62no—
67#3673—62no—
68C4×Dic9C2×D311yescharacteristic, complemented
69#3667—32no—
70#3667—62no—
71C22×C3×C9—22no—
72C22×C3×C9—42no—
73C22×C3×C9C2×D411yescharacteristic
74C2×D3×C9—22no—
75C2×D3×C9—82no—
76C2×C3×D9—42no—
77C2×C3×D9—41no—
78C3×C4×C9C2×D412yes—
79C9×Dic3—22no—
80C3×Dic9—22no—
81C3×H(5,𝔽2)—31no—
82#1004D911yescharacteristic, complemented
83C3×#127—91no—
84#949—92no—
85#940—31no—
86#932—92no—
87#837—92no—
88#833—91no—
89#833—91no—
90#833—181no—
91#805—92no—
92C23×C32#6712yes—
93C22×C3×D3—22no—
94C22×C3×D3—61no—
95C32×D4—22no—
96C2×C32×C4C22×D311yescharacteristic
97C3×#67—24no—
98C3×#67—22no—
99C3×#67—124no—
100C2×C3×Dic3—22no—
101C2×C3×Dic3—61no—
102C3×D12—62no—
103C3×D12#6712yes—
104C3×C4×D3—22no—
105C3×Dic6#6712yes—
106C23×C9—22no—
107C23×C9—62no—
108C23×C9#6712yes—
109C22×D9—21no—
110C3⋊(C3⋊C8)—62no—
111Q8×C9—32no—
112D4×C9—22no—
113D4×C9—32no—
114D4×C9—42no—
115D4×C9—64no—
116D4×C9—62no—
117C2×C4×C9—21no—
118C2×C4×C9—62no—
119C2×C4×C9—62no—
120C2×C4×C9C22×D311yescharacteristic
121#611—44no—
122C2×Dic9—21no—
123D36—22no—
124C9⋊C8—62no—
125C9⋊C8—122no—
126#453—271noSylow
127C2×C3×C9—21no—
128C2×C3×C9—42no—
129C2×C3×C9#12011yescharacteristic
130D3×C9—42no—
131C3×D9—81no—
132C3×#41—32no—
133C3×#41—62no—
134C2×C3×D4—21no—
135C2×C3×D4—62no—
136C2×C3×D4—62no—
137C2×C3×D4—91no—
138C2×C3×D4—182no—
139C2×C3×D4C2×D911yescharacteristic
140C2×C3×D4D3211yescharacteristic
141C2×#67—22no—
142C2×#67—62no—
143#237—22no—
144D3×D4—22no—
145#235C2×D912yes—
146C2×D12—31no—
147C3×C42—91no—
148#215—182no—
149#214—182no—
150#213—182no—
151#213—182no—
152C4×Dic3—31no—
153#208—92no—
154#208—92no—
155#208—182no—
156C22×C32—22no—
157C22×C32—42no—
158C22×C32C2×#6711yescharacteristic
159C2×C3×D3—22no—
160C2×C3×D3—82no—
161C2×C3×D3—122no—
162C2×C3×D3—121no—
163C32×C4C2×#6712yes—
164C3×Dic3—22no—
165C3×Dic3—62no—
166C22×C9—22no—
167C22×C9—21no—
168C22×C9—42no—
169C22×C9—42no—
170C22×C9—62no—
171C22×C9—82no—
172C22×C9—242no—
173C22×C9C2×#6711yescharacteristic
174C2×D9—42no—
175C2×D9—41no—
176C4×C9—22no—
177C4×C9—62no—
178C4×C9D3×D412yes—
179Dic9—22no—
180H(5,𝔽2)—31no—
181AGL(1,ℤ/8)—272no—
182#127—91no—
183#104—272no—
184#100—271no—
185C3×C9#45311yescharacteristic, complemented, Sylow
186C23×C3—22no—
187C23×C3—62no—
188C23×C3—181no—
189C23×C3#61112yes—
190C23×C3#62512yes—
191C22×D3—22no—
192C22×D3—61no—
193C3×Q8—32no—
194C3×D4—22no—
195C3×D4—22no—
196C3×D4—32no—
197C3×D4—42no—
198C3×D4—64no—
199C3×D4—62no—
200C3×D4—182no—
201C3×D4—364no—
202C2×C3×C4—21no—
203C2×C3×C4—62no—
204C2×C3×C4—62no—
205C2×C3×C4—181no—
206C2×C3×C4C22×D911yescharacteristic
207C2×C3×C4C2×D3211yescharacteristic
208#67—24no—
209#67—22no—
210#67—124no—
211C2×Dic3—22no—
212C2×Dic3—61no—
213D12—62no—
214D12#61112yescomplemented
215C4×D3—22no—
216Dic6#61112yescomplemented
217C3⋊C8—182no—
218C3⋊C8—182no—
219C3⋊C8—362no—
220C2×C32—21no—
221C2×C32—42no—
222C2×C32#95311yescharacteristic
223C3×D3—42no—
224C3×D3—241no—
225C2×C9—21no—
226C2×C9—21no—
227C2×C9—41no—
228C2×C9—42no—
229C2×C9—82no—
230C2×C9—122no—
231C2×C9#93711yescharacteristic
232D9—81no—
233#41—32no—
234#41—62no—
235C2×D4—62no—
236C2×D4—62no—
237C2×D4—91no—
238C2×D4—182no—
239C2×D4D3×D911yescharacteristic
240SD16—542no—
241D8—542no—
242M16—272no—
243C42—91no—
244C22×C3—22no—
245C22×C3—22no—
246C22×C3—21no—
247C22×C3—42no—
248C22×C3—42no—
249C22×C3—42no—
250C22×C3—62no—
251C22×C3—82no—
252C22×C3—242no—
253C22×C3—362no—
254C22×C3—361no—
255C22×C3C2×#61111yescharacteristic
256C22×C3C2×#62511yescharacteristic
257C2×D3—22no—
258C2×D3—82no—
259C2×D3—122no—
260C2×D3—121no—
261C3×C4—22no—
262C3×C4—62no—
263C3×C4—182no—
264C3×C4C2×#61112yes—
265C3×C4D3×#6712yes—
266Dic3—22no—
267Dic3—62no—
268C32#524111yescharacteristic
269C9—21no—
270C9#507711yescharacteristic, complemented
271C23—62no—
272C23—181no—
273C23#619412yes—
274Q8—32no—
275D4—22no—
276D4—32no—
277D4—64no—
278D4—62no—
279D4—182no—
280D4—364no—
281C2×C4—62no—
282C2×C4—62no—
283C2×C4—181no—
284C2×C4C2×D3×D911yescharacteristic
285C8—542no—
286C2×C3—21no—
287C2×C3—21no—
288C2×C3—21no—
289C2×C3—41no—
290C2×C3—42no—
291C2×C3—42no—
292C2×C3—82no—
293C2×C3—122no—
294C2×C3—721no—
295C2×C3#6351511yescharacteristic
296C2×C3#6397611yescharacteristic
297D3—42no—
298D3—241no—
299C22—22no—
300C22—42no—
301C22—62no—
302C22—242no—
303C22—362no—
304C22—361no—
305C22C2×#619411yescharacteristic
306C4—62no—
307C4—182no—
308C4D3×#61112yes—
309C3—21no—
310C3#1058892211yescharacteristic, complemented
311C3#1059150911yescharacteristic
312C2—21no—
313C2—42no—
314C2—122no—
315C2—721no—
316C2#1172128611yescharacteristic
subgroupG/H
centerC2#11721286
commutator, cosocleC2×C3×C4×C9C23
FrattiniC2×C3×C4C2×D32
FittingC2×C3×D4×C9C22
socleC2×C32#953
hypercenterC2×D4D3×D9
hypocenterC3×C9#453
LCSG ▷ C2×C3×C4×C9 ▷ C2×C3×C9 ▷ C3×C9
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
1. C9×#1004C2semidirect
2. C3×#63502C2semidirect
3. #11720882C2semidirect
4. #11720575C2semidirect
5. #11720301C2semidirect
14. C2×C3×D36C22semidirect
15. #88532C22semidirect
16. C3×C4×Dic9C22semidirect
26. #63502D3semidirect
40. C9×D12D4semidirect
43. C9×Dic6D4semidirect
65. C2×D36C2×D3semidirect
68. C4×Dic9C2×D3semidirect
82. #1004D9semidirect
185. C3×C9#453semidirect
214. D12#611semidirect
216. Dic6#611semidirect
270. C9#5077semidirect
310. C3#10588922semidirect

Maximal subgroups

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

subgroupindex
C9×#10042
C3×#635022
#117208822
#117205752
#117203012
#105915093
#105889223