Finite Group Databasebeta

← #13140391

Subgroups — #13140391

Order 1728.

612 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 612 conjugacy classes of subgroups — too many to draw. Only its 85 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
1C3×#63710C211yesmaximal, characteristic, complemented
2#11721840C211yesmaximal, characteristic, complemented
3#11720873C212yesmaximal, complemented
4C9×#833C211yesmaximal, characteristic, complemented
5#11720291C212yesmaximal, complemented
6#10591510—31nomaximal
7#10588427—31nomaximal
8C2×C3×D4×C9C2211yescharacteristic
9C2×#6233—22no—
10C2×C4⋊(C3⋊D9)C2211yescharacteristic, complemented
11C2×C3×#611—22no—
12C3×#3699—22no—
13C3×D4×D9—22no—
14C3×#3697C2212yes—
15C4×C3⋊Dic9C2211yescharacteristic, complemented
16C9×#208C2212yescomplemented
17#88417—22no—
18#88416—22no—
19C3×#1004—31no—
20#64149—31no—
21#63825—32no—
22#63710D311yescharacteristic, complemented
23C3×#833D311yescharacteristic
24#63568—32no—
25#63502—31no—
26#63502—61no—
27#63470—32no—
28C9×#100—31no—
29#63384—32no—
30C23×C3×C9D412yes—
31C22×C3⋊D9—21no—
32C22×C3×D9—22no—
33C3×D4×C9—22no—
34C2×C3×C4×C9C2311yescharacteristic
35#6233—44no—
36C2×C3⋊Dic9—21no—
37C4⋊(C3⋊D9)—22no—
38C3×#611—24no—
39C3×#611—22no—
40C2×C3×Dic9—22no—
41C3×D36D412yescomplemented
42C3×C4×D9—22no—
43C3×Dic18D412yescomplemented
44C9×C3⋊C8—22no—
45#5241—91no—
46#4747—91no—
47C2×C32×D4C2×D311yescharacteristic
48C2×#638—62no—
49C2×C4⋊(C3⋊D3)—31no—
50C2×C3×#67—62no—
51C3×#237—62no—
52C3×D3×D4—62no—
53C3×#235—32no—
54C4×C3⋊Dic3—31no—
55C3×#208C2×D312yes—
56#3717—62no—
57#3714—62no—
58C2×D4×C9—21no—
59C2×D4×C9C2×D311yescharacteristic
60C2×#611—22no—
61C2×#611—64no—
62C2×#611—62no—
63#3699—22no—
64D4×D9—22no—
65#3697C2×D312yes—
66C2×D36—31no—
67C2×D36—61no—
68C9×M16—32no—
69C4×Dic9—31no—
70C4×Dic9—61no—
71D72—62no—
72#3664—62no—
73C22×C3×C9—22no—
74C22×C3×C9—42no—
75C22×C3×C9C2×D411yescharacteristic
76C2×C3⋊D9—42no—
77C2×C3⋊D9—41no—
78C2×C3×D9—22no—
79C2×C3×D9—82no—
80C3×C4×C9C2×D412yes—
81C3⋊Dic9—22no—
82C3×Dic9—22no—
83C3×H(5,𝔽2)—91no—
84#1004—31no—
85#949—92no—
86#940—91no—
87#940—91no—
88#940—181no—
89#908—92no—
90C3×#100—31no—
91#837—92no—
92#833D911yescharacteristic, complemented
93#825—92no—
94C23×C32D1212yes—
95C22×C3⋊D3—61no—
96C22×C3×D3—62no—
97C32×D4—22no—
98C2×C32×C4C22×D311yescharacteristic
99#638—124no—
100C2×C3⋊Dic3—61no—
101C4⋊(C3⋊D3)—62no—
102C3×#67—64no—
103C3×#67—62no—
104C2×C3×Dic3—62no—
105C3×D12—32no—
106C3×C4×D3—62no—
107C3×Dic6—32no—
108C23×C9—22no—
109C23×C9#6712yes—
110C22×D9—22no—
111C22×D9—62no—
112C22×D9—61no—
113C3×C3⋊C8—22no—
114D4×C9—22no—
115D4×C9—42no—
116C2×C4×C9—21no—
117C2×C4×C9C22×D311yescharacteristic
118#611—24no—
119#611—22no—
120#611—124no—
121#611—128no—
122C2×Dic9—22no—
123C2×Dic9—62no—
124C2×Dic9—61no—
125D36—62no—
126D36—122no—
127D36#6712yescomplemented
128C4×D9—22no—
129Dic18#6712yescomplemented
130C8×C9—62no—
131#453—271noSylow
132C2×C3×C9—21no—
133C2×C3×C9—42no—
134C2×C3×C9#12011yescharacteristic
135C3⋊D9—81no—
136C3×D9—42no—
137C3×#41—92no—
138C3×#41—182no—
139C2×C3×D4—21no—
140C2×C3×D4—182no—
141C2×C3×D4—182no—
142C2×C3×D4C2×D911yescharacteristic
143C2×C3×D4D3211yescharacteristic
144C2×#67—62no—
145C2×#67—184no—
146C2×#67—182no—
147C2×#67—182no—
148#237—62no—
149D3×D4—62no—
150#235—32no—
151C2×D12—91no—
152C2×D12—91no—
153C2×D12—181no—
154C3×M16—32no—
155#215—182no—
156#213—182no—
157C4×Dic3—91no—
158C4×Dic3—91no—
159C4×Dic3—181no—
160#208C2×D912yescomplemented
161D24—182no—
162#204—182no—
163C22×C32—22no—
164C22×C32—42no—
165C22×C32C2×D1211yescharacteristic
166C2×C3⋊D3—122no—
167C2×C3⋊D3—121no—
168C2×C3×D3—62no—
169C2×C3×D3—242no—
170C32×C4D3×D412yes—
171C3⋊Dic3—62no—
172C3×Dic3—62no—
173C22×C9—22no—
174C22×C9—21no—
175C22×C9—42no—
176C22×C9—42no—
177C22×C9—44no—
178C22×C9C2×#6711yescharacteristic
179C2×D9—22no—
180C2×D9—82no—
181C2×D9—122no—
182C2×D9—124no—
183C2×D9—122no—
184C2×D9—121no—
185C4×C9—22no—
186C4×C9C2×#6712yes—
187Dic9—22no—
188Dic9—64no—
189Dic9—62no—
190H(5,𝔽2)—91no—
191AGL(1,ℤ/8)—272no—
192#127—271no—
193#104—272no—
194#100—31no—
195C3×C9#45311yescharacteristic, complemented, Sylow
196C23×C3—22no—
197C23×C3—182no—
198C23×C3D3612yes—
199C23×C3#62612yes—
200C22×D3—62no—
201C22×D3—182no—
202C22×D3—181no—
203C22×D3—181no—
204C3×Q8—92no—
205C3×D4—22no—
206C3×D4—22no—
207C3×D4—42no—
208C3×D4—92no—
209C3×D4—184no—
210C3×D4—182no—
211C2×C3×C4—21no—
212C2×C3×C4—182no—
213C2×C3×C4—182no—
214C2×C3×C4C22×D911yescharacteristic
215C2×C3×C4C2×D3211yescharacteristic
216#67—64no—
217#67—62no—
218#67—364no—
219#67—364no—
220#67—368no—
221C2×Dic3—62no—
222C2×Dic3—182no—
223C2×Dic3—181no—
224C2×Dic3—181no—
225D12—32no—
226D12—182no—
227D12—182no—
228D12—362no—
229C4×D3—62no—
230Dic6—32no—
231C3×C8—62no—
232C3⋊C8—22no—
233C2×C32—21no—
234C2×C32—42no—
235C2×C32#88211yescharacteristic
236C3⋊D3—241no—
237C3×D3—122no—
238C2×C9—22no—
239C2×C9—21no—
240C2×C9—21no—
241C2×C9—42no—
242C2×C9—82no—
243C2×C9#95311yescharacteristic
244D9—42no—
245D9—242no—
246D9—241no—
247#41—92no—
248#41—182no—
249C2×D4—182no—
250C2×D4—182no—
251C2×D4—271no—
252C2×D4—542no—
253C2×D4D3×D911yescharacteristic
254SD16—542no—
255D8—542no—
256M16—32no—
257C42—271no—
258C22×C3—22no—
259C22×C3—22no—
260C22×C3—21no—
261C22×C3—42no—
262C22×C3—42no—
263C22×C3—42no—
264C22×C3—44no—
265C22×C3—182no—
266C22×C3—722no—
267C22×C3C2×D3611yescharacteristic
268C22×C3C2×#62611yescharacteristic
269C2×D3—62no—
270C2×D3—242no—
271C2×D3—362no—
272C2×D3—362no—
273C2×D3—364no—
274C2×D3—362no—
275C2×D3—361no—
276C2×D3—361no—
277C3×C4—22no—
278C3×C4—182no—
279C3×C4D4×D912yes—
280C3×C4D3×#6712yes—
281Dic3—62no—
282Dic3—182no—
283Dic3—184no—
284Dic3—182no—
285C32#474711yescharacteristic
286C9—21no—
287C9#524111yescharacteristic, complemented
288C23—182no—
289C23—541no—
290C23#619212yes—
291Q8—92no—
292D4—22no—
293D4—92no—
294D4—184no—
295D4—182no—
296D4—542no—
297D4—1084no—
298C2×C4—182no—
299C2×C4—182no—
300C2×C4—541no—
301C2×C4C2×D3×D911yescharacteristic
302C8—62no—
303C2×C3—22no—
304C2×C3—21no—
305C2×C3—21no—
306C2×C3—21no—
307C2×C3—42no—
308C2×C3—42no—
309C2×C3—82no—
310C2×C3—362no—
311C2×C3#6344411yescharacteristic
312C2×C3#6397711yescharacteristic
313D3—122no—
314D3—722no—
315D3—721no—
316D3—721no—
317C22—22no—
318C22—42no—
319C22—182no—
320C22—722no—
321C22—1082no—
322C22—1081no—
323C22C2×#619211yescharacteristic
324C4—182no—
325C4—542no—
326C4D9×#6712yes—
327C3—21no—
328C3#1058842711yescharacteristic, complemented
329C3#1059151011yescharacteristic
330C2—21no—
331C2—42no—
332C2—362no—
333C2—2161no—
334C2#1172127611yescharacteristic
subgroupG/H
centerC2#11721276
commutator, cosocleC2×C3×C4×C9C23
FrattiniC2×C3×C4C2×D32
FittingC2×C3×D4×C9C22
socleC2×C32#882
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. C3×#63710C2semidirect
2. #11721840C2semidirect
3. #11720873C2semidirect
4. C9×#833C2semidirect
5. #11720291C2semidirect
10. C2×C4⋊(C3⋊D9)C22semidirect
15. C4×C3⋊Dic9C22semidirect
16. C9×#208C22semidirect
22. #63710D3semidirect
41. C3×D36D4semidirect
43. C3×Dic18D4semidirect
92. #833D9semidirect
127. D36#67semidirect
129. Dic18#67semidirect
160. #208C2×D9semidirect
195. C3×C9#453semidirect
287. C9#5241semidirect
328. C3#10588427semidirect

Maximal subgroups

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

subgroupindex
C3×#637102
#117218402
#117208732
C9×#8332
#117202912
#105915103
#105884273