Finite Group Databasebeta

← #13140488

Subgroups — #13140488

Order 1728.

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

normal subgroups only

This group has 612 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
2#11721840C211yesmaximal, characteristic, complemented
3#11720885C212yesmaximal, complemented
4C3×#63391C211yesmaximal, characteristic, complemented
5#11720300C212yesmaximal, complemented
6#10591510—31nomaximal
7#10588922—31nomaximal
8C2×C3×D4×C9C2211yescharacteristic
9C2×#6233—22no—
10C2×C4⋊(C3⋊D9)C2211yescharacteristic, complemented
11C2×C9×#67—22no—
12C9×#237—22no—
13D3×D4×C9—22no—
14C9×#235C2212yes—
15C4×C3⋊Dic9C2211yescharacteristic, complemented
16C3×#3667C2212yescomplemented
17#88426—22no—
18#88422—22no—
19C3×#1004D311yescharacteristic
20#64149—31no—
21#63825—32no—
22C9×H(5,𝔽2)—31no—
23C3×#833—31no—
24#63568—32no—
25#63512—32no—
26#63502—31no—
27#63502—61no—
28#63396—32no—
29#63391D311yescharacteristic, complemented
30C23×C3×C9D412yes—
31C22×C3⋊D9—21no—
32C22×D3×C9—22no—
33C3×D4×C9—22no—
34C2×C3×C4×C9C2311yescharacteristic
35#6233—44no—
36C2×C3⋊Dic9—21no—
37C4⋊(C3⋊D9)—22no—
38C9×#67—24no—
39C9×#67—22no—
40C2×C9×Dic3—22no—
41C9×D12D412yescomplemented
42C4×D3×C9—22no—
43C9×Dic6D412yescomplemented
44C3×C9⋊C8—22no—
45#5241—91no—
46#4747—91no—
47C2×C32×D4C2×D311yescharacteristic
48C2×#638—62no—
49C2×C4⋊(C3⋊D3)—31no—
50C2×C3×#67—22no—
51C3×#237—22no—
52C3×D3×D4—22no—
53C3×#235C2×D312yes—
54C4×C3⋊Dic3—31no—
55C3×#208—32no—
56#3717—62no—
57#3714—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—64no—
65C2×#611—62no—
66C2×D36—31no—
67C2×D36—61no—
68#3675—62no—
69#3673—62no—
70C4×Dic9—31no—
71C4×Dic9—61no—
72#3667C2×D312yescomplemented
73C22×C3×C9—22no—
74C22×C3×C9—42no—
75C22×C3×C9C2×D411yescharacteristic
76C2×C3⋊D9—42no—
77C2×C3⋊D9—41no—
78C2×D3×C9—22no—
79C2×D3×C9—82no—
80C3×C4×C9C2×D412yes—
81C3⋊Dic9—22no—
82C9×Dic3—22no—
83C3×H(5,𝔽2)—31no—
84#1004D911yescharacteristic, complemented
85#949—92no—
86#940—91no—
87#940—91no—
88#940—181no—
89#908—92no—
90C3×#100—91no—
91#837—92no—
92#833—31no—
93#825—92no—
94C23×C32#6712yes—
95C22×C3⋊D3—61no—
96C22×C3×D3—22no—
97C32×D4—22no—
98C2×C32×C4C22×D311yescharacteristic
99#638—124no—
100C2×C3⋊Dic3—61no—
101C4⋊(C3⋊D3)—62no—
102C3×#67—24no—
103C3×#67—22no—
104C2×C3×Dic3—22no—
105C3×D12#6712yes—
106C3×C4×D3—22no—
107C3×Dic6#6712yes—
108C23×C9—22no—
109C23×C9—62no—
110C23×C9D1212yes—
111C22×D9—62no—
112C22×D9—61no—
113C3×C3⋊C8—62no—
114Q8×C9—32no—
115D4×C9—22no—
116D4×C9—32no—
117D4×C9—42no—
118D4×C9—64no—
119D4×C9—62no—
120C2×C4×C9—21no—
121C2×C4×C9—62no—
122C2×C4×C9—62no—
123C2×C4×C9C22×D311yescharacteristic
124#611—124no—
125#611—128no—
126C2×Dic9—62no—
127C2×Dic9—61no—
128D36—62no—
129D36—122no—
130C9⋊C8—22no—
131#453—271noSylow
132C2×C3×C9—21no—
133C2×C3×C9—42no—
134C2×C3×C9#12011yescharacteristic
135C3⋊D9—81no—
136D3×C9—42no—
137C3×#41—32no—
138C3×#41—62no—
139C2×C3×D4—21no—
140C2×C3×D4—62no—
141C2×C3×D4—62no—
142C2×C3×D4C2×D911yescharacteristic
143C2×C3×D4D3211yescharacteristic
144C2×#67—22no—
145C2×#67—184no—
146C2×#67—182no—
147C2×#67—182no—
148#237—22no—
149D3×D4—22no—
150#235C2×D912yes—
151C2×D12—91no—
152C2×D12—91no—
153C2×D12—181no—
154C3×M16—92no—
155#215—182no—
156#213—182no—
157C4×Dic3—91no—
158C4×Dic3—91no—
159C4×Dic3—181no—
160#208—32no—
161D24—182no—
162#204—182no—
163C22×C32—22no—
164C22×C32—42no—
165C22×C32C2×#6711yescharacteristic
166C2×C3⋊D3—122no—
167C2×C3⋊D3—121no—
168C2×C3×D3—22no—
169C2×C3×D3—82no—
170C32×C4C2×#6712yes—
171C3⋊Dic3—62no—
172C3×Dic3—22no—
173C22×C9—22no—
174C22×C9—21no—
175C22×C9—42no—
176C22×C9—42no—
177C22×C9—44no—
178C22×C9—62no—
179C22×C9—242no—
180C22×C9C2×D1211yescharacteristic
181C2×D9—122no—
182C2×D9—124no—
183C2×D9—122no—
184C2×D9—121no—
185C4×C9—22no—
186C4×C9—62no—
187C4×C9D3×D412yes—
188Dic9—64no—
189Dic9—62no—
190H(5,𝔽2)—31no—
191AGL(1,ℤ/8)—272no—
192#127—271no—
193#104—272no—
194#100—91no—
195C3×C9#45311yescharacteristic, complemented, Sylow
196C23×C3—22no—
197C23×C3—62no—
198C23×C3#61112yes—
199C23×C3#62612yes—
200C22×D3—22no—
201C22×D3—182no—
202C22×D3—181no—
203C22×D3—181no—
204C3×Q8—32no—
205C3×D4—22no—
206C3×D4—22no—
207C3×D4—32no—
208C3×D4—42no—
209C3×D4—64no—
210C3×D4—62no—
211C2×C3×C4—21no—
212C2×C3×C4—62no—
213C2×C3×C4—62no—
214C2×C3×C4C22×D911yescharacteristic
215C2×C3×C4C2×D3211yescharacteristic
216#67—24no—
217#67—22no—
218#67—364no—
219#67—364no—
220#67—368no—
221C2×Dic3—22no—
222C2×Dic3—182no—
223C2×Dic3—181no—
224C2×Dic3—181no—
225D12—182no—
226D12—182no—
227D12—362no—
228D12#61112yescomplemented
229C4×D3—22no—
230Dic6#61112yescomplemented
231C3×C8—182no—
232C3⋊C8—62no—
233C2×C32—21no—
234C2×C32—42no—
235C2×C32#95311yescharacteristic
236C3⋊D3—241no—
237C3×D3—42no—
238C2×C9—22no—
239C2×C9—21no—
240C2×C9—21no—
241C2×C9—42no—
242C2×C9—82no—
243C2×C9—122no—
244C2×C9#88211yescharacteristic
245D9—242no—
246D9—241no—
247#41—32no—
248#41—62no—
249C2×D4—62no—
250C2×D4—62no—
251C2×D4—271no—
252C2×D4—542no—
253C2×D4D3×D911yescharacteristic
254SD16—542no—
255D8—542no—
256M16—92no—
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—62no—
266C22×C3—242no—
267C22×C3C2×#61111yescharacteristic
268C22×C3C2×#62611yescharacteristic
269C2×D3—22no—
270C2×D3—82no—
271C2×D3—362no—
272C2×D3—362no—
273C2×D3—364no—
274C2×D3—362no—
275C2×D3—361no—
276C2×D3—361no—
277C3×C4—22no—
278C3×C4—62no—
279C3×C4C2×#61112yes—
280C3×C4D3×#6712yes—
281Dic3—22no—
282Dic3—182no—
283Dic3—184no—
284Dic3—182no—
285C32#524111yescharacteristic
286C9—21no—
287C9#474711yescharacteristic, complemented
288C23—62no—
289C23—541no—
290C23#619512yes—
291Q8—32no—
292D4—22no—
293D4—32no—
294D4—64no—
295D4—62no—
296D4—542no—
297D4—1084no—
298C2×C4—62no—
299C2×C4—62no—
300C2×C4—541no—
301C2×C4C2×D3×D911yescharacteristic
302C8—182no—
303C2×C3—22no—
304C2×C3—21no—
305C2×C3—21no—
306C2×C3—21no—
307C2×C3—42no—
308C2×C3—42no—
309C2×C3—82no—
310C2×C3—122no—
311C2×C3#6351511yescharacteristic
312C2×C3#6397711yescharacteristic
313D3—42no—
314D3—722no—
315D3—721no—
316D3—721no—
317C22—22no—
318C22—42no—
319C22—62no—
320C22—242no—
321C22—1082no—
322C22—1081no—
323C22C2×#619511yescharacteristic
324C4—62no—
325C4—542no—
326C4D3×#61112yes—
327C3—21no—
328C3#1058892211yescharacteristic, complemented
329C3#1059151011yescharacteristic
330C2—21no—
331C2—42no—
332C2—122no—
333C2—2161no—
334C2#1172128711yescharacteristic
subgroupG/H
centerC2#11721287
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. #11721840C2semidirect
3. #11720885C2semidirect
4. C3×#63391C2semidirect
5. #11720300C2semidirect
10. C2×C4⋊(C3⋊D9)C22semidirect
15. C4×C3⋊Dic9C22semidirect
16. C3×#3667C22semidirect
29. #63391D3semidirect
41. C9×D12D4semidirect
43. C9×Dic6D4semidirect
72. #3667C2×D3semidirect
84. #1004D9semidirect
195. C3×C9#453semidirect
228. D12#611semidirect
230. Dic6#611semidirect
287. C9#4747semidirect
328. C3#10588922semidirect

Maximal subgroups

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

subgroupindex
C9×#10042
#117218402
#117208852
C3×#633912
#117203002
#105915103
#105889223