Finite Group Databasebeta

← #13138506

Subgroups — #13138506

Order 1728.

400 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 400 conjugacy classes of subgroups — too many to draw. Only its 95 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
1#11721025C211yesmaximal, characteristic
2#11720910C211yesmaximal, characteristic, complemented
3C2×#88423C211yesmaximal, characteristic
4#11720538C211yesmaximal, characteristic, complemented
5C9×#821C211yesmaximal, characteristic
6C3×#63376C211yesmaximal, characteristic, complemented
7#11720263C211yesmaximal, characteristic, complemented
8#10590743—31nomaximal
9#10588437—31nomaximal
10C2×C9×D12C2211yescharacteristic
11C2×C3×Dic18C2211yescharacteristic
12C2×D3×Dic9—21no—
13C2×C3×C8×C9C2211yescharacteristic
14C4×C3⋊Dic9C2211yescharacteristic
15C2×C3⋊(C9⋊C8)C2211yescharacteristic
16C9×C4⋊Dic3C2211yescharacteristic
17C3×C4⋊Dic9C2211yescharacteristic
18(C2×D3)⋊Dic9—21no—
19Dic9⋊Dic3—21no—
20Dic3×Dic9—21no—
21#88423C414yescomplemented
22#63898—31no—
23#63843—31no—
24C2×#3715—31no—
25#63642—31no—
26C3×#821D311yescharacteristic
27C3×#816—31no—
28#63576—31no—
29D4×Dic9—31no—
30C2×#3672—31no—
31#63449—31no—
32C9×#102—31no—
33#63376D311yescharacteristic, complemented
34#63375—31no—
35#63375—61no—
36#63366—31no—
37C22×D3×C9—21no—
38C2×C3×C4×C9C2311yescharacteristic
39C2×C3⋊Dic9D411yescharacteristic
40C2×C3⋊Dic9D411yescharacteristic
41C2×C9×Dic3—21no—
42C2×C3×Dic9—21no—
43C2×C3×Dic9—21no—
44C9×D12—21no—
45C9×D12C2×C412yes—
46C3×Dic18—21no—
47C3×Dic18C2×C412yescomplemented
48D3×Dic9—42no—
49C3×C8×C9—21no—
50C3⋊(C9⋊C8)C2×C412yescomplemented
51#4817—91no—
52#4757—91no—
53C2×C3×D12C2×D311yescharacteristic
54C2×C3×Dic6—31no—
55C2×D3×Dic3—61no—
56C2×C32×C8C2×D311yescharacteristic
57C4×C3⋊Dic3—31no—
58C2×C3⋊(C3⋊C8)—31no—
59C3×C4⋊Dic3—31no—
60C3×C4⋊Dic3C2×D311yescharacteristic
61Dic3⋊Dic3—61no—
62(C2×D3)⋊Dic3—61no—
63Dic32—61no—
64#3715—34no—
65C2×D4×C9—31no—
66C22×Dic9—61no—
67C2×Dic18C2×D311yescharacteristic
68C2×C8×C9—21no—
69C2×C8×C9C2×D311yescharacteristic
70C9×C4⋊C4—31no—
71#3676—61no—
72#3672—34no—
73C4⋊Dic9C2×D311yescharacteristic
74Dic9⋊C4—61no—
75C4×Dic9—31no—
76C4×Dic9—61no—
77C4×Dic9—61no—
78C2×C9⋊C8—31no—
79C2×C9⋊C8—61no—
80C22×C3×C9C2×D411yescharacteristic
81C2×D3×C9—22no—
82C2×D3×C9—41no—
83C2×D3×C9—41no—
84C3×C4×C9C22×C411yescharacteristic
85C3×C4×C9#4111yescharacteristic
86C3⋊Dic9—21no—
87C3⋊Dic9—21no—
88C9×Dic3—41no—
89C3×Dic9—22no—
90C3×Dic9—41no—
91Q8×Dic3—91no—
92C2×#215—91no—
93D4×Dic3—91no—
94C2×#214—91no—
95#893—91no—
96#887—91no—
97C3×#103—91no—
98C3×#102—31no—
99#821D911yescharacteristic
100#816—31no—
101#815—91no—
102#815—91no—
103#815—181no—
104#807—91no—
105#807—91no—
106C22×C3×D3—21no—
107C2×C32×C4C22×D311yescharacteristic
108C2×C3⋊Dic3—31no—
109C2×C3⋊Dic3—31no—
110C2×C3×Dic3—21no—
111C2×C3×Dic3—61no—
112C2×C3×Dic3—61no—
113C3×D12—21no—
114C3×D12C4×D312yes—
115C3×Dic6—32no—
116C3×Dic6—61no—
117D3×Dic3—122no—
118C23×C9—61no—
119C32×C8—21no—
120C3⋊(C3⋊C8)—32no—
121D4×C9—32no—
122D4×C9—61no—
123C2×C4×C9—21no—
124C2×C4×C9—61no—
125C2×C4×C9C22×D311yescharacteristic
126C2×Dic9—21no—
127C2×Dic9—21no—
128C2×Dic9—31no—
129C2×Dic9—31no—
130C2×Dic9—61no—
131C2×Dic9—61no—
132C2×Dic9—122no—
133Dic18—21no—
134Dic18C4×D312yescomplemented
135C8×C9—22no—
136C8×C9—21no—
137C9⋊C8—32no—
138C9⋊C8—62no—
139#440—271noSylow
140C2×C3×C9C4×D411yescharacteristic
141C2×C3×C9AGL(1,ℤ/8)11yescharacteristic
142C2×C3×C9#13711yescharacteristic
143D3×C9—42no—
144C2×C3×Q8—91no—
145C2×C3×D4—31no—
146C22×Dic3—181no—
147C2×D12C2×D911yescharacteristic
148C2×C4×D3—181no—
149C2×Dic6—31no—
150C2×C3×C8—21no—
151C2×C3×C8C2×D911yescharacteristic
152C2×C3×C8D3211yescharacteristic
153C3×C4⋊C4—31no—
154C3×C4⋊C4—91no—
155#217—181no—
156#215—94no—
157#214—94no—
158(C2×D3)⋊C4—181no—
159C4⋊Dic3—31no—
160C4⋊Dic3—181no—
161C4⋊Dic3C2×D911yescharacteristic
162Dic3⋊C4—181no—
163C4×Dic3—91no—
164C4×Dic3—91no—
165C4×Dic3—181no—
166C4×Dic3—181no—
167C4×Dic3—181no—
168C2×C3⋊C8—91no—
169C2×C3⋊C8—91no—
170C2×C3⋊C8—181no—
171C22×C32D3×D411yescharacteristic
172C2×C3×D3—22no—
173C2×C3×D3—41no—
174C2×C3×D3—41no—
175C32×C4C2×C4×D311yescharacteristic
176C32×C4#23711yescharacteristic
177C3⋊Dic3—61no—
178C3⋊Dic3—61no—
179C3×Dic3—41no—
180C3×Dic3—62no—
181C3×Dic3—121no—
182C22×C9—21no—
183C22×C9—62no—
184C22×C9—121no—
185C22×C9—121no—
186C22×C9D3×D411yescharacteristic
187C4×C9—21no—
188C4×C9—21no—
189C4×C9—121no—
190C4×C9C2×C4×D311yescharacteristic
191C4×C9#23711yescharacteristic
192Dic9—22no—
193Dic9—41no—
194Dic9—61no—
195Dic9—61no—
196Dic9—121no—
197Dic9—121no—
198C2×SD16—271no—
199C4×Q8—271no—
200C4×D4—271no—
201#107—271no—
202#103—91no—
203#102—31no—
204#97—271no—
205C3×C9#44011yescharacteristic, complemented, Sylow
206C23×C3—61no—
207C22×D3—21no—
208C3×Q8—92no—
209C3×Q8—181no—
210C3×D4—32no—
211C3×D4—61no—
212C2×C3×C4—21no—
213C2×C3×C4—61no—
214C2×C3×C4—181no—
215C2×C3×C4—181no—
216C2×C3×C4C22×D911yescharacteristic
217C2×C3×C4C2×D3211yescharacteristic
218C2×Dic3—21no—
219C2×Dic3—61no—
220C2×Dic3—61no—
221C2×Dic3—91no—
222C2×Dic3—91no—
223C2×Dic3—91no—
224C2×Dic3—91no—
225C2×Dic3—181no—
226C2×Dic3—181no—
227C2×Dic3—362no—
228D12—21no—
229D12C4×D912yes—
230C4×D3—362no—
231Dic6—32no—
232Dic6—61no—
233C3×C8—22no—
234C3×C8—21no—
235C3×C8—21no—
236C3⋊C8—92no—
237C3⋊C8—92no—
238C3⋊C8—182no—
239C2×C32#67⋊C411yescharacteristic
240C2×C32#91111yescharacteristic
241C2×C32#91511yescharacteristic
242C3×D3—42no—
243C2×C9—21no—
244C2×C9—21no—
245C2×C9—21no—
246C2×C9—122no—
247C2×C9#67⋊C411yescharacteristic
248C2×C9#91411yescharacteristic
249C2×C9#91811yescharacteristic
250C2×Q8—91no—
251C2×D4—31no—
252C22×C4—541no—
253SD16—274no—
254C2×C8—271no—
255C2×C8D3×D911yescharacteristic
256C4⋊C4—31no—
257C4⋊C4—91no—
258C4⋊C4—541no—
259C22⋊C4—541no—
260C42—271no—
261C42—541no—
262C22×C3—21no—
263C22×C3—62no—
264C22×C3—121no—
265C22×C3—121no—
266C22×C3D4×D911yescharacteristic
267C22×C3#380211yescharacteristic
268C2×D3—22no—
269C2×D3—41no—
270C2×D3—41no—
271C3×C4—21no—
272C3×C4—21no—
273C3×C4—121no—
274C3×C4—182no—
275C3×C4—361no—
276C3×C4C2×C4×D911yescharacteristic
277C3×C4#369911yescharacteristic
278C3×C4C4×D3211yescharacteristic
279C3×C4#380511yescharacteristic
280Dic3—41no—
281Dic3—62no—
282Dic3—121no—
283Dic3—181no—
284Dic3—181no—
285Dic3—181no—
286Dic3—181no—
287Dic3—361no—
288Dic3—361no—
289C32#475711yescharacteristic
290C9—21no—
291C9#481711yescharacteristic, complemented
292C23—61no—
293Q8—92no—
294Q8—181no—
295D4—32no—
296D4—61no—
297C2×C4—61no—
298C2×C4—181no—
299C2×C4—181no—
300C2×C4—271no—
301C2×C4—271no—
302C2×C4—1082no—
303C2×C4C2×D3×D911yescharacteristic
304C8—21no—
305C8—272no—
306C2×C3—21no—
307C2×C3—21no—
308C2×C3—21no—
309C2×C3—122no—
310C2×C3#611⋊C411yescharacteristic
311C2×C3#6347311yescharacteristic
312C2×C3#6347711yescharacteristic
313C2×C3#6379811yescharacteristic
314C2×C3#6381111yescharacteristic
315C2×C3#6390211yescharacteristic
316D3—42no—
317C22—62no—
318C22—121no—
319C22—121no—
320C22#8864611yescharacteristic
321C4—121no—
322C4—182no—
323C4—361no—
324C4—541no—
325C4—541no—
326C4C4×D3×D911yescharacteristic
327C4#8866311yescharacteristic
328C3—21no—
329C3#1058843711yescharacteristic, complemented
330C3#1059074311yescharacteristic
331C2—122no—
332C2#1172081911yescharacteristic
333C2#1172082511yescharacteristic
334C2#1172102811yescharacteristic
subgroupG/H
centerC22#88646
commutatorC3×C4×C9C22×C4
FrattiniC2×C3×C4C2×D32
FittingC2×C3×C8×C9C22
socleC22×C32D3×D4
prime radicalC2×C3×D12C2×D3
cosocleC2×C3×C4×C9C23
hypercenterC2×C8D3×D9
hypocenterC3×C9#440
LCSG ▷ C3×C4×C9 ▷ C2×C3×C9 ▷ C3×C9
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×C8

Complements

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

NH ≅ G/Ntype
2. #11720910C2semidirect
4. #11720538C2semidirect
6. C3×#63376C2semidirect
7. #11720263C2semidirect
21. #88423C4semidirect
33. #63376D3semidirect
47. C3×Dic18C2×C4semidirect
50. C3⋊(C9⋊C8)C2×C4semidirect
134. Dic18C4×D3semidirect
205. C3×C9#440semidirect
291. C9#4817semidirect
329. C3#10588437semidirect

Maximal subgroups

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

subgroupindex
#117210252
#117209102
C2×#884232
#117205382
C9×#8212
C3×#633762
#117202632
#105907433
#105884373