Finite Group Databasebeta

← #13140725

Subgroups — #13140725

Order 1728.

428 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 428 conjugacy classes of subgroups — too many to draw. Only its 101 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×#88548C211yesmaximal, characteristic
2C3×Q8×Dic9C211yesmaximal, characteristic, complemented
3#11720992C211yesmaximal, characteristic
4C9×#835C211yesmaximal, characteristic, complemented
5#11720284C211yesmaximal, characteristic
6#11720274C211yesmaximal, characteristic, complemented
7#11720263C211yesmaximal, characteristic, complemented
8#10591540—31nomaximal
9#10588497—31nomaximal
10C2×C3×Q8×C9C2211yescharacteristic
11C2×C4⋊(C3⋊D9)C2211yescharacteristic
12C2×(C3⋊D9)⋊C4—21no—
13#88548C414yescomplemented
14C2×C3⋊(C9⋊C8)C2211yescharacteristic
15C9×C4⋊Dic3C2211yescharacteristic
16C3×C4⋊Dic9—21no—
17C3×C4⋊Dic9C2211yescharacteristic
18C3×C4×Dic9—21no—
19C3×C4×Dic9C2211yescharacteristic
20C2×C9×C3⋊C8C2211yescharacteristic
21#88445—21no—
22C2×#3755—31no—
23C3×Q8×Dic3—31no—
24#63894—31no—
25C3×#835D311yescharacteristic
26#63576—31no—
27#63564—31no—
28#63554—31no—
29Q8×Dic9D311yescharacteristic, complemented
30C2×#3675—31no—
31C2×#3675—61no—
32#63455—31no—
33C9×#103—31no—
34#63380—31no—
35#63375—31no—
36#63366—31no—
37C22×C3⋊D9—21no—
38C3×Q8×C9—21no—
39C3×Q8×C9C2×C412yescomplemented
40C2×C3×C4×C9—21no—
41C2×C3×C4×C9C2311yescharacteristic
42C4⋊(C3⋊D9)—21no—
43C4⋊(C3⋊D9)C2×C412yes—
44C2×C9×Dic3—21no—
45C2×C3×Dic9—21no—
46C2×C3×Dic9D411yescharacteristic
47C2×C3×Dic9D411yescharacteristic
48(C3⋊D9)⋊C4—42no—
49C3⋊(C9⋊C8)C2×C412yescomplemented
50C9×C3⋊C8—21no—
51#5026—91no—
52#4817—91no—
53C2×C32×Q8C2×D311yescharacteristic
54C2×C4⋊(C3⋊D3)—31no—
55C2×#624—61no—
56#3755—34no—
57C2×C3⋊(C3⋊C8)—31no—
58C3×C4⋊Dic3—31no—
59C3×C4⋊Dic3—61no—
60C3×C4⋊Dic3C2×D311yescharacteristic
61C3×C4×Dic3—31no—
62C3×C4×Dic3—61no—
63C2×C3×C3⋊C8C2×D311yescharacteristic
64#3722—61no—
65C2×Q8×C9—21no—
66C2×Q8×C9C2×D311yescharacteristic
67C2×D36—31no—
68C2×D36—61no—
69C2×C4×D9—61no—
70C2×C8×C9—31no—
71C9×C4⋊C4—31no—
72#3675—34no—
73#3675—64no—
74(C2×D9)⋊C4—61no—
75C4⋊Dic9—21no—
76C4⋊Dic9C2×D311yescharacteristic
77C4×Dic9—21no—
78C4×Dic9C2×D311yescharacteristic
79C2×C9⋊C8—31no—
80C2×C9⋊C8—61no—
81C22×C3×C9C2×D411yescharacteristic
82C2×C3⋊D9—22no—
83C2×C3⋊D9—41no—
84C2×C3⋊D9—41no—
85C3×C4×C9—22no—
86C3×C4×C9C22×C411yescharacteristic
87C3×C4×C9#4111yescharacteristic
88C9×Dic3—41no—
89C3×Dic9—21no—
90C3×Dic9—21no—
91C3×Dic9—41no—
92C3×C4×Q8—91no—
93Q8×Dic3—31no—
94C2×#215—91no—
95C2×#215—91no—
96C2×#215—181no—
97#893—91no—
98C4×D12—91no—
99C3×#103—31no—
100#835D911yescharacteristic, complemented
101#821—91no—
102#815—91no—
103#809—91no—
104#807—91no—
105#807—91no—
106#803—91no—
107C22×C3⋊D3—61no—
108C32×Q8—21no—
109C32×Q8C4×D312yes—
110C2×C32×C4—21no—
111C2×C32×C4C22×D311yescharacteristic
112C4⋊(C3⋊D3)—32no—
113C4⋊(C3⋊D3)—61no—
114C2×C3×Dic3—21no—
115C2×C3×Dic3—31no—
116C2×C3×Dic3—31no—
117C2×C3×Dic3—61no—
118#624—122no—
119C22×D9—61no—
120C22×D9—121no—
121C3⋊(C3⋊C8)—32no—
122C3×C3⋊C8—21no—
123Q8×C9—21no—
124Q8×C9—22no—
125Q8×C9—41no—
126Q8×C9C4×D312yescomplemented
127C2×C4×C9—21no—
128C2×C4×C9—21no—
129C2×C4×C9—41no—
130C2×C4×C9—61no—
131C2×C4×C9C22×D311yescharacteristic
132C2×Dic9—21no—
133C2×Dic9#6711yescharacteristic
134C2×Dic9#6711yescharacteristic
135D36—32no—
136D36—62no—
137D36—61no—
138D36—121no—
139C4×D9—122no—
140C8×C9—61no—
141C9⋊C8—32no—
142C9⋊C8—62no—
143#440—271noSylow
144C2×C3×C9C4×D411yescharacteristic
145C2×C3×C9AGL(1,ℤ/8)11yescharacteristic
146C2×C3×C9#13711yescharacteristic
147C3⋊D9—42no—
148C2×C3×Q8—21no—
149C2×C3×Q8C2×D911yescharacteristic
150C2×C3×Q8D3211yescharacteristic
151C2×D12—91no—
152C2×D12—91no—
153C2×D12—181no—
154C2×C4×D3—181no—
155C2×C4×D3—181no—
156C2×C3×C8—31no—
157C3×C4⋊C4—31no—
158C3×C4⋊C4—91no—
159C3×C4⋊C4—181no—
160C3×C42—91no—
161C3×C42—181no—
162#215—94no—
163#215—94no—
164#215—184no—
165(C2×D3)⋊C4—181no—
166(C2×D3)⋊C4—181no—
167C4⋊Dic3—31no—
168C4⋊Dic3—61no—
169C4⋊Dic3C2×D911yescharacteristic
170C4×Dic3—31no—
171C4×Dic3—61no—
172C2×C3⋊C8—91no—
173C2×C3⋊C8—91no—
174C2×C3⋊C8—181no—
175C2×C3⋊C8C2×D911yescharacteristic
176C22×C32D3×D411yescharacteristic
177C2×C3⋊D3—62no—
178C2×C3⋊D3—121no—
179C2×C3⋊D3—121no—
180C32×C4—22no—
181C32×C4C2×C4×D311yescharacteristic
182C32×C4#23711yescharacteristic
183C3×Dic3—41no—
184C3×Dic3—61no—
185C3×Dic3—61no—
186C3×Dic3—121no—
187C22×C9—21no—
188C22×C9C2×#6711yescharacteristic
189C2×D9—62no—
190C2×D9—122no—
191C2×D9—121no—
192C2×D9—121no—
193C2×D9—241no—
194C2×D9—241no—
195C4×C9—22no—
196C4×C9—21no—
197C4×C9—21no—
198C4×C9—42no—
199C4×C9—121no—
200C4×C9C2×C4×D311yescharacteristic
201C4×C9#23511yescharacteristic
202Dic9—21no—
203Dic9—21no—
204Dic9—41no—
205C2×SD16—271no—
206C4×Q8—91no—
207C4×D4—271no—
208#107—271no—
209#103—31no—
210#102—271no—
211#97—271no—
212C3×C9#44011yescharacteristic, complemented, Sylow
213C22×D3—181no—
214C22×D3—181no—
215C22×D3—361no—
216C3×Q8—21no—
217C3×Q8—21no—
218C3×Q8—22no—
219C3×Q8—41no—
220C3×Q8C4×D912yescomplemented
221C3×Q8#62412yes—
222C2×C3×C4—21no—
223C2×C3×C4—21no—
224C2×C3×C4—21no—
225C2×C3×C4—41no—
226C2×C3×C4—61no—
227C2×C3×C4—91no—
228C2×C3×C4—91no—
229C2×C3×C4—181no—
230C2×C3×C4C22×D911yescharacteristic
231C2×C3×C4C2×D3211yescharacteristic
232C2×Dic3—21no—
233C2×Dic3—31no—
234C2×Dic3—31no—
235C2×Dic3—61no—
236D12—92no—
237D12—92no—
238D12—182no—
239D12—181no—
240D12—181no—
241D12—361no—
242C4×D3—362no—
243C4×D3—362no—
244C3×C8—61no—
245C3⋊C8—21no—
246C3⋊C8—92no—
247C3⋊C8—92no—
248C3⋊C8—182no—
249C2×C32#67⋊C411yescharacteristic
250C2×C32#91411yescharacteristic
251C2×C32#91811yescharacteristic
252C3⋊D3—122no—
253C2×C9—21no—
254C2×C9—21no—
255C2×C9—21no—
256C2×C9C4×#6711yescharacteristic
257C2×C9#94211yescharacteristic
258C2×C9#94911yescharacteristic
259D9—122no—
260D9—242no—
261C2×Q8D3×D911yescharacteristic
262C2×D4—271no—
263C22×C4—541no—
264SD16—274no—
265C2×C8—31no—
266C2×C8—271no—
267C4⋊C4—31no—
268C4⋊C4—91no—
269C4⋊C4—181no—
270C22⋊C4—541no—
271C42—91no—
272C42—181no—
273C22×C3—21no—
274C22×C3D4×D911yescharacteristic
275C22×C3D3×#6711yescharacteristic
276C2×D3—182no—
277C2×D3—182no—
278C2×D3—362no—
279C2×D3—361no—
280C2×D3—361no—
281C2×D3—361no—
282C2×D3—361no—
283C2×D3—721no—
284C2×D3—721no—
285C3×C4—22no—
286C3×C4—22no—
287C3×C4—21no—
288C3×C4—21no—
289C3×C4—42no—
290C3×C4—121no—
291C3×C4—181no—
292C3×C4—181no—
293C3×C4—361no—
294C3×C4C2×C4×D911yescharacteristic
295C3×C4#369911yescharacteristic
296C3×C4#380411yescharacteristic
297C3×C4C2×#62411yescharacteristic
298Dic3—41no—
299Dic3—61no—
300Dic3—61no—
301Dic3—121no—
302C32#481711yescharacteristic
303C9—21no—
304C9#502611yescharacteristic, complemented
305C23—541no—
306Q8—21no—
307Q8(C3⋊D9)⋊C412yescomplemented
308D4—272no—
309D4—541no—
310C2×C4—21no—
311C2×C4—61no—
312C2×C4—91no—
313C2×C4—91no—
314C2×C4—181no—
315C2×C4—1082no—
316C2×C4C2×D3×D911yescharacteristic
317C8—61no—
318C8—272no—
319C2×C3—21no—
320C2×C3—21no—
321C2×C3—21no—
322C2×C3#611⋊C411yescharacteristic
323C2×C3#6347611yescharacteristic
324C2×C3#6348011yescharacteristic
325C2×C3#6393911yescharacteristic
326C2×C3#6394811yescharacteristic
327C2×C3#6395211yescharacteristic
328D3—362no—
329D3—362no—
330D3—722no—
331C22—542no—
332C22—1081no—
333C22—1081no—
334C22D9×#6711yescharacteristic
335C4—22no—
336C4—121no—
337C4—181no—
338C4—181no—
339C4—361no—
340C4C2×(C3⋊D9)⋊C411yescharacteristic
341C4#8866211yescharacteristic
342C3—21no—
343C3#1058849711yescharacteristic, complemented
344C3#1059154011yescharacteristic
345C2—1082no—
346C2#1172121111yescharacteristic
347C2#1172122211yescharacteristic
348C2#1172124111yescharacteristic
subgroupG/H
centerC22D9×#67
commutatorC3×C4×C9C22×C4
FrattiniC2×C3×C4C2×D32
FittingC2×C3×Q8×C9C22
socleC22×C32D3×D4
prime radicalC2×C4⋊(C3⋊D9)C22
cosocleC2×C3×C4×C9C23
hypercenterC2×Q8D3×D9
hypocenterC3×C9#440
LCSG ▷ C3×C4×C9 ▷ C2×C3×C9 ▷ C3×C9
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×Q8

Complements

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

NH ≅ G/Ntype
2. C3×Q8×Dic9C2semidirect
4. C9×#835C2semidirect
6. #11720274C2semidirect
7. #11720263C2semidirect
13. #88548C4semidirect
29. Q8×Dic9D3semidirect
39. C3×Q8×C9C2×C4semidirect
49. C3⋊(C9⋊C8)C2×C4semidirect
100. #835D9semidirect
126. Q8×C9C4×D3semidirect
212. C3×C9#440semidirect
220. C3×Q8C4×D9semidirect
304. C9#5026semidirect
307. Q8(C3⋊D9)⋊C4semidirect
343. C3#10588497semidirect

Maximal subgroups

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

subgroupindex
C2×#885482
C3×Q8×Dic92
#117209922
C9×#8352
#117202842
#117202742
#117202632
#105915403
#105884973