Finite Group Databasebeta

← #5831⋊C9 #13147291

Subgroups — #5831⋊C9 #13147291

Order 1728.

359 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 359 conjugacy classes of subgroups — too many to draw. Only its 61 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×C3×#41⋊C9C211yesmaximal, characteristic, complemented
2C2×D3×Q8⋊C9C211yesmaximal, characteristic, complemented
3Dic3×Q8⋊C9C211yesmaximal, characteristic, complemented
4C3×#5831C311yesmaximal, characteristic
5#548⋊C9—31nomaximal
6C3×#41⋊C9—21no—
7C2×C3×Q8⋊C9C2211yescharacteristic
8D3×Q8⋊C9—21no—
9C9×(C2×D3)⋊C4—41nomaximal
10C2×C32×#41C2×C311yescharacteristic
11C2×C3×D3×Q8C2×C311yescharacteristic
12C3×Q8×Dic3C2×C311yescharacteristic
13C3×#938—31no—
14C3×#936—31no—
15C3×C4×#67—31no—
16C3×#924—31no—
17C2×#41⋊C9—21no—
18C2×#41⋊C9D311yescharacteristic, complemented
19C22×Q8⋊C9—31no—
20C4×Q8⋊C9—31no—
21C22×D3×C9—41no—
22C2×C3×C4×C9—41no—
23C2×C9×Dic3—41no—
24C3×Q8⋊C9D411yescharacteristic, complemented
25C3×#548—31no—
26#5831C911yescharacteristic, complemented
27C32×#41—21no—
28C32×#41—61no—
29C2×C32×Q8C22×C311yescharacteristic
30C2×C32×D4—31no—
31C22×C32×C4—31no—
32C2×C3×#67—31no—
33C3×D3×Q8—21no—
34C3×D3×Q8—61no—
35C2×C3×C4×D3—31no—
36C2×C3×Dic6—31no—
37C3×#217—32no—
38C3×#217—31no—
39C3×(C2×D3)⋊C4A411yescharacteristic
40C3×C4⋊Dic3—31no—
41C3×Dic3⋊C4—31no—
42C3×C4×Dic3—31no—
43#41⋊C9—22no—
44#41⋊C9—21no—
45C2×Q8⋊C9—21no—
46C2×Q8⋊C9—61no—
47C2×Q8⋊C9C2×D311yescharacteristic
48C9×C22⋊C4—121no—
49C22×C3×C9—41no—
50C2×D3×C9—42no—
51C2×D3×C9—81no—
52C2×D3×C9—81no—
53C3×C4×C9—81no—
54C9×Dic3—81no—
55C2×C3×#41—21no—
56C2×C3×#41C2×C911yescharacteristic, complemented
57C2×C3×#41C3×D311yescharacteristic
58C22×C3×Q8—31no—
59C2×D3×Q8C2×C911yescharacteristic
60C3×#124—91no—
61C3×C22⋊Q8—91no—
62C3×#121—91no—
63C3×C4×Q8—31no—
64C3×C4×D4—91no—
65Q8×Dic3C2×C911yescharacteristic, complemented
66#938—31no—
67#936—31no—
68C4×#67—31no—
69#924—31no—
70C23×C32—31no—
71C22×C3×D3C2×A411yescharacteristic
72C32×Q8—31no—
73C32×Q8C3×D411yescharacteristic
74C32×D4—61no—
75C32×D4—61no—
76C2×C32×C4—31no—
77C2×C32×C4—61no—
78C2×C32×C4—61no—
79C2×C32×C4C2×A411yescharacteristic
80C3×#67—62no—
81C2×C3×Dic3—31no—
82C2×C3×Dic3—31no—
83C2×C3×Dic3C2×A411yescharacteristic
84C3×C4×D3—61no—
85C3×C4×D3—61no—
86C3×Dic6—61no—
87C3×Dic6—61no—
88C23×C9—121no—
89C2×C4×C9—41no—
90C2×C4×C9—81no—
91C2×C4×C9—121no—
92Q8⋊C9—21no—
93Q8⋊C9#6711yescharacteristic, complemented
94#548—31noSylow
95C2×C3×C9—41no—
96C2×C3×C9—41no—
97C2×C3×C9—41no—
98D3×C9—82no—
99C3×#41—22no—
100C3×#41—21no—
101C3×#41—21no—
102C3×#41—62no—
103C3×#41—61no—
104C3×#41—61no—
105C2×C3×Q8—21no—
106C2×C3×Q8—61no—
107C2×C3×Q8—91no—
108C2×C3×Q8—181no—
109C2×C3×Q8C22×C911yescharacteristic
110C2×C3×Q8C2×C3×D311yescharacteristic
111C2×C3×D4—31no—
112C2×C3×D4—31no—
113C2×C3×D4—61no—
114C2×C3×D4—91no—
115C22×C3×C4—31no—
116C22×C3×C4—31no—
117C22×C3×C4—61no—
118C22×C3×C4—91no—
119C2×#67—31no—
120D3×Q8—21no—
121D3×Q8—61no—
122C2×C4×D3—31no—
123C2×Dic6—31no—
124C3×C4⋊C4—91no—
125C3×C4⋊C4—91no—
126C3×C22⋊C4—31no—
127C3×C22⋊C4—91no—
128C3×C22⋊C4—92no—
129C3×C42—91no—
130#217—32no—
131#217—31no—
132(C2×D3)⋊C4C22⋊C911yescharacteristic
133C4⋊Dic3—31no—
134Dic3⋊C4—31no—
135C4×Dic3—31no—
136C22×C32—61no—
137C22×C32—61no—
138C22×C32—61no—
139C22×C32C22×A411yescharacteristic
140C2×C3×D3—21no—
141C2×C3×D3—21no—
142C2×C3×D3#41⋊C312yes—
143C32×C4—21no—
144C32×C4—31no—
145C32×C4—31no—
146C3×Dic3—21no—
147C3×Dic3—61no—
148C3×Dic3—61no—
149C22×C9—41no—
150C22×C9—81no—
151C22×C9—122no—
152C22×C9—241no—
153C22×C9—241no—
154C4×C9—81no—
155C4×C9—82no—
156C4×C9—241no—
157C2×#41D3×C911yescharacteristic, complemented
158C22×Q8—31no—
159#124—91no—
160C22⋊Q8—91no—
161#121—91no—
162C4×Q8—31no—
163C4×D4—91no—
164C3×C9—41noSylow
165C23×C3—31no—
166C23×C3—31no—
167C23×C3—31no—
168C23×C3—61no—
169C22×D3C2×C22⋊C911yescharacteristic
170C3×Q8—21no—
171C3×Q8—31no—
172C3×Q8—31no—
173C3×Q8—61no—
174C3×Q8—181no—
175C3×Q8—181no—
176C3×Q8D4×C911yescharacteristic
177C3×Q8C3×#6711yescharacteristic
178C3×D4—61no—
179C3×D4—61no—
180C3×D4—61no—
181C3×D4—61no—
182C3×D4—62no—
183C3×D4—62no—
184C3×D4—182no—
185C2×C3×C4—21no—
186C2×C3×C4—31no—
187C2×C3×C4—31no—
188C2×C3×C4—31no—
189C2×C3×C4—61no—
190C2×C3×C4—61no—
191C2×C3×C4—61no—
192C2×C3×C4—61no—
193C2×C3×C4—61no—
194C2×C3×C4—62no—
195C2×C3×C4—62no—
196C2×C3×C4—91no—
197C2×C3×C4—91no—
198C2×C3×C4—181no—
199C2×C3×C4—181no—
200C2×C3×C4C2×C22⋊C911yescharacteristic
201C2×C3×C4D3×A411yescharacteristic
202#67—62no—
203C2×Dic3—31no—
204C2×Dic3—31no—
205C2×Dic3C2×C22⋊C911yescharacteristic
206C4×D3—61no—
207C4×D3—61no—
208Dic6—61no—
209Dic6—61no—
210C2×C32—121no—
211C2×C32D4×A411yescharacteristic
212C2×C32C2×#41⋊C311yescharacteristic
213C2×C322−1+4⋊C311yescharacteristic
214C3×D3—22no—
215C2×C9—41no—
216C2×C9—41no—
217C2×C9—41no—
218C2×C9—81no—
219C2×C9—81no—
220C2×C9—81no—
221C2×C9—242no—
222#41—21no—
223#41—61no—
224C2×Q8—61no—
225C2×Q8—91no—
226C2×Q8—181no—
227C2×Q8C2×D3×C911yescharacteristic
228C2×D4—31no—
229C2×D4—91no—
230C22×C4—31no—
231C22×C4—91no—
232C4⋊C4—91no—
233C4⋊C4—91no—
234C22⋊C4—31no—
235C22⋊C4—91no—
236C22⋊C4—92no—
237C42—91no—
238C22×C3—21no—
239C22×C3—32no—
240C22×C3—62no—
241C22×C3—61no—
242C22×C3—61no—
243C22×C3—61no—
244C22×C3—61no—
245C22×C3—61no—
246C22×C3—61no—
247C22×C3—61no—
248C22×C3—61no—
249C22×C3—121no—
250C22×C3—121no—
251C22×C3C22×C22⋊C911yescharacteristic
252C22×C3C2×D3×A411yescharacteristic
253C2×D3—21no—
254C2×D3—21no—
255C2×D3#41⋊C912yescomplemented
256C3×C4—22no—
257C3×C4—21no—
258C3×C4—21no—
259C3×C4—31no—
260C3×C4—31no—
261C3×C4—31no—
262C3×C4—31no—
263C3×C4—61no—
264C3×C4—61no—
265C3×C4—61no—
266C3×C4—181no—
267C3×C4—181no—
268Dic3—21no—
269Dic3—61no—
270Dic3—61no—
271C32#548⋊C311yescharacteristic
272C9—41no—
273C9—81no—
274C23—31no—
275C23—31no—
276Q8—31no—
277Q8—181no—
278Q8—181no—
279Q8C9×#6711yescharacteristic
280D4—61no—
281D4—61no—
282D4—182no—
283C2×C4—31no—
284C2×C4—31no—
285C2×C4—61no—
286C2×C4—61no—
287C2×C4—91no—
288C2×C4—91no—
289C2×C4—181no—
290C2×C4—181no—
291C2×C4D3×C22⋊C911yescharacteristic
292C2×C3—21no—
293C2×C3—21no—
294C2×C3—21no—
295C2×C3—62no—
296C2×C3—122no—
297C2×C3—121no—
298C2×C3—121no—
299C2×C3D4×C22⋊C911yescharacteristic
300C2×C3C2×#41⋊C911yescharacteristic
301C2×C32−1+4⋊C911yescharacteristic
302C2×C3D3×#41⋊C311yescharacteristic
303C2×C3#1010⋊C311yescharacteristic
304C2×C3A4×#6711yescharacteristic
305D3—22no—
306C22—32no—
307C22—61no—
308C22—61no—
309C22—61no—
310C22—61no—
311C22—61no—
312C22C2×D3×C22⋊C911yescharacteristic
313C4—21no—
314C4—31no—
315C4—31no—
316C4—61no—
317C4—181no—
318C4—181no—
319C3—21no—
320C3#548⋊C911yescharacteristic, complemented
321C3#5831⋊C311yescharacteristic
322C2—62no—
323C2—121no—
324C2D3×#41⋊C911yescharacteristic
325C2#1010⋊C911yescharacteristic
326C2#67×C22⋊C911yescharacteristic
subgroupG/H
center, FrattiniC22×C3C2×D3×A4
commutatorC2×C3×Q8C22×C9
FittingC2×C32×#41C2×C3
socleC22×C32C22×A4
prime radicalC3×#5831C3
cosocleC2×C32×Q8C22×C3
hypercenterC2×C3×C4D3×A4
hypocenterC3×Q8D4×C9
LCSG ▷ C2×C3×Q8 ▷ C3×Q8
UCSC1 ◁ C22×C3 ◁ C2×C3×C4

Complements

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

NH ≅ G/Ntype
1. C2×C3×#41⋊C9C2semidirect
2. C2×D3×Q8⋊C9C2semidirect
3. Dic3×Q8⋊C9C2semidirect
18. C2×#41⋊C9D3semidirect
24. C3×Q8⋊C9D4semidirect
26. #5831C9semidirect
56. C2×C3×#41C2×C9semidirect
65. Q8×Dic3C2×C9semidirect
93. Q8⋊C9#67semidirect
157. C2×#41D3×C9semidirect
255. C2×D3#41⋊C9semidirect
320. C3#548⋊C9semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#41⋊C92
C2×D3×Q8⋊C92
Dic3×Q8⋊C92
C3×#58313
#548⋊C93
C9×(C2×D3)⋊C44