Finite Group Databasebeta

← #72228955

Subgroups — #72228955

Order 3888.

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

normal subgroups only

This group has 693 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
1C22×#92309C211yesmaximal, characteristic, complemented
2C2×#92264⋊C2C211yesmaximal, characteristic, complemented
3#14517437C212yesmaximal, complemented
4#14516824C212yesmaximal, complemented
5#14516822C211yesmaximal, characteristic, complemented
6#88668⋊C3—32nomaximal
7#13047160C311yesmaximal, characteristic, complemented
8C22×#6896C2211yescharacteristic, complemented
9C2×#92309—22no—
10C2×#92309C2211yescharacteristic
11C2×#92309C2211yescharacteristic, complemented
12C2×#92264C2212yescomplemented
13#92264⋊C2—22no—
14#11757568C2212yescomplemented
15C22×C3⋊AGL(1,ℤ/9)—32no—
16C2×D3×AGL(1,ℤ/9)—32no—
17#67×3−1+2—32no—
18C3×#611⋊C3D312yes—
19#6195⋊C3—32no—
20#6192⋊C3—32no—
21(C3⋊AGL(1,ℤ/9))⋊C4—32no—
22C22×C9⋊D9C2×C311yescharacteristic, complemented
23C22×#4169—61no—
24C2×D92C2×C311yescharacteristic, complemented
25#611⋊C9—22no—
26C9×#611C2×C312yescomplemented
27#10619530C2×C312yescomplemented
28#10619528C2×C311yescharacteristic, complemented
29C2×#6896—21no—
30C2×#6896C2311yescharacteristic
31#92309—21no—
32#92309D412yescomplemented
33#92264—22no—
34C3×#3811—91no—
35D4×AGL(1,ℤ/9)—92no—
36#88668—32no—
37C22×C3×3−1+2C2×D312yes—
38C2×C3⋊AGL(1,ℤ/9)—32no—
39C2×C3⋊AGL(1,ℤ/9)—32no—
40C2×C3⋊AGL(1,ℤ/9)—64no—
41C2×D3×3−1+2—32no—
42C2×C3×AGL(1,ℤ/9)C2×D312yes—
43D3×AGL(1,ℤ/9)—64no—
44Dic3×3−1+2—32no—
45C3×3−1+2⋊C4C2×D312yes—
46C22×C9⋊C9—22no—
47C22×C9⋊C9—41no—
48C22×C32⋊C9—21no—
49C22×C92C22×C311yescharacteristic, complemented
50C2×C9⋊D9—22no—
51C2×C9⋊D9C22×C311yescharacteristic
52C2×C9⋊D9C22×C311yescharacteristic, complemented
53C2×D9⋊C9—22no—
54C2×#4169—61no—
55C2×#4169—61no—
56C2×#4169—64no—
57C2×C9×D9C22×C312yescomplemented
58D92—22no—
59Dic9⋊C9—22no—
60C9×Dic9C22×C312yescomplemented
61#6896C2×D411yescharacteristic, complemented, Sylow
62C22×C3×C3⋊D3—91no—
63C2×C3×D32—91no—
64C32×#67—32no—
65C3×#626—92no—
66C3×#624—91no—
67C22×C3⋊D9—32no—
68C22×C3⋊D9—61no—
69C22×AGL(1,ℤ/9)—92no—
70C22×AGL(1,ℤ/9)—92no—
71C22×AGL(1,ℤ/9)—182no—
72C2×D3×D9—32no—
73D4×3−1+2—92no—
74#611⋊C3—32no—
75#611⋊C3—92no—
76C9×#67—32no—
77C9×#67—62no—
78C3×#611C3×D312yes—
79C4⋊AGL(1,ℤ/9)—92no—
80C4×AGL(1,ℤ/9)—92no—
81#6195—32no—
82#6192—32no—
83(C3⋊D9)⋊C4—32no—
84C2×C3×3−1+2—22no—
85C2×C3×3−1+2C22×D312yes—
86C3⋊AGL(1,ℤ/9)—34no—
87C3⋊AGL(1,ℤ/9)—62no—
88D3×3−1+2—62no—
89C3×AGL(1,ℤ/9)—22no—
90C2×C9⋊C9—22no—
91C2×C9⋊C9—41no—
92C2×C9⋊C9—42no—
93C2×C9⋊C9—42no—
94C2×C32⋊C9—22no—
95C2×C32⋊C9—21no—
96C2×C92—21no—
97C2×C92C23×C311yescharacteristic
98C9⋊D9—21no—
99C9⋊D9C3×D412yescomplemented
100D9⋊C9—42no—
101#4169—62no—
102#4169—62no—
103C9×D9—22no—
104C3×D3×D4—272no—
105#3811—91no—
106D4×D9—92no—
107C22×C33D3211yescharacteristic
108C2×C3×C3⋊D3—91no—
109C2×C3×C3⋊D3—91no—
110C2×C3×C3⋊D3—182no—
111C2×C32×D3—32no—
112C3×D32—182no—
113C32×Dic3—32no—
114C22×3−1+2—32no—
115C22×3−1+2—62no—
116C22×3−1+2—92no—
117C22×C3×C9—21no—
118C22×C3×C9—22no—
119C22×C3×C9—41no—
120C22×C3×C9C2×C3×D312yes—
121C2×C3⋊D9—32no—
122C2×C3⋊D9—32no—
123C2×C3⋊D9—64no—
124C2×C3⋊D9—61no—
125C2×C3⋊D9—61no—
126C2×C3⋊D9—64no—
127C2×AGL(1,ℤ/9)—32no—
128C2×AGL(1,ℤ/9)—92no—
129C2×AGL(1,ℤ/9)—92no—
130C2×AGL(1,ℤ/9)—184no—
131C2×AGL(1,ℤ/9)—184no—
132C2×AGL(1,ℤ/9)—182no—
133C2×AGL(1,ℤ/9)—182no—
134C2×AGL(1,ℤ/9)—188no—
135C2×D3×C9—32no—
136C2×D3×C9—62no—
137C2×C3×D9C2×C3×D312yes—
138D3×D9—64no—
139C4×3−1+2—92no—
1403−1+2⋊C4—32no—
141C9×Dic3—32no—
142C9×Dic3—62no—
143C3×Dic9C2×C3×D312yes—
144C3×3−1+2D3×D412yes—
145C9⋊C9—22no—
146C9⋊C9—41no—
147C32⋊C9—21no—
148C92C2×C3×D411yescharacteristic, complemented
149C22×C3⋊D3—91no—
150C22×C3×D3—272no—
151C22×C3×D3—272no—
152C22×C3×D3—541no—
153C2×D32—91no—
154C32×D4—92no—
155C3×#67—32no—
156C3×#67—92no—
157C3×#67—272no—
158C3×D12—272no—
159C3×C4×D3—272no—
160#626—92no—
161#624—91no—
162C22×D9—92no—
163C22×D9—92no—
164C22×D9—182no—
165C22×D9—541no—
166D4×C9—92no—
167D4×C9—182no—
168#611—92no—
169#611AGL(1,ℤ/9)12yescomplemented
170D36—92no—
171C4×D9—92no—
172C2×C33—21no—
173C2×C33C2×D3211yescharacteristic
174C3×C3⋊D3—92no—
175C3×C3⋊D3—181no—
176C32×D3—62no—
177C2×3−1+2—32no—
178C2×3−1+2—62no—
179C2×3−1+2—62no—
180C2×3−1+2—64no—
181C2×3−1+2—182no—
182C2×C3×C9—22no—
183C2×C3×C9—22no—
184C2×C3×C9—22no—
185C2×C3×C9—21no—
186C2×C3×C9—42no—
187C2×C3×C9—42no—
188C2×C3×C9—41no—
189C2×C3×C9C22×C3×D312yes—
190C3⋊D9—34no—
191C3⋊D9—62no—
192C3⋊D9—62no—
193C3⋊D9—62no—
194AGL(1,ℤ/9)—62no—
195AGL(1,ℤ/9)—94no—
196AGL(1,ℤ/9)—184no—
197AGL(1,ℤ/9)—182no—
198AGL(1,ℤ/9)—184no—
199D3×C9—62no—
200D3×C9—122no—
201C3×D9—22no—
202C2×C3×D4—811no—
203D3×D4—272no—
204C22×C32—32no—
205C22×C32—61no—
206C22×C32—92no—
207C22×C32C3×D3211yescharacteristic
208C2×C3⋊D3—91no—
209C2×C3⋊D3—91no—
210C2×C3⋊D3—182no—
211C2×C3×D3—32no—
212C2×C3×D3—92no—
213C2×C3×D3—272no—
214C2×C3×D3—272no—
215C2×C3×D3—541no—
216C2×C3×D3—541no—
217C2×C3×D3—544no—
218C2×C3×D3—544no—
219C2×C3×D3—544no—
220D32—182no—
221C32×C4—92no—
222C3×Dic3—32no—
223C3×Dic3—92no—
224C22×C9—22no—
225C22×C9—61no—
226C22×C9—62no—
227C22×C9—92no—
228C22×C9—121no—
229C22×C9—182no—
230C22×C9C2×AGL(1,ℤ/9)12yescomplemented
231C2×D9—92no—
232C2×D9—92no—
233C2×D9—182no—
234C2×D9—182no—
235C2×D9—184no—
236C2×D9—184no—
237C2×D9—188no—
238C2×D9—541no—
239C2×D9—541no—
240C2×D9—544no—
241C2×D9C2×AGL(1,ℤ/9)12yescomplemented
242C4×C9—92no—
243C4×C9—182no—
244Dic9C2×AGL(1,ℤ/9)12yescomplemented
245C33#381111yescharacteristic
2463−1+2—32no—
2473−1+2—62no—
248C3×C9—22no—
249C3×C9—21no—
250C3×C9—41no—
251C3×C9C3×D3×D412yes—
252C23×C3—811no—
253C23×C3—811no—
254C22×D3—272no—
255C22×D3—272no—
256C22×D3—541no—
257C3×D4—92no—
258C3×D4—272no—
259C3×D4—812no—
260C2×C3×C4—811no—
261#67—32no—
262#67—272no—
263D12—272no—
264C4×D3—272no—
265C2×C32—21no—
266C2×C32—32no—
267C2×C32—61no—
268C2×C32—62no—
269C2×C32—62no—
270C2×C32—182no—
271C2×C32C2×C3×D3211yescharacteristic
272C3⋊D3—92no—
273C3⋊D3—181no—
274C3×D3—62no—
275C3×D3—182no—
276C3×D3—274no—
277C3×D3—542no—
278C3×D3—542no—
279C3×D3—542no—
280C2×C9—24no—
281C2×C9—22no—
282C2×C9—22no—
283C2×C9—62no—
284C2×C9—61no—
285C2×C9—62no—
286C2×C9—122no—
287C2×C9—122no—
288C2×C9—121no—
289C2×C9—182no—
290C2×C9—362no—
291C2×C9C22×AGL(1,ℤ/9)12yes—
292D9—22no—
293D9—94no—
294D9—184no—
295D9—184no—
296D9—182no—
297D9—542no—
298D9—542no—
299C2×D4—811noSylow
300C22×C3—21no—
301C22×C3—92no—
302C22×C3—91no—
303C22×C3—272no—
304C22×C3—811no—
305C22×C3—811no—
306C22×C3—1622no—
307C22×C3—1622no—
308C22×C3D3×AGL(1,ℤ/9)12yes—
309C2×D3—32no—
310C2×D3—272no—
311C2×D3—272no—
312C2×D3—541no—
313C2×D3—541no—
314C2×D3—544no—
315C2×D3—544no—
316C2×D3—544no—
317C3×C4—92no—
318C3×C4—272no—
319Dic3—32no—
320C32—32no—
321C32—61no—
322C32C3×#381111yescharacteristic
323C9—22no—
324C9—61no—
325C9—62no—
326C9—121no—
327C9D4×AGL(1,ℤ/9)12yescomplemented
328C23—811no—
329C23—811no—
330D4—92no—
331D4—812no—
332C2×C4—811no—
333C2×C3—22no—
334C2×C3—22no—
335C2×C3—21no—
336C2×C3—91no—
337C2×C3—181no—
338C2×C3—182no—
339C2×C3—542no—
340C2×C3—812no—
341C2×C3—1621no—
342C2×C3C2×D3×AGL(1,ℤ/9)12yes—
343D3—62no—
344D3—274no—
345D3—542no—
346D3—542no—
347D3—542no—
348C22—92no—
349C22—811no—
350C22—811no—
351C22—1622no—
352C22—1622no—
353C22#92264⋊C211yescharacteristic, complemented
354C4—92no—
355C3—21no—
356C3—91no—
357C3#88668⋊C312yes—
358C2—21no—
359C2—182no—
360C2—812no—
361C2—1621no—
362C2C2×#92264⋊C211yescharacteristic
subgroupG/H
centerC2C2×#92264⋊C2
commutator, cosocleC2×C92C23×C3
Frattini, socleC2×C32C2×C3×D32
FittingC22×#6896C22
hypercenterC22#92264⋊C2
hypocenterC92C2×C3×D4
LCSG ▷ C2×C92 ▷ C92
UCSC1 ◁ C2 ◁ C22

Complements

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

NH ≅ G/Ntype
1. C22×#92309C2semidirect
2. C2×#92264⋊C2C2semidirect
3. #14517437C2semidirect
4. #14516824C2semidirect
5. #14516822C2semidirect
7. #13047160C3semidirect
8. C22×#6896C22semidirect
11. C2×#92309C22semidirect
12. C2×#92264C22semidirect
14. #11757568C22semidirect
22. C22×C9⋊D9C2×C3semidirect
24. C2×D92C2×C3semidirect
26. C9×#611C2×C3semidirect
27. #10619530C2×C3semidirect
28. #10619528C2×C3semidirect
32. #92309D4semidirect
49. C22×C92C22×C3semidirect
52. C2×C9⋊D9C22×C3semidirect
57. C2×C9×D9C22×C3semidirect
60. C9×Dic9C22×C3semidirect
61. #6896C2×D4semidirect
99. C9⋊D9C3×D4semidirect
148. C92C2×C3×D4semidirect
169. #611AGL(1,ℤ/9)semidirect
230. C22×C9C2×AGL(1,ℤ/9)semidirect
241. C2×D9C2×AGL(1,ℤ/9)semidirect
244. Dic9C2×AGL(1,ℤ/9)semidirect
327. C9D4×AGL(1,ℤ/9)semidirect
353. C22#92264⋊C2semidirect

Maximal subgroups

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

subgroupindex
C22×#923092
C2×#92264⋊C22
#145174372
#145168242
#145168222
#88668⋊C33
#130471603