Finite Group Databasebeta

← #3487⋊C9 #11919739

Subgroups — #3487⋊C9 #11919739

Order 1152.

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

normal subgroups only

This group has 666 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
1C2×2−1+4⋊C9C211yesmaximal, characteristic, complemented
2C22×#41⋊C9C211yesmaximal, characteristic, complemented
3C4⋊(#41⋊C9)C212yesmaximal, complemented
4#548⋊C9C212yesmaximal, complemented
5#519⋊C9C211yesmaximal, characteristic, complemented
6C3×#3487C311yesmaximal, characteristic
72−1+4⋊C9—22no—
8C2×#41⋊C9—22no—
9C2×#41⋊C9C2212yes—
10C2×#41⋊C9C2211yescharacteristic, complemented
11C22×Q8⋊C9C2211yescharacteristic
12C22×Q8⋊C9C2211yescharacteristic, complemented
13C9×#123—41nomaximal
14C4×Q8⋊C9C2212yescomplemented
15C2×C3×2−1+4C2×C311yescharacteristic
16C22×C3×#41C2×C311yescharacteristic
17C3×#550—32no—
18C3×#550C2×C312yes—
19C3×#548—32no—
20C3×#548C2×C312yes—
21C3×#535—31no—
22C3×#535—31no—
23C3×#525—31no—
24C3×#525—31no—
25C3×#519C2×C311yescharacteristic
26C2×D4×C9—41no—
27C22×C4×C9—41no—
28#41⋊C9—22no—
29#41⋊C9—21no—
30C2×Q8⋊C9—21no—
31C2×Q8⋊C9D412yescomplemented
32C2×Q8⋊C9C2311yescharacteristic
33C9×C4⋊C4—42no—
34C9×C22⋊C4—41no—
35C9×C22⋊C4—42no—
36#3487C911yescharacteristic, complemented, Sylow
37C3×2−1+4—22no—
38C3×2−1+4—62no—
39C2×C3×#41—22no—
40C2×C3×#41—31no—
41C2×C3×#41—31no—
42C2×C3×#41—31no—
43C2×C3×#41—32no—
44C2×C3×#41—62no—
45C2×C3×#41C22×C312yes—
46C2×C3×#41C22×C311yescharacteristic
47C22×C3×Q8—31no—
48C22×C3×Q8C22×C311yescharacteristic
49C22×C3×Q8C22×C311yescharacteristic
50C22×C3×D4—31no—
51C23×C3×C4—31no—
52C3×#127—32no—
53C3×#127—32no—
54C3×#124—32no—
55C3×#124—32no—
56C3×#123—31no—
57C3×#123A411yescharacteristic
58C3×C22⋊Q8—32no—
59C3×C22⋊Q8—32no—
60C3×#121—32no—
61C3×#121—34no—
62C3×#121—32no—
63C3×#121—32no—
64C3×#121—32no—
65C3×#120—32no—
66C3×#120—31no—
67C3×#120—31no—
68C3×C4×Q8—32no—
69C3×C4×Q8C22×C312yes—
70C3×C4×D4—34no—
71C3×C4×D4—32no—
72C3×C4×D4—32no—
73C3×#117—31no—
74C3×#117—31no—
75C23×C9—41no—
76C23×C9—41no—
77D4×C9—82no—
78C2×C4×C9—42no—
79C2×C4×C9—42no—
80C2×C4×C9—41no—
81C2×C4×C9—82no—
82Q8⋊C9C2×D411yescharacteristic
83C2×2−1+4C2×C911yescharacteristic
84C22×#41C2×C911yescharacteristic, complemented
85#550—32no—
86#550C2×C912yescomplemented
87#548—32no—
88#548C2×C912yescomplemented
89#535—31no—
90#535—31no—
91#525—31no—
92#525—31no—
93#519C2×C911yescharacteristic, complemented
94C24×C3—31no—
95C3×#41—22no—
96C3×#41—21no—
97C3×#41—62no—
98C3×#41—62no—
99C3×#41—62no—
100C3×#41—62no—
101C3×#41—64no—
102C3×#41—62no—
103C3×#41—62no—
104C3×#41—64no—
105C3×#41—62no—
106C3×#41—61no—
107C2×C3×Q8—21no—
108C2×C3×Q8—31no—
109C2×C3×Q8—31no—
110C2×C3×Q8—32no—
111C2×C3×Q8—31no—
112C2×C3×Q8—62no—
113C2×C3×Q8—62no—
114C2×C3×Q8—61no—
115C2×C3×Q8C3×D412yes—
116C2×C3×Q8C23×C311yescharacteristic
117C2×C3×D4—32no—
118C2×C3×D4—31no—
119C2×C3×D4—32no—
120C2×C3×D4—31no—
121C2×C3×D4—31no—
122C2×C3×D4—34no—
123C2×C3×D4—32no—
124C2×C3×D4—32no—
125C2×C3×D4—62no—
126C2×C3×D4—62no—
127C2×C3×D4C2×A411yescharacteristic
128C22×C3×C4—31no—
129C22×C3×C4—32no—
130C22×C3×C4—31no—
131C22×C3×C4—31no—
132C22×C3×C4—32no—
133C22×C3×C4—32no—
134C22×C3×C4—62no—
135C22×C3×C4—62no—
136C22×C3×C4C2×A411yescharacteristic
137C3×C4⋊C4—32no—
138C3×C4⋊C4—32no—
139C3×C4⋊C4—32no—
140C3×C4⋊C4C2×A412yes—
141C3×C22⋊C4—34no—
142C3×C22⋊C4—31no—
143C3×C22⋊C4—32no—
144C3×C22⋊C4—32no—
145C3×C22⋊C4—32no—
146C3×C22⋊C4C2×A411yescharacteristic
147C3×C22⋊C4C2×A412yes—
148C3×C42—32no—
149C3×C42—32no—
150C22×C9—42no—
151C22×C9—41no—
152C22×C9—82no—
153C22×C9—81no—
154C22×C9—82no—
155C4×C9—82no—
156C4×C9—82no—
157C4×C9—81no—
1582−1+4—22no—
1592−1+4—62no—
160C2×#41—22no—
161C2×#41—31no—
162C2×#41—31no—
163C2×#41—31no—
164C2×#41—32no—
165C2×#41—62no—
166C2×#41C22×C912yes—
167C2×#41C22×C911yescharacteristic
168C22×Q8—31no—
169C22×Q8C22×C911yescharacteristic
170C22×Q8C22×C911yescharacteristic
171C22×D4—31no—
172C23×C4—31no—
173#127—32no—
174#127—32no—
175#124—32no—
176#124—32no—
177#123—31no—
178#123C22⋊C911yescharacteristic
179C22⋊Q8—32no—
180C22⋊Q8—32no—
181#121—32no—
182#121—34no—
183#121—32no—
184#121—32no—
185#121—32no—
186#120—32no—
187#120—31no—
188#120—31no—
189C4×Q8—32no—
190C4×Q8C22×C912yes—
191C4×D4—34no—
192C4×D4—32no—
193C4×D4—32no—
194#117—31no—
195#117—31no—
196C23×C3—32no—
197C23×C3—31no—
198C23×C3—62no—
199C23×C3—122no—
200C23×C3C22×A411yescharacteristic
201C23×C3C22×A411yescharacteristic
202C3×Q8—32no—
203C3×Q8—31no—
204C3×Q8—32no—
205C3×Q8—62no—
206C3×Q8—61no—
207C3×Q8—61no—
208C3×Q8C2×C3×D411yescharacteristic
209C3×D4—22no—
210C3×D4—62no—
211C3×D4—62no—
212C3×D4—62no—
213C3×D4—61no—
214C3×D4—61no—
215C3×D4—64no—
216C3×D4—62no—
217C3×D4—122no—
218C3×D4—122no—
219C3×D4—124no—
220C2×C3×C4—22no—
221C2×C3×C4—32no—
222C2×C3×C4—32no—
223C2×C3×C4—31no—
224C2×C3×C4—31no—
225C2×C3×C4—32no—
226C2×C3×C4—31no—
227C2×C3×C4—62no—
228C2×C3×C4—64no—
229C2×C3×C4—62no—
230C2×C3×C4—61no—
231C2×C3×C4—61no—
232C2×C3×C4—61no—
233C2×C3×C4—61no—
234C2×C3×C4—64no—
235C2×C3×C4C22×A412yes—
236C2×C3×C4C22×A411yescharacteristic
237C2×C3×C4C22×A412yes—
238C2×C9—41no—
239C2×C9—42no—
240C2×C9—82no—
241C2×C9—161no—
242C24—31no—
243#41—22no—
244#41—21no—
245#41—62no—
246#41—62no—
247#41—62no—
248#41—62no—
249#41—64no—
250#41—62no—
251#41—62no—
252#41—64no—
253#41—62no—
254#41—61no—
255C2×Q8—21no—
256C2×Q8—31no—
257C2×Q8—31no—
258C2×Q8—32no—
259C2×Q8—31no—
260C2×Q8—62no—
261C2×Q8—62no—
262C2×Q8—61no—
263C2×Q8D4×C912yes—
264C2×Q8C23×C911yescharacteristic
265C2×D4—32no—
266C2×D4—31no—
267C2×D4—32no—
268C2×D4—31no—
269C2×D4—31no—
270C2×D4—34no—
271C2×D4—32no—
272C2×D4—32no—
273C2×D4—62no—
274C2×D4—62no—
275C2×D4C2×C22⋊C911yescharacteristic
276C22×C4—31no—
277C22×C4—32no—
278C22×C4—31no—
279C22×C4—31no—
280C22×C4—32no—
281C22×C4—32no—
282C22×C4—62no—
283C22×C4—62no—
284C22×C4C2×C22⋊C911yescharacteristic
285C4⋊C4—32no—
286C4⋊C4—32no—
287C4⋊C4—32no—
288C4⋊C4C2×C22⋊C912yes—
289C22⋊C4—34no—
290C22⋊C4—31no—
291C22⋊C4—32no—
292C22⋊C4—32no—
293C22⋊C4—32no—
294C22⋊C4C2×C22⋊C911yescharacteristic
295C22⋊C4C2×C22⋊C912yes—
296C42—32no—
297C42—32no—
298C22×C3—22no—
299C22×C3—21no—
300C22×C3—22no—
301C22×C3—61no—
302C22×C3—62no—
303C22×C3—62no—
304C22×C3—62no—
305C22×C3—62no—
306C22×C3—124no—
307C22×C3D4×A412yes—
308C22×C3C23×A411yescharacteristic
309C3×C4—22no—
310C3×C4—21no—
311C3×C4—22no—
312C3×C4—32no—
313C3×C4—31no—
314C3×C4—31no—
315C3×C4—61no—
316C3×C4—62no—
317C9—41noSylow
318C23—32no—
319C23—31no—
320C23—62no—
321C23—122no—
322C23C22×C22⋊C911yescharacteristic
323C23C22×C22⋊C911yescharacteristic
324Q8—32no—
325Q8—31no—
326Q8—32no—
327Q8—62no—
328Q8—61no—
329Q8—61no—
330Q8C2×D4×C911yescharacteristic
331D4—22no—
332D4—62no—
333D4—62no—
334D4—62no—
335D4—61no—
336D4—61no—
337D4—64no—
338D4—62no—
339D4—122no—
340D4—122no—
341D4—124no—
342C2×C4—22no—
343C2×C4—31no—
344C2×C4—31no—
345C2×C4—32no—
346C2×C4—32no—
347C2×C4—32no—
348C2×C4—31no—
349C2×C4—62no—
350C2×C4—62no—
351C2×C4—64no—
352C2×C4—61no—
353C2×C4—61no—
354C2×C4—61no—
355C2×C4—61no—
356C2×C4—64no—
357C2×C4C22×C22⋊C912yes—
358C2×C4C22×C22⋊C911yescharacteristic
359C2×C4C22×C22⋊C912yes—
360C2×C3—22no—
361C2×C3—41no—
362C2×C3—121no—
363C2×C3—122no—
364C2×C3C2×D4×A411yescharacteristic
365C2×C3#594512yes—
366C22—22no—
367C22—21no—
368C22—22no—
369C22—61no—
370C22—62no—
371C22—62no—
372C22—62no—
373C22—62no—
374C22—124no—
375C22D4×C22⋊C912yes—
376C22C23×C22⋊C911yescharacteristic
377C4—22no—
378C4—21no—
379C4—22no—
380C4—31no—
381C4—31no—
382C4—32no—
383C4—61no—
384C4—62no—
385C3#3487⋊C311yescharacteristic
386C2—22no—
387C2—41no—
388C2—122no—
389C2—121no—
390C2C2×D4×C22⋊C911yescharacteristic
391C2#1059312012yes—
subgroupG/H
center, Frattini, socleC22×C3C23×A4
commutatorC2×Q8C23×C9
Fitting, prime radicalC3×#3487C3
cosocleC2×C3×Q8C23×C3
hypercenterC3×#123A4
hypocenterQ8C2×D4×C9
LCSG ▷ C2×Q8 ▷ Q8
UCSC1 ◁ C22×C3 ◁ C3×#123

Complements

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

NH ≅ G/Ntype
1. C2×2−1+4⋊C9C2semidirect
2. C22×#41⋊C9C2semidirect
3. C4⋊(#41⋊C9)C2semidirect
4. #548⋊C9C2semidirect
5. #519⋊C9C2semidirect
10. C2×#41⋊C9C22semidirect
12. C22×Q8⋊C9C22semidirect
14. C4×Q8⋊C9C22semidirect
31. C2×Q8⋊C9D4semidirect
36. #3487C9semidirect
84. C22×#41C2×C9semidirect
86. #550C2×C9semidirect
88. #548C2×C9semidirect
93. #519C2×C9semidirect

Maximal subgroups

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

subgroupindex
C2×2−1+4⋊C92
C22×#41⋊C92
C4⋊(#41⋊C9)2
#548⋊C92
#519⋊C92
C3×#34873
C9×#1234