Finite Group Databasebeta

← #3488⋊C9 #11919741

Subgroups — #3488⋊C9 #11919741

Order 1152.

702 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 702 conjugacy classes of subgroups — too many to draw. Only its 105 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×#63700C211yesmaximal, characteristic, complemented
2C22×#41⋊C9C211yesmaximal, characteristic, complemented
3C4⋊(#41⋊C9)C212yesmaximal, complemented
4#548⋊C9C212yesmaximal, complemented
5#520⋊C9C211yesmaximal, characteristic, complemented
6C3×#3488C311yesmaximal, characteristic
7#63700—22no—
8C2×#41⋊C9—22no—
9C2×#41⋊C9C2211yescharacteristic
10C2×#41⋊C9C2211yescharacteristic, complemented
11C2×#41⋊C9C2212yescomplemented
12C22×Q8⋊C9C2211yescharacteristic, complemented
13C9×C22⋊Q8—41nomaximal
14C4×Q8⋊C9C2212yescomplemented
15C2×C3×H(5,𝔽2)C2×C311yescharacteristic
16C22×C3×#41C2×C311yescharacteristic
17C3×#550C2×C312yes—
18C3×#548C2×C312yes—
19C3×#546—32no—
20C3×D42—32no—
21C3×#537—31no—
22C3×#535—31no—
23C3×#525—31no—
24C2×C3×#121—31no—
25C3×#520C2×C311yescharacteristic
26C2×Q8×C9—41no—
27C22×C4×C9—41no—
28#41⋊C9—21no—
29#41⋊C9—22no—
30#41⋊C9D412yescomplemented
31C2×Q8⋊C9—21no—
32C2×Q8⋊C9C2311yescharacteristic
33C9×C4⋊C4—41no—
34C9×C4⋊C4—42no—
35C9×C22⋊C4—42no—
36#3488C911yescharacteristic, complemented, Sylow
37C3×H(5,𝔽2)—22no—
38C3×H(5,𝔽2)—62no—
39C2×C3×#41—22no—
40C2×C3×#41—31no—
41C2×C3×#41—31no—
42C2×C3×#41—31no—
43C2×C3×#41—62no—
44C2×C3×#41C22×C311yescharacteristic
45C2×C3×#41C22×C312yes—
46C2×C3×#41C22×C311yescharacteristic
47C22×C3×Q8C22×C311yescharacteristic
48C22×C3×D4—31no—
49C22×C3×D4—32no—
50C22×C3×D4—31no—
51C22×C3×D4—32no—
52C23×C3×C4—31no—
53C3×#127—32no—
54C3×#124—32no—
55C3×#123—32no—
56C3×C22⋊Q8—31no—
57C3×C22⋊Q8—32no—
58C3×C22⋊Q8A411yescharacteristic
59C3×#121—32no—
60C3×#121—32no—
61C3×#121—34no—
62C3×#121—31no—
63C3×#121—31no—
64C3×#121—32no—
65C3×#121—32no—
66C3×#120—34no—
67C3×#120—32no—
68C3×C4×Q8C22×C312yes—
69C3×C4×D4—32no—
70C3×C4×D4—32no—
71C3×C4×D4—32no—
72C3×#117—31no—
73C2×C3×C4⋊C4—31no—
74C2×C3×C22⋊C4—32no—
75C23×C9—41no—
76Q8×C9—82no—
77C2×C4×C9—41no—
78C2×C4×C9—42no—
79C2×C4×C9—42no—
80C2×C4×C9—41no—
81C2×C4×C9—82no—
82Q8⋊C9C2×D411yescharacteristic, complemented
83C2×H(5,𝔽2)C2×C911yescharacteristic, complemented
84C22×#41C2×C911yescharacteristic
85#550C2×C912yescomplemented
86#548C2×C912yes—
87#546—32no—
88D42—32no—
89#537—31no—
90#535—31no—
91#525—31no—
92C2×#121—31no—
93#520C2×C911yescharacteristic, complemented
94C24×C3—32no—
95C24×C3—31no—
96C3×#41—21no—
97C3×#41—22no—
98C3×#41—32no—
99C3×#41—62no—
100C3×#41—62no—
101C3×#41—62no—
102C3×#41—62no—
103C3×#41—62no—
104C3×#41—62no—
105C3×#41—61no—
106C3×#41—62no—
107C3×#41C3×D412yes—
108C2×C3×Q8—21no—
109C2×C3×Q8—31no—
110C2×C3×Q8—61no—
111C2×C3×Q8C2×A411yescharacteristic
112C2×C3×Q8C23×C311yescharacteristic
113C2×C3×D4—31no—
114C2×C3×D4—32no—
115C2×C3×D4—31no—
116C2×C3×D4—32no—
117C2×C3×D4—32no—
118C2×C3×D4—34no—
119C2×C3×D4—32no—
120C2×C3×D4—31no—
121C2×C3×D4—32no—
122C2×C3×D4—62no—
123C2×C3×D4—62no—
124C2×C3×D4—62no—
125C2×C3×D4—62no—
126C2×C3×D4—64no—
127C2×C3×D4—64no—
128C2×C3×D4—124no—
129C22×C3×C4—31no—
130C22×C3×C4—31no—
131C22×C3×C4—32no—
132C22×C3×C4—31no—
133C22×C3×C4—31no—
134C22×C3×C4—32no—
135C22×C3×C4—31no—
136C22×C3×C4—62no—
137C22×C3×C4—62no—
138C22×C3×C4C2×A411yescharacteristic
139C3×C4⋊C4—32no—
140C3×C4⋊C4—31no—
141C3×C4⋊C4—32no—
142C3×C4⋊C4C2×A411yescharacteristic
143C3×C4⋊C4C2×A412yes—
144C3×C22⋊C4—34no—
145C3×C22⋊C4—32no—
146C3×C22⋊C4—32no—
147C3×C22⋊C4—32no—
148C3×C22⋊C4—32no—
149C3×C22⋊C4C2×A412yes—
150C3×C42—32no—
151C22×C9—42no—
152C22×C9—41no—
153C22×C9—81no—
154C22×C9—81no—
155C4×C9—42no—
156C4×C9—82no—
157C4×C9—81no—
158C4×C9—82no—
159H(5,𝔽2)—22no—
160H(5,𝔽2)—62no—
161C2×#41—22no—
162C2×#41—31no—
163C2×#41—31no—
164C2×#41—31no—
165C2×#41—62no—
166C2×#41C22×C912yes—
167C2×#41C22×C911yescharacteristic
168C2×#41C22×C911yescharacteristic
169C22×Q8C22×C911yescharacteristic
170C22×D4—31no—
171C22×D4—32no—
172C22×D4—31no—
173C22×D4—32no—
174C23×C4—31no—
175#127—32no—
176#124—32no—
177#123—32no—
178C22⋊Q8—31no—
179C22⋊Q8—32no—
180C22⋊Q8C22⋊C911yescharacteristic
181#121—32no—
182#121—32no—
183#121—34no—
184#121—31no—
185#121—31no—
186#121—32no—
187#121—32no—
188#120—34no—
189#120—32no—
190C4×Q8C22×C912yes—
191C4×D4—32no—
192C4×D4—32no—
193C4×D4—32no—
194#117—31no—
195C2×C4⋊C4—31no—
196C2×C22⋊C4—32no—
197C23×C3—31no—
198C23×C3—32no—
199C23×C3—31no—
200C23×C3—62no—
201C23×C3—64no—
202C23×C3—62no—
203C23×C3—121no—
204C23×C3—122no—
205C23×C3—122no—
206C23×C3C22×A411yescharacteristic
207C3×Q8—22no—
208C3×Q8—31no—
209C3×Q8—61no—
210C3×Q8—61no—
211C3×Q8C2×C3×D411yescharacteristic
212C3×D4—32no—
213C3×D4—32no—
214C3×D4—64no—
215C3×D4—62no—
216C3×D4—62no—
217C3×D4—61no—
218C3×D4—61no—
219C3×D4—64no—
220C3×D4—64no—
221C3×D4—62no—
222C3×D4—122no—
223C3×D4—124no—
224C2×C3×C4—22no—
225C2×C3×C4—31no—
226C2×C3×C4—32no—
227C2×C3×C4—32no—
228C2×C3×C4—32no—
229C2×C3×C4—31no—
230C2×C3×C4—61no—
231C2×C3×C4—61no—
232C2×C3×C4—62no—
233C2×C3×C4—62no—
234C2×C3×C4—62no—
235C2×C3×C4—61no—
236C2×C3×C4—61no—
237C2×C3×C4—62no—
238C2×C3×C4—122no—
239C2×C3×C4C22×A412yes—
240C2×C3×C4C22×A411yescharacteristic
241C2×C3×C4C22×A412yes—
242C2×C3×C4C22×A411yescharacteristic
243C2×C9—41no—
244C2×C9—41no—
245C2×C9—41no—
246C2×C9—82no—
247C24—32no—
248C24—31no—
249#41—21no—
250#41—22no—
251#41—32no—
252#41—62no—
253#41—62no—
254#41—62no—
255#41—62no—
256#41—62no—
257#41—62no—
258#41—61no—
259#41—62no—
260#41D4×C912yes—
261C2×Q8—21no—
262C2×Q8—31no—
263C2×Q8—61no—
264C2×Q8C2×C22⋊C911yescharacteristic
265C2×Q8C23×C911yescharacteristic
266C2×D4—31no—
267C2×D4—32no—
268C2×D4—31no—
269C2×D4—32no—
270C2×D4—32no—
271C2×D4—34no—
272C2×D4—32no—
273C2×D4—31no—
274C2×D4—32no—
275C2×D4—62no—
276C2×D4—62no—
277C2×D4—62no—
278C2×D4—62no—
279C2×D4—64no—
280C2×D4—64no—
281C2×D4—124no—
282C22×C4—31no—
283C22×C4—31no—
284C22×C4—32no—
285C22×C4—31no—
286C22×C4—31no—
287C22×C4—32no—
288C22×C4—31no—
289C22×C4—62no—
290C22×C4—62no—
291C22×C4C2×C22⋊C911yescharacteristic
292C4⋊C4—32no—
293C4⋊C4—31no—
294C4⋊C4—32no—
295C4⋊C4C2×C22⋊C911yescharacteristic
296C4⋊C4C2×C22⋊C912yes—
297C22⋊C4—34no—
298C22⋊C4—32no—
299C22⋊C4—32no—
300C22⋊C4—32no—
301C22⋊C4—32no—
302C22⋊C4C2×C22⋊C912yes—
303C42—32no—
304C22×C3—21no—
305C22×C3—21no—
306C22×C3—32no—
307C22×C3—61no—
308C22×C3—61no—
309C22×C3—61no—
310C22×C3—62no—
311C22×C3—62no—
312C22×C3—61no—
313C22×C3—61no—
314C22×C3—62no—
315C22×C3—124no—
316C22×C3—244no—
317C22×C3#99412yes—
318C22×C3C23×A411yescharacteristic
319C3×C4—21no—
320C3×C4—22no—
321C3×C4—22no—
322C3×C4—31no—
323C3×C4—31no—
324C3×C4—61no—
325C3×C4—62no—
326C3×C4D4×A412yes—
327C9—41noSylow
328C23—31no—
329C23—32no—
330C23—31no—
331C23—62no—
332C23—64no—
333C23—62no—
334C23—121no—
335C23—122no—
336C23—122no—
337C23C22×C22⋊C911yescharacteristic
338Q8—22no—
339Q8—31no—
340Q8—61no—
341Q8—61no—
342Q8C2×D4×C911yescharacteristic
343D4—32no—
344D4—32no—
345D4—64no—
346D4—62no—
347D4—62no—
348D4—61no—
349D4—61no—
350D4—64no—
351D4—64no—
352D4—62no—
353D4—122no—
354D4—124no—
355C2×C4—22no—
356C2×C4—31no—
357C2×C4—32no—
358C2×C4—32no—
359C2×C4—32no—
360C2×C4—31no—
361C2×C4—61no—
362C2×C4—61no—
363C2×C4—62no—
364C2×C4—62no—
365C2×C4—62no—
366C2×C4—61no—
367C2×C4—61no—
368C2×C4—62no—
369C2×C4—122no—
370C2×C4C22×C22⋊C912yes—
371C2×C4C22×C22⋊C911yescharacteristic
372C2×C4C22×C22⋊C912yes—
373C2×C4C22×C22⋊C911yescharacteristic
374C2×C3—22no—
375C2×C3—62no—
376C2×C3—121no—
377C2×C3—122no—
378C2×C3C2×D4×A411yescharacteristic
379C2×C3C2×#99411yescharacteristic
380C2×C3#594511yescharacteristic
381C22—21no—
382C22—21no—
383C22—32no—
384C22—61no—
385C22—61no—
386C22—61no—
387C22—62no—
388C22—62no—
389C22—61no—
390C22—61no—
391C22—62no—
392C22—124no—
393C22—244no—
394C22#6370012yescomplemented
395C22C23×C22⋊C911yescharacteristic
396C4—21no—
397C4—22no—
398C4—22no—
399C4—31no—
400C4—31no—
401C4—61no—
402C4—62no—
403C4D4×C22⋊C912yes—
404C3#3488⋊C311yescharacteristic
405C2—22no—
406C2—62no—
407C2—121no—
408C2—122no—
409C2C2×D4×C22⋊C911yescharacteristic
410C2C2×#6370011yescharacteristic
411C2#1059312011yescharacteristic
subgroupG/H
center, Frattini, socleC22×C3C23×A4
commutatorC2×Q8C23×C9
Fitting, prime radicalC3×#3488C3
cosocleC2×C3×Q8C23×C3
hypercenterC3×C22⋊Q8A4
hypocenterQ8C2×D4×C9
LCSG ▷ C2×Q8 ▷ Q8
UCSC1 ◁ C22×C3 ◁ C3×C22⋊Q8

Complements

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

NH ≅ G/Ntype
1. C2×#63700C2semidirect
2. C22×#41⋊C9C2semidirect
3. C4⋊(#41⋊C9)C2semidirect
4. #548⋊C9C2semidirect
5. #520⋊C9C2semidirect
10. C2×#41⋊C9C22semidirect
11. C2×#41⋊C9C22semidirect
12. C22×Q8⋊C9C22semidirect
14. C4×Q8⋊C9C22semidirect
30. #41⋊C9D4semidirect
36. #3488C9semidirect
82. Q8⋊C9C2×D4semidirect
83. C2×H(5,𝔽2)C2×C9semidirect
85. #550C2×C9semidirect
93. #520C2×C9semidirect
394. C22#63700semidirect

Maximal subgroups

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

subgroupindex
C2×#637002
C22×#41⋊C92
C4⋊(#41⋊C9)2
#548⋊C92
#520⋊C92
C3×#34883
C9×C22⋊Q84