Finite Group Databasebeta

← #3486⋊C9 #11919737

Subgroups — #3486⋊C9 #11919737

Order 1152.

712 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 712 conjugacy classes of subgroups — too many to draw. Only its 109 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
2C23×Q8⋊C9C211yesmaximal, characteristic, complemented
3D4×Q8⋊C9C212yesmaximal, complemented
4#548⋊C9C212yesmaximal, complemented
5#519⋊C9C211yesmaximal, characteristic, complemented
6C3×#3486C311yesmaximal, characteristic
72−1+4⋊C9—22no—
8C2×#41⋊C9C2211yescharacteristic, complemented
9C22×Q8⋊C9—22no—
10C22×Q8⋊C9C2212yes—
11C22×Q8⋊C9C2211yescharacteristic
12C22×Q8⋊C9C2211yescharacteristic, complemented
13C9×#120—41nomaximal
14C4×Q8⋊C9C2212yescomplemented
15C2×C3×2−1+4C2×C311yescharacteristic
16C23×C3×Q8C2×C311yescharacteristic
17C3×D4×Q8—32no—
18C3×D4×Q8C2×C312yes—
19C3×#548—32no—
20C3×#548C2×C312yes—
21C3×#536—31no—
22C3×#536—31no—
23C3×#525—31no—
24C3×#525—31no—
25C3×#519C2×C311yescharacteristic
26C24×C9—41no—
27C2×D4×C9—41no—
28C2×D4×C9—42no—
29#41⋊C9—21no—
30C2×Q8⋊C9—22no—
31C2×Q8⋊C9—21no—
32C2×Q8⋊C9D412yescomplemented
33C2×Q8⋊C9C2311yescharacteristic
34C9×C22⋊C4—41no—
35C9×C22⋊C4—42no—
36#3486C911yescharacteristic, complemented, Sylow
37C3×2−1+4—22no—
38C3×2−1+4—62no—
39C2×C3×#41—31no—
40C2×C3×#41—31no—
41C2×C3×#41—31no—
42C2×C3×#41C22×C311yescharacteristic
43C22×C3×Q8—22no—
44C22×C3×Q8—31no—
45C22×C3×Q8—31no—
46C22×C3×Q8—32no—
47C22×C3×Q8—62no—
48C22×C3×Q8C22×C312yes—
49C22×C3×Q8C22×C311yescharacteristic
50C22×C3×Q8C22×C311yescharacteristic
51C23×C3×C4—31no—
52C3×C4⋊Q8—32no—
53C3×C4⋊Q8—32no—
54C3×#124—32no—
55C3×#124—32no—
56C3×#123—32no—
57C3×#123—31no—
58C3×#123—31no—
59C3×C22⋊Q8—34no—
60C3×C22⋊Q8—32no—
61C3×C22⋊Q8—32no—
62C3×C22⋊Q8—32no—
63C3×C22⋊Q8—32no—
64C3×#121—32no—
65C3×#121—32no—
66C3×#120—31no—
67C3×#120A411yescharacteristic
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—42no—
77C23×C9—41no—
78C23×C9—84no—
79C23×C9—82no—
80D4×C9—84no—
81D4×C9—82no—
82C2×C4×C9—42no—
83C2×C4×C9—41no—
84Q8⋊C9C2×D411yescharacteristic
85C2×2−1+4C2×C911yescharacteristic, complemented
86C23×Q8C2×C911yescharacteristic, complemented
87D4×Q8—32no—
88D4×Q8C2×C912yescomplemented
89#548—32no—
90#548C2×C912yescomplemented
91#536—31no—
92#536—31no—
93#525—31no—
94#525—31no—
95#519C2×C911yescharacteristic, complemented
96C24×C3C2×A411yescharacteristic
97C3×#41—21no—
98C3×#41—62no—
99C3×#41—62no—
100C3×#41—62no—
101C3×#41—62no—
102C3×#41—62no—
103C3×#41—62no—
104C3×#41—61no—
105C2×C3×Q8—22no—
106C2×C3×Q8—21no—
107C2×C3×Q8—32no—
108C2×C3×Q8—31no—
109C2×C3×Q8—31no—
110C2×C3×Q8—32no—
111C2×C3×Q8—32no—
112C2×C3×Q8—31no—
113C2×C3×Q8—62no—
114C2×C3×Q8—62no—
115C2×C3×Q8—62no—
116C2×C3×Q8—64no—
117C2×C3×Q8—62no—
118C2×C3×Q8—62no—
119C2×C3×Q8—61no—
120C2×C3×Q8—64no—
121C2×C3×Q8C3×D412yes—
122C2×C3×Q8C23×C311yescharacteristic
123C2×C3×D4—31no—
124C2×C3×D4—32no—
125C2×C3×D4—31no—
126C2×C3×D4—31no—
127C2×C3×D4C2×A412yes—
128C2×C3×D4C2×A411yescharacteristic
129C22×C3×C4—31no—
130C22×C3×C4—32no—
131C22×C3×C4—32no—
132C22×C3×C4—32no—
133C22×C3×C4—31no—
134C22×C3×C4—31no—
135C22×C3×C4—31no—
136C22×C3×C4—62no—
137C22×C3×C4—62no—
138C3×C4⋊C4—32no—
139C3×C4⋊C4—34no—
140C3×C4⋊C4—32no—
141C3×C4⋊C4—32no—
142C3×C4⋊C4—32no—
143C3×C22⋊C4—34no—
144C3×C22⋊C4—31no—
145C3×C22⋊C4—32no—
146C3×C22⋊C4—32no—
147C3×C22⋊C4—32no—
148C3×C22⋊C4C2×A411yescharacteristic
149C3×C22⋊C4C2×A412yes—
150C3×C42—32no—
151C3×C42—32no—
152C22×C9—42no—
153C22×C9—44no—
154C22×C9—41no—
155C22×C9—82no—
156C22×C9—82no—
157C22×C9—82no—
158C22×C9—81no—
159C22×C9—82no—
160C22×C9—88no—
161C4×C9—82no—
162C4×C9—81no—
1632−1+4—22no—
1642−1+4—62no—
165C2×#41—31no—
166C2×#41—31no—
167C2×#41—31no—
168C2×#41C22×C911yescharacteristic, complemented
169C22×Q8—22no—
170C22×Q8—31no—
171C22×Q8—31no—
172C22×Q8—32no—
173C22×Q8—62no—
174C22×Q8C22×C912yes—
175C22×Q8C22×C911yescharacteristic
176C22×Q8C22×C911yescharacteristic, complemented
177C23×C4—31no—
178C4⋊Q8—32no—
179C4⋊Q8—32no—
180#124—32no—
181#124—32no—
182#123—32no—
183#123—31no—
184#123—31no—
185C22⋊Q8—34no—
186C22⋊Q8—32no—
187C22⋊Q8—32no—
188C22⋊Q8—32no—
189C22⋊Q8—32no—
190#121—32no—
191#121—32no—
192#120—31no—
193#120C22⋊C911yescharacteristic
194C4×Q8—32no—
195C4×Q8C22×C912yescomplemented
196C4×D4—34no—
197C4×D4—32no—
198C4×D4—32no—
199#117—31no—
200#117—31no—
201C23×C3—22no—
202C23×C3—24no—
203C23×C3—31no—
204C23×C3C22×A411yescharacteristic
205C23×C3C22×A411yescharacteristic
206C23×C3C22×A412yes—
207C3×Q8—32no—
208C3×Q8—31no—
209C3×Q8—32no—
210C3×Q8—62no—
211C3×Q8—62no—
212C3×Q8—61no—
213C3×Q8—62no—
214C3×Q8—61no—
215C3×Q8—64no—
216C3×Q8C2×C3×D411yescharacteristic
217C3×D4—22no—
218C3×D4—24no—
219C3×D4—62no—
220C3×D4—61no—
221C3×D4—61no—
222C3×D4—62no—
223C3×D4—64no—
224C2×C3×C4—32no—
225C2×C3×C4—32no—
226C2×C3×C4—31no—
227C2×C3×C4—31no—
228C2×C3×C4—32no—
229C2×C3×C4—32no—
230C2×C3×C4—31no—
231C2×C3×C4—61no—
232C2×C3×C4—61no—
233C2×C3×C4—62no—
234C2×C3×C4—64no—
235C2×C3×C4—62no—
236C2×C3×C4—61no—
237C2×C3×C4—61no—
238C2×C3×C4—64no—
239C2×C3×C4—62no—
240C2×C3×C4C22×A412yes—
241C2×C3×C4C22×A411yescharacteristic
242C2×C9—41no—
243C2×C9—42no—
244C2×C9—82no—
245C2×C9—84no—
246C2×C9—161no—
247C24C2×C22⋊C911yescharacteristic
248#41—21no—
249#41—62no—
250#41—62no—
251#41—62no—
252#41—62no—
253#41—62no—
254#41—62no—
255#41—61no—
256C2×Q8—22no—
257C2×Q8—21no—
258C2×Q8—32no—
259C2×Q8—31no—
260C2×Q8—31no—
261C2×Q8—32no—
262C2×Q8—32no—
263C2×Q8—31no—
264C2×Q8—62no—
265C2×Q8—62no—
266C2×Q8—62no—
267C2×Q8—64no—
268C2×Q8—62no—
269C2×Q8—62no—
270C2×Q8—61no—
271C2×Q8—64no—
272C2×Q8D4×C912yescomplemented
273C2×Q8C23×C911yescharacteristic
274C2×D4—31no—
275C2×D4—32no—
276C2×D4—31no—
277C2×D4—31no—
278C2×D4C2×C22⋊C912yes—
279C2×D4C2×C22⋊C911yescharacteristic
280C22×C4—31no—
281C22×C4—32no—
282C22×C4—32no—
283C22×C4—32no—
284C22×C4—31no—
285C22×C4—31no—
286C22×C4—31no—
287C22×C4—62no—
288C22×C4—62no—
289C4⋊C4—32no—
290C4⋊C4—34no—
291C4⋊C4—32no—
292C4⋊C4—32no—
293C4⋊C4—32no—
294C22⋊C4—34no—
295C22⋊C4—31no—
296C22⋊C4—32no—
297C22⋊C4—32no—
298C22⋊C4—32no—
299C22⋊C4C2×C22⋊C911yescharacteristic
300C22⋊C4C2×C22⋊C912yes—
301C42—32no—
302C42—32no—
303C22×C3—28no—
304C22×C3—22no—
305C22×C3—22no—
306C22×C3—22no—
307C22×C3—21no—
308C22×C3—22no—
309C22×C3—61no—
310C22×C3—62no—
311C22×C3D4×A412yes—
312C22×C32−1+4⋊C314yes—
313C22×C3C23×A411yescharacteristic
314C3×C4—21no—
315C3×C4—22no—
316C3×C4—32no—
317C3×C4—31no—
318C3×C4—31no—
319C3×C4—61no—
320C3×C4—62no—
321C3×C4—62no—
322C9—41noSylow
323C23—22no—
324C23—24no—
325C23—31no—
326C23C22×C22⋊C911yescharacteristic
327C23C22×C22⋊C911yescharacteristic
328C23C22×C22⋊C912yes—
329Q8—32no—
330Q8—31no—
331Q8—32no—
332Q8—62no—
333Q8—62no—
334Q8—61no—
335Q8—62no—
336Q8—61no—
337Q8—64no—
338Q8C2×D4×C911yescharacteristic
339D4—22no—
340D4—24no—
341D4—62no—
342D4—61no—
343D4—61no—
344D4—62no—
345D4—64no—
346C2×C4—31no—
347C2×C4—31no—
348C2×C4—32no—
349C2×C4—32no—
350C2×C4—32no—
351C2×C4—32no—
352C2×C4—31no—
353C2×C4—61no—
354C2×C4—61no—
355C2×C4—62no—
356C2×C4—62no—
357C2×C4—64no—
358C2×C4—61no—
359C2×C4—61no—
360C2×C4—64no—
361C2×C4—62no—
362C2×C4C22×C22⋊C912yes—
363C2×C4C22×C22⋊C911yescharacteristic
364C2×C3—22no—
365C2×C3—24no—
366C2×C3—41no—
367C2×C3—121no—
368C2×C3C2×D4×A411yescharacteristic
369C2×C3C2×2−1+4⋊C312yes—
370C22—28no—
371C22—22no—
372C22—22no—
373C22—22no—
374C22—21no—
375C22—22no—
376C22—61no—
377C22—62no—
378C22D4×C22⋊C912yes—
379C222−1+4⋊C914yescomplemented
380C22C23×C22⋊C911yescharacteristic
381C4—21no—
382C4—22no—
383C4—31no—
384C4—31no—
385C4—32no—
386C4—61no—
387C4—62no—
388C4—62no—
389C3#3486⋊C311yescharacteristic
390C2—22no—
391C2—24no—
392C2—41no—
393C2—121no—
394C2C2×D4×C22⋊C911yescharacteristic
395C2C2×2−1+4⋊C912yes—
subgroupG/H
center, Frattini, socleC22×C3C23×A4
commutatorC2×Q8C23×C9
Fitting, prime radicalC3×#3486C3
cosocleC2×C3×Q8C23×C3
hypercenterC3×#120A4
hypocenterQ8C2×D4×C9
LCSG ▷ C2×Q8 ▷ Q8
UCSC1 ◁ C22×C3 ◁ C3×#120

Complements

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

NH ≅ G/Ntype
1. C2×2−1+4⋊C9C2semidirect
2. C23×Q8⋊C9C2semidirect
3. D4×Q8⋊C9C2semidirect
4. #548⋊C9C2semidirect
5. #519⋊C9C2semidirect
8. C2×#41⋊C9C22semidirect
12. C22×Q8⋊C9C22semidirect
14. C4×Q8⋊C9C22semidirect
32. C2×Q8⋊C9D4semidirect
36. #3486C9semidirect
85. C2×2−1+4C2×C9semidirect
86. C23×Q8C2×C9semidirect
88. D4×Q8C2×C9semidirect
90. #548C2×C9semidirect
95. #519C2×C9semidirect
168. C2×#41C22×C9semidirect
176. C22×Q8C22×C9semidirect
195. C4×Q8C22×C9semidirect
272. C2×Q8D4×C9semidirect
379. C222−1+4⋊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
C23×Q8⋊C92
D4×Q8⋊C92
#548⋊C92
#519⋊C92
C3×#34863
C9×#1204