Finite Group Databasebeta

← C4×2−1+4⋊C9 #11919689

Subgroups — C4×2−1+4⋊C9 #11919689

Order 1152.

670 conjugacy classes of subgroups; only the 145 normal subgroups are shown.

normal subgroups only

This group has 670 conjugacy classes of subgroups — too many to draw. Only its 145 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
2#520⋊C9C211yesmaximal, characteristic, complemented
3#519⋊C9C212yesmaximal, complemented
4C4×#41⋊C9C211yesmaximal, characteristic, complemented
5C2×C4×Q8⋊C9C212yesmaximal, complemented
6C3×C4×2−1+4C311yesmaximal, characteristic
72−1+4⋊C9C414yesdirect product factor
8C2×#41⋊C9C2211yescharacteristic
9C22×Q8⋊C9C2212yes—
10C4×D4×C9—41nomaximal
11C4×Q8⋊C9C2211yescharacteristic
12C4×Q8⋊C9C2212yes—
13C4×Q8⋊C9C2211yescharacteristic
14C2×C3×2−1+4C2×C311yescharacteristic
15C3×#520—31no—
16C3×#520—32no—
17C3×#520C2×C311yescharacteristic
18C3×#519—31no—
19C3×#519C2×C312yes—
20C3×C4×#41—31no—
21C3×C4×#41—32no—
22C3×C4×#41C2×C311yescharacteristic
23C2×C3×C4×Q8—31no—
24C2×C3×C4×Q8C2×C312yes—
25C2×D4×C9—41no—
26C22×C4×C9—42no—
27#41⋊C9C2×C412yescomplemented
28C2×Q8⋊C9C2×C414yescomplemented
29C2×Q8⋊C9C2311yescharacteristic
30C9×C4⋊C4—41no—
31C9×C22⋊C4—42no—
32C42×C9—41no—
33C4×2−1+4C911yescharacteristic, complemented, Sylow
34C3×2−1+4—34no—
35C3×2−1+4C3×C414yes—
36C2×C3×#41—31no—
37C2×C3×#41—32no—
38C2×C3×#41C22×C311yescharacteristic
39C22×C3×Q8—31no—
40C22×C3×Q8C22×C312yes—
41C3×C4×Q8—32no—
42C3×C4×Q8—31no—
43C3×C4×Q8—32no—
44C3×C4×Q8—32no—
45C3×C4×Q8—31no—
46C3×C4×Q8—32no—
47C3×C4×Q8—32no—
48C3×C4×Q8C22×C311yescharacteristic
49C3×C4×Q8C22×C312yes—
50C3×C4×Q8C22×C311yescharacteristic
51C3×C4×D4—34no—
52C3×C4×D4—31no—
53C3×C4×D4—32no—
54C3×C4×D4—31no—
55C3×C4×D4—31no—
56C3×C4×D4—32no—
57C3×C4×D4—32no—
58C3×C4×D4A411yescharacteristic
59C3×#117—34no—
60C3×#117—31no—
61C3×#117—32no—
62C3×#117—32no—
63C3×#117—31no—
64C2×C3×C4⋊C4—32no—
65C2×C3×C4⋊C4—32no—
66C2×C3×C4⋊C4—31no—
67C2×C3×C42—32no—
68C2×C3×C42—32no—
69C2×C3×C42—31no—
70C23×C9—42no—
71D4×C9—44no—
72C2×C4×C9—41no—
73C2×C4×C9—41no—
74C2×C4×C9—42no—
75C2×C4×C9—41no—
76C2×C4×C9—84no—
77Q8⋊C9C22×C411yescharacteristic
78C2×2−1+4C2×C911yescharacteristic
79#520—31no—
80#520—32no—
81#520C2×C911yescharacteristic, complemented
82#519—31no—
83#519C2×C912yescomplemented
84C4×#41—31no—
85C4×#41—32no—
86C4×#41C2×C911yescharacteristic, complemented
87C2×C4×Q8—31no—
88C2×C4×Q8C2×C912yescomplemented
89C3×#41—38no—
90C3×#41—34no—
91C3×#41—34no—
92C3×#41—38no—
93C3×#41—32no—
94C3×#41C2×C3×C412yes—
95C2×C3×Q8—34no—
96C2×C3×Q8—34no—
97C2×C3×Q8—31no—
98C2×C3×Q8—32no—
99C2×C3×Q8—34no—
100C2×C3×Q8C2×C3×C414yes—
101C2×C3×Q8C23×C311yescharacteristic
102C2×C3×D4—31no—
103C2×C3×D4—32no—
104C2×C3×D4C2×A411yescharacteristic
105C22×C3×C4—32no—
106C22×C3×C4—32no—
107C22×C3×C4—31no—
108C22×C3×C4—32no—
109C22×C3×C4—32no—
110C22×C3×C4—31no—
111C22×C3×C4—31no—
112C22×C3×C4C2×A412yes—
113C3×C4⋊C4—31no—
114C3×C4⋊C4—32no—
115C3×C4⋊C4—31no—
116C3×C4⋊C4—34no—
117C3×C4⋊C4—32no—
118C3×C4⋊C4—31no—
119C3×C4⋊C4—32no—
120C3×C4⋊C4C2×A411yescharacteristic
121C3×C22⋊C4—32no—
122C3×C22⋊C4—32no—
123C3×C22⋊C4—32no—
124C3×C22⋊C4C2×A412yes—
125C3×C42—34no—
126C3×C42—32no—
127C3×C42—31no—
128C3×C42—32no—
129C3×C42—31no—
130C3×C42—32no—
131C3×C42—31no—
132C3×C42C2×A411yescharacteristic
133C22×C9—44no—
134C22×C9—41no—
135C22×C9—82no—
136C22×C9—82no—
137C4×C9—42no—
138C4×C9—42no—
139C4×C9—81no—
140C4×C9—82no—
1412−1+4—34no—
1422−1+4C4×C914yescomplemented
143C2×#41—31no—
144C2×#41—32no—
145C2×#41C22×C911yescharacteristic
146C22×Q8—31no—
147C22×Q8C22×C912yes—
148C4×Q8—32no—
149C4×Q8—31no—
150C4×Q8—32no—
151C4×Q8—32no—
152C4×Q8—31no—
153C4×Q8—32no—
154C4×Q8—32no—
155C4×Q8C22×C911yescharacteristic
156C4×Q8C22×C912yes—
157C4×Q8C22×C911yescharacteristic
158C4×D4—34no—
159C4×D4—31no—
160C4×D4—32no—
161C4×D4—31no—
162C4×D4—31no—
163C4×D4—32no—
164C4×D4—32no—
165C4×D4C22⋊C911yescharacteristic
166#117—34no—
167#117—31no—
168#117—32no—
169#117—32no—
170#117—31no—
171C2×C4⋊C4—32no—
172C2×C4⋊C4—32no—
173C2×C4⋊C4—31no—
174C2×C42—32no—
175C2×C42—32no—
176C2×C42—31no—
177C23×C3—31no—
178C23×C3C22×A412yes—
179C3×Q8—34no—
180C3×Q8—31no—
181C3×Q8—34no—
182C3×Q8—34no—
183C3×Q8C22×C3×C411yescharacteristic
184C3×D4—32no—
185C3×D4—32no—
186C3×D4—38no—
187C3×D4C4×A414yes—
188C2×C3×C4—24no—
189C2×C3×C4—38no—
190C2×C3×C4—34no—
191C2×C3×C4—34no—
192C2×C3×C4—32no—
193C2×C3×C4—31no—
194C2×C3×C4—32no—
195C2×C3×C4—32no—
196C2×C3×C4—31no—
197C2×C3×C4—32no—
198C2×C3×C4—31no—
199C2×C3×C4—62no—
200C2×C3×C4—121no—
201C2×C3×C4—122no—
202C2×C3×C4—122no—
203C2×C3×C4C22×A412yes—
204C2×C3×C4C22×A411yescharacteristic
205C2×C3×C4C22×A411yescharacteristic
206C2×C3×C4C22×A411yescharacteristic
207C2×C9—41no—
208C2×C9—41no—
209C2×C9—41no—
210C2×C9—84no—
211#41—38no—
212#41—34no—
213#41—34no—
214#41—38no—
215#41—32no—
216#41C2×C4×C912yescomplemented
217C2×Q8—34no—
218C2×Q8—34no—
219C2×Q8—31no—
220C2×Q8—32no—
221C2×Q8—34no—
222C2×Q8C2×C4×C914yescomplemented
223C2×Q8C23×C911yescharacteristic
224C2×D4—31no—
225C2×D4—32no—
226C2×D4C2×C22⋊C911yescharacteristic
227C22×C4—32no—
228C22×C4—32no—
229C22×C4—31no—
230C22×C4—32no—
231C22×C4—32no—
232C22×C4—31no—
233C22×C4—31no—
234C22×C4C2×C22⋊C912yes—
235C4⋊C4—31no—
236C4⋊C4—32no—
237C4⋊C4—31no—
238C4⋊C4—34no—
239C4⋊C4—32no—
240C4⋊C4—31no—
241C4⋊C4—32no—
242C4⋊C4C2×C22⋊C911yescharacteristic
243C22⋊C4—32no—
244C22⋊C4—32no—
245C22⋊C4—32no—
246C22⋊C4C2×C22⋊C912yes—
247C42—34no—
248C42—31no—
249C42—32no—
250C42—31no—
251C42—32no—
252C42—32no—
253C42—31no—
254C42C2×C22⋊C911yescharacteristic
255C22×C3—22no—
256C22×C3—22no—
257C22×C3—32no—
258C22×C3—61no—
259C22×C3—61no—
260C22×C3C2×C4×A414yes—
261C22×C3C23×A411yescharacteristic
262C3×C4—22no—
263C3×C4—21no—
264C3×C4—34no—
265C3×C4—31no—
266C3×C4—31no—
267C3×C4—61no—
268C3×C4—61no—
269C3×C4—62no—
270C3×C4C2×C4×A412yes—
271C3×C42−1+4⋊C312yes—
272C9—41noSylow
273C23—31no—
274C23C22×C22⋊C912yes—
275Q8—34no—
276Q8—31no—
277Q8—34no—
278Q8—34no—
279Q8C22×C4×C911yescharacteristic
280D4—32no—
281D4—32no—
282D4—38no—
283D4C4×C22⋊C914yes—
284C2×C4—24no—
285C2×C4—38no—
286C2×C4—31no—
287C2×C4—32no—
288C2×C4—34no—
289C2×C4—34no—
290C2×C4—32no—
291C2×C4—32no—
292C2×C4—31no—
293C2×C4—32no—
294C2×C4—31no—
295C2×C4—62no—
296C2×C4—121no—
297C2×C4—122no—
298C2×C4—122no—
299C2×C4C22×C22⋊C912yes—
300C2×C4C22×C22⋊C911yescharacteristic
301C2×C4C22×C22⋊C911yescharacteristic
302C2×C4C22×C22⋊C911yescharacteristic
303C2×C3—24no—
304C2×C3—62no—
305C2×C3C22×C4×A411yescharacteristic
306C2×C3C2×2−1+4⋊C311yescharacteristic
307C2×C3#594511yescharacteristic
308C22—22no—
309C22—22no—
310C22—32no—
311C22—61no—
312C22—61no—
313C22C2×C4×C22⋊C914yes—
314C22C23×C22⋊C911yescharacteristic
315C4—22no—
316C4—21no—
317C4—31no—
318C4—34no—
319C4—31no—
320C4—61no—
321C4—61no—
322C4—62no—
323C4C2×C4×C22⋊C912yes—
324C42−1+4⋊C912yesdirect product factor
325C3C4×2−1+4⋊C311yescharacteristic
326C2—24no—
327C2—62no—
328C2C22×C4×C22⋊C911yescharacteristic
329C2C2×2−1+4⋊C911yescharacteristic
330C2#1059312011yescharacteristic
subgroupG/H
centerC2×C3×C4C22×A4
commutator, hypocenterQ8C22×C4×C9
Frattini, socleC22×C3C23×A4
FittingC3×C4×2−1+4C3
prime radicalC2×C3×2−1+4C2×C3
cosocleC2×C3×Q8C23×C3
hypercenterC3×C4×D4A4
LCSG ▷ Q8
UCSC1 ◁ C2×C3×C4 ◁ C3×C4×D4

Complements

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

NH ≅ G/Ntype
2. #520⋊C9C2semidirect
3. #519⋊C9C2semidirect
4. C4×#41⋊C9C2semidirect
5. C2×C4×Q8⋊C9C2semidirect
7. 2−1+4⋊C9C4direct (N × H)
27. #41⋊C9C2×C4semidirect
28. C2×Q8⋊C9C2×C4semidirect
33. C4×2−1+4C9semidirect
81. #520C2×C9semidirect
83. #519C2×C9semidirect
86. C4×#41C2×C9semidirect
88. C2×C4×Q8C2×C9semidirect
142. 2−1+4C4×C9semidirect
216. #41C2×C4×C9semidirect
222. C2×Q8C2×C4×C9semidirect
324. C42−1+4⋊C9direct (N × H)

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
#520⋊C92
#519⋊C92
C4×#41⋊C92
C2×C4×Q8⋊C92
C3×C4×2−1+43
C4×D4×C94