Finite Group Databasebeta

← #37543358

Subgroups — #37543358

Order 23168.

492 conjugacy classes of subgroups.

The subgroup lattice has 408 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×SD16×D181C211yesmaximal, characteristic, complemented
2C2×#19153824C211yesmaximal, characteristic, complemented
3#26041852C211yesmaximal, characteristic, complemented
4#26041845C211yesmaximal, characteristic
5#26041526C211yesmaximal, characteristic, complemented
6#26041525C211yesmaximal, characteristic, complemented
7#26041522C211yesmaximal, characteristic, complemented
8#26041511C211yesmaximal, characteristic, complemented
9#26041510C211yesmaximal, characteristic, complemented
10#26041507C211yesmaximal, characteristic, complemented
11#26041506C211yesmaximal, characteristic, complemented
12#26041316C211yesmaximal, characteristic, complemented
13#26041286C211yesmaximal, characteristic, complemented
14#26041282C211yesmaximal, characteristic, complemented
15#468×C181C211yesmaximal, characteristic, complemented
16C2×#15668449C2211yescharacteristic
17C2×Q8×D181C2211yescharacteristic
18C2×D4×D181C2211yescharacteristic
19C2×#15668438C2211yescharacteristic
20C2×#15668437C2211yescharacteristic
21C2×#15668436C2211yescharacteristic
22SD16×D181—24no—
23#19153824—24no—
24C2×#15668432C2211yescharacteristic
25C2×#15668432C2211yescharacteristic, complemented
26C22×#13089678C2211yescharacteristic, complemented
27C2×C8×D181C2211yescharacteristic
28C2×C8×D181C2211yescharacteristic, complemented
29#19153815—21no—
30#19153812—21no—
31#19153802—21no—
32#19153802—21no—
33#19153801—21no—
34#19153798—21no—
35#19153796C2211yescharacteristic, complemented
36#19153795C2211yescharacteristic, complemented
37#19153792C2211yescharacteristic, complemented
38#19153791C2211yescharacteristic
39#19153790C2211yescharacteristic
40#19153787C2211yescharacteristic
41#19153786—21no—
42#19153784—21no—
43#19153774C2211yescharacteristic
44#19153774C2211yescharacteristic
45#19153750C2211yescharacteristic, complemented
46#19153747C2211yescharacteristic, complemented
47#19153747C2211yescharacteristic, complemented
48#19153741C2211yescharacteristic, complemented
49#19153740C2211yescharacteristic, complemented
50#19153739C2211yescharacteristic, complemented
51#19153738C2211yescharacteristic
52#19153737C2211yescharacteristic
53#19153736C2211yescharacteristic
54C2×SD16×C181C2211yescharacteristic
55C2×M16×C181C2211yescharacteristic, complemented
56C22⋊Q8×C181C2211yescharacteristic, complemented
57#121×C181C2211yescharacteristic
58#107×C181C2211yescharacteristic, complemented
59#103×C181C2211yescharacteristic, complemented
60#102×C181C2211yescharacteristic
61C23×D181—21no—
62#15668449—22no—
63#15668449—22no—
64C2×Dic362—21no—
65C2×Dic362D411yescharacteristic
66C2×Dic362C2311yescharacteristic
67Q8×D181—22no—
68Q8×D181—22no—
69C2×D724—21no—
70C2×D724D411yescharacteristic
71C2×D724C2311yescharacteristic
72C2×C22⋊D181—21no—
73C2×C22⋊D181—21no—
74C2×C22⋊D181—21no—
75C2×C22⋊D181D411yescharacteristic
76C2×C22⋊D181D411yescharacteristic
77D4×D181—22no—
78D4×D181—22no—
79C2×C4×D181—21no—
80C2×C4×D181C2311yescharacteristic
81C2×C4×D181C2311yescharacteristic
82#15668438—22no—
83#15668437—22no—
84#15668436—22no—
85#15668432—22no—
86#15668432—22no—
87C2×#13089678—22no—
88C2×#13089678C2311yescharacteristic
89C2×#13089678C2311yescharacteristic
90C8×D181—22no—
91C8×D181D414yescomplemented
92C4⋊Dic181—21no—
93C4⋊Dic181C2311yescharacteristic
94C4⋊Dic181C2311yescharacteristic
95#15668428—21no—
96#15668428—21no—
97C22⋊Dic181—21no—
98C22⋊Dic181—21no—
99#15668426—21no—
100#15668426—21no—
101#15668426—21no—
102#15668426—21no—
103#15668426—21no—
104#15668426—21no—
105C4×Dic181—21no—
106C4×Dic181—21no—
107C2×Q8×C181C2311yescharacteristic
108C2×D4×C181—21no—
109C2×D4×C181C2311yescharacteristic
110C22×C4×C181C2311yescharacteristic
111SD16×C181—22no—
112M16×C181—22no—
113C2×C8×C181C2311yescharacteristic
114C2×C8×C181C2311yescharacteristic
115C4⋊C4×C181—21no—
116C4⋊C4×C181C2311yescharacteristic
117C4⋊C4×C181C2311yescharacteristic
118C22⋊C4×C181—21no—
119C22⋊C4×C181—21no—
120C22×D181—21no—
121C22×D181—42no—
122C22×D181—81no—
123C22×D181—81no—
124C22×D181C2×D411yescharacteristic
125C22×D181C2×D411yescharacteristic
126Dic362—22no—
127Dic362—21no—
128Dic362—21no—
129Dic362—42no—
130D724—22no—
131D724—21no—
132D724—21no—
133D724—81no—
134C22⋊D181—22no—
135C22⋊D181—22no—
136C22⋊D181—42no—
137C22⋊D181—81no—
138C22⋊D181—81no—
139C2×Dic181—21no—
140C2×Dic181—21no—
141C2×Dic181—21no—
142C2×Dic181C2×D411yescharacteristic
143C2×Dic181C2×D411yescharacteristic
144C4×D181—21no—
145C4×D181—21no—
146C4×D181—42no—
147C4×D181C2×D412yes—
148C4×D181#4112yes—
149#13089678—21no—
150#13089678C2×D412yes—
151C23×C181—21no—
152C23×C181C2×D411yescharacteristic
153Q8×C181—21no—
154Q8×C181—21no—
155D4×C181—21no—
156D4×C181—21no—
157D4×C181—81no—
158C2×C4×C181—21no—
159C2×C4×C181—21no—
160C2×C4×C181—21no—
161C2×C4×C181—21no—
162C2×C4×C181—21no—
163C2×C4×C181C2×D411yescharacteristic
164C2×C4×C181C2411yescharacteristic
165C8×C181—21no—
166C8×C181C2×D412yes—
167C22×C181—21no—
168C22×C181—21no—
169C22×C181—21no—
170C22×C181—41no—
171C22×C181—41no—
172C22×C181—41no—
173C22×C181C22×D411yescharacteristic
174C2×D181—21no—
175C2×D181—21no—
176C2×D181—21no—
177C2×D181—21no—
178C2×D181—21no—
179C2×D181—41no—
180C2×D181—41no—
181C2×D181—41no—
182C2×D181—84no—
183C2×D181#12112yes—
184C4×C181—21no—
185C4×C181—41no—
186C4×C181—41no—
187C4×C181—41no—
188C4×C181C22×D411yescharacteristic
189C4×C181C2×#4111yescharacteristic
190Dic181—21no—
191Dic181—41no—
192Dic181—41no—
193Dic181—41no—
194Dic181#12112yes—
195C2×C181—41no—
196C2×C181—81no—
197C2×C181C2×#12111yescharacteristic
198C2×C181#57711yescharacteristic
199C2×C181#57711yescharacteristic
200D181—22no—
201D181—41no—
202D181—81no—
203C181#305711yescharacteristic, complemented, Sylow
204#3057—1811nomaximal, Sylow
205C22×SD16—1811no—
206C2×#131—1811no—
207#536—1811no—
208#535—1811no—
209#468—1811no—
210#468—1811no—
211#468—1811no—
212#468D18111yescharacteristic, complemented
213#465—1811no—
214#465—1811no—
215#465—1811no—
216#465—1811no—
217#427—1811no—
218#419—1811no—
219#418—1811no—
220C2×#41—1811no—
221C22×Q8—1811no—
222C22×D4—1811no—
223C2×SD16—3624no—
224C2×SD16—1811no—
225C2×SD16—1811no—
226C2×SD16—1811no—
227C2×SD16C2×D18111yescharacteristic
228#131—3624no—
229C2×M16—1811no—
230C2×M16—1811no—
231C2×M16C2×D18111yescharacteristic, complemented
232C22×C8—1811no—
233C22×C8—1811no—
234C22×C8—1811no—
235C4⋊Q8—3621no—
236#127—3621no—
237#124—3621no—
238#124—3621no—
239#123—3621no—
240#123—3621no—
241C22⋊Q8—1811no—
242C22⋊Q8—1811no—
243C22⋊Q8—1811no—
244C22⋊Q8C2×D18111yescharacteristic, complemented
245#121—1811no—
246#121—1811no—
247#121—1811no—
248#121C2×D18111yescharacteristic
249#120—3621no—
250#120—3621no—
251#117—1811no—
252#117—1811no—
253#107—1811no—
254#107C2×D18111yescharacteristic, complemented
255#106—1811no—
256#106—1811no—
257#103—1811no—
258#103—1811no—
259#103—1811no—
260#103C2×D18111yescharacteristic, complemented
261#102—1811no—
262#102—1811no—
263#102—1811no—
264#102C2×D18111yescharacteristic
265C24—3621no—
266#41—3622no—
267#41—3622no—
268C2×Q8—3621no—
269C2×Q8—3622no—
270C2×Q8—3622no—
271C2×Q8—1811no—
272C2×Q8—1811no—
273C2×Q8C22×D18111yescharacteristic
274C2×D4—21no—
275C2×D4—3621no—
276C2×D4—3621no—
277C2×D4—3621no—
278C2×D4—3622no—
279C2×D4—3622no—
280C2×D4—3621no—
281C2×D4—1811no—
282C2×D4—1811no—
283C2×D4—1811no—
284C2×D4—1811no—
285C2×D4C22×D18111yescharacteristic
286C22×C4—3621no—
287C22×C4—1811no—
288C22×C4—1811no—
289C22×C4C22×D18111yescharacteristic
290SD16—22no—
291SD16—3622no—
292SD16—3622no—
293SD16—3622no—
294M16—22no—
295M16—3622no—
296M16—3622no—
297C2×C8—3622no—
298C2×C8—3622no—
299C2×C8—1811no—
300C2×C8—1811no—
301C2×C8—1814no—
302C2×C8C22×D18111yescharacteristic
303C2×C8C22×D18111yescharacteristic
304C4⋊C4—21no—
305C4⋊C4—3621no—
306C4⋊C4—3621no—
307C4⋊C4—3621no—
308C4⋊C4—1811no—
309C4⋊C4—1811no—
310C4⋊C4C22×D18111yescharacteristic
311C4⋊C4C22×D18111yescharacteristic
312C22⋊C4—21no—
313C22⋊C4—21no—
314C22⋊C4—3621no—
315C22⋊C4—3621no—
316C22⋊C4—3621no—
317C22⋊C4—3621no—
318C22⋊C4—3621no—
319C22⋊C4—3621no—
320C22⋊C4—3621no—
321C22⋊C4—3621no—
322C42—3621no—
323C42—3621no—
324C23—21no—
325C23—3621no—
326C23—14481no—
327C23—14481no—
328C23—1811no—
329C23—1811no—
330C23—7242no—
331C23D4×D18111yescharacteristic
332Q8—21no—
333Q8—21no—
334Q8—3622no—
335Q8—3621no—
336Q8—3621no—
337Q8—7242no—
338D4—21no—
339D4—21no—
340D4—81no—
341D4—3622no—
342D4—3622no—
343D4—3622no—
344D4—3621no—
345D4—3621no—
346D4—14481no—
347D4—14481no—
348D4—14481no—
349D4—7242no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—3621no—
356C2×C4—3621no—
357C2×C4—3621no—
358C2×C4—3621no—
359C2×C4—3621no—
360C2×C4—1811no—
361C2×C4—1811no—
362C2×C4—1812no—
363C2×C4—1812no—
364C2×C4—7242no—
365C2×C4D4×D18111yescharacteristic
366C2×C4C23×D18111yescharacteristic
367C8—21no—
368C8—3621no—
369C8—1812no—
370C8D4×D18112yes—
371C22—21no—
372C22—21no—
373C22—21no—
374C22—41no—
375C22—41no—
376C22—41no—
377C22—3621no—
378C22—3621no—
379C22—3621no—
380C22—3621no—
381C22—3621no—
382C22—14484no—
383C22—1812no—
384C22—7241no—
385C22—7241no—
386C22—7241no—
387C22C2×D4×D18111yescharacteristic
388C4—21no—
389C4—41no—
390C4—41no—
391C4—41no—
392C4—3621no—
393C4—1812no—
394C4—7241no—
395C4—7241no—
396C4—7241no—
397C4C2×D4×D18111yescharacteristic
398C4#41×D18111yescharacteristic
399C2—41no—
400C2—81no—
401C2—3622no—
402C2—14481no—
403C2—7241no—
404C2#121×D18111yescharacteristic
405C2#2604208811yescharacteristic
406C2#2604208911yescharacteristic
subgroupG/H
centerC22C2×D4×D181
commutator, cosocleC2×C4×C181C24
FrattiniC2×C4C23×D181
Fitting#468×C181C2
socleC22×C181C22×D4
prime radical#26041845C2
hypercenter#468D181
hypocenterC181#3057
LCSG ▷ C2×C4×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C22 ◁ C22×C4 ◁ #468

Complements

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

NH ≅ G/Ntype
1. C2×SD16×D181C2semidirect
2. C2×#19153824C2semidirect
3. #26041852C2semidirect
5. #26041526C2semidirect
6. #26041525C2semidirect
7. #26041522C2semidirect
8. #26041511C2semidirect
9. #26041510C2semidirect
10. #26041507C2semidirect
11. #26041506C2semidirect
12. #26041316C2semidirect
13. #26041286C2semidirect
14. #26041282C2semidirect
15. #468×C181C2semidirect
25. C2×#15668432C22semidirect
26. C22×#13089678C22semidirect
28. C2×C8×D181C22semidirect
35. #19153796C22semidirect
36. #19153795C22semidirect
37. #19153792C22semidirect
45. #19153750C22semidirect
46. #19153747C22semidirect
47. #19153747C22semidirect
48. #19153741C22semidirect
49. #19153740C22semidirect
50. #19153739C22semidirect
55. C2×M16×C181C22semidirect
56. C22⋊Q8×C181C22semidirect
58. #107×C181C22semidirect
59. #103×C181C22semidirect
91. C8×D181D4semidirect
203. C181#3057semidirect
212. #468D181semidirect
231. C2×M16C2×D181semidirect
244. C22⋊Q8C2×D181semidirect
254. #107C2×D181semidirect
260. #103C2×D181semidirect

Maximal subgroups

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

subgroupindex
C2×SD16×D1812
C2×#191538242
#260418522
#260418452
#260415262
#260415252
#260415222
#260415112
#260415102
#260415072
#260415062
#260413162
#260412862
#260412822
#468×C1812
#3057181