Finite Group Databasebeta

← #37542980

Subgroups — #37542980

Order 23168.

754 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 754 conjugacy classes of subgroups — too many to draw. Only its 141 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
1C22×#15668450C211yesmaximal, characteristic
2C2×#19153848C212yesmaximal, complemented
3C22⋊Q8×D181C211yesmaximal, characteristic, complemented
4#26041444C212yesmaximal, complemented
5#26041440C212yesmaximal, complemented
6#26041435C211yesmaximal, characteristic, complemented
7#26041432C211yesmaximal, characteristic, complemented
8#26041429C211yesmaximal, characteristic, complemented
9#26041279C212yesmaximal, complemented
10#26041241C211yesmaximal, characteristic, complemented
11#451×C181C211yesmaximal, characteristic, complemented
12C2×#15668450—28no—
13C2×#15668450—21no—
14C2×#15668450—21no—
15C2×#15668450C2212yes—
16C2×#15668450C2212yes—
17C2×Q8×D181C2211yescharacteristic, complemented
18C22×D724—21no—
19C22×D724C2211yescharacteristic
20C22×C4×D181—21no—
21C22×C4×D181C2211yescharacteristic
22#19153848—28no—
23C2×#15668439C2212yescomplemented
24C2×#15668438C2212yescomplemented
25C2×#15668436C2212yescomplemented
26C2×#15668432C2212yescomplemented
27#19153796C2211yescharacteristic, complemented
28#19153795C2211yescharacteristic
29#19153794—21no—
30#19153793—21no—
31#19153792C2211yescharacteristic, complemented
32C4⋊C4×D181—21no—
33C4⋊C4×D181C2211yescharacteristic, complemented
34C22⋊C4×D181—21no—
35#19153740C2212yescomplemented
36#19153738C2212yescomplemented
37#19153736C2212yescomplemented
38C22⋊#13089678C2211yescharacteristic
39#19153728C2212yescomplemented
40C22×Q8×C181C2211yescharacteristic
41C2×Q16×C181C2212yescomplemented
42C22⋊Q8×C181C2211yescharacteristic
43#103×C181C2212yescomplemented
44C22⋊C8×C181C2211yescharacteristic
45C23×D181—21no—
46C23×D181D411yescharacteristic
47#15668450—28no—
48#15668450—28no—
49#15668450—24no—
50#15668450—24no—
51#15668450—24no—
52#15668450D418yes—
53C2×Dic362—21no—
54C2×Dic362C2311yescharacteristic
55Q8×D181—24no—
56C2×D724—28no—
57C2×D724—22no—
58C2×D724—22no—
59C2×D724—24no—
60C2×D724—21no—
61C2×D724C2312yes—
62C22×Dic181D411yescharacteristic
63C2×C4×D181—28no—
64C2×C4×D181—22no—
65C2×C4×D181—22no—
66C2×C4×D181—24no—
67C2×C4×D181—21no—
68C2×C4×D181—21no—
69C2×C4×D181D411yescharacteristic
70C2×C4×D181D411yescharacteristic
71C2×C4×D181C2311yescharacteristic
72C2×C4×D181C2311yescharacteristic
73#15668439—24no—
74#15668438—24no—
75#15668436—24no—
76#15668432—24no—
77C2×#13089678C2312yes—
78C4⋊Dic181—21no—
79C4⋊Dic181C2311yescharacteristic
80#15668428—21no—
81#15668428—21no—
82#15668428—21no—
83C22⋊Dic181—21no—
84#15668426—21no—
85#15668426—21no—
86C2×Q8×C181—24no—
87C2×Q8×C181—21no—
88C2×Q8×C181C2311yescharacteristic
89C2×Q8×C181C2312yes—
90C22×C4×C181—21no—
91C22×C4×C181C2311yescharacteristic
92Q16×C181—24no—
93C2×C8×C181C2312yes—
94C4⋊C4×C181—21no—
95C4⋊C4×C181C2311yescharacteristic
96C22⋊C4×C181—21no—
97C22×D181—24no—
98C22×D181—22no—
99C22×D181—22no—
100C22×D181—22no—
101C22×D181—41no—
102C22×D181—42no—
103C22×D181—42no—
104C22×D181C2×D411yescharacteristic
105C22×D181C2×D411yescharacteristic
106Dic362—22no—
107Dic362—42no—
108D724—28no—
109D724—24no—
110D724—28no—
111D724—22no—
112D724C2×D414yes—
113C2×Dic181—22no—
114C2×Dic181—21no—
115C2×Dic181—21no—
116C2×Dic181C2×D411yescharacteristic
117C2×Dic181C2×D411yescharacteristic
118C4×D181—28no—
119C4×D181—22no—
120C4×D181—22no—
121C4×D181—28no—
122C4×D181—22no—
123C4×D181—42no—
124C4×D181—42no—
125C4×D181C2×D414yes—
126#13089678—22no—
127C23×C181C2×D411yescharacteristic
128Q8×C181—24no—
129Q8×C181—22no—
130Q8×C181—22no—
131Q8×C181C2×D414yes—
132C2×C4×C181—24no—
133C2×C4×C181—22no—
134C2×C4×C181—22no—
135C2×C4×C181—21no—
136C2×C4×C181—21no—
137C2×C4×C181C2×D411yescharacteristic
138C2×C4×C181C2411yescharacteristic
139C8×C181—22no—
140C22×C181—21no—
141C22×C181—21no—
142C22×C181#13712yescomplemented
143C22×C181C22×D411yescharacteristic
144C2×D181—21no—
145C2×D181—21no—
146C2×D181—21no—
147C2×D181—24no—
148C2×D181—21no—
149C2×D181—21no—
150C2×D181—44no—
151C2×D181—48no—
152C2×D181—42no—
153C2×D181—42no—
154C2×D181#12012yes—
155C4×C181—24no—
156C4×C181—21no—
157C4×C181—41no—
158C4×C181—41no—
159C4×C181C22×D412yes—
160Dic181—21no—
161Dic181—41no—
162Dic181—41no—
163Dic181#12012yes—
164C2×C181—22no—
165C2×C181C2×#12011yescharacteristic
166C2×C181C2×#13711yescharacteristic
167C2×C181#57511yescharacteristic
168D181—22no—
169D181—41no—
170D181—44no—
171C181#300711yescharacteristic, complemented, Sylow
172#3007—1811nomaximal, Sylow
173C22×#41—1811no—
174C2×#137—1812no—
175C2×C22⋊Q8—1811no—
176#452—1812no—
177#452—1812no—
178#451—1811no—
179#451D18111yescharacteristic, complemented
180#450—1811no—
181#450—1811no—
182#417—1812no—
183#407—1811no—
184C2×#41—3628no—
185C2×#41—3621no—
186C2×#41—3621no—
187C2×#41—1812no—
188C2×#41—1812no—
189C22×Q8—1811no—
190C22×Q8C2×D18111yescharacteristic
191C22×D4—3621no—
192C22×D4—1811no—
193C23×C4—3621no—
194C23×C4—1811no—
195#137—3628no—
196C2×Q16—1812no—
197C2×Q16C2×D18112yescomplemented
198C2×SD16—1812no—
199C2×SD16—1812no—
200C2×M16—1812no—
201C22⋊Q8—3621no—
202C22⋊Q8—3621no—
203C22⋊Q8—1811no—
204C22⋊Q8—1811no—
205C22⋊Q8—1811no—
206C22⋊Q8C2×D18111yescharacteristic
207C2×C4⋊C4—3621no—
208C2×C4⋊C4—1811no—
209C2×C22⋊C4—3621no—
210#103—1812no—
211#103C2×D18112yescomplemented
212#102—1812no—
213#102—1812no—
214C22⋊C8—1811no—
215C22⋊C8—1812no—
216C22⋊C8C2×D18111yescharacteristic
217C24—3621no—
218C24—1811no—
219#41—3628no—
220#41—3628no—
221#41—3624no—
222#41—3624no—
223#41—3624no—
224#41—1818no—
225C2×Q8—24no—
226C2×Q8—21no—
227C2×Q8—3621no—
228C2×Q8—3624no—
229C2×Q8—1811no—
230C2×Q8C22×D18111yescharacteristic
231C2×Q8C22×D18112yes—
232C2×D4—3628no—
233C2×D4—3622no—
234C2×D4—3622no—
235C2×D4—3624no—
236C2×D4—3621no—
237C2×D4—1812no—
238C22×C4—21no—
239C22×C4—3628no—
240C22×C4—3622no—
241C22×C4—3622no—
242C22×C4—3624no—
243C22×C4—3621no—
244C22×C4—3621no—
245C22×C4—1811no—
246C22×C4—1811no—
247C22×C4—1811no—
248C22×C4—1811no—
249C22×C4—1811no—
250C22×C4C22×D18111yescharacteristic
251Q16—24no—
252Q16—3624no—
253SD16—3624no—
254SD16—3624no—
255M16—3624no—
256C2×C8—1812no—
257C2×C8C22×D18112yes—
258C4⋊C4—21no—
259C4⋊C4—3621no—
260C4⋊C4—3621no—
261C4⋊C4—3621no—
262C4⋊C4—3621no—
263C4⋊C4—1811no—
264C4⋊C4C22×D18111yescharacteristic
265C22⋊C4—21no—
266C22⋊C4—3621no—
267C22⋊C4—3621no—
268C22⋊C4—3621no—
269C23—3624no—
270C23—3622no—
271C23—3622no—
272C23—3622no—
273C23—1811no—
274C23—1811no—
275C23—7241no—
276C23—7242no—
277C23—7242no—
278C23D4×D18111yescharacteristic
279Q8—24no—
280Q8—22no—
281Q8—22no—
282Q8—3622no—
283Q8—7242no—
284Q8D4×D18114yes—
285D4—3628no—
286D4—3624no—
287D4—3628no—
288D4—3622no—
289D4—1814no—
290C2×C4—24no—
291C2×C4—22no—
292C2×C4—22no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—3628no—
296C2×C4—3622no—
297C2×C4—3628no—
298C2×C4—3622no—
299C2×C4—3622no—
300C2×C4—3622no—
301C2×C4—3621no—
302C2×C4—3621no—
303C2×C4—1811no—
304C2×C4—1811no—
305C2×C4—1814no—
306C2×C4—7242no—
307C2×C4—7242no—
308C2×C4D4×D18111yescharacteristic
309C2×C4C23×D18111yescharacteristic
310C8—22no—
311C8—3622no—
312C22—21no—
313C22—21no—
314C22—3621no—
315C22—3621no—
316C22—3621no—
317C22—3624no—
318C22—3621no—
319C22—3621no—
320C22—1812no—
321C22—7244no—
322C22—7248no—
323C22—7242no—
324C22—7242no—
325C22#1915384812yescomplemented
326C22C2×D4×D18111yescharacteristic
327C4—24no—
328C4—21no—
329C4—41no—
330C4—41no—
331C4—3621no—
332C4—1812no—
333C4—7241no—
334C4—7241no—
335C4C2×D4×D18112yes—
336C2—22no—
337C2—3622no—
338C2—7241no—
339C2—7244no—
340C2#120×D18111yescharacteristic
341C2C2×#1915384811yescharacteristic
342C2#2604207811yescharacteristic
subgroupG/H
centerC22C2×D4×D181
commutator, cosocleC2×C4×C181C24
FrattiniC2×C4C23×D181
Fitting#451×C181C2
socleC22×C181C22×D4
prime radicalC22×#15668450C2
hypercenter#451D181
hypocenterC181#3007
LCSG ▷ C2×C4×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C22 ◁ C22×C4 ◁ #451

Complements

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

NH ≅ G/Ntype
2. C2×#19153848C2semidirect
3. C22⋊Q8×D181C2semidirect
4. #26041444C2semidirect
5. #26041440C2semidirect
6. #26041435C2semidirect
7. #26041432C2semidirect
8. #26041429C2semidirect
9. #26041279C2semidirect
10. #26041241C2semidirect
11. #451×C181C2semidirect
17. C2×Q8×D181C22semidirect
23. C2×#15668439C22semidirect
24. C2×#15668438C22semidirect
25. C2×#15668436C22semidirect
26. C2×#15668432C22semidirect
27. #19153796C22semidirect
31. #19153792C22semidirect
33. C4⋊C4×D181C22semidirect
35. #19153740C22semidirect
36. #19153738C22semidirect
37. #19153736C22semidirect
39. #19153728C22semidirect
41. C2×Q16×C181C22semidirect
43. #103×C181C22semidirect
142. C22×C181#137semidirect
171. C181#3007semidirect
179. #451D181semidirect
197. C2×Q16C2×D181semidirect
211. #103C2×D181semidirect
325. C22#19153848semidirect

Maximal subgroups

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

subgroupindex
C22×#156684502
C2×#191538482
C22⋊Q8×D1812
#260414442
#260414402
#260414352
#260414322
#260414292
#260412792
#260412412
#451×C1812
#3007181