Finite Group Databasebeta

← #37543027

Subgroups — #37543027

Order 23168.

764 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 764 conjugacy classes of subgroups — too many to draw. Only its 149 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×#19153862C211yesmaximal, characteristic, complemented
2C2×#19153844C211yesmaximal, characteristic, complemented
3C2×#19153836C211yesmaximal, characteristic, complemented
4#26041860C211yesmaximal, characteristic, complemented
5#26041444C212yesmaximal, complemented
6#26041431C212yesmaximal, complemented
7#26041427C212yesmaximal, complemented
8#26041414C212yesmaximal, complemented
9C2×#19153736C211yesmaximal, characteristic, complemented
10#26041247C211yesmaximal, characteristic, complemented
11#417×C181C211yesmaximal, characteristic, complemented
12#19153862—24no—
13#19153862—24no—
14C2×#15668450C2212yes—
15C2×#15668449—21no—
16C2×#15668449C2211yescharacteristic
17C2×#15668449C2211yescharacteristic, complemented
18C22×D724—21no—
19C22×D724C2211yescharacteristic
20C2×D4×D181—22no—
21C2×D4×D181C2212yes—
22#19153844—24no—
23#19153836—24no—
24C2×#15668439C2211yescharacteristic, complemented
25C2×#15668438C2212yescomplemented
26C2×#15668437C2211yescharacteristic, complemented
27C2×#15668436C2211yescharacteristic, complemented
28C2×D1448C2212yescomplemented
29C2×#15668434C2211yescharacteristic, complemented
30C22×#13089678C2211yescharacteristic, complemented
31#19153796—21no—
32#19153796C2212yescomplemented
33#19153791—21no—
34#19153791C2212yescomplemented
35#19153790—22no—
36#19153736C2212yescomplemented
37#19153736C2212yescomplemented
38#19153728C2212yescomplemented
39#19153728C2212yescomplemented
40C2×#41×C181C2211yescharacteristic
41C2×M16×C181C2211yescharacteristic, complemented
42C2×C4⋊C4×C181C2211yescharacteristic, complemented
43#103×C181C2212yescomplemented
44#102×C181C2212yescomplemented
45C23×D181—22no—
46C23×D181—21no—
47#15668450—24no—
48#15668450—24no—
49#15668449—28no—
50#15668449—22no—
51#15668449—22no—
52#15668449—22no—
53#15668449—22no—
54C2×Dic362D411yescharacteristic
55C2×Dic362C2311yescharacteristic
56C2×D724—28no—
57C2×D724—22no—
58C2×D724—22no—
59C2×D724—21no—
60C2×D724—22no—
61C2×D724D411yescharacteristic
62C2×D724D414yescomplemented
63C2×D724C2311yescharacteristic
64C2×D724C2312yes—
65C2×C22⋊D181—22no—
66C2×C22⋊D181—21no—
67C2×C22⋊D181D412yes—
68D4×D181—216no—
69D4×D181—24no—
70D4×D181—24no—
71C2×C4×D181—22no—
72C2×C4×D181—21no—
73C2×C4×D181C2312yes—
74#15668439—22no—
75#15668438—24no—
76#15668437—22no—
77#15668436—22no—
78D1448—24no—
79#15668434—22no—
80C2×#13089678—22no—
81C2×#13089678C2311yescharacteristic
82C2×#13089678C2311yescharacteristic
83C4⋊Dic181—22no—
84#15668426—22no—
85#15668426—22no—
86#41×C181—22no—
87#41×C181D414yescomplemented
88C2×Q8×C181C2311yescharacteristic
89C2×D4×C181—21no—
90C2×D4×C181C2311yescharacteristic
91C22×C4×C181—21no—
92C22×C4×C181—21no—
93C22×C4×C181C2311yescharacteristic
94M16×C181—22no—
95C2×C8×C181C2312yes—
96C4⋊C4×C181—21no—
97C4⋊C4×C181C2312yes—
98C22×D181—22no—
99C22×D181—28no—
100C22×D181—24no—
101C22×D181—21no—
102C22×D181—82no—
103C22×D181—82no—
104C22×D181—81no—
105C22×D181—81no—
106C22×D181C2×D412yes—
107Dic362—22no—
108Dic362—21no—
109Dic362—21no—
110D724—28no—
111D724—22no—
112D724—24no—
113D724—22no—
114D724—21no—
115D724—21no—
116D724—82no—
117D724C2×D414yes—
118C22⋊D181—28no—
119C22⋊D181—24no—
120C22⋊D181—42no—
121C2×Dic181—21no—
122C2×Dic181C2×D412yes—
123C4×D181—28no—
124C4×D181—22no—
125C4×D181—22no—
126C4×D181—42no—
127#13089678—21no—
128#13089678—21no—
129C23×C181—21no—
130C23×C181C2×D411yescharacteristic
131Q8×C181—21no—
132Q8×C181C2×D412yes—
133D4×C181—24no—
134D4×C181—21no—
135D4×C181C2×D412yes—
136C2×C4×C181—21no—
137C2×C4×C181—24no—
138C2×C4×C181—22no—
139C2×C4×C181—81no—
140C2×C4×C181C2×D411yescharacteristic
141C2×C4×C181C2×D412yes—
142C2×C4×C181C2×D412yes—
143C2×C4×C181C2411yescharacteristic
144C8×C181—22no—
145C22×C181—21no—
146C22×C181—21no—
147C22×C181—22no—
148C22×C181—41no—
149C22×C181—41no—
150C22×C181#12012yes—
151C22×C181C22×D411yescharacteristic
152C2×D181—22no—
153C2×D181—24no—
154C2×D181—22no—
155C2×D181—22no—
156C2×D181—42no—
157C2×D181—42no—
158C2×D181—41no—
159C2×D181—41no—
160C2×D181—41no—
161C2×D181—84no—
162C2×D181—88no—
163C4×C181—22no—
164C4×C181—42no—
165C4×C181#12012yes—
166C4×C181C22×D411yescharacteristic
167C4×C181C22×D411yescharacteristic
168Dic181—22no—
169Dic181—41no—
170C2×C181—22no—
171C2×C181—42no—
172C2×C181C2×#12011yescharacteristic
173C2×C181#57611yescharacteristic
174C2×C181#57711yescharacteristic
175D181—42no—
176D181—44no—
177D181—81no—
178C181#301211yescharacteristic, complemented, Sylow
179#3012—1811nomaximal, Sylow
180C2×H(5,𝔽2)—1811no—
181C2×AGL(1,ℤ/8)—1811no—
182C2×#135—1811no—
183#537—1811no—
184#452—1812no—
185#450—1812no—
186#449—1812no—
187#447—1812no—
188#417D18111yescharacteristic, complemented
189C2×#102—1811no—
190#408—1811no—
191H(5,𝔽2)—3624no—
192H(5,𝔽2)—3624no—
193C2×#41—3621no—
194C2×#41—1812no—
195C2×#41—1811no—
196C2×#41—1811no—
197C2×#41C2×D18111yescharacteristic
198C22×D4—3621no—
199C22×D4—3622no—
200C22×D4—1812no—
201C22×D4—1811no—
202AGL(1,ℤ/8)—3624no—
203#135—3624no—
204C2×Q16—1811no—
205C2×SD16—1811no—
206C2×SD16—1811no—
207C2×SD16—1812no—
208C2×D8—1811no—
209C2×D8—1812no—
210C2×M16C2×D18111yescharacteristic, complemented
211C22×C8—1811no—
212C22⋊Q8—3621no—
213C22⋊Q8—1812no—
214#121—3621no—
215#121—3622no—
216#121—1812no—
217C2×C4⋊C4C2×D18111yescharacteristic, complemented
218#103C2×D18112yescomplemented
219#102—1812no—
220#102—1812no—
221#102C2×D18112yescomplemented
222C22⋊C8—1812no—
223C22⋊C8—1812no—
224C24—3622no—
225C24—3621no—
226#41—22no—
227#41—3628no—
228#41—3624no—
229#41—3624no—
230#41—3622no—
231#41—3622no—
232#41—3622no—
233#41—3622no—
234#41D72414yescomplemented
235C2×Q8—1811no—
236C2×Q8—1811no—
237C2×Q8C22×D18111yescharacteristic
238C2×D4—21no—
239C2×D4—3628no—
240C2×D4—3622no—
241C2×D4—3622no—
242C2×D4—36216no—
243C2×D4—3624no—
244C2×D4—3624no—
245C2×D4—3621no—
246C2×D4—3622no—
247C2×D4—3621no—
248C2×D4—3622no—
249C2×D4—1812no—
250C2×D4—1811no—
251C2×D4—1814no—
252C2×D4—1811no—
253C2×D4—1812no—
254C2×D4C22×D18111yescharacteristic
255C22×C4—21no—
256C22×C4—21no—
257C22×C4—3622no—
258C22×C4—3621no—
259C22×C4—1812no—
260C22×C4C22×D18111yescharacteristic
261Q16—3622no—
262SD16—3624no—
263SD16—3622no—
264SD16—3622no—
265D8—3624no—
266D8—3622no—
267M16—22no—
268C2×C8—3622no—
269C2×C8—1811no—
270C2×C8—1811no—
271C2×C8C22×D18112yes—
272C4⋊C4—21no—
273C4⋊C4—3622no—
274C4⋊C4C22×D18112yes—
275C22⋊C4—3622no—
276C22⋊C4—3622no—
277C23—21no—
278C23—3628no—
279C23—3624no—
280C23—3622no—
281C23—3621no—
282C23—14481no—
283C23—14481no—
284C23—14482no—
285C23—14482no—
286C23—1812no—
287C23D4×D18111yescharacteristic
288Q8—21no—
289Q8—3622no—
290Q8—3621no—
291Q8—3621no—
292Q8C2×D72412yes—
293D4—24no—
294D4—21no—
295D4—3628no—
296D4—3628no—
297D4—3624no—
298D4—3622no—
299D4—3624no—
300D4—3621no—
301D4—3622no—
302D4—3621no—
303D4—14482no—
304D4—1814no—
305D4—7242no—
306D4C2×D72412yes—
307C2×C4—21no—
308C2×C4—24no—
309C2×C4—22no—
310C2×C4—81no—
311C2×C4—3628no—
312C2×C4—3622no—
313C2×C4—3622no—
314C2×C4—3621no—
315C2×C4—1812no—
316C2×C4—7242no—
317C2×C4D4×D18111yescharacteristic
318C2×C4D4×D18112yes—
319C2×C4C2×D72412yes—
320C2×C4C23×D18111yescharacteristic
321C8—22no—
322C8—3621no—
323C8—3621no—
324C22—21no—
325C22—21no—
326C22—22no—
327C22—41no—
328C22—41no—
329C22—3622no—
330C22—3624no—
331C22—3622no—
332C22—3622no—
333C22—14484no—
334C22—14488no—
335C22—7241no—
336C22—7241no—
337C22—7241no—
338C22—7242no—
339C22—7242no—
340C22#1915378612yes—
341C22C2×D4×D18111yescharacteristic
342C4—22no—
343C4—42no—
344C4—3622no—
345C4—7241no—
346C4#1915378612yes—
347C4C2×D4×D18111yescharacteristic
348C4C22×D72411yescharacteristic
349C2—22no—
350C2—42no—
351C2—14481no—
352C2—7242no—
353C2—7244no—
354C2C2×#1915378611yescharacteristic
355C2#2604208311yescharacteristic
356C2#2604208811yescharacteristic
subgroupG/H
centerC22C2×D4×D181
commutator, cosocleC2×C4×C181C24
FrattiniC2×C4C23×D181
Fitting#417×C181C2
socleC22×C181C22×D4
hypercenter#417D181
hypocenterC181#3012
LCSG ▷ C2×C4×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C22 ◁ C22×C4 ◁ #417

Complements

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

NH ≅ G/Ntype
1. C2×#19153862C2semidirect
2. C2×#19153844C2semidirect
3. C2×#19153836C2semidirect
4. #26041860C2semidirect
5. #26041444C2semidirect
6. #26041431C2semidirect
7. #26041427C2semidirect
8. #26041414C2semidirect
9. C2×#19153736C2semidirect
10. #26041247C2semidirect
11. #417×C181C2semidirect
17. C2×#15668449C22semidirect
24. C2×#15668439C22semidirect
25. C2×#15668438C22semidirect
26. C2×#15668437C22semidirect
27. C2×#15668436C22semidirect
28. C2×D1448C22semidirect
29. C2×#15668434C22semidirect
30. C22×#13089678C22semidirect
32. #19153796C22semidirect
34. #19153791C22semidirect
36. #19153736C22semidirect
37. #19153736C22semidirect
38. #19153728C22semidirect
39. #19153728C22semidirect
41. C2×M16×C181C22semidirect
42. C2×C4⋊C4×C181C22semidirect
43. #103×C181C22semidirect
44. #102×C181C22semidirect
62. C2×D724D4semidirect
87. #41×C181D4semidirect
178. C181#3012semidirect
188. #417D181semidirect
210. C2×M16C2×D181semidirect
217. C2×C4⋊C4C2×D181semidirect
218. #103C2×D181semidirect
221. #102C2×D181semidirect
234. #41D724semidirect

Maximal subgroups

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

subgroupindex
C2×#191538622
C2×#191538442
C2×#191538362
#260418602
#260414442
#260414312
#260414272
#260414142
C2×#191537362
#260412472
#417×C1812
#3012181