Finite Group Databasebeta

← #37543002

Subgroups — #37543002

Order 23168.

732 conjugacy classes of subgroups.

The subgroup lattice has 474 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19153864C211yesmaximal, characteristic, complemented
2C2×#19153843C211yesmaximal, characteristic, complemented
3C2×SD16×D181C211yesmaximal, characteristic, complemented
4#26041845C211yesmaximal, characteristic, complemented
5#26041427C211yesmaximal, characteristic, complemented
6#26041426C211yesmaximal, characteristic, complemented
7#26041423C211yesmaximal, characteristic, complemented
8#26041422C211yesmaximal, characteristic, complemented
9#26041419C211yesmaximal, characteristic, complemented
10#26041418C211yesmaximal, characteristic, complemented
11#26041416C211yesmaximal, characteristic, complemented
12#26041286C211yesmaximal, characteristic, complemented
13#26041274C211yesmaximal, characteristic, complemented
14#26041244C211yesmaximal, characteristic, complemented
15#448×C181C211yesmaximal, characteristic, complemented
16#19153864—24no—
17#19153864—24no—
18C2×#15668450—21no—
19C2×#15668450C2211yescharacteristic
20C2×#15668450C2211yescharacteristic, complemented
21C2×#15668449—21no—
22C2×#15668449—21no—
23C2×#15668449C2211yescharacteristic
24C2×#15668449C2211yescharacteristic, complemented
25C2×Q8×D181—21no—
26C2×Q8×D181C2211yescharacteristic
27C2×Q8×D181C2211yescharacteristic, complemented
28C2×D4×D181C2211yescharacteristic
29#19153843—24no—
30C2×#15668438C2211yescharacteristic
31C2×#15668437C2211yescharacteristic
32C2×#15668436C2211yescharacteristic
33C2×#15668436C2211yescharacteristic
34SD16×D181—24no—
35C2×D1448C2211yescharacteristic
36C2×#15668434C2211yescharacteristic
37C2×#15668432C2211yescharacteristic
38C2×C8×D181C2211yescharacteristic
39#19153812—21no—
40#19153802—21no—
41#19153791C2211yescharacteristic
42#19153790C2211yescharacteristic
43#19153787C2211yescharacteristic
44#19153786—21no—
45#19153784—21no—
46#19153774C2211yescharacteristic
47#19153741C2211yescharacteristic, complemented
48#19153740C2211yescharacteristic, complemented
49#19153740C2211yescharacteristic, complemented
50#19153739C2211yescharacteristic, complemented
51#19153738C2211yescharacteristic, complemented
52#19153736C2211yescharacteristic, complemented
53C22⋊#13089678C2211yescharacteristic, complemented
54#19153728C2211yescharacteristic, complemented
55#19153728C2211yescharacteristic, complemented
56C22×Q8×C181C2211yescharacteristic, complemented
57C2×SD16×C181C2211yescharacteristic
58C2×SD16×C181C2211yescharacteristic
59#121×C181C2211yescharacteristic
60#103×C181C2211yescharacteristic, complemented
61#103×C181C2211yescharacteristic, complemented
62C22⋊C8×C181C2211yescharacteristic, complemented
63C23×D181—21no—
64#15668450—28no—
65#15668450—22no—
66#15668450—22no—
67#15668450—22no—
68#15668450D414yescomplemented
69#15668449—28no—
70#15668449—28no—
71#15668449—22no—
72#15668449—22no—
73#15668449—22no—
74#15668449—22no—
75C2×Dic362—21no—
76C2×Dic362—21no—
77C2×Dic362D411yescharacteristic
78C2×Dic362C2311yescharacteristic
79Q8×D181—28no—
80Q8×D181—22no—
81Q8×D181—22no—
82Q8×D181—22no—
83Q8×D181D414yescomplemented
84C2×D724—21no—
85C2×D724—21no—
86C2×D724—21no—
87C2×D724D411yescharacteristic
88C2×D724C2311yescharacteristic
89C2×D724C2311yescharacteristic
90C2×C22⋊D181—21no—
91C2×C22⋊D181—21no—
92C2×C22⋊D181—21no—
93C2×C22⋊D181—21no—
94C2×C22⋊D181D411yescharacteristic
95C2×C22⋊D181D411yescharacteristic
96D4×D181—22no—
97D4×D181—22no—
98C2×C4×D181—21no—
99C2×C4×D181—21no—
100C2×C4×D181—21no—
101C2×C4×D181—21no—
102C2×C4×D181—21no—
103C2×C4×D181C2311yescharacteristic
104C2×C4×D181C2311yescharacteristic
105#15668438—22no—
106#15668437—22no—
107#15668436—22no—
108#15668436—22no—
109D1448—22no—
110#15668434—22no—
111#15668432—22no—
112C2×#13089678C2311yescharacteristic
113C2×#13089678C2311yescharacteristic
114C8×D181—22no—
115C4⋊Dic181C2311yescharacteristic
116C22⋊Dic181—21no—
117#15668426—21no—
118#15668426—21no—
119#15668426—21no—
120C4×Dic181—21no—
121C2×Q8×C181—22no—
122C2×Q8×C181—22no—
123C2×Q8×C181—21no—
124C2×Q8×C181C2311yescharacteristic
125C2×Q8×C181C2311yescharacteristic
126C2×D4×C181—21no—
127C2×D4×C181C2311yescharacteristic
128C22×C4×C181—21no—
129C22×C4×C181C2311yescharacteristic
130SD16×C181—22no—
131SD16×C181—22no—
132C2×C8×C181C2311yescharacteristic
133C2×C8×C181C2311yescharacteristic
134C4⋊C4×C181C2311yescharacteristic
135C22⋊C4×C181—21no—
136C22×D181—21no—
137C22×D181—21no—
138C22×D181—42no—
139C22×D181—81no—
140C22×D181—81no—
141C22×D181C2×D411yescharacteristic
142C22×D181C2×D411yescharacteristic
143Dic362—24no—
144Dic362—21no—
145Dic362—22no—
146Dic362—24no—
147Dic362C2×D412yes—
148D724—24no—
149D724—24no—
150D724—22no—
151D724—21no—
152D724—21no—
153D724—21no—
154D724—81no—
155D724C2×D412yes—
156C22⋊D181—24no—
157C22⋊D181—22no—
158C22⋊D181—22no—
159C22⋊D181—42no—
160C22⋊D181—81no—
161C22⋊D181—81no—
162C2×Dic181—21no—
163C2×Dic181—21no—
164C2×Dic181C2×D411yescharacteristic
165C2×Dic181C2×D411yescharacteristic
166C4×D181—24no—
167C4×D181—24no—
168C4×D181—24no—
169C4×D181—24no—
170C4×D181—21no—
171C4×D181—21no—
172C4×D181—24no—
173C4×D181C2×D412yes—
174C4×D181C2×D412yes—
175#13089678—21no—
176#13089678—21no—
177C23×C181—21no—
178C23×C181C2×D411yescharacteristic
179Q8×C181—24no—
180Q8×C181—21no—
181Q8×C181—21no—
182Q8×C181C2×D412yes—
183Q8×C181C2×D412yes—
184D4×C181—21no—
185D4×C181—21no—
186D4×C181—81no—
187C2×C4×C181—24no—
188C2×C4×C181—21no—
189C2×C4×C181—21no—
190C2×C4×C181—21no—
191C2×C4×C181—21no—
192C2×C4×C181—21no—
193C2×C4×C181C2×D411yescharacteristic
194C2×C4×C181C2411yescharacteristic
195C8×C181—21no—
196C8×C181—21no—
197C22×C181—21no—
198C22×C181—21no—
199C22×C181—21no—
200C22×C181—41no—
201C22×C181—41no—
202C22×C181—41no—
203C22×C181C22×D411yescharacteristic
204C2×D181—21no—
205C2×D181—21no—
206C2×D181—21no—
207C2×D181—22no—
208C2×D181—21no—
209C2×D181—21no—
210C2×D181—41no—
211C2×D181—41no—
212C2×D181—41no—
213C2×D181—41no—
214C2×D181—41no—
215C2×D181—84no—
216C2×D181#12012yes—
217C4×C181—22no—
218C4×C181—22no—
219C4×C181—21no—
220C4×C181—41no—
221C4×C181C22×D411yescharacteristic
222C4×C181C22×D411yescharacteristic
223Dic181—22no—
224Dic181—21no—
225Dic181—41no—
226Dic181#12012yes—
227C2×C181—41no—
228C2×C181—81no—
229C2×C181C2×#12011yescharacteristic
230C2×C181#57711yescharacteristic
231C2×C181#57711yescharacteristic
232D181—22no—
233D181—41no—
234D181—42no—
235D181—81no—
236C181#301011yescharacteristic, complemented, Sylow
237#3010—1811nomaximal, Sylow
238C2×2−1+4—1811no—
239C2×AGL(1,ℤ/8)—1811no—
240C22×SD16—1811no—
241#535—1811no—
242#449—1811no—
243#449—1811no—
244#449—1811no—
245#449—1811no—
246#448—1811no—
247#448—1811no—
248#448—1811no—
249#448D18111yescharacteristic, complemented
250#419—1811no—
251#416—1811no—
252#408—1811no—
2532−1+4—3624no—
2542−1+4—3624no—
255C2×#41—3621no—
256C2×#41—3621no—
257C2×#41—3621no—
258C2×#41—1811no—
259C2×#41—1811no—
260C2×#41—1811no—
261C2×#41—1811no—
262C22×Q8—3621no—
263C22×Q8—1811no—
264C22×Q8—1811no—
265C22×Q8C2×D18111yescharacteristic, complemented
266C22×D4—1811no—
267AGL(1,ℤ/8)—3624no—
268C2×SD16—3624no—
269C2×SD16—1811no—
270C2×SD16—1811no—
271C2×SD16—1811no—
272C2×SD16—1811no—
273C2×SD16C2×D18111yescharacteristic
274C2×SD16C2×D18111yescharacteristic
275C2×D8—1811no—
276C2×D8—1811no—
277C2×M16—1811no—
278C22×C8—1811no—
279#127—3621no—
280#124—3621no—
281#121—1811no—
282#121—1811no—
283#121—1811no—
284#121C2×D18111yescharacteristic
285#120—3621no—
286#120—3621no—
287#117—1811no—
288#103—1811no—
289#103—1811no—
290#103—1811no—
291#103—1811no—
292#103C2×D18111yescharacteristic, complemented
293#103C2×D18111yescharacteristic, complemented
294#102—1811no—
295#102—1811no—
296C22⋊C8—1811no—
297C22⋊C8—1811no—
298C22⋊C8—1811no—
299C22⋊C8C2×D18111yescharacteristic, complemented
300C24—3621no—
301#41—3628no—
302#41—3628no—
303#41—3622no—
304#41—3622no—
305#41—3622no—
306#41—3622no—
307#41—3628no—
308#41—3622no—
309#41—3622no—
310#41—3622no—
311#41—1814no—
312C2×Q8—22no—
313C2×Q8—22no—
314C2×Q8—21no—
315C2×Q8—3621no—
316C2×Q8—3621no—
317C2×Q8—3628no—
318C2×Q8—3622no—
319C2×Q8—3622no—
320C2×Q8—3622no—
321C2×Q8—1814no—
322C2×Q8—1811no—
323C2×Q8—1811no—
324C2×Q8C22×D18111yescharacteristic
325C2×Q8C22×D18111yescharacteristic
326C2×D4—21no—
327C2×D4—3621no—
328C2×D4—3621no—
329C2×D4—3621no—
330C2×D4—3621no—
331C2×D4—3621no—
332C2×D4—3621no—
333C2×D4—3622no—
334C2×D4—3622no—
335C2×D4—3621no—
336C2×D4—1811no—
337C2×D4—1811no—
338C2×D4—1811no—
339C2×D4—1811no—
340C2×D4—1811no—
341C2×D4C22×D18111yescharacteristic
342C22×C4—21no—
343C22×C4—3621no—
344C22×C4—3621no—
345C22×C4—3621no—
346C22×C4—3621no—
347C22×C4—3621no—
348C22×C4—1811no—
349C22×C4—1811no—
350C22×C4C22×D18111yescharacteristic
351SD16—22no—
352SD16—22no—
353SD16—3622no—
354SD16—3622no—
355SD16—3622no—
356SD16—3622no—
357D8—3622no—
358D8—3622no—
359M16—3622no—
360C2×C8—3622no—
361C2×C8—1811no—
362C2×C8—1811no—
363C2×C8C22×D18111yescharacteristic
364C2×C8C22×D18111yescharacteristic
365C4⋊C4—1811no—
366C4⋊C4C22×D18111yescharacteristic
367C22⋊C4—21no—
368C22⋊C4—3621no—
369C22⋊C4—3621no—
370C22⋊C4—3621no—
371C22⋊C4—3621no—
372C42—3621no—
373C23—21no—
374C23—3621no—
375C23—3621no—
376C23—14481no—
377C23—14481no—
378C23—1811no—
379C23—1811no—
380C23—7242no—
381C23D4×D18111yescharacteristic
382Q8—24no—
383Q8—21no—
384Q8—21no—
385Q8—3624no—
386Q8—3624no—
387Q8—3622no—
388Q8—3621no—
389Q8—1812no—
390Q8D4×D18112yes—
391Q8D4×D18112yes—
392D4—21no—
393D4—21no—
394D4—81no—
395D4—3624no—
396D4—3624no—
397D4—3622no—
398D4—3624no—
399D4—3622no—
400D4—3622no—
401D4—3621no—
402D4—3621no—
403D4—3621no—
404D4—14481no—
405D4—14481no—
406D4—14481no—
407D4—1812no—
408D4—7242no—
409C2×C4—24no—
410C2×C4—21no—
411C2×C4—21no—
412C2×C4—21no—
413C2×C4—21no—
414C2×C4—21no—
415C2×C4—3624no—
416C2×C4—3624no—
417C2×C4—3624no—
418C2×C4—3624no—
419C2×C4—3624no—
420C2×C4—3621no—
421C2×C4—3621no—
422C2×C4—3621no—
423C2×C4—3621no—
424C2×C4—1811no—
425C2×C4—1811no—
426C2×C4—1812no—
427C2×C4—1812no—
428C2×C4D4×D18111yescharacteristic
429C2×C4C23×D18111yescharacteristic
430C8—21no—
431C8—21no—
432C8—3621no—
433C8—3621no—
434C22—21no—
435C22—21no—
436C22—21no—
437C22—41no—
438C22—41no—
439C22—41no—
440C22—3621no—
441C22—3621no—
442C22—3621no—
443C22—3622no—
444C22—3621no—
445C22—3621no—
446C22—14484no—
447C22—1812no—
448C22—7241no—
449C22—7241no—
450C22—7241no—
451C22—7241no—
452C22—7241no—
453C22C2×D4×D18111yescharacteristic
454C4—22no—
455C4—22no—
456C4—21no—
457C4—41no—
458C4—3622no—
459C4—3621no—
460C4—1812no—
461C4—7241no—
462C4C2×D4×D18111yescharacteristic
463C4C2×D4×D18111yescharacteristic
464C2—41no—
465C2—81no—
466C2—3622no—
467C2—14481no—
468C2—7241no—
469C2—7242no—
470C2#120×D18111yescharacteristic
471C2#2604208511yescharacteristic
472C2#2604208811yescharacteristic
subgroupG/H
centerC22C2×D4×D181
commutator, cosocleC2×C4×C181C24
FrattiniC2×C4C23×D181
Fitting#448×C181C2
socleC22×C181C22×D4
hypercenter#448D181
hypocenterC181#3010
LCSG ▷ C2×C4×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C22 ◁ C22×C4 ◁ #448

Complements

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

NH ≅ G/Ntype
1. C2×#19153864C2semidirect
2. C2×#19153843C2semidirect
3. C2×SD16×D181C2semidirect
4. #26041845C2semidirect
5. #26041427C2semidirect
6. #26041426C2semidirect
7. #26041423C2semidirect
8. #26041422C2semidirect
9. #26041419C2semidirect
10. #26041418C2semidirect
11. #26041416C2semidirect
12. #26041286C2semidirect
13. #26041274C2semidirect
14. #26041244C2semidirect
15. #448×C181C2semidirect
20. C2×#15668450C22semidirect
24. C2×#15668449C22semidirect
27. C2×Q8×D181C22semidirect
47. #19153741C22semidirect
48. #19153740C22semidirect
49. #19153740C22semidirect
50. #19153739C22semidirect
51. #19153738C22semidirect
52. #19153736C22semidirect
53. C22⋊#13089678C22semidirect
54. #19153728C22semidirect
55. #19153728C22semidirect
56. C22×Q8×C181C22semidirect
60. #103×C181C22semidirect
61. #103×C181C22semidirect
62. C22⋊C8×C181C22semidirect
68. #15668450D4semidirect
83. Q8×D181D4semidirect
236. C181#3010semidirect
249. #448D181semidirect
265. C22×Q8C2×D181semidirect
292. #103C2×D181semidirect
293. #103C2×D181semidirect
299. C22⋊C8C2×D181semidirect

Maximal subgroups

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

subgroupindex
C2×#191538642
C2×#191538432
C2×SD16×D1812
#260418452
#260414272
#260414262
#260414232
#260414222
#260414192
#260414182
#260414162
#260412862
#260412742
#260412442
#448×C1812
#3010181