Finite Group Databasebeta

← #37545255

Subgroups — #37545255

Order 23168.

510 conjugacy classes of subgroups.

The subgroup lattice has 430 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19153837C211yesmaximal, characteristic, complemented
2#26041948C211yesmaximal, characteristic, complemented
3#26041923C211yesmaximal, characteristic, complemented
4#26041667C211yesmaximal, characteristic, complemented
5#26041525C211yesmaximal, characteristic, complemented
6#26041512C211yesmaximal, characteristic, complemented
7#26041499C211yesmaximal, characteristic, complemented
8#26041472C211yesmaximal, characteristic, complemented
9#26041441C211yesmaximal, characteristic, complemented
10#26041423C211yesmaximal, characteristic, complemented
11#26041421C211yesmaximal, characteristic, complemented
12#26041413C211yesmaximal, characteristic, complemented
13#26041380C211yesmaximal, characteristic, complemented
14#26041348C211yesmaximal, characteristic, complemented
15C2×AGL(1,ℤ/8)×C181C211yesmaximal, characteristic, complemented
16C2×#15668450C2211yescharacteristic, complemented
17C2×#15668448C2211yescharacteristic
18#19153837—24no—
19C2×#15668439C2211yescharacteristic, complemented
20C2×#15668438C2211yescharacteristic, complemented
21C2×#15668437C2211yescharacteristic, complemented
22C2×#15668434C2211yescharacteristic
23C2×#15668434C2211yescharacteristic
24C2×C8×D181C2211yescharacteristic, complemented
25#19153812—21no—
26#19153812C2211yescharacteristic, complemented
27#19153803C2211yescharacteristic
28#19153797—21no—
29#19153795C2211yescharacteristic
30#19153790—21no—
31#19153790C2211yescharacteristic
32#19153788—21no—
33#19153787—21no—
34#19153787C2211yescharacteristic
35#19153787C2211yescharacteristic, complemented
36#19153785—21no—
37Q8×Dic181C2211yescharacteristic, complemented
38D4×Dic181C2211yescharacteristic
39C4×C22⋊D181—21no—
40C4×C22⋊D181C2211yescharacteristic
41C2×C22⋊Dic181—21no—
42#19153750C2211yescharacteristic, complemented
43#19153745C2211yescharacteristic, complemented
44#19153741C2211yescharacteristic, complemented
45#19153740C2211yescharacteristic, complemented
46#19153738C2211yescharacteristic, complemented
47#19153737C2211yescharacteristic
48#19153737C2211yescharacteristic
49#19153737C2211yescharacteristic, complemented
50C22⋊#13089678C2211yescharacteristic
51#19153728C2211yescharacteristic, complemented
52#19153726C2211yescharacteristic, complemented
53C2×#41×C181C2211yescharacteristic
54C22×D4×C181C2211yescharacteristic
55AGL(1,ℤ/8)×C181—24no—
56C2×SD16×C181C2211yescharacteristic
57C2×SD16×C181C2211yescharacteristic
58C2×D8×C181C2211yescharacteristic
59C2×D8×C181C2211yescharacteristic
60C2×M16×C181C2211yescharacteristic
61#15668450—22no—
62#15668450—41no—
63#15668448—22no—
64#15668448—41no—
65C2×Dic362C2311yescharacteristic
66C2×D724—21no—
67C2×D724C2311yescharacteristic
68C2×C22⋊D181—21no—
69C2×C22⋊D181—21no—
70C2×C22⋊D181—21no—
71C2×C22⋊D181—21no—
72C2×C22⋊D181—21no—
73C2×C22⋊D181—21no—
74C2×C22⋊D181D411yescharacteristic
75C22×Dic181—21no—
76C22×Dic181—21no—
77C2×C4×D181—21no—
78C2×C4×D181C2311yescharacteristic
79#15668439—22no—
80#15668438—22no—
81#15668437—41no—
82#15668434—22no—
83#15668434—41no—
84C2×#13089678C2311yescharacteristic
85C2×#13089678C2311yescharacteristic
86C8×D181—22no—
87C4⋊Dic181—21no—
88C4⋊Dic181C2311yescharacteristic
89C4⋊Dic181C2311yescharacteristic
90#15668428—21no—
91#15668428D411yescharacteristic
92C22⋊Dic181—21no—
93C22⋊Dic181—21no—
94C22⋊Dic181—21no—
95C22⋊Dic181—21no—
96C22⋊Dic181—21no—
97C22⋊Dic181—21no—
98C22⋊Dic181D411yescharacteristic
99#15668426—21no—
100#15668426D411yescharacteristic
101C4×Dic181—21no—
102C4×Dic181C2311yescharacteristic
103C24×C181—21no—
104#41×C181—22no—
105#41×C181—41no—
106C2×Q8×C181C2311yescharacteristic
107C2×D4×C181—22no—
108C2×D4×C181—21no—
109C2×D4×C181—21no—
110C2×D4×C181—41no—
111C2×D4×C181C2311yescharacteristic
112C2×D4×C181C2311yescharacteristic
113C2×D4×C181C2311yescharacteristic
114C22×C4×C181—21no—
115C22×C4×C181C2311yescharacteristic
116SD16×C181—22no—
117SD16×C181D414yescomplemented
118D8×C181—22no—
119D8×C181—22no—
120M16×C181—22no—
121C2×C8×C181C2311yescharacteristic
122C2×C8×C181C2311yescharacteristic
123C22×D181—21no—
124C22×D181C2×D411yescharacteristic
125Dic362—21no—
126Dic362—21no—
127D724—21no—
128D724—21no—
129D724—42no—
130C22⋊D181—41no—
131C22⋊D181—42no—
132C22⋊D181—42no—
133C22⋊D181—81no—
134C22⋊D181—81no—
135C22⋊D181—81no—
136C22⋊D181—81no—
137C2×Dic181—21no—
138C2×Dic181—21no—
139C2×Dic181—21no—
140C2×Dic181—42no—
141C2×Dic181—81no—
142C2×Dic181C2×D411yescharacteristic
143C2×Dic181C2×D411yescharacteristic
144C2×Dic181C2×D411yescharacteristic
145C4×D181—21no—
146C4×D181—21no—
147C4×D181—42no—
148#13089678—21no—
149#13089678—21no—
150C23×C181—21no—
151C23×C181—21no—
152C23×C181—21no—
153C23×C181—42no—
154C23×C181—42no—
155C23×C181—81no—
156C23×C181C2×D411yescharacteristic
157Q8×C181—21no—
158Q8×C181C2×D412yes—
159D4×C181—21no—
160D4×C181—21no—
161D4×C181—21no—
162D4×C181—21no—
163D4×C181—21no—
164D4×C181—42no—
165D4×C181—42no—
166D4×C181C2×D412yes—
167C2×C4×C181—21no—
168C2×C4×C181—21no—
169C2×C4×C181—21no—
170C2×C4×C181—42no—
171C2×C4×C181C2×D411yescharacteristic
172C2×C4×C181C2411yescharacteristic
173C8×C181—21no—
174C8×C181C2×D412yes—
175C22×C181—21no—
176C22×C181—22no—
177C22×C181—21no—
178C22×C181—21no—
179C22×C181—41no—
180C22×C181—41no—
181C22×C181—41no—
182C22×C181—41no—
183C22×C181—41no—
184C22×C181—41no—
185C22×C181—41no—
186C22×C181—41no—
187C22×C181—84no—
188C22×C181C22×D411yescharacteristic
189C2×D181—21no—
190C2×D181—21no—
191C2×D181—21no—
192C2×D181—41no—
193C2×D181—41no—
194C2×D181—41no—
195C4×C181—21no—
196C4×C181—22no—
197C4×C181C22×D411yescharacteristic
198C4×C181H(5,𝔽2)11yescharacteristic
199Dic181—21no—
200Dic181—21no—
201Dic181—21no—
202Dic181—41no—
203Dic181—41no—
204Dic181—41no—
205C2×C181—41no—
206C2×C181—42no—
207C2×C181—81no—
208C2×C181—81no—
209C2×C181D4211yescharacteristic
210C2×C181#57511yescharacteristic
211C2×C181#57611yescharacteristic
212D181—41no—
213D181—81no—
214C181#326811yescharacteristic, complemented, Sylow
215#3268—1811nomaximal, Sylow
216C2×AGL(1,ℤ/8)D18111yescharacteristic, complemented
217C2×#135—1811no—
218#550—1811no—
219#546—1811no—
220#496—1811no—
221#468—1811no—
222#466—1811no—
223#464—1811no—
224#459—1811no—
225#452—1811no—
226#449—1811no—
227#449—1811no—
228#447—1811no—
229#440—1811no—
230#435—1811no—
231C2×#41—1811no—
232C2×#41—1811no—
233C2×#41C2×D18111yescharacteristic
234C22×D4C2×D18111yescharacteristic
235AGL(1,ℤ/8)—24no—
236#135—3624no—
237C2×Q16—1811no—
238C2×SD16—1811no—
239C2×SD16—1811no—
240C2×SD16C2×D18111yescharacteristic
241C2×SD16C2×D18111yescharacteristic
242C2×D8—1811no—
243C2×D8—1811no—
244C2×D8C2×D18111yescharacteristic
245C2×D8C2×D18111yescharacteristic
246C2×M16C2×D18111yescharacteristic
247C22×C8—1811no—
248#127—3621no—
249#127—1811no—
250#124—1811no—
251#123—3621no—
252C22⋊Q8—1811no—
253#121—3621no—
254#121—3621no—
255#121—3621no—
256#121—1811no—
257#121—1811no—
258#121—1811no—
259#120—3621no—
260C4×Q8—1811no—
261C4×D4—3621no—
262C4×D4—1811no—
263C4×D4—1811no—
264C2×C22⋊C4—3621no—
265#107—1811no—
266C4⋊C8—1811no—
267#103—1811no—
268#103—1811no—
269#102—1811no—
270#102—1811no—
271#102—1811no—
272#102—1811no—
273C22⋊C8—1811no—
274C22⋊C8—1811no—
275#97—1811no—
276C24—21no—
277#41—22no—
278#41—41no—
279#41—3622no—
280#41—3622no—
281#41—7241no—
282#41—7241no—
283C2×Q8—1811no—
284C2×Q8C22×D18111yescharacteristic
285C2×D4—22no—
286C2×D4—21no—
287C2×D4—21no—
288C2×D4—41no—
289C2×D4—3621no—
290C2×D4—3621no—
291C2×D4—3621no—
292C2×D4—3621no—
293C2×D4—3621no—
294C2×D4—3621no—
295C2×D4—3621no—
296C2×D4—1811no—
297C2×D4—1811no—
298C2×D4C22×D18111yescharacteristic
299C2×D4C22×D18111yescharacteristic
300C2×D4C22×D18111yescharacteristic
301C22×C4—21no—
302C22×C4—3621no—
303C22×C4—3621no—
304C22×C4—3621no—
305C22×C4—1811no—
306C22×C4C22×D18111yescharacteristic
307Q16—3622no—
308SD16—22no—
309SD16—3622no—
310SD16—7241no—
311SD16C22⋊D18114yescomplemented
312D8—22no—
313D8—22no—
314D8—3622no—
315D8—7241no—
316M16—22no—
317C2×C8—3622no—
318C2×C8—1811no—
319C2×C8—1811no—
320C2×C8C22×D18111yescharacteristic
321C2×C8C22×D18111yescharacteristic
322C4⋊C4—3621no—
323C4⋊C4—3621no—
324C4⋊C4—1811no—
325C4⋊C4—1811no—
326C4⋊C4—1811no—
327C22⋊C4—3621no—
328C22⋊C4—3621no—
329C22⋊C4—3621no—
330C22⋊C4—3621no—
331C22⋊C4—3621no—
332C22⋊C4—3621no—
333C22⋊C4—3621no—
334C22⋊C4—1811no—
335C22⋊C4—1811no—
336C42—3621no—
337C42—1811no—
338C23—21no—
339C23—21no—
340C23—21no—
341C23—42no—
342C23—42no—
343C23—81no—
344C23—3621no—
345C23—1811no—
346C23D4×D18111yescharacteristic
347Q8—21no—
348Q8—3621no—
349Q8—3621no—
350Q8C2×C22⋊D18112yes—
351D4—21no—
352D4—21no—
353D4—21no—
354D4—21no—
355D4—21no—
356D4—42no—
357D4—42no—
358D4—3621no—
359D4—3621no—
360D4—14481no—
361D4—14481no—
362D4—14481no—
363D4—14481no—
364D4—7242no—
365D4—7241no—
366D4—7242no—
367D4—7242no—
368D4C2×C22⋊D18112yes—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—42no—
373C2×C4—3621no—
374C2×C4—3621no—
375C2×C4—3621no—
376C2×C4—3621no—
377C2×C4—3621no—
378C2×C4—14481no—
379C2×C4—1811no—
380C2×C4—1811no—
381C2×C4—1811no—
382C2×C4—7242no—
383C2×C4—7242no—
384C2×C4D4×D18111yescharacteristic
385C2×C4C23×D18111yescharacteristic
386C8—21no—
387C8—3621no—
388C8—3621no—
389C8C2×C22⋊D18112yes—
390C22—21no—
391C22—22no—
392C22—21no—
393C22—21no—
394C22—41no—
395C22—41no—
396C22—41no—
397C22—41no—
398C22—41no—
399C22—41no—
400C22—41no—
401C22—41no—
402C22—84no—
403C22—3621no—
404C22—3621no—
405C22—3621no—
406C22—7241no—
407C22—7241no—
408C22—7241no—
409C22C2×D4×D18111yescharacteristic
410C4—21no—
411C4—22no—
412C4—3621no—
413C4—3621no—
414C4—3621no—
415C4—7241no—
416C4—7241no—
417C4—7241no—
418C4C22×C22⋊D18111yescharacteristic
419C4#1915386111yescharacteristic
420C2—41no—
421C2—42no—
422C2—81no—
423C2—81no—
424C2—14481no—
425C2—7241no—
426C2D4×C22⋊D18111yescharacteristic
427C2#2604207511yescharacteristic
428C2#2604208311yescharacteristic
subgroupG/H
centerC22C2×D4×D181
commutator, cosocleC2×C4×C181C24
FrattiniC2×C4C23×D181
FittingC2×AGL(1,ℤ/8)×C181C2
socleC22×C181C22×D4
hypercenterC2×AGL(1,ℤ/8)D181
hypocenterC181#3268
LCSG ▷ C2×C4×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
1. C2×#19153837C2semidirect
2. #26041948C2semidirect
3. #26041923C2semidirect
4. #26041667C2semidirect
5. #26041525C2semidirect
6. #26041512C2semidirect
7. #26041499C2semidirect
8. #26041472C2semidirect
9. #26041441C2semidirect
10. #26041423C2semidirect
11. #26041421C2semidirect
12. #26041413C2semidirect
13. #26041380C2semidirect
14. #26041348C2semidirect
15. C2×AGL(1,ℤ/8)×C181C2semidirect
16. C2×#15668450C22semidirect
19. C2×#15668439C22semidirect
20. C2×#15668438C22semidirect
21. C2×#15668437C22semidirect
24. C2×C8×D181C22semidirect
26. #19153812C22semidirect
35. #19153787C22semidirect
37. Q8×Dic181C22semidirect
42. #19153750C22semidirect
43. #19153745C22semidirect
44. #19153741C22semidirect
45. #19153740C22semidirect
46. #19153738C22semidirect
49. #19153737C22semidirect
51. #19153728C22semidirect
52. #19153726C22semidirect
117. SD16×C181D4semidirect
214. C181#3268semidirect
216. C2×AGL(1,ℤ/8)D181semidirect
311. SD16C22⋊D181semidirect

Maximal subgroups

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

subgroupindex
C2×#191538372
#260419482
#260419232
#260416672
#260415252
#260415122
#260414992
#260414722
#260414412
#260414232
#260414212
#260414132
#260413802
#260413482
C2×AGL(1,ℤ/8)×C1812
#3268181