Finite Group Databasebeta

← #37545352

Subgroups — #37545352

Order 23168.

498 conjugacy classes of subgroups.

The subgroup lattice has 418 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19153845C211yesmaximal, characteristic, complemented
2#26041936C211yesmaximal, characteristic, complemented
3#26041923C211yesmaximal, characteristic, complemented
4#26041662C211yesmaximal, characteristic, complemented
5#26041537C211yesmaximal, characteristic, complemented
6#26041529C211yesmaximal, characteristic, complemented
7#26041499C211yesmaximal, characteristic, complemented
8#26041492C211yesmaximal, characteristic, complemented
9#26041441C211yesmaximal, characteristic, complemented
10#26041430C211yesmaximal, characteristic, complemented
11#26041421C211yesmaximal, characteristic, complemented
12#26041416C211yesmaximal, characteristic, complemented
13SD16×Dic181C211yesmaximal, characteristic, complemented
14#26041355C211yesmaximal, characteristic, complemented
15C2×AGL(1,ℤ/8)×C181C211yesmaximal, characteristic, complemented
16C2×#15668448C2211yescharacteristic
17C2×Q8×D181C2211yescharacteristic, complemented
18#19153845—24no—
19C2×Dic724C2211yescharacteristic, complemented
20C2×#15668439C2211yescharacteristic, complemented
21C2×#15668437C2211yescharacteristic
22C2×#15668437C2211yescharacteristic
23C2×#15668437C2211yescharacteristic, complemented
24C2×#15668432C2211yescharacteristic, complemented
25#19153803—21no—
26#19153803C2211yescharacteristic
27#19153803C2211yescharacteristic, complemented
28#19153797—21no—
29#19153795—21no—
30#19153795C2211yescharacteristic
31#19153795C2211yescharacteristic
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#19153748C2211yescharacteristic, complemented
43#19153745C2211yescharacteristic, complemented
44#19153741C2211yescharacteristic, complemented
45#19153741C2211yescharacteristic, complemented
46#19153740C2211yescharacteristic, complemented
47#19153737C2211yescharacteristic
48#19153737C2211yescharacteristic
49#19153737C2211yescharacteristic, complemented
50C22⋊#13089678C2211yescharacteristic
51#19153728C2211yescharacteristic, complemented
52C8×Dic181C2211yescharacteristic, 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#15668448—22no—
62#15668448—41no—
63C2×Dic362—21no—
64C2×Dic362C2311yescharacteristic
65C2×Dic362C2311yescharacteristic
66Q8×D181—22no—
67Q8×D181—41no—
68C2×C22⋊D181—21no—
69C2×C22⋊D181—21no—
70C2×C22⋊D181—21no—
71C2×C22⋊D181D411yescharacteristic
72C22×Dic181—21no—
73C22×Dic181—21no—
74C2×C4×D181—21no—
75C2×C4×D181C2311yescharacteristic
76Dic724—22no—
77#15668439—22no—
78#15668437—22no—
79#15668437—41no—
80#15668437—41no—
81#15668432—22no—
82C2×#13089678C2311yescharacteristic
83C2×#13089678C2311yescharacteristic
84C4⋊Dic181—21no—
85C4⋊Dic181C2311yescharacteristic
86C4⋊Dic181C2311yescharacteristic
87#15668428—21no—
88#15668428—21no—
89#15668428D411yescharacteristic
90C22⋊Dic181—21no—
91C22⋊Dic181—21no—
92C22⋊Dic181—21no—
93C22⋊Dic181—21no—
94C22⋊Dic181—21no—
95C22⋊Dic181—21no—
96C22⋊Dic181—21no—
97C22⋊Dic181—21no—
98C22⋊Dic181—21no—
99C22⋊Dic181D411yescharacteristic
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×D181C2×D411yescharacteristic
124Dic362—21no—
125Dic362—21no—
126Dic362—21no—
127Dic362—21no—
128Dic362—42no—
129C22⋊D181—41no—
130C22⋊D181—42no—
131C22⋊D181—42no—
132C22⋊D181—81no—
133C2×Dic181—21no—
134C2×Dic181—21no—
135C2×Dic181—21no—
136C2×Dic181—21no—
137C2×Dic181—42no—
138C2×Dic181—81no—
139C2×Dic181C2×D411yescharacteristic
140C2×Dic181C2×D411yescharacteristic
141C2×Dic181C2×D411yescharacteristic
142C4×D181—21no—
143C4×D181—21no—
144C4×D181—42no—
145#13089678—21no—
146#13089678—21no—
147C23×C181—21no—
148C23×C181—21no—
149C23×C181—21no—
150C23×C181—42no—
151C23×C181—42no—
152C23×C181—81no—
153C23×C181C2×D411yescharacteristic
154Q8×C181—21no—
155Q8×C181C2×D412yes—
156D4×C181—21no—
157D4×C181—21no—
158D4×C181—21no—
159D4×C181—21no—
160D4×C181—21no—
161D4×C181—42no—
162D4×C181—42no—
163D4×C181C2×D412yes—
164C2×C4×C181—21no—
165C2×C4×C181—21no—
166C2×C4×C181—21no—
167C2×C4×C181—42no—
168C2×C4×C181C2×D411yescharacteristic
169C2×C4×C181C2411yescharacteristic
170C8×C181—21no—
171C8×C181C2×D412yes—
172C22×C181—21no—
173C22×C181—22no—
174C22×C181—21no—
175C22×C181—21no—
176C22×C181—41no—
177C22×C181—41no—
178C22×C181—41no—
179C22×C181—41no—
180C22×C181—41no—
181C22×C181—41no—
182C22×C181—41no—
183C22×C181—41no—
184C22×C181—84no—
185C22×C181C22×D411yescharacteristic
186C2×D181—21no—
187C2×D181—21no—
188C2×D181—21no—
189C4×C181—21no—
190C4×C181—22no—
191C4×C181C22×D411yescharacteristic
192C4×C181H(5,𝔽2)11yescharacteristic
193Dic181—21no—
194Dic181—21no—
195Dic181—21no—
196Dic181—41no—
197Dic181—41no—
198Dic181—41no—
199Dic181—41no—
200C2×C181—41no—
201C2×C181—42no—
202C2×C181—81no—
203C2×C181—81no—
204C2×C181D4211yescharacteristic
205C2×C181#57511yescharacteristic
206C2×C181#57711yescharacteristic
207D181—41no—
208C181#327511yescharacteristic, complemented, Sylow
209#3275—1811nomaximal, Sylow
210C2×#137—1811no—
211C2×AGL(1,ℤ/8)D18111yescharacteristic, complemented
212#548—1811no—
213#546—1811no—
214#495—1811no—
215#470—1811no—
216#469—1811no—
217#464—1811no—
218#463—1811no—
219#452—1811no—
220#450—1811no—
221#449—1811no—
222#448—1811no—
223C4×SD16—1811no—
224#436—1811no—
225C2×#41—1811no—
226C2×#41C2×D18111yescharacteristic
227C22×Q8—1811no—
228C22×D4C2×D18111yescharacteristic
229#137—3624no—
230AGL(1,ℤ/8)—24no—
231C2×Q16—1811no—
232C2×Q16—1811no—
233C2×SD16—1811no—
234C2×SD16—1811no—
235C2×SD16—1811no—
236C2×SD16C2×D18111yescharacteristic
237C2×SD16C2×D18111yescharacteristic
238C2×D8C2×D18111yescharacteristic
239C2×D8C2×D18111yescharacteristic
240C2×M16—1811no—
241C2×M16C2×D18111yescharacteristic
242#124—3621no—
243#124—1811no—
244#124—1811no—
245#123—3621no—
246C22⋊Q8—3621no—
247C22⋊Q8—1811no—
248C22⋊Q8—1811no—
249#121—3621no—
250#121—3621no—
251#121—1811no—
252#121—1811no—
253#120—3621no—
254C4×Q8—1811no—
255C4×D4—3621no—
256C4×D4—1811no—
257C4×D4—1811no—
258C2×C22⋊C4—3621no—
259#106—1811no—
260C4⋊C8—1811no—
261#103—1811no—
262#103—1811no—
263#103—1811no—
264#102—1811no—
265#102—1811no—
266#102—1811no—
267C22⋊C8—1811no—
268C22⋊C8—1811no—
269C4×C8—1811no—
270C24—21no—
271#41—22no—
272#41—41no—
273#41—3622no—
274#41—7241no—
275C2×Q8—3621no—
276C2×Q8—3622no—
277C2×Q8—1811no—
278C2×Q8—1811no—
279C2×Q8—7241no—
280C2×Q8C22×D18111yescharacteristic
281C2×D4—22no—
282C2×D4—21no—
283C2×D4—21no—
284C2×D4—41no—
285C2×D4—3621no—
286C2×D4—3621no—
287C2×D4—3621no—
288C2×D4—1811no—
289C2×D4C22×D18111yescharacteristic
290C2×D4C22×D18111yescharacteristic
291C2×D4C22×D18111yescharacteristic
292C22×C4—21no—
293C22×C4—3621no—
294C22×C4—3621no—
295C22×C4—3621no—
296C22×C4—1811no—
297C22×C4C22×D18111yescharacteristic
298Q16—3622no—
299Q16—3622no—
300SD16—22no—
301SD16—3622no—
302SD16—7241no—
303SD16—7241no—
304SD16C22⋊D18114yescomplemented
305D8—22no—
306D8—22no—
307M16—22no—
308M16—3622no—
309C2×C8—1811no—
310C2×C8—1811no—
311C2×C8C22×D18111yescharacteristic
312C2×C8C22×D18111yescharacteristic
313C4⋊C4—3621no—
314C4⋊C4—3621no—
315C4⋊C4—3621no—
316C4⋊C4—1811no—
317C4⋊C4—1811no—
318C4⋊C4—1811no—
319C22⋊C4—3621no—
320C22⋊C4—3621no—
321C22⋊C4—3621no—
322C22⋊C4—3621no—
323C22⋊C4—3621no—
324C22⋊C4—3621no—
325C22⋊C4—3621no—
326C22⋊C4—3621no—
327C22⋊C4—3621no—
328C22⋊C4—1811no—
329C22⋊C4—1811no—
330C42—3621no—
331C42—1811no—
332C23—21no—
333C23—21no—
334C23—21no—
335C23—42no—
336C23—42no—
337C23—81no—
338C23—1811no—
339C23D4×D18111yescharacteristic
340Q8—21no—
341Q8—3621no—
342Q8—3621no—
343Q8—3621no—
344Q8—3621no—
345Q8—7242no—
346Q8C2×C22⋊D18112yes—
347D4—21no—
348D4—21no—
349D4—21no—
350D4—21no—
351D4—21no—
352D4—42no—
353D4—42no—
354D4—14481no—
355D4—7241no—
356D4—7242no—
357D4—7242no—
358D4C2×C22⋊D18112yes—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—42no—
363C2×C4—3621no—
364C2×C4—3621no—
365C2×C4—3621no—
366C2×C4—3621no—
367C2×C4—3621no—
368C2×C4—3621no—
369C2×C4—14481no—
370C2×C4—1811no—
371C2×C4—1811no—
372C2×C4—1811no—
373C2×C4—7242no—
374C2×C4—7242no—
375C2×C4D4×D18111yescharacteristic
376C2×C4C23×D18111yescharacteristic
377C8—21no—
378C8—3621no—
379C8—3621no—
380C8C2×C22⋊D18112yes—
381C22—21no—
382C22—22no—
383C22—21no—
384C22—21no—
385C22—41no—
386C22—41no—
387C22—41no—
388C22—41no—
389C22—41no—
390C22—41no—
391C22—41no—
392C22—41no—
393C22—84no—
394C22—3621no—
395C22—3621no—
396C22—3621no—
397C22C2×D4×D18111yescharacteristic
398C4—21no—
399C4—22no—
400C4—3621no—
401C4—3621no—
402C4—3621no—
403C4—7241no—
404C4—7241no—
405C4—7241no—
406C4—7241no—
407C4C22×C22⋊D18111yescharacteristic
408C4#1915386111yescharacteristic
409C2—41no—
410C2—42no—
411C2—81no—
412C2—81no—
413C2—7241no—
414C2D4×C22⋊D18111yescharacteristic
415C2#2604207511yescharacteristic
416C2#2604208911yescharacteristic
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#3275
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×#19153845C2semidirect
2. #26041936C2semidirect
3. #26041923C2semidirect
4. #26041662C2semidirect
5. #26041537C2semidirect
6. #26041529C2semidirect
7. #26041499C2semidirect
8. #26041492C2semidirect
9. #26041441C2semidirect
10. #26041430C2semidirect
11. #26041421C2semidirect
12. #26041416C2semidirect
13. SD16×Dic181C2semidirect
14. #26041355C2semidirect
15. C2×AGL(1,ℤ/8)×C181C2semidirect
17. C2×Q8×D181C22semidirect
19. C2×Dic724C22semidirect
20. C2×#15668439C22semidirect
23. C2×#15668437C22semidirect
24. C2×#15668432C22semidirect
27. #19153803C22semidirect
35. #19153787C22semidirect
37. Q8×Dic181C22semidirect
42. #19153748C22semidirect
43. #19153745C22semidirect
44. #19153741C22semidirect
45. #19153741C22semidirect
46. #19153740C22semidirect
49. #19153737C22semidirect
51. #19153728C22semidirect
52. C8×Dic181C22semidirect
117. SD16×C181D4semidirect
208. C181#3275semidirect
211. C2×AGL(1,ℤ/8)D181semidirect
304. SD16C22⋊D181semidirect

Maximal subgroups

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

subgroupindex
C2×#191538452
#260419362
#260419232
#260416622
#260415372
#260415292
#260414992
#260414922
#260414412
#260414302
#260414212
#260414162
SD16×Dic1812
#260413552
C2×AGL(1,ℤ/8)×C1812
#3275181