Finite Group Databasebeta

← #34078724

Subgroups — #34078724

Order 19328.

548 conjugacy classes of subgroups.

The subgroup lattice has 428 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#18012766C211yesmaximal, characteristic, complemented
2C2×M16×D151C211yesmaximal, characteristic, complemented
3#23752725C211yesmaximal, characteristic, complemented
4#121×D151C211yesmaximal, characteristic, complemented
5#23752465C211yesmaximal, characteristic, complemented
6#23752463C211yesmaximal, characteristic, complemented
7#23752460C211yesmaximal, characteristic, complemented
8#23752449C211yesmaximal, characteristic, complemented
9#23752447C211yesmaximal, characteristic, complemented
10#23752444C211yesmaximal, characteristic, complemented
11#23752438C211yesmaximal, characteristic, complemented
12#23752253C211yesmaximal, characteristic, complemented
13#23752215C211yesmaximal, characteristic, complemented
14#23752212C211yesmaximal, characteristic, complemented
15#468×C151C211yesmaximal, characteristic, complemented
16C2×#14559005C2211yescharacteristic
17C2×D4×D151—21no—
18C2×D4×D151C2211yescharacteristic
19C22×C4×D151C2211yescharacteristic
20#18012766—24no—
21C2×#14558991C2211yescharacteristic, complemented
22C2×D1208C2211yescharacteristic, complemented
23C2×#14558989C2211yescharacteristic, complemented
24C2×#14558988C2211yescharacteristic, complemented
25C2×#14558987C2211yescharacteristic
26C2×#14558987C2211yescharacteristic
27M16×D151—24no—
28C2×C8×D151C2211yescharacteristic, complemented
29C2×C8×D151C2211yescharacteristic, complemented
30#18012724—21no—
31#18012715C2211yescharacteristic, complemented
32#18012714C2211yescharacteristic
33#18012714C2211yescharacteristic, complemented
34#18012713C2211yescharacteristic
35#18012713C2211yescharacteristic, complemented
36#18012712—21no—
37#18012711—21no—
38#18012710C2211yescharacteristic
39#18012709—21no—
40#18012701—21no—
41(C22⋊D151)⋊C4—21no—
42#18012697C2211yescharacteristic
43C4⋊C4×D151C2211yescharacteristic
44C22⋊C4×D151—21no—
45#18012672C2211yescharacteristic, complemented
46#18012671C2211yescharacteristic, complemented
47#18012670C2211yescharacteristic, complemented
48#18012663C2211yescharacteristic, complemented
49#18012661C2211yescharacteristic, complemented
50#18012661C2211yescharacteristic, complemented
51#18012660C2211yescharacteristic, complemented
52#18012659C2211yescharacteristic, complemented
53#18012659C2211yescharacteristic, complemented
54C2×SD16×C151C2211yescharacteristic, complemented
55C2×M16×C151C2211yescharacteristic, complemented
56C22⋊Q8×C151C2211yescharacteristic, complemented
57#121×C151C2211yescharacteristic
58#107×C151C2211yescharacteristic, complemented
59#103×C151C2211yescharacteristic, complemented
60#102×C151C2211yescharacteristic, complemented
61C23×D151—21no—
62C23×D151D411yescharacteristic
63#14559005—22no—
64#14559005—22no—
65C2×D604—21no—
66C2×D604—21no—
67C2×D604—21no—
68C2×D604C2311yescharacteristic
69C2×D604C2311yescharacteristic
70C2×C22⋊D151—21no—
71C2×C22⋊D151—21no—
72C2×C22⋊D151—21no—
73C2×C22⋊D151—21no—
74D4×D151—22no—
75D4×D151—22no—
76D4×D151—44no—
77C22×Dic151D411yescharacteristic
78C2×C4×D151—22no—
79C2×C4×D151—22no—
80C2×C4×D151—21no—
81C2×C4×D151—21no—
82C2×C4×D151D411yescharacteristic
83C2×C4×D151D411yescharacteristic
84C2×C4×D151C2311yescharacteristic
85C2×C4×D151C2311yescharacteristic
86#14558991—22no—
87D1208—22no—
88#14558989—22no—
89#14558988—22no—
90#14558987—22no—
91#14558987D414yescomplemented
92C2×C151⋊C8C2311yescharacteristic
93C2×C151⋊C8C2311yescharacteristic
94C8×D151—22no—
95C8×D151—22no—
96C4⋊Dic151C2311yescharacteristic
97C4⋊Dic151C2311yescharacteristic
98Dic151⋊C4—21no—
99Dic151⋊C4—21no—
100C22⋊Dic151—21no—
101(C2×D151)⋊C4—21no—
102(C2×D151)⋊C4—21no—
103(C2×D151)⋊C4—21no—
104(C2×D151)⋊C4—21no—
105(C2×D151)⋊C4—21no—
106(C2×D151)⋊C4—21no—
107C4×Dic151—21no—
108C2×Q8×C151C2311yescharacteristic
109C2×D4×C151—21no—
110C2×D4×C151C2311yescharacteristic
111C22×C4×C151C2311yescharacteristic
112SD16×C151—22no—
113M16×C151—22no—
114C2×C8×C151C2311yescharacteristic
115C2×C8×C151C2311yescharacteristic
116C4⋊C4×C151—21no—
117C4⋊C4×C151C2311yescharacteristic
118C4⋊C4×C151C2311yescharacteristic
119C22⋊C4×C151—21no—
120C22⋊C4×C151—21no—
121C22×D151—22no—
122C22×D151—22no—
123C22×D151—21no—
124C22×D151—21no—
125C22×D151—41no—
126C22×D151—42no—
127C22×D151—42no—
128C22×D151—42no—
129C22×D151C2×D411yescharacteristic
130C22×D151C2×D411yescharacteristic
131D604—21no—
132D604—21no—
133D604—21no—
134D604—21no—
135D604—42no—
136D604—42no—
137D604—42no—
138C22⋊D151—42no—
139C22⋊D151—42no—
140C22⋊D151—42no—
141C22⋊D151—42no—
142C2×Dic151—22no—
143C2×Dic151—21no—
144C2×Dic151—21no—
145C2×Dic151C2×D411yescharacteristic
146C2×Dic151C2×D411yescharacteristic
147C4×D151—22no—
148C4×D151—21no—
149C4×D151—21no—
150C4×D151—22no—
151C4×D151—42no—
152C4×D151—42no—
153C4×D151C2×D412yes—
154C4×D151#4112yes—
155C151⋊C8—21no—
156C151⋊C8C2×D412yescomplemented
157C23×C151—21no—
158C23×C151C2×D411yescharacteristic
159Q8×C151—21no—
160Q8×C151—21no—
161D4×C151—21no—
162D4×C151—21no—
163D4×C151—42no—
164C2×C4×C151—21no—
165C2×C4×C151—21no—
166C2×C4×C151—21no—
167C2×C4×C151—21no—
168C2×C4×C151—21no—
169C2×C4×C151C2×D411yescharacteristic
170C2×C4×C151C2411yescharacteristic
171C8×C151—21no—
172C8×C151C2×D412yescomplemented
173C22×C151—21no—
174C22×C151—21no—
175C22×C151—21no—
176C22×C151—41no—
177C22×C151—41no—
178C22×C151—41no—
179C22×C151C22×D411yescharacteristic
180C2×D151—21no—
181C2×D151—21no—
182C2×D151—21no—
183C2×D151—21no—
184C2×D151—44no—
185C2×D151—41no—
186C2×D151—41no—
187C2×D151—41no—
188C2×D151—41no—
189C2×D151—41no—
190C2×D151—41no—
191C2×D151—84no—
192C2×D151#12112yes—
193C2×D151AGL(1,ℤ/8)12yescomplemented
194C4×C151—21no—
195C4×C151—41no—
196C4×C151—41no—
197C4×C151—41no—
198C4×C151C22×D411yescharacteristic
199C4×C151C2×#4111yescharacteristic
200Dic151—21no—
201Dic151—41no—
202Dic151—41no—
203Dic151#12112yes—
204C2×C151—41no—
205C2×C151—81no—
206C2×C151C2×#12111yescharacteristic
207C2×C151C2×AGL(1,ℤ/8)11yescharacteristic
208C2×C151#57511yescharacteristic
209D151—22no—
210D151—22no—
211D151—81no—
212D151—81no—
213C151#305511yescharacteristic, complemented, Sylow
214#3055—1511nomaximal, Sylow
215C2×AGL(1,ℤ/8)—1511no—
216C22×M16—1511no—
217#525—1511no—
218C2×#121—1511no—
219#469—1511no—
220#469—1511no—
221#468—1511no—
222#468D15111yescharacteristic, complemented
223#466—1511no—
224#466—1511no—
225#465—1511no—
226#465—1511no—
227#428—1511no—
228#418—1511no—
229#417—1511no—
230C2×#41—1511no—
231C22×D4—3021no—
232C22×D4—1511no—
233C23×C4—1511no—
234AGL(1,ℤ/8)—3024no—
235C2×SD16—1511no—
236C2×SD16C2×D15111yescharacteristic, complemented
237C2×D8—1511no—
238C2×D8—1511no—
239C2×M16—3024no—
240C2×M16—1511no—
241C2×M16—1511no—
242C2×M16—1511no—
243C2×M16C2×D15111yescharacteristic, complemented
244C22×C8—1511no—
245C22×C8—1511no—
246#123—3021no—
247C22⋊Q8—1511no—
248C22⋊Q8C2×D15111yescharacteristic, complemented
249#121—3021no—
250#121—3021no—
251#121—1511no—
252#121—1511no—
253#121—1511no—
254#121—1511no—
255#121—1511no—
256#121C2×D15111yescharacteristic
257#120—3021no—
258C4×D4—3021no—
259C4×D4—3021no—
260#117—1511no—
261C2×C4⋊C4—1511no—
262C2×C22⋊C4—3021no—
263#107—1511no—
264#107C2×D15111yescharacteristic, complemented
265#106—1511no—
266#106—1511no—
267#103—1511no—
268#103C2×D15111yescharacteristic, complemented
269#102—1511no—
270#102—1511no—
271#102—1511no—
272#102—1511no—
273#102—1511no—
274#102C2×D15111yescharacteristic, complemented
275C24—3021no—
276C24—1511no—
277#41—3022no—
278#41—3022no—
279C2×Q8C22×D15111yescharacteristic
280C2×D4—21no—
281C2×D4—3021no—
282C2×D4—3021no—
283C2×D4—3021no—
284C2×D4—3021no—
285C2×D4—3021no—
286C2×D4—3021no—
287C2×D4—3022no—
288C2×D4—3022no—
289C2×D4—3021no—
290C2×D4—6044no—
291C2×D4—1511no—
292C2×D4—1511no—
293C2×D4C22×D15111yescharacteristic
294C22×C4—3022no—
295C22×C4—3022no—
296C22×C4—3021no—
297C22×C4—3021no—
298C22×C4—1511no—
299C22×C4—1511no—
300C22×C4—1511no—
301C22×C4—1511no—
302C22×C4—1511no—
303C22×C4C22×D15111yescharacteristic
304SD16—22no—
305SD16—3022no—
306D8—3022no—
307D8—3022no—
308M16—22no—
309M16—3022no—
310M16—3022no—
311M16—1514no—
312C2×C8—3022no—
313C2×C8—3022no—
314C2×C8—1511no—
315C2×C8—1511no—
316C2×C8C22×D15111yescharacteristic
317C2×C8C22×D15111yescharacteristic
318C4⋊C4—21no—
319C4⋊C4—3021no—
320C4⋊C4—3021no—
321C4⋊C4—1511no—
322C4⋊C4—1511no—
323C4⋊C4C22×D15111yescharacteristic
324C4⋊C4C22×D15111yescharacteristic
325C22⋊C4—21no—
326C22⋊C4—21no—
327C22⋊C4—3021no—
328C22⋊C4—3021no—
329C22⋊C4—3021no—
330C22⋊C4—3021no—
331C22⋊C4—3021no—
332C22⋊C4—3021no—
333C22⋊C4—3021no—
334C42—3021no—
335C23—21no—
336C23—3022no—
337C23—3022no—
338C23—3021no—
339C23—3021no—
340C23—6041no—
341C23—6042no—
342C23—6042no—
343C23—6042no—
344C23—1511no—
345C23—1511no—
346C23D4×D15111yescharacteristic
347Q8—21no—
348Q8—21no—
349D4—21no—
350D4—21no—
351D4—42no—
352D4—3021no—
353D4—3021no—
354D4—3021no—
355D4—3021no—
356D4—6042no—
357D4—6042no—
358D4—6042no—
359D4—6042no—
360D4—6042no—
361D4—6042no—
362D4—6042no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—21no—
367C2×C4—21no—
368C2×C4—3022no—
369C2×C4—3022no—
370C2×C4—3022no—
371C2×C4—3021no—
372C2×C4—3021no—
373C2×C4—3021no—
374C2×C4—3021no—
375C2×C4—6042no—
376C2×C4—6042no—
377C2×C4—1511no—
378C2×C4—1511no—
379C2×C4—1512no—
380C2×C4—1512no—
381C2×C4D4×D15111yescharacteristic
382C2×C4C23×D15111yescharacteristic
383C8—21no—
384C8—3021no—
385C8—1512no—
386C8D4×D15112yescomplemented
387C22—21no—
388C22—21no—
389C22—21no—
390C22—41no—
391C22—41no—
392C22—41no—
393C22—3021no—
394C22—3021no—
395C22—3021no—
396C22—3021no—
397C22—6044no—
398C22—6041no—
399C22—6041no—
400C22—6041no—
401C22—6041no—
402C22—6041no—
403C22—6041no—
404C22—1512no—
405C22—1512no—
406C22—12084no—
407C22C2×D4×D15111yescharacteristic
408C4—21no—
409C4—41no—
410C4—41no—
411C4—41no—
412C4—3021no—
413C4—6041no—
414C4—6041no—
415C4—1512no—
416C4C2×D4×D15111yescharacteristic
417C4#41×D15111yescharacteristic
418C2—41no—
419C2—81no—
420C2—3022no—
421C2—3022no—
422C2—12081no—
423C2—12081no—
424C2#121×D15111yescharacteristic
425C2AGL(1,ℤ/8)×D15111yescharacteristic
426C2#2375301111yescharacteristic
subgroupG/H
centerC22C2×D4×D151
commutator, cosocleC2×C4×C151C24
FrattiniC2×C4C23×D151
Fitting#468×C151C2
socleC22×C151C22×D4
hypercenter#468D151
hypocenterC151#3055
LCSG ▷ C2×C4×C151 ▷ C2×C151 ▷ C151
UCSC1 ◁ C22 ◁ C22×C4 ◁ #468

Complements

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

NH ≅ G/Ntype
1. C2×#18012766C2semidirect
2. C2×M16×D151C2semidirect
3. #23752725C2semidirect
4. #121×D151C2semidirect
5. #23752465C2semidirect
6. #23752463C2semidirect
7. #23752460C2semidirect
8. #23752449C2semidirect
9. #23752447C2semidirect
10. #23752444C2semidirect
11. #23752438C2semidirect
12. #23752253C2semidirect
13. #23752215C2semidirect
14. #23752212C2semidirect
15. #468×C151C2semidirect
21. C2×#14558991C22semidirect
22. C2×D1208C22semidirect
23. C2×#14558989C22semidirect
24. C2×#14558988C22semidirect
28. C2×C8×D151C22semidirect
29. C2×C8×D151C22semidirect
31. #18012715C22semidirect
33. #18012714C22semidirect
35. #18012713C22semidirect
45. #18012672C22semidirect
46. #18012671C22semidirect
47. #18012670C22semidirect
48. #18012663C22semidirect
49. #18012661C22semidirect
50. #18012661C22semidirect
51. #18012660C22semidirect
52. #18012659C22semidirect
53. #18012659C22semidirect
54. C2×SD16×C151C22semidirect
55. C2×M16×C151C22semidirect
56. C22⋊Q8×C151C22semidirect
58. #107×C151C22semidirect
59. #103×C151C22semidirect
60. #102×C151C22semidirect
91. #14558987D4semidirect
156. C151⋊C8C2×D4semidirect
172. C8×C151C2×D4semidirect
193. C2×D151AGL(1,ℤ/8)semidirect
213. C151#3055semidirect
222. #468D151semidirect
236. C2×SD16C2×D151semidirect
243. C2×M16C2×D151semidirect
248. C22⋊Q8C2×D151semidirect
264. #107C2×D151semidirect
268. #103C2×D151semidirect
274. #102C2×D151semidirect
386. C8D4×D151semidirect

Maximal subgroups

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

subgroupindex
C2×#180127662
C2×M16×D1512
#237527252
#121×D1512
#237524652
#237524632
#237524602
#237524492
#237524472
#237524442
#237524382
#237522532
#237522152
#237522122
#468×C1512
#3055151