Finite Group Databasebeta

← #34078611

Subgroups — #34078611

Order 19328.

526 conjugacy classes of subgroups.

The subgroup lattice has 424 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#18012766C211yesmaximal, characteristic, complemented
2C2×D8×D151C211yesmaximal, characteristic, complemented
3#23752780C211yesmaximal, characteristic, complemented
4#23752684C211yesmaximal, characteristic, complemented
5#23752430C211yesmaximal, characteristic, complemented
6#23752429C211yesmaximal, characteristic, complemented
7#23752424C211yesmaximal, characteristic, complemented
8#23752408C211yesmaximal, characteristic, complemented
9#23752407C211yesmaximal, characteristic, complemented
10#23752406C211yesmaximal, characteristic, complemented
11#23752405C211yesmaximal, characteristic, complemented
12#23752237C211yesmaximal, characteristic, complemented
13#23752215C211yesmaximal, characteristic, complemented
14#23752207C211yesmaximal, characteristic, complemented
15#463×C151C211yesmaximal, characteristic, complemented
16C2×#14559005C2211yescharacteristic
17C2×D4×D151C2211yescharacteristic
18C2×D4×D151C2211yescharacteristic
19#18012766—24no—
20C2×#14558991C2211yescharacteristic, complemented
21C2×D1208C2211yescharacteristic
22C2×D1208C2211yescharacteristic, complemented
23C2×#14558989C2211yescharacteristic
24C2×#14558989C2211yescharacteristic
25C2×#14558989C2211yescharacteristic, complemented
26D8×D151—24no—
27C2×#14558987C2211yescharacteristic
28C2×C8×D151C2211yescharacteristic
29#18012736C2211yescharacteristic, complemented
30#18012735C2211yescharacteristic, complemented
31#18012728C2211yescharacteristic, complemented
32#18012712—21no—
33#18012712—21no—
34#18012709—21no—
35#18012709—21no—
36C4×D604C2211yescharacteristic, complemented
37D4×Dic151C2211yescharacteristic, complemented
38#18012701C2211yescharacteristic, complemented
39C4×C22⋊D151—21no—
40(C22⋊D151)⋊C4—21no—
41#18012697C2211yescharacteristic
42#18012696C2211yescharacteristic
43C2×(C2×D151)⋊C4—21no—
44C22⋊C4×D151—21no—
45C4⋊(C151⋊C8)C2211yescharacteristic, complemented
46Dic151⋊C8C2211yescharacteristic, complemented
47Dic151⋊C8C2211yescharacteristic, complemented
48#18012663C2211yescharacteristic, complemented
49#18012661C2211yescharacteristic, complemented
50#18012661C2211yescharacteristic, complemented
51#18012660C2211yescharacteristic, complemented
52#18012659C2211yescharacteristic, complemented
53#18012659C2211yescharacteristic, complemented
54C2×SD16×C151C2211yescharacteristic, complemented
55C2×D8×C151C2211yescharacteristic
56#124×C151C2211yescharacteristic, complemented
57C4×D4×C151C2211yescharacteristic, complemented
58C4⋊C8×C151C2211yescharacteristic, complemented
59#103×C151C2211yescharacteristic, complemented
60#102×C151C2211yescharacteristic, complemented
61C23×D151—21no—
62C23×D151—21no—
63#14559005—22no—
64#14559005—22no—
65C2×D604—21no—
66C2×D604—21no—
67C2×D604—21no—
68C2×D604C2311yescharacteristic
69C2×D604C2311yescharacteristic
70C2×D604C2311yescharacteristic
71C2×C22⋊D151—21no—
72C2×C22⋊D151—21no—
73C2×C22⋊D151—21no—
74C2×C22⋊D151—21no—
75D4×D151—22no—
76D4×D151—22no—
77D4×D151—22no—
78D4×D151D414yescomplemented
79C22×Dic151—21no—
80C2×C4×D151—21no—
81C2×C4×D151—21no—
82C2×C4×D151—21no—
83C2×C4×D151C2311yescharacteristic
84#14558991—22no—
85D1208—22no—
86D1208—22no—
87#14558989—22no—
88#14558989—22no—
89#14558989—22no—
90#14558987—22no—
91C2×C151⋊C8C2311yescharacteristic
92C2×C151⋊C8C2311yescharacteristic
93C8×D151—22no—
94C4⋊Dic151C2311yescharacteristic
95Dic151⋊C4D411yescharacteristic
96Dic151⋊C4D411yescharacteristic
97C22⋊Dic151—21no—
98(C2×D151)⋊C4—21no—
99(C2×D151)⋊C4—21no—
100(C2×D151)⋊C4—21no—
101(C2×D151)⋊C4—21no—
102(C2×D151)⋊C4—21no—
103(C2×D151)⋊C4D411yescharacteristic
104(C2×D151)⋊C4D411yescharacteristic
105C4×Dic151—21no—
106C4×Dic151C2311yescharacteristic
107C2×Q8×C151C2311yescharacteristic
108C2×D4×C151C2311yescharacteristic
109C2×D4×C151C2311yescharacteristic
110C22×C4×C151—21no—
111SD16×C151—22no—
112D8×C151—22no—
113C2×C8×C151C2311yescharacteristic
114C2×C8×C151C2311yescharacteristic
115C4⋊C4×C151C2311yescharacteristic
116C22⋊C4×C151—21no—
117C22⋊C4×C151—21no—
118C22⋊C4×C151—21no—
119C42×C151C2311yescharacteristic
120C22×D151—21no—
121C22×D151—24no—
122C22×D151—21no—
123C22×D151—21no—
124C22×D151—42no—
125C22×D151—81no—
126C22×D151—81no—
127C22×D151—81no—
128C22×D151—81no—
129C22×D151C2×D411yescharacteristic
130D604—21no—
131D604—21no—
132D604—21no—
133D604—21no—
134D604—21no—
135D604—42no—
136D604—81no—
137D604—81no—
138D604C2×D412yes—
139C22⋊D151—24no—
140C22⋊D151—42no—
141C22⋊D151—81no—
142C22⋊D151—81no—
143C2×Dic151—21no—
144C2×Dic151—81no—
145C2×Dic151C2×D411yescharacteristic
146C2×Dic151C2×D411yescharacteristic
147C2×Dic151C2×D411yescharacteristic
148C4×D151—42no—
149C4×D151—81no—
150C4×D151—81no—
151C4×D151C2×D412yes—
152C4×D151#4112yes—
153C151⋊C8—21no—
154C151⋊C8—21no—
155C23×C151—21no—
156C23×C151—21no—
157Q8×C151—21no—
158Q8×C151—21no—
159D4×C151—21no—
160D4×C151—21no—
161D4×C151—21no—
162D4×C151C2×D412yes—
163C2×C4×C151—21no—
164C2×C4×C151—21no—
165C2×C4×C151—81no—
166C2×C4×C151C2×D411yescharacteristic
167C2×C4×C151C2×D411yescharacteristic
168C2×C4×C151C2411yescharacteristic
169C8×C151—21no—
170C8×C151—21no—
171C22×C151—22no—
172C22×C151—41no—
173C22×C151—41no—
174C22×C151—41no—
175C22×C151—41no—
176C22×C151—41no—
177C22×C151C22×D411yescharacteristic
178C2×D151—21no—
179C2×D151—21no—
180C2×D151—22no—
181C2×D151—41no—
182C2×D151—41no—
183C2×D151—41no—
184C2×D151—41no—
185C2×D151—41no—
186C2×D151—41no—
187C2×D151—41no—
188C2×D151—41no—
189C2×D151—84no—
190C2×D151—84no—
191C2×D151#12112yes—
192C4×C151—21no—
193C4×C151—21no—
194C4×C151—41no—
195C4×C151—41no—
196C4×C151C22×D411yescharacteristic
197C4×C151C2×#4111yescharacteristic
198Dic151—21no—
199Dic151—21no—
200Dic151—41no—
201Dic151#12112yes—
202C2×C151—42no—
203C2×C151—81no—
204C2×C151C2×#12111yescharacteristic
205C2×C151#57611yescharacteristic
206C2×C151#57611yescharacteristic
207D151—22no—
208D151—42no—
209D151—81no—
210D151—81no—
211C151#304111yescharacteristic, complemented, Sylow
212#3041—1511nomaximal, Sylow
213C2×AGL(1,ℤ/8)—1511no—
214C22×D8—1511no—
215#535—1511no—
216#518—1511no—
217#463—1511no—
218#463—1511no—
219#463—1511no—
220#463D15111yescharacteristic, complemented
221#459—1511no—
222#459—1511no—
223#459—1511no—
224#459—1511no—
225#424—1511no—
226#418—1511no—
227#416—1511no—
228C2×#41—1511no—
229C22×D4—1511no—
230C22×D4—1511no—
231AGL(1,ℤ/8)—3024no—
232C2×SD16—1511no—
233C2×SD16C2×D15111yescharacteristic, complemented
234C2×D8—3024no—
235C2×D8—1511no—
236C2×D8—1511no—
237C2×D8—1511no—
238C2×D8—1511no—
239C2×D8—1511no—
240C2×D8C2×D15111yescharacteristic
241C2×M16—1511no—
242C22×C8—1511no—
243#127—1511no—
244#127—1511no—
245#124—1511no—
246#124C2×D15111yescharacteristic, complemented
247#121—3021no—
248#121—3021no—
249#120—3021no—
250#120—3021no—
251C4×D4—3021no—
252C4×D4—3021no—
253C4×D4—1511no—
254C4×D4—1511no—
255C4×D4—1511no—
256C4×D4C2×D15111yescharacteristic, complemented
257#117—1511no—
258#117—1511no—
259C2×C22⋊C4—3021no—
260C2×C22⋊C4—3021no—
261C4⋊C8—1511no—
262C4⋊C8—1511no—
263C4⋊C8—1511no—
264C4⋊C8C2×D15111yescharacteristic, complemented
265#103—1511no—
266#103C2×D15111yescharacteristic, complemented
267#102—1511no—
268#102—1511no—
269#102—1511no—
270#102—1511no—
271#102—1511no—
272#102C2×D15111yescharacteristic, complemented
273C24—3021no—
274C24—3021no—
275#41—3022no—
276#41—3022no—
277C2×Q8C22×D15111yescharacteristic
278C2×D4—3021no—
279C2×D4—3022no—
280C2×D4—3022no—
281C2×D4—3022no—
282C2×D4—3021no—
283C2×D4—3021no—
284C2×D4—3021no—
285C2×D4—3021no—
286C2×D4—3021no—
287C2×D4—3021no—
288C2×D4—1514no—
289C2×D4—1511no—
290C2×D4—1511no—
291C2×D4—1511no—
292C2×D4C22×D15111yescharacteristic
293C2×D4C22×D15111yescharacteristic
294C22×C4—21no—
295C22×C4—3021no—
296C22×C4—3021no—
297C22×C4—3021no—
298C22×C4—3021no—
299C22×C4—1511no—
300SD16—22no—
301SD16—3022no—
302D8—22no—
303D8—3022no—
304D8—3022no—
305D8—3022no—
306D8—3022no—
307D8—3022no—
308M16—3022no—
309C2×C8—3022no—
310C2×C8—1511no—
311C2×C8—1511no—
312C2×C8C22×D15111yescharacteristic
313C2×C8C22×D15111yescharacteristic
314C4⋊C4—1511no—
315C4⋊C4—1511no—
316C4⋊C4—1511no—
317C4⋊C4C22×D15111yescharacteristic
318C22⋊C4—21no—
319C22⋊C4—21no—
320C22⋊C4—21no—
321C22⋊C4—3021no—
322C22⋊C4—3021no—
323C22⋊C4—3021no—
324C22⋊C4—3021no—
325C22⋊C4—3021no—
326C22⋊C4—3021no—
327C22⋊C4—1511no—
328C22⋊C4—1511no—
329C42—3021no—
330C42—1511no—
331C42C22×D15111yescharacteristic
332C23—21no—
333C23—21no—
334C23—3024no—
335C23—3021no—
336C23—3021no—
337C23—3021no—
338C23—6042no—
339C23—1511no—
340C23—12081no—
341C23—12081no—
342C23—12081no—
343C23—12081no—
344Q8—21no—
345Q8—21no—
346D4—21no—
347D4—21no—
348D4—21no—
349D4—3024no—
350D4—3021no—
351D4—3021no—
352D4—3021no—
353D4—3021no—
354D4—3021no—
355D4—6042no—
356D4—6042no—
357D4—1512no—
358D4—12081no—
359D4—12081no—
360D4—12081no—
361D4—12081no—
362D4D4×D15112yes—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—81no—
366C2×C4—3021no—
367C2×C4—6042no—
368C2×C4—1511no—
369C2×C4—1511no—
370C2×C4—1511no—
371C2×C4—1512no—
372C2×C4—1512no—
373C2×C4—12081no—
374C2×C4—12081no—
375C2×C4—12081no—
376C2×C4D4×D15111yescharacteristic
377C2×C4D4×D15111yescharacteristic
378C2×C4C23×D15111yescharacteristic
379C8—21no—
380C8—21no—
381C8—3021no—
382C8—3021no—
383C22—22no—
384C22—41no—
385C22—41no—
386C22—41no—
387C22—41no—
388C22—41no—
389C22—3021no—
390C22—3021no—
391C22—3022no—
392C22—6041no—
393C22—6041no—
394C22—6041no—
395C22—6041no—
396C22—6041no—
397C22—6041no—
398C22—6041no—
399C22—6041no—
400C22—1512no—
401C22—12084no—
402C22—12084no—
403C22C2×D4×D15111yescharacteristic
404C4—21no—
405C4—21no—
406C4—41no—
407C4—41no—
408C4—3021no—
409C4—3021no—
410C4—6041no—
411C4—1512no—
412C4C2×D4×D15111yescharacteristic
413C4#41×D15111yescharacteristic
414C2—42no—
415C2—81no—
416C2—3022no—
417C2—6042no—
418C2—12081no—
419C2—12081no—
420C2#121×D15111yescharacteristic
421C2#2375301411yescharacteristic
422C2#2375301611yescharacteristic
subgroupG/H
centerC22C2×D4×D151
commutator, cosocleC2×C4×C151C24
FrattiniC2×C4C23×D151
Fitting#463×C151C2
socleC22×C151C22×D4
hypercenter#463D151
hypocenterC151#3041
LCSG ▷ C2×C4×C151 ▷ C2×C151 ▷ C151
UCSC1 ◁ C22 ◁ C42 ◁ #463

Complements

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

NH ≅ G/Ntype
1. C2×#18012766C2semidirect
2. C2×D8×D151C2semidirect
3. #23752780C2semidirect
4. #23752684C2semidirect
5. #23752430C2semidirect
6. #23752429C2semidirect
7. #23752424C2semidirect
8. #23752408C2semidirect
9. #23752407C2semidirect
10. #23752406C2semidirect
11. #23752405C2semidirect
12. #23752237C2semidirect
13. #23752215C2semidirect
14. #23752207C2semidirect
15. #463×C151C2semidirect
20. C2×#14558991C22semidirect
22. C2×D1208C22semidirect
25. C2×#14558989C22semidirect
29. #18012736C22semidirect
30. #18012735C22semidirect
31. #18012728C22semidirect
36. C4×D604C22semidirect
37. D4×Dic151C22semidirect
38. #18012701C22semidirect
45. C4⋊(C151⋊C8)C22semidirect
46. Dic151⋊C8C22semidirect
47. Dic151⋊C8C22semidirect
48. #18012663C22semidirect
49. #18012661C22semidirect
50. #18012661C22semidirect
51. #18012660C22semidirect
52. #18012659C22semidirect
53. #18012659C22semidirect
54. C2×SD16×C151C22semidirect
56. #124×C151C22semidirect
57. C4×D4×C151C22semidirect
58. C4⋊C8×C151C22semidirect
59. #103×C151C22semidirect
60. #102×C151C22semidirect
78. D4×D151D4semidirect
211. C151#3041semidirect
220. #463D151semidirect
233. C2×SD16C2×D151semidirect
246. #124C2×D151semidirect
256. C4×D4C2×D151semidirect
264. C4⋊C8C2×D151semidirect
266. #103C2×D151semidirect
272. #102C2×D151semidirect

Maximal subgroups

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

subgroupindex
C2×#180127662
C2×D8×D1512
#237527802
#237526842
#237524302
#237524292
#237524242
#237524082
#237524072
#237524062
#237524052
#237522372
#237522152
#237522072
#463×C1512
#3041151