Finite Group Databasebeta

← #34078330

Subgroups — #34078330

Order 19328.

764 conjugacy classes of subgroups.

The subgroup lattice has 476 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#41×D151C211yesmaximal, characteristic, complemented
2C2×#18012761C211yesmaximal, characteristic, complemented
3C2×#18012759C211yesmaximal, characteristic, complemented
4#23752724C211yesmaximal, characteristic, complemented
5#23752377C211yesmaximal, characteristic, complemented
6#23752375C211yesmaximal, characteristic, complemented
7#23752369C211yesmaximal, characteristic, complemented
8#23752364C211yesmaximal, characteristic, complemented
9#23752360C211yesmaximal, characteristic, complemented
10#23752358C211yesmaximal, characteristic, complemented
11#23752352C211yesmaximal, characteristic, complemented
12#23752347C211yesmaximal, characteristic, complemented
13#23752206C211yesmaximal, characteristic, complemented
14C2×(C2×D151)⋊C8C211yesmaximal, characteristic, complemented
15#416×C151C211yesmaximal, characteristic, complemented
16C2×#14559005C2211yescharacteristic
17C2×#14559004—21no—
18C2×#14559004C2211yescharacteristic
19C2×#14559004C2211yescharacteristic, complemented
20C2×#14559003—21no—
21C2×#14559003C2211yescharacteristic
22#41×D151—24no—
23#41×D151—24no—
24C2×Q8×D151C2211yescharacteristic
25C2×D4×D151—21no—
26C2×D4×D151C2211yescharacteristic
27C22×C4×D151—21no—
28C22×C4×D151C2211yescharacteristic
29#18012761—24no—
30#18012759—24no—
31C2×Dic604C2211yescharacteristic, complemented
32C2×#14558994C2211yescharacteristic, complemented
33C2×#14558993C2211yescharacteristic, complemented
34C2×#14558993C2211yescharacteristic, complemented
35C2×#14558992C2211yescharacteristic, complemented
36C2×#14558991C2211yescharacteristic, complemented
37C2×D1208C2211yescharacteristic, complemented
38C2×#14558989C2211yescharacteristic, complemented
39C22×C151⋊C8C2211yescharacteristic, complemented
40#18012723—21no—
41#18012719C2211yescharacteristic
42#18012719C2211yescharacteristic, complemented
43#18012714C2211yescharacteristic
44#18012714C2211yescharacteristic, complemented
45#18012709—21no—
46C4×D604—21no—
47C4×D604—21no—
48#18012662C2211yescharacteristic, complemented
49#18012662C2211yescharacteristic, complemented
50#18012659C2211yescharacteristic, complemented
51#18012659C2211yescharacteristic, complemented
52(C2×D151)⋊C8C2211yescharacteristic
53(C2×D151)⋊C8C2211yescharacteristic
54(C2×D151)⋊C8C2211yescharacteristic, complemented
55(C2×D151)⋊C8C2211yescharacteristic, complemented
56C2×#41×C151C2211yescharacteristic
57C22×C8×C151C2211yescharacteristic, complemented
58#117×C151C2211yescharacteristic, complemented
59#103×C151C2211yescharacteristic, complemented
60#103×C151C2211yescharacteristic, complemented
61#102×C151C2211yescharacteristic, complemented
62#102×C151C2211yescharacteristic, complemented
63C23×D151—21no—
64C23×D151D411yescharacteristic
65#14559005—22no—
66#14559005—22no—
67#14559004—28no—
68#14559004—22no—
69#14559004—22no—
70#14559004—22no—
71#14559004D414yescomplemented
72#14559003—28no—
73#14559003—22no—
74#14559003—22no—
75C2×Dic302—21no—
76C2×Dic302C2311yescharacteristic
77C2×Dic302C2311yescharacteristic
78Q8×D151—22no—
79Q8×D151—22no—
80C2×D604—21no—
81C2×D604—21no—
82C2×D604—21no—
83C2×D604C2311yescharacteristic
84C2×D604C2311yescharacteristic
85C2×C22⋊D151—21no—
86C2×C22⋊D151—21no—
87C2×C22⋊D151—21no—
88C2×C22⋊D151—21no—
89D4×D151—28no—
90D4×D151—22no—
91D4×D151—22no—
92C22×Dic151—21no—
93C22×Dic151D411yescharacteristic
94C2×C4×D151—28no—
95C2×C4×D151—21no—
96C2×C4×D151—21no—
97C2×C4×D151—21no—
98C2×C4×D151—22no—
99C2×C4×D151—22no—
100C2×C4×D151—21no—
101C2×C4×D151D411yescharacteristic
102C2×C4×D151D411yescharacteristic
103C2×C4×D151C2311yescharacteristic
104C2×C4×D151C2311yescharacteristic
105Dic604—22no—
106#14558994—22no—
107#14558993—22no—
108#14558993—22no—
109#14558992—22no—
110#14558991—22no—
111D1208—22no—
112#14558989—22no—
113C2×C151⋊C8—22no—
114C2×C151⋊C8C2311yescharacteristic
115C2×C151⋊C8C2311yescharacteristic
116C4⋊Dic151—21no—
117C4⋊Dic151—21no—
118(C2×D151)⋊C4—21no—
119(C2×D151)⋊C4—21no—
120(C2×D151)⋊C4—21no—
121(C2×D151)⋊C4—21no—
122#41×C151—22no—
123#41×C151D414yescomplemented
124C2×Q8×C151C2311yescharacteristic
125C2×D4×C151—21no—
126C2×D4×C151C2311yescharacteristic
127C22×C4×C151—21no—
128C22×C4×C151C2311yescharacteristic
129C2×C8×C151—22no—
130C2×C8×C151C2311yescharacteristic
131C2×C8×C151C2311yescharacteristic
132C4⋊C4×C151C2311yescharacteristic
133C4⋊C4×C151C2311yescharacteristic
134C22⋊C4×C151—21no—
135C42×C151—21no—
136C22×D151—24no—
137C22×D151—22no—
138C22×D151—21no—
139C22×D151—22no—
140C22×D151—21no—
141C22×D151—41no—
142C22×D151—42no—
143C22×D151—42no—
144C22×D151C2×D411yescharacteristic
145C22×D151C2×D411yescharacteristic
146Dic302—24no—
147Dic302—21no—
148Dic302—21no—
149Dic302—21no—
150Dic302C2×D412yes—
151D604—24no—
152D604—21no—
153D604—21no—
154D604—21no—
155D604—42no—
156D604—42no—
157D604C2×D412yes—
158C22⋊D151—24no—
159C22⋊D151—24no—
160C22⋊D151—24no—
161C22⋊D151—42no—
162C2×Dic151—24no—
163C2×Dic151—22no—
164C2×Dic151—21no—
165C2×Dic151—21no—
166C2×Dic151C2×D411yescharacteristic
167C2×Dic151C2×D411yescharacteristic
168C4×D151—24no—
169C4×D151—24no—
170C4×D151—22no—
171C4×D151—24no—
172C4×D151—22no—
173C4×D151—21no—
174C4×D151—21no—
175C4×D151—21no—
176C4×D151—21no—
177C4×D151—42no—
178C151⋊C8—21no—
179C151⋊C8—21no—
180C23×C151—21no—
181C23×C151C2×D411yescharacteristic
182Q8×C151—21no—
183Q8×C151C2×D412yes—
184D4×C151—24no—
185D4×C151—21no—
186D4×C151C2×D412yes—
187C2×C4×C151—21no—
188C2×C4×C151—24no—
189C2×C4×C151—21no—
190C2×C4×C151—21no—
191C2×C4×C151C2×D412yes—
192C2×C4×C151C2×D411yescharacteristic
193C2×C4×C151C2×D412yes—
194C2×C4×C151C2411yescharacteristic
195C8×C151—21no—
196C8×C151—21no—
197C22×C151—21no—
198C22×C151—21no—
199C22×C151—22no—
200C22×C151—41no—
201C22×C151—41no—
202C22×C151#12012yes—
203C22×C151C22×D411yescharacteristic
204C2×D151—21no—
205C2×D151—21no—
206C2×D151—21no—
207C2×D151—22no—
208C2×D151—21no—
209C2×D151—21no—
210C2×D151—44no—
211C2×D151—48no—
212C2×D151—41no—
213C2×D151—41no—
214C2×D151—41no—
215C2×D151—41no—
216C2×D151—41no—
217C2×D151#13512yescomplemented
218C4×C151—22no—
219C4×C151—41no—
220C4×C151—41no—
221C4×C151#12012yes—
222C4×C151C22×D411yescharacteristic
223C4×C151C22×D411yescharacteristic
224Dic151—21no—
225Dic151—22no—
226Dic151—21no—
227Dic151—41no—
228C2×C151—22no—
229C2×C151—42no—
230C2×C151C2×#12011yescharacteristic
231C2×C151C2×#13511yescharacteristic
232C2×C151#57511yescharacteristic
233D151—22no—
234D151—41no—
235D151—42no—
236D151—81no—
237C151#300211yescharacteristic, complemented, Sylow
238#3002—1511nomaximal, Sylow
239C22×#41—1511no—
240C2×#135—1511no—
241C2×#135—1511no—
242#525—1511no—
243#452—1511no—
244#452—1511no—
245#451—1511no—
246#450—1511no—
247#449—1511no—
248#449—1511no—
249#448—1511no—
250#447—1511no—
251#416—1511no—
252#416D15111yescharacteristic, complemented
253C2×C22⋊C8—1511no—
254C2×#41—3021no—
255C2×#41—3024no—
256C2×#41—3024no—
257C2×#41—3021no—
258C2×#41—1511no—
259C2×#41—1511no—
260C2×#41—1511no—
261C2×#41—1511no—
262C2×#41C2×D15111yescharacteristic
263C22×Q8—1511no—
264C22×D4—3021no—
265C22×D4—1511no—
266C23×C4—3021no—
267C23×C4—1511no—
268#135—3024no—
269#135—3024no—
270C2×Q16—1511no—
271C2×Q16—1511no—
272C2×SD16—1511no—
273C2×SD16—1511no—
274C2×SD16—1511no—
275C2×SD16—1511no—
276C2×D8—1511no—
277C2×D8—1511no—
278C22×C8—1511no—
279C22×C8C2×D15111yescharacteristic, complemented
280#123—3021no—
281C22⋊Q8—1511no—
282C22⋊Q8—1511no—
283#121—1511no—
284#121—1511no—
285#120—3021no—
286C4×D4—3021no—
287C4×D4—3021no—
288#117C2×D15111yescharacteristic, complemented
289#103—1511no—
290#103—1511no—
291#103C2×D15111yescharacteristic, complemented
292#103C2×D15111yescharacteristic, complemented
293#102—1511no—
294#102—1511no—
295#102C2×D15111yescharacteristic, complemented
296#102C2×D15111yescharacteristic, complemented
297C22⋊C8—1511no—
298C22⋊C8—1511no—
299C22⋊C8—1511no—
300C22⋊C8—1511no—
301C24—3021no—
302C24—1511no—
303#41—22no—
304#41—3028no—
305#41—3028no—
306#41—3022no—
307#41—3022no—
308#41—3022no—
309#41—3022no—
310#41—3022no—
311#41—3022no—
312#41—3022no—
313#41—1514no—
314#41D60414yescomplemented
315C2×Q8—3021no—
316C2×Q8—3022no—
317C2×Q8—3022no—
318C2×Q8—1511no—
319C2×Q8—1511no—
320C2×Q8C22×D15111yescharacteristic
321C2×D4—21no—
322C2×D4—3021no—
323C2×D4—3021no—
324C2×D4—3021no—
325C2×D4—3028no—
326C2×D4—3022no—
327C2×D4—3021no—
328C2×D4—3021no—
329C2×D4—3022no—
330C2×D4—3021no—
331C2×D4—3021no—
332C2×D4—1511no—
333C2×D4—1511no—
334C2×D4C22×D15111yescharacteristic
335C22×C4—21no—
336C22×C4—3028no—
337C22×C4—3021no—
338C22×C4—3021no—
339C22×C4—3021no—
340C22×C4—3022no—
341C22×C4—3021no—
342C22×C4—3022no—
343C22×C4—3021no—
344C22×C4—1511no—
345C22×C4—1511no—
346C22×C4—1511no—
347C22×C4—1511no—
348C22×C4—1511no—
349C22×C4C22×D15111yescharacteristic
350Q16—3022no—
351Q16—3022no—
352SD16—3022no—
353SD16—3022no—
354SD16—3022no—
355SD16—3022no—
356D8—3022no—
357D8—3022no—
358C2×C8—22no—
359C2×C8—3022no—
360C2×C8—1511no—
361C2×C8—1511no—
362C2×C8C22×D15111yescharacteristic
363C2×C8C22×D15111yescharacteristic
364C4⋊C4—3021no—
365C4⋊C4—3021no—
366C4⋊C4C22×D15111yescharacteristic
367C4⋊C4C22×D15111yescharacteristic
368C22⋊C4—21no—
369C22⋊C4—3021no—
370C22⋊C4—3021no—
371C22⋊C4—3021no—
372C22⋊C4—3021no—
373C42—21no—
374C23—21no—
375C23—3024no—
376C23—3022no—
377C23—3022no—
378C23—3021no—
379C23—3021no—
380C23—6041no—
381C23—6042no—
382C23—6042no—
383C23—1511no—
384C23—1511no—
385C23D4×D15111yescharacteristic
386Q8—21no—
387Q8—3024no—
388Q8—3021no—
389Q8—3021no—
390Q8—3021no—
391Q8—1512no—
392Q8C2×D60412yes—
393D4—24no—
394D4—21no—
395D4—3024no—
396D4—3024no—
397D4—3024no—
398D4—3024no—
399D4—3021no—
400D4—3021no—
401D4—3021no—
402D4—6042no—
403D4—6042no—
404D4—6042no—
405D4—1512no—
406D4C2×D60412yes—
407C2×C4—21no—
408C2×C4—24no—
409C2×C4—21no—
410C2×C4—21no—
411C2×C4—3024no—
412C2×C4—3024no—
413C2×C4—3024no—
414C2×C4—3022no—
415C2×C4—3021no—
416C2×C4—3022no—
417C2×C4—3024no—
418C2×C4—3022no—
419C2×C4—3021no—
420C2×C4—3021no—
421C2×C4—3021no—
422C2×C4—3021no—
423C2×C4—3021no—
424C2×C4—6042no—
425C2×C4—1511no—
426C2×C4—1511no—
427C2×C4D4×D15112yes—
428C2×C4D4×D15111yescharacteristic
429C2×C4C2×D60412yes—
430C2×C4C23×D15111yescharacteristic
431C8—21no—
432C8—21no—
433C8—3021no—
434C8—3021no—
435C22—21no—
436C22—21no—
437C22—22no—
438C22—41no—
439C22—41no—
440C22—3021no—
441C22—3021no—
442C22—3021no—
443C22—3021no—
444C22—3021no—
445C22—3022no—
446C22—6044no—
447C22—6048no—
448C22—6041no—
449C22—6041no—
450C22—6041no—
451C22—6041no—
452C22—6041no—
453C22—1512no—
454C22#1801270912yes—
455C22C2×D4×D15111yescharacteristic
456C4—22no—
457C4—41no—
458C4—41no—
459C4—3021no—
460C4—3022no—
461C4—3021no—
462C4—6041no—
463C4#1801270912yes—
464C4C2×D4×D15111yescharacteristic
465C4C22×D60411yescharacteristic
466C2—22no—
467C2—42no—
468C2—3022no—
469C2—6041no—
470C2—6042no—
471C2—12081no—
472C2C2×#1801270911yescharacteristic
473C2#135×D15111yescharacteristic
474C2#2375300911yescharacteristic
subgroupG/H
centerC2×C4D4×D151
commutator, cosocleC2×C4×C151C24
FrattiniC2×C4C23×D151
Fitting#416×C151C2
socleC22×C151C22×D4
hypercenter#416D151
hypocenterC151#3002
LCSG ▷ C2×C4×C151 ▷ C2×C151 ▷ C151
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ #416

Complements

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

NH ≅ G/Ntype
1. C2×#41×D151C2semidirect
2. C2×#18012761C2semidirect
3. C2×#18012759C2semidirect
4. #23752724C2semidirect
5. #23752377C2semidirect
6. #23752375C2semidirect
7. #23752369C2semidirect
8. #23752364C2semidirect
9. #23752360C2semidirect
10. #23752358C2semidirect
11. #23752352C2semidirect
12. #23752347C2semidirect
13. #23752206C2semidirect
14. C2×(C2×D151)⋊C8C2semidirect
15. #416×C151C2semidirect
19. C2×#14559004C22semidirect
31. C2×Dic604C22semidirect
32. C2×#14558994C22semidirect
33. C2×#14558993C22semidirect
34. C2×#14558993C22semidirect
35. C2×#14558992C22semidirect
36. C2×#14558991C22semidirect
37. C2×D1208C22semidirect
38. C2×#14558989C22semidirect
39. C22×C151⋊C8C22semidirect
42. #18012719C22semidirect
44. #18012714C22semidirect
48. #18012662C22semidirect
49. #18012662C22semidirect
50. #18012659C22semidirect
51. #18012659C22semidirect
54. (C2×D151)⋊C8C22semidirect
55. (C2×D151)⋊C8C22semidirect
57. C22×C8×C151C22semidirect
58. #117×C151C22semidirect
59. #103×C151C22semidirect
60. #103×C151C22semidirect
61. #102×C151C22semidirect
62. #102×C151C22semidirect
71. #14559004D4semidirect
123. #41×C151D4semidirect
217. C2×D151#135semidirect
237. C151#3002semidirect
252. #416D151semidirect
279. C22×C8C2×D151semidirect
288. #117C2×D151semidirect
291. #103C2×D151semidirect
292. #103C2×D151semidirect
295. #102C2×D151semidirect
296. #102C2×D151semidirect
314. #41D604semidirect

Maximal subgroups

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

subgroupindex
C2×#41×D1512
C2×#180127612
C2×#180127592
#237527242
#237523772
#237523752
#237523692
#237523642
#237523602
#237523582
#237523522
#237523472
#237522062
C2×(C2×D151)⋊C82
#416×C1512
#3002151