Finite Group Databasebeta

← #34078384

Subgroups — #34078384

Order 19328.

752 conjugacy classes of subgroups.

The subgroup lattice has 478 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#18012785C211yesmaximal, characteristic
2C2×#18012771C211yesmaximal, characteristic, complemented
3C2×SD16×D151C211yesmaximal, characteristic, complemented
4#23752785C211yesmaximal, characteristic, complemented
5#23752377C211yesmaximal, characteristic, complemented
6#23752374C211yesmaximal, characteristic, complemented
7#23752373C211yesmaximal, characteristic, complemented
8#23752365C211yesmaximal, characteristic, complemented
9#23752364C211yesmaximal, characteristic, complemented
10#23752362C211yesmaximal, characteristic, complemented
11#23752361C211yesmaximal, characteristic, complemented
12#23752215C211yesmaximal, characteristic, complemented
13#23752207C211yesmaximal, characteristic, complemented
14#23752177C211yesmaximal, characteristic, complemented
15#452×C151C211yesmaximal, characteristic, complemented
16#18012785—24no—
17#18012785—24no—
18C2×#14559005C2211yescharacteristic
19C2×#14559005C2211yescharacteristic
20C2×#14559004—21no—
21C2×#14559004C2211yescharacteristic
22C2×Q8×D151C2211yescharacteristic, complemented
23C22×D604—21no—
24C22×D604C2211yescharacteristic
25C2×D4×D151—21no—
26C2×D4×D151—21no—
27C2×D4×D151C2211yescharacteristic
28C2×D4×D151C2211yescharacteristic
29#18012771—24no—
30C2×#14558994C2211yescharacteristic, complemented
31C2×#14558993C2211yescharacteristic, complemented
32C2×#14558993C2211yescharacteristic, complemented
33C2×#14558992C2211yescharacteristic, complemented
34C2×#14558991C2211yescharacteristic, complemented
35C2×#14558991C2211yescharacteristic, complemented
36SD16×D151—24no—
37C2×#14558987C2211yescharacteristic, complemented
38C2×C8×D151C2211yescharacteristic, complemented
39#18012738—21no—
40#18012725—21no—
41#18012724—21no—
42#18012721—21no—
43#18012719C2211yescharacteristic, complemented
44#18012718C2211yescharacteristic, complemented
45#18012715C2211yescharacteristic, complemented
46#18012697C2211yescharacteristic, complemented
47#18012663C2211yescharacteristic, complemented
48#18012661C2211yescharacteristic, complemented
49#18012661C2211yescharacteristic, complemented
50#18012660C2211yescharacteristic, complemented
51#18012659C2211yescharacteristic, complemented
52#18012659C2211yescharacteristic, complemented
53C22⋊(C151⋊C8)C2211yescharacteristic, complemented
54(C2×D151)⋊C8C2211yescharacteristic, complemented
55(C2×D151)⋊C8C2211yescharacteristic, complemented
56C2×#41×C151C2211yescharacteristic
57C2×Q16×C151C2211yescharacteristic, complemented
58C2×SD16×C151C2211yescharacteristic, complemented
59C22⋊Q8×C151C2211yescharacteristic, complemented
60#103×C151C2211yescharacteristic, complemented
61#102×C151C2211yescharacteristic, complemented
62C22⋊C8×C151C2211yescharacteristic, complemented
63C23×D151—21no—
64C23×D151—21no—
65C23×D151—21no—
66#14559005—22no—
67#14559005—22no—
68#14559005—22no—
69#14559005D414yes—
70#14559004—28no—
71#14559004—22no—
72#14559004—22no—
73C2×Dic302—21no—
74C2×Dic302D411yescharacteristic
75C2×Dic302C2311yescharacteristic
76Q8×D151—22no—
77Q8×D151—22no—
78C2×D604—28no—
79C2×D604—21no—
80C2×D604—21no—
81C2×D604—22no—
82C2×D604—22no—
83C2×D604—21no—
84C2×D604D411yescharacteristic
85C2×D604C2311yescharacteristic
86C2×D604C2311yescharacteristic
87C2×C22⋊D151—21no—
88C2×C22⋊D151—21no—
89C2×C22⋊D151D411yescharacteristic
90C2×C22⋊D151D411yescharacteristic
91D4×D151—28no—
92D4×D151—28no—
93D4×D151—22no—
94D4×D151—22no—
95D4×D151—22no—
96D4×D151D414yes—
97C2×C4×D151—21no—
98C2×C4×D151—21no—
99C2×C4×D151—21no—
100C2×C4×D151C2311yescharacteristic
101C2×C4×D151C2311yescharacteristic
102#14558994—22no—
103#14558993—22no—
104#14558993—22no—
105#14558992—22no—
106#14558991—22no—
107#14558991—22no—
108#14558987—22no—
109C2×C151⋊C8C2311yescharacteristic
110C2×C151⋊C8C2311yescharacteristic
111C8×D151—22no—
112C4⋊Dic151—21no—
113C4⋊Dic151C2311yescharacteristic
114Dic151⋊C4—21no—
115Dic151⋊C4—21no—
116C22⋊Dic151—21no—
117(C2×D151)⋊C4—21no—
118(C2×D151)⋊C4—21no—
119(C2×D151)⋊C4—21no—
120C4×Dic151—21no—
121#41×C151—22no—
122#41×C151—22no—
123C2×Q8×C151C2311yescharacteristic
124C2×Q8×C151C2311yescharacteristic
125C2×D4×C151—21no—
126C2×D4×C151C2311yescharacteristic
127C22×C4×C151—21no—
128C22×C4×C151C2311yescharacteristic
129Q16×C151—22no—
130SD16×C151—22no—
131C2×C8×C151C2311yescharacteristic
132C2×C8×C151C2311yescharacteristic
133C4⋊C4×C151—21no—
134C4⋊C4×C151C2311yescharacteristic
135C22⋊C4×C151—21no—
136C22×D151—21no—
137C22×D151—21no—
138C22×D151—24no—
139C22×D151—24no—
140C22×D151—24no—
141C22×D151—81no—
142C22×D151—81no—
143C22×D151—81no—
144C22×D151—81no—
145C22×D151—81no—
146C22×D151—81no—
147C22×D151C2×D411yescharacteristic
148C22×D151C2×D411yescharacteristic
149Dic302—22no—
150Dic302—21no—
151Dic302—21no—
152Dic302—42no—
153D604—24no—
154D604—24no—
155D604—24no—
156D604—22no—
157D604—21no—
158D604—21no—
159D604C2×D412yes—
160D604C2×D412yes—
161C22⋊D151—24no—
162C22⋊D151—22no—
163C22⋊D151—22no—
164C22⋊D151—24no—
165C2×Dic151—21no—
166C2×Dic151—21no—
167C2×Dic151C2×D411yescharacteristic
168C2×Dic151C2×D411yescharacteristic
169C4×D151—24no—
170C4×D151—21no—
171C4×D151—21no—
172C4×D151—24no—
173C4×D151—42no—
174C4×D151C2×D412yes—
175C4×D151C2×D412yes—
176C151⋊C8—21no—
177C151⋊C8—21no—
178C23×C151—21no—
179C23×C151C2×D411yescharacteristic
180Q8×C151—21no—
181Q8×C151—21no—
182Q8×C151—21no—
183Q8×C151C2×D412yes—
184D4×C151—24no—
185D4×C151—21no—
186D4×C151C2×D412yes—
187C2×C4×C151—24no—
188C2×C4×C151—21no—
189C2×C4×C151—21no—
190C2×C4×C151—21no—
191C2×C4×C151—21no—
192C2×C4×C151—21no—
193C2×C4×C151C2×D411yescharacteristic
194C2×C4×C151C2411yescharacteristic
195C8×C151—21no—
196C8×C151—21no—
197C22×C151—21no—
198C22×C151—22no—
199C22×C151—21no—
200C22×C151—21no—
201C22×C151—41no—
202C22×C151—41no—
203C22×C151C22×D411yescharacteristic
204C2×D151—21no—
205C2×D151—21no—
206C2×D151—21no—
207C2×D151—22no—
208C2×D151—22no—
209C2×D151—21no—
210C2×D151—21no—
211C2×D151—41no—
212C2×D151—41no—
213C2×D151—41no—
214C2×D151—41no—
215C2×D151—84no—
216C2×D151—84no—
217C2×D151—84no—
218C2×D151#12012yes—
219C4×C151—21no—
220C4×C151—22no—
221C4×C151—41no—
222C4×C151—41no—
223C4×C151C22×D411yescharacteristic
224C4×C151C22×D411yescharacteristic
225Dic151—21no—
226Dic151—41no—
227Dic151—41no—
228Dic151#12012yes—
229C2×C151—41no—
230C2×C151—42no—
231C2×C151C2×#12011yescharacteristic
232C2×C151#57711yescharacteristic
233C2×C151#57711yescharacteristic
234D151—22no—
235D151—41no—
236D151—42no—
237D151—42no—
238C151#300911yescharacteristic, complemented, Sylow
239#3009—1511nomaximal, Sylow
240C2×H(5,𝔽2)—1511no—
241C2×#137—1511no—
242C22×SD16—1511no—
243#536—1511no—
244#452—1511no—
245#452—1511no—
246#452—1511no—
247#452D15111yescharacteristic, complemented
248#450—1511no—
249#450—1511no—
250#450—1511no—
251#450—1511no—
252#418—1511no—
253#416—1511no—
254#408—1511no—
255H(5,𝔽2)—3024no—
256H(5,𝔽2)—3024no—
257C2×#41—3021no—
258C2×#41—1511no—
259C2×#41—1511no—
260C2×#41—1511no—
261C2×#41C2×D15111yescharacteristic
262C22×Q8—1511no—
263C22×D4—3021no—
264C22×D4—3021no—
265C22×D4—3021no—
266C22×D4—1511no—
267C22×D4—1511no—
268C22×D4—1511no—
269#137—3024no—
270C2×Q16—1511no—
271C2×Q16C2×D15111yescharacteristic, complemented
272C2×SD16—3024no—
273C2×SD16—1511no—
274C2×SD16—1511no—
275C2×SD16—1511no—
276C2×SD16—1511no—
277C2×SD16—1511no—
278C2×SD16C2×D15111yescharacteristic, complemented
279C2×M16—1511no—
280C22×C8—1511no—
281C4⋊Q8—3021no—
282#124—3021no—
283#123—3021no—
284#123—3021no—
285C22⋊Q8—1511no—
286C22⋊Q8—1511no—
287C22⋊Q8—1511no—
288C22⋊Q8C2×D15111yescharacteristic, complemented
289#117—1511no—
290#103—1511no—
291#103C2×D15111yescharacteristic, complemented
292#102—1511no—
293#102—1511no—
294#102—1511no—
295#102—1511no—
296#102—1511no—
297#102C2×D15111yescharacteristic, complemented
298C22⋊C8—1511no—
299C22⋊C8—1511no—
300C22⋊C8—1511no—
301C22⋊C8C2×D15111yescharacteristic, complemented
302C24—3021no—
303C24—3021no—
304C24—3021no—
305#41—22no—
306#41—22no—
307#41—3028no—
308#41—3022no—
309#41—3022no—
310#41—3022no—
311#41—3022no—
312#41—3022no—
313#41—1514no—
314C2×Q8—3021no—
315C2×Q8—3022no—
316C2×Q8—3022no—
317C2×Q8—1511no—
318C2×Q8—1511no—
319C2×Q8C22×D15111yescharacteristic
320C2×Q8C22×D15111yescharacteristic
321C2×D4—21no—
322C2×D4—3028no—
323C2×D4—3021no—
324C2×D4—3021no—
325C2×D4—3022no—
326C2×D4—3022no—
327C2×D4—3028no—
328C2×D4—3028no—
329C2×D4—3022no—
330C2×D4—3022no—
331C2×D4—3022no—
332C2×D4—3021no—
333C2×D4—3021no—
334C2×D4—3021no—
335C2×D4—1511no—
336C2×D4—1511no—
337C2×D4—1514no—
338C2×D4—1511no—
339C2×D4—1511no—
340C2×D4—1511no—
341C2×D4C22×D15111yescharacteristic
342C22×C4—21no—
343C22×C4—3021no—
344C22×C4—3021no—
345C22×C4—3021no—
346C22×C4—1511no—
347C22×C4—1511no—
348C22×C4C22×D15111yescharacteristic
349Q16—22no—
350Q16—3022no—
351SD16—22no—
352SD16—3022no—
353SD16—3022no—
354SD16—3022no—
355SD16—3022no—
356SD16—3022no—
357M16—3022no—
358C2×C8—3022no—
359C2×C8—1511no—
360C2×C8—1511no—
361C2×C8C22×D15111yescharacteristic
362C2×C8C22×D15111yescharacteristic
363C4⋊C4—21no—
364C4⋊C4—3021no—
365C4⋊C4—3021no—
366C4⋊C4—3021no—
367C4⋊C4—1511no—
368C4⋊C4C22×D15111yescharacteristic
369C22⋊C4—21no—
370C22⋊C4—3021no—
371C22⋊C4—3021no—
372C22⋊C4—3021no—
373C22⋊C4—3021no—
374C42—3021no—
375C23—21no—
376C23—3024no—
377C23—3024no—
378C23—3024no—
379C23—3021no—
380C23—3021no—
381C23—1511no—
382C23—1511no—
383C23—12081no—
384C23—12081no—
385C23—12081no—
386C23—12081no—
387C23—12081no—
388C23—12081no—
389C23D4×D15111yescharacteristic
390Q8—21no—
391Q8—21no—
392Q8—21no—
393Q8—3022no—
394Q8—3021no—
395Q8—3021no—
396Q8—6042no—
397Q8D4×D15112yes—
398D4—24no—
399D4—21no—
400D4—3024no—
401D4—3024no—
402D4—3024no—
403D4—3022no—
404D4—3024no—
405D4—3022no—
406D4—3022no—
407D4—3024no—
408D4—3021no—
409D4—3021no—
410D4—1512no—
411D4—1512no—
412D4D4×D15112yes—
413C2×C4—24no—
414C2×C4—21no—
415C2×C4—21no—
416C2×C4—21no—
417C2×C4—21no—
418C2×C4—21no—
419C2×C4—3024no—
420C2×C4—3024no—
421C2×C4—3021no—
422C2×C4—3021no—
423C2×C4—3021no—
424C2×C4—3021no—
425C2×C4—6042no—
426C2×C4—1511no—
427C2×C4—1511no—
428C2×C4—1512no—
429C2×C4—1512no—
430C2×C4D4×D15111yescharacteristic
431C2×C4C23×D15111yescharacteristic
432C8—21no—
433C8—21no—
434C8—3021no—
435C8—3021no—
436C22—21no—
437C22—21no—
438C22—21no—
439C22—22no—
440C22—41no—
441C22—41no—
442C22—3021no—
443C22—3021no—
444C22—3021no—
445C22—3021no—
446C22—3021no—
447C22—3022no—
448C22—3022no—
449C22—6041no—
450C22—6041no—
451C22—6041no—
452C22—6041no—
453C22—1512no—
454C22—12084no—
455C22—12084no—
456C22—12084no—
457C22C2×D4×D15111yescharacteristic
458C4—21no—
459C4—22no—
460C4—41no—
461C4—41no—
462C4—3021no—
463C4—6041no—
464C4—6041no—
465C4—1512no—
466C4C2×D4×D15111yescharacteristic
467C4C2×D4×D15111yescharacteristic
468C2—41no—
469C2—42no—
470C2—3022no—
471C2—6041no—
472C2—6042no—
473C2—6042no—
474C2#120×D15111yescharacteristic
475C2#2375301911yescharacteristic
476C2#2375302111yescharacteristic
subgroupG/H
centerC22C2×D4×D151
commutator, cosocleC2×C4×C151C24
FrattiniC2×C4C23×D151
Fitting#452×C151C2
socleC22×C151C22×D4
prime radicalC2×#18012785C2
hypercenter#452D151
hypocenterC151#3009
LCSG ▷ C2×C4×C151 ▷ C2×C151 ▷ C151
UCSC1 ◁ C22 ◁ C22×C4 ◁ #452

Complements

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

NH ≅ G/Ntype
2. C2×#18012771C2semidirect
3. C2×SD16×D151C2semidirect
4. #23752785C2semidirect
5. #23752377C2semidirect
6. #23752374C2semidirect
7. #23752373C2semidirect
8. #23752365C2semidirect
9. #23752364C2semidirect
10. #23752362C2semidirect
11. #23752361C2semidirect
12. #23752215C2semidirect
13. #23752207C2semidirect
14. #23752177C2semidirect
15. #452×C151C2semidirect
22. C2×Q8×D151C22semidirect
30. C2×#14558994C22semidirect
31. C2×#14558993C22semidirect
32. C2×#14558993C22semidirect
33. C2×#14558992C22semidirect
34. C2×#14558991C22semidirect
35. C2×#14558991C22semidirect
37. C2×#14558987C22semidirect
38. C2×C8×D151C22semidirect
43. #18012719C22semidirect
44. #18012718C22semidirect
45. #18012715C22semidirect
46. #18012697C22semidirect
47. #18012663C22semidirect
48. #18012661C22semidirect
49. #18012661C22semidirect
50. #18012660C22semidirect
51. #18012659C22semidirect
52. #18012659C22semidirect
53. C22⋊(C151⋊C8)C22semidirect
54. (C2×D151)⋊C8C22semidirect
55. (C2×D151)⋊C8C22semidirect
57. C2×Q16×C151C22semidirect
58. C2×SD16×C151C22semidirect
59. C22⋊Q8×C151C22semidirect
60. #103×C151C22semidirect
61. #102×C151C22semidirect
62. C22⋊C8×C151C22semidirect
238. C151#3009semidirect
247. #452D151semidirect
271. C2×Q16C2×D151semidirect
278. C2×SD16C2×D151semidirect
288. C22⋊Q8C2×D151semidirect
291. #103C2×D151semidirect
297. #102C2×D151semidirect
301. C22⋊C8C2×D151semidirect

Maximal subgroups

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

subgroupindex
C2×#180127852
C2×#180127712
C2×SD16×D1512
#237527852
#237523772
#237523742
#237523732
#237523652
#237523642
#237523622
#237523612
#237522152
#237522072
#237521772
#452×C1512
#3009151