Finite Group Databasebeta

← #34053948

Subgroups — #34053948

Order 19072.

754 conjugacy classes of subgroups.

The subgroup lattice has 468 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#41×D149C211yesmaximal, characteristic
2C2×#18011943C211yesmaximal, characteristic, complemented
3C2×#18011940C211yesmaximal, characteristic, complemented
4C22⋊Q8×D149C211yesmaximal, characteristic, complemented
5#23749789C211yesmaximal, characteristic, complemented
6#23749787C211yesmaximal, characteristic, complemented
7#23749785C211yesmaximal, characteristic, complemented
8#23749782C211yesmaximal, characteristic, complemented
9#23749779C211yesmaximal, characteristic, complemented
10#23749777C211yesmaximal, characteristic, complemented
11#23749774C211yesmaximal, characteristic, complemented
12#23749625C211yesmaximal, characteristic, complemented
13#23749623C211yesmaximal, characteristic, complemented
14#23749587C211yesmaximal, characteristic, complemented
15#452×C149C211yesmaximal, characteristic, complemented
16C2×#14558737C2211yescharacteristic
17C2×#14558736—21no—
18C2×#14558736C2211yescharacteristic
19C2×#14558735—21no—
20C2×#14558735C2211yescharacteristic
21#41×D149—24no—
22#41×D149—24no—
23C2×Q8×D149C2211yescharacteristic
24C2×Q8×D149C2211yescharacteristic, complemented
25C2×D4×D149—21no—
26C2×D4×D149C2211yescharacteristic
27C22×C4×D149—21no—
28C22×C4×D149C2211yescharacteristic
29#18011943—24no—
30#18011940—24no—
31C2×Dic596C2211yescharacteristic, complemented
32C2×#14558726C2211yescharacteristic, complemented
33C2×#14558726C2211yescharacteristic, complemented
34C2×#14558725C2211yescharacteristic, complemented
35C2×#14558724C2211yescharacteristic, complemented
36C2×#14558723C2211yescharacteristic, complemented
37C2×#14558719C2211yescharacteristic, complemented
38C2×#14558719C2211yescharacteristic, complemented
39#18011891C2211yescharacteristic
40#18011890C2211yescharacteristic, complemented
41#18011889—21no—
42#18011888—21no—
43#18011887C2211yescharacteristic
44C4⋊C4×D149—21no—
45C4⋊C4×D149C2211yescharacteristic, complemented
46C22⋊C4×D149—21no—
47#18011836C2211yescharacteristic, complemented
48#18011835C2211yescharacteristic, complemented
49#18011834C2211yescharacteristic, complemented
50#18011833C2211yescharacteristic, complemented
51#18011832C2211yescharacteristic, complemented
52#18011831C2211yescharacteristic, complemented
53C22⋊#11924746C2211yescharacteristic, complemented
54#18011823C2211yescharacteristic
55#18011823C2211yescharacteristic
56C2×#41×C149C2211yescharacteristic
57C2×Q16×C149C2211yescharacteristic, complemented
58C2×SD16×C149C2211yescharacteristic, complemented
59C22⋊Q8×C149C2211yescharacteristic, complemented
60#103×C149C2211yescharacteristic, complemented
61#102×C149C2211yescharacteristic, complemented
62C22⋊C8×C149C2211yescharacteristic, complemented
63C23×D149—21no—
64C23×D149D411yescharacteristic
65#14558737—22no—
66#14558737D414yes—
67#14558736—28no—
68#14558736—22no—
69#14558736—22no—
70#14558735—28no—
71#14558735—22no—
72#14558735D414yes—
73C2×Dic298—21no—
74C2×Dic298—21no—
75C2×Dic298C2311yescharacteristic
76C2×Dic298C2311yescharacteristic
77Q8×D149—22no—
78Q8×D149—22no—
79Q8×D149—22no—
80Q8×D149—22no—
81C2×D596—21no—
82C2×D596C2311yescharacteristic
83C2×C22⋊D149—21no—
84C2×C22⋊D149—21no—
85C2×C22⋊D149—21no—
86D4×D149—28no—
87D4×D149—22no—
88D4×D149—22no—
89C22×Dic149—21no—
90C22×Dic149D411yescharacteristic
91C2×C4×D149—28no—
92C2×C4×D149—21no—
93C2×C4×D149—21no—
94C2×C4×D149—22no—
95C2×C4×D149—22no—
96C2×C4×D149—21no—
97C2×C4×D149—21no—
98C2×C4×D149—21no—
99C2×C4×D149D411yescharacteristic
100C2×C4×D149D411yescharacteristic
101C2×C4×D149C2311yescharacteristic
102C2×C4×D149C2311yescharacteristic
103Dic596—22no—
104#14558726—22no—
105#14558726—22no—
106#14558725—22no—
107#14558724—22no—
108#14558723—22no—
109#14558719—22no—
110#14558719—22no—
111C2×#11924746C2311yescharacteristic
112C2×#11924746C2311yescharacteristic
113C4⋊Dic149—21no—
114C4⋊Dic149C2311yescharacteristic
115#14558715—21no—
116#14558715—21no—
117#14558715—21no—
118C22⋊Dic149—21no—
119#14558713—21no—
120#14558713—21no—
121#41×C149—22no—
122#41×C149—22no—
123C2×Q8×C149C2311yescharacteristic
124C2×Q8×C149C2311yescharacteristic
125C2×D4×C149—21no—
126C2×D4×C149C2311yescharacteristic
127C22×C4×C149—21no—
128C22×C4×C149C2311yescharacteristic
129Q16×C149—22no—
130SD16×C149—22no—
131C2×C8×C149C2311yescharacteristic
132C2×C8×C149C2311yescharacteristic
133C4⋊C4×C149—21no—
134C4⋊C4×C149C2311yescharacteristic
135C22⋊C4×C149—21no—
136C22×D149—24no—
137C22×D149—21no—
138C22×D149—22no—
139C22×D149—22no—
140C22×D149—41no—
141C22×D149—42no—
142C22×D149—42no—
143C22×D149C2×D411yescharacteristic
144C22×D149C2×D411yescharacteristic
145Dic298—24no—
146Dic298—21no—
147Dic298—21no—
148Dic298—21no—
149Dic298—42no—
150Dic298C2×D412yes—
151D596—24no—
152D596—21no—
153D596C2×D412yes—
154C22⋊D149—24no—
155C22⋊D149—24no—
156C22⋊D149—24no—
157C2×Dic149—21no—
158C2×Dic149—24no—
159C2×Dic149—22no—
160C2×Dic149—21no—
161C2×Dic149—21no—
162C2×Dic149C2×D411yescharacteristic
163C2×Dic149C2×D411yescharacteristic
164C4×D149—24no—
165C4×D149—24no—
166C4×D149—22no—
167C4×D149—21no—
168C4×D149—21no—
169C4×D149—22no—
170C4×D149—24no—
171C4×D149—42no—
172C4×D149—42no—
173C4×D149C2×D412yes—
174C4×D149C2×D412yes—
175#11924746—21no—
176#11924746—21no—
177C23×C149—21no—
178C23×C149C2×D411yescharacteristic
179Q8×C149—21no—
180Q8×C149—21no—
181Q8×C149—21no—
182Q8×C149C2×D412yes—
183D4×C149—24no—
184D4×C149—21no—
185D4×C149C2×D412yes—
186C2×C4×C149—24no—
187C2×C4×C149—21no—
188C2×C4×C149—21no—
189C2×C4×C149—21no—
190C2×C4×C149—21no—
191C2×C4×C149—21no—
192C2×C4×C149C2×D411yescharacteristic
193C2×C4×C149C2411yescharacteristic
194C8×C149—21no—
195C8×C149—21no—
196C22×C149—21no—
197C22×C149—22no—
198C22×C149—21no—
199C22×C149—21no—
200C22×C149—41no—
201C22×C149—41no—
202C22×C149C22×D411yescharacteristic
203C2×D149—21no—
204C2×D149—21no—
205C2×D149—21no—
206C2×D149—21no—
207C2×D149—22no—
208C2×D149—44no—
209C2×D149—48no—
210C2×D149—41no—
211C2×D149—41no—
212C2×D149#12012yes—
213C2×D149#13712yescomplemented
214C4×C149—21no—
215C4×C149—22no—
216C4×C149—41no—
217C4×C149—41no—
218C4×C149C22×D411yescharacteristic
219C4×C149C22×D411yescharacteristic
220Dic149—22no—
221Dic149—21no—
222Dic149—41no—
223Dic149—41no—
224Dic149#12012yes—
225C2×C149—41no—
226C2×C149—42no—
227C2×C149C2×#12011yescharacteristic
228C2×C149C2×#13711yescharacteristic
229C2×C149#57511yescharacteristic
230D149—22no—
231D149—22no—
232D149—42no—
233C149#300711yescharacteristic, complemented, Sylow
234#3007—1491nomaximal, Sylow
235C22×#41—1491no—
236C2×#137—1491no—
237C2×#137—1491no—
238C2×C22⋊Q8—1491no—
239#452—1491no—
240#452—1491no—
241#452—1491no—
242#452D14911yescharacteristic, complemented
243#451—1491no—
244#451—1491no—
245#450—1491no—
246#450—1491no—
247#417—1491no—
248#417—1491no—
249#407—1491no—
250C2×#41—2981no—
251C2×#41—2984no—
252C2×#41—2984no—
253C2×#41—2981no—
254C2×#41—1491no—
255C2×#41—1491no—
256C2×#41—1491no—
257C2×#41C2×D14911yescharacteristic
258C22×Q8—1491no—
259C22×Q8—1491no—
260C22×D4—2981no—
261C22×D4—1491no—
262C23×C4—2981no—
263C23×C4—1491no—
264#137—2984no—
265#137—2984no—
266C2×Q16—1491no—
267C2×Q16—1491no—
268C2×Q16—1491no—
269C2×Q16C2×D14911yescharacteristic, complemented
270C2×SD16—1491no—
271C2×SD16—1491no—
272C2×SD16—1491no—
273C2×SD16C2×D14911yescharacteristic, complemented
274C2×M16—1491no—
275C2×M16—1491no—
276C22⋊Q8—2981no—
277C22⋊Q8—2981no—
278C22⋊Q8—1491no—
279C22⋊Q8—1491no—
280C22⋊Q8—1491no—
281C22⋊Q8C2×D14911yescharacteristic, complemented
282C2×C4⋊C4—2981no—
283C2×C4⋊C4—1491no—
284C2×C22⋊C4—2981no—
285#103—1491no—
286#103—1491no—
287#103—1491no—
288#103C2×D14911yescharacteristic, complemented
289#102—1491no—
290#102—1491no—
291#102—1491no—
292#102C2×D14911yescharacteristic, complemented
293C22⋊C8—1491no—
294C22⋊C8—1491no—
295C22⋊C8—1491no—
296C22⋊C8C2×D14911yescharacteristic, complemented
297C24—2981no—
298C24—1491no—
299#41—22no—
300#41—22no—
301#41—2988no—
302#41—2982no—
303#41—2982no—
304#41—2988no—
305#41—2982no—
306#41—2982no—
307#41—1494no—
308#41—1494no—
309C2×Q8—2981no—
310C2×Q8—2981no—
311C2×Q8—2982no—
312C2×Q8—2982no—
313C2×Q8—2982no—
314C2×Q8—2982no—
315C2×Q8—1491no—
316C2×Q8—1491no—
317C2×Q8C22×D14911yescharacteristic
318C2×Q8C22×D14911yescharacteristic
319C2×D4—21no—
320C2×D4—2981no—
321C2×D4—2988no—
322C2×D4—2982no—
323C2×D4—2981no—
324C2×D4—2981no—
325C2×D4—2982no—
326C2×D4—2981no—
327C2×D4—1491no—
328C2×D4C22×D14911yescharacteristic
329C22×C4—21no—
330C22×C4—2988no—
331C22×C4—2981no—
332C22×C4—2981no—
333C22×C4—2982no—
334C22×C4—2982no—
335C22×C4—2981no—
336C22×C4—2981no—
337C22×C4—2981no—
338C22×C4—2981no—
339C22×C4—1491no—
340C22×C4—1491no—
341C22×C4—1491no—
342C22×C4—1491no—
343C22×C4—1491no—
344C22×C4C22×D14911yescharacteristic
345Q16—22no—
346Q16—2982no—
347Q16—2982no—
348Q16—2982no—
349SD16—22no—
350SD16—2982no—
351SD16—2982no—
352SD16—2982no—
353M16—2982no—
354M16—2982no—
355C2×C8—1491no—
356C2×C8—1491no—
357C2×C8C22×D14911yescharacteristic
358C2×C8C22×D14911yescharacteristic
359C4⋊C4—21no—
360C4⋊C4—2981no—
361C4⋊C4—2981no—
362C4⋊C4—2981no—
363C4⋊C4—2981no—
364C4⋊C4—1491no—
365C4⋊C4C22×D14911yescharacteristic
366C22⋊C4—21no—
367C22⋊C4—2981no—
368C22⋊C4—2981no—
369C22⋊C4—2981no—
370C23—21no—
371C23—2984no—
372C23—2982no—
373C23—2982no—
374C23—2981no—
375C23—5961no—
376C23—5962no—
377C23—5962no—
378C23—1491no—
379C23—1491no—
380C23D4×D14911yescharacteristic
381Q8—21no—
382Q8—21no—
383Q8—21no—
384Q8—2984no—
385Q8—2981no—
386Q8—2981no—
387Q8—2981no—
388Q8—5962no—
389Q8—1492no—
390Q8D4×D14912yes—
391D4—24no—
392D4—21no—
393D4—2984no—
394D4—2984no—
395D4—2984no—
396D4—2984no—
397D4—2981no—
398D4—1492no—
399D4D4×D14912yes—
400C2×C4—24no—
401C2×C4—21no—
402C2×C4—21no—
403C2×C4—21no—
404C2×C4—21no—
405C2×C4—21no—
406C2×C4—2984no—
407C2×C4—2984no—
408C2×C4—2982no—
409C2×C4—2982no—
410C2×C4—2984no—
411C2×C4—2981no—
412C2×C4—2984no—
413C2×C4—2982no—
414C2×C4—2981no—
415C2×C4—2981no—
416C2×C4—2981no—
417C2×C4—2981no—
418C2×C4—5962no—
419C2×C4—5962no—
420C2×C4—1491no—
421C2×C4—1491no—
422C2×C4—1492no—
423C2×C4—1492no—
424C2×C4D4×D14911yescharacteristic
425C2×C4C23×D14911yescharacteristic
426C8—21no—
427C8—21no—
428C8—2981no—
429C8—2981no—
430C22—21no—
431C22—21no—
432C22—21no—
433C22—22no—
434C22—41no—
435C22—41no—
436C22—2981no—
437C22—2981no—
438C22—2981no—
439C22—2981no—
440C22—2982no—
441C22—5964no—
442C22—5968no—
443C22—5961no—
444C22—5961no—
445C22—1492no—
446C22—1492no—
447C22C2×D4×D14911yescharacteristic
448C4—21no—
449C4—22no—
450C4—41no—
451C4—41no—
452C4—2982no—
453C4—2981no—
454C4—5961no—
455C4—5961no—
456C4—1492no—
457C4C2×D4×D14911yescharacteristic
458C4C2×D4×D14911yescharacteristic
459C2—41no—
460C2—42no—
461C2—2982no—
462C2—2982no—
463C2—5962no—
464C2#120×D14911yescharacteristic
465C2#137×D14911yescharacteristic
466C2#2375042211yescharacteristic
subgroupG/H
centerC22C2×D4×D149
commutator, cosocleC2×C4×C149C24
FrattiniC2×C4C23×D149
Fitting#452×C149C2
socleC22×C149C22×D4
prime radicalC2×#41×D149C2
hypercenter#452D149
hypocenterC149#3007
LCSG ▷ C2×C4×C149 ▷ C2×C149 ▷ C149
UCSC1 ◁ C22 ◁ C22×C4 ◁ #452

Complements

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

NH ≅ G/Ntype
2. C2×#18011943C2semidirect
3. C2×#18011940C2semidirect
4. C22⋊Q8×D149C2semidirect
5. #23749789C2semidirect
6. #23749787C2semidirect
7. #23749785C2semidirect
8. #23749782C2semidirect
9. #23749779C2semidirect
10. #23749777C2semidirect
11. #23749774C2semidirect
12. #23749625C2semidirect
13. #23749623C2semidirect
14. #23749587C2semidirect
15. #452×C149C2semidirect
24. C2×Q8×D149C22semidirect
31. C2×Dic596C22semidirect
32. C2×#14558726C22semidirect
33. C2×#14558726C22semidirect
34. C2×#14558725C22semidirect
35. C2×#14558724C22semidirect
36. C2×#14558723C22semidirect
37. C2×#14558719C22semidirect
38. C2×#14558719C22semidirect
40. #18011890C22semidirect
45. C4⋊C4×D149C22semidirect
47. #18011836C22semidirect
48. #18011835C22semidirect
49. #18011834C22semidirect
50. #18011833C22semidirect
51. #18011832C22semidirect
52. #18011831C22semidirect
53. C22⋊#11924746C22semidirect
57. C2×Q16×C149C22semidirect
58. C2×SD16×C149C22semidirect
59. C22⋊Q8×C149C22semidirect
60. #103×C149C22semidirect
61. #102×C149C22semidirect
62. C22⋊C8×C149C22semidirect
213. C2×D149#137semidirect
233. C149#3007semidirect
242. #452D149semidirect
269. C2×Q16C2×D149semidirect
273. C2×SD16C2×D149semidirect
281. C22⋊Q8C2×D149semidirect
288. #103C2×D149semidirect
292. #102C2×D149semidirect
296. C22⋊C8C2×D149semidirect

Maximal subgroups

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

subgroupindex
C2×#41×D1492
C2×#180119432
C2×#180119402
C22⋊Q8×D1492
#237497892
#237497872
#237497852
#237497822
#237497792
#237497772
#237497742
#237496252
#237496232
#237495872
#452×C1492
#3007149