Finite Group Databasebeta

← #16828766

Subgroups — #16828766

Order 3712.

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×#11745364C211yesmaximal, characteristic
2C2×#11745280C211yesmaximal, characteristic, complemented
3C2×SD16×D29C211yesmaximal, characteristic, complemented
4#14271736C211yesmaximal, characteristic, complemented
5#14271328C211yesmaximal, characteristic, complemented
6#14271325C211yesmaximal, characteristic, complemented
7#14271324C211yesmaximal, characteristic, complemented
8#14271316C211yesmaximal, characteristic, complemented
9#14271315C211yesmaximal, characteristic, complemented
10#14271313C211yesmaximal, characteristic, complemented
11#14271312C211yesmaximal, characteristic, complemented
12#14271166C211yesmaximal, characteristic, complemented
13#14271158C211yesmaximal, characteristic, complemented
14#14271128C211yesmaximal, characteristic, complemented
15C29×#452C211yesmaximal, characteristic, complemented
16C2×#41×C29C2211yescharacteristic
17#11745364—24no—
18#11745364—24no—
19C2×#90809C2211yescharacteristic
20C2×#90809C2211yescharacteristic
21C2×Q8×D29C2211yescharacteristic, complemented
22C2×D4×D29—21no—
23C2×D4×D29—21no—
24C2×D4×D29C2211yescharacteristic
25C2×D4×D29C2211yescharacteristic
26C2×#90805—21no—
27C2×#90805C2211yescharacteristic
28C22×D116—21no—
29C22×D116C2211yescharacteristic
30C2×Q16×C29C2211yescharacteristic, complemented
31C2×SD16×C29C2211yescharacteristic, complemented
32C29×C22⋊Q8C2211yescharacteristic, complemented
33#11745308C2211yescharacteristic, complemented
34C2×#90785C2211yescharacteristic, complemented
35C2×#90784C2211yescharacteristic, complemented
36C2×#90784C2211yescharacteristic, complemented
37C2×#90783C2211yescharacteristic, complemented
38#11745286C2211yescharacteristic, complemented
39#11745280—24no—
40SD16×D29—24no—
41C2×#90773C2211yescharacteristic, complemented
42C2×#90773C2211yescharacteristic, complemented
43C2×#90772C2211yescharacteristic, complemented
44C2×C8×D29C2211yescharacteristic, complemented
45#11745259C2211yescharacteristic, complemented
46#11745256—21no—
47#11745254C2211yescharacteristic, complemented
48#11745250—21no—
49#11745247—21no—
50#11745245—21no—
51C29×#103C2211yescharacteristic, complemented
52C29×#102C2211yescharacteristic, complemented
53C29×C22⋊C8C2211yescharacteristic, complemented
54#11745183C2211yescharacteristic, complemented
55#11745180C2211yescharacteristic, complemented
56C22⋊#6597C2211yescharacteristic, complemented
57#11745169C2211yescharacteristic, complemented
58#11745169C2211yescharacteristic, complemented
59#11745168C2211yescharacteristic, complemented
60#11745168C2211yescharacteristic, complemented
61#11745157C2211yescharacteristic, complemented
62#11745157C2211yescharacteristic, complemented
63C23×D29—21no—
64C23×D29—21no—
65C23×D29—21no—
66#41×C29—22no—
67#41×C29—22no—
68C2×Q8×C29C2311yescharacteristic
69C2×Q8×C29C2311yescharacteristic
70C2×D4×C29—21no—
71C2×D4×C29C2311yescharacteristic
72C22×C4×C29—21no—
73C22×C4×C29C2311yescharacteristic
74C2×C22⋊D29—21no—
75C2×C22⋊D29—21no—
76C2×C22⋊D29D411yescharacteristic
77C2×C22⋊D29D411yescharacteristic
78#90809—22no—
79#90809—22no—
80#90809—22no—
81#90809D414yes—
82Q8×D29—22no—
83Q8×D29—22no—
84D4×D29—28no—
85D4×D29—28no—
86D4×D29—22no—
87D4×D29—22no—
88D4×D29—22no—
89D4×D29D414yes—
90#90805—28no—
91#90805—22no—
92#90805—22no—
93C2×D116—28no—
94C2×D116—21no—
95C2×D116—21no—
96C2×D116—21no—
97C2×D116—22no—
98C2×D116—22no—
99C2×D116D411yescharacteristic
100C2×D116C2311yescharacteristic
101C2×D116C2311yescharacteristic
102C2×C4×D29—21no—
103C2×C4×D29—21no—
104C2×C4×D29—21no—
105C2×C4×D29C2311yescharacteristic
106C2×C4×D29C2311yescharacteristic
107C2×Dic58—21no—
108C2×Dic58D411yescharacteristic
109C2×Dic58C2311yescharacteristic
110Q16×C29—22no—
111SD16×C29—22no—
112C2×C8×C29C2311yescharacteristic
113C2×C8×C29C2311yescharacteristic
114C4⋊C4×C29—21no—
115C4⋊C4×C29C2311yescharacteristic
116C22⋊C4×C29—21no—
117C22⋊Dic29—21no—
118#90785—22no—
119#90784—22no—
120#90784—22no—
121#90783—22no—
122#90781—21no—
123#90781—21no—
124#90781—21no—
125C4⋊Dic29—21no—
126C4⋊Dic29C2311yescharacteristic
127#90779—21no—
128#90779—21no—
129C4×Dic29—21no—
130C2×#6597C2311yescharacteristic
131C2×#6597C2311yescharacteristic
132#90773—22no—
133#90773—22no—
134#90772—22no—
135C8×D29—22no—
136C23×C29—21no—
137C23×C29C2×D411yescharacteristic
138C22×D29—21no—
139C22×D29—21no—
140C22×D29—24no—
141C22×D29—24no—
142C22×D29—24no—
143C22×D29—81no—
144C22×D29—81no—
145C22×D29—81no—
146C22×D29—81no—
147C22×D29—81no—
148C22×D29—81no—
149C22×D29C2×D411yescharacteristic
150C22×D29C2×D411yescharacteristic
151Q8×C29—21no—
152Q8×C29—21no—
153Q8×C29—21no—
154Q8×C29C2×D412yes—
155D4×C29—24no—
156D4×C29—21no—
157D4×C29C2×D412yes—
158C2×C4×C29—24no—
159C2×C4×C29—21no—
160C2×C4×C29—21no—
161C2×C4×C29—21no—
162C2×C4×C29—21no—
163C2×C4×C29—21no—
164C2×C4×C29C2×D411yescharacteristic
165C2×C4×C29C2411yescharacteristic
166C22⋊D29—24no—
167C22⋊D29—22no—
168C22⋊D29—22no—
169C22⋊D29—24no—
170C2×Dic29—21no—
171C2×Dic29—21no—
172C2×Dic29C2×D411yescharacteristic
173C2×Dic29C2×D411yescharacteristic
174D116—24no—
175D116—24no—
176D116—22no—
177D116—24no—
178D116—21no—
179D116—21no—
180D116C2×D412yes—
181D116C2×D412yes—
182C4×D29—24no—
183C4×D29—21no—
184C4×D29—21no—
185C4×D29—24no—
186C4×D29—42no—
187C4×D29C2×D412yes—
188C4×D29C2×D412yes—
189Dic58—22no—
190Dic58—21no—
191Dic58—21no—
192Dic58—42no—
193C8×C29—21no—
194C8×C29—21no—
195#6597—21no—
196#6597—21no—
197#3009—291nomaximal, Sylow
198C22×C29—21no—
199C22×C29—22no—
200C22×C29—21no—
201C22×C29—21no—
202C22×C29—41no—
203C22×C29—41no—
204C22×C29C22×D411yescharacteristic
205C2×D29—21no—
206C2×D29—21no—
207C2×D29—21no—
208C2×D29—22no—
209C2×D29—22no—
210C2×D29—21no—
211C2×D29—21no—
212C2×D29—41no—
213C2×D29—41no—
214C2×D29—41no—
215C2×D29—41no—
216C2×D29—84no—
217C2×D29—84no—
218C2×D29—84no—
219C2×D29#12012yes—
220C4×C29—21no—
221C4×C29—22no—
222C4×C29—41no—
223C4×C29—41no—
224C4×C29C22×D411yescharacteristic
225C4×C29C22×D411yescharacteristic
226Dic29—21no—
227Dic29—41no—
228Dic29—41no—
229Dic29#12012yes—
230C2×H(5,𝔽2)—291no—
231C2×#137—291no—
232C22×SD16—291no—
233#536—291no—
234#452—291no—
235#452—291no—
236#452—291no—
237#452D2911yescharacteristic, complemented
238#450—291no—
239#450—291no—
240#450—291no—
241#450—291no—
242#418—291no—
243#416—291no—
244#408—291no—
245C2×C29—41no—
246C2×C29—42no—
247C2×C29C2×#12011yescharacteristic
248C2×C29#57711yescharacteristic
249C2×C29#57711yescharacteristic
250D29—22no—
251D29—41no—
252D29—42no—
253D29—42no—
254H(5,𝔽2)—584no—
255H(5,𝔽2)—584no—
256C2×#41—291no—
257C2×#41—291no—
258C2×#41—291no—
259C2×#41—581no—
260C2×#41C2×D2911yescharacteristic
261C22×Q8—291no—
262C22×D4—291no—
263C22×D4—291no—
264C22×D4—291no—
265C22×D4—581no—
266C22×D4—581no—
267C22×D4—581no—
268#137—584no—
269C2×Q16—291no—
270C2×Q16C2×D2911yescharacteristic, complemented
271C2×SD16—291no—
272C2×SD16—291no—
273C2×SD16—291no—
274C2×SD16—291no—
275C2×SD16—291no—
276C2×SD16—584no—
277C2×SD16C2×D2911yescharacteristic, complemented
278C2×M16—291no—
279C22×C8—291no—
280C4⋊Q8—581no—
281#124—581no—
282#123—581no—
283#123—581no—
284C22⋊Q8—291no—
285C22⋊Q8—291no—
286C22⋊Q8—291no—
287C22⋊Q8C2×D2911yescharacteristic, complemented
288#117—291no—
289#103—291no—
290#103C2×D2911yescharacteristic, complemented
291#102—291no—
292#102—291no—
293#102—291no—
294#102—291no—
295#102—291no—
296#102C2×D2911yescharacteristic, complemented
297C22⋊C8—291no—
298C22⋊C8—291no—
299C22⋊C8—291no—
300C22⋊C8C2×D2911yescharacteristic, complemented
301C29#300911yescharacteristic, complemented, Sylow
302C24—581no—
303C24—581no—
304C24—581no—
305#41—22no—
306#41—22no—
307#41—294no—
308#41—582no—
309#41—582no—
310#41—588no—
311#41—582no—
312#41—582no—
313#41—582no—
314C2×Q8—291no—
315C2×Q8—291no—
316C2×Q8—581no—
317C2×Q8—582no—
318C2×Q8—582no—
319C2×Q8C22×D2911yescharacteristic
320C2×Q8C22×D2911yescharacteristic
321C2×D4—21no—
322C2×D4—294no—
323C2×D4—291no—
324C2×D4—291no—
325C2×D4—291no—
326C2×D4—291no—
327C2×D4—291no—
328C2×D4—582no—
329C2×D4—582no—
330C2×D4—581no—
331C2×D4—581no—
332C2×D4—588no—
333C2×D4—582no—
334C2×D4—581no—
335C2×D4—581no—
336C2×D4—582no—
337C2×D4—588no—
338C2×D4—588no—
339C2×D4—582no—
340C2×D4—581no—
341C2×D4C22×D2911yescharacteristic
342C22×C4—21no—
343C22×C4—291no—
344C22×C4—291no—
345C22×C4—581no—
346C22×C4—581no—
347C22×C4—581no—
348C22×C4C22×D2911yescharacteristic
349Q16—22no—
350Q16—582no—
351SD16—22no—
352SD16—582no—
353SD16—582no—
354SD16—582no—
355SD16—582no—
356SD16—582no—
357M16—582no—
358C2×C8—291no—
359C2×C8—291no—
360C2×C8—582no—
361C2×C8C22×D2911yescharacteristic
362C2×C8C22×D2911yescharacteristic
363C4⋊C4—21no—
364C4⋊C4—291no—
365C4⋊C4—581no—
366C4⋊C4—581no—
367C4⋊C4—581no—
368C4⋊C4C22×D2911yescharacteristic
369C22⋊C4—21no—
370C22⋊C4—581no—
371C22⋊C4—581no—
372C22⋊C4—581no—
373C22⋊C4—581no—
374C42—581no—
375C23—21no—
376C23—291no—
377C23—291no—
378C23—584no—
379C23—584no—
380C23—584no—
381C23—581no—
382C23—581no—
383C23—2321no—
384C23—2321no—
385C23—2321no—
386C23—2321no—
387C23—2321no—
388C23—2321no—
389C23D4×D2911yescharacteristic
390Q8—21no—
391Q8—21no—
392Q8—21no—
393Q8—582no—
394Q8—581no—
395Q8—581no—
396Q8—1162no—
397Q8D4×D2912yes—
398D4—24no—
399D4—21no—
400D4—292no—
401D4—292no—
402D4—582no—
403D4—584no—
404D4—584no—
405D4—584no—
406D4—582no—
407D4—584no—
408D4—582no—
409D4—584no—
410D4—581no—
411D4—581no—
412D4D4×D2912yes—
413C2×C4—24no—
414C2×C4—21no—
415C2×C4—21no—
416C2×C4—21no—
417C2×C4—21no—
418C2×C4—21no—
419C2×C4—291no—
420C2×C4—292no—
421C2×C4—292no—
422C2×C4—291no—
423C2×C4—584no—
424C2×C4—584no—
425C2×C4—581no—
426C2×C4—581no—
427C2×C4—581no—
428C2×C4—581no—
429C2×C4—1162no—
430C2×C4D4×D2911yescharacteristic
431C2×C4C23×D2911yescharacteristic
432C8—21no—
433C8—21no—
434C8—581no—
435C8—581no—
436C22—21no—
437C22—22no—
438C22—21no—
439C22—21no—
440C22—41no—
441C22—41no—
442C22—292no—
443C22—581no—
444C22—581no—
445C22—582no—
446C22—582no—
447C22—581no—
448C22—581no—
449C22—581no—
450C22—1161no—
451C22—1161no—
452C22—1161no—
453C22—1161no—
454C22—2324no—
455C22—2324no—
456C22—2324no—
457C22C2×D4×D2911yescharacteristic
458C4—21no—
459C4—22no—
460C4—41no—
461C4—41no—
462C4—292no—
463C4—581no—
464C4—1161no—
465C4—1161no—
466C4C2×D4×D2911yescharacteristic
467C4C2×D4×D2911yescharacteristic
468C2—41no—
469C2—42no—
470C2—582no—
471C2—1161no—
472C2—1162no—
473C2—1162no—
474C2#120×D2911yescharacteristic
475C2#1427197011yescharacteristic
476C2#1427197211yescharacteristic
subgroupG/H
centerC22C2×D4×D29
commutator, cosocleC2×C4×C29C24
FrattiniC2×C4C23×D29
FittingC29×#452C2
socleC22×C29C22×D4
prime radicalC2×#11745364C2
hypercenter#452D29
hypocenterC29#3009
LCSG ▷ C2×C4×C29 ▷ C2×C29 ▷ C29
UCSC1 ◁ C22 ◁ C22×C4 ◁ #452

Complements

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

NH ≅ G/Ntype
2. C2×#11745280C2semidirect
3. C2×SD16×D29C2semidirect
4. #14271736C2semidirect
5. #14271328C2semidirect
6. #14271325C2semidirect
7. #14271324C2semidirect
8. #14271316C2semidirect
9. #14271315C2semidirect
10. #14271313C2semidirect
11. #14271312C2semidirect
12. #14271166C2semidirect
13. #14271158C2semidirect
14. #14271128C2semidirect
15. C29×#452C2semidirect
21. C2×Q8×D29C22semidirect
30. C2×Q16×C29C22semidirect
31. C2×SD16×C29C22semidirect
32. C29×C22⋊Q8C22semidirect
33. #11745308C22semidirect
34. C2×#90785C22semidirect
35. C2×#90784C22semidirect
36. C2×#90784C22semidirect
37. C2×#90783C22semidirect
38. #11745286C22semidirect
41. C2×#90773C22semidirect
42. C2×#90773C22semidirect
43. C2×#90772C22semidirect
44. C2×C8×D29C22semidirect
45. #11745259C22semidirect
47. #11745254C22semidirect
51. C29×#103C22semidirect
52. C29×#102C22semidirect
53. C29×C22⋊C8C22semidirect
54. #11745183C22semidirect
55. #11745180C22semidirect
56. C22⋊#6597C22semidirect
57. #11745169C22semidirect
58. #11745169C22semidirect
59. #11745168C22semidirect
60. #11745168C22semidirect
61. #11745157C22semidirect
62. #11745157C22semidirect
237. #452D29semidirect
270. C2×Q16C2×D29semidirect
277. C2×SD16C2×D29semidirect
287. C22⋊Q8C2×D29semidirect
290. #103C2×D29semidirect
296. #102C2×D29semidirect
300. C22⋊C8C2×D29semidirect
301. C29#3009semidirect

Maximal subgroups

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

subgroupindex
C2×#117453642
C2×#117452802
C2×SD16×D292
#142717362
#142713282
#142713252
#142713242
#142713162
#142713152
#142713132
#142713122
#142711662
#142711582
#142711282
C29×#4522
#300929