Finite Group Databasebeta

← #16854985

Subgroups — #16854985

Order 3968.

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×#11758566C211yesmaximal, characteristic
2C2×#11758495C211yesmaximal, characteristic, complemented
3C2×SD16×D31C211yesmaximal, characteristic, complemented
4#14521622C211yesmaximal, characteristic, complemented
5#14521214C211yesmaximal, characteristic, complemented
6#14521211C211yesmaximal, characteristic, complemented
7#14521210C211yesmaximal, characteristic, complemented
8#14521202C211yesmaximal, characteristic, complemented
9#14521201C211yesmaximal, characteristic, complemented
10#14521199C211yesmaximal, characteristic, complemented
11#14521198C211yesmaximal, characteristic, complemented
12#14521052C211yesmaximal, characteristic, complemented
13#14521044C211yesmaximal, characteristic, complemented
14#14521014C211yesmaximal, characteristic, complemented
15C31×#452C211yesmaximal, characteristic, complemented
16C2×#41×C31C2211yescharacteristic
17#11758566—24no—
18#11758566—24no—
19C2×#92499C2211yescharacteristic
20C2×#92499C2211yescharacteristic
21C2×Q8×D31C2211yescharacteristic, complemented
22C2×D4×D31—21no—
23C2×D4×D31—21no—
24C2×D4×D31C2211yescharacteristic
25C2×D4×D31C2211yescharacteristic
26C2×#92495—21no—
27C2×#92495C2211yescharacteristic
28C22×D124—21no—
29C22×D124C2211yescharacteristic
30C2×Q16×C31C2211yescharacteristic, complemented
31C2×SD16×C31C2211yescharacteristic, complemented
32C31×C22⋊Q8C2211yescharacteristic, complemented
33#11758523C2211yescharacteristic, complemented
34C2×#92482C2211yescharacteristic, complemented
35C2×#92481C2211yescharacteristic, complemented
36C2×#92481C2211yescharacteristic, complemented
37C2×#92480C2211yescharacteristic, complemented
38#11758501C2211yescharacteristic, complemented
39#11758495—24no—
40SD16×D31—24no—
41C2×#92470C2211yescharacteristic, complemented
42C2×#92470C2211yescharacteristic, complemented
43C2×#92469C2211yescharacteristic, complemented
44C2×C8×D31C2211yescharacteristic, complemented
45#11758474C2211yescharacteristic, complemented
46#11758471—21no—
47#11758469C2211yescharacteristic, complemented
48#11758465—21no—
49#11758462—21no—
50#11758460—21no—
51C31×#103C2211yescharacteristic, complemented
52C31×#102C2211yescharacteristic, complemented
53C31×C22⋊C8C2211yescharacteristic, complemented
54#11758423C2211yescharacteristic, complemented
55#11758420C2211yescharacteristic, complemented
56C22⋊(C31⋊C8)C2211yescharacteristic, complemented
57#11758409C2211yescharacteristic, complemented
58#11758409C2211yescharacteristic, complemented
59(C2×D31)⋊C8C2211yescharacteristic, complemented
60(C2×D31)⋊C8C2211yescharacteristic, complemented
61#11758397C2211yescharacteristic, complemented
62#11758397C2211yescharacteristic, complemented
63C23×D31—21no—
64C23×D31—21no—
65C23×D31—21no—
66#41×C31—22no—
67#41×C31—22no—
68C2×Q8×C31C2311yescharacteristic
69C2×Q8×C31C2311yescharacteristic
70C2×D4×C31—21no—
71C2×D4×C31C2311yescharacteristic
72C22×C4×C31—21no—
73C22×C4×C31C2311yescharacteristic
74C2×C22⋊D31—21no—
75C2×C22⋊D31—21no—
76C2×C22⋊D31D411yescharacteristic
77C2×C22⋊D31D411yescharacteristic
78#92499—22no—
79#92499—22no—
80#92499—22no—
81#92499D414yes—
82Q8×D31—22no—
83Q8×D31—22no—
84D4×D31—28no—
85D4×D31—28no—
86D4×D31—22no—
87D4×D31—22no—
88D4×D31—22no—
89D4×D31D414yes—
90#92495—28no—
91#92495—22no—
92#92495—22no—
93C2×D124—28no—
94C2×D124—21no—
95C2×D124—21no—
96C2×D124—21no—
97C2×D124—22no—
98C2×D124—22no—
99C2×D124D411yescharacteristic
100C2×D124C2311yescharacteristic
101C2×D124C2311yescharacteristic
102C2×C4×D31—21no—
103C2×C4×D31—21no—
104C2×C4×D31—21no—
105C2×C4×D31C2311yescharacteristic
106C2×C4×D31C2311yescharacteristic
107C2×Dic62—21no—
108C2×Dic62D411yescharacteristic
109C2×Dic62C2311yescharacteristic
110Q16×C31—22no—
111SD16×C31—22no—
112C2×C8×C31C2311yescharacteristic
113C2×C8×C31C2311yescharacteristic
114C4⋊C4×C31—21no—
115C4⋊C4×C31C2311yescharacteristic
116C22⋊C4×C31—21no—
117C22⋊Dic31—21no—
118#92482—22no—
119#92481—22no—
120#92481—22no—
121#92480—22no—
122(C2×D31)⋊C4—21no—
123(C2×D31)⋊C4—21no—
124(C2×D31)⋊C4—21no—
125C4⋊Dic31—21no—
126C4⋊Dic31C2311yescharacteristic
127Dic31⋊C4—21no—
128Dic31⋊C4—21no—
129C4×Dic31—21no—
130C2×C31⋊C8C2311yescharacteristic
131C2×C31⋊C8C2311yescharacteristic
132#92470—22no—
133#92470—22no—
134#92469—22no—
135C8×D31—22no—
136C23×C31—21no—
137C23×C31C2×D411yescharacteristic
138C22×D31—21no—
139C22×D31—21no—
140C22×D31—24no—
141C22×D31—24no—
142C22×D31—24no—
143C22×D31—81no—
144C22×D31—81no—
145C22×D31—81no—
146C22×D31—81no—
147C22×D31—81no—
148C22×D31—81no—
149C22×D31C2×D411yescharacteristic
150C22×D31C2×D411yescharacteristic
151Q8×C31—21no—
152Q8×C31—21no—
153Q8×C31—21no—
154Q8×C31C2×D412yes—
155D4×C31—24no—
156D4×C31—21no—
157D4×C31C2×D412yes—
158C2×C4×C31—24no—
159C2×C4×C31—21no—
160C2×C4×C31—21no—
161C2×C4×C31—21no—
162C2×C4×C31—21no—
163C2×C4×C31—21no—
164C2×C4×C31C2×D411yescharacteristic
165C2×C4×C31C2411yescharacteristic
166C22⋊D31—24no—
167C22⋊D31—22no—
168C22⋊D31—22no—
169C22⋊D31—24no—
170C2×Dic31—21no—
171C2×Dic31—21no—
172C2×Dic31C2×D411yescharacteristic
173C2×Dic31C2×D411yescharacteristic
174D124—24no—
175D124—24no—
176D124—22no—
177D124—24no—
178D124—21no—
179D124—21no—
180D124C2×D412yes—
181D124C2×D412yes—
182C4×D31—24no—
183C4×D31—21no—
184C4×D31—21no—
185C4×D31—24no—
186C4×D31—42no—
187C4×D31C2×D412yes—
188C4×D31C2×D412yes—
189Dic62—22no—
190Dic62—21no—
191Dic62—21no—
192Dic62—42no—
193C8×C31—21no—
194C8×C31—21no—
195C31⋊C8—21no—
196C31⋊C8—21no—
197#3009—311nomaximal, Sylow
198C22×C31—21no—
199C22×C31—22no—
200C22×C31—21no—
201C22×C31—21no—
202C22×C31—41no—
203C22×C31—41no—
204C22×C31C22×D411yescharacteristic
205C2×D31—21no—
206C2×D31—21no—
207C2×D31—21no—
208C2×D31—22no—
209C2×D31—22no—
210C2×D31—21no—
211C2×D31—21no—
212C2×D31—41no—
213C2×D31—41no—
214C2×D31—41no—
215C2×D31—41no—
216C2×D31—84no—
217C2×D31—84no—
218C2×D31—84no—
219C2×D31#12012yes—
220C4×C31—21no—
221C4×C31—22no—
222C4×C31—41no—
223C4×C31—41no—
224C4×C31C22×D411yescharacteristic
225C4×C31C22×D411yescharacteristic
226Dic31—21no—
227Dic31—41no—
228Dic31—41no—
229Dic31#12012yes—
230C2×H(5,𝔽2)—311no—
231C2×#137—311no—
232C22×SD16—311no—
233#536—311no—
234#452—311no—
235#452—311no—
236#452—311no—
237#452D3111yescharacteristic, complemented
238#450—311no—
239#450—311no—
240#450—311no—
241#450—311no—
242#418—311no—
243#416—311no—
244#408—311no—
245C2×C31—41no—
246C2×C31—42no—
247C2×C31C2×#12011yescharacteristic
248C2×C31#57711yescharacteristic
249C2×C31#57711yescharacteristic
250D31—22no—
251D31—41no—
252D31—42no—
253D31—42no—
254H(5,𝔽2)—624no—
255H(5,𝔽2)—624no—
256C2×#41—311no—
257C2×#41—311no—
258C2×#41—311no—
259C2×#41—621no—
260C2×#41C2×D3111yescharacteristic
261C22×Q8—311no—
262C22×D4—311no—
263C22×D4—311no—
264C22×D4—311no—
265C22×D4—621no—
266C22×D4—621no—
267C22×D4—621no—
268#137—624no—
269C2×Q16—311no—
270C2×Q16C2×D3111yescharacteristic, complemented
271C2×SD16—311no—
272C2×SD16—311no—
273C2×SD16—311no—
274C2×SD16—311no—
275C2×SD16—311no—
276C2×SD16—624no—
277C2×SD16C2×D3111yescharacteristic, complemented
278C2×M16—311no—
279C22×C8—311no—
280C4⋊Q8—621no—
281#124—621no—
282#123—621no—
283#123—621no—
284C22⋊Q8—311no—
285C22⋊Q8—311no—
286C22⋊Q8—311no—
287C22⋊Q8C2×D3111yescharacteristic, complemented
288#117—311no—
289#103—311no—
290#103C2×D3111yescharacteristic, complemented
291#102—311no—
292#102—311no—
293#102—311no—
294#102—311no—
295#102—311no—
296#102C2×D3111yescharacteristic, complemented
297C22⋊C8—311no—
298C22⋊C8—311no—
299C22⋊C8—311no—
300C22⋊C8C2×D3111yescharacteristic, complemented
301C31#300911yescharacteristic, complemented, Sylow
302C24—621no—
303C24—621no—
304C24—621no—
305#41—22no—
306#41—22no—
307#41—314no—
308#41—622no—
309#41—622no—
310#41—628no—
311#41—622no—
312#41—622no—
313#41—622no—
314C2×Q8—311no—
315C2×Q8—311no—
316C2×Q8—621no—
317C2×Q8—622no—
318C2×Q8—622no—
319C2×Q8C22×D3111yescharacteristic
320C2×Q8C22×D3111yescharacteristic
321C2×D4—21no—
322C2×D4—314no—
323C2×D4—311no—
324C2×D4—311no—
325C2×D4—311no—
326C2×D4—311no—
327C2×D4—311no—
328C2×D4—622no—
329C2×D4—622no—
330C2×D4—621no—
331C2×D4—621no—
332C2×D4—628no—
333C2×D4—622no—
334C2×D4—621no—
335C2×D4—621no—
336C2×D4—622no—
337C2×D4—628no—
338C2×D4—628no—
339C2×D4—622no—
340C2×D4—621no—
341C2×D4C22×D3111yescharacteristic
342C22×C4—21no—
343C22×C4—311no—
344C22×C4—311no—
345C22×C4—621no—
346C22×C4—621no—
347C22×C4—621no—
348C22×C4C22×D3111yescharacteristic
349Q16—22no—
350Q16—622no—
351SD16—22no—
352SD16—622no—
353SD16—622no—
354SD16—622no—
355SD16—622no—
356SD16—622no—
357M16—622no—
358C2×C8—311no—
359C2×C8—311no—
360C2×C8—622no—
361C2×C8C22×D3111yescharacteristic
362C2×C8C22×D3111yescharacteristic
363C4⋊C4—21no—
364C4⋊C4—311no—
365C4⋊C4—621no—
366C4⋊C4—621no—
367C4⋊C4—621no—
368C4⋊C4C22×D3111yescharacteristic
369C22⋊C4—21no—
370C22⋊C4—621no—
371C22⋊C4—621no—
372C22⋊C4—621no—
373C22⋊C4—621no—
374C42—621no—
375C23—21no—
376C23—311no—
377C23—311no—
378C23—624no—
379C23—624no—
380C23—624no—
381C23—621no—
382C23—621no—
383C23—2481no—
384C23—2481no—
385C23—2481no—
386C23—2481no—
387C23—2481no—
388C23—2481no—
389C23D4×D3111yescharacteristic
390Q8—21no—
391Q8—21no—
392Q8—21no—
393Q8—622no—
394Q8—621no—
395Q8—621no—
396Q8—1242no—
397Q8D4×D3112yes—
398D4—24no—
399D4—21no—
400D4—312no—
401D4—312no—
402D4—622no—
403D4—624no—
404D4—624no—
405D4—624no—
406D4—622no—
407D4—624no—
408D4—622no—
409D4—624no—
410D4—621no—
411D4—621no—
412D4D4×D3112yes—
413C2×C4—24no—
414C2×C4—21no—
415C2×C4—21no—
416C2×C4—21no—
417C2×C4—21no—
418C2×C4—21no—
419C2×C4—311no—
420C2×C4—312no—
421C2×C4—312no—
422C2×C4—311no—
423C2×C4—624no—
424C2×C4—624no—
425C2×C4—621no—
426C2×C4—621no—
427C2×C4—621no—
428C2×C4—621no—
429C2×C4—1242no—
430C2×C4D4×D3111yescharacteristic
431C2×C4C23×D3111yescharacteristic
432C8—21no—
433C8—21no—
434C8—621no—
435C8—621no—
436C22—21no—
437C22—21no—
438C22—21no—
439C22—22no—
440C22—41no—
441C22—41no—
442C22—312no—
443C22—621no—
444C22—621no—
445C22—621no—
446C22—621no—
447C22—622no—
448C22—622no—
449C22—621no—
450C22—1241no—
451C22—1241no—
452C22—1241no—
453C22—1241no—
454C22—2484no—
455C22—2484no—
456C22—2484no—
457C22C2×D4×D3111yescharacteristic
458C4—21no—
459C4—22no—
460C4—41no—
461C4—41no—
462C4—312no—
463C4—621no—
464C4—1241no—
465C4—1241no—
466C4C2×D4×D3111yescharacteristic
467C4C2×D4×D3111yescharacteristic
468C2—41no—
469C2—42no—
470C2—622no—
471C2—1241no—
472C2—1242no—
473C2—1242no—
474C2#120×D3111yescharacteristic
475C2#1452185611yescharacteristic
476C2#1452185811yescharacteristic
subgroupG/H
centerC22C2×D4×D31
commutator, cosocleC2×C4×C31C24
FrattiniC2×C4C23×D31
FittingC31×#452C2
socleC22×C31C22×D4
prime radicalC2×#11758566C2
hypercenter#452D31
hypocenterC31#3009
LCSG ▷ C2×C4×C31 ▷ C2×C31 ▷ C31
UCSC1 ◁ C22 ◁ C22×C4 ◁ #452

Complements

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

NH ≅ G/Ntype
2. C2×#11758495C2semidirect
3. C2×SD16×D31C2semidirect
4. #14521622C2semidirect
5. #14521214C2semidirect
6. #14521211C2semidirect
7. #14521210C2semidirect
8. #14521202C2semidirect
9. #14521201C2semidirect
10. #14521199C2semidirect
11. #14521198C2semidirect
12. #14521052C2semidirect
13. #14521044C2semidirect
14. #14521014C2semidirect
15. C31×#452C2semidirect
21. C2×Q8×D31C22semidirect
30. C2×Q16×C31C22semidirect
31. C2×SD16×C31C22semidirect
32. C31×C22⋊Q8C22semidirect
33. #11758523C22semidirect
34. C2×#92482C22semidirect
35. C2×#92481C22semidirect
36. C2×#92481C22semidirect
37. C2×#92480C22semidirect
38. #11758501C22semidirect
41. C2×#92470C22semidirect
42. C2×#92470C22semidirect
43. C2×#92469C22semidirect
44. C2×C8×D31C22semidirect
45. #11758474C22semidirect
47. #11758469C22semidirect
51. C31×#103C22semidirect
52. C31×#102C22semidirect
53. C31×C22⋊C8C22semidirect
54. #11758423C22semidirect
55. #11758420C22semidirect
56. C22⋊(C31⋊C8)C22semidirect
57. #11758409C22semidirect
58. #11758409C22semidirect
59. (C2×D31)⋊C8C22semidirect
60. (C2×D31)⋊C8C22semidirect
61. #11758397C22semidirect
62. #11758397C22semidirect
237. #452D31semidirect
270. C2×Q16C2×D31semidirect
277. C2×SD16C2×D31semidirect
287. C22⋊Q8C2×D31semidirect
290. #103C2×D31semidirect
296. #102C2×D31semidirect
300. C22⋊C8C2×D31semidirect
301. C31#3009semidirect

Maximal subgroups

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

subgroupindex
C2×#117585662
C2×#117584952
C2×SD16×D312
#145216222
#145212142
#145212112
#145212102
#145212022
#145212012
#145211992
#145211982
#145210522
#145210442
#145210142
C31×#4522
#300931