Finite Group Databasebeta

← #28399848

Subgroups — #28399848

Order 14464.

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×#16813668C211yesmaximal, characteristic
2C2×#16813654C211yesmaximal, characteristic, complemented
3C2×SD16×D113C211yesmaximal, characteristic, complemented
4#20299171C211yesmaximal, characteristic, complemented
5#20298763C211yesmaximal, characteristic, complemented
6#20298760C211yesmaximal, characteristic, complemented
7#20298759C211yesmaximal, characteristic, complemented
8#20298751C211yesmaximal, characteristic, complemented
9#20298750C211yesmaximal, characteristic, complemented
10#20298748C211yesmaximal, characteristic, complemented
11#20298747C211yesmaximal, characteristic, complemented
12#20298601C211yesmaximal, characteristic, complemented
13#20298593C211yesmaximal, characteristic, complemented
14#20298563C211yesmaximal, characteristic, complemented
15#452×C113C211yesmaximal, characteristic, complemented
16#16813668—24no—
17#16813668—24no—
18C2×#14268648C2211yescharacteristic
19C2×#14268648C2211yescharacteristic
20C2×#14268647—21no—
21C2×#14268647C2211yescharacteristic
22C2×Q8×D113C2211yescharacteristic, complemented
23C22×D452—21no—
24C22×D452C2211yescharacteristic
25C2×D4×D113—21no—
26C2×D4×D113—21no—
27C2×D4×D113C2211yescharacteristic
28C2×D4×D113C2211yescharacteristic
29#16813654—24no—
30C2×#14268637C2211yescharacteristic, complemented
31C2×#14268636C2211yescharacteristic, complemented
32C2×#14268636C2211yescharacteristic, complemented
33C2×#14268635C2211yescharacteristic, complemented
34C2×#14268634C2211yescharacteristic, complemented
35C2×#14268634C2211yescharacteristic, complemented
36SD16×D113—24no—
37C2×#14268630C2211yescharacteristic, complemented
38C2×C8×D113C2211yescharacteristic, complemented
39#16813621—21no—
40#16813608—21no—
41#16813607—21no—
42#16813604—21no—
43#16813602C2211yescharacteristic, complemented
44#16813601C2211yescharacteristic, complemented
45#16813598C2211yescharacteristic, complemented
46#16813580C2211yescharacteristic, complemented
47#16813546C2211yescharacteristic, complemented
48#16813544C2211yescharacteristic, complemented
49#16813544C2211yescharacteristic, complemented
50#16813543C2211yescharacteristic, complemented
51#16813542C2211yescharacteristic, complemented
52#16813542C2211yescharacteristic, complemented
53C22⋊#11744751C2211yescharacteristic, complemented
54#16813534C2211yescharacteristic, complemented
55#16813534C2211yescharacteristic, complemented
56C2×#41×C113C2211yescharacteristic
57C2×Q16×C113C2211yescharacteristic, complemented
58C2×SD16×C113C2211yescharacteristic, complemented
59C22⋊Q8×C113C2211yescharacteristic, complemented
60#103×C113C2211yescharacteristic, complemented
61#102×C113C2211yescharacteristic, complemented
62C22⋊C8×C113C2211yescharacteristic, complemented
63C23×D113—21no—
64C23×D113—21no—
65C23×D113—21no—
66#14268648—22no—
67#14268648—22no—
68#14268648—22no—
69#14268648D414yes—
70#14268647—28no—
71#14268647—22no—
72#14268647—22no—
73C2×Dic226—21no—
74C2×Dic226D411yescharacteristic
75C2×Dic226C2311yescharacteristic
76Q8×D113—22no—
77Q8×D113—22no—
78C2×D452—28no—
79C2×D452—21no—
80C2×D452—21no—
81C2×D452—21no—
82C2×D452—22no—
83C2×D452—22no—
84C2×D452D411yescharacteristic
85C2×D452C2311yescharacteristic
86C2×D452C2311yescharacteristic
87C2×C22⋊D113—21no—
88C2×C22⋊D113—21no—
89C2×C22⋊D113D411yescharacteristic
90C2×C22⋊D113D411yescharacteristic
91D4×D113—28no—
92D4×D113—28no—
93D4×D113—22no—
94D4×D113—22no—
95D4×D113—22no—
96D4×D113D414yes—
97C2×C4×D113—21no—
98C2×C4×D113—21no—
99C2×C4×D113—21no—
100C2×C4×D113C2311yescharacteristic
101C2×C4×D113C2311yescharacteristic
102#14268637—22no—
103#14268636—22no—
104#14268636—22no—
105#14268635—22no—
106#14268634—22no—
107#14268634—22no—
108#14268630—22no—
109C2×#11744751C2311yescharacteristic
110C2×#11744751C2311yescharacteristic
111C8×D113—22no—
112C4⋊Dic113—21no—
113C4⋊Dic113C2311yescharacteristic
114#14268626—21no—
115#14268626—21no—
116C22⋊Dic113—21no—
117#14268624—21no—
118#14268624—21no—
119#14268624—21no—
120C4×Dic113—21no—
121#41×C113—22no—
122#41×C113—22no—
123C2×Q8×C113C2311yescharacteristic
124C2×Q8×C113C2311yescharacteristic
125C2×D4×C113—21no—
126C2×D4×C113C2311yescharacteristic
127C22×C4×C113—21no—
128C22×C4×C113C2311yescharacteristic
129Q16×C113—22no—
130SD16×C113—22no—
131C2×C8×C113C2311yescharacteristic
132C2×C8×C113C2311yescharacteristic
133C4⋊C4×C113—21no—
134C4⋊C4×C113C2311yescharacteristic
135C22⋊C4×C113—21no—
136C23×C113—21no—
137C23×C113C2×D411yescharacteristic
138C22×D113—21no—
139C22×D113—21no—
140C22×D113—24no—
141C22×D113—24no—
142C22×D113—24no—
143C22×D113—81no—
144C22×D113—81no—
145C22×D113—81no—
146C22×D113—81no—
147C22×D113—81no—
148C22×D113—81no—
149C22×D113C2×D411yescharacteristic
150C22×D113C2×D411yescharacteristic
151Q8×C113—21no—
152Q8×C113—21no—
153Q8×C113—21no—
154Q8×C113C2×D412yes—
155D4×C113—24no—
156D4×C113—21no—
157D4×C113C2×D412yes—
158C2×C4×C113—24no—
159C2×C4×C113—21no—
160C2×C4×C113—21no—
161C2×C4×C113—21no—
162C2×C4×C113—21no—
163C2×C4×C113—21no—
164C2×C4×C113C2×D411yescharacteristic
165C2×C4×C113C2411yescharacteristic
166C22⋊D113—24no—
167C22⋊D113—22no—
168C22⋊D113—22no—
169C22⋊D113—24no—
170C2×Dic113—21no—
171C2×Dic113—21no—
172C2×Dic113C2×D411yescharacteristic
173C2×Dic113C2×D411yescharacteristic
174D452—24no—
175D452—24no—
176D452—24no—
177D452—22no—
178D452—21no—
179D452—21no—
180D452C2×D412yes—
181D452C2×D412yes—
182C4×D113—24no—
183C4×D113—21no—
184C4×D113—21no—
185C4×D113—24no—
186C4×D113—42no—
187C4×D113C2×D412yes—
188C4×D113C2×D412yes—
189Dic226—22no—
190Dic226—21no—
191Dic226—21no—
192Dic226—42no—
193C8×C113—21no—
194C8×C113—21no—
195#11744751—21no—
196#11744751—21no—
197C22×C113—21no—
198C22×C113—22no—
199C22×C113—21no—
200C22×C113—21no—
201C22×C113—41no—
202C22×C113—41no—
203C22×C113C22×D411yescharacteristic
204C2×D113—21no—
205C2×D113—21no—
206C2×D113—21no—
207C2×D113—22no—
208C2×D113—22no—
209C2×D113—21no—
210C2×D113—21no—
211C2×D113—41no—
212C2×D113—41no—
213C2×D113—41no—
214C2×D113—41no—
215C2×D113—84no—
216C2×D113—84no—
217C2×D113—84no—
218C2×D113#12012yes—
219C4×C113—21no—
220C4×C113—22no—
221C4×C113—41no—
222C4×C113—41no—
223C4×C113C22×D411yescharacteristic
224C4×C113C22×D411yescharacteristic
225Dic113—21no—
226Dic113—41no—
227Dic113—41no—
228Dic113#12012yes—
229C2×C113—41no—
230C2×C113—42no—
231C2×C113C2×#12011yescharacteristic
232C2×C113#57711yescharacteristic
233C2×C113#57711yescharacteristic
234D113—22no—
235D113—41no—
236D113—42no—
237D113—42no—
238#3009—1131nomaximal, Sylow
239C113#300911yescharacteristic, complemented, Sylow
240C2×H(5,𝔽2)—1131no—
241C2×#137—1131no—
242C22×SD16—1131no—
243#536—1131no—
244#452—1131no—
245#452—1131no—
246#452—1131no—
247#452D11311yescharacteristic, complemented
248#450—1131no—
249#450—1131no—
250#450—1131no—
251#450—1131no—
252#418—1131no—
253#416—1131no—
254#408—1131no—
255H(5,𝔽2)—2264no—
256H(5,𝔽2)—2264no—
257C2×#41—1131no—
258C2×#41—1131no—
259C2×#41—1131no—
260C2×#41—2261no—
261C2×#41C2×D11311yescharacteristic
262C22×Q8—1131no—
263C22×D4—1131no—
264C22×D4—1131no—
265C22×D4—1131no—
266C22×D4—2261no—
267C22×D4—2261no—
268C22×D4—2261no—
269#137—2264no—
270C2×Q16—1131no—
271C2×Q16C2×D11311yescharacteristic, complemented
272C2×SD16—1131no—
273C2×SD16—1131no—
274C2×SD16—1131no—
275C2×SD16—1131no—
276C2×SD16—1131no—
277C2×SD16—2264no—
278C2×SD16C2×D11311yescharacteristic, complemented
279C2×M16—1131no—
280C22×C8—1131no—
281C4⋊Q8—2261no—
282#124—2261no—
283#123—2261no—
284#123—2261no—
285C22⋊Q8—1131no—
286C22⋊Q8—1131no—
287C22⋊Q8—1131no—
288C22⋊Q8C2×D11311yescharacteristic, complemented
289#117—1131no—
290#103—1131no—
291#103C2×D11311yescharacteristic, complemented
292#102—1131no—
293#102—1131no—
294#102—1131no—
295#102—1131no—
296#102—1131no—
297#102C2×D11311yescharacteristic, complemented
298C22⋊C8—1131no—
299C22⋊C8—1131no—
300C22⋊C8—1131no—
301C22⋊C8C2×D11311yescharacteristic, complemented
302C24—2261no—
303C24—2261no—
304C24—2261no—
305#41—22no—
306#41—22no—
307#41—1134no—
308#41—2262no—
309#41—2262no—
310#41—2268no—
311#41—2262no—
312#41—2262no—
313#41—2262no—
314C2×Q8—1131no—
315C2×Q8—1131no—
316C2×Q8—2261no—
317C2×Q8—2262no—
318C2×Q8—2262no—
319C2×Q8C22×D11311yescharacteristic
320C2×Q8C22×D11311yescharacteristic
321C2×D4—21no—
322C2×D4—1134no—
323C2×D4—1131no—
324C2×D4—1131no—
325C2×D4—1131no—
326C2×D4—1131no—
327C2×D4—1131no—
328C2×D4—2262no—
329C2×D4—2262no—
330C2×D4—2261no—
331C2×D4—2261no—
332C2×D4—2268no—
333C2×D4—2262no—
334C2×D4—2261no—
335C2×D4—2261no—
336C2×D4—2262no—
337C2×D4—2268no—
338C2×D4—2268no—
339C2×D4—2262no—
340C2×D4—2261no—
341C2×D4C22×D11311yescharacteristic
342C22×C4—21no—
343C22×C4—1131no—
344C22×C4—1131no—
345C22×C4—2261no—
346C22×C4—2261no—
347C22×C4—2261no—
348C22×C4C22×D11311yescharacteristic
349Q16—22no—
350Q16—2262no—
351SD16—22no—
352SD16—2262no—
353SD16—2262no—
354SD16—2262no—
355SD16—2262no—
356SD16—2262no—
357M16—2262no—
358C2×C8—1131no—
359C2×C8—1131no—
360C2×C8—2262no—
361C2×C8C22×D11311yescharacteristic
362C2×C8C22×D11311yescharacteristic
363C4⋊C4—21no—
364C4⋊C4—1131no—
365C4⋊C4—2261no—
366C4⋊C4—2261no—
367C4⋊C4—2261no—
368C4⋊C4C22×D11311yescharacteristic
369C22⋊C4—21no—
370C22⋊C4—2261no—
371C22⋊C4—2261no—
372C22⋊C4—2261no—
373C22⋊C4—2261no—
374C42—2261no—
375C23—21no—
376C23—1131no—
377C23—1131no—
378C23—9041no—
379C23—9041no—
380C23—9041no—
381C23—9041no—
382C23—9041no—
383C23—9041no—
384C23—2264no—
385C23—2264no—
386C23—2264no—
387C23—2261no—
388C23—2261no—
389C23D4×D11311yescharacteristic
390Q8—21no—
391Q8—21no—
392Q8—21no—
393Q8—4522no—
394Q8—2262no—
395Q8—2261no—
396Q8—2261no—
397Q8D4×D11312yes—
398D4—24no—
399D4—21no—
400D4—1132no—
401D4—1132no—
402D4—2262no—
403D4—2264no—
404D4—2264no—
405D4—2264no—
406D4—2262no—
407D4—2264no—
408D4—2262no—
409D4—2264no—
410D4—2261no—
411D4—2261no—
412D4D4×D11312yes—
413C2×C4—24no—
414C2×C4—21no—
415C2×C4—21no—
416C2×C4—21no—
417C2×C4—21no—
418C2×C4—21no—
419C2×C4—1131no—
420C2×C4—1132no—
421C2×C4—1132no—
422C2×C4—1131no—
423C2×C4—4522no—
424C2×C4—2264no—
425C2×C4—2264no—
426C2×C4—2261no—
427C2×C4—2261no—
428C2×C4—2261no—
429C2×C4—2261no—
430C2×C4D4×D11311yescharacteristic
431C2×C4C23×D11311yescharacteristic
432C8—21no—
433C8—21no—
434C8—2261no—
435C8—2261no—
436C22—21no—
437C22—22no—
438C22—21no—
439C22—21no—
440C22—41no—
441C22—41no—
442C22—1132no—
443C22—9044no—
444C22—9044no—
445C22—9044no—
446C22—4521no—
447C22—4521no—
448C22—4521no—
449C22—4521no—
450C22—2262no—
451C22—2262no—
452C22—2261no—
453C22—2261no—
454C22—2261no—
455C22—2261no—
456C22—2261no—
457C22C2×D4×D11311yescharacteristic
458C4—21no—
459C4—22no—
460C4—41no—
461C4—41no—
462C4—1132no—
463C4—4521no—
464C4—4521no—
465C4—2261no—
466C4C2×D4×D11311yescharacteristic
467C4C2×D4×D11311yescharacteristic
468C2—41no—
469C2—42no—
470C2—4522no—
471C2—4522no—
472C2—4521no—
473C2—2262no—
474C2#120×D11311yescharacteristic
475C2#2029940511yescharacteristic
476C2#2029940711yescharacteristic
subgroupG/H
centerC22C2×D4×D113
commutator, cosocleC2×C4×C113C24
FrattiniC2×C4C23×D113
Fitting#452×C113C2
socleC22×C113C22×D4
prime radicalC2×#16813668C2
hypercenter#452D113
hypocenterC113#3009
LCSG ▷ C2×C4×C113 ▷ C2×C113 ▷ C113
UCSC1 ◁ C22 ◁ C22×C4 ◁ #452

Complements

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

NH ≅ G/Ntype
2. C2×#16813654C2semidirect
3. C2×SD16×D113C2semidirect
4. #20299171C2semidirect
5. #20298763C2semidirect
6. #20298760C2semidirect
7. #20298759C2semidirect
8. #20298751C2semidirect
9. #20298750C2semidirect
10. #20298748C2semidirect
11. #20298747C2semidirect
12. #20298601C2semidirect
13. #20298593C2semidirect
14. #20298563C2semidirect
15. #452×C113C2semidirect
22. C2×Q8×D113C22semidirect
30. C2×#14268637C22semidirect
31. C2×#14268636C22semidirect
32. C2×#14268636C22semidirect
33. C2×#14268635C22semidirect
34. C2×#14268634C22semidirect
35. C2×#14268634C22semidirect
37. C2×#14268630C22semidirect
38. C2×C8×D113C22semidirect
43. #16813602C22semidirect
44. #16813601C22semidirect
45. #16813598C22semidirect
46. #16813580C22semidirect
47. #16813546C22semidirect
48. #16813544C22semidirect
49. #16813544C22semidirect
50. #16813543C22semidirect
51. #16813542C22semidirect
52. #16813542C22semidirect
53. C22⋊#11744751C22semidirect
54. #16813534C22semidirect
55. #16813534C22semidirect
57. C2×Q16×C113C22semidirect
58. C2×SD16×C113C22semidirect
59. C22⋊Q8×C113C22semidirect
60. #103×C113C22semidirect
61. #102×C113C22semidirect
62. C22⋊C8×C113C22semidirect
239. C113#3009semidirect
247. #452D113semidirect
271. C2×Q16C2×D113semidirect
278. C2×SD16C2×D113semidirect
288. C22⋊Q8C2×D113semidirect
291. #103C2×D113semidirect
297. #102C2×D113semidirect
301. C22⋊C8C2×D113semidirect

Maximal subgroups

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

subgroupindex
C2×#168136682
C2×#168136542
C2×SD16×D1132
#202991712
#202987632
#202987602
#202987592
#202987512
#202987502
#202987482
#202987472
#202986012
#202985932
#202985632
#452×C1132
#3009113