Finite Group Databasebeta

← #46820142

Subgroups — #46820142

Order 34432.

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×D269C211yesmaximal, characteristic
2C2×#22575485C211yesmaximal, characteristic, complemented
3C2×#22575482C211yesmaximal, characteristic, complemented
4C22⋊Q8×D269C211yesmaximal, characteristic, complemented
5#31770769C211yesmaximal, characteristic, complemented
6#31770767C211yesmaximal, characteristic, complemented
7#31770765C211yesmaximal, characteristic, complemented
8#31770762C211yesmaximal, characteristic, complemented
9#31770759C211yesmaximal, characteristic, complemented
10#31770757C211yesmaximal, characteristic, complemented
11#31770754C211yesmaximal, characteristic, complemented
12#31770605C211yesmaximal, characteristic, complemented
13#31770603C211yesmaximal, characteristic, complemented
14#31770567C211yesmaximal, characteristic, complemented
15#452×C269C211yesmaximal, characteristic, complemented
16C2×#16864403C2211yescharacteristic
17C2×#16864402—21no—
18C2×#16864402C2211yescharacteristic
19C2×#16864401—21no—
20C2×#16864401C2211yescharacteristic
21#41×D269—24no—
22#41×D269—24no—
23C2×Q8×D269C2211yescharacteristic
24C2×Q8×D269C2211yescharacteristic, complemented
25C2×D4×D269—21no—
26C2×D4×D269C2211yescharacteristic
27C22×C4×D269—21no—
28C22×C4×D269C2211yescharacteristic
29#22575485—24no—
30#22575482—24no—
31C2×Dic1076C2211yescharacteristic, complemented
32C2×#16864392C2211yescharacteristic, complemented
33C2×#16864392C2211yescharacteristic, complemented
34C2×#16864391C2211yescharacteristic, complemented
35C2×#16864390C2211yescharacteristic, complemented
36C2×#16864389C2211yescharacteristic, complemented
37C2×#16864385C2211yescharacteristic, complemented
38C2×#16864385C2211yescharacteristic, complemented
39#22575433C2211yescharacteristic
40#22575432C2211yescharacteristic, complemented
41#22575431—21no—
42#22575430—21no—
43#22575429C2211yescharacteristic
44C4⋊C4×D269—21no—
45C4⋊C4×D269C2211yescharacteristic, complemented
46C22⋊C4×D269—21no—
47#22575378C2211yescharacteristic, complemented
48#22575377C2211yescharacteristic, complemented
49#22575376C2211yescharacteristic, complemented
50#22575375C2211yescharacteristic, complemented
51#22575374C2211yescharacteristic, complemented
52#22575373C2211yescharacteristic, complemented
53C22⋊#14524130C2211yescharacteristic, complemented
54#22575365C2211yescharacteristic
55#22575365C2211yescharacteristic
56C2×#41×C269C2211yescharacteristic
57C2×Q16×C269C2211yescharacteristic, complemented
58C2×SD16×C269C2211yescharacteristic, complemented
59C22⋊Q8×C269C2211yescharacteristic, complemented
60#103×C269C2211yescharacteristic, complemented
61#102×C269C2211yescharacteristic, complemented
62C22⋊C8×C269C2211yescharacteristic, complemented
63C23×D269—21no—
64C23×D269D411yescharacteristic
65#16864403—22no—
66#16864403D414yes—
67#16864402—28no—
68#16864402—22no—
69#16864402—22no—
70#16864401—28no—
71#16864401—22no—
72#16864401D414yes—
73C2×Dic538—21no—
74C2×Dic538—21no—
75C2×Dic538C2311yescharacteristic
76C2×Dic538C2311yescharacteristic
77Q8×D269—22no—
78Q8×D269—22no—
79Q8×D269—22no—
80Q8×D269—22no—
81C2×D1076—21no—
82C2×D1076C2311yescharacteristic
83C2×C22⋊D269—21no—
84C2×C22⋊D269—21no—
85C2×C22⋊D269—21no—
86D4×D269—28no—
87D4×D269—22no—
88D4×D269—22no—
89C22×Dic269—21no—
90C22×Dic269D411yescharacteristic
91C2×C4×D269—28no—
92C2×C4×D269—21no—
93C2×C4×D269—21no—
94C2×C4×D269—22no—
95C2×C4×D269—22no—
96C2×C4×D269—21no—
97C2×C4×D269—21no—
98C2×C4×D269—21no—
99C2×C4×D269D411yescharacteristic
100C2×C4×D269D411yescharacteristic
101C2×C4×D269C2311yescharacteristic
102C2×C4×D269C2311yescharacteristic
103Dic1076—22no—
104#16864392—22no—
105#16864392—22no—
106#16864391—22no—
107#16864390—22no—
108#16864389—22no—
109#16864385—22no—
110#16864385—22no—
111C2×#14524130C2311yescharacteristic
112C2×#14524130C2311yescharacteristic
113C4⋊Dic269—21no—
114C4⋊Dic269C2311yescharacteristic
115#16864381—21no—
116#16864381—21no—
117#16864381—21no—
118C22⋊Dic269—21no—
119#16864379—21no—
120#16864379—21no—
121#41×C269—22no—
122#41×C269—22no—
123C2×Q8×C269C2311yescharacteristic
124C2×Q8×C269C2311yescharacteristic
125C2×D4×C269—21no—
126C2×D4×C269C2311yescharacteristic
127C22×C4×C269—21no—
128C22×C4×C269C2311yescharacteristic
129Q16×C269—22no—
130SD16×C269—22no—
131C2×C8×C269C2311yescharacteristic
132C2×C8×C269C2311yescharacteristic
133C4⋊C4×C269—21no—
134C4⋊C4×C269C2311yescharacteristic
135C22⋊C4×C269—21no—
136C22×D269—24no—
137C22×D269—21no—
138C22×D269—22no—
139C22×D269—22no—
140C22×D269—41no—
141C22×D269—42no—
142C22×D269—42no—
143C22×D269C2×D411yescharacteristic
144C22×D269C2×D411yescharacteristic
145Dic538—24no—
146Dic538—21no—
147Dic538—21no—
148Dic538—21no—
149Dic538—42no—
150Dic538C2×D412yes—
151D1076—24no—
152D1076—21no—
153D1076C2×D412yes—
154C22⋊D269—24no—
155C22⋊D269—24no—
156C22⋊D269—24no—
157C2×Dic269—21no—
158C2×Dic269—24no—
159C2×Dic269—22no—
160C2×Dic269—21no—
161C2×Dic269—21no—
162C2×Dic269C2×D411yescharacteristic
163C2×Dic269C2×D411yescharacteristic
164C4×D269—24no—
165C4×D269—24no—
166C4×D269—22no—
167C4×D269—21no—
168C4×D269—21no—
169C4×D269—22no—
170C4×D269—24no—
171C4×D269—42no—
172C4×D269—42no—
173C4×D269C2×D412yes—
174C4×D269C2×D412yes—
175#14524130—21no—
176#14524130—21no—
177C23×C269—21no—
178C23×C269C2×D411yescharacteristic
179Q8×C269—21no—
180Q8×C269—21no—
181Q8×C269—21no—
182Q8×C269C2×D412yes—
183D4×C269—24no—
184D4×C269—21no—
185D4×C269C2×D412yes—
186C2×C4×C269—24no—
187C2×C4×C269—21no—
188C2×C4×C269—21no—
189C2×C4×C269—21no—
190C2×C4×C269—21no—
191C2×C4×C269—21no—
192C2×C4×C269C2×D411yescharacteristic
193C2×C4×C269C2411yescharacteristic
194C8×C269—21no—
195C8×C269—21no—
196C22×C269—21no—
197C22×C269—22no—
198C22×C269—21no—
199C22×C269—21no—
200C22×C269—41no—
201C22×C269—41no—
202C22×C269C22×D411yescharacteristic
203C2×D269—21no—
204C2×D269—21no—
205C2×D269—21no—
206C2×D269—21no—
207C2×D269—22no—
208C2×D269—44no—
209C2×D269—48no—
210C2×D269—41no—
211C2×D269—41no—
212C2×D269#12012yes—
213C2×D269#13712yescomplemented
214C4×C269—21no—
215C4×C269—22no—
216C4×C269—41no—
217C4×C269—41no—
218C4×C269C22×D411yescharacteristic
219C4×C269C22×D411yescharacteristic
220Dic269—22no—
221Dic269—21no—
222Dic269—41no—
223Dic269—41no—
224Dic269#12012yes—
225C2×C269—41no—
226C2×C269—42no—
227C2×C269C2×#12011yescharacteristic
228C2×C269C2×#13711yescharacteristic
229C2×C269#57511yescharacteristic
230D269—22no—
231D269—22no—
232D269—42no—
233C269#300711yescharacteristic, complemented, Sylow
234#3007—2691nomaximal, Sylow
235C22×#41—2691no—
236C2×#137—2691no—
237C2×#137—2691no—
238C2×C22⋊Q8—2691no—
239#452—2691no—
240#452—2691no—
241#452—2691no—
242#452D26911yescharacteristic, complemented
243#451—2691no—
244#451—2691no—
245#450—2691no—
246#450—2691no—
247#417—2691no—
248#417—2691no—
249#407—2691no—
250C2×#41—2691no—
251C2×#41—2691no—
252C2×#41—2691no—
253C2×#41—5381no—
254C2×#41—5381no—
255C2×#41—5384no—
256C2×#41—5384no—
257C2×#41C2×D26911yescharacteristic
258C22×Q8—2691no—
259C22×Q8—2691no—
260C22×D4—2691no—
261C22×D4—5381no—
262C23×C4—2691no—
263C23×C4—5381no—
264#137—5384no—
265#137—5384no—
266C2×Q16—2691no—
267C2×Q16—2691no—
268C2×Q16—2691no—
269C2×Q16C2×D26911yescharacteristic, complemented
270C2×SD16—2691no—
271C2×SD16—2691no—
272C2×SD16—2691no—
273C2×SD16C2×D26911yescharacteristic, complemented
274C2×M16—2691no—
275C2×M16—2691no—
276C22⋊Q8—2691no—
277C22⋊Q8—2691no—
278C22⋊Q8—2691no—
279C22⋊Q8—5381no—
280C22⋊Q8—5381no—
281C22⋊Q8C2×D26911yescharacteristic, complemented
282C2×C4⋊C4—2691no—
283C2×C4⋊C4—5381no—
284C2×C22⋊C4—5381no—
285#103—2691no—
286#103—2691no—
287#103—2691no—
288#103C2×D26911yescharacteristic, complemented
289#102—2691no—
290#102—2691no—
291#102—2691no—
292#102C2×D26911yescharacteristic, complemented
293C22⋊C8—2691no—
294C22⋊C8—2691no—
295C22⋊C8—2691no—
296C22⋊C8C2×D26911yescharacteristic, complemented
297C24—2691no—
298C24—5381no—
299#41—22no—
300#41—22no—
301#41—2694no—
302#41—2694no—
303#41—5382no—
304#41—5382no—
305#41—5388no—
306#41—5382no—
307#41—5382no—
308#41—5388no—
309C2×Q8—2691no—
310C2×Q8—2691no—
311C2×Q8—5381no—
312C2×Q8—5381no—
313C2×Q8—5382no—
314C2×Q8—5382no—
315C2×Q8—5382no—
316C2×Q8—5382no—
317C2×Q8C22×D26911yescharacteristic
318C2×Q8C22×D26911yescharacteristic
319C2×D4—21no—
320C2×D4—2691no—
321C2×D4—5381no—
322C2×D4—5381no—
323C2×D4—5382no—
324C2×D4—5388no—
325C2×D4—5381no—
326C2×D4—5381no—
327C2×D4—5382no—
328C2×D4C22×D26911yescharacteristic
329C22×C4—21no—
330C22×C4—2691no—
331C22×C4—2691no—
332C22×C4—2691no—
333C22×C4—2691no—
334C22×C4—2691no—
335C22×C4—5382no—
336C22×C4—5382no—
337C22×C4—5381no—
338C22×C4—5381no—
339C22×C4—5388no—
340C22×C4—5381no—
341C22×C4—5381no—
342C22×C4—5381no—
343C22×C4—5381no—
344C22×C4C22×D26911yescharacteristic
345Q16—22no—
346Q16—5382no—
347Q16—5382no—
348Q16—5382no—
349SD16—22no—
350SD16—5382no—
351SD16—5382no—
352SD16—5382no—
353M16—5382no—
354M16—5382no—
355C2×C8—2691no—
356C2×C8—2691no—
357C2×C8C22×D26911yescharacteristic
358C2×C8C22×D26911yescharacteristic
359C4⋊C4—21no—
360C4⋊C4—2691no—
361C4⋊C4—5381no—
362C4⋊C4—5381no—
363C4⋊C4—5381no—
364C4⋊C4—5381no—
365C4⋊C4C22×D26911yescharacteristic
366C22⋊C4—21no—
367C22⋊C4—5381no—
368C22⋊C4—5381no—
369C22⋊C4—5381no—
370C23—21no—
371C23—2691no—
372C23—2691no—
373C23—5382no—
374C23—5382no—
375C23—5384no—
376C23—5381no—
377C23—10761no—
378C23—10762no—
379C23—10762no—
380C23D4×D26911yescharacteristic
381Q8—21no—
382Q8—21no—
383Q8—21no—
384Q8—2692no—
385Q8—5384no—
386Q8—5381no—
387Q8—5381no—
388Q8—5381no—
389Q8—10762no—
390Q8D4×D26912yes—
391D4—24no—
392D4—21no—
393D4—2692no—
394D4—5384no—
395D4—5384no—
396D4—5384no—
397D4—5384no—
398D4—5381no—
399D4D4×D26912yes—
400C2×C4—24no—
401C2×C4—21no—
402C2×C4—21no—
403C2×C4—21no—
404C2×C4—21no—
405C2×C4—21no—
406C2×C4—2691no—
407C2×C4—2692no—
408C2×C4—2692no—
409C2×C4—2691no—
410C2×C4—5382no—
411C2×C4—5384no—
412C2×C4—5382no—
413C2×C4—5384no—
414C2×C4—5384no—
415C2×C4—5382no—
416C2×C4—5384no—
417C2×C4—5381no—
418C2×C4—5381no—
419C2×C4—5381no—
420C2×C4—5381no—
421C2×C4—5381no—
422C2×C4—10762no—
423C2×C4—10762no—
424C2×C4D4×D26911yescharacteristic
425C2×C4C23×D26911yescharacteristic
426C8—21no—
427C8—21no—
428C8—5381no—
429C8—5381no—
430C22—21no—
431C22—21no—
432C22—21no—
433C22—22no—
434C22—41no—
435C22—41no—
436C22—2692no—
437C22—2692no—
438C22—5381no—
439C22—5381no—
440C22—5381no—
441C22—5381no—
442C22—5382no—
443C22—10764no—
444C22—10768no—
445C22—10761no—
446C22—10761no—
447C22C2×D4×D26911yescharacteristic
448C4—21no—
449C4—22no—
450C4—41no—
451C4—41no—
452C4—2692no—
453C4—5381no—
454C4—5382no—
455C4—10761no—
456C4—10761no—
457C4C2×D4×D26911yescharacteristic
458C4C2×D4×D26911yescharacteristic
459C2—41no—
460C2—42no—
461C2—5382no—
462C2—5382no—
463C2—10762no—
464C2#120×D26911yescharacteristic
465C2#137×D26911yescharacteristic
466C2#3177140211yescharacteristic
subgroupG/H
centerC22C2×D4×D269
commutator, cosocleC2×C4×C269C24
FrattiniC2×C4C23×D269
Fitting#452×C269C2
socleC22×C269C22×D4
prime radicalC2×#41×D269C2
hypercenter#452D269
hypocenterC269#3007
LCSG ▷ C2×C4×C269 ▷ C2×C269 ▷ C269
UCSC1 ◁ C22 ◁ C22×C4 ◁ #452

Complements

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

NH ≅ G/Ntype
2. C2×#22575485C2semidirect
3. C2×#22575482C2semidirect
4. C22⋊Q8×D269C2semidirect
5. #31770769C2semidirect
6. #31770767C2semidirect
7. #31770765C2semidirect
8. #31770762C2semidirect
9. #31770759C2semidirect
10. #31770757C2semidirect
11. #31770754C2semidirect
12. #31770605C2semidirect
13. #31770603C2semidirect
14. #31770567C2semidirect
15. #452×C269C2semidirect
24. C2×Q8×D269C22semidirect
31. C2×Dic1076C22semidirect
32. C2×#16864392C22semidirect
33. C2×#16864392C22semidirect
34. C2×#16864391C22semidirect
35. C2×#16864390C22semidirect
36. C2×#16864389C22semidirect
37. C2×#16864385C22semidirect
38. C2×#16864385C22semidirect
40. #22575432C22semidirect
45. C4⋊C4×D269C22semidirect
47. #22575378C22semidirect
48. #22575377C22semidirect
49. #22575376C22semidirect
50. #22575375C22semidirect
51. #22575374C22semidirect
52. #22575373C22semidirect
53. C22⋊#14524130C22semidirect
57. C2×Q16×C269C22semidirect
58. C2×SD16×C269C22semidirect
59. C22⋊Q8×C269C22semidirect
60. #103×C269C22semidirect
61. #102×C269C22semidirect
62. C22⋊C8×C269C22semidirect
213. C2×D269#137semidirect
233. C269#3007semidirect
242. #452D269semidirect
269. C2×Q16C2×D269semidirect
273. C2×SD16C2×D269semidirect
281. C22⋊Q8C2×D269semidirect
288. #103C2×D269semidirect
292. #102C2×D269semidirect
296. C22⋊C8C2×D269semidirect

Maximal subgroups

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

subgroupindex
C2×#41×D2692
C2×#225754852
C2×#225754822
C22⋊Q8×D2692
#317707692
#317707672
#317707652
#317707622
#317707592
#317707572
#317707542
#317706052
#317706032
#317705672
#452×C2692
#3007269