Finite Group Databasebeta

← #46820166

Subgroups — #46820166

Order 34432.

732 conjugacy classes of subgroups.

The subgroup lattice has 474 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#22575501C211yesmaximal, characteristic, complemented
2C2×#22575480C211yesmaximal, characteristic, complemented
3C2×SD16×D269C211yesmaximal, characteristic, complemented
4#31771171C211yesmaximal, characteristic, complemented
5#31770753C211yesmaximal, characteristic, complemented
6#31770752C211yesmaximal, characteristic, complemented
7#31770749C211yesmaximal, characteristic, complemented
8#31770748C211yesmaximal, characteristic, complemented
9#31770745C211yesmaximal, characteristic, complemented
10#31770744C211yesmaximal, characteristic, complemented
11#31770742C211yesmaximal, characteristic, complemented
12#31770612C211yesmaximal, characteristic, complemented
13#31770600C211yesmaximal, characteristic, complemented
14#31770570C211yesmaximal, characteristic, complemented
15#448×C269C211yesmaximal, characteristic, complemented
16#22575501—24no—
17#22575501—24no—
18C2×#16864403—21no—
19C2×#16864403C2211yescharacteristic
20C2×#16864403C2211yescharacteristic, complemented
21C2×#16864402—21no—
22C2×#16864402—21no—
23C2×#16864402C2211yescharacteristic
24C2×#16864402C2211yescharacteristic, complemented
25C2×Q8×D269—21no—
26C2×Q8×D269C2211yescharacteristic
27C2×Q8×D269C2211yescharacteristic, complemented
28C2×D4×D269C2211yescharacteristic
29#22575480—24no—
30C2×#16864391C2211yescharacteristic
31C2×#16864390C2211yescharacteristic
32C2×#16864389C2211yescharacteristic
33C2×#16864389C2211yescharacteristic
34SD16×D269—24no—
35C2×D2152C2211yescharacteristic
36C2×#16864387C2211yescharacteristic
37C2×#16864385C2211yescharacteristic
38C2×C8×D269C2211yescharacteristic
39#22575449—21no—
40#22575439—21no—
41#22575428C2211yescharacteristic
42#22575427C2211yescharacteristic
43#22575424C2211yescharacteristic
44#22575423—21no—
45#22575421—21no—
46#22575411C2211yescharacteristic
47#22575378C2211yescharacteristic, complemented
48#22575377C2211yescharacteristic, complemented
49#22575377C2211yescharacteristic, complemented
50#22575376C2211yescharacteristic, complemented
51#22575375C2211yescharacteristic, complemented
52#22575373C2211yescharacteristic, complemented
53C22⋊#14524130C2211yescharacteristic, complemented
54#22575365C2211yescharacteristic, complemented
55#22575365C2211yescharacteristic, complemented
56C22×Q8×C269C2211yescharacteristic, complemented
57C2×SD16×C269C2211yescharacteristic
58C2×SD16×C269C2211yescharacteristic
59#121×C269C2211yescharacteristic
60#103×C269C2211yescharacteristic, complemented
61#103×C269C2211yescharacteristic, complemented
62C22⋊C8×C269C2211yescharacteristic, complemented
63C23×D269—21no—
64#16864403—28no—
65#16864403—22no—
66#16864403—22no—
67#16864403—22no—
68#16864403D414yescomplemented
69#16864402—28no—
70#16864402—28no—
71#16864402—22no—
72#16864402—22no—
73#16864402—22no—
74#16864402—22no—
75C2×Dic538—21no—
76C2×Dic538—21no—
77C2×Dic538D411yescharacteristic
78C2×Dic538C2311yescharacteristic
79Q8×D269—28no—
80Q8×D269—22no—
81Q8×D269—22no—
82Q8×D269—22no—
83Q8×D269D414yescomplemented
84C2×D1076—21no—
85C2×D1076—21no—
86C2×D1076—21no—
87C2×D1076D411yescharacteristic
88C2×D1076C2311yescharacteristic
89C2×D1076C2311yescharacteristic
90C2×C22⋊D269—21no—
91C2×C22⋊D269—21no—
92C2×C22⋊D269—21no—
93C2×C22⋊D269—21no—
94C2×C22⋊D269D411yescharacteristic
95C2×C22⋊D269D411yescharacteristic
96D4×D269—22no—
97D4×D269—22no—
98C2×C4×D269—21no—
99C2×C4×D269—21no—
100C2×C4×D269—21no—
101C2×C4×D269—21no—
102C2×C4×D269—21no—
103C2×C4×D269C2311yescharacteristic
104C2×C4×D269C2311yescharacteristic
105#16864391—22no—
106#16864390—22no—
107#16864389—22no—
108#16864389—22no—
109D2152—22no—
110#16864387—22no—
111#16864385—22no—
112C2×#14524130C2311yescharacteristic
113C2×#14524130C2311yescharacteristic
114C8×D269—22no—
115C4⋊Dic269C2311yescharacteristic
116C22⋊Dic269—21no—
117#16864379—21no—
118#16864379—21no—
119#16864379—21no—
120C4×Dic269—21no—
121C2×Q8×C269—22no—
122C2×Q8×C269—22no—
123C2×Q8×C269—21no—
124C2×Q8×C269C2311yescharacteristic
125C2×Q8×C269C2311yescharacteristic
126C2×D4×C269—21no—
127C2×D4×C269C2311yescharacteristic
128C22×C4×C269—21no—
129C22×C4×C269C2311yescharacteristic
130SD16×C269—22no—
131SD16×C269—22no—
132C2×C8×C269C2311yescharacteristic
133C2×C8×C269C2311yescharacteristic
134C4⋊C4×C269C2311yescharacteristic
135C22⋊C4×C269—21no—
136C22×D269—21no—
137C22×D269—21no—
138C22×D269—42no—
139C22×D269—81no—
140C22×D269—81no—
141C22×D269C2×D411yescharacteristic
142C22×D269C2×D411yescharacteristic
143Dic538—24no—
144Dic538—21no—
145Dic538—22no—
146Dic538—24no—
147Dic538C2×D412yes—
148D1076—24no—
149D1076—24no—
150D1076—22no—
151D1076—21no—
152D1076—21no—
153D1076—21no—
154D1076—81no—
155D1076C2×D412yes—
156C22⋊D269—24no—
157C22⋊D269—22no—
158C22⋊D269—22no—
159C22⋊D269—42no—
160C22⋊D269—81no—
161C22⋊D269—81no—
162C2×Dic269—21no—
163C2×Dic269—21no—
164C2×Dic269C2×D411yescharacteristic
165C2×Dic269C2×D411yescharacteristic
166C4×D269—24no—
167C4×D269—24no—
168C4×D269—24no—
169C4×D269—24no—
170C4×D269—21no—
171C4×D269—21no—
172C4×D269—24no—
173C4×D269C2×D412yes—
174C4×D269C2×D412yes—
175#14524130—21no—
176#14524130—21no—
177C23×C269—21no—
178C23×C269C2×D411yescharacteristic
179Q8×C269—24no—
180Q8×C269—21no—
181Q8×C269—21no—
182Q8×C269C2×D412yes—
183Q8×C269C2×D412yes—
184D4×C269—21no—
185D4×C269—21no—
186D4×C269—81no—
187C2×C4×C269—24no—
188C2×C4×C269—21no—
189C2×C4×C269—21no—
190C2×C4×C269—21no—
191C2×C4×C269—21no—
192C2×C4×C269—21no—
193C2×C4×C269C2×D411yescharacteristic
194C2×C4×C269C2411yescharacteristic
195C8×C269—21no—
196C8×C269—21no—
197C22×C269—21no—
198C22×C269—21no—
199C22×C269—21no—
200C22×C269—41no—
201C22×C269—41no—
202C22×C269—41no—
203C22×C269C22×D411yescharacteristic
204C2×D269—21no—
205C2×D269—21no—
206C2×D269—21no—
207C2×D269—22no—
208C2×D269—21no—
209C2×D269—21no—
210C2×D269—41no—
211C2×D269—41no—
212C2×D269—41no—
213C2×D269—41no—
214C2×D269—41no—
215C2×D269—84no—
216C2×D269#12012yes—
217C4×C269—22no—
218C4×C269—22no—
219C4×C269—21no—
220C4×C269—41no—
221C4×C269C22×D411yescharacteristic
222C4×C269C22×D411yescharacteristic
223Dic269—22no—
224Dic269—21no—
225Dic269—41no—
226Dic269#12012yes—
227C2×C269—41no—
228C2×C269—81no—
229C2×C269C2×#12011yescharacteristic
230C2×C269#57711yescharacteristic
231C2×C269#57711yescharacteristic
232D269—22no—
233D269—41no—
234D269—42no—
235D269—81no—
236C269#301011yescharacteristic, complemented, Sylow
237#3010—2691nomaximal, Sylow
238C2×2−1+4—2691no—
239C2×AGL(1,ℤ/8)—2691no—
240C22×SD16—2691no—
241#535—2691no—
242#449—2691no—
243#449—2691no—
244#449—2691no—
245#449—2691no—
246#448—2691no—
247#448—2691no—
248#448—2691no—
249#448D26911yescharacteristic, complemented
250#419—2691no—
251#416—2691no—
252#408—2691no—
2532−1+4—5384no—
2542−1+4—5384no—
255C2×#41—2691no—
256C2×#41—2691no—
257C2×#41—2691no—
258C2×#41—2691no—
259C2×#41—5381no—
260C2×#41—5381no—
261C2×#41—5381no—
262C22×Q8—2691no—
263C22×Q8—2691no—
264C22×Q8—5381no—
265C22×Q8C2×D26911yescharacteristic, complemented
266C22×D4—2691no—
267AGL(1,ℤ/8)—5384no—
268C2×SD16—2691no—
269C2×SD16—2691no—
270C2×SD16—2691no—
271C2×SD16—2691no—
272C2×SD16—5384no—
273C2×SD16C2×D26911yescharacteristic
274C2×SD16C2×D26911yescharacteristic
275C2×D8—2691no—
276C2×D8—2691no—
277C2×M16—2691no—
278C22×C8—2691no—
279#127—5381no—
280#124—5381no—
281#121—2691no—
282#121—2691no—
283#121—2691no—
284#121C2×D26911yescharacteristic
285#120—5381no—
286#120—5381no—
287#117—2691no—
288#103—2691no—
289#103—2691no—
290#103—2691no—
291#103—2691no—
292#103C2×D26911yescharacteristic, complemented
293#103C2×D26911yescharacteristic, complemented
294#102—2691no—
295#102—2691no—
296C22⋊C8—2691no—
297C22⋊C8—2691no—
298C22⋊C8—2691no—
299C22⋊C8C2×D26911yescharacteristic, complemented
300C24—5381no—
301#41—2694no—
302#41—5382no—
303#41—5382no—
304#41—5382no—
305#41—5382no—
306#41—5388no—
307#41—5388no—
308#41—5382no—
309#41—5382no—
310#41—5388no—
311#41—5382no—
312C2×Q8—22no—
313C2×Q8—22no—
314C2×Q8—21no—
315C2×Q8—2691no—
316C2×Q8—2694no—
317C2×Q8—2691no—
318C2×Q8—5381no—
319C2×Q8—5381no—
320C2×Q8—5382no—
321C2×Q8—5382no—
322C2×Q8—5388no—
323C2×Q8—5382no—
324C2×Q8C22×D26911yescharacteristic
325C2×Q8C22×D26911yescharacteristic
326C2×D4—21no—
327C2×D4—2691no—
328C2×D4—2691no—
329C2×D4—2691no—
330C2×D4—2691no—
331C2×D4—2691no—
332C2×D4—5381no—
333C2×D4—5381no—
334C2×D4—5381no—
335C2×D4—5382no—
336C2×D4—5382no—
337C2×D4—5381no—
338C2×D4—5381no—
339C2×D4—5381no—
340C2×D4—5381no—
341C2×D4C22×D26911yescharacteristic
342C22×C4—21no—
343C22×C4—2691no—
344C22×C4—2691no—
345C22×C4—5381no—
346C22×C4—5381no—
347C22×C4—5381no—
348C22×C4—5381no—
349C22×C4—5381no—
350C22×C4C22×D26911yescharacteristic
351SD16—22no—
352SD16—22no—
353SD16—5382no—
354SD16—5382no—
355SD16—5382no—
356SD16—5382no—
357D8—5382no—
358D8—5382no—
359M16—5382no—
360C2×C8—2691no—
361C2×C8—2691no—
362C2×C8—5382no—
363C2×C8C22×D26911yescharacteristic
364C2×C8C22×D26911yescharacteristic
365C4⋊C4—2691no—
366C4⋊C4C22×D26911yescharacteristic
367C22⋊C4—21no—
368C22⋊C4—5381no—
369C22⋊C4—5381no—
370C22⋊C4—5381no—
371C22⋊C4—5381no—
372C42—5381no—
373C23—21no—
374C23—2691no—
375C23—2691no—
376C23—5381no—
377C23—5381no—
378C23—10762no—
379C23—21521no—
380C23—21521no—
381C23D4×D26911yescharacteristic
382Q8—24no—
383Q8—21no—
384Q8—21no—
385Q8—2692no—
386Q8—5384no—
387Q8—5382no—
388Q8—5384no—
389Q8—5381no—
390Q8D4×D26912yes—
391Q8D4×D26912yes—
392D4—21no—
393D4—21no—
394D4—81no—
395D4—2692no—
396D4—5382no—
397D4—5384no—
398D4—5384no—
399D4—5382no—
400D4—5382no—
401D4—5384no—
402D4—5381no—
403D4—5381no—
404D4—5381no—
405D4—10762no—
406D4—21521no—
407D4—21521no—
408D4—21521no—
409C2×C4—24no—
410C2×C4—21no—
411C2×C4—21no—
412C2×C4—21no—
413C2×C4—21no—
414C2×C4—21no—
415C2×C4—2691no—
416C2×C4—2692no—
417C2×C4—2692no—
418C2×C4—2691no—
419C2×C4—5384no—
420C2×C4—5384no—
421C2×C4—5384no—
422C2×C4—5384no—
423C2×C4—5384no—
424C2×C4—5381no—
425C2×C4—5381no—
426C2×C4—5381no—
427C2×C4—5381no—
428C2×C4D4×D26911yescharacteristic
429C2×C4C23×D26911yescharacteristic
430C8—21no—
431C8—21no—
432C8—5381no—
433C8—5381no—
434C22—21no—
435C22—21no—
436C22—21no—
437C22—41no—
438C22—41no—
439C22—41no—
440C22—2692no—
441C22—5381no—
442C22—5381no—
443C22—5381no—
444C22—5381no—
445C22—5382no—
446C22—5381no—
447C22—10761no—
448C22—10761no—
449C22—10761no—
450C22—10761no—
451C22—10761no—
452C22—21524no—
453C22C2×D4×D26911yescharacteristic
454C4—22no—
455C4—22no—
456C4—21no—
457C4—41no—
458C4—2692no—
459C4—5381no—
460C4—5382no—
461C4—10761no—
462C4C2×D4×D26911yescharacteristic
463C4C2×D4×D26911yescharacteristic
464C2—41no—
465C2—81no—
466C2—5382no—
467C2—10761no—
468C2—10762no—
469C2—21521no—
470C2#120×D26911yescharacteristic
471C2#3177141111yescharacteristic
472C2#3177141411yescharacteristic
subgroupG/H
centerC22C2×D4×D269
commutator, cosocleC2×C4×C269C24
FrattiniC2×C4C23×D269
Fitting#448×C269C2
socleC22×C269C22×D4
hypercenter#448D269
hypocenterC269#3010
LCSG ▷ C2×C4×C269 ▷ C2×C269 ▷ C269
UCSC1 ◁ C22 ◁ C22×C4 ◁ #448

Complements

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

NH ≅ G/Ntype
1. C2×#22575501C2semidirect
2. C2×#22575480C2semidirect
3. C2×SD16×D269C2semidirect
4. #31771171C2semidirect
5. #31770753C2semidirect
6. #31770752C2semidirect
7. #31770749C2semidirect
8. #31770748C2semidirect
9. #31770745C2semidirect
10. #31770744C2semidirect
11. #31770742C2semidirect
12. #31770612C2semidirect
13. #31770600C2semidirect
14. #31770570C2semidirect
15. #448×C269C2semidirect
20. C2×#16864403C22semidirect
24. C2×#16864402C22semidirect
27. C2×Q8×D269C22semidirect
47. #22575378C22semidirect
48. #22575377C22semidirect
49. #22575377C22semidirect
50. #22575376C22semidirect
51. #22575375C22semidirect
52. #22575373C22semidirect
53. C22⋊#14524130C22semidirect
54. #22575365C22semidirect
55. #22575365C22semidirect
56. C22×Q8×C269C22semidirect
60. #103×C269C22semidirect
61. #103×C269C22semidirect
62. C22⋊C8×C269C22semidirect
68. #16864403D4semidirect
83. Q8×D269D4semidirect
236. C269#3010semidirect
249. #448D269semidirect
265. C22×Q8C2×D269semidirect
292. #103C2×D269semidirect
293. #103C2×D269semidirect
299. C22⋊C8C2×D269semidirect

Maximal subgroups

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

subgroupindex
C2×#225755012
C2×#225754802
C2×SD16×D2692
#317711712
#317707532
#317707522
#317707492
#317707482
#317707452
#317707442
#317707422
#317706122
#317706002
#317705702
#448×C2692
#3010269