Finite Group Databasebeta

← #46820159

Subgroups — #46820159

Order 34432.

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×#22575499C211yesmaximal, characteristic
2C2×#22575485C211yesmaximal, characteristic, complemented
3C2×SD16×D269C211yesmaximal, characteristic, complemented
4#31771178C211yesmaximal, characteristic, complemented
5#31770770C211yesmaximal, characteristic, complemented
6#31770767C211yesmaximal, characteristic, complemented
7#31770766C211yesmaximal, characteristic, complemented
8#31770758C211yesmaximal, characteristic, complemented
9#31770757C211yesmaximal, characteristic, complemented
10#31770755C211yesmaximal, characteristic, complemented
11#31770754C211yesmaximal, characteristic, complemented
12#31770608C211yesmaximal, characteristic, complemented
13#31770600C211yesmaximal, characteristic, complemented
14#31770570C211yesmaximal, characteristic, complemented
15#452×C269C211yesmaximal, characteristic, complemented
16#22575499—24no—
17#22575499—24no—
18C2×#16864403C2211yescharacteristic
19C2×#16864403C2211yescharacteristic
20C2×#16864402—21no—
21C2×#16864402C2211yescharacteristic
22C2×Q8×D269C2211yescharacteristic, complemented
23C22×D1076—21no—
24C22×D1076C2211yescharacteristic
25C2×D4×D269—21no—
26C2×D4×D269—21no—
27C2×D4×D269C2211yescharacteristic
28C2×D4×D269C2211yescharacteristic
29#22575485—24no—
30C2×#16864392C2211yescharacteristic, complemented
31C2×#16864391C2211yescharacteristic, complemented
32C2×#16864391C2211yescharacteristic, complemented
33C2×#16864390C2211yescharacteristic, complemented
34C2×#16864389C2211yescharacteristic, complemented
35C2×#16864389C2211yescharacteristic, complemented
36SD16×D269—24no—
37C2×#16864385C2211yescharacteristic, complemented
38C2×C8×D269C2211yescharacteristic, complemented
39#22575452—21no—
40#22575439—21no—
41#22575438—21no—
42#22575435—21no—
43#22575433C2211yescharacteristic, complemented
44#22575432C2211yescharacteristic, complemented
45#22575429C2211yescharacteristic, complemented
46#22575411C2211yescharacteristic, complemented
47#22575377C2211yescharacteristic, complemented
48#22575375C2211yescharacteristic, complemented
49#22575375C2211yescharacteristic, complemented
50#22575374C2211yescharacteristic, complemented
51#22575373C2211yescharacteristic, complemented
52#22575373C2211yescharacteristic, complemented
53C22⋊#14524130C2211yescharacteristic, complemented
54#22575365C2211yescharacteristic, complemented
55#22575365C2211yescharacteristic, complemented
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×D269—21no—
65C23×D269—21no—
66#16864403—22no—
67#16864403—22no—
68#16864403—22no—
69#16864403D414yes—
70#16864402—28no—
71#16864402—22no—
72#16864402—22no—
73C2×Dic538—21no—
74C2×Dic538D411yescharacteristic
75C2×Dic538C2311yescharacteristic
76Q8×D269—22no—
77Q8×D269—22no—
78C2×D1076—28no—
79C2×D1076—21no—
80C2×D1076—21no—
81C2×D1076—22no—
82C2×D1076—22no—
83C2×D1076—21no—
84C2×D1076D411yescharacteristic
85C2×D1076C2311yescharacteristic
86C2×D1076C2311yescharacteristic
87C2×C22⋊D269—21no—
88C2×C22⋊D269—21no—
89C2×C22⋊D269D411yescharacteristic
90C2×C22⋊D269D411yescharacteristic
91D4×D269—28no—
92D4×D269—28no—
93D4×D269—22no—
94D4×D269—22no—
95D4×D269—22no—
96D4×D269D414yes—
97C2×C4×D269—21no—
98C2×C4×D269—21no—
99C2×C4×D269—21no—
100C2×C4×D269C2311yescharacteristic
101C2×C4×D269C2311yescharacteristic
102#16864392—22no—
103#16864391—22no—
104#16864391—22no—
105#16864390—22no—
106#16864389—22no—
107#16864389—22no—
108#16864385—22no—
109C2×#14524130C2311yescharacteristic
110C2×#14524130C2311yescharacteristic
111C8×D269—22no—
112C4⋊Dic269—21no—
113C4⋊Dic269C2311yescharacteristic
114#16864381—21no—
115#16864381—21no—
116C22⋊Dic269—21no—
117#16864379—21no—
118#16864379—21no—
119#16864379—21no—
120C4×Dic269—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—21no—
137C22×D269—21no—
138C22×D269—24no—
139C22×D269—24no—
140C22×D269—24no—
141C22×D269—81no—
142C22×D269—81no—
143C22×D269—81no—
144C22×D269—81no—
145C22×D269—81no—
146C22×D269—81no—
147C22×D269C2×D411yescharacteristic
148C22×D269C2×D411yescharacteristic
149Dic538—22no—
150Dic538—21no—
151Dic538—21no—
152Dic538—42no—
153D1076—24no—
154D1076—24no—
155D1076—24no—
156D1076—22no—
157D1076—21no—
158D1076—21no—
159D1076C2×D412yes—
160D1076C2×D412yes—
161C22⋊D269—24no—
162C22⋊D269—22no—
163C22⋊D269—22no—
164C22⋊D269—24no—
165C2×Dic269—21no—
166C2×Dic269—21no—
167C2×Dic269C2×D411yescharacteristic
168C2×Dic269C2×D411yescharacteristic
169C4×D269—24no—
170C4×D269—21no—
171C4×D269—21no—
172C4×D269—24no—
173C4×D269—42no—
174C4×D269C2×D412yes—
175C4×D269C2×D412yes—
176#14524130—21no—
177#14524130—21no—
178C23×C269—21no—
179C23×C269C2×D411yescharacteristic
180Q8×C269—21no—
181Q8×C269—21no—
182Q8×C269—21no—
183Q8×C269C2×D412yes—
184D4×C269—24no—
185D4×C269—21no—
186D4×C269C2×D412yes—
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—22no—
199C22×C269—21no—
200C22×C269—21no—
201C22×C269—41no—
202C22×C269—41no—
203C22×C269C22×D411yescharacteristic
204C2×D269—21no—
205C2×D269—21no—
206C2×D269—21no—
207C2×D269—22no—
208C2×D269—22no—
209C2×D269—21no—
210C2×D269—21no—
211C2×D269—41no—
212C2×D269—41no—
213C2×D269—41no—
214C2×D269—41no—
215C2×D269—84no—
216C2×D269—84no—
217C2×D269—84no—
218C2×D269#12012yes—
219C4×C269—21no—
220C4×C269—22no—
221C4×C269—41no—
222C4×C269—41no—
223C4×C269C22×D411yescharacteristic
224C4×C269C22×D411yescharacteristic
225Dic269—21no—
226Dic269—41no—
227Dic269—41no—
228Dic269#12012yes—
229C2×C269—41no—
230C2×C269—42no—
231C2×C269C2×#12011yescharacteristic
232C2×C269#57711yescharacteristic
233C2×C269#57711yescharacteristic
234D269—22no—
235D269—41no—
236D269—42no—
237D269—42no—
238C269#300911yescharacteristic, complemented, Sylow
239#3009—2691nomaximal, Sylow
240C2×H(5,𝔽2)—2691no—
241C2×#137—2691no—
242C22×SD16—2691no—
243#536—2691no—
244#452—2691no—
245#452—2691no—
246#452—2691no—
247#452D26911yescharacteristic, complemented
248#450—2691no—
249#450—2691no—
250#450—2691no—
251#450—2691no—
252#418—2691no—
253#416—2691no—
254#408—2691no—
255H(5,𝔽2)—5384no—
256H(5,𝔽2)—5384no—
257C2×#41—2691no—
258C2×#41—2691no—
259C2×#41—2691no—
260C2×#41—5381no—
261C2×#41C2×D26911yescharacteristic
262C22×Q8—2691no—
263C22×D4—2691no—
264C22×D4—2691no—
265C22×D4—2691no—
266C22×D4—5381no—
267C22×D4—5381no—
268C22×D4—5381no—
269#137—5384no—
270C2×Q16—2691no—
271C2×Q16C2×D26911yescharacteristic, complemented
272C2×SD16—2691no—
273C2×SD16—2691no—
274C2×SD16—2691no—
275C2×SD16—2691no—
276C2×SD16—2691no—
277C2×SD16—5384no—
278C2×SD16C2×D26911yescharacteristic, complemented
279C2×M16—2691no—
280C22×C8—2691no—
281C4⋊Q8—5381no—
282#124—5381no—
283#123—5381no—
284#123—5381no—
285C22⋊Q8—2691no—
286C22⋊Q8—2691no—
287C22⋊Q8—2691no—
288C22⋊Q8C2×D26911yescharacteristic, complemented
289#117—2691no—
290#103—2691no—
291#103C2×D26911yescharacteristic, complemented
292#102—2691no—
293#102—2691no—
294#102—2691no—
295#102—2691no—
296#102—2691no—
297#102C2×D26911yescharacteristic, complemented
298C22⋊C8—2691no—
299C22⋊C8—2691no—
300C22⋊C8—2691no—
301C22⋊C8C2×D26911yescharacteristic, complemented
302C24—5381no—
303C24—5381no—
304C24—5381no—
305#41—22no—
306#41—22no—
307#41—2694no—
308#41—5382no—
309#41—5382no—
310#41—5388no—
311#41—5382no—
312#41—5382no—
313#41—5382no—
314C2×Q8—2691no—
315C2×Q8—2691no—
316C2×Q8—5381no—
317C2×Q8—5382no—
318C2×Q8—5382no—
319C2×Q8C22×D26911yescharacteristic
320C2×Q8C22×D26911yescharacteristic
321C2×D4—21no—
322C2×D4—2694no—
323C2×D4—2691no—
324C2×D4—2691no—
325C2×D4—2691no—
326C2×D4—2691no—
327C2×D4—2691no—
328C2×D4—5382no—
329C2×D4—5382no—
330C2×D4—5381no—
331C2×D4—5381no—
332C2×D4—5388no—
333C2×D4—5382no—
334C2×D4—5381no—
335C2×D4—5381no—
336C2×D4—5382no—
337C2×D4—5388no—
338C2×D4—5388no—
339C2×D4—5382no—
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×C4C22×D26911yescharacteristic
349Q16—22no—
350Q16—5382no—
351SD16—22no—
352SD16—5382no—
353SD16—5382no—
354SD16—5382no—
355SD16—5382no—
356SD16—5382no—
357M16—5382no—
358C2×C8—2691no—
359C2×C8—2691no—
360C2×C8—5382no—
361C2×C8C22×D26911yescharacteristic
362C2×C8C22×D26911yescharacteristic
363C4⋊C4—21no—
364C4⋊C4—2691no—
365C4⋊C4—5381no—
366C4⋊C4—5381no—
367C4⋊C4—5381no—
368C4⋊C4C22×D26911yescharacteristic
369C22⋊C4—21no—
370C22⋊C4—5381no—
371C22⋊C4—5381no—
372C22⋊C4—5381no—
373C22⋊C4—5381no—
374C42—5381no—
375C23—21no—
376C23—2691no—
377C23—2691no—
378C23—5384no—
379C23—5384no—
380C23—5384no—
381C23—5381no—
382C23—5381no—
383C23—21521no—
384C23—21521no—
385C23—21521no—
386C23—21521no—
387C23—21521no—
388C23—21521no—
389C23D4×D26911yescharacteristic
390Q8—21no—
391Q8—21no—
392Q8—21no—
393Q8—5382no—
394Q8—5381no—
395Q8—5381no—
396Q8—10762no—
397Q8D4×D26912yes—
398D4—24no—
399D4—21no—
400D4—2692no—
401D4—2692no—
402D4—5382no—
403D4—5384no—
404D4—5384no—
405D4—5384no—
406D4—5382no—
407D4—5384no—
408D4—5382no—
409D4—5384no—
410D4—5381no—
411D4—5381no—
412D4D4×D26912yes—
413C2×C4—24no—
414C2×C4—21no—
415C2×C4—21no—
416C2×C4—21no—
417C2×C4—21no—
418C2×C4—21no—
419C2×C4—2691no—
420C2×C4—2692no—
421C2×C4—2692no—
422C2×C4—2691no—
423C2×C4—5384no—
424C2×C4—5384no—
425C2×C4—5381no—
426C2×C4—5381no—
427C2×C4—5381no—
428C2×C4—5381no—
429C2×C4—10762no—
430C2×C4D4×D26911yescharacteristic
431C2×C4C23×D26911yescharacteristic
432C8—21no—
433C8—21no—
434C8—5381no—
435C8—5381no—
436C22—21no—
437C22—21no—
438C22—21no—
439C22—22no—
440C22—41no—
441C22—41no—
442C22—2692no—
443C22—5381no—
444C22—5381no—
445C22—5381no—
446C22—5381no—
447C22—5382no—
448C22—5382no—
449C22—5381no—
450C22—10761no—
451C22—10761no—
452C22—10761no—
453C22—10761no—
454C22—21524no—
455C22—21524no—
456C22—21524no—
457C22C2×D4×D26911yescharacteristic
458C4—21no—
459C4—22no—
460C4—41no—
461C4—41no—
462C4—2692no—
463C4—5381no—
464C4—10761no—
465C4—10761no—
466C4C2×D4×D26911yescharacteristic
467C4C2×D4×D26911yescharacteristic
468C2—41no—
469C2—42no—
470C2—5382no—
471C2—10761no—
472C2—10762no—
473C2—10762no—
474C2#120×D26911yescharacteristic
475C2#3177141211yescharacteristic
476C2#3177141411yescharacteristic
subgroupG/H
centerC22C2×D4×D269
commutator, cosocleC2×C4×C269C24
FrattiniC2×C4C23×D269
Fitting#452×C269C2
socleC22×C269C22×D4
prime radicalC2×#22575499C2
hypercenter#452D269
hypocenterC269#3009
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×SD16×D269C2semidirect
4. #31771178C2semidirect
5. #31770770C2semidirect
6. #31770767C2semidirect
7. #31770766C2semidirect
8. #31770758C2semidirect
9. #31770757C2semidirect
10. #31770755C2semidirect
11. #31770754C2semidirect
12. #31770608C2semidirect
13. #31770600C2semidirect
14. #31770570C2semidirect
15. #452×C269C2semidirect
22. C2×Q8×D269C22semidirect
30. C2×#16864392C22semidirect
31. C2×#16864391C22semidirect
32. C2×#16864391C22semidirect
33. C2×#16864390C22semidirect
34. C2×#16864389C22semidirect
35. C2×#16864389C22semidirect
37. C2×#16864385C22semidirect
38. C2×C8×D269C22semidirect
43. #22575433C22semidirect
44. #22575432C22semidirect
45. #22575429C22semidirect
46. #22575411C22semidirect
47. #22575377C22semidirect
48. #22575375C22semidirect
49. #22575375C22semidirect
50. #22575374C22semidirect
51. #22575373C22semidirect
52. #22575373C22semidirect
53. C22⋊#14524130C22semidirect
54. #22575365C22semidirect
55. #22575365C22semidirect
57. C2×Q16×C269C22semidirect
58. C2×SD16×C269C22semidirect
59. C22⋊Q8×C269C22semidirect
60. #103×C269C22semidirect
61. #102×C269C22semidirect
62. C22⋊C8×C269C22semidirect
238. C269#3009semidirect
247. #452D269semidirect
271. C2×Q16C2×D269semidirect
278. C2×SD16C2×D269semidirect
288. C22⋊Q8C2×D269semidirect
291. #103C2×D269semidirect
297. #102C2×D269semidirect
301. C22⋊C8C2×D269semidirect

Maximal subgroups

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

subgroupindex
C2×#225754992
C2×#225754852
C2×SD16×D2692
#317711782
#317707702
#317707672
#317707662
#317707582
#317707572
#317707552
#317707542
#317706082
#317706002
#317705702
#452×C2692
#3009269