Finite Group Databasebeta

← #46820100

Subgroups — #46820100

Order 34432.

764 conjugacy classes of subgroups.

The subgroup lattice has 476 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#41×D269C211yesmaximal, characteristic, complemented
2C2×#22575474C211yesmaximal, characteristic, complemented
3C2×#22575472C211yesmaximal, characteristic, complemented
4#31771115C211yesmaximal, characteristic, complemented
5#31770769C211yesmaximal, characteristic, complemented
6#31770767C211yesmaximal, characteristic, complemented
7#31770762C211yesmaximal, characteristic, complemented
8#31770757C211yesmaximal, characteristic, complemented
9#31770748C211yesmaximal, characteristic, complemented
10#31770742C211yesmaximal, characteristic, complemented
11#31770737C211yesmaximal, characteristic, complemented
12#31770600C211yesmaximal, characteristic, complemented
13#31770598C211yesmaximal, characteristic, complemented
14C22⋊C8×D269C211yesmaximal, characteristic, complemented
15#449×C269C211yesmaximal, characteristic, complemented
16C2×#16864403C2211yescharacteristic
17C2×#16864402—21no—
18C2×#16864402C2211yescharacteristic
19C2×#16864401—21no—
20C2×#16864401C2211yescharacteristic
21C2×#16864401C2211yescharacteristic, complemented
22#41×D269—24no—
23#41×D269—24no—
24C2×Q8×D269C2211yescharacteristic
25C2×D4×D269—21no—
26C2×D4×D269C2211yescharacteristic
27C22×C4×D269—21no—
28C22×C4×D269C2211yescharacteristic
29#22575474—24no—
30#22575472—24no—
31C2×Dic1076C2211yescharacteristic, complemented
32C2×#16864392C2211yescharacteristic, complemented
33C2×#16864391C2211yescharacteristic, complemented
34C2×#16864390C2211yescharacteristic, complemented
35C2×#16864390C2211yescharacteristic, complemented
36C2×#16864387C2211yescharacteristic, complemented
37C2×C8×D269C2211yescharacteristic, complemented
38C2×C8×D269C2211yescharacteristic, complemented
39#22575435—21no—
40#22575433C2211yescharacteristic
41#22575432C2211yescharacteristic, complemented
42#22575424C2211yescharacteristic
43#22575421—21no—
44D4×Dic269—21no—
45#22575413—21no—
46#22575411C2211yescharacteristic, complemented
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×SD16×C269C2211yescharacteristic, complemented
58C2×D8×C269C2211yescharacteristic, complemented
59#121×C269C2211yescharacteristic, complemented
60#103×C269C2211yescharacteristic, complemented
61#102×C269C2211yescharacteristic, complemented
62C22⋊C8×C269C2211yescharacteristic, complemented
63C23×D269—21no—
64C23×D269D411yescharacteristic
65#16864403—22no—
66#16864403D414yescomplemented
67#16864402—28no—
68#16864402—22no—
69#16864402—22no—
70#16864401—28no—
71#16864401—22no—
72#16864401—22no—
73#16864401—22no—
74#16864401D414yescomplemented
75C2×Dic538—21no—
76C2×Dic538C2311yescharacteristic
77C2×Dic538C2311yescharacteristic
78Q8×D269—22no—
79Q8×D269—22no—
80C2×D1076—21no—
81C2×D1076C2311yescharacteristic
82C2×C22⋊D269—21no—
83C2×C22⋊D269—21no—
84C2×C22⋊D269—21no—
85C2×C22⋊D269—21no—
86C2×C22⋊D269—21no—
87D4×D269—28no—
88D4×D269—22no—
89D4×D269—22no—
90C22×Dic269—21no—
91C22×Dic269—21no—
92C22×Dic269D411yescharacteristic
93C2×C4×D269—28no—
94C2×C4×D269—21no—
95C2×C4×D269—21no—
96C2×C4×D269—22no—
97C2×C4×D269—22no—
98C2×C4×D269—21no—
99C2×C4×D269D411yescharacteristic
100C2×C4×D269D411yescharacteristic
101C2×C4×D269C2311yescharacteristic
102C2×C4×D269C2311yescharacteristic
103Dic1076—22no—
104#16864392—22no—
105#16864391—22no—
106#16864390—22no—
107#16864390—22no—
108#16864387—22no—
109C2×#14524130C2311yescharacteristic
110C2×#14524130C2311yescharacteristic
111C8×D269—22no—
112C8×D269—22no—
113C4⋊Dic269—21no—
114C4⋊Dic269C2311yescharacteristic
115#16864381—21no—
116C22⋊Dic269—21no—
117C22⋊Dic269—21no—
118#16864379—21no—
119#16864379—21no—
120C4×Dic269—21no—
121#41×C269—22no—
122#41×C269—22no—
123C2×Q8×C269C2311yescharacteristic
124C2×D4×C269—21no—
125C2×D4×C269—21no—
126C2×D4×C269C2311yescharacteristic
127C2×D4×C269C2311yescharacteristic
128C22×C4×C269—21no—
129C22×C4×C269C2311yescharacteristic
130SD16×C269—22no—
131D8×C269—22no—
132C2×C8×C269C2311yescharacteristic
133C2×C8×C269C2311yescharacteristic
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—
149Dic538C2×D412yes—
150D1076—24no—
151D1076—21no—
152D1076C2×D412yes—
153C22⋊D269—24no—
154C22⋊D269—24no—
155C22⋊D269—24no—
156C22⋊D269—42no—
157C22⋊D269—42no—
158C2×Dic269—21no—
159C2×Dic269—24no—
160C2×Dic269—22no—
161C2×Dic269—21no—
162C2×Dic269—21no—
163C2×Dic269—42no—
164C2×Dic269C2×D411yescharacteristic
165C2×Dic269C2×D411yescharacteristic
166C4×D269—24no—
167C4×D269—24no—
168C4×D269—22no—
169C4×D269—21no—
170C4×D269—21no—
171C4×D269—22no—
172C4×D269—24no—
173C4×D269C2×D412yes—
174C4×D269C2×D412yes—
175#14524130—21no—
176#14524130—21no—
177C23×C269—21no—
178C23×C269—21no—
179C23×C269C2×D411yescharacteristic
180Q8×C269—21no—
181Q8×C269C2×D412yes—
182D4×C269—24no—
183D4×C269—21no—
184D4×C269—21no—
185D4×C269—21no—
186D4×C269—42no—
187D4×C269C2×D412yes—
188C2×C4×C269—24no—
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×C269—41no—
204C22×C269—41no—
205C22×C269—41no—
206C22×C269C22×D411yescharacteristic
207C2×D269—21no—
208C2×D269—21no—
209C2×D269—21no—
210C2×D269—21no—
211C2×D269—22no—
212C2×D269—44no—
213C2×D269—48no—
214C2×D269—41no—
215C2×D269—41no—
216C2×D269#12012yes—
217C2×D269#13512yescomplemented
218C4×C269—21no—
219C4×C269—22no—
220C4×C269—41no—
221C4×C269C22×D411yescharacteristic
222C4×C269C22×D411yescharacteristic
223Dic269—22no—
224Dic269—21no—
225Dic269—41no—
226Dic269—41no—
227Dic269#12012yes—
228C2×C269—41no—
229C2×C269—42no—
230C2×C269—81no—
231C2×C269C2×#12011yescharacteristic
232C2×C269C2×#13511yescharacteristic
233C2×C269#57511yescharacteristic
234D269—22no—
235D269—22no—
236D269—42no—
237C269#300211yescharacteristic, complemented, Sylow
238#3002—2691nomaximal, Sylow
239C22×#41—2691no—
240C2×#135—2691no—
241C2×#135—2691no—
242#525—2691no—
243#452—2691no—
244#452—2691no—
245#451—2691no—
246#450—2691no—
247#449—2691no—
248#449D26911yescharacteristic, complemented
249#448—2691no—
250#447—2691no—
251#416—2691no—
252#416—2691no—
253C2×C22⋊C8—2691no—
254C2×#41—2691no—
255C2×#41—2691no—
256C2×#41—2691no—
257C2×#41—2691no—
258C2×#41—5381no—
259C2×#41—5381no—
260C2×#41—5384no—
261C2×#41—5384no—
262C2×#41C2×D26911yescharacteristic
263C22×Q8—2691no—
264C22×D4—2691no—
265C22×D4—5381no—
266C23×C4—2691no—
267C23×C4—5381no—
268#135—5384no—
269#135—5384no—
270C2×Q16—2691no—
271C2×Q16—2691no—
272C2×SD16—2691no—
273C2×SD16—2691no—
274C2×SD16—2691no—
275C2×SD16C2×D26911yescharacteristic, complemented
276C2×D8—2691no—
277C2×D8C2×D26911yescharacteristic, complemented
278C22×C8—2691no—
279C22×C8—2691no—
280#123—5381no—
281C22⋊Q8—2691no—
282C22⋊Q8—2691no—
283#121—2691no—
284#121C2×D26911yescharacteristic, complemented
285#120—5381no—
286C4×D4—5381no—
287C4×D4—5381no—
288#117—2691no—
289#103—2691no—
290#103—2691no—
291#103—2691no—
292#103C2×D26911yescharacteristic, complemented
293#102—2691no—
294#102—2691no—
295#102—2691no—
296#102C2×D26911yescharacteristic, complemented
297C22⋊C8—2691no—
298C22⋊C8—2691no—
299C22⋊C8—2691no—
300C22⋊C8C2×D26911yescharacteristic, complemented
301C24—2691no—
302C24—5381no—
303#41—22no—
304#41—22no—
305#41—2694no—
306#41—2694no—
307#41—5382no—
308#41—5382no—
309#41—5388no—
310#41—5382no—
311#41—5382no—
312#41—5382no—
313#41—5382no—
314#41—5388no—
315C2×Q8—2691no—
316C2×Q8—2691no—
317C2×Q8—5381no—
318C2×Q8—5382no—
319C2×Q8—5382no—
320C2×Q8C22×D26911yescharacteristic
321C2×D4—21no—
322C2×D4—21no—
323C2×D4—2691no—
324C2×D4—5381no—
325C2×D4—5381no—
326C2×D4—5382no—
327C2×D4—5388no—
328C2×D4—5381no—
329C2×D4—5381no—
330C2×D4—5382no—
331C2×D4—5381no—
332C2×D4—5381no—
333C2×D4C22×D26911yescharacteristic
334C2×D4C22×D26911yescharacteristic
335C22×C4—21no—
336C22×C4—2691no—
337C22×C4—2691no—
338C22×C4—2691no—
339C22×C4—2691no—
340C22×C4—2691no—
341C22×C4—5382no—
342C22×C4—5382no—
343C22×C4—5381no—
344C22×C4—5381no—
345C22×C4—5388no—
346C22×C4—5381no—
347C22×C4—5381no—
348C22×C4—5381no—
349C22×C4C22×D26911yescharacteristic
350Q16—5382no—
351Q16—5382no—
352SD16—22no—
353SD16—5382no—
354SD16—5382no—
355SD16—5382no—
356D8—22no—
357D8—5382no—
358C2×C8—2691no—
359C2×C8—2691no—
360C2×C8—5382no—
361C2×C8—5382no—
362C2×C8C22×D26911yescharacteristic
363C2×C8C22×D26911yescharacteristic
364C4⋊C4—2691no—
365C4⋊C4—5381no—
366C4⋊C4—5381no—
367C4⋊C4C22×D26911yescharacteristic
368C22⋊C4—21no—
369C22⋊C4—5381no—
370C22⋊C4—5381no—
371C22⋊C4—5381no—
372C22⋊C4—5381no—
373C42—5381no—
374C23—21no—
375C23—21no—
376C23—2691no—
377C23—2691no—
378C23—5382no—
379C23—5382no—
380C23—5384no—
381C23—5381no—
382C23—10761no—
383C23—10762no—
384C23—10762no—
385C23D4×D26911yescharacteristic
386Q8—21no—
387Q8—2692no—
388Q8—5384no—
389Q8—5381no—
390Q8—5381no—
391Q8—5381no—
392Q8D4×D26912yes—
393D4—24no—
394D4—21no—
395D4—21no—
396D4—21no—
397D4—42no—
398D4—2692no—
399D4—5384no—
400D4—5384no—
401D4—5384no—
402D4—5384no—
403D4—5381no—
404D4—10762no—
405D4—10762no—
406D4D4×D26912yes—
407C2×C4—24no—
408C2×C4—21no—
409C2×C4—21no—
410C2×C4—21no—
411C2×C4—21no—
412C2×C4—2691no—
413C2×C4—2692no—
414C2×C4—2692no—
415C2×C4—2691no—
416C2×C4—5382no—
417C2×C4—5384no—
418C2×C4—5382no—
419C2×C4—5384no—
420C2×C4—5384no—
421C2×C4—5382no—
422C2×C4—5384no—
423C2×C4—5381no—
424C2×C4—5381no—
425C2×C4—5381no—
426C2×C4—5381no—
427C2×C4—5381no—
428C2×C4—10762no—
429C2×C4D4×D26911yescharacteristic
430C2×C4C23×D26911yescharacteristic
431C8—21no—
432C8—21no—
433C8—5381no—
434C8—5381no—
435C22—21no—
436C22—21no—
437C22—22no—
438C22—21no—
439C22—41no—
440C22—41no—
441C22—41no—
442C22—41no—
443C22—41no—
444C22—2692no—
445C22—2692no—
446C22—5381no—
447C22—5381no—
448C22—5381no—
449C22—5381no—
450C22—5382no—
451C22—10764no—
452C22—10768no—
453C22—10761no—
454C22—10761no—
455C22C2×D4×D26911yescharacteristic
456C4—21no—
457C4—22no—
458C4—41no—
459C4—2692no—
460C4—5381no—
461C4—5382no—
462C4—10761no—
463C4—10761no—
464C4C2×D4×D26911yescharacteristic
465C4C2×D4×D26911yescharacteristic
466C2—41no—
467C2—42no—
468C2—81no—
469C2—5382no—
470C2—5382no—
471C2—10762no—
472C2#120×D26911yescharacteristic
473C2#135×D26911yescharacteristic
474C2#3177140111yescharacteristic
subgroupG/H
centerC22C2×D4×D269
commutator, cosocleC2×C4×C269C24
FrattiniC2×C4C23×D269
Fitting#449×C269C2
socleC22×C269C22×D4
hypercenter#449D269
hypocenterC269#3002
LCSG ▷ C2×C4×C269 ▷ C2×C269 ▷ C269
UCSC1 ◁ C22 ◁ C22×C4 ◁ #449

Complements

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

NH ≅ G/Ntype
1. C2×#41×D269C2semidirect
2. C2×#22575474C2semidirect
3. C2×#22575472C2semidirect
4. #31771115C2semidirect
5. #31770769C2semidirect
6. #31770767C2semidirect
7. #31770762C2semidirect
8. #31770757C2semidirect
9. #31770748C2semidirect
10. #31770742C2semidirect
11. #31770737C2semidirect
12. #31770600C2semidirect
13. #31770598C2semidirect
14. C22⋊C8×D269C2semidirect
15. #449×C269C2semidirect
21. C2×#16864401C22semidirect
31. C2×Dic1076C22semidirect
32. C2×#16864392C22semidirect
33. C2×#16864391C22semidirect
34. C2×#16864390C22semidirect
35. C2×#16864390C22semidirect
36. C2×#16864387C22semidirect
37. C2×C8×D269C22semidirect
38. C2×C8×D269C22semidirect
41. #22575432C22semidirect
46. #22575411C22semidirect
47. #22575378C22semidirect
48. #22575377C22semidirect
49. #22575376C22semidirect
50. #22575375C22semidirect
51. #22575374C22semidirect
52. #22575373C22semidirect
53. C22⋊#14524130C22semidirect
57. C2×SD16×C269C22semidirect
58. C2×D8×C269C22semidirect
59. #121×C269C22semidirect
60. #103×C269C22semidirect
61. #102×C269C22semidirect
62. C22⋊C8×C269C22semidirect
66. #16864403D4semidirect
74. #16864401D4semidirect
217. C2×D269#135semidirect
237. C269#3002semidirect
248. #449D269semidirect
275. C2×SD16C2×D269semidirect
277. C2×D8C2×D269semidirect
284. #121C2×D269semidirect
292. #103C2×D269semidirect
296. #102C2×D269semidirect
300. 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×#225754742
C2×#225754722
#317711152
#317707692
#317707672
#317707622
#317707572
#317707482
#317707422
#317707372
#317706002
#317705982
C22⋊C8×D2692
#449×C2692
#3002269