Finite Group Databasebeta

← #46820105

Subgroups — #46820105

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×#22575475C211yesmaximal, characteristic, complemented
3C2×#22575473C211yesmaximal, characteristic, complemented
4#31771117C211yesmaximal, characteristic, complemented
5#31770770C211yesmaximal, characteristic, complemented
6#31770768C211yesmaximal, characteristic, complemented
7#31770762C211yesmaximal, characteristic, complemented
8#31770757C211yesmaximal, characteristic, complemented
9#31770753C211yesmaximal, characteristic, complemented
10#31770751C211yesmaximal, characteristic, complemented
11#31770745C211yesmaximal, characteristic, complemented
12#31770740C211yesmaximal, characteristic, complemented
13#31770599C211yesmaximal, characteristic, complemented
14C2×#22575365C211yesmaximal, characteristic, complemented
15#416×C269C211yesmaximal, characteristic, complemented
16C2×#16864403C2211yescharacteristic
17C2×#16864402—21no—
18C2×#16864402C2211yescharacteristic
19C2×#16864402C2211yescharacteristic, complemented
20C2×#16864401—21no—
21C2×#16864401C2211yescharacteristic
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#22575475—24no—
30#22575473—24no—
31C2×Dic1076C2211yescharacteristic, complemented
32C2×#16864392C2211yescharacteristic, complemented
33C2×#16864391C2211yescharacteristic, complemented
34C2×#16864391C2211yescharacteristic, complemented
35C2×#16864390C2211yescharacteristic, complemented
36C2×#16864389C2211yescharacteristic, complemented
37C2×D2152C2211yescharacteristic, complemented
38C2×#16864387C2211yescharacteristic, complemented
39C22×#14524130C2211yescharacteristic, complemented
40#22575437—21no—
41#22575433C2211yescharacteristic
42#22575433C2211yescharacteristic, complemented
43#22575428C2211yescharacteristic
44#22575428C2211yescharacteristic, complemented
45#22575423—21no—
46C4×D1076—21no—
47C4×D1076—21no—
48#22575376C2211yescharacteristic, complemented
49#22575376C2211yescharacteristic, complemented
50#22575373C2211yescharacteristic, complemented
51#22575373C2211yescharacteristic, complemented
52#22575365C2211yescharacteristic
53#22575365C2211yescharacteristic
54#22575365C2211yescharacteristic, complemented
55#22575365C2211yescharacteristic, complemented
56C2×#41×C269C2211yescharacteristic
57C22×C8×C269C2211yescharacteristic, complemented
58#117×C269C2211yescharacteristic, complemented
59#103×C269C2211yescharacteristic, complemented
60#103×C269C2211yescharacteristic, complemented
61#102×C269C2211yescharacteristic, complemented
62#102×C269C2211yescharacteristic, complemented
63C23×D269—21no—
64C23×D269D411yescharacteristic
65#16864403—22no—
66#16864403—22no—
67#16864402—28no—
68#16864402—22no—
69#16864402—22no—
70#16864402—22no—
71#16864402D414yescomplemented
72#16864401—28no—
73#16864401—22no—
74#16864401—22no—
75C2×Dic538—21no—
76C2×Dic538C2311yescharacteristic
77C2×Dic538C2311yescharacteristic
78Q8×D269—22no—
79Q8×D269—22no—
80C2×D1076—21no—
81C2×D1076—21no—
82C2×D1076—21no—
83C2×D1076C2311yescharacteristic
84C2×D1076C2311yescharacteristic
85C2×C22⋊D269—21no—
86C2×C22⋊D269—21no—
87C2×C22⋊D269—21no—
88C2×C22⋊D269—21no—
89D4×D269—28no—
90D4×D269—22no—
91D4×D269—22no—
92C22×Dic269—21no—
93C22×Dic269D411yescharacteristic
94C2×C4×D269—28no—
95C2×C4×D269—21no—
96C2×C4×D269—21no—
97C2×C4×D269—21no—
98C2×C4×D269—22no—
99C2×C4×D269—22no—
100C2×C4×D269—21no—
101C2×C4×D269D411yescharacteristic
102C2×C4×D269D411yescharacteristic
103C2×C4×D269C2311yescharacteristic
104C2×C4×D269C2311yescharacteristic
105Dic1076—22no—
106#16864392—22no—
107#16864391—22no—
108#16864391—22no—
109#16864390—22no—
110#16864389—22no—
111D2152—22no—
112#16864387—22no—
113C2×#14524130—22no—
114C2×#14524130C2311yescharacteristic
115C2×#14524130C2311yescharacteristic
116C4⋊Dic269—21no—
117C4⋊Dic269—21no—
118#16864379—21no—
119#16864379—21no—
120#16864379—21no—
121#16864379—21no—
122#41×C269—22no—
123#41×C269D414yescomplemented
124C2×Q8×C269C2311yescharacteristic
125C2×D4×C269—21no—
126C2×D4×C269C2311yescharacteristic
127C22×C4×C269—21no—
128C22×C4×C269C2311yescharacteristic
129C2×C8×C269—22no—
130C2×C8×C269C2311yescharacteristic
131C2×C8×C269C2311yescharacteristic
132C4⋊C4×C269C2311yescharacteristic
133C4⋊C4×C269C2311yescharacteristic
134C22⋊C4×C269—21no—
135C42×C269—21no—
136C22×D269—24no—
137C22×D269—22no—
138C22×D269—21no—
139C22×D269—22no—
140C22×D269—21no—
141C22×D269—41no—
142C22×D269—42no—
143C22×D269—42no—
144C22×D269C2×D411yescharacteristic
145C22×D269C2×D411yescharacteristic
146Dic538—24no—
147Dic538—21no—
148Dic538—21no—
149Dic538—21no—
150Dic538C2×D412yes—
151D1076—24no—
152D1076—21no—
153D1076—21no—
154D1076—21no—
155D1076—42no—
156D1076—42no—
157D1076C2×D412yes—
158C22⋊D269—24no—
159C22⋊D269—24no—
160C22⋊D269—24no—
161C22⋊D269—42no—
162C2×Dic269—24no—
163C2×Dic269—22no—
164C2×Dic269—21no—
165C2×Dic269—21no—
166C2×Dic269C2×D411yescharacteristic
167C2×Dic269C2×D411yescharacteristic
168C4×D269—24no—
169C4×D269—24no—
170C4×D269—22no—
171C4×D269—24no—
172C4×D269—22no—
173C4×D269—21no—
174C4×D269—21no—
175C4×D269—21no—
176C4×D269—21no—
177C4×D269—42no—
178#14524130—21no—
179#14524130—21no—
180C23×C269—21no—
181C23×C269C2×D411yescharacteristic
182Q8×C269—21no—
183Q8×C269C2×D412yes—
184D4×C269—24no—
185D4×C269—21no—
186D4×C269C2×D412yes—
187C2×C4×C269—21no—
188C2×C4×C269—24no—
189C2×C4×C269—21no—
190C2×C4×C269—21no—
191C2×C4×C269C2×D412yes—
192C2×C4×C269C2×D411yescharacteristic
193C2×C4×C269C2×D412yes—
194C2×C4×C269C2411yescharacteristic
195C8×C269—21no—
196C8×C269—21no—
197C22×C269—21no—
198C22×C269—21no—
199C22×C269—22no—
200C22×C269—41no—
201C22×C269—41no—
202C22×C269#12012yes—
203C22×C269C22×D411yescharacteristic
204C2×D269—21no—
205C2×D269—21no—
206C2×D269—21no—
207C2×D269—22no—
208C2×D269—21no—
209C2×D269—21no—
210C2×D269—44no—
211C2×D269—48no—
212C2×D269—41no—
213C2×D269—41no—
214C2×D269—41no—
215C2×D269—41no—
216C2×D269—41no—
217C2×D269#13512yescomplemented
218C4×C269—22no—
219C4×C269—41no—
220C4×C269—41no—
221C4×C269#12012yes—
222C4×C269C22×D411yescharacteristic
223C4×C269C22×D411yescharacteristic
224Dic269—21no—
225Dic269—22no—
226Dic269—21no—
227Dic269—41no—
228C2×C269—22no—
229C2×C269—42no—
230C2×C269C2×#12011yescharacteristic
231C2×C269C2×#13511yescharacteristic
232C2×C269#57511yescharacteristic
233D269—22no—
234D269—41no—
235D269—42no—
236D269—81no—
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#449—2691no—
249#448—2691no—
250#447—2691no—
251#416—2691no—
252#416D26911yescharacteristic, complemented
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×SD16—2691no—
276C2×D8—2691no—
277C2×D8—2691no—
278C22×C8—2691no—
279C22×C8C2×D26911yescharacteristic, complemented
280#123—5381no—
281C22⋊Q8—2691no—
282C22⋊Q8—2691no—
283#121—2691no—
284#121—2691no—
285#120—5381no—
286C4×D4—5381no—
287C4×D4—5381no—
288#117C2×D26911yescharacteristic, complemented
289#103—2691no—
290#103—2691no—
291#103C2×D26911yescharacteristic, complemented
292#103C2×D26911yescharacteristic, complemented
293#102—2691no—
294#102—2691no—
295#102C2×D26911yescharacteristic, complemented
296#102C2×D26911yescharacteristic, complemented
297C22⋊C8—2691no—
298C22⋊C8—2691no—
299C22⋊C8—2691no—
300C22⋊C8—2691no—
301C24—2691no—
302C24—5381no—
303#41—22no—
304#41—2694no—
305#41—5388no—
306#41—5382no—
307#41—5388no—
308#41—5382no—
309#41—5382no—
310#41—5382no—
311#41—5382no—
312#41—5382no—
313#41—5382no—
314#41D107614yescomplemented
315C2×Q8—2691no—
316C2×Q8—2691no—
317C2×Q8—5381no—
318C2×Q8—5382no—
319C2×Q8—5382no—
320C2×Q8C22×D26911yescharacteristic
321C2×D4—21no—
322C2×D4—2691no—
323C2×D4—2691no—
324C2×D4—5381no—
325C2×D4—5381no—
326C2×D4—5381no—
327C2×D4—5381no—
328C2×D4—5382no—
329C2×D4—5388no—
330C2×D4—5381no—
331C2×D4—5381no—
332C2×D4—5382no—
333C2×D4—5381no—
334C2×D4C22×D26911yescharacteristic
335C22×C4—21no—
336C22×C4—2691no—
337C22×C4—2691no—
338C22×C4—2691no—
339C22×C4—2691no—
340C22×C4—2691no—
341C22×C4—5381no—
342C22×C4—5381no—
343C22×C4—5388no—
344C22×C4—5381no—
345C22×C4—5382no—
346C22×C4—5381no—
347C22×C4—5382no—
348C22×C4—5381no—
349C22×C4C22×D26911yescharacteristic
350Q16—5382no—
351Q16—5382no—
352SD16—5382no—
353SD16—5382no—
354SD16—5382no—
355SD16—5382no—
356D8—5382no—
357D8—5382no—
358C2×C8—22no—
359C2×C8—2691no—
360C2×C8—2691no—
361C2×C8—5382no—
362C2×C8C22×D26911yescharacteristic
363C2×C8C22×D26911yescharacteristic
364C4⋊C4—5381no—
365C4⋊C4—5381no—
366C4⋊C4C22×D26911yescharacteristic
367C4⋊C4C22×D26911yescharacteristic
368C22⋊C4—21no—
369C22⋊C4—5381no—
370C22⋊C4—5381no—
371C22⋊C4—5381no—
372C22⋊C4—5381no—
373C42—21no—
374C23—21no—
375C23—2691no—
376C23—2691no—
377C23—5382no—
378C23—5384no—
379C23—5382no—
380C23—5381no—
381C23—5381no—
382C23—10761no—
383C23—10762no—
384C23—10762no—
385C23D4×D26911yescharacteristic
386Q8—21no—
387Q8—2692no—
388Q8—5384no—
389Q8—5381no—
390Q8—5381no—
391Q8—5381no—
392Q8C2×D107612yes—
393D4—24no—
394D4—21no—
395D4—2692no—
396D4—5384no—
397D4—5384no—
398D4—5384no—
399D4—5384no—
400D4—5381no—
401D4—5381no—
402D4—5381no—
403D4—10762no—
404D4—10762no—
405D4—10762no—
406D4C2×D107612yes—
407C2×C4—21no—
408C2×C4—24no—
409C2×C4—21no—
410C2×C4—21no—
411C2×C4—2691no—
412C2×C4—2691no—
413C2×C4—5384no—
414C2×C4—5384no—
415C2×C4—5384no—
416C2×C4—5382no—
417C2×C4—5381no—
418C2×C4—5382no—
419C2×C4—5382no—
420C2×C4—5384no—
421C2×C4—5381no—
422C2×C4—5381no—
423C2×C4—5381no—
424C2×C4—5381no—
425C2×C4—5381no—
426C2×C4—10762no—
427C2×C4D4×D26912yes—
428C2×C4D4×D26911yescharacteristic
429C2×C4C2×D107612yes—
430C2×C4C23×D26911yescharacteristic
431C8—21no—
432C8—21no—
433C8—5381no—
434C8—5381no—
435C22—21no—
436C22—21no—
437C22—22no—
438C22—41no—
439C22—41no—
440C22—2692no—
441C22—5381no—
442C22—5381no—
443C22—5381no—
444C22—5382no—
445C22—5381no—
446C22—5381no—
447C22—10764no—
448C22—10768no—
449C22—10761no—
450C22—10761no—
451C22—10761no—
452C22—10761no—
453C22—10761no—
454C22#2257542312yes—
455C22C2×D4×D26911yescharacteristic
456C4—22no—
457C4—41no—
458C4—41no—
459C4—5381no—
460C4—5381no—
461C4—5382no—
462C4—10761no—
463C4#2257542312yes—
464C4C2×D4×D26911yescharacteristic
465C4C22×D107611yescharacteristic
466C2—22no—
467C2—42no—
468C2—5382no—
469C2—10761no—
470C2—10762no—
471C2—21521no—
472C2C2×#2257542311yescharacteristic
473C2#135×D26911yescharacteristic
474C2#3177140211yescharacteristic
subgroupG/H
centerC2×C4D4×D269
commutator, cosocleC2×C4×C269C24
FrattiniC2×C4C23×D269
Fitting#416×C269C2
socleC22×C269C22×D4
hypercenter#416D269
hypocenterC269#3002
LCSG ▷ C2×C4×C269 ▷ C2×C269 ▷ C269
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ #416

Complements

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

NH ≅ G/Ntype
1. C2×#41×D269C2semidirect
2. C2×#22575475C2semidirect
3. C2×#22575473C2semidirect
4. #31771117C2semidirect
5. #31770770C2semidirect
6. #31770768C2semidirect
7. #31770762C2semidirect
8. #31770757C2semidirect
9. #31770753C2semidirect
10. #31770751C2semidirect
11. #31770745C2semidirect
12. #31770740C2semidirect
13. #31770599C2semidirect
14. C2×#22575365C2semidirect
15. #416×C269C2semidirect
19. C2×#16864402C22semidirect
31. C2×Dic1076C22semidirect
32. C2×#16864392C22semidirect
33. C2×#16864391C22semidirect
34. C2×#16864391C22semidirect
35. C2×#16864390C22semidirect
36. C2×#16864389C22semidirect
37. C2×D2152C22semidirect
38. C2×#16864387C22semidirect
39. C22×#14524130C22semidirect
42. #22575433C22semidirect
44. #22575428C22semidirect
48. #22575376C22semidirect
49. #22575376C22semidirect
50. #22575373C22semidirect
51. #22575373C22semidirect
54. #22575365C22semidirect
55. #22575365C22semidirect
57. C22×C8×C269C22semidirect
58. #117×C269C22semidirect
59. #103×C269C22semidirect
60. #103×C269C22semidirect
61. #102×C269C22semidirect
62. #102×C269C22semidirect
71. #16864402D4semidirect
123. #41×C269D4semidirect
217. C2×D269#135semidirect
237. C269#3002semidirect
252. #416D269semidirect
279. C22×C8C2×D269semidirect
288. #117C2×D269semidirect
291. #103C2×D269semidirect
292. #103C2×D269semidirect
295. #102C2×D269semidirect
296. #102C2×D269semidirect
314. #41D1076semidirect

Maximal subgroups

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

subgroupindex
C2×#41×D2692
C2×#225754752
C2×#225754732
#317711172
#317707702
#317707682
#317707622
#317707572
#317707532
#317707512
#317707452
#317707402
#317705992
C2×#225753652
#416×C2692
#3002269