Finite Group Databasebeta

← #35559090

Subgroups — #35559090

Order 21248.

556 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
1#24918000C211yesmaximal, characteristic, complemented
2#24916310C211yesmaximal, characteristic
3#24909623C211yesmaximal, characteristic, complemented
4#24909618C211yesmaximal, characteristic, complemented
5#24909617C211yesmaximal, characteristic, complemented
6#24909610C211yesmaximal, characteristic, complemented
7#24909609C211yesmaximal, characteristic, complemented
8#24909601C211yesmaximal, characteristic, complemented
9#24909598C211yesmaximal, characteristic, complemented
10#24909250C211yesmaximal, characteristic, complemented
11#24909244C211yesmaximal, characteristic, complemented
12#24909220C211yesmaximal, characteristic, complemented
13#24909177C211yesmaximal, characteristic
14#357×D83C211yesmaximal, characteristic
15C83×#2228C211yesmaximal, characteristic, complemented
16C2×D8×D83C2211yescharacteristic
17#19128331—21no—
18#19128173—21no—
19C2×#14581731C2211yescharacteristic
20C2×C16×D83C2211yescharacteristic
21#19128111—21no—
22#19128108—21no—
23#19128106C2211yescharacteristic, complemented
24#19128105C2211yescharacteristic, complemented
25#19128102C2211yescharacteristic, complemented
26#19127811—21no—
27#19127810—21no—
28C4×D664C2211yescharacteristic
29D8×Dic83C2211yescharacteristic
30#19127783C2211yescharacteristic
31#19127740—21no—
32#19127738C2211yescharacteristic
33#19127738C2211yescharacteristic
34#107×D83C2211yescharacteristic
35#19127704—21no—
36#19127655C2211yescharacteristic
37#19127528C2211yescharacteristic, complemented
38#19127528C2211yescharacteristic, complemented
39#19127528C2211yescharacteristic, complemented
40#19127527C2211yescharacteristic, complemented
41#19127527C2211yescharacteristic, complemented
42#19127526C2211yescharacteristic, complemented
43C4⋊(C83⋊C16)C2211yescharacteristic, complemented
44Dic83⋊C16C2211yescharacteristic, complemented
45Dic83⋊C16C2211yescharacteristic, complemented
46#19127503C2211yescharacteristic
47#19127503C2211yescharacteristic
48#19127502C2211yescharacteristic
49#19127502C2211yescharacteristic
50#19127501C2211yescharacteristic
51#19127501C2211yescharacteristic
52#499×C83C2211yescharacteristic, complemented
53C4×D8×C83C2211yescharacteristic
54#367×C83C2211yescharacteristic, complemented
55#366×C83C2211yescharacteristic, complemented
56C4⋊C16×C83C2211yescharacteristic, complemented
57#357×C83C2211yescharacteristic
58#357×C83C2211yescharacteristic
59C2×D4×D83—21no—
60C2×D664C2311yescharacteristic
61C2×#13050575—21no—
62D8×D83—22no—
63D8×D83Q814yescomplemented
64C2×#13050573D411yescharacteristic
65C2×C8×D83C2311yescharacteristic
66#14581792—21no—
67#14581791—21no—
68#14581791—41no—
69#14581783—21no—
70#14581782—41no—
71#14581772—41no—
72#14581770—41no—
73C4×D332—21no—
74D4×Dic83—21no—
75#14581754—21no—
76C4×C22⋊D83—41no—
77(C22⋊D83)⋊C4—41no—
78#14581750—21no—
79#14581750—21no—
80#14581749D411yescharacteristic
81C4⋊C4×D83—21no—
82C2×(C2×D83)⋊C4—41no—
83C22⋊C4×D83—41no—
84#14581731—22no—
85C2×C83⋊C16C2311yescharacteristic
86C2×C83⋊C16C2311yescharacteristic
87C16×D83—22no—
88#14581726—21no—
89#14581726C2311yescharacteristic
90#14581726C2311yescharacteristic
91#14581726C2311yescharacteristic
92#14581725—21no—
93#14581725—21no—
94#14581723—21no—
95#14581723—21no—
96#14581723—21no—
97#14581714—21no—
98#14581713—21no—
99#14581712—21no—
100#14581703D411yescharacteristic
101#14581702D411yescharacteristic
102C8×Dic83C2311yescharacteristic
103C2×D8×C83C2311yescharacteristic
104#125×C83—21no—
105C4×D4×C83—21no—
106C2×C16×C83C2311yescharacteristic
107C2×C16×C83C2311yescharacteristic
108#107×C83C2311yescharacteristic
109#107×C83C2311yescharacteristic
110#107×C83C2311yescharacteristic
111#106×C83—21no—
112#102×C83—21no—
113C4×C8×C83C2311yescharacteristic
114C23×D83—41no—
115C2×Dic166—41no—
116C2×Dic166—41no—
117C2×D332—21no—
118C2×C22⋊D83—41no—
119D4×D83—24no—
120D4×D83—42no—
121C22×Dic83—41no—
122C2×C4×D83—41no—
123C2×C4×D83—41no—
124C2×C4×D83—41no—
125C2×C4×D83C2×D411yescharacteristic
126D664—21no—
127D664C2×Q812yes—
128#13050575—24no—
129#13050573—22no—
130C2×C83⋊C8C2×D411yescharacteristic
131C2×C83⋊C8C2×D411yescharacteristic
132C8×D83C2×Q812yes—
133C8×D83#4112yes—
134C4⋊Dic83—21no—
135C4⋊Dic83—21no—
136C4⋊Dic83—21no—
137C4⋊Dic83—41no—
138C4⋊Dic83—41no—
139Dic83⋊C4—41no—
140Dic83⋊C4—41no—
141Dic83⋊C4—41no—
142Dic83⋊C4D811yescharacteristic
143Dic83⋊C4D811yescharacteristic
144C22⋊Dic83—41no—
145(C2×D83)⋊C4—41no—
146(C2×D83)⋊C4—41no—
147(C2×D83)⋊C4—41no—
148(C2×D83)⋊C4—41no—
149(C2×D83)⋊C4D811yescharacteristic
150(C2×D83)⋊C4D811yescharacteristic
151C4×Dic83—41no—
152C4×Dic83—41no—
153C4×Dic83C2×D411yescharacteristic
154C83⋊C16—21no—
155C83⋊C16—21no—
156C2×D4×C83—21no—
157C22×C4×C83—41no—
158D8×C83—21no—
159D8×C83C2×Q812yes—
160C2×C8×C83C2×D411yescharacteristic
161C2×C8×C83C2411yescharacteristic
162C4⋊C4×C83—21no—
163C4⋊C4×C83—21no—
164C4⋊C4×C83—21no—
165C4⋊C4×C83—41no—
166C4⋊C4×C83—41no—
167C22⋊C4×C83—41no—
168C42×C83C2×D411yescharacteristic
169C16×C83—21no—
170C16×C83—21no—
171C23×C83—41no—
172C22×D83—41no—
173C22×D83—44no—
174C22×D83—161no—
175C22×D83—161no—
176C22×D83C2×D811yescharacteristic
177D4×C83—22no—
178D4×C83—41no—
179C2×C4×C83—41no—
180C2×C4×C83—41no—
181C2×C4×C83—41no—
182C2×C4×C83—161no—
183C2×C4×C83C2×D811yescharacteristic
184C2×C4×C83C2×D811yescharacteristic
185C2×C4×C83C22×D411yescharacteristic
186C22⋊D83—44no—
187C2×Dic83—41no—
188C2×Dic83—41no—
189C2×Dic83—41no—
190C2×Dic83—161no—
191C2×Dic83C2×D811yescharacteristic
192C2×Dic83C2×D811yescharacteristic
193C2×Dic83C2×D811yescharacteristic
194D332—22no—
195D332—41no—
196C4×D83—161no—
197C4×D83—161no—
198C4×D83—161no—
199C4×D83C22⋊Q812yes—
200C4×D83#13712yes—
201Dic166—161no—
202Dic166—161no—
203C8×C83—21no—
204C8×C83C22×Q811yescharacteristic
205C8×C83C2×#4111yescharacteristic
206C83⋊C8—21no—
207C83⋊C8C22⋊Q812yes—
208C22×C83—42no—
209C22×C83—81no—
210C22×C83—81no—
211C22×C83C22×D811yescharacteristic
212C2×D83—21no—
213C2×D83—21no—
214C2×D83—42no—
215C2×D83—81no—
216C2×D83—81no—
217C2×D83—164no—
218C2×D83#47412yes—
219C4×C83—21no—
220C4×C83—21no—
221C4×C83—81no—
222C4×C83—81no—
223C4×C83—81no—
224C4×C83C2×C22⋊Q811yescharacteristic
225C4×C83C2×#13711yescharacteristic
226Dic83—21no—
227Dic83—21no—
228Dic83—81no—
229Dic83—81no—
230Dic83—81no—
231Dic83#47412yes—
232#33631—831nomaximal, Sylow
233C2×C83—82no—
234C2×C83C2×#47411yescharacteristic
235C2×C83#341511yescharacteristic
236C2×C83#341611yescharacteristic
237D83—22no—
238D83—82no—
239#3164—831no—
240#2948—831no—
241#2228—831no—
242#2228—831no—
243#2228—831no—
244#2228D8311yescharacteristic, complemented
245#2226—831no—
246#2226—831no—
247#2224—831no—
248#2224—831no—
249#2159—831no—
250#2158—831no—
251#2151—831no—
252#2140—831no—
253C2×#357—831no—
254C83#3363111yescharacteristic, complemented, Sylow
255C22×D8—831no—
256#544—1661no—
257#518—1661no—
258C2×M32—831no—
259C22×C16—831no—
260#501—1661no—
261#501—1661no—
262#500—831no—
263#500—831no—
264#499—831no—
265#499C2×D8311yescharacteristic, complemented
266#442—1661no—
267#442—1661no—
268C4×D8—831no—
269C4×D8—831no—
270C4×D8—831no—
271C4×D8C2×D8311yescharacteristic
272#428—1661no—
273#427—831no—
274#427—831no—
275C2×#107—831no—
276#418—1661no—
277#405—831no—
278#367—831no—
279#367—831no—
280#367—831no—
281#367C2×D8311yescharacteristic, complemented
282#366—831no—
283#366—831no—
284#366—831no—
285#366C2×D8311yescharacteristic, complemented
286C4⋊C16—831no—
287C4⋊C16—831no—
288C4⋊C16—831no—
289C4⋊C16C2×D8311yescharacteristic, complemented
290#357—831no—
291#357—831no—
292#357—831no—
293#357—831no—
294#357—831no—
295#357—831no—
296#357C2×D8311yescharacteristic
297#357C2×D8311yescharacteristic
298C22×D4—1661no—
299C2×D8—834no—
300C2×D8—831no—
301C2×D8—1662no—
302C2×D8—1661no—
303C2×D8C22×D8311yescharacteristic
304C2×M16—831no—
305C22×C8—831no—
306C4⋊Q8—3321no—
307C4⋊Q8—1661no—
308C4⋊Q8—1661no—
309#125—21no—
310#125—3321no—
311#125—1661no—
312C22⋊Q8—3321no—
313C22⋊Q8—3321no—
314C4×D4—21no—
315C4×D4—3321no—
316C4×D4—3321no—
317C4×D4—1661no—
318C4×D4—1661no—
319C4×D4—1661no—
320#117—831no—
321#117—1661no—
322#117—1661no—
323C2×C4⋊C4—1661no—
324C2×C22⋊C4—3321no—
325C2×C22⋊C4—3321no—
326M32—1662no—
327C2×C16—831no—
328C2×C16—831no—
329C2×C16—1662no—
330C2×C16C22×D8311yescharacteristic
331C2×C16C22×D8311yescharacteristic
332#107—831no—
333#107—831no—
334#107—831no—
335#107—1661no—
336#107—1661no—
337#107—1661no—
338#107C22×D8311yescharacteristic
339#107C22×D8311yescharacteristic
340#107C22×D8311yescharacteristic
341#106—21no—
342#106—1661no—
343#106—1661no—
344#106—1661no—
345#102—21no—
346#102—1661no—
347#102—1661no—
348#102—1661no—
349#97—831no—
350#97—831no—
351C4×C8—831no—
352C4×C8C22×D8311yescharacteristic
353C24—3321no—
354C2×Q8—3321no—
355C2×Q8—3321no—
356C2×D4—21no—
357C2×D4—3322no—
358C2×D4—3321no—
359C2×D4—1664no—
360C2×D4—1661no—
361C22×C4—41no—
362C22×C4—3321no—
363C22×C4—3321no—
364C22×C4—3321no—
365C22×C4—3321no—
366C22×C4—831no—
367D8—21no—
368D8—832no—
369D8—1664no—
370D8—1661no—
371D8Q8×D8312yes—
372M16—1662no—
373C2×C8—831no—
374C2×C8—832no—
375C2×C8—832no—
376C2×C8—831no—
377C2×C8D4×D8311yescharacteristic
378C2×C8C23×D8311yescharacteristic
379C4⋊C4—21no—
380C4⋊C4—21no—
381C4⋊C4—21no—
382C4⋊C4—41no—
383C4⋊C4—41no—
384C4⋊C4—3321no—
385C4⋊C4—3321no—
386C4⋊C4—3321no—
387C4⋊C4—3321no—
388C4⋊C4—3321no—
389C4⋊C4—831no—
390C4⋊C4—831no—
391C4⋊C4—1661no—
392C4⋊C4—1661no—
393C4⋊C4—1661no—
394C22⋊C4—41no—
395C22⋊C4—3321no—
396C22⋊C4—3321no—
397C22⋊C4—3321no—
398C22⋊C4—3321no—
399C22⋊C4—3321no—
400C22⋊C4—831no—
401C22⋊C4—831no—
402C42—3321no—
403C42—3321no—
404C42—831no—
405C42D4×D8311yescharacteristic
406C16—21no—
407C16—21no—
408C16—1661no—
409C16—1661no—
410C23—41no—
411C23—13281no—
412C23—13281no—
413C23—3324no—
414C23—3321no—
415C23—831no—
416Q8—13281no—
417Q8—13281no—
418D4—22no—
419D4—41no—
420D4—3324no—
421D4—3321no—
422D4—1662no—
423C2×C4—41no—
424C2×C4—41no—
425C2×C4—41no—
426C2×C4—161no—
427C2×C4—13281no—
428C2×C4—13281no—
429C2×C4—13281no—
430C2×C4—13281no—
431C2×C4—3321no—
432C2×C4—3321no—
433C2×C4—3321no—
434C2×C4—831no—
435C2×C4—832no—
436C2×C4—832no—
437C2×C4—831no—
438C2×C4—831no—
439C2×C4D8×D8311yescharacteristic
440C2×C4D8×D8311yescharacteristic
441C2×C4C2×D4×D8311yescharacteristic
442C8—21no—
443C8—832no—
444C8—1661no—
445C8C2×Q8×D8311yescharacteristic
446C8#41×D8311yescharacteristic
447C22—42no—
448C22—81no—
449C22—81no—
450C22—13284no—
451C22—3322no—
452C22—832no—
453C22—6641no—
454C22—6641no—
455C22—1661no—
456C22—1661no—
457C22C2×D8×D8311yescharacteristic
458C4—21no—
459C4—21no—
460C4—81no—
461C4—81no—
462C4—81no—
463C4—832no—
464C4—6641no—
465C4—6641no—
466C4—6641no—
467C4—1661no—
468C4—1661no—
469C4C22⋊Q8×D8311yescharacteristic
470C4#137×D8311yescharacteristic
471C2—82no—
472C2—6642no—
473C2—1662no—
474C2#474×D8311yescharacteristic
475C2#2492060111yescharacteristic
476C2#2492060911yescharacteristic
subgroupG/H
centerC22C2×D8×D83
commutator, cosocleC2×C8×C83C24
FrattiniC2×C8C23×D83
FittingC83×#2228C2
socleC22×C83C22×D8
prime radicalC2×D8×D83C22
hypercenter#2228D83
hypocenterC83#33631
LCSG ▷ C2×C8×C83 ▷ C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ #2228

Complements

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

NH ≅ G/Ntype
1. #24918000C2semidirect
3. #24909623C2semidirect
4. #24909618C2semidirect
5. #24909617C2semidirect
6. #24909610C2semidirect
7. #24909609C2semidirect
8. #24909601C2semidirect
9. #24909598C2semidirect
10. #24909250C2semidirect
11. #24909244C2semidirect
12. #24909220C2semidirect
15. C83×#2228C2semidirect
23. #19128106C22semidirect
24. #19128105C22semidirect
25. #19128102C22semidirect
37. #19127528C22semidirect
38. #19127528C22semidirect
39. #19127528C22semidirect
40. #19127527C22semidirect
41. #19127527C22semidirect
42. #19127526C22semidirect
43. C4⋊(C83⋊C16)C22semidirect
44. Dic83⋊C16C22semidirect
45. Dic83⋊C16C22semidirect
52. #499×C83C22semidirect
54. #367×C83C22semidirect
55. #366×C83C22semidirect
56. C4⋊C16×C83C22semidirect
63. D8×D83Q8semidirect
244. #2228D83semidirect
254. C83#33631semidirect
265. #499C2×D83semidirect
281. #367C2×D83semidirect
285. #366C2×D83semidirect
289. C4⋊C16C2×D83semidirect

Maximal subgroups

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

subgroupindex
#249180002
#249163102
#249096232
#249096182
#249096172
#249096102
#249096092
#249096012
#249095982
#249092502
#249092442
#249092202
#249091772
#357×D832
C83×#22282
#3363183