Finite Group Databasebeta

← #35374239

Subgroups — #35374239

Order 21248.

662 conjugacy classes of subgroups.

The subgroup lattice has 382 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#24916546C211yesmaximal, characteristic, complemented
2#24916503C211yesmaximal, characteristic
3C2×C4×#13050579C211yesmaximal, characteristic
4#24906364C214yesmaximal, complemented
5#24906352C214yesmaximal, complemented
6#24905857C211yesmaximal, characteristic, complemented
7#24905853C211yesmaximal, characteristic
8#24905746C211yesmaximal, characteristic, complemented
9C83×#1564C211yesmaximal, characteristic, complemented
10C22×#13050579C411yescharacteristic
11C2×C4×Dic166C2211yescharacteristic
12C2×C4×D332C2211yescharacteristic
13#19127831—22no—
14#19127828C2214yescomplemented
15C4×#13050579C412yes—
16C4×#13050579C412yes—
17C4×#13050579C2212yes—
18C4×#13050579C2212yes—
19#19127780—22no—
20#19127778C2214yes—
21#19127755—22no—
22#19127754C2211yescharacteristic
23#19127754C2211yescharacteristic
24C2×#14581724C411yescharacteristic
25C2×#14581717C411yescharacteristic
26C2×#14581714C411yescharacteristic
27#19127663—22no—
28#19127431C2214yescomplemented
29#19127428C2214yes—
30#19127422C2212yescomplemented
31C8×C83⋊C8C2212yescomplemented
32#431×C83C2211yescharacteristic
33#431×C83C2211yescharacteristic
34C2×C4×C8×C83C2211yescharacteristic
35#335×C83C2212yescomplemented
36#334×C83C2212yescomplemented
37C22×Dic166C2×C411yescharacteristic
38C22×D332C2×C411yescharacteristic
39C22×C4×D83—21no—
40C2×#13050579—24no—
41C2×#13050579C2×C412yes—
42C2×#13050579C2×C412yes—
43C2×#13050573—24no—
44C4×Dic166—21no—
45C4×Dic166—21no—
46C4×Dic166C2×C412yes—
47C4×Dic166C2312yes—
48C4×D332—21no—
49C4×D332—21no—
50C4×D332C2×C412yes—
51C4×D332C2312yes—
52C2×C4⋊Dic83C2×C411yescharacteristic
53C2×C4⋊Dic83C2×C411yescharacteristic
54C2×Dic83⋊C4—21no—
55C2×(C2×D83)⋊C4—21no—
56C2×C4×Dic83—21no—
57#14581724C2×C412yes—
58#14581724C2×C412yes—
59C4⋊(C83⋊C8)—21no—
60C4⋊(C83⋊C8)D412yes—
61Dic83⋊C8—24no—
62#14581717C2×C412yes—
63#14581717C2×C412yes—
64#14581714C2×C412yes—
65#14581714C2×C412yes—
66C22⋊(C83⋊C8)—21no—
67C22⋊(C83⋊C8)D412yes—
68(C2×D83)⋊C8—24no—
69#14581702—24no—
70C4×C83⋊C8C2312yes—
71C4×C83⋊C8C2312yes—
72C22×C8×C83C2×C411yescharacteristic
73C22×C8×C83C2×C411yescharacteristic
74C2×C42×C83C2311yescharacteristic
75C4⋊C8×C83C2312yes—
76C4⋊C8×C83C2312yes—
77C22⋊C8×C83—21no—
78C22⋊C8×C83—21no—
79C4×C8×C83—21no—
80C4×C8×C83—21no—
81C4×C8×C83C2×C411yescharacteristic
82C4×C8×C83C2×C411yescharacteristic
83C4×C8×C83C2311yescharacteristic
84C4×C8×C83C2311yescharacteristic
85C23×D83—21no—
86C2×Dic166—22no—
87C2×Dic166—22no—
88C2×Dic166—21no—
89C2×Dic166C22×C412yes—
90C2×D332—22no—
91C2×D332—22no—
92C2×D332—21no—
93C2×D332C22×C412yes—
94C22×Dic83—21no—
95C22×Dic83—21no—
96C22×Dic83—21no—
97C2×C4×D83—22no—
98C2×C4×D83—22no—
99C2×C4×D83—82no—
100#13050579—24no—
101#13050579—24no—
102#13050573—84no—
103C2×C83⋊C8—22no—
104C2×C83⋊C8C2×D414yes—
105C4⋊Dic83—21no—
106C4⋊Dic83—21no—
107C4⋊Dic83C22×C412yes—
108C4⋊Dic83C22×C412yes—
109Dic83⋊C4—22no—
110Dic83⋊C4—22no—
111(C2×D83)⋊C4—22no—
112(C2×D83)⋊C4—22no—
113C4×Dic83—22no—
114C4×Dic83—22no—
115C22×C4×C83C22×C411yescharacteristic
116C22×C4×C83C2×D411yescharacteristic
117C22×C4×C83#4111yescharacteristic
118C2×C8×C83—22no—
119C2×C8×C83—22no—
120C2×C8×C83—22no—
121C2×C8×C83M1614yescomplemented
122C2×C8×C83C22×C411yescharacteristic
123C2×C8×C83C22×C411yescharacteristic
124C2×C8×C83C22×C411yescharacteristic
125C2×C8×C83C22×C411yescharacteristic
126C42×C83C22×C411yescharacteristic
127C42×C83C2×D411yescharacteristic
128C42×C83#4111yescharacteristic
129C42×C83C2411yescharacteristic
130C23×C83C4×D411yescharacteristic
131C22×D83—22no—
132C22×D83—42no—
133C22×D83—81no—
134C22×D83—81no—
135C2×C4×C83—22no—
136C2×C4×C83—41no—
137C2×C4×C83C4×D411yescharacteristic
138C2×C4×C83C4×D411yescharacteristic
139C2×C4×C83C4×D411yescharacteristic
140C2×C4×C83C2×M1612yes—
141C2×C4×C83#13112yes—
142C2×C4×C83C23×C411yescharacteristic
143C2×C4×C83C22×D411yescharacteristic
144C2×C4×C83C2×#4111yescharacteristic
145C2×Dic83—22no—
146C2×Dic83—22no—
147C2×Dic83—22no—
148C2×Dic83—42no—
149C2×Dic83—81no—
150C2×Dic83—81no—
151D332—22no—
152D332—22no—
153D332—42no—
154C4×D83—44no—
155C4×D83—82no—
156Dic166—22no—
157Dic166—22no—
158Dic166—42no—
159C8×C83—21no—
160C8×C83—21no—
161C8×C83—42no—
162C8×C83—42no—
163C8×C83C2×M1612yes—
164C8×C83C2×M1612yes—
165C83⋊C8—24no—
166C83⋊C8—42no—
167C22×C83—21no—
168C22×C83—21no—
169C22×C83#43612yes—
170C22×C83C2×C4×D411yescharacteristic
171C2×D83—42no—
172C2×D83—42no—
173C2×D83—42no—
174C2×D83—84no—
175C4×C83—21no—
176C4×C83—21no—
177C4×C83—21no—
178C4×C83#43612yes—
179C4×C83C22×M1611yescharacteristic
180C4×C83C2×#13111yescharacteristic
181C4×C83#57712yes—
182Dic83—42no—
183Dic83—42no—
184Dic83—42no—
185#17896—831nomaximal, Sylow
186C2×C83—22no—
187C2×C83C2×#43611yescharacteristic
188C2×C83#294911yescharacteristic
189C2×C83#295711yescharacteristic
190D83—82no—
191#2983—831no—
192#2975—831no—
193C2×C4×SD16—831no—
194#1592—834no—
195#1590—834no—
196#1564D8311yescharacteristic, complemented
197#1479—831no—
198#1478—831no—
199#1450—831no—
200C83#1789611yescharacteristic, complemented, Sylow
201C22×SD16—831no—
202C2×C4×Q8—831no—
203C2×C4×D4—831no—
204#446—834no—
205#446—1662no—
206C4×SD16—832no—
207C4×SD16—832no—
208C4×SD16—832no—
209C4×SD16—832no—
210#436—834no—
211#436—1662no—
212#432—1662no—
213#431—831no—
214#431—831no—
215#431C2×D8311yescharacteristic
216#431C2×D8311yescharacteristic
217C2×#106—831no—
218C2×#103—831no—
219C2×#102—831no—
220#407—1662no—
221C2×C4×C8C2×D8311yescharacteristic
222#335C2×D8312yescomplemented
223#334C2×D8312yescomplemented
224Q8⋊C8—834no—
225D4⋊C8—834no—
226#322—832no—
227C82—832no—
228C22×Q8—831no—
229C22×D4—831no—
230C23×C4—1661no—
231C2×SD16—832no—
232C2×SD16—832no—
233C2×SD16—1664no—
234C2×M16—1664no—
235C22×C8C4×D8311yescharacteristic
236C22×C8C4×D8311yescharacteristic
237C4×Q8—832no—
238C4×Q8—832no—
239C4×Q8—1661no—
240C4×Q8—1661no—
241C4×D4—832no—
242C4×D4—832no—
243C4×D4—1661no—
244C4×D4—1661no—
245C2×C4⋊C4—831no—
246C2×C4⋊C4—831no—
247C2×C4⋊C4—1661no—
248C2×C22⋊C4—1661no—
249C2×C42—1661no—
250C2×C42C22×D8311yescharacteristic
251#106—832no—
252#106—832no—
253C4⋊C8—832no—
254C4⋊C8—1664no—
255C4⋊C8—1661no—
256C4⋊C8C22×D8312yes—
257C4⋊C8C22×D8312yes—
258#103—832no—
259#103—832no—
260#102—832no—
261#102—832no—
262C22⋊C8—21no—
263C22⋊C8—21no—
264C22⋊C8—832no—
265C22⋊C8—1664no—
266C22⋊C8—1661no—
267#97—1664no—
268C4×C8—21no—
269C4×C8—21no—
270C4×C8—832no—
271C4×C8—832no—
272C4×C8C4×D8311yescharacteristic
273C4×C8C4×D8311yescharacteristic
274C4×C8C22×D8311yescharacteristic
275C4×C8C22×D8311yescharacteristic
276C24—1661no—
277C2×Q8—832no—
278C2×Q8—1662no—
279C2×Q8—1662no—
280C2×Q8—1661no—
281C2×D4—832no—
282C2×D4—1662no—
283C2×D4—1662no—
284C2×D4—1661no—
285C22×C4—6642no—
286C22×C4—1661no—
287C22×C4—1662no—
288C22×C4—1661no—
289C22×C4—1661no—
290C22×C4—1662no—
291C22×C4C2×C4×D8311yescharacteristic
292C22×C4D4×D8311yescharacteristic
293C22×C4#1305059111yescharacteristic
294SD16—1664no—
295SD16—1664no—
296M16—6644no—
297C2×C8—22no—
298C2×C8—22no—
299C2×C8—22no—
300C2×C8—834no—
301C2×C8—1662no—
302C2×C8#1305057314yescomplemented
303C2×C8C2×C4×D8311yescharacteristic
304C2×C8C2×C4×D8311yescharacteristic
305C2×C8C2×C4×D8311yescharacteristic
306C2×C8C2×C4×D8311yescharacteristic
307C4⋊C4—832no—
308C4⋊C4—832no—
309C4⋊C4—1662no—
310C4⋊C4—1661no—
311C4⋊C4—1661no—
312C4⋊C4—1662no—
313C22⋊C4—1662no—
314C22⋊C4—1662no—
315C42—1662no—
316C42—1662no—
317C42C2×C4×D8311yescharacteristic
318C42D4×D8311yescharacteristic
319C42#1305059111yescharacteristic
320C42C23×D8311yescharacteristic
321C23—3322no—
322C23—6641no—
323C23—6641no—
324C23—1662no—
325C23#1458175411yescharacteristic
326Q8—3322no—
327Q8—1662no—
328Q8—1662no—
329D4—3322no—
330D4—1662no—
331D4—1662no—
332C2×C4—22no—
333C2×C4—41no—
334C2×C4—3322no—
335C2×C4—3324no—
336C2×C4—6642no—
337C2×C4—6641no—
338C2×C4—6641no—
339C2×C4—1662no—
340C2×C4—1662no—
341C2×C4—1662no—
342C2×C4#1458175411yescharacteristic
343C2×C4#1458175411yescharacteristic
344C2×C4#1458175411yescharacteristic
345C2×C4C2×#1305057312yes—
346C2×C4#1458180012yes—
347C2×C4C22×C4×D8311yescharacteristic
348C2×C4C2×D4×D8311yescharacteristic
349C2×C4C2×#1305059111yescharacteristic
350C8—21no—
351C8—21no—
352C8—42no—
353C8—42no—
354C8—3322no—
355C8—1664no—
356C8C2×#1305057312yes—
357C8C2×#1305057312yes—
358C22—21no—
359C22—21no—
360C22—3322no—
361C22—3322no—
362C22—3322no—
363C22—6644no—
364C22#1912777812yes—
365C22C2×#1458175411yescharacteristic
366C4—21no—
367C4—21no—
368C4—21no—
369C4—3322no—
370C4—3322no—
371C4—3322no—
372C4#1912777812yes—
373C4C22×#1305057311yescharacteristic
374C4C2×#1458180011yescharacteristic
375C4#1912850812yes—
376C2—22no—
377C2—6642no—
378C2C2×#1912777811yescharacteristic
379C2#2491632111yescharacteristic
380C2#2491637911yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D83
commutatorC2×C4×C83C23×C4
FrattiniC42C23×D83
FittingC83×#1564C2
socleC22×C83C2×C4×D4
prime radicalC22×D332C2×C4
cosocleC42×C83C24
hypercenter#1564D83
hypocenterC83#17896
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1564

Complements

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

NH ≅ G/Ntype
1. #24916546C2semidirect
4. #24906364C2semidirect
5. #24906352C2semidirect
6. #24905857C2semidirect
8. #24905746C2semidirect
9. C83×#1564C2semidirect
14. #19127828C22semidirect
28. #19127431C22semidirect
30. #19127422C22semidirect
31. C8×C83⋊C8C22semidirect
35. #335×C83C22semidirect
36. #334×C83C22semidirect
121. C2×C8×C83M16semidirect
196. #1564D83semidirect
200. C83#17896semidirect
222. #335C2×D83semidirect
223. #334C2×D83semidirect
302. C2×C8#13050573semidirect

Maximal subgroups

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

subgroupindex
#249165462
#249165032
C2×C4×#130505792
#249063642
#249063522
#249058572
#249058532
#249057462
C83×#15642
#1789683