Finite Group Databasebeta

← #35374236

Subgroups — #35374236

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#24916548C211yesmaximal, characteristic, complemented
2#24916503C211yesmaximal, characteristic
3C2×C4×#13050577C211yesmaximal, characteristic
4#24906368C214yesmaximal, complemented
5#24906350C214yesmaximal, complemented
6#24906209C211yesmaximal, characteristic, complemented
7#24905855C211yesmaximal, characteristic
8#24905747C211yesmaximal, characteristic, complemented
9C83×#1479C211yesmaximal, characteristic, complemented
10C22×#13050577C411yescharacteristic
11C2×C4×D332C2211yescharacteristic
12#19127832—22no—
13#19127832C2214yescomplemented
14C4×#13050577C412yes—
15C4×#13050577C412yes—
16C4×#13050577C2212yes—
17C4×#13050577C2212yes—
18#19127780—22no—
19#19127778C2214yes—
20#19127758—22no—
21#19127754C2211yescharacteristic
22#19127754C2211yescharacteristic
23C2×#14581723C411yescharacteristic
24C2×#14581716C411yescharacteristic
25C2×#14581712C411yescharacteristic
26#19127663—22no—
27C2×C4×C83⋊C8C2211yescharacteristic
28#19127453C2212yescomplemented
29#19127451C2212yescomplemented
30#19127430C2214yes—
31#19127423C2212yescomplemented
32C8×C83⋊C8C2212yescomplemented
33C2×C4×Q8×C83C2211yescharacteristic
34#431×C83C2211yescharacteristic
35#431×C83C2211yescharacteristic
36Q8⋊C8×C83C2214yescomplemented
37C22×D332C2×C411yescharacteristic
38C22×C4×D83—21no—
39C2×#13050577—24no—
40C2×#13050577C2×C412yes—
41C2×#13050577C2×C412yes—
42C2×#13050573—24no—
43C22×C83⋊C8C2×C411yescharacteristic
44C22×C83⋊C8C2×C411yescharacteristic
45C4×D332—21no—
46C4×D332—21no—
47C4×D332C2×C412yes—
48C4×D332C2312yes—
49C2×C4⋊Dic83C2×C411yescharacteristic
50C2×(C2×D83)⋊C4—21no—
51#14581723C2×C412yes—
52#14581723C2×C412yes—
53C4⋊(C83⋊C8)—24no—
54C4⋊(C83⋊C8)D412yes—
55C4⋊(C83⋊C8)C2312yes—
56#14581716C2×C412yes—
57#14581716C2×C412yes—
58#14581712C2×C412yes—
59#14581712C2×C412yes—
60C22⋊(C83⋊C8)—21no—
61C22⋊(C83⋊C8)D412yes—
62(C2×D83)⋊C8—24no—
63#14581703—24no—
64C4×C83⋊C8—21no—
65C4×C83⋊C8C2×C411yescharacteristic
66C4×C83⋊C8C2×C411yescharacteristic
67C4×C83⋊C8C2311yescharacteristic
68C4×C83⋊C8C2311yescharacteristic
69C4×C83⋊C8C2312yes—
70C22×Q8×C83C2×C411yescharacteristic
71C4×Q8×C83—21no—
72C4×Q8×C83—21no—
73C4×Q8×C83C2×C412yes—
74C4×Q8×C83C2312yes—
75C2×C4⋊C4×C83—21no—
76C2×C4⋊C4×C83C2×C411yescharacteristic
77C2×C42×C83—21no—
78C2×C42×C83C2311yescharacteristic
79C4⋊C8×C83—21no—
80C4⋊C8×C83C2312yes—
81C22⋊C8×C83—21no—
82C22⋊C8×C83—21no—
83C4×C8×C83—21no—
84C4×C8×C83C2312yes—
85C23×D83—21no—
86C2×D332—22no—
87C2×D332—22no—
88C2×D332—21no—
89C2×D332C22×C412yes—
90C22×Dic83—21no—
91C2×C4×D83—22no—
92C2×C4×D83—22no—
93C2×C4×D83—82no—
94#13050577—24no—
95#13050577—24no—
96#13050573—84no—
97C2×C83⋊C8—22no—
98C2×C83⋊C8—22no—
99C2×C83⋊C8M1614yescomplemented
100C2×C83⋊C8C22×C411yescharacteristic
101C2×C83⋊C8C22×C411yescharacteristic
102C2×C83⋊C8C22×C411yescharacteristic
103C2×C83⋊C8C22×C411yescharacteristic
104C2×C83⋊C8C2×D414yes—
105C4⋊Dic83—21no—
106C4⋊Dic83C22×C412yes—
107(C2×D83)⋊C4—22no—
108(C2×D83)⋊C4—22no—
109C2×Q8×C83—21no—
110C2×Q8×C83—22no—
111C2×Q8×C83—22no—
112C2×Q8×C83C22×C412yes—
113C22×C4×C83—21no—
114C22×C4×C83—21no—
115C22×C4×C83C22×C411yescharacteristic
116C22×C4×C83C2×D411yescharacteristic
117C22×C4×C83#4111yescharacteristic
118C2×C8×C83—22no—
119C2×C8×C83—22no—
120C4⋊C4×C83—22no—
121C4⋊C4×C83—21no—
122C4⋊C4×C83—22no—
123C4⋊C4×C83C22×C412yes—
124C42×C83—22no—
125C42×C83—22no—
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—
135Q8×C83—22no—
136Q8×C83—22no—
137Q8×C83—42no—
138C2×C4×C83—22no—
139C2×C4×C83—22no—
140C2×C4×C83—22no—
141C2×C4×C83—41no—
142C2×C4×C83—42no—
143C2×C4×C83—81no—
144C2×C4×C83C4×D411yescharacteristic
145C2×C4×C83C4×D411yescharacteristic
146C2×C4×C83C4×D411yescharacteristic
147C2×C4×C83C2×M1612yes—
148C2×C4×C83#13112yes—
149C2×C4×C83C23×C411yescharacteristic
150C2×C4×C83C22×D411yescharacteristic
151C2×C4×C83C2×#4111yescharacteristic
152C2×Dic83—22no—
153C2×Dic83—81no—
154D332—22no—
155D332—22no—
156D332—42no—
157C4×D83—44no—
158C4×D83—82no—
159C8×C83—42no—
160C8×C83—42no—
161C83⋊C8—21no—
162C83⋊C8—21no—
163C83⋊C8—24no—
164C83⋊C8—42no—
165C83⋊C8C2×M1612yes—
166C83⋊C8C2×M1612yes—
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—42no—
179C4×C83—42no—
180C4×C83#43612yes—
181C4×C83C22×M1611yescharacteristic
182C4×C83C2×#13111yescharacteristic
183C4×C83#57712yes—
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#1564—831no—
197#1479D8311yescharacteristic, complemented
198#1478—831no—
199#1450—831no—
200C83#1789611yescharacteristic, complemented, Sylow
201C22×SD16—831no—
202C2×C4×Q8C2×D8311yescharacteristic
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×C8—831no—
222#335—832no—
223#334—832no—
224Q8⋊C8C2×D8314yescomplemented
225D4⋊C8—834no—
226#322—832no—
227C82—832no—
228C22×Q8C4×D8311yescharacteristic
229C22×D4—831no—
230C23×C4—1661no—
231C2×SD16—832no—
232C2×SD16—832no—
233C2×SD16—1664no—
234C2×M16—1664no—
235C22×C8—831no—
236C22×C8—831no—
237C4×Q8—21no—
238C4×Q8—21no—
239C4×Q8C4×D8312yes—
240C4×Q8C22×D8312yes—
241C4×D4—832no—
242C4×D4—832no—
243C4×D4—1661no—
244C4×D4—1661no—
245C2×C4⋊C4—21no—
246C2×C4⋊C4—831no—
247C2×C4⋊C4C4×D8311yescharacteristic
248C2×C22⋊C4—1661no—
249C2×C42—21no—
250C2×C42C22×D8311yescharacteristic
251#106—832no—
252#106—832no—
253C4⋊C8—21no—
254C4⋊C8—832no—
255C4⋊C8—832no—
256C4⋊C8—1664no—
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—831no—
270C4×C8—831no—
271C4×C8—831no—
272C4×C8—832no—
273C4×C8—831no—
274C4×C8—1661no—
275C4×C8C22×D8312yes—
276C24—1661no—
277C2×Q8—21no—
278C2×Q8—22no—
279C2×Q8—22no—
280C2×Q8C2×C4×D8312yes—
281C2×D4—832no—
282C2×D4—1662no—
283C2×D4—1662no—
284C2×D4—1661no—
285C22×C4—21no—
286C22×C4—21no—
287C22×C4—6642no—
288C22×C4—1662no—
289C22×C4—1661no—
290C22×C4—1662no—
291C22×C4C2×C4×D8311yescharacteristic
292C22×C4D4×D8311yescharacteristic
293C22×C4#1305058911yescharacteristic
294SD16—1664no—
295SD16—1664no—
296M16—6644no—
297C2×C8—22no—
298C2×C8—22no—
299C2×C8—834no—
300C2×C8—831no—
301C2×C8—831no—
302C2×C8—831no—
303C2×C8—831no—
304C2×C8—834no—
305C2×C8—1662no—
306C2×C8—1662no—
307C4⋊C4—22no—
308C4⋊C4—22no—
309C4⋊C4—21no—
310C4⋊C4—832no—
311C4⋊C4—1661no—
312C4⋊C4C2×C4×D8312yes—
313C22⋊C4—1662no—
314C22⋊C4—1662no—
315C42—22no—
316C42—22no—
317C42C2×C4×D8311yescharacteristic
318C42D4×D8311yescharacteristic
319C42#1305058911yescharacteristic
320C42C23×D8311yescharacteristic
321C23—3322no—
322C23—6641no—
323C23—6641no—
324C23—1662no—
325C23(C22⋊D83)⋊C411yescharacteristic
326Q8—22no—
327Q8—22no—
328Q8—42no—
329D4—3322no—
330D4—1662no—
331D4—1662no—
332C2×C4—22no—
333C2×C4—22no—
334C2×C4—22no—
335C2×C4—41no—
336C2×C4—42no—
337C2×C4—81no—
338C2×C4—3324no—
339C2×C4—6642no—
340C2×C4—6641no—
341C2×C4—1662no—
342C2×C4(C22⋊D83)⋊C411yescharacteristic
343C2×C4(C22⋊D83)⋊C411yescharacteristic
344C2×C4(C22⋊D83)⋊C411yescharacteristic
345C2×C4M16×D8312yes—
346C2×C4#1458179812yes—
347C2×C4C22×C4×D8311yescharacteristic
348C2×C4C2×D4×D8311yescharacteristic
349C2×C4C2×#1305058911yescharacteristic
350C8—42no—
351C8—42no—
352C8—3322no—
353C8—832no—
354C8—832no—
355C8—1661no—
356C8—1661no—
357C8—1664no—
358C22—21no—
359C22—21no—
360C22—3322no—
361C22—3322no—
362C22—3322no—
363C22—6644no—
364C22#1912777612yes—
365C22C2×(C22⋊D83)⋊C411yescharacteristic
366C4—21no—
367C4—21no—
368C4—21no—
369C4—42no—
370C4—42no—
371C4—3322no—
372C4#1912777612yes—
373C4C2×M16×D8311yescharacteristic
374C4C2×#1458179811yescharacteristic
375C4#1912851012yes—
376C2—22no—
377C2—6642no—
378C2C2×#1912777611yescharacteristic
379C2#2491631811yescharacteristic
380C2#2491637611yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D83
commutatorC2×C4×C83C23×C4
FrattiniC42C23×D83
FittingC83×#1479C2
socleC22×C83C2×C4×D4
prime radicalC22×D332C2×C4
cosocleC42×C83C24
hypercenter#1479D83
hypocenterC83#17896
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1479

Complements

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

NH ≅ G/Ntype
1. #24916548C2semidirect
4. #24906368C2semidirect
5. #24906350C2semidirect
6. #24906209C2semidirect
8. #24905747C2semidirect
9. C83×#1479C2semidirect
13. #19127832C22semidirect
28. #19127453C22semidirect
29. #19127451C22semidirect
31. #19127423C22semidirect
32. C8×C83⋊C8C22semidirect
36. Q8⋊C8×C83C22semidirect
99. C2×C83⋊C8M16semidirect
197. #1479D83semidirect
200. C83#17896semidirect
224. Q8⋊C8C2×D83semidirect

Maximal subgroups

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

subgroupindex
#249165482
#249165032
C2×C4×#130505772
#249063682
#249063502
#249062092
#249058552
#249057472
C83×#14792
#1789683