Finite Group Databasebeta

← #35374243

Subgroups — #35374243

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#24916504C211yesmaximal, characteristic
3C2×C4×#13050578C211yesmaximal, characteristic
4#24906362C214yesmaximal, complemented
5#24906356C214yesmaximal, complemented
6#24906209C211yesmaximal, characteristic, complemented
7#24905859C211yesmaximal, characteristic, complemented
8#24905747C211yesmaximal, characteristic, complemented
9C83×#1478C211yesmaximal, characteristic
10C22×#13050578C411yescharacteristic
11C2×C4×Dic166C2211yescharacteristic
12#19127831—22no—
13#19127828C2214yescomplemented
14C4×#13050578C412yes—
15C4×#13050578C412yes—
16C4×#13050578C2212yes—
17C4×#13050578C2212yes—
18#19127779—22no—
19#19127779C2214yes—
20#19127755—22no—
21#19127754C2211yescharacteristic
22#19127754C2211yescharacteristic
23C2×#14581723C411yescharacteristic
24C2×#14581715C411yescharacteristic
25C2×#14581713C411yescharacteristic
26#19127665—22no—
27C2×C4×C83⋊C8C2211yescharacteristic
28#19127453C2212yescomplemented
29#19127451C2212yescomplemented
30#19127433C2214yescomplemented
31#19127423C2212yescomplemented
32C8×C83⋊C8C2212yescomplemented
33C2×C4×D4×C83C2211yescharacteristic
34#431×C83C2211yescharacteristic
35#431×C83C2211yescharacteristic
36D4⋊C8×C83C2214yes—
37C22×Dic166C2×C411yescharacteristic
38C2×#13050578—24no—
39C2×#13050578C2×C412yes—
40C2×#13050578C2×C412yes—
41C2×#13050574—24no—
42C22×C83⋊C8C2×C411yescharacteristic
43C22×C83⋊C8C2×C411yescharacteristic
44C4×Dic166—21no—
45C4×Dic166—21no—
46C4×Dic166C2×C412yes—
47C4×Dic166C2312yes—
48C2×C4⋊Dic83C2×C411yescharacteristic
49C2×Dic83⋊C4—21no—
50C2×C4×Dic83—21no—
51#14581723C2×C412yes—
52#14581723C2×C412yes—
53C4⋊(C83⋊C8)D412yes—
54C4⋊(C83⋊C8)C2312yes—
55Dic83⋊C8—24no—
56#14581715C2×C412yes—
57#14581715C2×C412yes—
58#14581713C2×C412yes—
59#14581713C2×C412yes—
60C22⋊(C83⋊C8)—24no—
61C22⋊(C83⋊C8)—21no—
62C22⋊(C83⋊C8)D412yes—
63#14581702—24no—
64C4×C83⋊C8—21no—
65C4×C83⋊C8C2×C411yescharacteristic
66C4×C83⋊C8C2×C411yescharacteristic
67C4×C83⋊C8C2311yescharacteristic
68C4×C83⋊C8C2311yescharacteristic
69C4×C83⋊C8C2312yes—
70C22×D4×C83C2×C411yescharacteristic
71C23×C4×C83—21no—
72C4×D4×C83—21no—
73C4×D4×C83—21no—
74C4×D4×C83C2×C412yes—
75C4×D4×C83C2312yes—
76C2×C4⋊C4×C83C2×C411yescharacteristic
77C2×C22⋊C4×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—
85C2×Dic166—22no—
86C2×Dic166—22no—
87C2×Dic166—21no—
88C2×Dic166C22×C412yes—
89C22×Dic83—21no—
90C22×Dic83—21no—
91#13050578—24no—
92#13050578—24no—
93#13050574—84no—
94C2×C83⋊C8—22no—
95C2×C83⋊C8—22no—
96C2×C83⋊C8M1614yescomplemented
97C2×C83⋊C8C22×C411yescharacteristic
98C2×C83⋊C8C22×C411yescharacteristic
99C2×C83⋊C8C22×C411yescharacteristic
100C2×C83⋊C8C22×C411yescharacteristic
101C2×C83⋊C8C2×D414yes—
102C4⋊Dic83—21no—
103C4⋊Dic83C22×C412yes—
104Dic83⋊C4—22no—
105Dic83⋊C4—22no—
106C4×Dic83—22no—
107C4×Dic83—22no—
108C24×C83—21no—
109C2×D4×C83—21no—
110C2×D4×C83—22no—
111C2×D4×C83—22no—
112C2×D4×C83C22×C412yes—
113C22×C4×C83—21no—
114C22×C4×C83—22no—
115C22×C4×C83—22no—
116C22×C4×C83—82no—
117C22×C4×C83C22×C411yescharacteristic
118C22×C4×C83C2×D411yescharacteristic
119C22×C4×C83#4111yescharacteristic
120C2×C8×C83—22no—
121C2×C8×C83—22no—
122C4⋊C4×C83—21no—
123C4⋊C4×C83C22×C412yes—
124C22⋊C4×C83—22no—
125C22⋊C4×C83—22no—
126C42×C83C22×C411yescharacteristic
127C42×C83C2×D411yescharacteristic
128C42×C83#4111yescharacteristic
129C42×C83C2411yescharacteristic
130C23×C83—22no—
131C23×C83—42no—
132C23×C83—81no—
133C23×C83—81no—
134C23×C83C4×D411yescharacteristic
135D4×C83—22no—
136D4×C83—22no—
137D4×C83—42no—
138C2×C4×C83—22no—
139C2×C4×C83—22no—
140C2×C4×C83—41no—
141C2×C4×C83—44no—
142C2×C4×C83—82no—
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—22no—
154C2×Dic83—42no—
155C2×Dic83—81no—
156Dic166—22no—
157Dic166—22no—
158Dic166—42no—
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—42no—
170C22×C83—42no—
171C22×C83—42no—
172C22×C83—84no—
173C22×C83#43612yes—
174C22×C83C2×C4×D411yescharacteristic
175C4×C83—21no—
176C4×C83—21no—
177C4×C83—21no—
178C4×C83—42no—
179C4×C83#43612yes—
180C4×C83C22×M1611yescharacteristic
181C4×C83C2×#13111yescharacteristic
182C4×C83#57712yes—
183Dic83—42no—
184Dic83—42no—
185#17896—831nomaximal, Sylow
186C2×C83—22no—
187C2×C83—82no—
188C2×C83C2×#43611yescharacteristic
189C2×C83#294911yescharacteristic
190C2×C83#295711yescharacteristic
191#2983—831no—
192#2975—831no—
193C2×C4×SD16—831no—
194#1592—834no—
195#1590—834no—
196#1564—831no—
197#1479—831no—
198#1478D8311yescharacteristic
199#1450—831no—
200C83#1789611yescharacteristic, complemented, Sylow
201C22×SD16—831no—
202C2×C4×Q8—831no—
203C2×C4×D4C2×D8311yescharacteristic
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⋊C8—834no—
225D4⋊C8C2×D8314yes—
226#322—832no—
227C82—832no—
228C22×Q8—831no—
229C22×D4C4×D8311yescharacteristic
230C23×C4—21no—
231C2×SD16—832no—
232C2×SD16—832no—
233C2×SD16—1664no—
234C2×M16—1664no—
235C22×C8—831no—
236C22×C8—831no—
237C4×Q8—832no—
238C4×Q8—832no—
239C4×Q8—1661no—
240C4×Q8—1661no—
241C4×D4—21no—
242C4×D4—21no—
243C4×D4C4×D8312yes—
244C4×D4C22×D8312yes—
245C2×C4⋊C4—831no—
246C2×C4⋊C4—1661no—
247C2×C4⋊C4C4×D8311yescharacteristic
248C2×C22⋊C4—21no—
249C2×C42—1661no—
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—21no—
277C2×Q8—832no—
278C2×Q8—1662no—
279C2×Q8—1662no—
280C2×Q8—1661no—
281C2×D4—22no—
282C2×D4—22no—
283C2×D4—21no—
284C2×D4C2×C4×D8312yes—
285C22×C4—21no—
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—82no—
289C22×C4—1661no—
290C22×C4—1661no—
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—21no—
308C4⋊C4—832no—
309C4⋊C4—1662no—
310C4⋊C4—1661no—
311C4⋊C4—1662no—
312C4⋊C4C2×C4×D8312yes—
313C22⋊C4—22no—
314C22⋊C4—22no—
315C42—1662no—
316C42—1662no—
317C42C2×C4×D8311yescharacteristic
318C42D4×D8311yescharacteristic
319C42#1305058911yescharacteristic
320C42C23×D8311yescharacteristic
321C23—22no—
322C23—42no—
323C23—81no—
324C23—81no—
325C23(C22⋊D83)⋊C411yescharacteristic
326Q8—3322no—
327Q8—1662no—
328Q8—1662no—
329D4—22no—
330D4—22no—
331D4—42no—
332C2×C4—22no—
333C2×C4—22no—
334C2×C4—41no—
335C2×C4—44no—
336C2×C4—82no—
337C2×C4—81no—
338C2×C4—3322no—
339C2×C4—6641no—
340C2×C4—1662no—
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—42no—
361C22—42no—
362C22—42no—
363C22—84no—
364C22#1912777612yes—
365C22C2×(C22⋊D83)⋊C411yescharacteristic
366C4—21no—
367C4—21no—
368C4—21no—
369C4—42no—
370C4—3322no—
371C4—3322no—
372C4#1912777612yes—
373C4C2×M16×D8311yescharacteristic
374C4C2×#1458179811yescharacteristic
375C4#1912851112yes—
376C2—22no—
377C2—82no—
378C2C2×#1912777611yescharacteristic
379C2#2491632511yescharacteristic
380C2#2491637611yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D83
commutatorC2×C4×C83C23×C4
FrattiniC42C23×D83
FittingC83×#1478C2
socleC22×C83C2×C4×D4
prime radicalC22×D4×C83C2×C4
cosocleC42×C83C24
hypercenter#1478D83
hypocenterC83#17896
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1478

Complements

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

NH ≅ G/Ntype
1. #24916546C2semidirect
4. #24906362C2semidirect
5. #24906356C2semidirect
6. #24906209C2semidirect
7. #24905859C2semidirect
8. #24905747C2semidirect
13. #19127828C22semidirect
28. #19127453C22semidirect
29. #19127451C22semidirect
30. #19127433C22semidirect
31. #19127423C22semidirect
32. C8×C83⋊C8C22semidirect
96. C2×C83⋊C8M16semidirect
200. C83#17896semidirect

Maximal subgroups

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

subgroupindex
#249165462
#249165042
C2×C4×#130505782
#249063622
#249063562
#249062092
#249058592
#249057472
C83×#14782
#1789683