Finite Group Databasebeta

← #35374540

Subgroups — #35374540

Order 21248.

638 conjugacy classes of subgroups.

The subgroup lattice has 388 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#24917471C211yesmaximal, characteristic
2#24916560C211yesmaximal, characteristic, complemented
3#24916487C211yesmaximal, characteristic
4#24906317C212yesmaximal, complemented
5#24906296C212yesmaximal, complemented
6#24906272C212yesmaximal, complemented
7#24906266C212yesmaximal, complemented
8#24906244C211yesmaximal, characteristic, complemented
9#24905885C211yesmaximal, characteristic, complemented
10#24905760C211yesmaximal, characteristic, complemented
11C83×#1490C211yesmaximal, characteristic
12#19128258C2211yescharacteristic
13#19128031C412yes—
14#19127991C412yes—
15#19127906C412yes—
16#19127852C412yes—
17#19127829—22no—
18#19127828C2212yescomplemented
19#19127825C2212yescomplemented
20C4×#13050578C2212yes—
21C4×#13050578C2212yes—
22D4×C83⋊C8—22no—
23D4×C83⋊C8C2212yes—
24D4×C83⋊C8C2212yes—
25#19127759—22no—
26#19127754C2211yescharacteristic
27#19127754C2211yescharacteristic
28#19127754C2211yescharacteristic
29C2×C22⋊(C83⋊C8)—22no—
30#19127453C2212yescomplemented
31#19127451C2212yescomplemented
32#19127433C2212yescomplemented
33#19127433C2212yescomplemented
34#19127423C2211yescharacteristic, complemented
35#19127423C2211yescharacteristic, complemented
36#19127422C2211yescharacteristic, complemented
37C8×C83⋊C8C2211yescharacteristic, complemented
38C2×C4×D4×C83C2211yescharacteristic
39#432×C83C2211yescharacteristic
40C4×M16×C83C2211yescharacteristic
41D4⋊C8×C83C2212yes—
42D4⋊C8×C83C2212yes—
43C2×#13050578C2×C412yes—
44C2×#13050578C2×C412yes—
45C22×C83⋊C8—22no—
46C22×C83⋊C8—22no—
47#14581792C2×C411yescharacteristic
48#14581792C2×C411yescharacteristic
49#14581784—21no—
50#14581771—21no—
51#14581771C2×C412yes—
52C4×Dic166—21no—
53C4×Dic166C2312yes—
54#14581751—21no—
55#14581723C2×C412yes—
56#14581723C2×C412yes—
57C4⋊(C83⋊C8)C2×C412yes—
58C4⋊(C83⋊C8)D412yes—
59C4⋊(C83⋊C8)C2312yes—
60Dic83⋊C8—22no—
61Dic83⋊C8—22no—
62#14581715C2×C412yes—
63#14581715C2×C412yes—
64#14581713C2×C412yes—
65#14581713C2×C412yes—
66C22⋊(C83⋊C8)—22no—
67C22⋊(C83⋊C8)—22no—
68C22⋊(C83⋊C8)—21no—
69C22⋊(C83⋊C8)C2×C412yes—
70C22⋊(C83⋊C8)D412yes—
71#14581702—22no—
72C4×C83⋊C8—21no—
73C4×C83⋊C8C2311yescharacteristic
74C4×C83⋊C8C2311yescharacteristic
75C4×C83⋊C8C2311yescharacteristic
76C4×C83⋊C8C2311yescharacteristic
77C8×Dic83—22no—
78C22×D4×C83C2×C411yescharacteristic
79C23×C4×C83—21no—
80C2×M16×C83—21no—
81C4×D4×C83—21no—
82C4×D4×C83—21no—
83C4×D4×C83C2×C412yes—
84C4×D4×C83C2312yes—
85C2×C4⋊C4×C83C2×C411yescharacteristic
86C2×C22⋊C4×C83—21no—
87C2×C42×C83C2311yescharacteristic
88C4⋊C8×C83C2312yes—
89C22⋊C8×C83—21no—
90#97×C83—21no—
91#97×C83—21no—
92C4×C8×C83C2311yescharacteristic
93C4×C8×C83C2311yescharacteristic
94C2×Dic166—21no—
95C2×Dic166C22×C412yes—
96#13050578—24no—
97#13050578—24no—
98C2×C83⋊C8—22no—
99C2×C83⋊C8—82no—
100C2×C83⋊C8—82no—
101C2×C83⋊C8C22×C412yes—
102C2×C83⋊C8C22×C412yes—
103C2×C83⋊C8C2×D412yes—
104C2×C83⋊C8C2×D412yes—
105C4⋊Dic83—21no—
106C4⋊Dic83—21no—
107C4⋊Dic83—21no—
108C4⋊Dic83C22×C412yes—
109Dic83⋊C4—22no—
110Dic83⋊C4—22no—
111C22⋊Dic83—22no—
112C4×Dic83—22no—
113C24×C83—21no—
114C2×D4×C83—24no—
115C2×D4×C83—21no—
116C2×D4×C83C22×C412yes—
117C22×C4×C83—21no—
118C22×C4×C83—22no—
119C22×C4×C83—22no—
120C22×C4×C83—82no—
121C22×C4×C83C22×C411yescharacteristic
122C22×C4×C83C2×D411yescharacteristic
123C22×C4×C83#4111yescharacteristic
124M16×C83—42no—
125C2×C8×C83—21no—
126C2×C8×C83—22no—
127C2×C8×C83—21no—
128C4⋊C4×C83—21no—
129C4⋊C4×C83C22×C412yes—
130C22⋊C4×C83—22no—
131C22⋊C4×C83—22no—
132C42×C83C22×C411yescharacteristic
133C42×C83C2×D411yescharacteristic
134C42×C83#4111yescharacteristic
135C42×C83C2411yescharacteristic
136C23×C83—24no—
137C23×C83—22no—
138C23×C83—81no—
139C23×C83—81no—
140C23×C83C4×D411yescharacteristic
141D4×C83—24no—
142D4×C83—22no—
143D4×C83#13114yes—
144C2×C4×C83—22no—
145C2×C4×C83—22no—
146C2×C4×C83—22no—
147C2×C4×C83—41no—
148C2×C4×C83—42no—
149C2×C4×C83—42no—
150C2×C4×C83—82no—
151C2×C4×C83—81no—
152C2×C4×C83C4×D411yescharacteristic
153C2×C4×C83C4×D411yescharacteristic
154C2×C4×C83C4×D411yescharacteristic
155C2×C4×C83C23×C411yescharacteristic
156C2×C4×C83C22×D411yescharacteristic
157C2×C4×C83C2×#4111yescharacteristic
158C2×Dic83—22no—
159C2×Dic83—22no—
160Dic166—22no—
161Dic166—22no—
162Dic166—81no—
163C8×C83—22no—
164C8×C83—41no—
165C8×C83—42no—
166C83⋊C8—22no—
167C83⋊C8—22no—
168C83⋊C8—22no—
169C83⋊C8—22no—
170C83⋊C8—42no—
171C22×C83—21no—
172C22×C83—24no—
173C22×C83—21no—
174C22×C83—21no—
175C22×C83—42no—
176C22×C83—42no—
177C22×C83—84no—
178C22×C83C2×C4×D411yescharacteristic
179C4×C83—21no—
180C4×C83—21no—
181C4×C83—21no—
182C4×C83—21no—
183C4×C83—42no—
184C4×C83C2×#13112yes—
185C4×C83#57712yes—
186Dic83—42no—
187Dic83—42no—
188#17924—831nomaximal, Sylow
189C2×C83—41no—
190C2×C83—44no—
191C2×C83#292911yescharacteristic
192C2×C83#294911yescharacteristic
193C2×C83#295611yescharacteristic
194#3101—831no—
195#2985—831no—
196#2972—831no—
197#1585—832no—
198#1582—832no—
199SD16⋊C8—832no—
200C8×SD16—832no—
201#1572—831no—
202#1490D8311yescharacteristic
203#1485—831no—
204#1453—831no—
205C83#1792411yescharacteristic, complemented, Sylow
206#533—831no—
207C2×C4×D4C2×D8311yescharacteristic
208#483—832no—
209#476—832no—
210#461—832no—
211#450—832no—
212#446—832no—
213#446—1662no—
214C8×Q8—832no—
215C4×SD16—832no—
216C4×SD16—832no—
217C8×D4—832no—
218C8×D4—832no—
219C8×D4—1662no—
220#433—1662no—
221#432C2×D8311yescharacteristic
222#431—831no—
223#431—831no—
224#431—831no—
225C2×C22⋊C8—1662no—
226C4×M16C2×D8311yescharacteristic
227#335—832no—
228#334—832no—
229Q8⋊C8—832no—
230Q8⋊C8—832no—
231D4⋊C8C2×D8312yes—
232D4⋊C8C2×D8312yes—
233#322—831no—
234#322—831no—
235#322—831no—
236C82—831no—
237C22×D4C4×D8311yescharacteristic
238C23×C4—21no—
239C2×SD16—832no—
240C2×SD16—832no—
241C2×M16—21no—
242C22×C8—1662no—
243C22×C8—1662no—
244C4⋊Q8—831no—
245C4⋊Q8—831no—
246#125—1661no—
247C22⋊Q8—832no—
248C22⋊Q8—1661no—
249C4×Q8—832no—
250C4×Q8—1661no—
251C4×D4—21no—
252C4×D4—21no—
253C4×D4C4×D8312yes—
254C4×D4C22×D8312yes—
255#117—1661no—
256C2×C4⋊C4C4×D8311yescharacteristic
257C2×C22⋊C4—21no—
258C2×C42C22×D8311yescharacteristic
259#106—832no—
260#106—832no—
261C4⋊C8—832no—
262C4⋊C8—832no—
263C4⋊C8—832no—
264C4⋊C8—1662no—
265C4⋊C8—1662no—
266C4⋊C8C22×D8312yes—
267#103—832no—
268#103—832no—
269#102—832no—
270#102—832no—
271C22⋊C8—21no—
272C22⋊C8—832no—
273C22⋊C8—832no—
274C22⋊C8—1662no—
275C22⋊C8—1662no—
276C22⋊C8—1661no—
277#97—21no—
278#97—21no—
279#97—1662no—
280C4×C8—831no—
281C4×C8—831no—
282C4×C8—831no—
283C4×C8—831no—
284C4×C8—1662no—
285C4×C8—1661no—
286C4×C8C22×D8311yescharacteristic
287C4×C8C22×D8311yescharacteristic
288C24—21no—
289C2×Q8—832no—
290C2×Q8—1661no—
291C2×D4—24no—
292C2×D4—21no—
293C2×D4C2×C4×D8312yes—
294C22×C4—21no—
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—82no—
298C22×C4C2×C4×D8311yescharacteristic
299C22×C4D4×D8311yescharacteristic
300C22×C4#1305058911yescharacteristic
301SD16—1664no—
302SD16—1664no—
303M16—42no—
304C2×C8—21no—
305C2×C8—22no—
306C2×C8—21no—
307C2×C8—832no—
308C2×C8—832no—
309C2×C8—832no—
310C2×C8—832no—
311C2×C8—6642no—
312C2×C8—6642no—
313C2×C8—1662no—
314C4⋊C4—21no—
315C4⋊C4—832no—
316C4⋊C4—1662no—
317C4⋊C4—1661no—
318C4⋊C4—1661no—
319C4⋊C4—1662no—
320C4⋊C4—1661no—
321C4⋊C4C2×C4×D8312yes—
322C22⋊C4—22no—
323C22⋊C4—22no—
324C22⋊C4—1662no—
325C42—1662no—
326C42C2×C4×D8311yescharacteristic
327C42D4×D8311yescharacteristic
328C42#1305058911yescharacteristic
329C42C23×D8311yescharacteristic
330C23—24no—
331C23—22no—
332C23—81no—
333C23—81no—
334C23(C22⋊D83)⋊C411yescharacteristic
335Q8—6641no—
336Q8—1662no—
337Q8—1662no—
338D4—24no—
339D4—22no—
340D4#1458180014yes—
341C2×C4—22no—
342C2×C4—22no—
343C2×C4—22no—
344C2×C4—41no—
345C2×C4—42no—
346C2×C4—42no—
347C2×C4—82no—
348C2×C4—81no—
349C2×C4—1662no—
350C2×C4—1662no—
351C2×C4(C22⋊D83)⋊C411yescharacteristic
352C2×C4(C22⋊D83)⋊C411yescharacteristic
353C2×C4(C22⋊D83)⋊C411yescharacteristic
354C2×C4C22×C4×D8311yescharacteristic
355C2×C4C2×D4×D8311yescharacteristic
356C2×C4C2×#1305058911yescharacteristic
357C8—22no—
358C8—41no—
359C8—42no—
360C8—3322no—
361C8—1662no—
362C8—1662no—
363C8—1662no—
364C8—1662no—
365C22—21no—
366C22—21no—
367C22—21no—
368C22—24no—
369C22—42no—
370C22—42no—
371C22—84no—
372C22C2×(C22⋊D83)⋊C411yescharacteristic
373C4—21no—
374C4—21no—
375C4—21no—
376C4—21no—
377C4—42no—
378C4—3322no—
379C4—3322no—
380C4C2×#1458180012yes—
381C4#1912851112yes—
382C2—41no—
383C2—44no—
384C2#2491615311yescharacteristic
385C2#2491632511yescharacteristic
386C2#2491636911yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D83
commutatorC2×C4×C83C23×C4
FrattiniC42C23×D83
FittingC83×#1490C2
socleC22×C83C2×C4×D4
prime radicalC22×D4×C83C2×C4
cosocleC42×C83C24
hypercenter#1490D83
hypocenterC83#17924
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1490

Complements

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

NH ≅ G/Ntype
2. #24916560C2semidirect
4. #24906317C2semidirect
5. #24906296C2semidirect
6. #24906272C2semidirect
7. #24906266C2semidirect
8. #24906244C2semidirect
9. #24905885C2semidirect
10. #24905760C2semidirect
18. #19127828C22semidirect
19. #19127825C22semidirect
30. #19127453C22semidirect
31. #19127451C22semidirect
32. #19127433C22semidirect
33. #19127433C22semidirect
34. #19127423C22semidirect
35. #19127423C22semidirect
36. #19127422C22semidirect
37. C8×C83⋊C8C22semidirect
205. C83#17924semidirect

Maximal subgroups

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

subgroupindex
#249174712
#249165602
#249164872
#249063172
#249062962
#249062722
#249062662
#249062442
#249058852
#249057602
C83×#14902
#1792483