Finite Group Databasebeta

← #35375105

Subgroups — #35375105

Order 21248.

592 conjugacy classes of subgroups.

The subgroup lattice has 376 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#24916612C212yesmaximal, complemented
2#24906377C211yesmaximal, characteristic, complemented
3#24906372C211yesmaximal, characteristic, complemented
4#24906354C212yesmaximal, complemented
5#24906287C211yesmaximal, characteristic, complemented
6#24906282C211yesmaximal, characteristic, complemented
7SD16×C83⋊C8C212yesmaximal, complemented
8#24906219C211yesmaximal, characteristic, complemented
9#24905928C212yesmaximal, complemented
10#24905746C211yesmaximal, characteristic, complemented
11C83×#3168C211yesmaximal, characteristic
12#19127832C2212yes—
13Q8×C83⋊C8C2212yes—
14#19127779—22no—
15#19127779C2212yes—
16#19127773—22no—
17D4×C83⋊C8C2212yes—
18#19127762—22no—
19#19127754C2211yescharacteristic
20#19127754C2211yescharacteristic
21#19127728—22no—
22#19127725C2212yes—
23#19127452C2212yes—
24#19127452C2211yescharacteristic
25#19127452C2211yescharacteristic
26Q8⋊(C83⋊C8)C2212yes—
27Q8⋊(C83⋊C8)C2212yes—
28D4⋊(C83⋊C8)C2212yes—
29D4⋊(C83⋊C8)C2212yes—
30#19127422C2211yescharacteristic
31#19127422C2211yescharacteristic
32C8×C83⋊C8C2211yescharacteristic
33C8×C83⋊C8C2211yescharacteristic
34#529×C83C2212yes—
35#499×C83C411yescharacteristic
36#495×C83C411yescharacteristic
37#488×C83C412yes—
38#467×C83C411yescharacteristic
39#466×C83C411yescharacteristic
40#465×C83C412yes—
41C4×Q16×C83C2211yescharacteristic
42C4×SD16×C83C2212yes—
43C4×D8×C83C2211yescharacteristic
44C2×C4×C8×C83C2211yescharacteristic
45C2×#13050574—22no—
46C2×#13050574—22no—
47C22×C83⋊C8—22no—
48C4⋊(C83⋊C8)—22no—
49C4⋊(C83⋊C8)—22no—
50C4⋊(C83⋊C8)—22no—
51C4⋊(C83⋊C8)—21no—
52C4⋊(C83⋊C8)D412yes—
53C4⋊(C83⋊C8)C2312yes—
54C4⋊(C83⋊C8)C2312yes—
55C22⋊(C83⋊C8)—22no—
56C22⋊(C83⋊C8)—22no—
57C22⋊(C83⋊C8)—21no—
58C22⋊(C83⋊C8)D412yes—
59#14581703—22no—
60C4×C83⋊C8—22no—
61C4×C83⋊C8C2311yescharacteristic
62C4×C83⋊C8C2311yescharacteristic
63C4×C83⋊C8C2311yescharacteristic
64C4×C83⋊C8C2311yescharacteristic
65C2×Q16×C83C2×C411yescharacteristic
66C2×SD16×C83C2×C412yes—
67C2×D8×C83C2×C411yescharacteristic
68C22×C8×C83C2×C411yescharacteristic
69C22×C8×C83C2×C411yescharacteristic
70#126×C83—22no—
71#125×C83C2×C412yes—
72#124×C83C2×C412yes—
73#123×C83—22no—
74C22⋊Q8×C83C2×C412yes—
75#121×C83C2×C412yes—
76C4×Q8×C83C2312yes—
77C4×D4×C83—22no—
78C4×D4×C83C2312yes—
79#117×C83—22no—
80C2×C42×C83C2311yescharacteristic
81#107×C83C2×C411yescharacteristic
82#107×C83C2×C411yescharacteristic
83#106×C83C2×C412yes—
84#103×C83C2×C412yes—
85#103×C83C2×C412yes—
86#102×C83C2×C412yes—
87#102×C83C2×C412yes—
88C4×C8×C83C2×C411yescharacteristic
89C4×C8×C83C2×C411yescharacteristic
90C4×C8×C83C2311yescharacteristic
91C4×C8×C83C2311yescharacteristic
92#13050574—82no—
93#13050574—82no—
94C2×C83⋊C8—22no—
95C2×C83⋊C8—22no—
96C2×C83⋊C8—21no—
97C2×C83⋊C8—21no—
98C2×C83⋊C8—82no—
99C2×C83⋊C8C2×D412yes—
100C2×C83⋊C8C2×D412yes—
101C2×Q8×C83C22×C412yes—
102C2×D4×C83—22no—
103C2×D4×C83C22×C412yes—
104C22×C4×C83—22no—
105C22×C4×C83C22×C411yescharacteristic
106C22×C4×C83C2×D411yescharacteristic
107C22×C4×C83#4111yescharacteristic
108Q16×C83—22no—
109SD16×C83—24no—
110D8×C83—22no—
111C2×C8×C83—22no—
112C2×C8×C83—22no—
113C2×C8×C83C22×C411yescharacteristic
114C2×C8×C83C22×C411yescharacteristic
115C2×C8×C83C22×C411yescharacteristic
116C2×C8×C83C22×C411yescharacteristic
117C4⋊C4×C83—22no—
118C4⋊C4×C83—22no—
119C4⋊C4×C83—22no—
120C4⋊C4×C83—22no—
121C4⋊C4×C83C22×C412yes—
122C4⋊C4×C83C22×C412yes—
123C22⋊C4×C83—22no—
124C22⋊C4×C83—22no—
125C22⋊C4×C83—22no—
126C22⋊C4×C83—22no—
127C22⋊C4×C83—22no—
128C42×C83—22no—
129C42×C83C22×C411yescharacteristic
130C42×C83C2×D411yescharacteristic
131C42×C83#4111yescharacteristic
132C42×C83C2411yescharacteristic
133C23×C83—22no—
134C23×C83C4×D411yescharacteristic
135Q8×C83—22no—
136Q8×C83—22no—
137D4×C83—22no—
138D4×C83—22no—
139D4×C83—82no—
140C2×C4×C83—21no—
141C2×C4×C83—21no—
142C2×C4×C83—22no—
143C2×C4×C83—22no—
144C2×C4×C83—22no—
145C2×C4×C83—22no—
146C2×C4×C83—41no—
147C2×C4×C83—42no—
148C2×C4×C83—42no—
149C2×C4×C83C4×D411yescharacteristic
150C2×C4×C83C4×D411yescharacteristic
151C2×C4×C83C4×D411yescharacteristic
152C2×C4×C83C23×C411yescharacteristic
153C2×C4×C83C22×D411yescharacteristic
154C2×C4×C83C2×#4111yescharacteristic
155C8×C83—21no—
156C8×C83—21no—
157C8×C83#13112yes—
158C8×C83#13112yes—
159C83⋊C8—22no—
160C83⋊C8—22no—
161C83⋊C8—41no—
162C83⋊C8—41no—
163C83⋊C8—42no—
164C83⋊C8—42no—
165C22×C83—21no—
166C22×C83—21no—
167C22×C83—21no—
168C22×C83—42no—
169C22×C83—42no—
170C22×C83—42no—
171C22×C83C2×C4×D411yescharacteristic
172C4×C83—21no—
173C4×C83—21no—
174C4×C83—21no—
175C4×C83—21no—
176C4×C83—42no—
177C4×C83—42no—
178C4×C83—42no—
179C4×C83C2×#13111yescharacteristic
180C4×C83C2×#13111yescharacteristic
181C4×C83#57712yes—
182#17974—831nomaximal, Sylow
183C2×C83—41no—
184C2×C83—82no—
185C2×C83#293111yescharacteristic
186C2×C83#295211yescharacteristic
187C2×C83#295711yescharacteristic
188#3168D8311yescharacteristic
189#2993—832no—
190#1594—831no—
191#1593—831no—
192#1590—832no—
193#1580—831no—
194#1579—831no—
195C8×SD16—832no—
196#1566—831no—
197#1495—832no—
198#1450—831no—
199C83#1797411yescharacteristic, complemented, Sylow
200#529C2×D8312yes—
201#499Dic8311yescharacteristic
202#495Dic8311yescharacteristic
203#488Dic8312yes—
204#467Dic8311yescharacteristic
205#466Dic8311yescharacteristic
206#465Dic8312yes—
207#446—832no—
208C8×Q8—832no—
209C4×Q16C2×D8311yescharacteristic
210C4×SD16C2×D8312yes—
211C4×D8C2×D8311yescharacteristic
212#436—832no—
213#436—1662no—
214#435—1662no—
215C8×D4—832no—
216#433—1662no—
217#431—831no—
218#431—831no—
219#424—1662no—
220#423—832no—
221C2×C4×C8C2×D8311yescharacteristic
222#334—832no—
223#334—831no—
224#334—831no—
225Q8⋊C8—832no—
226Q8⋊C8—832no—
227D4⋊C8—832no—
228D4⋊C8—832no—
229#322—831no—
230#322—831no—
231C82—831no—
232C82—831no—
233C2×Q16C2×Dic8311yescharacteristic
234C2×SD16C2×Dic8312yes—
235C2×D8C2×Dic8311yescharacteristic
236C2×M16—1662no—
237C2×M16—1662no—
238C22×C8—1662no—
239C22×C8C2×Dic8311yescharacteristic
240C22×C8C2×Dic8311yescharacteristic
241#126—22no—
242#125C2×Dic8312yes—
243#124C2×Dic8312yes—
244#123—22no—
245C22⋊Q8C2×Dic8312yes—
246#121C2×Dic8312yes—
247C4×Q8C22×D8312yes—
248C4×D4—22no—
249C4×D4C22×D8312yes—
250#117—22no—
251C2×C42C22×D8311yescharacteristic
252#107C2×Dic8311yescharacteristic
253#107C2×Dic8311yescharacteristic
254#106C2×Dic8312yes—
255C4⋊C8—832no—
256C4⋊C8—832no—
257C4⋊C8—832no—
258C4⋊C8—1662no—
259C4⋊C8—1662no—
260C4⋊C8—1662no—
261C4⋊C8—1661no—
262#103C2×Dic8312yes—
263#103C2×Dic8312yes—
264#102C2×Dic8312yes—
265#102C2×Dic8312yes—
266C22⋊C8—832no—
267C22⋊C8—1662no—
268C22⋊C8—1662no—
269C22⋊C8—1661no—
270#97—1662no—
271C4×C8—831no—
272C4×C8—831no—
273C4×C8—831no—
274C4×C8—831no—
275C4×C8—1662no—
276C4×C8C2×Dic8311yescharacteristic
277C4×C8C2×Dic8311yescharacteristic
278C4×C8C22×D8311yescharacteristic
279C4×C8C22×D8311yescharacteristic
280C2×Q8C22×Dic8312yes—
281C2×D4—22no—
282C2×D4C22×Dic8312yes—
283C22×C4—22no—
284C22×C4C22×Dic8311yescharacteristic
285C22×C4D4×D8311yescharacteristic
286C22×C4#1305058911yescharacteristic
287Q16—22no—
288SD16—24no—
289D8—22no—
290M16—6642no—
291M16—6642no—
292C2×C8—22no—
293C2×C8—22no—
294C2×C8—832no—
295C2×C8—832no—
296C2×C8—6642no—
297C2×C8—1662no—
298C2×C8—1662no—
299C2×C8—1661no—
300C2×C8—1661no—
301C2×C8C22×Dic8311yescharacteristic
302C2×C8C22×Dic8311yescharacteristic
303C2×C8C22×Dic8311yescharacteristic
304C2×C8C22×Dic8311yescharacteristic
305C4⋊C4—22no—
306C4⋊C4—22no—
307C4⋊C4—22no—
308C4⋊C4—22no—
309C4⋊C4C22×Dic8312yes—
310C4⋊C4C22×Dic8312yes—
311C22⋊C4—22no—
312C22⋊C4—22no—
313C22⋊C4—22no—
314C22⋊C4—22no—
315C22⋊C4—22no—
316C42—22no—
317C42C22×Dic8311yescharacteristic
318C42D4×D8311yescharacteristic
319C42#1305058911yescharacteristic
320C42C23×D8311yescharacteristic
321C23—22no—
322C23D4×Dic8311yescharacteristic
323Q8—22no—
324Q8—22no—
325D4—22no—
326D4—22no—
327D4—82no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—22no—
331C2×C4—22no—
332C2×C4—22no—
333C2×C4—22no—
334C2×C4—41no—
335C2×C4—42no—
336C2×C4—42no—
337C2×C4D4×Dic8311yescharacteristic
338C2×C4D4×Dic8311yescharacteristic
339C2×C4D4×Dic8311yescharacteristic
340C2×C4C23×Dic8311yescharacteristic
341C2×C4C2×D4×D8311yescharacteristic
342C2×C4C2×#1305058911yescharacteristic
343C8—21no—
344C8—21no—
345C8—3321no—
346C8—3321no—
347C8—3322no—
348C8—3322no—
349C8—1662no—
350C8—1662no—
351C8#1458179912yes—
352C8#1458179912yes—
353C22—21no—
354C22—21no—
355C22—21no—
356C22—42no—
357C22—42no—
358C22—42no—
359C22C2×D4×Dic8311yescharacteristic
360C4—21no—
361C4—21no—
362C4—21no—
363C4—21no—
364C4—42no—
365C4—42no—
366C4—42no—
367C4C2×#1458179911yescharacteristic
368C4C2×#1458179911yescharacteristic
369C4#1912850812yes—
370C2—41no—
371C2—82no—
372C2#2491617511yescharacteristic
373C2#2491634711yescharacteristic
374C2#2491638011yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D83
commutatorC2×C4×C83C23×C4
FrattiniC42C23×D83
FittingC83×#3168C2
socleC22×C83C2×C4×D4
prime radical#466×C83C4
cosocleC42×C83C24
hypercenter#3168D83
hypocenterC83#17974
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #3168

Complements

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

NH ≅ G/Ntype
1. #24916612C2semidirect
2. #24906377C2semidirect
3. #24906372C2semidirect
4. #24906354C2semidirect
5. #24906287C2semidirect
6. #24906282C2semidirect
7. SD16×C83⋊C8C2semidirect
8. #24906219C2semidirect
9. #24905928C2semidirect
10. #24905746C2semidirect
199. C83#17974semidirect

Maximal subgroups

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

subgroupindex
#249166122
#249063772
#249063722
#249063542
#249062872
#249062822
SD16×C83⋊C82
#249062192
#249059282
#249057462
C83×#31682
#1797483