Finite Group Databasebeta

← #35408688

Subgroups — #35408688

Order 21248.

766 conjugacy classes of subgroups.

The subgroup lattice has 400 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×#13050580C211yesmaximal, characteristic
2#24910115C211yesmaximal, characteristic
3#24908333C212yesmaximal
4#24908198C212yesmaximal
5#24908142C212yesmaximal
6#24908138C212yesmaximal
7C4×#14581723C211yesmaximal, characteristic
8#24907296C211yesmaximal, characteristic
9#24907293C211yesmaximal, characteristic
10#24907150C211yesmaximal, characteristic
11C83×#2320C211yesmaximal, characteristic
12C22×#13050580C2211yescharacteristic
13C2×C4×Dic166C2211yescharacteristic
14C4×#13050580C418yescomplemented
15C2×#14581725C2211yescharacteristic
16C2×#14581723C2212yes—
17C2×#14581716C2212yes—
18C2×#14581716C2211yescharacteristic
19C2×#14581715C2212yes—
20C2×#14581715C2211yescharacteristic
21C2×#14581703C2211yescharacteristic
22C2×#14581703C2212yes—
23C2×C4×C83⋊C8C2212yes—
24C2×C4×C83⋊C8C2211yescharacteristic
25#19127607C2212yes—
26#19127599C2212yes—
27#19127583—22no—
28#19127568—21no—
29#19127565—21no—
30C4×C4⋊Dic83C2211yescharacteristic
31#19127463C2212yes—
32#19127463C2212yes—
33#19127463C2212yes—
34C2×C4×Q8×C83C2211yescharacteristic
35#391×C83C2212yes—
36#389×C83C2212yes—
37#387×C83—22no—
38#383×C83—21no—
39#382×C83—21no—
40C4×C4⋊C4×C83C2211yescharacteristic
41#376×C83C2211yescharacteristic
42C22×Dic166C2311yescharacteristic
43C2×#13050580—24no—
44C2×#13050580C2×C414yes—
45C22×C83⋊C8C2311yescharacteristic
46C22×C83⋊C8C2312yes—
47C22×C83⋊C8C2311yescharacteristic
48C4×Dic166—22no—
49C4×Dic166C2×C414yes—
50C2×C4⋊Dic83C2312yes—
51C2×C4⋊Dic83C2311yescharacteristic
52C2×Dic83⋊C4—21no—
53C2×Dic83⋊C4—22no—
54C2×C4×Dic83—21no—
55C2×C4×Dic83—21no—
56#14581725C2×C414yes—
57#14581723—24no—
58#14581716—24no—
59#14581716C2×C414yes—
60#14581715—24no—
61#14581715C2×C414yes—
62#14581703—24no—
63#14581703—22no—
64C4×C83⋊C8—24no—
65C4×C83⋊C8C2×C414yes—
66C83⋊#95—21no—
67C83⋊#95—22no—
68C83⋊#95—22no—
69C83⋊#95—22no—
70C22×Q8×C83C2311yescharacteristic
71C4×Q8×C83—22no—
72C4×Q8×C83C2×C414yes—
73C2×C4⋊C4×C83—21no—
74C2×C4⋊C4×C83—22no—
75C2×C4⋊C4×C83C2312yes—
76C2×C4⋊C4×C83C2311yescharacteristic
77C2×C42×C83—21no—
78C2×C42×C83—21no—
79C2×C42×C83C2312yes—
80C2×C42×C83C2311yescharacteristic
81#95×C83—21no—
82#95×C83—22no—
83#95×C83—22no—
84#95×C83—22no—
85#95×C83D411yescharacteristic
86#95×C83D411yescharacteristic
87#95×C83D412yes—
88C2×Dic166—22no—
89C2×Dic166—22no—
90C2×Dic166—21no—
91C2×Dic166C22×C412yes—
92C22×Dic83—21no—
93C22×Dic83—21no—
94C22×Dic83—22no—
95#13050580—28no—
96C2×C83⋊C8—22no—
97C2×C83⋊C8—22no—
98C2×C83⋊C8—22no—
99C2×C83⋊C8C22×C412yes—
100C2×C83⋊C8C22×C412yes—
101C2×C83⋊C8#4114yes—
102C2×C83⋊C8#4114yes—
103C4⋊Dic83—21no—
104C4⋊Dic83—22no—
105C4⋊Dic83—22no—
106C4⋊Dic83C22×C412yes—
107Dic83⋊C4—24no—
108Dic83⋊C4—82no—
109C4×Dic83—24no—
110C4×Dic83—81no—
111C2×Q8×C83—21no—
112C2×Q8×C83—22no—
113C2×Q8×C83—22no—
114C2×Q8×C83C22×C412yes—
115C22×C4×C83—21no—
116C22×C4×C83—21no—
117C22×C4×C83—22no—
118C22×C4×C83C2×D412yes—
119C22×C4×C83C2×D412yes—
120C22×C4×C83C2×D411yescharacteristic
121C22×C4×C83C2×D411yescharacteristic
122C22×C4×C83C2411yescharacteristic
123C4⋊C4×C83—24no—
124C4⋊C4×C83—21no—
125C4⋊C4×C83—22no—
126C4⋊C4×C83—22no—
127C4⋊C4×C83—82no—
128C4⋊C4×C83C22×C412yes—
129C42×C83—24no—
130C42×C83—22no—
131C42×C83—22no—
132C42×C83—81no—
133C42×C83C22×C412yes—
134C42×C83#4112yes—
135C23×C83C22×D411yescharacteristic
136Q8×C83—22no—
137Q8×C83—22no—
138Q8×C83—42no—
139C2×C4×C83—21no—
140C2×C4×C83—21no—
141C2×C4×C83—22no—
142C2×C4×C83—22no—
143C2×C4×C83—22no—
144C2×C4×C83—22no—
145C2×C4×C83—22no—
146C2×C4×C83—22no—
147C2×C4×C83—22no—
148C2×C4×C83—22no—
149C2×C4×C83—22no—
150C2×C4×C83—42no—
151C2×C4×C83—41no—
152C2×C4×C83—41no—
153C2×C4×C83—42no—
154C2×C4×C83—42no—
155C2×C4×C83—42no—
156C2×C4×C83C4×D412yes—
157C2×C4×C83C4×D412yes—
158C2×C4×C83C23×C411yescharacteristic
159C2×C4×C83C2×#4112yes—
160C2×C4×C83C2×#4112yes—
161C2×C4×C83C2×#4111yescharacteristic
162C2×Dic83—22no—
163C2×Dic83—22no—
164C2×Dic83—42no—
165C2×Dic83—41no—
166C2×Dic83—41no—
167C2×Dic83—42no—
168C2×Dic83—42no—
169C2×Dic83—42no—
170Dic166—22no—
171Dic166—22no—
172Dic166—42no—
173C83⋊C8—22no—
174C83⋊C8—24no—
175C83⋊C8#11714yes—
176C22×C83C2×C4×D411yescharacteristic
177C22×C83#53212yes—
178C22×C83#57711yescharacteristic
179C22×C83#57711yescharacteristic
180C22×C83#57812yes—
181C4×C83—22no—
182C4×C83—22no—
183C4×C83—42no—
184C4×C83—42no—
185C4×C83—42no—
186C4×C83—42no—
187C4×C83—82no—
188C4×C83C2×#11711yescharacteristic
189C4×C83C4×#4111yescharacteristic
190C4×C83#52912yes—
191Dic83—42no—
192Dic83—42no—
193Dic83—82no—
194#21283—831nomaximal, Sylow
195C2×C83#229311yescharacteristic
196C2×C83#295011yescharacteristic
197C2×C83#295211yescharacteristic
198C2×C83#317812yes—
199C2×C83#317812yes—
200C2×C4×Q16—831no—
201#2320—831no—
202#2320D8311yescharacteristic
203#1971—832no—
204#1943—832no—
205#1933—832no—
206#1933—832no—
207C4×#106—831no—
208#1763—831no—
209#1763—831no—
210#1726—831no—
211C83#2128311yescharacteristic, complemented, Sylow
212C22×Q16—831no—
213C2×C4×Q8—831no—
214C2×C4×Q8C2×D8311yescharacteristic
215C4×Q16—838no—
216C2×#107—831no—
217C2×#106—832no—
218C2×#103—831no—
219C2×#103—831no—
220C2×#103—832no—
221C2×#103—832no—
222C2×#97—832no—
223C2×#97—831no—
224C2×C4×C8—831no—
225C2×C4×C8—832no—
226#391—832no—
227#391C2×D8312yes—
228#389—832no—
229#389C2×D8312yes—
230#387—22no—
231#387—1662no—
232#383—21no—
233#383—1661no—
234#382—21no—
235#382—1661no—
236C4×C4⋊C4—831no—
237C4×C4⋊C4C2×D8311yescharacteristic
238#376C2×D8311yescharacteristic
239#340—832no—
240#340—832no—
241#340—832no—
242C22×Q8—831no—
243C22×Q8C22×D8311yescharacteristic
244C2×Q16—834no—
245C2×Q16—1664no—
246C22×C8—831no—
247C22×C8—831no—
248C22×C8—832no—
249C4×Q8—22no—
250C4×Q8—834no—
251C4×Q8—1662no—
252C4×Q8C4×D8314yes—
253C2×C4⋊C4—21no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—831no—
256C2×C4⋊C4—832no—
257C2×C4⋊C4—1662no—
258C2×C4⋊C4—1661no—
259C2×C4⋊C4C22×D8312yes—
260C2×C4⋊C4C22×D8311yescharacteristic
261C2×C42—21no—
262C2×C42—21no—
263C2×C42—1661no—
264C2×C42—1661no—
265C2×C42C22×D8312yes—
266C2×C42C22×D8311yescharacteristic
267#107—834no—
268#106—1664no—
269#103—834no—
270#103—834no—
271#103—1664no—
272#103—1664no—
273#97—1664no—
274#97—1662no—
275C4×C8—834no—
276C4×C8—1664no—
277#95—21no—
278#95—22no—
279#95—22no—
280#95—22no—
281#95—1661no—
282#95—1662no—
283#95—1662no—
284#95—1662no—
285#95C22⋊D8311yescharacteristic
286#95C22⋊D8311yescharacteristic
287#95C22⋊D8312yes—
288C2×Q8—21no—
289C2×Q8—22no—
290C2×Q8—22no—
291C2×Q8—832no—
292C2×Q8—1662no—
293C2×Q8—1662no—
294C2×Q8—1661no—
295C2×Q8C2×C4×D8312yes—
296C22×C4—21no—
297C22×C4—21no—
298C22×C4—22no—
299C22×C4—1662no—
300C22×C4—1661no—
301C22×C4—1661no—
302C22×C4C2×C22⋊D8312yes—
303C22×C4C2×C22⋊D8312yes—
304C22×C4C2×C22⋊D8311yescharacteristic
305C22×C4C2×C22⋊D8311yescharacteristic
306C22×C4C23×D8311yescharacteristic
307Q16—1668no—
308C2×C8—832no—
309C2×C8—834no—
310C2×C8—832no—
311C2×C8—834no—
312C2×C8—1662no—
313C2×C8—1662no—
314C2×C8—1662no—
315C4⋊C4—21no—
316C4⋊C4—24no—
317C4⋊C4—22no—
318C4⋊C4—22no—
319C4⋊C4—82no—
320C4⋊C4—832no—
321C4⋊C4—6642no—
322C4⋊C4—1662no—
323C4⋊C4—1662no—
324C4⋊C4—1661no—
325C4⋊C4—1664no—
326C4⋊C4C2×C4×D8312yes—
327C42—22no—
328C42—22no—
329C42—24no—
330C42—81no—
331C42—6641no—
332C42—1664no—
333C42C2×C4×D8312yes—
334C42#1305059012yes—
335C23C22×C22⋊D8311yescharacteristic
336Q8—22no—
337Q8—22no—
338Q8—42no—
339Q8—3322no—
340Q8—1662no—
341Q8—1662no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—22no—
345C2×C4—22no—
346C2×C4—22no—
347C2×C4—22no—
348C2×C4—22no—
349C2×C4—22no—
350C2×C4—22no—
351C2×C4—22no—
352C2×C4—22no—
353C2×C4—42no—
354C2×C4—41no—
355C2×C4—41no—
356C2×C4—42no—
357C2×C4—42no—
358C2×C4—42no—
359C2×C4—3322no—
360C2×C4—3321no—
361C2×C4—3321no—
362C2×C4—3322no—
363C2×C4—3322no—
364C2×C4—3322no—
365C2×C4—1662no—
366C2×C4—1662no—
367C2×C4C4×C22⋊D8312yes—
368C2×C4C4×C22⋊D8312yes—
369C2×C4C22×C4×D8311yescharacteristic
370C2×C4#41×D8312yes—
371C2×C4#41×D8312yes—
372C2×C4C2×#1305059011yescharacteristic
373C8—834no—
374C8—1662no—
375C8—1664no—
376C22C2×C4×C22⋊D8311yescharacteristic
377C22#1912824712yes—
378C22#1912850911yescharacteristic
379C22#1912851211yescharacteristic
380C22#1912851612yes—
381C4—22no—
382C4—22no—
383C4—42no—
384C4—42no—
385C4—42no—
386C4—42no—
387C4—82no—
388C4—3322no—
389C4—3322no—
390C4—6642no—
391C4#117×D8311yescharacteristic
392C4#1912816911yescharacteristic
393C4#1912822912yes—
394C2#2490992211yescharacteristic
395C2#2491632911yescharacteristic
396C2#2491634411yescharacteristic
397C2#2491812512yes—
398C2#2491813312yes—
subgroupG/H
centerC23C22×C22⋊D83
commutatorC2×C4×C83C23×C4
FrattiniC22×C4C23×D83
FittingC83×#2320C2
socle, prime radicalC23×C83C22×D4
cosocleC22×C4×C83C24
hypercenter#2320D83
hypocenterC83#21283
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C23 ◁ #376 ◁ #2320

Complements

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

NH ≅ G/Ntype
14. C4×#13050580C4semidirect
211. C83#21283semidirect

Maximal subgroups

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

subgroupindex
C2×C4×#130505802
#249101152
#249083332
#249081982
#249081422
#249081382
C4×#145817232
#249072962
#249072932
#249071502
C83×#23202
#2128383