Finite Group Databasebeta

← #35407609

Subgroups — #35407609

Order 21248.

814 conjugacy classes of subgroups.

The subgroup lattice has 404 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×#13050580C211yesmaximal, characteristic
2#24910114C211yesmaximal, characteristic
3#24909827C211yesmaximal, characteristic
4#24908107C212yesmaximal
5#24908037C212yesmaximal
6#24907801C212yesmaximal
7#24907718C212yesmaximal
8C4×#14581725C211yesmaximal, characteristic
9#24907294C211yesmaximal, characteristic
10#24907216C211yesmaximal, characteristic
11C83×#1763C211yesmaximal, characteristic
12C22×#13050580C2211yescharacteristic
13C2×C4×Dic166C2211yescharacteristic
14C2×#14581757C2212yes—
15C4×#13050580C418yescomplemented
16C2×#14581725C2212yes—
17C2×#14581725C2211yescharacteristic
18C2×#14581717C2212yes—
19C2×#14581716C2211yescharacteristic
20C2×#14581715C2211yescharacteristic
21C2×C4×C83⋊C8C2211yescharacteristic
22#19127608C2212yes—
23#19127600C2212yes—
24#19127587—22no—
25#19127569—21no—
26#19127567—21no—
27C4⋊C4×Dic83C2212yes—
28C4×Dic83⋊C4—22no—
29C4×C4⋊Dic83C2211yescharacteristic
30#19127552—21no—
31#19127547C2211yescharacteristic
32#19127546—21no—
33#19127464C2212yes—
34#19127463C2212yes—
35#19127455C2212yes—
36#19127455C2212yes—
37C2×C4×Q8×C83C2211yescharacteristic
38C2×#103×C83C2212yes—
39C2×#97×C83C2211yescharacteristic
40C4×C4⋊C4×C83C2211yescharacteristic
41#340×C83C2212yes—
42C22×Dic166C2311yescharacteristic
43C2×#13050580—24no—
44C2×#13050580C2×C414yes—
45C22×C83⋊C8C2311yescharacteristic
46C22×C83⋊C8C2311yescharacteristic
47C4×Dic166—22no—
48C4×Dic166C2×C414yes—
49#14581757—24no—
50#14581757—24no—
51C2×C4⋊Dic83—22no—
52C2×C4⋊Dic83C2312yes—
53C2×C4⋊Dic83C2311yescharacteristic
54C2×Dic83⋊C4—22no—
55C2×Dic83⋊C4—21no—
56C2×C4×Dic83—22no—
57C2×C4×Dic83—22no—
58C2×C4×Dic83—21no—
59C2×C4×Dic83—21no—
60C2×C4×Dic83C2312yes—
61#14581725—24no—
62#14581725C2×C414yes—
63#14581717—24no—
64#14581716C2×C414yes—
65#14581715C2×C414yes—
66C4×C83⋊C8C2×C414yes—
67C83⋊#95—22no—
68C83⋊#95—22no—
69C83⋊#95—22no—
70C83⋊#95—22no—
71C83⋊#95—21no—
72C83⋊#95D411yescharacteristic
73C83⋊#95D412yes—
74C83⋊#95D411yescharacteristic
75C22×Q8×C83C2311yescharacteristic
76C22×C8×C83C2312yes—
77C4×Q8×C83—22no—
78C4×Q8×C83C2×C414yes—
79C2×C4⋊C4×C83—21no—
80C2×C4⋊C4×C83C2311yescharacteristic
81C2×C4⋊C4×C83C2312yes—
82C2×C42×C83—21no—
83C2×C42×C83—21no—
84C2×C42×C83C2311yescharacteristic
85#103×C83—24no—
86#97×C83—22no—
87#95×C83—21no—
88C2×Dic166—22no—
89C2×Dic166—21no—
90C2×Dic166C22×C412yes—
91C2×Dic166#4114yes—
92C22×Dic83—21no—
93C22×Dic83—21no—
94C22×Dic83—22no—
95C22×Dic83C2×D412yes—
96C22×Dic83C2×D412yes—
97#13050580—28no—
98C2×C83⋊C8—22no—
99C2×C83⋊C8—22no—
100C2×C83⋊C8C22×C412yes—
101C2×C83⋊C8C22×C412yes—
102C4⋊Dic83—21no—
103C4⋊Dic83—22no—
104C4⋊Dic83—22no—
105C4⋊Dic83—82no—
106C4⋊Dic83C22×C412yes—
107Dic83⋊C4—28no—
108Dic83⋊C4—24no—
109C4×Dic83—28no—
110C4×Dic83—24no—
111C4×Dic83—22no—
112C4×Dic83—22no—
113C4×Dic83—82no—
114C4×Dic83—81no—
115C2×Q8×C83—21no—
116C2×Q8×C83—22no—
117C2×Q8×C83—22no—
118C2×Q8×C83C22×C412yes—
119C22×C4×C83—22no—
120C22×C4×C83—21no—
121C22×C4×C83—21no—
122C22×C4×C83C2×D411yescharacteristic
123C22×C4×C83C2×D411yescharacteristic
124C22×C4×C83C2411yescharacteristic
125C2×C8×C83—22no—
126C2×C8×C83—22no—
127C2×C8×C83—22no—
128C4⋊C4×C83—24no—
129C4⋊C4×C83—21no—
130C4⋊C4×C83—22no—
131C4⋊C4×C83C22×C412yes—
132C4⋊C4×C83#4114yes—
133C42×C83—24no—
134C42×C83—81no—
135C42×C83C22×C412yes—
136C42×C83#4112yes—
137C23×C83C22×D411yescharacteristic
138Q8×C83—22no—
139Q8×C83—22no—
140Q8×C83—42no—
141C2×C4×C83—21no—
142C2×C4×C83—22no—
143C2×C4×C83—21no—
144C2×C4×C83—22no—
145C2×C4×C83—22no—
146C2×C4×C83—24no—
147C2×C4×C83—42no—
148C2×C4×C83—42no—
149C2×C4×C83—41no—
150C2×C4×C83—42no—
151C2×C4×C83—41no—
152C2×C4×C83C4×D412yes—
153C2×C4×C83C4×D412yes—
154C2×C4×C83C23×C411yescharacteristic
155C2×C4×C83C2×#4112yes—
156C2×C4×C83C2×#4111yescharacteristic
157C2×C4×C83C2×#4112yes—
158C2×Dic83—22no—
159C2×Dic83—24no—
160C2×Dic83—22no—
161C2×Dic83—22no—
162C2×Dic83—22no—
163C2×Dic83—22no—
164C2×Dic83—22no—
165C2×Dic83—22no—
166C2×Dic83—22no—
167C2×Dic83—41no—
168C2×Dic83—41no—
169C2×Dic83—42no—
170C2×Dic83—42no—
171C2×Dic83—42no—
172Dic166—22no—
173Dic166—24no—
174Dic166#11714yes—
175C8×C83—42no—
176C83⋊C8—22no—
177C83⋊C8—22no—
178C22×C83C2×C4×D411yescharacteristic
179C22×C83#52512yes—
180C22×C83#57611yescharacteristic
181C22×C83#57812yes—
182C22×C83#57811yescharacteristic
183C4×C83—22no—
184C4×C83—22no—
185C4×C83—44no—
186C4×C83—42no—
187C4×C83—42no—
188C4×C83C2×#11711yescharacteristic
189C4×C83C4×#4111yescharacteristic
190C4×C83#52912yes—
191Dic83—24no—
192Dic83—42no—
193Dic83—42no—
194Dic83—42no—
195Dic83—82no—
196#21135—831nomaximal, Sylow
197C2×C83#229011yescharacteristic
198C2×C83#295011yescharacteristic
199C2×C83#295111yescharacteristic
200C2×C83#312412yes—
201C2×C83#312712yes—
202C2×C4×Q16—831no—
203#2320—831no—
204#2276—831no—
205#1928—832no—
206#1918—832no—
207#1878—832no—
208#1863—832no—
209C4×#107—831no—
210#1763—831no—
211#1763D8311yescharacteristic
212#1746—831no—
213C83#2113511yescharacteristic, complemented, Sylow
214C22×Q16—831no—
215C2×C4×Q8—832no—
216C2×C4×Q8—831no—
217C2×C4×Q8C2×D8311yescharacteristic
218C4×Q16—838no—
219C2×#107—831no—
220C2×#107—832no—
221C2×#103—831no—
222C2×#103—831no—
223C2×#103—832no—
224C2×#103C2×D8312yes—
225C2×#97C2×D8311yescharacteristic
226C2×C4×C8—831no—
227#391—832no—
228#389—832no—
229#387—1662no—
230#383—1661no—
231#382—1661no—
232C4×C4⋊C4—832no—
233C4×C4⋊C4—831no—
234C4×C4⋊C4—1661no—
235C4×C4⋊C4—1662no—
236C4×C4⋊C4C2×D8311yescharacteristic
237#376—831no—
238#376—1661no—
239#340—832no—
240#340—832no—
241#340C2×D8312yes—
242#336—832no—
243#336—832no—
244C22×Q8—831no—
245C22×Q8C22×D8311yescharacteristic
246C2×Q16—834no—
247C2×Q16—1664no—
248C22×C8—831no—
249C22×C8—831no—
250C22×C8C22×D8312yes—
251C4×Q8—22no—
252C4×Q8—834no—
253C4×Q8—1664no—
254C4×Q8—1664no—
255C4×Q8—1662no—
256C4×Q8C4×D8314yes—
257C2×C4⋊C4—21no—
258C2×C4⋊C4—831no—
259C2×C4⋊C4—832no—
260C2×C4⋊C4—1662no—
261C2×C4⋊C4—1662no—
262C2×C4⋊C4—1661no—
263C2×C4⋊C4C22×D8311yescharacteristic
264C2×C4⋊C4C22×D8312yes—
265C2×C42—21no—
266C2×C42—21no—
267C2×C42—832no—
268C2×C42—1662no—
269C2×C42—1661no—
270C2×C42—1662no—
271C2×C42—1661no—
272C2×C42C22×D8311yescharacteristic
273#107—834no—
274#107—1664no—
275#103—24no—
276#103—834no—
277#103—834no—
278#103—1664no—
279#97—22no—
280C4×C8—834no—
281#95—21no—
282#95—831no—
283#95—831no—
284#95—832no—
285#95—1662no—
286#95—1662no—
287#95—1662no—
288#95—1662no—
289#95—1661no—
290C2×Q8—21no—
291C2×Q8—22no—
292C2×Q8—22no—
293C2×Q8—832no—
294C2×Q8—834no—
295C2×Q8—1661no—
296C2×Q8—1662no—
297C2×Q8C2×C4×D8312yes—
298C22×C4—22no—
299C22×C4—21no—
300C22×C4—21no—
301C22×C4—832no—
302C22×C4—832no—
303C22×C4—1662no—
304C22×C4—1661no—
305C22×C4—1661no—
306C22×C4D4×D8311yescharacteristic
307C22×C4D4×D8311yescharacteristic
308C22×C4C23×D8311yescharacteristic
309Q16—1668no—
310C2×C8—22no—
311C2×C8—22no—
312C2×C8—22no—
313C2×C8—832no—
314C2×C8—832no—
315C2×C8—1662no—
316C2×C8—1662no—
317C4⋊C4—22no—
318C4⋊C4—21no—
319C4⋊C4—24no—
320C4⋊C4—832no—
321C4⋊C4—6642no—
322C4⋊C4—1668no—
323C4⋊C4—1662no—
324C4⋊C4—1662no—
325C4⋊C4—1661no—
326C4⋊C4—1664no—
327C4⋊C4C2×C4×D8312yes—
328C4⋊C4#1305059014yes—
329C42—24no—
330C42—81no—
331C42—6641no—
332C42—6642no—
333C42—1668no—
334C42—1662no—
335C42—1662no—
336C42—1664no—
337C42C2×C4×D8312yes—
338C42#1305058912yes—
339C23C2×D4×D8311yescharacteristic
340Q8—22no—
341Q8—22no—
342Q8—42no—
343Q8—834no—
344Q8—1662no—
345Q8—1664no—
346C2×C4—21no—
347C2×C4—22no—
348C2×C4—21no—
349C2×C4—22no—
350C2×C4—22no—
351C2×C4—24no—
352C2×C4—42no—
353C2×C4—42no—
354C2×C4—41no—
355C2×C4—41no—
356C2×C4—42no—
357C2×C4—3321no—
358C2×C4—3321no—
359C2×C4—3322no—
360C2×C4—3322no—
361C2×C4—3322no—
362C2×C4—1662no—
363C2×C4—1662no—
364C2×C4—1662no—
365C2×C4—1662no—
366C2×C4—1662no—
367C2×C4—1662no—
368C2×C4—1662no—
369C2×C4—1662no—
370C2×C4—1664no—
371C2×C4(C22⋊D83)⋊C412yes—
372C2×C4(C22⋊D83)⋊C412yes—
373C2×C4C22×C4×D8311yescharacteristic
374C2×C4#41×D8312yes—
375C2×C4C2×#1305058911yescharacteristic
376C2×C4C2×#1305059012yes—
377C8—42no—
378C8—1662no—
379C8—1662no—
380C22C2×(C22⋊D83)⋊C411yescharacteristic
381C22#1912821012yes—
382C22#1912850511yescharacteristic
383C22#1912851711yescharacteristic
384C22#1912851712yes—
385C4—22no—
386C4—22no—
387C4—42no—
388C4—42no—
389C4—44no—
390C4—3322no—
391C4—3322no—
392C4—3322no—
393C4—6642no—
394C4—1664no—
395C4#117×D8311yescharacteristic
396C4C4×#1305059011yescharacteristic
397C4#1912823312yes—
398C2#2490990911yescharacteristic
399C2#2491632711yescharacteristic
400C2#2491633411yescharacteristic
401C2#2491772912yes—
402C2#2491776212yes—
subgroupG/H
centerC23C2×D4×D83
commutatorC2×C4×C83C23×C4
FrattiniC22×C4C23×D83
FittingC83×#1763C2
socle, prime radicalC23×C83C22×D4
cosocleC22×C4×C83C24
hypercenter#1763D83
hypocenterC83#21135
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1763

Complements

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

NH ≅ G/Ntype
15. C4×#13050580C4semidirect
213. C83#21135semidirect

Maximal subgroups

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

subgroupindex
C2×C4×#130505802
#249101142
#249098272
#249081072
#249080372
#249078012
#249077182
C4×#145817252
#249072942
#249072162
C83×#17632
#2113583