Finite Group Databasebeta

← #35408078

Subgroups — #35408078

Order 21248.

1638 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×#19128249C211yesmaximal, characteristic
2C2×#19127699C214yesmaximal, complemented
3C2×#19127653C211yesmaximal, characteristic, complemented
4#382×D83C211yesmaximal, characteristic, complemented
5#24908288C211yesmaximal, characteristic, complemented
6#24908283C211yesmaximal, characteristic, complemented
7#24908282C211yesmaximal, characteristic, complemented
8#24908273C214yesmaximal, complemented
9C83×#1961C211yesmaximal, characteristic, complemented
10C22×#13050589C2212yes—
11C22×#13050573C2212yes—
12#19128249—24no—
13#19128249C418yescomplemented
14C2×#14581792C2211yescharacteristic
15C2×#14581779C2212yes—
16C2×#14581763—22no—
17C2×D4×Dic83—22no—
18C2×C4⋊C4×D83C2212yes—
19C2×C42×D83C2211yescharacteristic
20#19127699—216no—
21C2×#14581717C2214yes—
22C2×#14581715C2214yes—
23C2×#14581713C2214yes—
24#19127653—24no—
25C2×#14581703C2211yescharacteristic
26C2×C8×Dic83C2212yes—
27#19127567C2211yescharacteristic
28#19127566C2212yes—
29#19127565—22no—
30#95×D83—22no—
31C2×#127×C83C2211yescharacteristic
32C2×#102×C83C2214yes—
33C2×#97×C83C2211yescharacteristic
34#382×C83C2211yescharacteristic
35C2×#13050589—24no—
36C2×#13050589—28no—
37C2×#13050589—28no—
38C2×#13050589C2×C418yes—
39C22×Dic166C2312yes—
40C22×C22⋊D83—22no—
41C23×Dic83—22no—
42C22×C4×D83—22no—
43C22×C4×D83D412yes—
44C22×C4×D83C2311yescharacteristic
45C2×#13050573—24no—
46C2×#13050573—24no—
47C2×#13050573D418yescomplemented
48C22×C83⋊C8C2312yes—
49#14581792—22no—
50#14581792C2×C414yescomplemented
51#14581779—24no—
52#14581779C2×C418yescomplemented
53#14581763—216no—
54D4×Dic83—216no—
55C2×C4⋊Dic83—22no—
56C2×C4⋊Dic83C2312yes—
57C2×Dic83⋊C4—22no—
58C4⋊C4×D83—28no—
59C2×C22⋊Dic83—24no—
60C2×C4×Dic83D412yes—
61C2×C4×Dic83C2311yescharacteristic
62C42×D83C2×C414yes—
63C42×D83D414yes—
64#14581717—28no—
65#14581715—28no—
66#14581713—28no—
67#14581703—22no—
68C8×Dic83—24no—
69C83⋊#95—24no—
70C83⋊#95—22no—
71C22×D4×C83—22no—
72C22×D4×C83C2312yes—
73C22×C8×C83C2312yes—
74#127×C83—22no—
75#127×C83C2×C414yescomplemented
76C2×C4⋊C4×C83C2312yes—
77C2×C42×C83C2311yescharacteristic
78#102×C83—28no—
79#97×C83—22no—
80#95×C83—22no—
81C23×D83C2×D411yescharacteristic
82#13050589—232no—
83#13050589—416no—
84C2×Dic166—24no—
85C2×Dic166—24no—
86C2×Dic166—22no—
87C2×Dic166C22×C414yes—
88C2×C22⋊D83—28no—
89C2×C22⋊D83—44no—
90C2×C22⋊D83—44no—
91C22×Dic83—22no—
92C22×Dic83—28no—
93C22×Dic83—22no—
94C22×Dic83—44no—
95C22×Dic83—44no—
96C22×Dic83C2×D412yes—
97C22×Dic83C2×D411yescharacteristic
98C2×C4×D83—44no—
99C2×C4×D83—48no—
100C2×C4×D83—44no—
101C2×C4×D83C22⋊C418yes—
102C2×C4×D83C22×C412yes—
103C2×C4×D83C2×D414yes—
104C2×C4×D83C2×D412yes—
105C2×C4×D83#4114yes—
106#13050573—216no—
107C2×C83⋊C8—22no—
108C2×C83⋊C8—22no—
109C2×C83⋊C8C2×D414yes—
110C4⋊Dic83—28no—
111C4⋊Dic83—24no—
112Dic83⋊C4—44no—
113C22⋊Dic83—216no—
114C4×Dic83C22⋊C418yes—
115C4×Dic83C22×C412yes—
116C4×Dic83C2×D412yes—
117C24×C83—22no—
118C2×D4×C83—28no—
119C2×D4×C83—24no—
120C2×D4×C83—22no—
121C2×D4×C83—24no—
122C2×D4×C83—48no—
123C2×D4×C83C22×C414yes—
124C22×C4×C83—22no—
125C22×C4×C83C2×D412yes—
126C22×C4×C83C2411yescharacteristic
127C2×C8×C83—22no—
128C2×C8×C83—22no—
129C2×C8×C83C2×D414yes—
130C4⋊C4×C83—24no—
131C42×C83C22×C412yes—
132C42×C83C2×D412yes—
133C23×C83—24no—
134C23×C83—42no—
135C23×C83—42no—
136C23×C83—48no—
137C23×C83C22×D411yescharacteristic
138C22×D83—24no—
139C22×D83C2×C22⋊C412yes—
140C22×D83#12412yes—
141C22×D83#12712yes—
142D4×C83—28no—
143D4×C83—416no—
144D4×C83—44no—
145C2×C4×C83—24no—
146C2×C4×C83—42no—
147C2×C4×C83—44no—
148C2×C4×C83C2×C22⋊C414yes—
149C2×C4×C83C23×C411yescharacteristic
150C2×C4×C83C22×D412yes—
151C2×C4×C83C22×D411yescharacteristic
152C2×C4×C83C2×#4112yes—
153C22⋊D83—416no—
154C2×Dic83—24no—
155C2×Dic83—24no—
156C2×Dic83—42no—
157C2×Dic83—42no—
158C2×Dic83—416no—
159C2×Dic83—42no—
160C2×Dic83—44no—
161C2×Dic83C2×C22⋊C414yes—
162C2×Dic83C2×C22⋊C412yes—
163C2×Dic83#12412yes—
164C2×Dic83#12712yes—
165C4×D83—48no—
166C4×D83—88no—
167C4×D83C4×D418yes—
168C4×D83#12118yes—
169Dic166—28no—
170Dic166—44no—
171C8×C83—24no—
172C83⋊C8—24no—
173C22×C83—48no—
174C22×C83—44no—
175C22×C83—84no—
176C22×C83—84no—
177C22×C83C22×C22⋊C411yescharacteristic
178C22×C83C2×#12411yescharacteristic
179C22×C83C2×#12711yescharacteristic
180C22×C83#57514yes—
181C2×D83—24no—
182C2×D83—24no—
183C2×D83—24no—
184C2×D83#39014yes—
185C4×C83—24no—
186C4×C83—82no—
187C4×C83C2×C4×D412yes—
188C4×C83C2×#12112yes—
189Dic83—24no—
190Dic83—44no—
191Dic83—82no—
192Dic83#39014yes—
193#21201—831nomaximal, Sylow
194C2×C83—84no—
195C2×C83C2×#39011yescharacteristic
196C2×C83#289612yes—
197C2×C83#314212yes—
198C2×C83#315312yes—
199D83—24no—
200C2×#532—831no—
201C2×#417—834no—
202C2×C4×M16—831no—
203C2×#382—831no—
204#1961—831no—
205#1961D8311yescharacteristic, complemented
206#1960—831no—
207#1960—831no—
208#1958—834no—
209C83#2120111yescharacteristic, complemented, Sylow
210C22×#41—832no—
211C22×M16—832no—
212#532—838no—
213#532—1664no—
214C2×C4⋊Q8—831no—
215C2×#127C2×D8311yescharacteristic
216C2×#124—832no—
217C2×#121—1662no—
218C2×C4×D4—1662no—
219C22×C4⋊C4—832no—
220C22×C42—831no—
221#417—16616no—
222C2×#103—834no—
223C2×#103—834no—
224C2×#102—834no—
225C2×#102C2×D8314yes—
226C4×M16—1664no—
227C2×#97—831no—
228C2×#97C2×D8311yescharacteristic
229C2×C4×C8—832no—
230#382—832no—
231#382—831no—
232#382—1662no—
233#382C2×D8311yescharacteristic
234C2×#95—1662no—
235C2×#41—838no—
236C2×#41—1668no—
237C2×#41—1668no—
238C2×#41—1664no—
239C22×Q8—832no—
240C22×D4—22no—
241C22×D4—1662no—
242C22×D4C22×D8312yes—
243C23×C4—832no—
244C23×C4—831no—
245C23×C4—1662no—
246C23×C4—1662no—
247C2×M16—838no—
248C2×M16—1664no—
249C2×M16—1664no—
250C22×C8—832no—
251C22×C8C22×D8312yes—
252C4⋊Q8—834no—
253C4⋊Q8—1662no—
254#127—22no—
255#127C4×D8314yescomplemented
256#124—838no—
257#124—1664no—
258#121—16616no—
259C4×D4—16616no—
260C2×C4⋊C4—832no—
261C2×C4⋊C4—1668no—
262C2×C4⋊C4—1662no—
263C2×C4⋊C4—1662no—
264C2×C4⋊C4C22×D8312yes—
265C2×C22⋊C4—1664no—
266C2×C42—832no—
267C2×C42—831no—
268C2×C42—834no—
269C2×C42—834no—
270C2×C42C22×D8311yescharacteristic
271#103—1668no—
272#103—1668no—
273#102—28no—
274#102—1668no—
275#97—22no—
276#97—1662no—
277C4×C8—1664no—
278#95—22no—
279#95—1664no—
280#95—1662no—
281C24—22no—
282C24—831no—
283#41—33216no—
284#41—16632no—
285C2×Q8—834no—
286C2×Q8—1664no—
287C2×Q8—1664no—
288C2×Q8—1662no—
289C2×D4—24no—
290C2×D4—22no—
291C2×D4—24no—
292C2×D4—28no—
293C2×D4—48no—
294C2×D4—3324no—
295C2×D4—3324no—
296C2×D4—1668no—
297C2×D4C2×C4×D8314yes—
298C22×C4—22no—
299C22×C4—3324no—
300C22×C4—3324no—
301C22×C4—3328no—
302C22×C4—3324no—
303C22×C4—3324no—
304C22×C4—838no—
305C22×C4—831no—
306C22×C4—832no—
307C22×C4—832no—
308C22×C4—834no—
309C22×C4—834no—
310C22×C4—832no—
311C22×C4—1662no—
312C22×C4—1668no—
313C22×C4—1662no—
314C22×C4D4×D8312yes—
315C22×C4C23×D8311yescharacteristic
316M16—16616no—
317C2×C8—22no—
318C2×C8—22no—
319C2×C8—834no—
320C2×C8—1662no—
321C2×C8—1662no—
322C2×C8D4×D8314yes—
323C4⋊C4—24no—
324C4⋊C4—3324no—
325C4⋊C4—1664no—
326C4⋊C4—1668no—
327C22⋊C4—16616no—
328C42—832no—
329C42—832no—
330C42—838no—
331C42C2×C4×D8312yes—
332C42D4×D8312yes—
333C23—24no—
334C23—42no—
335C23—42no—
336C23—48no—
337C23—832no—
338C23—832no—
339C23—832no—
340C23—1664no—
341C23C2×D4×D8311yescharacteristic
342Q8—3324no—
343Q8—1668no—
344D4—28no—
345D4—416no—
346D4—44no—
347D4—33216no—
348C2×C4—24no—
349C2×C4—42no—
350C2×C4—44no—
351C2×C4—3328no—
352C2×C4—3322no—
353C2×C4—3322no—
354C2×C4—33216no—
355C2×C4—3322no—
356C2×C4—3324no—
357C2×C4—832no—
358C2×C4—832no—
359C2×C4—832no—
360C2×C4—838no—
361C2×C4—838no—
362C2×C4—834no—
363C2×C4—6648no—
364C2×C4—1664no—
365C2×C4—1664no—
366C2×C4C22⋊C4×D8314yes—
367C2×C4C22×C4×D8311yescharacteristic
368C2×C4C2×D4×D8312yes—
369C2×C4C2×D4×D8311yescharacteristic
370C2×C4#41×D8312yes—
371C8—24no—
372C8—1664no—
373C22—48no—
374C22—44no—
375C22—84no—
376C22—84no—
377C22—834no—
378C22—1664no—
379C22—1664no—
380C22—1664no—
381C22C2×C22⋊C4×D8311yescharacteristic
382C22#124×D8311yescharacteristic
383C22#127×D8311yescharacteristic
384C22#1912849714yes—
385C4—24no—
386C4—82no—
387C4—3324no—
388C4—834no—
389C4—6642no—
390C4—1664no—
391C4C4×D4×D8312yes—
392C4#121×D8312yes—
393C2—84no—
394C2—1664no—
395C2#390×D8311yescharacteristic
396C2#2491597712yes—
397C2#2491786212yes—
398C2#2491792812yes—
subgroupG/H
centerC23C2×D4×D83
commutatorC2×C4×C83C23×C4
FrattiniC22×C4C23×D83
FittingC83×#1961C2
socleC23×C83C22×D4
prime radicalC2×#19128249C2
cosocleC22×C4×C83C24
hypercenter#1961D83
hypocenterC83#21201
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1961

Complements

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

NH ≅ G/Ntype
2. C2×#19127699C2semidirect
3. C2×#19127653C2semidirect
4. #382×D83C2semidirect
5. #24908288C2semidirect
6. #24908283C2semidirect
7. #24908282C2semidirect
8. #24908273C2semidirect
9. C83×#1961C2semidirect
13. #19128249C4semidirect
47. C2×#13050573D4semidirect
50. #14581792C2×C4semidirect
52. #14581779C2×C4semidirect
75. #127×C83C2×C4semidirect
205. #1961D83semidirect
209. C83#21201semidirect
255. #127C4×D83semidirect

Maximal subgroups

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

subgroupindex
C2×#191282492
C2×#191276992
C2×#191276532
#382×D832
#249082882
#249082832
#249082822
#249082732
C83×#19612
#2120183