Finite Group Databasebeta

← #35408076

Subgroups — #35408076

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×#19128252C211yesmaximal, characteristic
2C2×#19127701C214yesmaximal, complemented
3C2×#19127653C211yesmaximal, characteristic, complemented
4#382×D83C211yesmaximal, characteristic, complemented
5#24908287C211yesmaximal, characteristic, complemented
6#24908286C211yesmaximal, characteristic, complemented
7#24908284C211yesmaximal, characteristic, complemented
8#24908271C214yesmaximal, complemented
9C83×#1960C211yesmaximal, characteristic, complemented
10C22×#13050591C2212yes—
11C22×#13050573C2212yes—
12#19128252—24no—
13#19128252C418yescomplemented
14C2×#14581789C2211yescharacteristic
15C2×#14581781C2212yes—
16C2×#14581767—22no—
17C2×#14581754—22no—
18C2×C4⋊C4×D83C2212yes—
19C2×C42×D83C2211yescharacteristic
20#19127701—216no—
21C2×#14581716C2214yes—
22C2×#14581714C2214yes—
23C2×#14581712C2214yes—
24#19127653—24no—
25C2×#14581703C2211yescharacteristic
26C2×C8×Dic83C2212yes—
27#19127567C2211yescharacteristic
28#19127566C2212yes—
29#19127565—22no—
30#95×D83—22no—
31C2×C4⋊Q8×C83C2211yescharacteristic
32C2×#103×C83C2214yes—
33C2×#97×C83C2211yescharacteristic
34#382×C83C2211yescharacteristic
35C2×#13050591—24no—
36C2×#13050591—28no—
37C2×#13050591—28no—
38C2×#13050591C2×C418yes—
39C22×D332—22no—
40C22×D332—22no—
41C22×D332C2312yes—
42C22×C4×D83—22no—
43C22×C4×D83—22no—
44C22×C4×D83D412yes—
45C22×C4×D83C2311yescharacteristic
46C2×#13050573—24no—
47C2×#13050573—24no—
48C2×#13050573D418yescomplemented
49C22×C83⋊C8C2312yes—
50#14581789—22no—
51#14581789C2×C414yescomplemented
52#14581781—24no—
53#14581781C2×C418yescomplemented
54#14581767—216no—
55#14581754—216no—
56C2×C4⋊Dic83C2312yes—
57C2×Dic83⋊C4—22no—
58C4⋊C4×D83—28no—
59C2×(C2×D83)⋊C4—24no—
60C2×C4×Dic83D412yes—
61C2×C4×Dic83C2311yescharacteristic
62C42×D83C2×C414yes—
63C42×D83D414yes—
64#14581716—28no—
65#14581714—28no—
66#14581712—28no—
67#14581703—22no—
68C8×Dic83—24no—
69C83⋊#95—24no—
70C83⋊#95—22no—
71C22×Q8×C83C2312yes—
72C22×C8×C83C2312yes—
73C4⋊Q8×C83—22no—
74C4⋊Q8×C83C2×C414yescomplemented
75C2×C4⋊C4×C83—22no—
76C2×C4⋊C4×C83C2312yes—
77C2×C42×C83C2311yescharacteristic
78#103×C83—28no—
79#97×C83—22no—
80#95×C83—22no—
81C23×D83—22no—
82C23×D83C2×D411yescharacteristic
83#13050591—232no—
84#13050591—416no—
85C2×D332—24no—
86C2×D332—24no—
87C2×D332—28no—
88C2×D332—28no—
89C2×D332—22no—
90C2×D332—48no—
91C2×D332—44no—
92C2×D332—44no—
93C2×D332C22×C414yes—
94C22×Dic83—22no—
95C22×Dic83C2×D412yes—
96C22×Dic83C2×D411yescharacteristic
97C2×C4×D83—28no—
98C2×C4×D83—44no—
99C2×C4×D83—44no—
100C2×C4×D83—44no—
101C2×C4×D83—48no—
102C2×C4×D83—44no—
103C2×C4×D83C22⋊C418yes—
104C2×C4×D83C22×C412yes—
105C2×C4×D83C2×D414yes—
106C2×C4×D83C2×D412yes—
107C2×C4×D83#4114yes—
108#13050573—216no—
109C2×C83⋊C8—22no—
110C2×C83⋊C8—22no—
111C2×C83⋊C8C2×D414yes—
112C4⋊Dic83—24no—
113Dic83⋊C4—44no—
114(C2×D83)⋊C4—216no—
115C4×Dic83C22⋊C418yes—
116C4×Dic83C22×C412yes—
117C4×Dic83C2×D412yes—
118C2×Q8×C83—22no—
119C2×Q8×C83—24no—
120C2×Q8×C83—24no—
121C2×Q8×C83C22×C414yes—
122C22×C4×C83—22no—
123C22×C4×C83—22no—
124C22×C4×C83C2×D412yes—
125C22×C4×C83C2411yescharacteristic
126C2×C8×C83—22no—
127C2×C8×C83—22no—
128C2×C8×C83C2×D414yes—
129C4⋊C4×C83—28no—
130C4⋊C4×C83—24no—
131C42×C83C22×C412yes—
132C42×C83C2×D412yes—
133C23×C83C22×D411yescharacteristic
134C22×D83—24no—
135C22×D83—24no—
136C22×D83—42no—
137C22×D83—42no—
138C22×D83—48no—
139C22×D83C2×C22⋊C412yes—
140C22×D83#12412yes—
141C22×D83#12712yes—
142Q8×C83—28no—
143Q8×C83—44no—
144C2×C4×C83—24no—
145C2×C4×C83—24no—
146C2×C4×C83—42no—
147C2×C4×C83—42no—
148C2×C4×C83—42no—
149C2×C4×C83—44no—
150C2×C4×C83C2×C22⋊C414yes—
151C2×C4×C83C23×C411yescharacteristic
152C2×C4×C83C22×D412yes—
153C2×C4×C83C22×D411yescharacteristic
154C2×C4×C83C2×#4112yes—
155C2×Dic83—24no—
156C2×Dic83—42no—
157C2×Dic83—44no—
158C2×Dic83C2×C22⋊C414yes—
159C2×Dic83C2×C22⋊C412yes—
160C2×Dic83#12412yes—
161C2×Dic83#12712yes—
162D332—28no—
163D332—416no—
164D332—416no—
165D332—44no—
166C4×D83—48no—
167C4×D83—416no—
168C4×D83—88no—
169C4×D83C4×D418yes—
170C4×D83#12118yes—
171C8×C83—24no—
172C83⋊C8—24no—
173C22×C83C22×C22⋊C411yescharacteristic
174C22×C83C2×#12411yescharacteristic
175C22×C83C2×#12711yescharacteristic
176C22×C83#57514yes—
177C2×D83—24no—
178C2×D83—24no—
179C2×D83—24no—
180C2×D83—48no—
181C2×D83—44no—
182C2×D83—84no—
183C2×D83—84no—
184C2×D83#39014yes—
185C4×C83—24no—
186C4×C83—44no—
187C4×C83—82no—
188C4×C83C2×C4×D412yes—
189C4×C83C2×#12112yes—
190Dic83—24no—
191Dic83—82no—
192Dic83#39014yes—
193#21201—831nomaximal, Sylow
194C2×C83C2×#39011yescharacteristic
195C2×C83#289612yes—
196C2×C83#314212yes—
197C2×C83#315312yes—
198D83—24no—
199D83—84no—
200C2×#532—831no—
201C2×#417—834no—
202C2×C4×M16—831no—
203C2×#382—831no—
204#1961—831no—
205#1961—831no—
206#1960—831no—
207#1960D8311yescharacteristic, complemented
208#1958—834no—
209C83#2120111yescharacteristic, complemented, Sylow
210C22×#41—832no—
211C22×M16—832no—
212#532—838no—
213#532—1664no—
214C2×C4⋊Q8C2×D8311yescharacteristic
215C2×#127—831no—
216C2×#124—832no—
217C2×#121—1662no—
218C2×C4×D4—1662no—
219C22×C4⋊C4—832no—
220C22×C42—831no—
221#417—16616no—
222C2×#103—834no—
223C2×#103C2×D8314yes—
224C2×#102—834no—
225C2×#102—834no—
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×Q8C22×D8312yes—
240C22×D4—832no—
241C22×D4—1662no—
242C22×D4—1662no—
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—22no—
253C4⋊Q8C4×D8314yescomplemented
254#127—834no—
255#127—1662no—
256#124—838no—
257#124—1664no—
258#121—16616no—
259C4×D4—16616no—
260C2×C4⋊C4—22no—
261C2×C4⋊C4—832no—
262C2×C4⋊C4—1668no—
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—28no—
272#103—1668no—
273#102—1668no—
274#102—1668no—
275#97—22no—
276#97—1662no—
277C4×C8—1664no—
278#95—22no—
279#95—1664no—
280#95—1662no—
281C24—831no—
282C24—1662no—
283#41—33216no—
284#41—16632no—
285C2×Q8—22no—
286C2×Q8—24no—
287C2×Q8—24no—
288C2×Q8C2×C4×D8314yes—
289C2×D4—3328no—
290C2×D4—3324no—
291C2×D4—3324no—
292C2×D4—834no—
293C2×D4—1668no—
294C2×D4—1662no—
295C2×D4—1664no—
296C2×D4—1664no—
297C2×D4—1668no—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—3324no—
301C22×C4—3324no—
302C22×C4—3324no—
303C22×C4—3324no—
304C22×C4—3328no—
305C22×C4—838no—
306C22×C4—831no—
307C22×C4—832no—
308C22×C4—832no—
309C22×C4—834no—
310C22×C4—834no—
311C22×C4—832no—
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—28no—
324C4⋊C4—24no—
325C4⋊C4—3324no—
326C4⋊C4—1664no—
327C22⋊C4—16616no—
328C42—832no—
329C42—832no—
330C42—838no—
331C42C2×C4×D8312yes—
332C42D4×D8312yes—
333C23—3322no—
334C23—3322no—
335C23—3328no—
336C23—832no—
337C23—832no—
338C23—832no—
339C23—1664no—
340C23—1664no—
341C23C2×D4×D8311yescharacteristic
342Q8—28no—
343Q8—44no—
344D4—33216no—
345D4—33216no—
346D4—3324no—
347D4—1668no—
348C2×C4—24no—
349C2×C4—24no—
350C2×C4—42no—
351C2×C4—42no—
352C2×C4—42no—
353C2×C4—44no—
354C2×C4—3328no—
355C2×C4—33216no—
356C2×C4—3322no—
357C2×C4—3324no—
358C2×C4—832no—
359C2×C4—832no—
360C2×C4—832no—
361C2×C4—838no—
362C2×C4—838no—
363C2×C4—834no—
364C2×C4—6648no—
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—3328no—
374C22—3324no—
375C22—834no—
376C22—6644no—
377C22—6644no—
378C22—1664no—
379C22—1664no—
380C22—1664no—
381C22C2×C22⋊C4×D8311yescharacteristic
382C22#124×D8311yescharacteristic
383C22#127×D8311yescharacteristic
384C22#1912850014yes—
385C4—24no—
386C4—44no—
387C4—82no—
388C4—834no—
389C4—6642no—
390C4—1664no—
391C4C4×D4×D8312yes—
392C4#121×D8312yes—
393C2—6644no—
394C2—1664no—
395C2#390×D8311yescharacteristic
396C2#2491598112yes—
397C2#2491786512yes—
398C2#2491792612yes—
subgroupG/H
centerC23C2×D4×D83
commutatorC2×C4×C83C23×C4
FrattiniC22×C4C23×D83
FittingC83×#1960C2
socleC23×C83C22×D4
prime radicalC2×#19128252C2
cosocleC22×C4×C83C24
hypercenter#1960D83
hypocenterC83#21201
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1960

Complements

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

NH ≅ G/Ntype
2. C2×#19127701C2semidirect
3. C2×#19127653C2semidirect
4. #382×D83C2semidirect
5. #24908287C2semidirect
6. #24908286C2semidirect
7. #24908284C2semidirect
8. #24908271C2semidirect
9. C83×#1960C2semidirect
13. #19128252C4semidirect
48. C2×#13050573D4semidirect
51. #14581789C2×C4semidirect
53. #14581781C2×C4semidirect
74. C4⋊Q8×C83C2×C4semidirect
207. #1960D83semidirect
209. C83#21201semidirect
253. C4⋊Q8C4×D83semidirect

Maximal subgroups

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

subgroupindex
C2×#191282522
C2×#191277012
C2×#191276532
#382×D832
#249082872
#249082862
#249082842
#249082712
C83×#19602
#2120183