Finite Group Databasebeta

← #35321775

Subgroups — #35321775

Order 21248.

558 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
1C22×#14581731C211yesmaximal, characteristic, complemented
2C2×#19127938C211yesmaximal, characteristic
3C2×#19127503C211yesmaximal, characteristic, complemented
4C2×#19127501C211yesmaximal, characteristic, complemented
5#24907489C211yesmaximal, characteristic, complemented
6#24905554C211yesmaximal, characteristic, complemented
7C83×#1380C211yesmaximal, characteristic, complemented
8C22×D664C2211yescharacteristic
9C22×C8×D83C2211yescharacteristic
10C2×#14581767—21no—
11C2×#14581731—22no—
12C2×#14581731—22no—
13C2×#14581731—22no—
14C22×C83⋊C16C2211yescharacteristic
15#19127938—22no—
16#19127938C414yescomplemented
17C2×#14581726C2211yescharacteristic
18C2×#14581712—21no—
19#19127558—21no—
20#19127503—22no—
21#19127501—22no—
22#19127463—21no—
23C22×C16×C83C2211yescharacteristic
24C2×#107×C83C2211yescharacteristic
25C2×#107×C83C2211yescharacteristic
26C22×D332—21no—
27C22×D332—41no—
28C22×C4×D83D411yescharacteristic
29C2×D664—21no—
30C2×D664—21no—
31C2×D664—21no—
32C2×D664—21no—
33C2×D664—21no—
34C2×D664C2×C412yes—
35C22×C83⋊C8D411yescharacteristic
36C2×C8×D83—21no—
37C2×C8×D83—21no—
38C2×C8×D83—21no—
39C2×C8×D83—21no—
40C2×C8×D83C2×C412yes—
41C2×C8×D83D412yes—
42#14581767—24no—
43#14581767—42no—
44C2×C4⋊Dic83—21no—
45C2×(C2×D83)⋊C4—41no—
46C2×(C2×D83)⋊C4—41no—
47#14581731—28no—
48C2×C83⋊C16—21no—
49C2×C83⋊C16—21no—
50C2×C83⋊C16D412yescomplemented
51#14581726—21no—
52#14581726—21no—
53#14581712—24no—
54C83⋊#95—41no—
55C22×C8×C83C2311yescharacteristic
56C2×C4⋊C4×C83—21no—
57C2×C4⋊C4×C83—21no—
58C2×C16×C83—21no—
59C2×C16×C83—21no—
60C2×C16×C83D412yescomplemented
61#107×C83—21no—
62#107×C83—21no—
63#107×C83—21no—
64#107×C83C2×C412yes—
65C23×D83—41no—
66C23×D83D811yescharacteristic
67C2×D332—22no—
68C2×D332—41no—
69C2×D332—41no—
70C2×D332—41no—
71C2×D332—41no—
72C2×D332—44no—
73C2×D332—41no—
74C2×D332—82no—
75C2×D332—82no—
76C22×Dic83—41no—
77C22×Dic83D811yescharacteristic
78C2×C4×D83—21no—
79C2×C4×D83—21no—
80C2×C4×D83—21no—
81C2×C4×D83—21no—
82C2×C4×D83C22⋊C412yes—
83C2×C4×D83SD1612yes—
84D664—22no—
85D664—22no—
86D664—41no—
87D664—41no—
88C2×C83⋊C8—21no—
89C2×C83⋊C8—21no—
90C2×C83⋊C8C22⋊C412yes—
91C8×D83—22no—
92C8×D83—22no—
93C8×D83—22no—
94C8×D83—22no—
95C4⋊Dic83—41no—
96C4⋊Dic83—41no—
97(C2×D83)⋊C4—44no—
98(C2×D83)⋊C4—82no—
99C83⋊C16—22no—
100C22×C4×C83—41no—
101C22×C4×C83—41no—
102C22×C4×C83C2×D411yescharacteristic
103C2×C8×C83C22×C411yescharacteristic
104C2×C8×C83C2×D411yescharacteristic
105C2×C8×C83C2×D411yescharacteristic
106C2×C8×C83C2×D411yescharacteristic
107C2×C8×C83#4111yescharacteristic
108C2×C8×C83#4111yescharacteristic
109C4⋊C4×C83—22no—
110C4⋊C4×C83—41no—
111C4⋊C4×C83—41no—
112C4⋊C4×C83—41no—
113C16×C83—22no—
114C23×C83C2×D811yescharacteristic
115C22×D83—21no—
116C22×D83—21no—
117C22×D83—21no—
118C22×D83—21no—
119C22×D83—21no—
120C22×D83—21no—
121C22×D83—42no—
122C22×D83—81no—
123C22×D83—81no—
124C22×D83—81no—
125C22×D83—81no—
126C22×D83—81no—
127C22×D83—81no—
128C22×D83#10212yes—
129C2×C4×C83—42no—
130C2×C4×C83—81no—
131C2×C4×C83—81no—
132C2×C4×C83—81no—
133C2×C4×C83—81no—
134C2×C4×C83—81no—
135C2×C4×C83C2×C22⋊C411yescharacteristic
136C2×C4×C83#12111yescharacteristic
137C2×C4×C83#12111yescharacteristic
138C2×C4×C83C2×SD1611yescharacteristic
139C2×C4×C83#13511yescharacteristic
140C2×C4×C83#13511yescharacteristic
141C2×Dic83—21no—
142C2×Dic83—21no—
143C2×Dic83—81no—
144C2×Dic83—81no—
145C2×Dic83—81no—
146C2×Dic83#10212yes—
147D332—42no—
148D332—42no—
149D332—88no—
150D332—81no—
151D332—81no—
152C4×D83—22no—
153C4×D83—22no—
154C4×D83—22no—
155C4×D83—22no—
156C8×C83C4×D411yescharacteristic
157C8×C83C4×D411yescharacteristic
158C8×C83#12011yescharacteristic
159C8×C83#12311yescharacteristic
160C83⋊C8—22no—
161C22×C83C2×#10211yescharacteristic
162C22×C83#46611yescharacteristic
163C22×C83#46611yescharacteristic
164C22×C83#50911yescharacteristic
165C22×C83#50911yescharacteristic
166C22×C83#50911yescharacteristic
167C22×C83#51011yescharacteristic
168C2×D83—22no—
169C2×D83—22no—
170C2×D83—22no—
171C2×D83—41no—
172C2×D83—41no—
173C2×D83—41no—
174C2×D83—41no—
175C2×D83—82no—
176C2×D83—82no—
177C2×D83—82no—
178C2×D83—161no—
179C2×D83—161no—
180C2×D83—161no—
181C2×D83—161no—
182C4×C83—82no—
183C4×C83—161no—
184C4×C83#38611yescharacteristic
185C4×C83#41611yescharacteristic
186C4×C83#46511yescharacteristic
187C4×C83#46511yescharacteristic
188Dic83—22no—
189Dic83—161no—
190#13061—831nomaximal, Sylow
191C2×C83#189311yescharacteristic
192C2×C83#213911yescharacteristic
193C2×C83#214011yescharacteristic
194C2×C83#221811yescharacteristic
195C2×C83#221811yescharacteristic
196C2×C83#222011yescharacteristic
197C2×C83#222011yescharacteristic
198D83—42no—
199D83—162no—
200C22×M32—831no—
201C2×#466—831no—
202C2×#357—831no—
203C2×#357—831no—
204#1808—831no—
205#1380—831no—
206#1380D8311yescharacteristic, complemented
207C83#1306111yescharacteristic, complemented, Sylow
208C22×D8—831no—
209C23×C8—831no—
210C2×#121—1661no—
211C2×M32—1662no—
212C2×M32—1662no—
213C2×M32—1662no—
214C22×C16—831no—
215C22×C16C2×D8311yescharacteristic
216#466—834no—
217#466—1662no—
218C2×#107—831no—
219C2×#107C2×D8311yescharacteristic
220C2×#107C2×D8311yescharacteristic
221C2×#102—1661no—
222#380—1661no—
223#357—1662no—
224#357—1662no—
225#340—1661no—
226C22×D4—3321no—
227C22×D4—1661no—
228C23×C4—831no—
229C2×D8—832no—
230C2×D8—1661no—
231C2×D8—1661no—
232C2×D8—1661no—
233C2×D8—1661no—
234C2×D8—1661no—
235C22×C8—831no—
236C22×C8—832no—
237C22×C8—832no—
238C22×C8—1661no—
239C22×C8—1661no—
240C22×C8—1661no—
241C22×C8—1661no—
242C22×C8C22×D8311yescharacteristic
243#121—3322no—
244#121—1664no—
245C2×C4⋊C4—21no—
246C2×C4⋊C4—21no—
247C2×C4⋊C4—1661no—
248C2×C22⋊C4—3321no—
249C2×C22⋊C4—3321no—
250M32—1668no—
251C2×C16—21no—
252C2×C16—21no—
253C2×C16—832no—
254C2×C16—1661no—
255C2×C16—1661no—
256C2×C16D33212yescomplemented
257#107—21no—
258#107—21no—
259#107—21no—
260#107—1661no—
261#107—1661no—
262#107C4×D8312yes—
263#102—1664no—
264#95—3321no—
265C24—3321no—
266C24—831no—
267C2×D4—3324no—
268C2×D4—3321no—
269C2×D4—3321no—
270C2×D4—3321no—
271C2×D4—3321no—
272C2×D4—3321no—
273C2×D4—6642no—
274C2×D4—6642no—
275C2×D4—1662no—
276C22×C4—41no—
277C22×C4—41no—
278C22×C4—3321no—
279C22×C4—831no—
280C22×C4—832no—
281C22×C4—832no—
282C22×C4—1661no—
283C22×C4—1661no—
284C22×C4—1661no—
285C22×C4—1661no—
286C22×C4D4×D8311yescharacteristic
287D8—3321no—
288D8—3321no—
289D8—1662no—
290D8—1662no—
291C2×C8—832no—
292C2×C8—1661no—
293C2×C8—1661no—
294C2×C8—1662no—
295C2×C8—1662no—
296C2×C8—1662no—
297C2×C8—1662no—
298C2×C8C2×C4×D8311yescharacteristic
299C2×C8D4×D8311yescharacteristic
300C2×C8D4×D8311yescharacteristic
301C2×C8C2×D33211yescharacteristic
302C2×C8#1305059011yescharacteristic
303C2×C8#1305059111yescharacteristic
304C4⋊C4—22no—
305C4⋊C4—41no—
306C4⋊C4—41no—
307C4⋊C4—41no—
308C4⋊C4—3321no—
309C4⋊C4—3321no—
310C22⋊C4—3324no—
311C22⋊C4—6642no—
312C16—22no—
313C16—1662no—
314C23—3322no—
315C23—832no—
316C23—6641no—
317C23—6641no—
318C23—6641no—
319C23—6641no—
320C23—6641no—
321C23—6641no—
322C23—1661no—
323C23—1661no—
324C23—1661no—
325C23—1661no—
326C23—1661no—
327C23—1661no—
328C23D8×D8311yescharacteristic
329D4—3322no—
330D4—3322no—
331D4—6648no—
332D4—6641no—
333D4—6641no—
334C2×C4—42no—
335C2×C4—81no—
336C2×C4—81no—
337C2×C4—81no—
338C2×C4—81no—
339C2×C4—81no—
340C2×C4—832no—
341C2×C4—6641no—
342C2×C4—6641no—
343C2×C4—6641no—
344C2×C4—1661no—
345C2×C4—1661no—
346C2×C4—1662no—
347C2×C4—1662no—
348C2×C4—1662no—
349C2×C4—1662no—
350C2×C4C22⋊C4×D8311yescharacteristic
351C2×C4#1458176511yescharacteristic
352C2×C4#1458176711yescharacteristic
353C2×C4SD16×D8311yescharacteristic
354C2×C4#1458181311yescharacteristic
355C2×C4#1458181511yescharacteristic
356C8—1662no—
357C8#1458175411yescharacteristic
358C8C4×D33211yescharacteristic
359C8#1458176211yescharacteristic
360C8#1458177711yescharacteristic
361C22—13281no—
362C22—13281no—
363C22—13281no—
364C22—13281no—
365C22—3321no—
366C22—3321no—
367C22—3321no—
368C22—3321no—
369C22—6642no—
370C22—6642no—
371C22—6642no—
372C22—1662no—
373C22—1662no—
374C22—1662no—
375C22#102×D8311yescharacteristic
376C22#1912793611yescharacteristic
377C22#1912793811yescharacteristic
378C22#1912813511yescharacteristic
379C22#1912813811yescharacteristic
380C22#1912813811yescharacteristic
381C22#1912814311yescharacteristic
382C4—82no—
383C4—161no—
384C4—13281no—
385C4—1662no—
386C4#1912757911yescharacteristic
387C4#1912769611yescharacteristic
388C4#1912793111yescharacteristic
389C4#1912793311yescharacteristic
390C2—13282no—
391C2—3322no—
392C2#2490789011yescharacteristic
393C2#2490917411yescharacteristic
394C2#2490917711yescharacteristic
395C2#2490956511yescharacteristic
396C2#2490956711yescharacteristic
397C2#2490957711yescharacteristic
398C2#2490957911yescharacteristic
subgroupG/H
centerC23D8×D83
commutatorC2×C8×C83C22×C4
FrattiniC22×C8C22×D83
FittingC83×#1380C2
socleC23×C83C2×D8
prime radicalC2×#19127938C2
cosocleC22×C8×C83C23
hypercenter#1380D83
hypocenterC83#13061
LCSG ▷ C2×C8×C83 ▷ C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C23 ◁ C22×C4 ◁ C22×C8 ◁ #1380

Complements

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

NH ≅ G/Ntype
1. C22×#14581731C2semidirect
3. C2×#19127503C2semidirect
4. C2×#19127501C2semidirect
5. #24907489C2semidirect
6. #24905554C2semidirect
7. C83×#1380C2semidirect
16. #19127938C4semidirect
50. C2×C83⋊C16D4semidirect
60. C2×C16×C83D4semidirect
206. #1380D83semidirect
207. C83#13061semidirect
256. C2×C16D332semidirect

Maximal subgroups

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

subgroupindex
C22×#145817312
C2×#191279382
C2×#191275032
C2×#191275012
#249074892
#249055542
C83×#13802
#1306183