Finite Group Databasebeta

← #1314×D83 #35313461

Subgroups — #1314×D83 #35313461

Order 21248.

682 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 682 conjugacy classes of subgroups — too many to draw. Only its 121 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C22⋊C8×D83C211yesmaximal, characteristic
2C22⋊C16×D83C212yesmaximal, complemented
3#24905387C211yesmaximal, characteristic, complemented
4#24905386C212yesmaximal, complemented
5C83×#1314C211yesmaximal, characteristic, complemented
6C23×C4×D83C411yescharacteristic
7C22×C8×D83C411yescharacteristic
8C22×C8×D83C2211yescharacteristic
9C2×C16×D83—22no—
10C2×C22⋊(C83⋊C8)C411yescharacteristic
11C2×(C2×D83)⋊C8C411yescharacteristic
12C2×(C2×D83)⋊C8C2211yescharacteristic
13C22⋊C8×D83—22no—
14C22⋊C8×D83—22no—
15C22⋊(C83⋊C16)C2212yescomplemented
16(C2×D83)⋊C16—22no—
17C2×C22⋊C8×C83C2211yescharacteristic
18C22⋊C16×C83C2212yescomplemented
19C24×D83C811yescharacteristic
20C23×Dic83C811yescharacteristic
21C22×C4×D83—21no—
22C22×C4×D83—21no—
23C22×C4×D83—21no—
24C22×C4×D83—21no—
25C22×C4×D83—42no—
26C22×C4×D83—42no—
27C22×C4×D83C811yescharacteristic
28C22×C4×D83C811yescharacteristic
29C22×C4×D83C2×C411yescharacteristic
30C22×C4×D83C2×C411yescharacteristic
31C22×C83⋊C8C2×C411yescharacteristic
32C22×C83⋊C8C2×C411yescharacteristic
33C2×C8×D83—21no—
34C2×C8×D83—21no—
35C2×C8×D83—42no—
36C2×C8×D83—42no—
37C2×C8×D83D412yes—
38C2×C8×D83D412yes—
39C2×C83⋊C16—22no—
40C16×D83—44no—
41C22⋊(C83⋊C8)—22no—
42(C2×D83)⋊C8—22no—
43(C2×D83)⋊C8—22no—
44C23×C4×C83C2×C411yescharacteristic
45C22×C8×C83C2×C411yescharacteristic
46C22×C8×C83C2311yescharacteristic
47C2×C16×C83—22no—
48C22⋊C8×C83—22no—
49C23×D83—21no—
50C23×D83—21no—
51C23×D83—22no—
52C23×D83—42no—
53C23×D83—42no—
54C23×D83C1614yescomplemented
55C23×D83C2×C811yescharacteristic
56C23×D83C2×C811yescharacteristic
57C22×Dic83—21no—
58C22×Dic83—21no—
59C22×Dic83—42no—
60C22×Dic83C2×C811yescharacteristic
61C22×Dic83C2×C811yescharacteristic
62C2×C4×D83—22no—
63C2×C4×D83—22no—
64C2×C4×D83—21no—
65C2×C4×D83—21no—
66C2×C4×D83—21no—
67C2×C4×D83—21no—
68C2×C4×D83—21no—
69C2×C4×D83—21no—
70C2×C4×D83—44no—
71C2×C4×D83—42no—
72C2×C4×D83—42no—
73C2×C4×D83—44no—
74C2×C4×D83—42no—
75C2×C4×D83—42no—
76C2×C4×D83—42no—
77C2×C4×D83—41no—
78C2×C4×D83—41no—
79C2×C4×D83—42no—
80C2×C4×D83C22⋊C411yescharacteristic
81C2×C4×D83C22⋊C411yescharacteristic
82C2×C4×D83M1611yescharacteristic
83C2×C4×D83M1611yescharacteristic
84C2×C83⋊C8—21no—
85C2×C83⋊C8—41no—
86C2×C83⋊C8—41no—
87C2×C83⋊C8C22⋊C412yes—
88C8×D83—24no—
89C8×D83—42no—
90C8×D83—42no—
91C8×D83—42no—
92C83⋊C16—42no—
93C24×C83C2×C811yescharacteristic
94C22×C4×C83—21no—
95C22×C4×C83—21no—
96C22×C4×C83—41no—
97C22×C4×C83—41no—
98C22×C4×C83C2×C811yescharacteristic
99C22×C4×C83C22×C411yescharacteristic
100C2×C8×C83—21no—
101C2×C8×C83—41no—
102C2×C8×C83—41no—
103C2×C8×C83C2×D412yes—
104C16×C83—42no—
105C23×C83—21no—
106C23×C83—21no—
107C23×C83—41no—
108C23×C83—41no—
109C23×C83C2×C1612yescomplemented
110C23×C83C22×C811yescharacteristic
111C22×D83—22no—
112C22×D83—22no—
113C22×D83—21no—
114C22×D83—21no—
115C22×D83—24no—
116C22×D83—24no—
117C22×D83—42no—
118C22×D83—42no—
119C22×D83—48no—
120C22×D83—48no—
121C22×D83—42no—
122C22×D83—42no—
123C22×D83—41no—
124C22×D83—41no—
125C22×D83C22⋊C811yescharacteristic
126C22×D83C22⋊C811yescharacteristic
127C22×D83C2×C1612yes—
128C22×D83C2×C1612yescomplemented
129C22×D83M3212yes—
130C2×C4×C83—21no—
131C2×C4×C83—21no—
132C2×C4×C83—21no—
133C2×C4×C83—42no—
134C2×C4×C83—41no—
135C2×C4×C83—41no—
136C2×C4×C83—41no—
137C2×C4×C83C2×C22⋊C411yescharacteristic
138C2×C4×C83C2×M1611yescharacteristic
139C2×Dic83—22no—
140C2×Dic83—21no—
141C2×Dic83—42no—
142C2×Dic83—42no—
143C2×Dic83—41no—
144C2×Dic83C22⋊C811yescharacteristic
145C2×Dic83C22⋊C811yescharacteristic
146C4×D83—22no—
147C4×D83—21no—
148C4×D83—21no—
149C4×D83—22no—
150C4×D83—42no—
151C4×D83—42no—
152C4×D83—42no—
153C4×D83—42no—
154C4×D83—41no—
155C4×D83—41no—
156C4×D83—42no—
157C4×D83#9912yes—
158C4×D83#10012yes—
159C8×C83—22no—
160C8×C83—41no—
161C83⋊C8—22no—
162C83⋊C8—41no—
163C22×C83—21no—
164C22×C83—21no—
165C22×C83—22no—
166C22×C83—42no—
167C22×C83—42no—
168C22×C83—41no—
169C22×C83—41no—
170C22×C83C2×C22⋊C811yescharacteristic
171C22×C83C22×C1611yescharacteristic
172C22×C83C2×M3211yescharacteristic
173C2×D83—21no—
174C2×D83—21no—
175C2×D83—24no—
176C2×D83—24no—
177C2×D83—22no—
178C2×D83—48no—
179C2×D83—42no—
180C2×D83—41no—
181C2×D83—41no—
182C2×D83—48no—
183C2×D83—42no—
184C2×D83#32312yes—
185C2×D83C22⋊C1612yes—
186C2×D83C22⋊C1612yes—
187C2×D83#34912yes—
188C4×C83—21no—
189C4×C83—41no—
190C4×C83C2×#9911yescharacteristic
191C4×C83C2×#10011yescharacteristic
192Dic83—21no—
193Dic83—41no—
194Dic83#32312yes—
195C2×#1314—831nomaximal, Sylow
196C2×C83—21no—
197C2×C83—21no—
198C2×C83—42no—
199C2×C83C2×#32311yescharacteristic
200C2×C83C2×C22⋊C1611yescharacteristic
201C2×C83C2×#34911yescharacteristic
202D83—22no—
203D83—42no—
204D83#131414yesdirect product factor
205C22×C22⋊C8—831no—
206C2×C22⋊C16—832no—
207#1314—831no—
208#1314—832no—
209#1314D8311yescharacteristic, direct product factor
210C83C2×#131411yescharacteristic, complemented, Sylow
211C24×C4—831no—
212C23×C8—831no—
213C23×C8—831no—
214C22×C16—1662no—
215C2×C22⋊C8—831no—
216C2×C22⋊C8—831no—
217C2×C22⋊C8—831no—
218C2×C22⋊C8—1662no—
219C2×C22⋊C8—1662no—
220C2×C22⋊C8C2×D8311yescharacteristic
221C22⋊C16—832no—
222C22⋊C16—1662no—
223C22⋊C16C2×D8312yescomplemented
224C25—831no—
225C23×C4—3322no—
226C23×C4—3322no—
227C23×C4—831no—
228C23×C4—831no—
229C23×C4—831no—
230C23×C4—831no—
231C23×C4—831no—
232C23×C4—1661no—
233C23×C4—1661no—
234C23×C4—1661no—
235C23×C4—1661no—
236C23×C4C4×D8311yescharacteristic
237C22×C8—3322no—
238C22×C8—3322no—
239C22×C8—831no—
240C22×C8—832no—
241C22×C8—831no—
242C22×C8—832no—
243C22×C8—1661no—
244C22×C8—1661no—
245C22×C8C4×D8311yescharacteristic
246C22×C8C22×D8311yescharacteristic
247C2×C16—22no—
248C2×C16—3324no—
249C2×C16—1662no—
250C22⋊C8—22no—
251C22⋊C8—1662no—
252C22⋊C8—1662no—
253C22⋊C8—1662no—
254C24—3322no—
255C24—3322no—
256C24—831no—
257C24—834no—
258C24—831no—
259C24—1661no—
260C24—1661no—
261C24—1662no—
262C24C8×D8311yescharacteristic
263C22×C4—21no—
264C22×C4—21no—
265C22×C4—41no—
266C22×C4—41no—
267C22×C4—3321no—
268C22×C4—3321no—
269C22×C4—3322no—
270C22×C4—3322no—
271C22×C4—3324no—
272C22×C4—3324no—
273C22×C4—3322no—
274C22×C4—3322no—
275C22×C4—3322no—
276C22×C4—3322no—
277C22×C4—3322no—
278C22×C4—831no—
279C22×C4—831no—
280C22×C4—831no—
281C22×C4—831no—
282C22×C4—831no—
283C22×C4—831no—
284C22×C4—1661no—
285C22×C4—1661no—
286C22×C4—1661no—
287C22×C4—1661no—
288C22×C4—1662no—
289C22×C4—1662no—
290C22×C4—1661no—
291C22×C4—1661no—
292C22×C4—1661no—
293C22×C4—1661no—
294C22×C4C8×D8311yescharacteristic
295C22×C4C2×C4×D8311yescharacteristic
296C2×C8—21no—
297C2×C8—41no—
298C2×C8—41no—
299C2×C8—3322no—
300C2×C8—3322no—
301C2×C8—3322no—
302C2×C8—3321no—
303C2×C8—3321no—
304C2×C8—832no—
305C2×C8—1664no—
306C2×C8—1661no—
307C2×C8D4×D8312yes—
308C16—42no—
309C16—3322no—
310C23—21no—
311C23—21no—
312C23—41no—
313C23—41no—
314C23—3321no—
315C23—3321no—
316C23—3322no—
317C23—3322no—
318C23—3322no—
319C23—3328no—
320C23—3322no—
321C23—3328no—
322C23—832no—
323C23—831no—
324C23—832no—
325C23—832no—
326C23—831no—
327C23—1664no—
328C23—1664no—
329C23—1662no—
330C23—1662no—
331C23—1661no—
332C23—1661no—
333C23C16×D8312yescomplemented
334C23C2×C8×D8311yescharacteristic
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—42no—
339C2×C4—41no—
340C2×C4—41no—
341C2×C4—41no—
342C2×C4—3322no—
343C2×C4—3322no—
344C2×C4—3322no—
345C2×C4—3322no—
346C2×C4—3322no—
347C2×C4—3322no—
348C2×C4—3322no—
349C2×C4—3321no—
350C2×C4—3321no—
351C2×C4—3321no—
352C2×C4—831no—
353C2×C4—832no—
354C2×C4—832no—
355C2×C4—831no—
356C2×C4—1662no—
357C2×C4—1662no—
358C2×C4—1661no—
359C2×C4—1662no—
360C2×C4—1661no—
361C2×C4—1661no—
362C2×C4C22⋊C4×D8311yescharacteristic
363C2×C4M16×D8311yescharacteristic
364C8—22no—
365C8—41no—
366C8—3321no—
367C8—1662no—
368C22—21no—
369C22—21no—
370C22—22no—
371C22—42no—
372C22—42no—
373C22—41no—
374C22—41no—
375C22—3322no—
376C22—3322no—
377C22—3328no—
378C22—3328no—
379C22—3321no—
380C22—3321no—
381C22—832no—
382C22—832no—
383C22—832no—
384C22—832no—
385C22—1664no—
386C22—1664no—
387C22—1662no—
388C22—1661no—
389C22—1661no—
390C22C22⋊C8×D8311yescharacteristic
391C22C2×C16×D8311yescharacteristic
392C22M32×D8311yescharacteristic
393C4—21no—
394C4—41no—
395C4—3321no—
396C4—832no—
397C4—1661no—
398C4#99×D8311yescharacteristic
399C4#100×D8311yescharacteristic
400C2—21no—
401C2—21no—
402C2—42no—
403C2—3322no—
404C2—834no—
405C2—1662no—
406C2#323×D8311yescharacteristic
407C2C22⋊C16×D8311yescharacteristic
408C2#349×D8311yescharacteristic
subgroupG/H
centerC2×C4C22⋊C4×D83
commutatorC22×C83C22×C16
FrattiniC22×C8C22×D83
FittingC83×#1314C2
socleC22×C83C2×C22⋊C8
prime radicalC24×D83C8
cosocleC22×C8×C83C23
hypercenter#1314D83
hypocenterC83C2×#1314
LCSG ▷ C22×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C2×C4 ◁ C22×C8 ◁ #1314

Complements

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

NH ≅ G/Ntype
2. C22⋊C16×D83C2semidirect
3. #24905387C2semidirect
4. #24905386C2semidirect
5. C83×#1314C2semidirect
15. C22⋊(C83⋊C16)C22semidirect
18. C22⋊C16×C83C22semidirect
54. C23×D83C16semidirect
109. C23×C83C2×C16semidirect
128. C22×D83C2×C16semidirect
204. D83#1314direct (N × H)
209. #1314D83direct (N × H)
210. C83C2×#1314semidirect
223. C22⋊C16C2×D83semidirect
333. C23C16×D83semidirect

Maximal subgroups

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

subgroupindex
C2×C22⋊C8×D832
C22⋊C16×D832
#249053872
#249053862
C83×#13142
C2×#131483