Finite Group Databasebeta

← #35412272

Subgroups — #35412272

Order 21248.

974 conjugacy classes of subgroups.

The subgroup lattice has 368 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19127825C211yesmaximal, characteristic, complemented
2C2×D4×C83⋊C8C211yesmaximal, characteristic
3C4×#14581779C211yesmaximal, characteristic
4#24908399C211yesmaximal, characteristic, complemented
5#24908066C214yesmaximal, complemented
6#24907579C211yesmaximal, characteristic, complemented
7#24907237C214yesmaximal, complemented
8C4×C8×Dic83C211yesmaximal, characteristic, complemented
9C83×#1954C211yesmaximal, characteristic
10C23×C83⋊C8C2212yes—
11C2×#14581779C411yescharacteristic
12C2×C4×Dic166C2211yescharacteristic
13#19127825—24no—
14D4×C83⋊C8—24no—
15C2×C4⋊(C83⋊C8)C2211yescharacteristic
16C2×Dic83⋊C8C2212yes—
17C2×C22⋊(C83⋊C8)C2212yes—
18C2×C4×C83⋊C8C2211yescharacteristic
19C2×C8×Dic83C2212yes—
20C2×C8×Dic83C2212yes—
21C2×C8×Dic83C2212yes—
22#19127606C411yescharacteristic
23#19127604C411yescharacteristic
24#19127595C414yes—
25#19127566C411yescharacteristic
26C4×C22⋊Dic83C2214yes—
27C42×Dic83C2211yescharacteristic
28#19127455C2214yes—
29#19127455C2214yes—
30C2×C4×D4×C83C2211yescharacteristic
31C2×C4⋊C8×C83C2211yescharacteristic
32C2×C22⋊C8×C83C2214yes—
33C2×C4×C8×C83C2211yescharacteristic
34C22×Dic166C2×C411yescharacteristic
35C22×C83⋊C8—24no—
36C22×C83⋊C8—28no—
37C22×C83⋊C8C2312yes—
38C22×C83⋊C8C2311yescharacteristic
39C22×C83⋊C8C2311yescharacteristic
40#14581779C818yescomplemented
41C4×Dic166—22no—
42C4×Dic166—22no—
43C2×C4⋊Dic83C2×C411yescharacteristic
44C2×Dic83⋊C4C2×C412yes—
45C2×C22⋊Dic83C2×C414yes—
46C2×C22⋊Dic83C2×C414yes—
47C2×C4×Dic83C2×C412yes—
48C2×C4×Dic83C2×C411yescharacteristic
49C2×C4×Dic83C2×C411yescharacteristic
50C2×C4×Dic83C2312yes—
51C2×C4×Dic83C2312yes—
52C4⋊(C83⋊C8)—22no—
53Dic83⋊C8—24no—
54C22⋊(C83⋊C8)—24no—
55C4×C83⋊C8D414yes—
56C8×Dic83—24no—
57C8×Dic83—24no—
58C8×Dic83—24no—
59C83⋊#95C2×C414yes—
60C83⋊#95C2×C414yes—
61C22×D4×C83C2×C411yescharacteristic
62C23×C4×C83C2312yes—
63C22×C8×C83C2312yes—
64C22×C8×C83C2312yes—
65C4×D4×C83—22no—
66C4×D4×C83—22no—
67C2×C4⋊C4×C83C2×C411yescharacteristic
68C2×C22⋊C4×C83C2×C412yes—
69C2×C42×C83C2311yescharacteristic
70C4⋊C8×C83—22no—
71C22⋊C8×C83—28no—
72C4×C8×C83—22no—
73C2×Dic166—22no—
74C2×Dic166—22no—
75C2×Dic166C2×C814yescomplemented
76C22×Dic83C22×C412yes—
77C22×Dic83C22×C412yes—
78C22×Dic83C22×C412yes—
79C22×Dic83C22×C412yes—
80C2×C83⋊C8—22no—
81C2×C83⋊C8—22no—
82C2×C83⋊C8—24no—
83C2×C83⋊C8—48no—
84C2×C83⋊C8C2×D412yes—
85C2×C83⋊C8C2×D412yes—
86C2×C83⋊C8#4114yes—
87C4⋊Dic83—22no—
88Dic83⋊C4—24no—
89C22⋊Dic83—28no—
90C22⋊Dic83C2×C8116yescomplemented
91C4×Dic83—22no—
92C4×Dic83—22no—
93C4×Dic83—22no—
94C4×Dic83—22no—
95C4×Dic83—22no—
96C4×Dic83—24no—
97C4×Dic83C2×C814yescomplemented
98C24×C83C22×C412yes—
99C2×D4×C83—22no—
100C2×D4×C83—22no—
101C2×D4×C83C2×C814yes—
102C22×C4×C83—22no—
103C22×C4×C83—24no—
104C22×C4×C83—24no—
105C22×C4×C83—22no—
106C22×C4×C83C22×C412yes—
107C22×C4×C83C22×C411yescharacteristic
108C22×C4×C83C22×C411yescharacteristic
109C22×C4×C83C2411yescharacteristic
110C2×C8×C83—22no—
111C2×C8×C83—22no—
112C2×C8×C83—22no—
113C2×C8×C83—24no—
114C2×C8×C83#4114yes—
115C4⋊C4×C83—22no—
116C22⋊C4×C83—24no—
117C42×C83C2×D412yes—
118C42×C83#4112yes—
119C23×C83—22no—
120C23×C83—24no—
121C23×C83—22no—
122C23×C83—24no—
123C23×C83C22×C814yes—
124C23×C83C23×C411yescharacteristic
125D4×C83—28no—
126C2×C4×C83—22no—
127C2×C4×C83—22no—
128C2×C4×C83—24no—
129C2×C4×C83—42no—
130C2×C4×C83—44no—
131C2×C4×C83—42no—
132C2×C4×C83C4×D412yes—
133C2×C4×C83C4×D412yes—
134C2×C4×C83C22×C812yes—
135C2×C4×C83#13112yes—
136C2×C4×C83C22×D411yescharacteristic
137C2×C4×C83C2×#4112yes—
138C2×C4×C83C2×#4111yescharacteristic
139C2×C4×C83C2×#4112yes—
140C2×Dic83—22no—
141C2×Dic83—22no—
142C2×Dic83—22no—
143C2×Dic83—22no—
144C2×Dic83—22no—
145C2×Dic83—22no—
146C2×Dic83—22no—
147C2×Dic83—24no—
148C2×Dic83C22×C814yes—
149C2×Dic83C22×C814yes—
150C2×Dic83#13114yes—
151Dic166—28no—
152C8×C83—24no—
153C8×C83—42no—
154C83⋊C8—22no—
155C83⋊C8—42no—
156C83⋊C8#12414yescomplemented
157C22×C83—24no—
158C22×C83—28no—
159C22×C83—42no—
160C22×C83—44no—
161C22×C83—42no—
162C22×C83C2×C4×D411yescharacteristic
163C22×C83C4×#4112yes—
164C22×C83C23×C811yescharacteristic
165C22×C83C2×#13112yes—
166C22×C83C2×#13111yescharacteristic
167C4×C83—22no—
168C4×C83—42no—
169C4×C83C8×D414yes—
170C4×C83C2×#12411yescharacteristic
171C4×C83#52912yes—
172C4×C83#53211yescharacteristic
173Dic83—24no—
174Dic83—24no—
175Dic83—24no—
176Dic83—42no—
177C8×#124—831nomaximal, Sylow
178C2×C83—44no—
179C2×C83C4×#12411yescharacteristic
180C2×C83C2×C8×D411yescharacteristic
181C2×C83#293111yescharacteristic
182C2×C83C8×#4112yes—
183C2×C83#296612yes—
184C2×C8×Q8—831no—
185C2×C8×D4—831no—
186C4×#124—831no—
187#1984—831no—
188#1954D8311yescharacteristic
189#1923—834no—
190#1831—831no—
191C8×C22⋊C4—834no—
192C42×C8—831no—
193C83C8×#12411yescharacteristic, complemented, Sylow
194C23×C8—832no—
195C2×#124—831no—
196C2×C4×Q8—831no—
197C2×C4×D4C2×D8311yescharacteristic
198C8×Q8—1664no—
199C8×D4—1664no—
200C2×C4⋊C8—832no—
201C2×C4⋊C8—831no—
202C2×C4⋊C8C2×D8311yescharacteristic
203C2×C22⋊C8—832no—
204C2×C22⋊C8C2×D8314yes—
205C2×C4×C8—831no—
206C2×C4×C8—832no—
207C2×C4×C8—832no—
208C2×C4×C8—832no—
209C2×C4×C8C2×D8311yescharacteristic
210#391—831no—
211#390—831no—
212#388—834no—
213#382—831no—
214C4×C22⋊C4—834no—
215C43—831no—
216#336—834no—
217#336—834no—
218C22×Q8—831no—
219C22×D4C4×D8311yescharacteristic
220C23×C4C22×D8312yes—
221C22×C8—831no—
222C22×C8—831no—
223C22×C8—832no—
224C22×C8—1668no—
225C22×C8—1664no—
226C22×C8C22×D8312yes—
227C22×C8C22×D8312yes—
228#124—838no—
229C4×Q8—1662no—
230C4×Q8—1662no—
231C4×D4—22no—
232C4×D4—22no—
233C2×C4⋊C4—831no—
234C2×C4⋊C4—832no—
235C2×C4⋊C4C4×D8311yescharacteristic
236C2×C22⋊C4—834no—
237C2×C22⋊C4—834no—
238C2×C22⋊C4C4×D8312yes—
239C2×C42—831no—
240C2×C42—832no—
241C2×C42—831no—
242C2×C42—832no—
243C2×C42—832no—
244C2×C42C22×D8311yescharacteristic
245C4⋊C8—22no—
246C4⋊C8—1664no—
247C4⋊C8—1662no—
248C22⋊C8—28no—
249C22⋊C8—1664no—
250C4×C8—22no—
251C4×C8—834no—
252C4×C8—1664no—
253C4×C8—1664no—
254C4×C8—1664no—
255#95—834no—
256#95—834no—
257C24C2×C4×D8312yes—
258C2×Q8—834no—
259C2×Q8—1662no—
260C2×Q8—1662no—
261C2×D4—22no—
262C2×D4—22no—
263C2×D4C8×D8314yes—
264C22×C4—22no—
265C22×C4—24no—
266C22×C4—24no—
267C22×C4—22no—
268C22×C4—832no—
269C22×C4—832no—
270C22×C4—832no—
271C22×C4—832no—
272C22×C4C2×C4×D8312yes—
273C22×C4C2×C4×D8311yescharacteristic
274C22×C4C2×C4×D8311yescharacteristic
275C22×C4C23×D8311yescharacteristic
276C2×C8—22no—
277C2×C8—22no—
278C2×C8—22no—
279C2×C8—24no—
280C2×C8—3328no—
281C2×C8—832no—
282C2×C8—834no—
283C2×C8—832no—
284C2×C8—1662no—
285C2×C8—1664no—
286C2×C8—1662no—
287C2×C8#1305058914yes—
288C4⋊C4—22no—
289C4⋊C4—1664no—
290C4⋊C4—1662no—
291C22⋊C4—24no—
292C22⋊C4—8316no—
293C22⋊C4—1668no—
294C42—834no—
295C42—1664no—
296C42—1662no—
297C42—1662no—
298C42—1662no—
299C42—1662no—
300C42—1662no—
301C42D4×D8312yes—
302C42#1305058912yes—
303C23—22no—
304C23—24no—
305C23—24no—
306C23—22no—
307C23C2×C8×D8314yes—
308C23C22×C4×D8311yescharacteristic
309Q8—1668no—
310D4—28no—
311C2×C4—22no—
312C2×C4—22no—
313C2×C4—24no—
314C2×C4—42no—
315C2×C4—44no—
316C2×C4—42no—
317C2×C4—834no—
318C2×C4—834no—
319C2×C4—834no—
320C2×C4—1662no—
321C2×C4—1664no—
322C2×C4—1662no—
323C2×C4—1662no—
324C2×C4—1662no—
325C2×C4—1662no—
326C2×C4—1662no—
327C2×C4—1662no—
328C2×C4(C22⋊D83)⋊C412yes—
329C2×C4(C22⋊D83)⋊C412yes—
330C2×C4C2×C8×D8312yes—
331C2×C4#1458180012yes—
332C2×C4C2×D4×D8311yescharacteristic
333C2×C4#41×D8312yes—
334C2×C4C2×#1305058911yescharacteristic
335C2×C4C2×#1305058912yes—
336C8—24no—
337C8—42no—
338C8—3322no—
339C8—834no—
340C8—1662no—
341C22—24no—
342C22—28no—
343C22—42no—
344C22—42no—
345C22—44no—
346C22C2×(C22⋊D83)⋊C411yescharacteristic
347C22C4×#1305058912yes—
348C22C22×C8×D8311yescharacteristic
349C22#131×D8312yes—
350C22C2×#1458180011yescharacteristic
351C4—22no—
352C4—42no—
353C4—3322no—
354C4—1664no—
355C4—1664no—
356C4—1664no—
357C4(C22⋊D83)⋊C814yes—
358C4#124×D8311yescharacteristic
359C4#1912823212yes—
360C4#1912824911yescharacteristic
361C2—44no—
362C2#2491001011yescharacteristic
363C2C2×(C22⋊D83)⋊C811yescharacteristic
364C2#2491617111yescharacteristic
365C2C8×#1305058912yes—
366C2#2491644112yes—
subgroupG/H
center, FrattiniC22×C4C23×D83
commutatorC22×C83C23×C8
FittingC83×#1954C2
socleC23×C83C23×C4
prime radicalC22×D4×C83C2×C4
cosocleC22×C4×C83C24
hypercenter#1954D83
hypocenterC83C8×#124
LCSG ▷ C22×C83 ▷ C83
UCSC1 ◁ C22×C4 ◁ #1954

Complements

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

NH ≅ G/Ntype
1. C2×#19127825C2semidirect
4. #24908399C2semidirect
5. #24908066C2semidirect
6. #24907579C2semidirect
7. #24907237C2semidirect
8. C4×C8×Dic83C2semidirect
40. #14581779C8semidirect
75. C2×Dic166C2×C8semidirect
90. C22⋊Dic83C2×C8semidirect
97. C4×Dic83C2×C8semidirect
156. C83⋊C8#124semidirect
193. C83C8×#124semidirect

Maximal subgroups

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

subgroupindex
C2×#191278252
C2×D4×C83⋊C82
C4×#145817792
#249083992
#249080662
#249075792
#249072372
C4×C8×Dic832
C83×#19542
C8×#12483