Finite Group Databasebeta

← #35311115

Subgroups — #35311115

Order 21248.

2118 conjugacy classes of subgroups.

The subgroup lattice has 394 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14581713C214yesmaximal, complemented
2C2×#19127456C211yesmaximal, characteristic, complemented
3#24905693C211yesmaximal, characteristic, complemented
4C22×#127×C83C211yesmaximal, characteristic
5C23×C83⋊C8C2212yes—
6C22×C4⋊Dic83C2212yes—
7C2×#14581713—28no—
8C2×#14581713—28no—
9C2×#14581713—28no—
10C2×C83⋊#95—22no—
11#19127456—22no—
12C23×D4×C83—22no—
13C23×D4×C83C2212yes—
14C2×#127×C83—28no—
15C2×#127×C83—22no—
16C2×#127×C83C414yescomplemented
17C22×C42×C83C2211yescharacteristic
18C23×Dic83—22no—
19C22×C83⋊C8—24no—
20C22×C83⋊C8—24no—
21C22×C83⋊C8—22no—
22C22×C83⋊C8—22no—
23C22×C83⋊C8D414yescomplemented
24C2×C4⋊Dic83—24no—
25C2×C4⋊Dic83—24no—
26C2×C4⋊Dic83—22no—
27C2×C4⋊Dic83D414yescomplemented
28#14581713—232no—
29C83⋊#95—44no—
30C25×C83—22no—
31C22×D4×C83—216no—
32C22×D4×C83—28no—
33C22×D4×C83—24no—
34C22×D4×C83—24no—
35C22×D4×C83—24no—
36C22×D4×C83—22no—
37C22×D4×C83—24no—
38C22×D4×C83—24no—
39C22×D4×C83—48no—
40C22×D4×C83—44no—
41C22×D4×C83—44no—
42C22×D4×C83C2×C414yes—
43C22×D4×C83D418yes—
44C23×C4×C83D412yes—
45C23×C4×C83C2311yescharacteristic
46#127×C83—216no—
47#127×C83—216no—
48C2×C42×C83—24no—
49C2×C42×C83C2×C412yes—
50C2×C42×C83D412yes—
51C22×Dic83—24no—
52C22×Dic83—42no—
53C22×Dic83—42no—
54C22×Dic83—44no—
55C22×Dic83—44no—
56C2×C83⋊C8—28no—
57C2×C83⋊C8—24no—
58C2×C83⋊C8—42no—
59C2×C83⋊C8—44no—
60C2×C83⋊C8—42no—
61C4⋊Dic83—28no—
62C4⋊Dic83—44no—
63C24×C83—24no—
64C24×C83—28no—
65C24×C83—44no—
66C24×C83—42no—
67C24×C83—42no—
68C24×C83—44no—
69C24×C83—44no—
70C24×C83—48no—
71C24×C83C2×D411yescharacteristic
72C2×D4×C83—232no—
73C2×D4×C83—24no—
74C2×D4×C83—28no—
75C2×D4×C83—28no—
76C2×D4×C83—28no—
77C2×D4×C83—28no—
78C2×D4×C83—28no—
79C2×D4×C83—28no—
80C2×D4×C83—28no—
81C2×D4×C83—416no—
82C2×D4×C83—432no—
83C2×D4×C83—432no—
84C2×D4×C83—416no—
85C2×D4×C83—44no—
86C2×D4×C83—44no—
87C2×D4×C83—44no—
88C2×D4×C83—44no—
89C2×D4×C83—44no—
90C2×D4×C83—42no—
91C2×D4×C83—42no—
92C2×D4×C83C22⋊C418yes—
93C22×C4×C83—24no—
94C22×C4×C83—24no—
95C22×C4×C83C22⋊C414yes—
96C22×C4×C83D814yes—
97C22×C4×C83SD1614yes—
98C22×C4×C83C22×C411yescharacteristic
99C22×C4×C83C2×D412yes—
100C22×C4×C83C2×D412yes—
101C22×C4×C83C2×D414yes—
102C22×C4×C83C2×D411yescharacteristic
103C22×C4×C83#4112yes—
104C22×C4×C83#4112yes—
105C42×C83—24no—
106C42×C83—24no—
107C23×C83—28no—
108C23×C83—44no—
109C23×C83—44no—
110C23×C83—48no—
111C23×C83—48no—
112C23×C83—48no—
113C23×C83—48no—
114C23×C83—416no—
115C23×C83—416no—
116C23×C83—88no—
117C23×C83—88no—
118C23×C83—84no—
119C23×C83—84no—
120C23×C83—82no—
121C23×C83—82no—
122C23×C83—84no—
123C23×C83C2×C22⋊C411yescharacteristic
124C23×C83#12012yes—
125C23×C83#12112yes—
126C23×C83#12411yescharacteristic
127C23×C83#12711yescharacteristic
128C23×C83C2×D812yes—
129C23×C83C2×SD1612yes—
130C23×C83AGL(1,ℤ/8)14yes—
131D4×C83—216no—
132D4×C83—464no—
133D4×C83—48no—
134D4×C83—48no—
135D4×C83—48no—
136C2×C4×C83—24no—
137C2×C4×C83—24no—
138C2×C4×C83—28no—
139C2×C4×C83—28no—
140C2×C4×C83#10218yescomplemented
141C2×C4×C83C2×C22⋊C412yes—
142C2×C4×C83C4×D412yes—
143C2×C4×C83C4×D412yes—
144C2×C4×C83#12014yes—
145C2×C4×C83#12012yes—
146C2×C4×C83#12112yes—
147C2×C4×C83#12114yes—
148C2×C4×C83#12114yes—
149C2×C4×C83#12314yes—
150C2×C4×C83#12411yescharacteristic
151C2×C4×C83#12711yescharacteristic
152C2×Dic83—44no—
153C2×Dic83—48no—
154C2×Dic83—84no—
155C2×Dic83—82no—
156C2×Dic83—82no—
157C83⋊C8—44no—
158C22×C83—48no—
159C22×C83—416no—
160C22×C83—84no—
161C22×C83—84no—
162C22×C83—88no—
163C22×C83—88no—
164C22×C83—88no—
165C22×C83—816no—
166C22×C83#37911yescharacteristic
167C22×C83#39011yescharacteristic
168C22×C83#39011yescharacteristic
169C22×C83#39212yes—
170C22×C83#39412yes—
171C22×C83C2×#10212yes—
172C22×C83#41812yes—
173C22×C83#44714yes—
174C22×C83#45014yes—
175C22×C83#45914yes—
176C22×C83#46014yes—
177C22×C83#48612yes—
178C22×C83#49012yes—
179C22×C83#49211yescharacteristic
180C22×C83#49311yescharacteristic
181C22×C83#49612yes—
182C4×C83—28no—
183C4×C83#38614yes—
184C4×C83#39212yes—
185C4×C83#39412yes—
186Dic83—84no—
187#11008—831nomaximal, Sylow
188C2×C83—88no—
189C2×C83#143811yescharacteristic
190C2×C83#178612yes—
191C2×C83#195711yescharacteristic
192C2×C83#196111yescharacteristic
193C2×C83#198212yes—
194C2×C83#201112yes—
195C2×C83#201412yes—
196C2×C83#204114yes—
197C22×#127D8311yescharacteristic
198C22×#102—834no—
199C2×#336—831no—
200C4⋊#95—831no—
201C83#1100811yescharacteristic, complemented, Sylow
202C23×D4—22no—
203C23×D4C2×D8312yes—
204C23×C8—832no—
205C2×#127—28no—
206C2×#127—22no—
207C2×#127Dic8314yescomplemented
208C22×C4⋊C4—832no—
209C22×C42C2×D8311yescharacteristic
210C2×#102—1668no—
211C2×#102—1668no—
212C2×#102—1668no—
213C2×#95—1662no—
214#336—1662no—
215C25—22no—
216C22×D4—216no—
217C22×D4—28no—
218C22×D4—24no—
219C22×D4—24no—
220C22×D4—24no—
221C22×D4—22no—
222C22×D4—24no—
223C22×D4—24no—
224C22×D4—48no—
225C22×D4—44no—
226C22×D4—44no—
227C22×D4C2×Dic8314yes—
228C22×D4C22⋊D8318yes—
229C23×C4—1662no—
230C23×C4C22⋊D8312yes—
231C23×C4C22×D8311yescharacteristic
232C22×C8—834no—
233C22×C8—1664no—
234C22×C8—1662no—
235C22×C8—1664no—
236C22×C8—1662no—
237#127—216no—
238#127—216no—
239C2×C4⋊C4—834no—
240C2×C4⋊C4—1662no—
241C2×C4⋊C4—1664no—
242C2×C4⋊C4—1664no—
243C2×C42—24no—
244C2×C42C2×Dic8312yes—
245C2×C42C22⋊D8312yes—
246#102—16632no—
247#95—3324no—
248C24—28no—
249C24—24no—
250C24—44no—
251C24—42no—
252C24—42no—
253C24—44no—
254C24—44no—
255C24—48no—
256C24C2×C22⋊D8311yescharacteristic
257C2×D4—232no—
258C2×D4—28no—
259C2×D4—28no—
260C2×D4—28no—
261C2×D4—28no—
262C2×D4—28no—
263C2×D4—24no—
264C2×D4—28no—
265C2×D4—28no—
266C2×D4—432no—
267C2×D4—432no—
268C2×D4—416no—
269C2×D4—44no—
270C2×D4—44no—
271C2×D4—44no—
272C2×D4—44no—
273C2×D4—44no—
274C2×D4—42no—
275C2×D4—42no—
276C2×D4—416no—
277C2×D4C22⋊Dic8318yes—
278C22×C4—24no—
279C22×C4—24no—
280C22×C4—3322no—
281C22×C4—3322no—
282C22×C4—3324no—
283C22×C4—3324no—
284C22×C4—1664no—
285C22×C4C22⋊Dic8314yes—
286C22×C4#1305057514yes—
287C22×C4#1305057814yes—
288C22×C4C22×Dic8311yescharacteristic
289C22×C4D4×D8312yes—
290C22×C4D4×D8312yes—
291C22×C4C2×C22⋊D8314yes—
292C22×C4C2×C22⋊D8311yescharacteristic
293C22×C4#1305058912yes—
294C22×C4#1305058912yes—
295C2×C8—3322no—
296C2×C8—3324no—
297C2×C8—3322no—
298C2×C8—1668no—
299C2×C8—1664no—
300C4⋊C4—3324no—
301C4⋊C4—1668no—
302C42—24no—
303C42—24no—
304C23—28no—
305C23—416no—
306C23—416no—
307C23—44no—
308C23—44no—
309C23—48no—
310C23—48no—
311C23—48no—
312C23—48no—
313C23—84no—
314C23—82no—
315C23—82no—
316C23—84no—
317C23—88no—
318C23—88no—
319C23—84no—
320C23C2×C22⋊Dic8311yescharacteristic
321C23#1458176112yes—
322C23#1458176312yes—
323C23#1458177911yescharacteristic
324C23#1458178811yescharacteristic
325C23C2×#1305057512yes—
326C23C2×#1305057812yes—
327C23#1458181814yes—
328D4—216no—
329D4—464no—
330D4—48no—
331D4—48no—
332D4—48no—
333C2×C4—24no—
334C2×C4—28no—
335C2×C4—28no—
336C2×C4—24no—
337C2×C4—3324no—
338C2×C4—3328no—
339C2×C4—6642no—
340C2×C4—6642no—
341C2×C4—6644no—
342C2×C4#1458171318yescomplemented
343C2×C4C2×C22⋊Dic8312yes—
344C2×C4D4×Dic8312yes—
345C2×C4D4×Dic8312yes—
346C2×C4#1458176014yes—
347C2×C4#1458176112yes—
348C2×C4#1458176312yes—
349C2×C4#1458176414yes—
350C2×C4#1458176414yes—
351C2×C4#1458177314yes—
352C2×C4#1458177911yescharacteristic
353C2×C4#1458178811yescharacteristic
354C8—3324no—
355C22—416no—
356C22—48no—
357C22—84no—
358C22—88no—
359C22—88no—
360C22—816no—
361C22—84no—
362C22—88no—
363C22#1912755711yescharacteristic
364C22#1912760411yescharacteristic
365C22#1912760411yescharacteristic
366C22#1912760912yes—
367C22#1912761512yes—
368C22C2×#1458171312yes—
369C22#1912770612yes—
370C22#1912783514yes—
371C22#1912785214yes—
372C22#1912789314yes—
373C22#1912789814yes—
374C22#1912804712yes—
375C22#1912806512yes—
376C22#1912807311yescharacteristic
377C22#1912807711yescharacteristic
378C22#1912808812yes—
379C4—28no—
380C4—6644no—
381C4#1912758114yes—
382C4#1912760912yes—
383C4#1912761512yes—
384C2—88no—
385C2#2490570411yescharacteristic
386C2#2490739412yes—
387C2#2490827011yescharacteristic
388C2#2490828811yescharacteristic
389C2#2490839112yes—
390C2#2490853412yes—
391C2#2490855112yes—
392C2#2490869414yes—
subgroupG/H
centerC24C2×C22⋊D83
commutatorC22×C4×C83C22×C4
FrattiniC23×C4C22×D83
Fitting, prime radicalC22×#127×C83C2
socleC24×C83C2×D4
cosocleC23×C4×C83C23
hypercenterC22×#127D83
hypocenterC83#11008
LCSG ▷ C22×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C24 ◁ C22×C42 ◁ C22×#127

Complements

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

NH ≅ G/Ntype
1. C22×#14581713C2semidirect
2. C2×#19127456C2semidirect
3. #24905693C2semidirect
16. C2×#127×C83C4semidirect
23. C22×C83⋊C8D4semidirect
27. C2×C4⋊Dic83D4semidirect
140. C2×C4×C83#102semidirect
201. C83#11008semidirect
207. C2×#127Dic83semidirect
342. C2×C4#14581713semidirect

Maximal subgroups

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

subgroupindex
C22×#145817132
C2×#191274562
#249056932
C22×#127×C832
#1100883