Finite Group Databasebeta

← #35300447

Subgroups — #35300447

Order 21248.

684 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 684 conjugacy classes of subgroups — too many to draw. Only its 131 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
1#24913268C211yesmaximal, characteristic
2#24906034C212yesmaximal, complemented
3C2×#19127425C211yesmaximal, characteristic
4#24905330C212yesmaximal, complemented
5C83×#2315C211yesmaximal, characteristic
6C2×#14581771C2211yescharacteristic
7C2×C22⋊(C83⋊C8)C2212yes—
8#19127637C412yescomplemented
9#19127624—22no—
10#19127613C412yescomplemented
11#19127574—21no—
12#19127561C2211yescharacteristic
13#19127557—21no—
14#19127437—24no—
15#19127425—22no—
16C22×C22⋊C4×C83—21no—
17C22×C22⋊C4×C83C2211yescharacteristic
18#385×C83—24no—
19#380×C83—21no—
20#380×C83C2212yes—
21C22×Dic166C2×C411yescharacteristic, complemented
22C22×C83⋊C8—22no—
23#14581771—22no—
24#14581771—22no—
25C2×C4⋊Dic83—21no—
26C2×C4⋊Dic83C2×C411yescharacteristic, complemented
27C2×Dic83⋊C4—22no—
28C2×Dic83⋊C4—21no—
29C2×C22⋊Dic83—22no—
30C2×C22⋊Dic83—22no—
31C2×C22⋊Dic83—21no—
32C2×C22⋊Dic83—21no—
33C22⋊(C83⋊C8)—22no—
34C22⋊(C83⋊C8)—22no—
35C83⋊#95—22no—
36C83⋊#95C2×C412yescomplemented
37C25×C83D411yescharacteristic
38C23×C4×C83—22no—
39C23×C4×C83D411yescharacteristic
40C23×C4×C83C2311yescharacteristic
41C2×C4⋊C4×C83—22no—
42C2×C4⋊C4×C83—22no—
43C2×C22⋊C4×C83—22no—
44C2×C22⋊C4×C83—22no—
45C2×C22⋊C4×C83—22no—
46C2×C22⋊C4×C83—22no—
47C2×C22⋊C4×C83—42no—
48C2×C22⋊C4×C83—42no—
49C2×C22⋊C4×C83C2×C412yes—
50C2×C22⋊C4×C83D412yes—
51#95×C83—22no—
52#95×C83D414yes—
53C2×Dic166—22no—
54C2×Dic166—81no—
55C22×Dic83—21no—
56C22×Dic83—22no—
57C22×Dic83—21no—
58C2×C83⋊C8—42no—
59C2×C83⋊C8—82no—
60C4⋊Dic83—22no—
61C4⋊Dic83—81no—
62Dic83⋊C4—42no—
63Dic83⋊C4—82no—
64C22⋊Dic83—41no—
65C22⋊Dic83—41no—
66C22⋊Dic83—41no—
67C22⋊Dic83—41no—
68C22⋊Dic83—82no—
69C22⋊Dic83—82no—
70C24×C83—21no—
71C24×C83—21no—
72C24×C83—21no—
73C24×C83—21no—
74C24×C83—81no—
75C24×C83—81no—
76C24×C83C22⋊C412yes—
77C24×C83SD1612yes—
78C24×C83Q1612yes—
79C24×C83C2×D411yescharacteristic
80C22×C4×C83—21no—
81C22×C4×C83—21no—
82C22×C4×C83—24no—
83C22×C4×C83—44no—
84C22×C4×C83—81no—
85C22×C4×C83—81no—
86C22×C4×C83—82no—
87C22×C4×C83—82no—
88C22×C4×C83C22⋊C412yes—
89C22×C4×C83C22×C411yescharacteristic
90C22×C4×C83C2×D412yes—
91C22×C4×C83C2×D411yescharacteristic
92C22×C4×C83#4112yes—
93C4⋊C4×C83—82no—
94C4⋊C4×C83—82no—
95C22⋊C4×C83—24no—
96C22⋊C4×C83—24no—
97C22⋊C4×C83—44no—
98C22⋊C4×C83—42no—
99C22⋊C4×C83—42no—
100C23×C83—22no—
101C23×C83—22no—
102C23×C83—22no—
103C23×C83—22no—
104C23×C83—22no—
105C23×C83—22no—
106C23×C83—41no—
107C23×C83—41no—
108C23×C83—42no—
109C23×C83—42no—
110C23×C83—42no—
111C23×C83—42no—
112C23×C83—81no—
113C23×C83—81no—
114C23×C83—82no—
115C23×C83—82no—
116C23×C83—82no—
117C23×C83—82no—
118C23×C83#10314yes—
119C23×C83C2×C22⋊C411yescharacteristic
120C23×C83#12011yescharacteristic
121C23×C83#12311yescharacteristic
122C23×C83C2×SD1611yescharacteristic
123C23×C83C2×Q1611yescharacteristic
124C23×C83AGL(1,ℤ/8)11yescharacteristic
125C23×C83#13711yescharacteristic
126C2×C4×C83—22no—
127C2×C4×C83—42no—
128C2×C4×C83—41no—
129C2×C4×C83—41no—
130C2×C4×C83—42no—
131C2×C4×C83—42no—
132C2×C4×C83—42no—
133C2×C4×C83—44no—
134C2×C4×C83—44no—
135C2×C4×C83—44no—
136C2×C4×C83—81no—
137C2×C4×C83—88no—
138C2×C4×C83C4×D412yes—
139C2×C4×C83#12112yes—
140C2×Dic83—42no—
141C2×Dic83—42no—
142C2×Dic83—42no—
143C2×Dic83—42no—
144C2×Dic83—41no—
145C2×Dic83—41no—
146C2×Dic83—42no—
147Dic166—82no—
148C83⋊C8—82no—
149C22×C83—21no—
150C22×C83—21no—
151C22×C83—21no—
152C22×C83—21no—
153C22×C83—21no—
154C22×C83—21no—
155C22×C83—24no—
156C22×C83—41no—
157C22×C83—41no—
158C22×C83—41no—
159C22×C83—41no—
160C22×C83—44no—
161C22×C83—44no—
162C22×C83—44no—
163C22×C83—42no—
164C22×C83—42no—
165C22×C83—82no—
166C22×C83—82no—
167C22×C83—82no—
168C22×C83#38611yescharacteristic
169C22×C83C2×#10311yescharacteristic
170C22×C83#41711yescharacteristic
171C22×C83#45011yescharacteristic
172C22×C83#45111yescharacteristic
173C22×C83#48311yescharacteristic
174C22×C83#48411yescharacteristic
175C4×C83—22no—
176C4×C83—22no—
177C4×C83—42no—
178C4×C83—84no—
179Dic83—82no—
180Dic83—81no—
181Dic83—81no—
182#9199—831nomaximal, Sylow
183C2×C83—21no—
184C2×C83—21no—
185C2×C83—21no—
186C2×C83—21no—
187C2×C83—21no—
188C2×C83—21no—
189C2×C83—44no—
190C2×C83#187211yescharacteristic
191C2×C83#191111yescharacteristic
192C2×C83#191311yescharacteristic
193#2626—831no—
194#2315D8311yescharacteristic
195#1520—832no—
196C2×#323—831no—
197#1289—832no—
198C83#919911yescharacteristic, complemented, Sylow
199C2×C22⋊Q8—831no—
200C22×C22⋊C4—21no—
201C22×C22⋊C4C2×D8311yescharacteristic
202C2×C22⋊C8—832no—
203#399—832no—
204#396—1662no—
205#393—832no—
206#385—24no—
207#385—1661no—
208#380—21no—
209#380—831no—
210#380C2×D8312yes—
211#379—1661no—
212#328—1664no—
213#323—1662no—
214C25C22⋊D8311yescharacteristic
215C22×Q8—831no—
216C23×C4—22no—
217C23×C4C22⋊D8311yescharacteristic
218C23×C4C22×D8311yescharacteristic
219C22×C8—1662no—
220C22⋊Q8—1662no—
221C22⋊Q8—1662no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—831no—
225C2×C4⋊C4—1661no—
226C2×C4⋊C4—1662no—
227C2×C4⋊C4—1661no—
228C2×C22⋊C4—22no—
229C2×C22⋊C4—22no—
230C2×C22⋊C4—22no—
231C2×C22⋊C4—22no—
232C2×C22⋊C4—42no—
233C2×C22⋊C4—42no—
234C2×C22⋊C4—1661no—
235C2×C22⋊C4—1661no—
236C2×C22⋊C4—1662no—
237C2×C22⋊C4—1662no—
238C2×C22⋊C4C4×D8312yes—
239C2×C22⋊C4D33212yes—
240C22⋊C8—1662no—
241C22⋊C8—1662no—
242#95—22no—
243#95—832no—
244#95—1662no—
245#95C22⋊D8314yes—
246C24—21no—
247C24—21no—
248C24—21no—
249C24—21no—
250C24—81no—
251C24—81no—
252C24(C2×D83)⋊C412yes—
253C24#1305057812yes—
254C24#1305058012yes—
255C24C2×C22⋊D8311yescharacteristic
256C2×Q8—6641no—
257C2×Q8—1662no—
258C22×C4—21no—
259C22×C4—21no—
260C22×C4—24no—
261C22×C4—44no—
262C22×C4—81no—
263C22×C4—81no—
264C22×C4—82no—
265C22×C4—82no—
266C22×C4—1662no—
267C22×C4—1661no—
268C22×C4—1661no—
269C22×C4(C2×D83)⋊C412yes—
270C22×C4C2×C4×D8311yescharacteristic
271C22×C4C2×C22⋊D8312yes—
272C22×C4C2×D33211yescharacteristic
273C22×C4#1305059012yes—
274C2×C8—3322no—
275C2×C8—6642no—
276C4⋊C4—82no—
277C4⋊C4—82no—
278C4⋊C4—3322no—
279C4⋊C4—6642no—
280C4⋊C4—6641no—
281C4⋊C4—1662no—
282C22⋊C4—24no—
283C22⋊C4—24no—
284C22⋊C4—44no—
285C22⋊C4—42no—
286C22⋊C4—42no—
287C22⋊C4—3321no—
288C22⋊C4—3321no—
289C22⋊C4—3321no—
290C22⋊C4—3321no—
291C22⋊C4—6642no—
292C22⋊C4—6642no—
293C23—22no—
294C23—22no—
295C23—22no—
296C23—22no—
297C23—22no—
298C23—22no—
299C23—41no—
300C23—41no—
301C23—42no—
302C23—42no—
303C23—42no—
304C23—42no—
305C23—81no—
306C23—81no—
307C23—82no—
308C23—82no—
309C23—82no—
310C23—82no—
311C23#1458171514yes—
312C23C2×(C2×D83)⋊C411yescharacteristic
313C23#1458176111yescharacteristic
314C23#1458177511yescharacteristic
315C23C2×#1305057811yescharacteristic
316C23C2×#1305058011yescharacteristic
317C23#1458181811yescharacteristic
318C23#1458182311yescharacteristic
319Q8—6642no—
320C2×C4—22no—
321C2×C4—42no—
322C2×C4—41no—
323C2×C4—42no—
324C2×C4—41no—
325C2×C4—42no—
326C2×C4—42no—
327C2×C4—44no—
328C2×C4—44no—
329C2×C4—44no—
330C2×C4—81no—
331C2×C4—88no—
332C2×C4—3322no—
333C2×C4—3322no—
334C2×C4—3322no—
335C2×C4—3322no—
336C2×C4—3321no—
337C2×C4—3321no—
338C2×C4—3322no—
339C2×C4C4×C22⋊D8312yes—
340C2×C4#1458176612yes—
341C8—6642no—
342C22—21no—
343C22—21no—
344C22—21no—
345C22—21no—
346C22—21no—
347C22—21no—
348C22—24no—
349C22—41no—
350C22—41no—
351C22—41no—
352C22—41no—
353C22—44no—
354C22—44no—
355C22—44no—
356C22—42no—
357C22—42no—
358C22—82no—
359C22—82no—
360C22—82no—
361C22#1912758011yescharacteristic
362C22C2×#1458171511yescharacteristic
363C22#1912770011yescharacteristic
364C22#1912785211yescharacteristic
365C22#1912785711yescharacteristic
366C22#1912803111yescharacteristic
367C22#1912803611yescharacteristic
368C4—22no—
369C4—22no—
370C4—42no—
371C4—84no—
372C4—6642no—
373C4—6641no—
374C4—6641no—
375C2—21no—
376C2—21no—
377C2—21no—
378C2—21no—
379C2—21no—
380C2—21no—
381C2—44no—
382C2#2490777011yescharacteristic
383C2#2490799511yescharacteristic
384C2#2490800511yescharacteristic
subgroupG/H
centerC22#19127580
commutatorC22×C4×C83C22×C4
FrattiniC23×C4C22×D83
FittingC83×#2315C2
socleC22×C83#386
prime radicalC25×C83D4
cosocleC23×C4×C83C23
hypercenter#2315D83
hypocenterC83#9199
LCSG ▷ C22×C4×C83 ▷ C22×C83 ▷ C83
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2315

Complements

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

NH ≅ G/Ntype
2. #24906034C2semidirect
4. #24905330C2semidirect
8. #19127637C4semidirect
10. #19127613C4semidirect
21. C22×Dic166C2×C4semidirect
26. C2×C4⋊Dic83C2×C4semidirect
36. C83⋊#95C2×C4semidirect
198. C83#9199semidirect

Maximal subgroups

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

subgroupindex
#249132682
#249060342
C2×#191274252
#249053302
C83×#23152
#919983