Finite Group Databasebeta

← #35299451

Subgroups — #35299451

Order 21248.

768 conjugacy classes of subgroups; only the 143 normal subgroups are shown.

normal subgroups only

This group has 768 conjugacy classes of subgroups — too many to draw. Only its 143 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×#19127626C211yesmaximal, characteristic
2#24910077C211yesmaximal, characteristic
3#24907550C212yesmaximal, complemented
4#24905343C212yesmaximal, complemented
5C2×#323×C83C211yesmaximal, characteristic
6C22×C4⋊Dic83C2211yescharacteristic
7C22×C22⋊Dic83—21no—
8C22×C22⋊Dic83—21no—
9#19127626—22no—
10#19127626C414yescomplemented
11#19127576—24no—
12#19127561—21no—
13#19127561C2212yes—
14C2×C83⋊#95—21no—
15C2×C83⋊#95C2211yescharacteristic
16#19127464—24no—
17C22×C22⋊C4×C83C2211yescharacteristic
18C2×C22⋊C8×C83C2212yes—
19#323×C83—22no—
20C23×Dic83—22no—
21C23×Dic83—21no—
22C23×Dic83—21no—
23C2×C4⋊Dic83—22no—
24C2×C4⋊Dic83—22no—
25C2×C4⋊Dic83—21no—
26C2×C4⋊Dic83—22no—
27C2×C4⋊Dic83C2×C412yescomplemented
28C2×C4⋊Dic83D414yes—
29C2×C22⋊Dic83—22no—
30C2×C22⋊Dic83—24no—
31C2×C22⋊Dic83—22no—
32C2×C22⋊Dic83—42no—
33C2×C22⋊Dic83—42no—
34C2×C22⋊Dic83—42no—
35C2×C22⋊Dic83—42no—
36C83⋊#95—22no—
37C83⋊#95—24no—
38C83⋊#95—22no—
39C83⋊#95—21no—
40C83⋊#95C2×C412yescomplemented
41C25×C83D411yescharacteristic
42C23×C4×C83D411yescharacteristic
43C23×C4×C83C2311yescharacteristic
44C22×C8×C83—22no—
45C2×C22⋊C4×C83—22no—
46C2×C22⋊C4×C83—22no—
47C2×C22⋊C4×C83C2×C412yes—
48C2×C22⋊C4×C83D412yes—
49C22⋊C8×C83—22no—
50C22⋊C8×C83—22no—
51C22×Dic83—22no—
52C22×Dic83—22no—
53C22×Dic83—24no—
54C22×Dic83—42no—
55C22×Dic83—42no—
56C22×Dic83—44no—
57C22×Dic83—41no—
58C22×Dic83—41no—
59C22×Dic83—41no—
60C22×Dic83—41no—
61C22×Dic83—41no—
62C22×Dic83—41no—
63C22×Dic83—41no—
64C22×Dic83—41no—
65C22×Dic83—82no—
66C22×Dic83—82no—
67C4⋊Dic83—24no—
68C4⋊Dic83—42no—
69C4⋊Dic83—44no—
70C4⋊Dic83—82no—
71C22⋊Dic83—44no—
72C22⋊Dic83—48no—
73C22⋊Dic83—42no—
74C22⋊Dic83—42no—
75C24×C83—21no—
76C24×C83—21no—
77C24×C83—21no—
78C24×C83—21no—
79C24×C83—42no—
80C24×C83—42no—
81C24×C83C22⋊C412yes—
82C24×C83D812yes—
83C24×C83SD1612yes—
84C24×C83C2×D411yescharacteristic
85C22×C4×C83—21no—
86C22×C4×C83—21no—
87C22×C4×C83—42no—
88C22×C4×C83—42no—
89C22×C4×C83C22⋊C412yes—
90C22×C4×C83C22×C411yescharacteristic
91C22×C4×C83C2×D412yes—
92C22×C4×C83C2×D411yescharacteristic
93C22×C4×C83#4112yes—
94C2×C8×C83—44no—
95C2×C8×C83—42no—
96C22⋊C4×C83—24no—
97C22⋊C4×C83—24no—
98C23×C83—22no—
99C23×C83—22no—
100C23×C83—22no—
101C23×C83—22no—
102C23×C83—22no—
103C23×C83—22no—
104C23×C83—22no—
105C23×C83—22no—
106C23×C83—42no—
107C23×C83—42no—
108C23×C83—44no—
109C23×C83—44no—
110C23×C83—42no—
111C23×C83—42no—
112C23×C83—82no—
113C23×C83—82no—
114C23×C83—81no—
115C23×C83—81no—
116C23×C83#10214yes—
117C23×C83C2×C22⋊C411yescharacteristic
118C23×C83#12011yescharacteristic
119C23×C83#12311yescharacteristic
120C23×C83C2×D811yescharacteristic
121C23×C83C2×SD1611yescharacteristic
122C23×C83AGL(1,ℤ/8)11yescharacteristic
123C23×C83#13711yescharacteristic
124C2×C4×C83—24no—
125C2×C4×C83—22no—
126C2×C4×C83—41no—
127C2×C4×C83—41no—
128C2×C4×C83—42no—
129C2×C4×C83—42no—
130C2×C4×C83—42no—
131C2×C4×C83—81no—
132C2×C4×C83C4×D412yes—
133C2×C4×C83#12112yes—
134C2×Dic83—42no—
135C2×Dic83—42no—
136C2×Dic83—42no—
137C2×Dic83—42no—
138C2×Dic83—42no—
139C2×Dic83—42no—
140C2×Dic83—44no—
141C2×Dic83—44no—
142C2×Dic83—44no—
143C2×Dic83—82no—
144C2×Dic83—82no—
145C2×Dic83—88no—
146C2×Dic83—81no—
147C2×Dic83—81no—
148C2×Dic83—81no—
149C2×Dic83—81no—
150C8×C83—82no—
151C22×C83—22no—
152C22×C83—22no—
153C22×C83—22no—
154C22×C83—22no—
155C22×C83—21no—
156C22×C83—21no—
157C22×C83—21no—
158C22×C83—21no—
159C22×C83—24no—
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×#10211yescharacteristic
170C22×C83#41711yescharacteristic
171C22×C83#44711yescharacteristic
172C22×C83#44811yescharacteristic
173C22×C83#45512yes—
174C22×C83#45812yes—
175C22×C83#48011yescharacteristic
176C22×C83#48111yescharacteristic
177C4×C83—22no—
178C4×C83—22no—
179C4×C83—42no—
180Dic83—82no—
181Dic83—84no—
182Dic83—82no—
183#9025—831nomaximal, Sylow
184C2×C83—21no—
185C2×C83—21no—
186C2×C83—21no—
187C2×C83—21no—
188C2×C83—44no—
189C2×C83#160712yes—
190C2×C83#160812yes—
191C2×C83#187411yescharacteristic
192C2×C83#188511yescharacteristic
193C2×C83#189011yescharacteristic
194C2×#397—831no—
195#2315—831no—
196#1823—832no—
197C2×#323D8311yescharacteristic
198#1294—832no—
199C83#902511yescharacteristic, complemented, Sylow
200C22×C4⋊C4—831no—
201C22×C22⋊C4—1661no—
202C22×C22⋊C4—1661no—
203C22×C22⋊C4C2×D8311yescharacteristic
204C2×C22⋊C8C2×D8312yes—
205#397—834no—
206#397—1662no—
207#385—1664no—
208#380—832no—
209#380—1661no—
210C2×#95—831no—
211C2×#95—1661no—
212#340—1664no—
213#323—22no—
214C25D33211yescharacteristic
215C23×C4—1661no—
216C23×C4—1661no—
217C23×C4—1662no—
218C23×C4D33211yescharacteristic
219C23×C4C22×D8311yescharacteristic
220C22×C8—22no—
221C2×C4⋊C4—834no—
222C2×C4⋊C4—832no—
223C2×C4⋊C4—1661no—
224C2×C4⋊C4—1662no—
225C2×C4⋊C4—1662no—
226C2×C4⋊C4—1662no—
227C2×C22⋊C4—22no—
228C2×C22⋊C4—22no—
229C2×C22⋊C4—3322no—
230C2×C22⋊C4—3322no—
231C2×C22⋊C4—3322no—
232C2×C22⋊C4—3322no—
233C2×C22⋊C4—1662no—
234C2×C22⋊C4—1664no—
235C2×C22⋊C4—1662no—
236C2×C22⋊C4C4×D8312yes—
237C2×C22⋊C4C22⋊D8312yes—
238C22⋊C8—22no—
239C22⋊C8—22no—
240#95—832no—
241#95—1662no—
242#95—1664no—
243#95—1662no—
244#95—1661no—
245C24—21no—
246C24—21no—
247C24—21no—
248C24—21no—
249C24—42no—
250C24—42no—
251C24(C2×D83)⋊C412yes—
252C24D66412yes—
253C24#1305057912yes—
254C24C2×D33211yescharacteristic
255C22×C4—21no—
256C22×C4—21no—
257C22×C4—42no—
258C22×C4—42no—
259C22×C4—3324no—
260C22×C4—3322no—
261C22×C4—3322no—
262C22×C4—3321no—
263C22×C4—3321no—
264C22×C4—3321no—
265C22×C4—3321no—
266C22×C4—3321no—
267C22×C4—3321no—
268C22×C4—3321no—
269C22×C4—3321no—
270C22×C4—6642no—
271C22×C4—6642no—
272C22×C4—1664no—
273C22×C4—1662no—
274C22×C4—1662no—
275C22×C4(C2×D83)⋊C412yes—
276C22×C4C2×C4×D8311yescharacteristic
277C22×C4D4×D8312yes—
278C22×C4C2×C22⋊D8311yescharacteristic
279C22×C4#1305058912yes—
280C2×C8—44no—
281C2×C8—42no—
282C4⋊C4—3324no—
283C4⋊C4—3322no—
284C4⋊C4—6642no—
285C4⋊C4—1664no—
286C22⋊C4—24no—
287C22⋊C4—24no—
288C22⋊C4—3324no—
289C22⋊C4—3322no—
290C22⋊C4—3322no—
291C22⋊C4—3328no—
292C23—22no—
293C23—22no—
294C23—22no—
295C23—22no—
296C23—22no—
297C23—22no—
298C23—22no—
299C23—22no—
300C23—42no—
301C23—42no—
302C23—44no—
303C23—44no—
304C23—42no—
305C23—42no—
306C23—81no—
307C23—81no—
308C23—82no—
309C23—82no—
310C23#1458171414yes—
311C23C2×(C2×D83)⋊C411yescharacteristic
312C23#1458176211yescharacteristic
313C23#1458177611yescharacteristic
314C23C2×D66411yescharacteristic
315C23C2×#1305057911yescharacteristic
316C23#1458182011yescharacteristic
317C23#1458182511yescharacteristic
318C2×C4—24no—
319C2×C4—22no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—42no—
323C2×C4—42no—
324C2×C4—42no—
325C2×C4—81no—
326C2×C4—3322no—
327C2×C4—3322no—
328C2×C4—3322no—
329C2×C4—3324no—
330C2×C4—3324no—
331C2×C4—3324no—
332C2×C4—3322no—
333C2×C4—3322no—
334C2×C4—3322no—
335C2×C4—6642no—
336C2×C4—6642no—
337C2×C4—6648no—
338C2×C4—6641no—
339C2×C4—6641no—
340C2×C4—6641no—
341C2×C4—6641no—
342C2×C4(C22⋊D83)⋊C412yes—
343C2×C4#1458176412yes—
344C8—82no—
345C22—21no—
346C22—21no—
347C22—21no—
348C22—21no—
349C22—22no—
350C22—22no—
351C22—22no—
352C22—22no—
353C22—24no—
354C22—44no—
355C22—42no—
356C22—42no—
357C22—44no—
358C22—44no—
359C22—82no—
360C22—82no—
361C22—82no—
362C22#1912757811yescharacteristic
363C22C2×#1458171411yescharacteristic
364C22#1912770311yescharacteristic
365C22#1912783711yescharacteristic
366C22#1912784211yescharacteristic
367C22#1912788112yes—
368C22#1912789212yes—
369C22#1912801511yescharacteristic
370C22#1912802011yescharacteristic
371C4—22no—
372C4—22no—
373C4—42no—
374C4—6642no—
375C4—6642no—
376C4—6644no—
377C2—21no—
378C2—21no—
379C2—21no—
380C2—21no—
381C2—44no—
382C2#2490645112yes—
383C2#2490645612yes—
384C2#2490777811yescharacteristic
385C2#2490784911yescharacteristic
386C2#2490787411yescharacteristic
subgroupG/H
centerC23#14581762
commutatorC22×C4×C83C22×C4
FrattiniC23×C4C22×D83
FittingC2×#323×C83C2
socleC23×C83#120
prime radicalC25×C83D4
cosocleC23×C4×C83C23
hypercenterC2×#323D83
hypocenterC83#9025
LCSG ▷ C22×C4×C83 ▷ C22×C83 ▷ C83
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#323

Complements

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

NH ≅ G/Ntype
3. #24907550C2semidirect
4. #24905343C2semidirect
10. #19127626C4semidirect
27. C2×C4⋊Dic83C2×C4semidirect
40. C83⋊#95C2×C4semidirect
199. C83#9025semidirect

Maximal subgroups

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

subgroupindex
C2×#191276262
#249100772
#249075502
#249053432
C2×#323×C832
#902583