Finite Group Databasebeta

← #35293447

Subgroups — #35293447

Order 21248.

446 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 446 conjugacy classes of subgroups — too many to draw. Only its 91 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#24908991C211yesmaximal, characteristic
2#24908934C211yesmaximal, characteristic, complemented
3#24908871C211yesmaximal, characteristic, complemented
4#24905289C211yesmaximal, characteristic, complemented
5#24905286C211yesmaximal, characteristic, complemented
6#24905284C211yesmaximal, characteristic
7C83×#2732C211yesmaximal, characteristic
8C2×#14581725—21no—
9C2×#14581725—21no—
10C2×#14581716—21no—
11C2×#14581713—21no—
12#19127571C2211yescharacteristic
13#19127463—21no—
14#19127463—21no—
15#19127456C2211yescharacteristic
16#19127456C2211yescharacteristic
17#19127456C2211yescharacteristic
18C2×C22⋊Q8×C83—21no—
19#399×C83—21no—
20#398×C83—21no—
21#396×C83—21no—
22#391×C83C2211yescharacteristic
23#390×C83C2211yescharacteristic
24#389×C83C2211yescharacteristic
25#386×C83—21no—
26#385×C83—21no—
27C22×C83⋊C8—21no—
28C22×C83⋊C8—21no—
29C22×C83⋊C8—21no—
30C2×C4⋊Dic83—21no—
31C2×C4⋊Dic83—21no—
32C2×C4⋊Dic83—21no—
33#14581725—42no—
34#14581725—42no—
35#14581716—42no—
36#14581713—42no—
37C22×Q8×C83D411yescharacteristic
38C22×D4×C83D411yescharacteristic
39C23×C4×C83—21no—
40C22⋊Q8×C83—42no—
41C22⋊Q8×C83—42no—
42C2×C4⋊C4×C83—21no—
43C2×C4⋊C4×C83—21no—
44C2×C4⋊C4×C83—21no—
45C2×C4⋊C4×C83—21no—
46C2×C4⋊C4×C83D411yescharacteristic
47C2×C4⋊C4×C83D411yescharacteristic
48C2×C4⋊C4×C83D411yescharacteristic
49C2×C4⋊C4×C83D411yescharacteristic
50C2×C22⋊C4×C83—21no—
51C2×C22⋊C4×C83—21no—
52C2×C22⋊C4×C83—21no—
53C2×C22⋊C4×C83—21no—
54C2×C22⋊C4×C83—21no—
55C2×C42×C83C2311yescharacteristic
56#95×C83—21no—
57#95×C83—21no—
58#95×C83—21no—
59#95×C83—21no—
60C22×Dic83—41no—
61C2×C83⋊C8—22no—
62C2×C83⋊C8—22no—
63C2×C83⋊C8—41no—
64C2×C83⋊C8—41no—
65C2×C83⋊C8—41no—
66C2×C83⋊C8—41no—
67C2×C83⋊C8—41no—
68C2×C83⋊C8—41no—
69C2×C83⋊C8—41no—
70C4⋊Dic83—41no—
71C4⋊Dic83—41no—
72C4⋊Dic83—41no—
73C4⋊Dic83—41no—
74C4⋊Dic83—41no—
75C4⋊Dic83—41no—
76C24×C83—21no—
77C2×Q8×C83—21no—
78C2×Q8×C83—41no—
79C2×Q8×C83—41no—
80C2×Q8×C83D812yes—
81C2×D4×C83—21no—
82C2×D4×C83—41no—
83C2×D4×C83—41no—
84C2×D4×C83Q1612yes—
85C22×C4×C83—21no—
86C22×C4×C83—21no—
87C22×C4×C83—21no—
88C22×C4×C83—21no—
89C22×C4×C83—21no—
90C22×C4×C83—42no—
91C22×C4×C83—81no—
92C22×C4×C83—81no—
93C22×C4×C83C2×D411yescharacteristic
94C22×C4×C83C2×D411yescharacteristic
95C22×C4×C83C2×D411yescharacteristic
96C4⋊C4×C83—21no—
97C4⋊C4×C83—21no—
98C4⋊C4×C83—41no—
99C4⋊C4×C83—41no—
100C4⋊C4×C83—41no—
101C4⋊C4×C83—42no—
102C4⋊C4×C83—42no—
103C4⋊C4×C83—41no—
104C4⋊C4×C83—41no—
105C4⋊C4×C83—81no—
106C22⋊C4×C83—42no—
107C22⋊C4×C83—41no—
108C22⋊C4×C83—41no—
109C22⋊C4×C83—41no—
110C22⋊C4×C83—41no—
111C22⋊C4×C83—81no—
112C22⋊C4×C83—81no—
113C42×C83—41no—
114C23×C83—22no—
115C23×C83—41no—
116C23×C83—41no—
117C23×C83—41no—
118C23×C83—41no—
119C23×C83—41no—
120C23×C83—41no—
121C23×C83#12011yescharacteristic
122Q8×C83—42no—
123Q8×C83—81no—
124D4×C83—42no—
125D4×C83—81no—
126C2×C4×C83—21no—
127C2×C4×C83—21no—
128C2×C4×C83—21no—
129C2×C4×C83—21no—
130C2×C4×C83—21no—
131C2×C4×C83—21no—
132C2×C4×C83—22no—
133C2×C4×C83—41no—
134C2×C4×C83—41no—
135C2×C4×C83—41no—
136C2×C4×C83—41no—
137C2×C4×C83—41no—
138C2×C4×C83—41no—
139C2×C4×C83—41no—
140C2×C4×C83—41no—
141C2×C4×C83—41no—
142C2×C4×C83—41no—
143C2×C4×C83—41no—
144C2×C4×C83—41no—
145C2×C4×C83—41no—
146C2×C4×C83—41no—
147C2×C4×C83—84no—
148C2×C4×C83C2×D811yescharacteristic
149C2×C4×C83C2×Q1611yescharacteristic
150C2×C4×C83#13511yescharacteristic
151C2×C4×C83AGL(1,ℤ/8)11yescharacteristic
152C2×C4×C83AGL(1,ℤ/8)11yescharacteristic
153C2×C4×C83#13711yescharacteristic
154C2×Dic83—81no—
155C2×Dic83—81no—
156C2×Dic83—81no—
157C83⋊C8—42no—
158C83⋊C8—42no—
159C83⋊C8—81no—
160C22×C83—42no—
161C22×C83—42no—
162C22×C83—42no—
163C22×C83—81no—
164C22×C83—81no—
165C22×C83—81no—
166C22×C83—81no—
167C22×C83#44711yescharacteristic
168C22×C83#44911yescharacteristic
169C22×C83#45111yescharacteristic
170C22×C83#45311yescharacteristic
171C22×C83#45511yescharacteristic
172C22×C83#45611yescharacteristic
173C22×C83#45811yescharacteristic
174C4×C83—21no—
175C4×C83—21no—
176C4×C83—21no—
177C4×C83—21no—
178C4×C83—21no—
179C4×C83—21no—
180C4×C83—42no—
181C4×C83—81no—
182C4×C83—81no—
183C4×C83—81no—
184C4×C83—81no—
185Dic83—161no—
186#7789—831nomaximal, Sylow
187C2×C83—82no—
188C2×C83#160711yescharacteristic
189C2×C83#160911yescharacteristic
190C2×C83#161411yescharacteristic
191C2×C83#162111yescharacteristic
192C2×C83#164411yescharacteristic
193C2×C83#164911yescharacteristic
194C2×C83#166111yescharacteristic
195#2732D8311yescharacteristic
196#2093—831no—
197#2083—831no—
198#2072—831no—
199#1272—831no—
200#1271—831no—
201#1270—831no—
202C83#778911yescharacteristic, complemented, Sylow
203C2×C22⋊Q8—21no—
204C2×#107—1661no—
205C2×#107—1661no—
206C2×#103—1661no—
207C2×#102—1661no—
208#399—21no—
209#398—21no—
210#396—21no—
211#391C2×D8311yescharacteristic
212#390C2×D8311yescharacteristic
213#389C2×D8311yescharacteristic
214#386—21no—
215#385—21no—
216#384—831no—
217#340—1661no—
218#340—1661no—
219#336—831no—
220#336—831no—
221#336—831no—
222C22×Q8C22⋊D8311yescharacteristic
223C22×D4C22⋊D8311yescharacteristic
224C23×C4—21no—
225C22×C8—1661no—
226C22×C8—1661no—
227C22×C8—1661no—
228C22⋊Q8—42no—
229C22⋊Q8—42no—
230C2×C4⋊C4—21no—
231C2×C4⋊C4—21no—
232C2×C4⋊C4—21no—
233C2×C4⋊C4—21no—
234C2×C4⋊C4—1661no—
235C2×C4⋊C4—1661no—
236C2×C4⋊C4—1661no—
237C2×C4⋊C4C22⋊D8311yescharacteristic
238C2×C4⋊C4C22⋊D8311yescharacteristic
239C2×C4⋊C4C22⋊D8311yescharacteristic
240C2×C4⋊C4C22⋊D8311yescharacteristic
241C2×C22⋊C4—21no—
242C2×C22⋊C4—21no—
243C2×C22⋊C4—21no—
244C2×C22⋊C4—21no—
245C2×C22⋊C4—21no—
246C2×C42C22×D8311yescharacteristic
247#107—3322no—
248#107—3322no—
249#103—3322no—
250#102—3322no—
251#95—21no—
252#95—21no—
253#95—21no—
254#95—21no—
255C24—21no—
256C2×Q8—21no—
257C2×Q8—41no—
258C2×Q8—41no—
259C2×Q8#1305057512yes—
260C2×D4—21no—
261C2×D4—41no—
262C2×D4—41no—
263C2×D4#1305058012yes—
264C22×C4—21no—
265C22×C4—21no—
266C22×C4—21no—
267C22×C4—21no—
268C22×C4—21no—
269C22×C4—42no—
270C22×C4—81no—
271C22×C4—81no—
272C22×C4—3321no—
273C22×C4C2×C22⋊D8311yescharacteristic
274C22×C4C2×C22⋊D8311yescharacteristic
275C22×C4C2×C22⋊D8311yescharacteristic
276C2×C8—3321no—
277C2×C8—3321no—
278C2×C8—3321no—
279C2×C8—3321no—
280C2×C8—3321no—
281C2×C8—3321no—
282C2×C8—3321no—
283C2×C8—1662no—
284C2×C8—1662no—
285C4⋊C4—21no—
286C4⋊C4—21no—
287C4⋊C4—41no—
288C4⋊C4—41no—
289C4⋊C4—41no—
290C4⋊C4—42no—
291C4⋊C4—42no—
292C4⋊C4—41no—
293C4⋊C4—41no—
294C4⋊C4—81no—
295C4⋊C4—3321no—
296C4⋊C4—3321no—
297C4⋊C4—3321no—
298C4⋊C4—3321no—
299C4⋊C4—3321no—
300C4⋊C4—3321no—
301C22⋊C4—42no—
302C22⋊C4—41no—
303C22⋊C4—41no—
304C22⋊C4—41no—
305C22⋊C4—41no—
306C22⋊C4—81no—
307C22⋊C4—81no—
308C42—41no—
309C23—22no—
310C23—41no—
311C23—41no—
312C23—41no—
313C23—41no—
314C23—41no—
315C23—41no—
316C23#1458176111yescharacteristic
317Q8—42no—
318Q8—81no—
319D4—42no—
320D4—81no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—22no—
328C2×C4—41no—
329C2×C4—41no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—41no—
334C2×C4—41no—
335C2×C4—41no—
336C2×C4—41no—
337C2×C4—41no—
338C2×C4—41no—
339C2×C4—41no—
340C2×C4—41no—
341C2×C4—41no—
342C2×C4—84no—
343C2×C4—6641no—
344C2×C4—6641no—
345C2×C4—6641no—
346C2×C4C2×#1305057511yescharacteristic
347C2×C4C2×#1305058011yescharacteristic
348C2×C4#1458181211yescharacteristic
349C2×C4#1458181711yescharacteristic
350C2×C4#1458181811yescharacteristic
351C2×C4#1458182311yescharacteristic
352C8—3322no—
353C8—3322no—
354C8—6641no—
355C22—42no—
356C22—42no—
357C22—42no—
358C22—81no—
359C22—81no—
360C22—81no—
361C22—81no—
362C22#1912783511yescharacteristic
363C22#1912784611yescharacteristic
364C22#1912785711yescharacteristic
365C22#1912786911yescharacteristic
366C22#1912787911yescharacteristic
367C22#1912788411yescharacteristic
368C22#1912789111yescharacteristic
369C4—21no—
370C4—21no—
371C4—21no—
372C4—21no—
373C4—21no—
374C4—21no—
375C4—42no—
376C4—81no—
377C4—81no—
378C4—81no—
379C4—81no—
380C4—13281no—
381C2—82no—
382C2#2490644911yescharacteristic
383C2#2490646011yescharacteristic
384C2#2490648811yescharacteristic
385C2#2490652811yescharacteristic
386C2#2490667711yescharacteristic
387C2#2490671211yescharacteristic
388C2#2490678011yescharacteristic
subgroupG/H
centerC23#14581761
commutator, cosocleC2×C42×C83C23
FrattiniC2×C42C22×D83
FittingC83×#2732C2
socleC23×C83#120
prime radicalC22×D4×C83D4
hypercenter#2732D83
hypocenterC83#7789
LCSG ▷ C2×C42×C83 ▷ C23×C83 ▷ C83
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2732

Complements

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

NH ≅ G/Ntype
2. #24908934C2semidirect
3. #24908871C2semidirect
4. #24905289C2semidirect
5. #24905286C2semidirect
202. C83#7789semidirect

Maximal subgroups

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

subgroupindex
#249089912
#249089342
#249088712
#249052892
#249052862
#249052842
C83×#27322
#778983