Finite Group Databasebeta

← #35292789

Subgroups — #35292789

Order 21248.

438 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 438 conjugacy classes of subgroups — too many to draw. Only its 101 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#24915831C211yesmaximal, characteristic
2#24907578C211yesmaximal, characteristic, complemented
3#24907207C211yesmaximal, characteristic, complemented
4#24905286C211yesmaximal, characteristic, complemented
5#24905283C211yesmaximal, characteristic
6#24905283C211yesmaximal, characteristic
7C83×#1271C211yesmaximal, characteristic, complemented
8C2×#14581787—21no—
9C2×Dic83⋊C8—21no—
10C2×C8×Dic83—21no—
11#19127603C2211yescharacteristic
12#19127602C411yescharacteristic
13#19127602C411yescharacteristic
14#19127600C411yescharacteristic
15#19127597C411yescharacteristic
16#19127593—21no—
17#19127591—21no—
18#19127590—21no—
19C42×Dic83C2211yescharacteristic
20#19127456C2211yescharacteristic
21#19127456C2211yescharacteristic
22#19127455—21no—
23#19127455—21no—
24#389×C83C2211yescharacteristic
25#336×C83C2211yescharacteristic
26#336×C83C2211yescharacteristic
27C22×D332C2×C411yescharacteristic
28C22×C83⋊C8—21no—
29C22×C83⋊C8—21no—
30#14581787—42no—
31#14581787—42no—
32C2×C4⋊Dic83—21no—
33C2×C4⋊Dic83C2×C411yescharacteristic
34C2×Dic83⋊C4—21no—
35C2×(C2×D83)⋊C4—21no—
36C2×(C2×D83)⋊C4—21no—
37C2×(C2×D83)⋊C4—21no—
38C2×(C2×D83)⋊C4—21no—
39C2×(C2×D83)⋊C4—21no—
40C2×(C2×D83)⋊C4—21no—
41C2×C4×Dic83C2×C411yescharacteristic
42C2×C4×Dic83C2×C411yescharacteristic
43C2×C4×Dic83D411yescharacteristic
44C2×C4×Dic83D411yescharacteristic
45C2×C4×Dic83D411yescharacteristic
46C2×C4×Dic83D411yescharacteristic
47Dic83⋊C8—42no—
48C8×Dic83—24no—
49C83⋊#95—21no—
50C83⋊#95—21no—
51C22×C8×C83—21no—
52C22×C8×C83—21no—
53C2×C4⋊C4×C83—21no—
54C2×C4⋊C4×C83C2×C411yescharacteristic
55C2×C4⋊C4×C83C2×C411yescharacteristic
56C2×C42×C83C2311yescharacteristic
57#95×C83—21no—
58C23×D83—21no—
59C2×D332—21no—
60C2×D332—21no—
61C2×D332—41no—
62C2×D332—41no—
63C22×Dic83—21no—
64C22×Dic83C22⋊C411yescharacteristic
65C22×Dic83C22⋊C411yescharacteristic
66C22×Dic83C22⋊C411yescharacteristic
67C22×Dic83C22⋊C411yescharacteristic
68C2×C83⋊C8—22no—
69C2×C83⋊C8—41no—
70C2×C83⋊C8—41no—
71C2×C83⋊C8—41no—
72C2×C83⋊C8—41no—
73C2×C83⋊C8—41no—
74C4⋊Dic83—42no—
75C4⋊Dic83—41no—
76Dic83⋊C4—42no—
77(C2×D83)⋊C4—42no—
78(C2×D83)⋊C4—41no—
79(C2×D83)⋊C4—41no—
80(C2×D83)⋊C4—41no—
81(C2×D83)⋊C4—41no—
82(C2×D83)⋊C4—42no—
83(C2×D83)⋊C4—42no—
84(C2×D83)⋊C4—42no—
85C4×Dic83—21no—
86C4×Dic83—21no—
87C4×Dic83—21no—
88C4×Dic83—21no—
89C4×Dic83—22no—
90C4×Dic83—41no—
91C4×Dic83—41no—
92C4×Dic83D812yes—
93C4×Dic83SD1612yes—
94C22×C4×C83—21no—
95C22×C4×C83—21no—
96C22×C4×C83C22×C411yescharacteristic
97C22×C4×C83C2×D411yescharacteristic
98C22×C4×C83C2×D411yescharacteristic
99C2×C8×C83—22no—
100C2×C8×C83—41no—
101C2×C8×C83—41no—
102C2×C8×C83—41no—
103C2×C8×C83—41no—
104C2×C8×C83—41no—
105C4⋊C4×C83—21no—
106C4⋊C4×C83—21no—
107C4⋊C4×C83—41no—
108C4⋊C4×C83—41no—
109C4⋊C4×C83—41no—
110C42×C83—41no—
111C23×C83C2×C22⋊C411yescharacteristic
112C22×D83—41no—
113C22×D83—41no—
114C22×D83—41no—
115C22×D83—41no—
116C22×D83—41no—
117C22×D83—41no—
118C22×D83—41no—
119C2×C4×C83—21no—
120C2×C4×C83—21no—
121C2×C4×C83—21no—
122C2×C4×C83—21no—
123C2×C4×C83—21no—
124C2×C4×C83—21no—
125C2×C4×C83—41no—
126C2×C4×C83—41no—
127C2×C4×C83—41no—
128C2×C4×C83—41no—
129C2×C4×C83—41no—
130C2×C4×C83—41no—
131C2×C4×C83#13111yescharacteristic
132C2×C4×C83#13111yescharacteristic
133C2×C4×C83C2×D811yescharacteristic
134C2×C4×C83C2×SD1611yescharacteristic
135C2×C4×C83#13511yescharacteristic
136C2×C4×C83#13511yescharacteristic
137C2×Dic83—21no—
138C2×Dic83—21no—
139C2×Dic83—21no—
140C2×Dic83—21no—
141C2×Dic83—21no—
142C2×Dic83—21no—
143C2×Dic83—21no—
144C2×Dic83—21no—
145C2×Dic83—21no—
146C2×Dic83—41no—
147C2×Dic83—41no—
148C2×Dic83—41no—
149C2×Dic83#10212yes—
150C2×Dic83#10212yes—
151C2×Dic83#10412yes—
152D332—81no—
153D332—81no—
154C8×C83—42no—
155C8×C83—81no—
156C83⋊C8—42no—
157C83⋊C8—81no—
158C22×C83#40811yescharacteristic
159C22×C83#41011yescharacteristic
160C22×C83#41311yescharacteristic
161C22×C83C2×#10211yescharacteristic
162C22×C83#41611yescharacteristic
163C22×C83C2×#10411yescharacteristic
164C22×C83#42111yescharacteristic
165C2×D83—81no—
166C2×D83—81no—
167C2×D83—81no—
168C2×D83—81no—
169C2×D83—81no—
170C2×D83—81no—
171C2×D83—81no—
172C4×C83—21no—
173C4×C83—21no—
174C4×C83—21no—
175C4×C83—21no—
176C4×C83—21no—
177C4×C83—21no—
178C4×C83—81no—
179C4×C83—81no—
180Dic83—22no—
181Dic83—22no—
182Dic83—22no—
183Dic83—41no—
184Dic83—81no—
185#7666—831nomaximal, Sylow
186C2×C83#147311yescharacteristic
187C2×C83#148211yescharacteristic
188C2×C83#149211yescharacteristic
189C2×C83#150011yescharacteristic
190C2×C83#151711yescharacteristic
191C2×C83#153211yescharacteristic
192C2×C83#154011yescharacteristic
193D83—161no—
194#2860—831no—
195#1831—831no—
196#1744—831no—
197#1271—831no—
198#1271D8311yescharacteristic, complemented
199#1270—831no—
200#1270—831no—
201C83#766611yescharacteristic, complemented, Sylow
202C2×#126—1661no—
203C2×C4⋊C8—1661no—
204C2×C4×C8—1661no—
205#390—831no—
206#390—831no—
207#390—831no—
208#389—831no—
209#389—831no—
210#389C2×D8311yescharacteristic
211#388—1661no—
212#388—1661no—
213#388—1661no—
214C43—831no—
215#336—831no—
216#336—831no—
217#336—1661no—
218#336—1661no—
219#336C2×D8311yescharacteristic
220#336C2×D8311yescharacteristic
221C22×D4—831no—
222C22×C8—21no—
223C22×C8—21no—
224C22×C8—1661no—
225C22×C8—1661no—
226#126—3322no—
227#126—3322no—
228C2×C4⋊C4—21no—
229C2×C4⋊C4—831no—
230C2×C4⋊C4—1661no—
231C2×C4⋊C4—1661no—
232C2×C4⋊C4C4×D8311yescharacteristic
233C2×C4⋊C4C4×D8311yescharacteristic
234C2×C22⋊C4—1661no—
235C2×C22⋊C4—1661no—
236C2×C22⋊C4—1661no—
237C2×C22⋊C4—1661no—
238C2×C22⋊C4—1661no—
239C2×C22⋊C4—1661no—
240C2×C42—831no—
241C2×C42—831no—
242C2×C42—831no—
243C2×C42—831no—
244C2×C42—831no—
245C2×C42—831no—
246C2×C42C22×D8311yescharacteristic
247C4⋊C8—3322no—
248C4×C8—1664no—
249#95—21no—
250#95—1661no—
251#95—1661no—
252C24—1661no—
253C2×D4—3321no—
254C2×D4—3321no—
255C2×D4—1661no—
256C2×D4—1661no—
257C22×C4—21no—
258C22×C4—21no—
259C22×C4—831no—
260C22×C4—831no—
261C22×C4—831no—
262C22×C4—831no—
263C22×C4—1661no—
264C22×C4C2×C4×D8311yescharacteristic
265C22×C4D4×D8311yescharacteristic
266C22×C4D4×D8311yescharacteristic
267C2×C8—22no—
268C2×C8—41no—
269C2×C8—41no—
270C2×C8—41no—
271C2×C8—41no—
272C2×C8—41no—
273C2×C8—3321no—
274C2×C8—3321no—
275C2×C8—3321no—
276C2×C8—3321no—
277C2×C8—3321no—
278C2×C8—1662no—
279C4⋊C4—21no—
280C4⋊C4—21no—
281C4⋊C4—41no—
282C4⋊C4—41no—
283C4⋊C4—41no—
284C4⋊C4—3321no—
285C4⋊C4—3322no—
286C4⋊C4—3322no—
287C22⋊C4—3322no—
288C22⋊C4—3322no—
289C22⋊C4—3322no—
290C22⋊C4—3322no—
291C22⋊C4—3321no—
292C22⋊C4—3321no—
293C22⋊C4—3321no—
294C22⋊C4—3321no—
295C42—41no—
296C42—3321no—
297C42—3321no—
298C42—832no—
299C42—832no—
300C42—1662no—
301C42—1661no—
302C42—1661no—
303C42—1661no—
304C42—1661no—
305C23—3321no—
306C23—3321no—
307C23—3321no—
308C23—3321no—
309C23—3321no—
310C23—3321no—
311C23—3321no—
312C23C22⋊C4×D8311yescharacteristic
313D4—6641no—
314D4—6641no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—41no—
327C2×C4—3321no—
328C2×C4—3321no—
329C2×C4—3321no—
330C2×C4—832no—
331C2×C4—832no—
332C2×C4—832no—
333C2×C4—1661no—
334C2×C4—1661no—
335C2×C4—1661no—
336C2×C4—1661no—
337C2×C4—1661no—
338C2×C4—1661no—
339C2×C4—1661no—
340C2×C4—1661no—
341C2×C4—1661no—
342C2×C4#1458179811yescharacteristic
343C2×C4#1458180011yescharacteristic
344C2×C4D8×D8311yescharacteristic
345C2×C4SD16×D8311yescharacteristic
346C2×C4#1458181311yescharacteristic
347C2×C4#1458181511yescharacteristic
348C8—42no—
349C8—81no—
350C8—3322no—
351C8—6641no—
352C22—6641no—
353C22—6641no—
354C22—6641no—
355C22—6641no—
356C22—6641no—
357C22—6641no—
358C22—6641no—
359C22#1912766611yescharacteristic
360C22#1912767311yescharacteristic
361C22#1912768411yescharacteristic
362C22#102×D8311yescharacteristic
363C22#1912769611yescharacteristic
364C22#104×D8311yescharacteristic
365C22#1912771611yescharacteristic
366C4—21no—
367C4—21no—
368C4—21no—
369C4—21no—
370C4—21no—
371C4—21no—
372C4—81no—
373C4—81no—
374C4—3321no—
375C4—6641no—
376C4—1662no—
377C4—1662no—
378C4—1662no—
379C2—13281no—
380C2#2490583111yescharacteristic
381C2#2490587011yescharacteristic
382C2#2490591211yescharacteristic
383C2#2490594811yescharacteristic
384C2#2490602111yescharacteristic
385C2#2490608211yescharacteristic
386C2#2490611111yescharacteristic
subgroupG/H
centerC23C22⋊C4×D83
commutatorC22×C4×C83C22×C4
FrattiniC2×C42C22×D83
FittingC83×#1271C2
socleC23×C83C2×C22⋊C4
prime radicalC22×D332C2×C4
cosocleC2×C42×C83C23
hypercenter#1271D83
hypocenterC83#7666
LCSG ▷ C22×C4×C83 ▷ C22×C83 ▷ C83
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1271

Complements

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

NH ≅ G/Ntype
2. #24907578C2semidirect
3. #24907207C2semidirect
4. #24905286C2semidirect
7. C83×#1271C2semidirect
198. #1271D83semidirect
201. C83#7666semidirect

Maximal subgroups

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

subgroupindex
#249158312
#249075782
#249072072
#249052862
#249052832
C83×#12712
#766683