Finite Group Databasebeta

← #35308809

Subgroups — #35308809

Order 21248.

560 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 560 conjugacy classes of subgroups — too many to draw. Only its 121 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#24907655C212yesmaximal
2#24905719C211yesmaximal, characteristic
3#24905341C212yesmaximal
4#24905306C211yesmaximal, characteristic
5C83×#2659C211yesmaximal, characteristic
6C22×C4⋊Dic83C2211yescharacteristic
7C2×#14581723—22no—
8C2×C22⋊(C83⋊C8)C2212yes—
9C2×C83⋊#95—21no—
10C2×C83⋊#95—21no—
11C2×C83⋊#95C2211yescharacteristic
12#19127463—22no—
13C2×C22⋊Q8×C83—21no—
14#398×C83—22no—
15#395×C83—22no—
16#380×C83—21no—
17#380×C83C2212yes—
18C2×#95×C83C2211yescharacteristic
19C23×Dic83—21no—
20C23×Dic83—21no—
21C22×C83⋊C8—22no—
22C2×C4⋊Dic83—22no—
23C2×C4⋊Dic83—42no—
24#14581723—44no—
25C22⋊(C83⋊C8)—22no—
26C22⋊(C83⋊C8)Q814yescomplemented
27C83⋊#95—21no—
28C83⋊#95—42no—
29C83⋊#95—42no—
30C83⋊#95Q812yescomplemented
31C22×Q8×C83—21no—
32C23×C4×C83D411yescharacteristic
33C23×C4×C83D411yescharacteristic
34C23×C4×C83C2311yescharacteristic
35C22⋊Q8×C83—42no—
36C22⋊Q8×C83—42no—
37C2×C4⋊C4×C83—22no—
38C2×C4⋊C4×C83—21no—
39C2×C4⋊C4×C83—21no—
40C2×C4⋊C4×C83—21no—
41C2×C4⋊C4×C83D412yes—
42C2×C4⋊C4×C83D412yes—
43C2×C22⋊C4×C83—22no—
44#95×C83—22no—
45#95×C83—22no—
46#95×C83—22no—
47C22×Dic83—22no—
48C22×Dic83—22no—
49C22×Dic83—41no—
50C22×Dic83—41no—
51C22×Dic83—41no—
52C22×Dic83—41no—
53C22×Dic83—41no—
54C22×Dic83—41no—
55C22×Dic83—41no—
56C22×Dic83—41no—
57C22×Dic83—41no—
58C22×Dic83—41no—
59C22×Dic83—41no—
60C22×Dic83—41no—
61C2×C83⋊C8—24no—
62C2×C83⋊C8—42no—
63C2×C83⋊C8—42no—
64C4⋊Dic83—24no—
65C4⋊Dic83—42no—
66C24×C83C2×D411yescharacteristic
67C2×Q8×C83—42no—
68C2×Q8×C83—81no—
69C2×Q8×C83—81no—
70C22×C4×C83—21no—
71C22×C4×C83—21no—
72C22×C4×C83—21no—
73C22×C4×C83—21no—
74C22×C4×C83—22no—
75C22×C4×C83—22no—
76C22×C4×C83—41no—
77C22×C4×C83—41no—
78C22×C4×C83—41no—
79C22×C4×C83—41no—
80C22×C4×C83—41no—
81C22×C4×C83—41no—
82C22×C4×C83SD1612yes—
83C22×C4×C83SD1612yes—
84C22×C4×C83C2×D412yes—
85C22×C4×C83C2×Q812yes—
86C22×C4×C83C2×Q811yescharacteristic
87C22×C4×C83#4111yescharacteristic
88C4⋊C4×C83—42no—
89C4⋊C4×C83—42no—
90C4⋊C4×C83—42no—
91C4⋊C4×C83—44no—
92C4⋊C4×C83—81no—
93C4⋊C4×C83—81no—
94C22⋊C4×C83—42no—
95C22⋊C4×C83—42no—
96C23×C83—21no—
97C23×C83—21no—
98C23×C83—21no—
99C23×C83—21no—
100C23×C83#12011yescharacteristic
101C23×C83C22⋊Q811yescharacteristic
102C23×C83C4⋊Q811yescharacteristic
103C23×C83C2×SD1611yescharacteristic
104C23×C83C2×SD1611yescharacteristic
105C23×C83AGL(1,ℤ/8)11yescharacteristic
106C23×C83#13711yescharacteristic
107Q8×C83—84no—
108C2×C4×C83—22no—
109C2×C4×C83—22no—
110C2×C4×C83—22no—
111C2×C4×C83—22no—
112C2×C4×C83—42no—
113C2×C4×C83—41no—
114C2×C4×C83—41no—
115C2×C4×C83—41no—
116C2×C4×C83—41no—
117C2×C4×C83—41no—
118C2×C4×C83—41no—
119C2×C4×C83—42no—
120C2×C4×C83—42no—
121C2×C4×C83—42no—
122C2×C4×C83—42no—
123C2×C4×C83—42no—
124C2×C4×C83—42no—
125C2×C4×C83—42no—
126C2×C4×C83—42no—
127C2×C4×C83—81no—
128C2×C4×C83—81no—
129C2×C4×C83C22⋊Q812yes—
130C2×C4×C83C4⋊Q812yes—
131C2×Dic83—42no—
132C2×Dic83—42no—
133C2×Dic83—42no—
134C2×Dic83—42no—
135C2×Dic83—42no—
136C2×Dic83—42no—
137C2×Dic83—81no—
138C2×Dic83—81no—
139C2×Dic83—81no—
140C2×Dic83—81no—
141C2×Dic83—81no—
142C2×Dic83—81no—
143C2×Dic83—81no—
144C2×Dic83—81no—
145C83⋊C8—44no—
146C22×C83—21no—
147C22×C83—21no—
148C22×C83—21no—
149C22×C83—21no—
150C22×C83—21no—
151C22×C83—21no—
152C22×C83—21no—
153C22×C83—21no—
154C22×C83—21no—
155C22×C83—21no—
156C22×C83—21no—
157C22×C83—21no—
158C22×C83#39511yescharacteristic
159C22×C83#44811yescharacteristic
160C22×C83#45011yescharacteristic
161C22×C83#45411yescharacteristic
162C22×C83#45511yescharacteristic
163C22×C83B(4,𝔽2)11yescharacteristic
164C22×C83#45811yescharacteristic
165C22×C83#47511yescharacteristic
166C22×C83#47611yescharacteristic
167C22×C83#49811yescharacteristic
168C22×C83#50111yescharacteristic
169C4×C83—22no—
170C4×C83—22no—
171C4×C83—42no—
172C4×C83—42no—
173C4×C83—82no—
174C4×C83—82no—
175Dic83—82no—
176Dic83—82no—
177#10568—831nomaximal, Sylow
178C2×C83—21no—
179C2×C83—21no—
180C2×C83—21no—
181C2×C83—21no—
182C2×C83#159911yescharacteristic
183C2×C83#160111yescharacteristic
184C2×C83#160811yescharacteristic
185C2×C83#161611yescharacteristic
186C2×C83#205811yescharacteristic
187C2×C83#206111yescharacteristic
188C2×C83#206211yescharacteristic
189#2659D8311yescharacteristic
190#1853—832no—
191#1442—831no—
192#1294—832no—
193#1280—831no—
194C83#1056811yescharacteristic, complemented, Sylow
195C2×C22⋊Q8—21no—
196C22×C4⋊C4—831no—
197C2×#106—1662no—
198C2×C22⋊C8—832no—
199#398—22no—
200#395—22no—
201#380—21no—
202#380C2×D8312yes—
203C2×#95—831no—
204C2×#95—1661no—
205C2×#95—1661no—
206C2×#95C2×D8311yescharacteristic
207#340—1662no—
208C22×Q8—21no—
209C23×C4—1661no—
210C23×C4—1661no—
211C23×C4C22⋊D8311yescharacteristic
212C23×C4C22⋊D8311yescharacteristic
213C23×C4C22×D8311yescharacteristic
214C22×C8—1662no—
215C22⋊Q8—42no—
216C22⋊Q8—42no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4—21no—
221C2×C4⋊C4—3322no—
222C2×C4⋊C4—1662no—
223C2×C4⋊C4C22⋊D8312yes—
224C2×C4⋊C4C22⋊D8312yes—
225C2×C22⋊C4—22no—
226#106—3324no—
227C22⋊C8—834no—
228C22⋊C8—1662no—
229#95—22no—
230#95—22no—
231#95—22no—
232#95—3322no—
233#95—3322no—
234#95—832no—
235#95—1661no—
236C24C2×C22⋊D8311yescharacteristic
237C2×Q8—42no—
238C2×Q8—81no—
239C2×Q8—81no—
240C22×C4—21no—
241C22×C4—21no—
242C22×C4—21no—
243C22×C4—21no—
244C22×C4—22no—
245C22×C4—22no—
246C22×C4—41no—
247C22×C4—41no—
248C22×C4—41no—
249C22×C4—41no—
250C22×C4—41no—
251C22×C4—41no—
252C22×C4—3321no—
253C22×C4—3321no—
254C22×C4—3321no—
255C22×C4—3321no—
256C22×C4—3321no—
257C22×C4—3321no—
258C22×C4—3321no—
259C22×C4—3321no—
260C22×C4—3321no—
261C22×C4—3321no—
262C22×C4—3321no—
263C22×C4—3321no—
264C22×C4—1662no—
265C22×C4—1662no—
266C22×C4#1305057712yes—
267C22×C4#1305057812yes—
268C22×C4C2×C22⋊D8312yes—
269C22×C4Q8×D8312yes—
270C22×C4Q8×D8311yescharacteristic
271C22×C4#1305059111yescharacteristic
272C2×C8—3322no—
273C2×C8—3322no—
274C2×C8—1664no—
275C4⋊C4—42no—
276C4⋊C4—42no—
277C4⋊C4—42no—
278C4⋊C4—44no—
279C4⋊C4—81no—
280C4⋊C4—81no—
281C4⋊C4—3322no—
282C4⋊C4—1664no—
283C22⋊C4—42no—
284C22⋊C4—42no—
285C23—21no—
286C23—21no—
287C23—21no—
288C23—21no—
289C23#1458176111yescharacteristic
290C23#1458176811yescharacteristic
291C23#1458179011yescharacteristic
292C23C2×#1305057711yescharacteristic
293C23C2×#1305057811yescharacteristic
294C23#1458181811yescharacteristic
295C23#1458182311yescharacteristic
296Q8—84no—
297C2×C4—22no—
298C2×C4—22no—
299C2×C4—22no—
300C2×C4—22no—
301C2×C4—42no—
302C2×C4—41no—
303C2×C4—41no—
304C2×C4—41no—
305C2×C4—41no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—42no—
309C2×C4—42no—
310C2×C4—42no—
311C2×C4—42no—
312C2×C4—42no—
313C2×C4—42no—
314C2×C4—42no—
315C2×C4—42no—
316C2×C4—81no—
317C2×C4—81no—
318C2×C4—3322no—
319C2×C4—3322no—
320C2×C4—3322no—
321C2×C4—3322no—
322C2×C4—3322no—
323C2×C4—3322no—
324C2×C4—6641no—
325C2×C4—6641no—
326C2×C4—6641no—
327C2×C4—6641no—
328C2×C4—6641no—
329C2×C4—6641no—
330C2×C4—6641no—
331C2×C4—6641no—
332C2×C4#1458176812yes—
333C2×C4#1458179012yes—
334C8—3324no—
335C22—21no—
336C22—21no—
337C22—21no—
338C22—21no—
339C22—21no—
340C22—21no—
341C22—21no—
342C22—21no—
343C22—21no—
344C22—21no—
345C22—21no—
346C22—21no—
347C22#1912762011yescharacteristic
348C22#1912784011yescharacteristic
349C22#1912785211yescharacteristic
350C22#1912787411yescharacteristic
351C22#1912787911yescharacteristic
352C22#1912788811yescharacteristic
353C22#1912789111yescharacteristic
354C22#1912798311yescharacteristic
355C22#1912798811yescharacteristic
356C22#1912809711yescharacteristic
357C22#1912810711yescharacteristic
358C4—22no—
359C4—22no—
360C4—42no—
361C4—42no—
362C4—82no—
363C4—82no—
364C4—6642no—
365C4—6642no—
366C2—21no—
367C2—21no—
368C2—21no—
369C2—21no—
370C2#2490640311yescharacteristic
371C2#2490641511yescharacteristic
372C2#2490645411yescharacteristic
373C2#2490650011yescharacteristic
374C2#2490879311yescharacteristic
375C2#2490881011yescharacteristic
376C2#2490881511yescharacteristic
subgroupG/H
centerC23#14581761
commutator, cosocleC23×C4×C83C23
FrattiniC23×C4C22×D83
FittingC83×#2659C2
socleC23×C83#120
prime radicalC24×C83C2×D4
hypercenter#2659D83
hypocenterC83#10568
LCSG ▷ C23×C4×C83 ▷ C22×C83 ▷ C83
UCSC1 ◁ C23 ◁ C23×C4 ◁ #2659

Complements

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

NH ≅ G/Ntype
26. C22⋊(C83⋊C8)Q8semidirect
30. C83⋊#95Q8semidirect
194. C83#10568semidirect

Maximal subgroups

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

subgroupindex
#249076552
#249057192
#249053412
#249053062
C83×#26592
#1056883