Finite Group Databasebeta

← #35308216

Subgroups — #35308216

Order 21248.

522 conjugacy classes of subgroups.

The subgroup lattice has 428 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19127901C211yesmaximal, characteristic
2#24908269C211yesmaximal, characteristic
3#24907468C211yesmaximal, characteristic, complemented
4C2×#19127452C211yesmaximal, characteristic, complemented
5C2×#19127444C211yesmaximal, characteristic
6#24905347C211yesmaximal, characteristic, complemented
7C83×#1984C211yesmaximal, characteristic, complemented
8C22×#13050577—21no—
9C2×#14581789C2211yescharacteristic
10#19127901—22no—
11#19127901—22no—
12C2×C4⋊(C83⋊C8)C2211yescharacteristic
13C2×C4⋊(C83⋊C8)C2211yescharacteristic
14C2×#14581716—21no—
15C2×#14581714—21no—
16C2×#14581714—21no—
17C2×#14581712—21no—
18#19127571C2211yescharacteristic
19#19127463—21no—
20#19127452—22no—
21#19127444—22no—
22C2×C4×Q8×C83C2211yescharacteristic
23C2×C4⋊C8×C83C2211yescharacteristic
24C2×C4×C8×C83C2211yescharacteristic
25#336×C83—21no—
26#336×C83—21no—
27C22×D332—21no—
28C22×D332—21no—
29C22×D332—21no—
30C2×#13050577—24no—
31C2×#13050577—42no—
32C2×#13050577—42no—
33C22×C83⋊C8—21no—
34C22×C83⋊C8—21no—
35#14581789—21no—
36#14581789—21no—
37#14581789—21no—
38#14581789D412yes—
39C2×C4⋊Dic83—21no—
40C2×C4⋊Dic83—21no—
41C2×C4⋊Dic83—21no—
42C4⋊(C83⋊C8)—21no—
43C4⋊(C83⋊C8)—21no—
44C4⋊(C83⋊C8)—21no—
45C4⋊(C83⋊C8)D412yes—
46#14581716—42no—
47#14581714—42no—
48#14581714—42no—
49#14581712—24no—
50C22×Q8×C83D411yescharacteristic
51C22×C8×C83—21no—
52C22×C8×C83D411yescharacteristic
53C22×C8×C83D411yescharacteristic
54C4×Q8×C83—21no—
55C4×Q8×C83—21no—
56C4×Q8×C83—21no—
57C4×Q8×C83—21no—
58C2×C4⋊C4×C83—21no—
59C2×C4⋊C4×C83D411yescharacteristic
60C2×C42×C83—21no—
61C2×C42×C83C2311yescharacteristic
62C4⋊C8×C83—21no—
63C4⋊C8×C83—21no—
64C4×C8×C83—21no—
65C4×C8×C83—21no—
66C23×D83—41no—
67C2×D332—22no—
68C2×D332—22no—
69C2×D332—22no—
70C2×D332—41no—
71C2×D332—41no—
72C2×D332—41no—
73C2×D332—41no—
74C2×D332—41no—
75C2×D332—41no—
76C2×D332—41no—
77C2×D332—41no—
78C2×D332—41no—
79C2×D332—41no—
80C2×D332—41no—
81C2×D332—81no—
82C2×D332—81no—
83C22×Dic83—41no—
84#13050577—48no—
85C2×C83⋊C8—22no—
86C2×C83⋊C8—41no—
87C2×C83⋊C8—41no—
88C2×C83⋊C8—41no—
89C2×C83⋊C8—41no—
90C2×C83⋊C8—41no—
91C4⋊Dic83—41no—
92C4⋊Dic83—41no—
93C4⋊Dic83—41no—
94C4⋊Dic83—41no—
95C4⋊Dic83—81no—
96C2×Q8×C83—21no—
97C2×Q8×C83—21no—
98C2×Q8×C83—21no—
99C2×Q8×C83—21no—
100C2×Q8×C83—21no—
101C2×Q8×C83SD1612yes—
102C22×C4×C83—21no—
103C22×C4×C83—21no—
104C22×C4×C83C2×D411yescharacteristic
105C22×C4×C83C2×D411yescharacteristic
106C22×C4×C83#4111yescharacteristic
107C2×C8×C83—21no—
108C2×C8×C83—21no—
109C2×C8×C83—21no—
110C2×C8×C83—21no—
111C2×C8×C83—21no—
112C2×C8×C83—41no—
113C2×C8×C83—41no—
114C2×C8×C83—41no—
115C2×C8×C83SD1612yescomplemented
116C4⋊C4×C83—21no—
117C4⋊C4×C83—21no—
118C4⋊C4×C83—41no—
119C4⋊C4×C83—41no—
120C42×C83—41no—
121C42×C83—41no—
122C42×C83C2×D411yescharacteristic
123C42×C83C2×D411yescharacteristic
124C42×C83#4111yescharacteristic
125C42×C83#4111yescharacteristic
126C23×C83#12111yescharacteristic
127C22×D83—42no—
128C22×D83—81no—
129C22×D83—81no—
130C22×D83—81no—
131C22×D83—81no—
132C22×D83—81no—
133C22×D83—81no—
134Q8×C83—22no—
135Q8×C83—22no—
136Q8×C83—41no—
137Q8×C83—41no—
138C2×C4×C83—21no—
139C2×C4×C83—21no—
140C2×C4×C83—22no—
141C2×C4×C83—41no—
142C2×C4×C83—41no—
143C2×C4×C83—41no—
144C2×C4×C83—41no—
145C2×C4×C83—41no—
146C2×C4×C83#12011yescharacteristic
147C2×C4×C83#12111yescharacteristic
148C2×C4×C83#12111yescharacteristic
149C2×C4×C83#12311yescharacteristic
150C2×C4×C83#12311yescharacteristic
151C2×C4×C83#12411yescharacteristic
152C2×C4×C83C2×SD1611yescharacteristic
153C2×C4×C83C2×SD1611yescharacteristic
154C2×C4×C83#13511yescharacteristic
155C2×C4×C83#13511yescharacteristic
156C2×C4×C83AGL(1,ℤ/8)11yescharacteristic
157C2×C4×C83AGL(1,ℤ/8)11yescharacteristic
158C2×C4×C83AGL(1,ℤ/8)11yescharacteristic
159C2×C4×C83#13711yescharacteristic
160C2×Dic83—81no—
161C2×Dic83—81no—
162C2×Dic83—81no—
163D332—42no—
164D332—42no—
165D332—42no—
166D332—42no—
167D332—82no—
168D332—82no—
169D332—81no—
170D332—81no—
171D332—81no—
172D332—81no—
173C8×C83—22no—
174C8×C83—41no—
175C8×C83—81no—
176C83⋊C8—42no—
177C83⋊C8—81no—
178C22×C83#39411yescharacteristic
179C22×C83#46011yescharacteristic
180C22×C83#46411yescharacteristic
181C22×C83#46511yescharacteristic
182C22×C83#46911yescharacteristic
183C22×C83#47111yescharacteristic
184C22×C83#47211yescharacteristic
185C2×D83—82no—
186C2×D83—82no—
187C2×D83—82no—
188C2×D83—161no—
189C2×D83—161no—
190C2×D83—161no—
191C2×D83—161no—
192C4×C83—21no—
193C4×C83—21no—
194C4×C83—42no—
195C4×C83—81no—
196C4×C83#44911yescharacteristic
197C4×C83#45011yescharacteristic
198C4×C83#46111yescharacteristic
199C4×C83#46311yescharacteristic
200C4×C83#46511yescharacteristic
201C4×C83#46811yescharacteristic
202C4×C83#48111yescharacteristic
203C4×C83#48211yescharacteristic
204Dic83—161no—
205#10475—831nomaximal, Sylow
206C2×C83#166611yescharacteristic
207C2×C83#167811yescharacteristic
208C2×C83#168811yescharacteristic
209C2×C83#170211yescharacteristic
210C2×C83#203411yescharacteristic
211C2×C83#205111yescharacteristic
212C2×C83#205511yescharacteristic
213D83—162no—
214C2×#460—831no—
215#1984D8311yescharacteristic, complemented
216#1957—831no—
217#1803—831no—
218C2×#334—831no—
219C2×#331—831no—
220#1296—831no—
221C83#1047511yescharacteristic, complemented, Sylow
222C22×SD16—1661no—
223C2×#127—831no—
224C2×C4×Q8C2×D8311yescharacteristic
225#460—1662no—
226#460—1662no—
227C2×C4⋊C8—831no—
228C2×C4⋊C8—831no—
229C2×C4⋊C8C2×D8311yescharacteristic
230C2×#103—1661no—
231C2×#102—1661no—
232C2×#102—1661no—
233C2×#102—1661no—
234C2×C4×C8C2×D8311yescharacteristic
235#384—831no—
236#340—1661no—
237#336—21no—
238#336—21no—
239#334—1662no—
240#331—1662no—
241C22×Q8D33211yescharacteristic
242C22×D4—1661no—
243C22×D4—1661no—
244C22×D4—1661no—
245C2×SD16—3322no—
246C2×SD16—3322no—
247C2×SD16—1664no—
248C22×C8—21no—
249C22×C8—1661no—
250C22×C8—1661no—
251C22×C8C22⋊D8311yescharacteristic
252C22×C8C22⋊D8311yescharacteristic
253#127—832no—
254#127—1661no—
255#127—1661no—
256#127—1661no—
257C4×Q8—21no—
258C4×Q8—21no—
259C4×Q8—21no—
260C4×Q8—21no—
261C2×C4⋊C4—21no—
262C2×C4⋊C4—1661no—
263C2×C4⋊C4—1661no—
264C2×C4⋊C4—1661no—
265C2×C4⋊C4D33211yescharacteristic
266C2×C42—21no—
267C2×C42C22×D8311yescharacteristic
268C4⋊C8—21no—
269C4⋊C8—21no—
270C4⋊C8—832no—
271C4⋊C8—1661no—
272C4⋊C8—1661no—
273C4⋊C8—1661no—
274#103—3322no—
275#102—3322no—
276#102—3322no—
277#102—1664no—
278C4×C8—21no—
279C4×C8—21no—
280C24—3321no—
281C2×Q8—21no—
282C2×Q8—21no—
283C2×Q8—21no—
284C2×Q8—21no—
285C2×Q8—21no—
286C2×Q8#1305057912yes—
287C2×D4—3321no—
288C2×D4—3321no—
289C2×D4—3321no—
290C2×D4—3321no—
291C2×D4—3321no—
292C2×D4—3321no—
293C2×D4—3321no—
294C2×D4—3321no—
295C2×D4—3321no—
296C2×D4—3321no—
297C2×D4—3321no—
298C2×D4—6641no—
299C2×D4—6641no—
300C2×D4—1662no—
301C2×D4—1662no—
302C2×D4—1662no—
303C22×C4—21no—
304C22×C4—21no—
305C22×C4—3321no—
306C22×C4C2×C22⋊D8311yescharacteristic
307C22×C4C2×D33211yescharacteristic
308C22×C4#1305059011yescharacteristic
309SD16—3328no—
310C2×C8—21no—
311C2×C8—21no—
312C2×C8—21no—
313C2×C8—21no—
314C2×C8—21no—
315C2×C8—41no—
316C2×C8—41no—
317C2×C8—41no—
318C2×C8—3321no—
319C2×C8—3321no—
320C2×C8—3321no—
321C2×C8—3321no—
322C2×C8—3321no—
323C2×C8—1662no—
324C2×C8#1305057712yescomplemented
325C4⋊C4—21no—
326C4⋊C4—21no—
327C4⋊C4—41no—
328C4⋊C4—41no—
329C4⋊C4—3321no—
330C4⋊C4—3321no—
331C4⋊C4—3321no—
332C4⋊C4—3321no—
333C4⋊C4—6641no—
334C42—41no—
335C42—41no—
336C42D4×D8311yescharacteristic
337C42D4×D8311yescharacteristic
338C42#1305058911yescharacteristic
339C42#1305058911yescharacteristic
340C23—3322no—
341C23—6641no—
342C23—6641no—
343C23—6641no—
344C23—6641no—
345C23—6641no—
346C23—6641no—
347C23#1458176611yescharacteristic
348Q8—22no—
349Q8—22no—
350Q8—41no—
351Q8—41no—
352D4—3322no—
353D4—3322no—
354D4—3322no—
355D4—3322no—
356D4—6642no—
357D4—6642no—
358D4—6641no—
359D4—6641no—
360D4—6641no—
361D4—6641no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—22no—
365C2×C4—41no—
366C2×C4—41no—
367C2×C4—41no—
368C2×C4—41no—
369C2×C4—41no—
370C2×C4—6641no—
371C2×C4—6641no—
372C2×C4—6641no—
373C2×C4#1458176211yescharacteristic
374C2×C4#1458176311yescharacteristic
375C2×C4#1458176411yescharacteristic
376C2×C4#1458177411yescharacteristic
377C2×C4#1458177611yescharacteristic
378C2×C4#1458177811yescharacteristic
379C2×C4C2×#1305057711yescharacteristic
380C2×C4C2×#1305057911yescharacteristic
381C2×C4#1458181211yescharacteristic
382C2×C4#1458181411yescharacteristic
383C2×C4#1458181711yescharacteristic
384C2×C4#1458182011yescharacteristic
385C2×C4#1458182011yescharacteristic
386C2×C4#1458182311yescharacteristic
387C8—22no—
388C8—41no—
389C8—81no—
390C8—3322no—
391C8—6641no—
392C22—13281no—
393C22—13281no—
394C22—13281no—
395C22—13281no—
396C22—6642no—
397C22—6642no—
398C22—6642no—
399C22#1912761711yescharacteristic
400C22#1912790111yescharacteristic
401C22#1912792311yescharacteristic
402C22#1912793011yescharacteristic
403C22#1912795411yescharacteristic
404C22#1912796411yescharacteristic
405C22#1912797111yescharacteristic
406C4—21no—
407C4—21no—
408C4—42no—
409C4—81no—
410C4—13281no—
411C4#1912784811yescharacteristic
412C4#1912785411yescharacteristic
413C4#1912790311yescharacteristic
414C4#1912791311yescharacteristic
415C4#1912793211yescharacteristic
416C4#1912794911yescharacteristic
417C4#1912802011yescharacteristic
418C4#1912802611yescharacteristic
419C2—13282no—
420C2#2490681111yescharacteristic
421C2#2490687911yescharacteristic
422C2#2490694711yescharacteristic
423C2#2490703111yescharacteristic
424C2#2490865311yescharacteristic
425C2#2490875411yescharacteristic
426C2#2490877811yescharacteristic
subgroupG/H
centerC23#14581766
commutator, cosocleC2×C42×C83C23
FrattiniC2×C42C22×D83
FittingC83×#1984C2
socleC23×C83#121
prime radicalC2×#14581789C22
hypercenter#1984D83
hypocenterC83#10475
LCSG ▷ C2×C42×C83 ▷ C22×C83 ▷ C83
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1984

Complements

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

NH ≅ G/Ntype
3. #24907468C2semidirect
4. C2×#19127452C2semidirect
6. #24905347C2semidirect
7. C83×#1984C2semidirect
115. C2×C8×C83SD16semidirect
215. #1984D83semidirect
221. C83#10475semidirect
324. C2×C8#13050577semidirect

Maximal subgroups

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

subgroupindex
C2×#191279012
#249082692
#249074682
C2×#191274522
C2×#191274442
#249053472
C83×#19842
#1047583