Finite Group Databasebeta

← #35308210

Subgroups — #35308210

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×#19127902C211yesmaximal, characteristic
2#24908399C211yesmaximal, characteristic, complemented
3#24908269C211yesmaximal, characteristic
4#24907469C211yesmaximal, characteristic, complemented
5C2×#19127444C211yesmaximal, characteristic
6#24905347C211yesmaximal, characteristic, complemented
7C2×#335×C83C211yesmaximal, characteristic, complemented
8C22×#13050579—21no—
9C2×#14581789C2211yescharacteristic
10C2×C4×Dic166C2211yescharacteristic
11#19127902—22no—
12#19127902—22no—
13C2×C4⋊(C83⋊C8)C2211yescharacteristic
14C2×#14581717—21no—
15C2×#14581714—21no—
16C2×#14581714—21no—
17C2×#14581714—21no—
18#19127571C2211yescharacteristic
19#19127464—21no—
20#19127455—21no—
21#19127455—21no—
22#19127444—22no—
23C2×C4⋊C8×C83C2211yescharacteristic
24C2×C4⋊C8×C83C2211yescharacteristic
25C2×C4×C8×C83C2211yescharacteristic
26#335×C83—22no—
27C22×Dic166D411yescharacteristic
28C22×D332—21no—
29C22×D332—21no—
30C22×D332—21no—
31C2×#13050579—24no—
32C2×#13050579—42no—
33C2×#13050579—42no—
34C22×C83⋊C8—21no—
35#14581789—21no—
36#14581789—21no—
37#14581789—21no—
38#14581789D412yes—
39C4×Dic166—21no—
40C4×Dic166—21no—
41C4×Dic166—21no—
42C4×Dic166—21no—
43C2×C4⋊Dic83—21no—
44C2×C4⋊Dic83—21no—
45C2×C4⋊Dic83—21no—
46C2×C4⋊Dic83D411yescharacteristic
47C2×Dic83⋊C4—21no—
48C2×C4×Dic83—21no—
49C4⋊(C83⋊C8)—21no—
50C4⋊(C83⋊C8)—21no—
51#14581717—42no—
52#14581714—24no—
53#14581714—42no—
54#14581714—42no—
55C22×C8×C83—21no—
56C22×C8×C83—21no—
57C22×C8×C83D411yescharacteristic
58C22×C8×C83D411yescharacteristic
59C2×C42×C83C2311yescharacteristic
60C4⋊C8×C83—21no—
61C4⋊C8×C83—21no—
62C4⋊C8×C83—21no—
63C4⋊C8×C83D412yes—
64C4×C8×C83—21no—
65C4×C8×C83—21no—
66C23×D83—41no—
67C2×Dic166—21no—
68C2×Dic166—21no—
69C2×Dic166—21no—
70C2×Dic166—21no—
71C2×Dic166—21no—
72C2×Dic166D812yes—
73C2×D332—22no—
74C2×D332—22no—
75C2×D332—22no—
76C2×D332—41no—
77C2×D332—41no—
78C2×D332—41no—
79C2×D332—41no—
80C2×D332—41no—
81C2×D332—41no—
82C2×D332—41no—
83C2×D332—41no—
84C2×D332—41no—
85C2×D332—41no—
86C2×D332—41no—
87C2×D332—81no—
88C2×D332—81no—
89C22×Dic83—21no—
90C22×Dic83—21no—
91C22×Dic83—41no—
92#13050579—48no—
93C2×C83⋊C8—41no—
94C2×C83⋊C8—41no—
95C2×C83⋊C8—41no—
96C4⋊Dic83—21no—
97C4⋊Dic83—21no—
98C4⋊Dic83—41no—
99C4⋊Dic83—41no—
100C4⋊Dic83—41no—
101C4⋊Dic83—41no—
102C4⋊Dic83—81no—
103Dic83⋊C4—41no—
104Dic83⋊C4—41no—
105C4×Dic83—41no—
106C4×Dic83—41no—
107C22×C4×C83C2×D411yescharacteristic
108C22×C4×C83C2×D411yescharacteristic
109C22×C4×C83#4111yescharacteristic
110C2×C8×C83—21no—
111C2×C8×C83—21no—
112C2×C8×C83—22no—
113C2×C8×C83—21no—
114C2×C8×C83—21no—
115C2×C8×C83—21no—
116C2×C8×C83—41no—
117C2×C8×C83—41no—
118C2×C8×C83—41no—
119C2×C8×C83—41no—
120C2×C8×C83—41no—
121C2×C8×C83SD1612yescomplemented
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—
134C2×C4×C83—21no—
135C2×C4×C83—21no—
136C2×C4×C83#12011yescharacteristic
137C2×C4×C83#12111yescharacteristic
138C2×C4×C83#12111yescharacteristic
139C2×C4×C83#12311yescharacteristic
140C2×C4×C83#12311yescharacteristic
141C2×C4×C83#12411yescharacteristic
142C2×C4×C83C2×D811yescharacteristic
143C2×C4×C83C2×SD1611yescharacteristic
144C2×C4×C83#13511yescharacteristic
145C2×C4×C83#13511yescharacteristic
146C2×C4×C83AGL(1,ℤ/8)11yescharacteristic
147C2×C4×C83AGL(1,ℤ/8)11yescharacteristic
148C2×C4×C83#13711yescharacteristic
149C2×C4×C83#13711yescharacteristic
150C2×Dic83—22no—
151C2×Dic83—41no—
152C2×Dic83—41no—
153C2×Dic83—41no—
154C2×Dic83—41no—
155C2×Dic83—41no—
156C2×Dic83—81no—
157C2×Dic83—81no—
158C2×Dic83—81no—
159D332—42no—
160D332—42no—
161D332—42no—
162D332—42no—
163D332—82no—
164D332—82no—
165D332—81no—
166D332—81no—
167D332—81no—
168D332—81no—
169Dic166—22no—
170Dic166—22no—
171Dic166—41no—
172Dic166—41no—
173C8×C83—22no—
174C8×C83—42no—
175C8×C83—41no—
176C8×C83—81no—
177C83⋊C8—81no—
178C22×C83#39411yescharacteristic
179C22×C83#46011yescharacteristic
180C22×C83#46411yescharacteristic
181C22×C83#46611yescharacteristic
182C22×C83#46811yescharacteristic
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#44711yescharacteristic
195C4×C83#45211yescharacteristic
196C4×C83#46111yescharacteristic
197C4×C83#46311yescharacteristic
198C4×C83#46511yescharacteristic
199C4×C83#46811yescharacteristic
200C4×C83#48011yescharacteristic
201C4×C83#48211yescharacteristic
202Dic83—42no—
203Dic83—81no—
204Dic83—161no—
205#10474—831nomaximal, Sylow
206C2×C83#166811yescharacteristic
207C2×C83#168211yescharacteristic
208C2×C83#168611yescharacteristic
209C2×C83#169811yescharacteristic
210C2×C83#203311yescharacteristic
211C2×C83#205111yescharacteristic
212C2×C83#205511yescharacteristic
213D83—162no—
214C2×#460—831no—
215#1984—831no—
216#1957—831no—
217#1803—831no—
218C2×#335D8311yescharacteristic, complemented
219C2×#331—831no—
220#1296—831no—
221C83#1047411yescharacteristic, complemented, Sylow
222C22×SD16—1661no—
223C2×#127—831no—
224C2×C4×Q8—831no—
225#460—1662no—
226#460—1662no—
227C2×C4⋊C8—831no—
228C2×C4⋊C8C2×D8311yescharacteristic
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—1661no—
238#336—1661no—
239#335—22no—
240#331—1662no—
241C22×Q8—831no—
242C22×D4—1661no—
243C22×D4—1661no—
244C22×D4—1661no—
245C2×SD16—3322no—
246C2×SD16—3322no—
247C2×SD16—1664no—
248C22×C8—21no—
249C22×C8—21no—
250C22×C8—1661no—
251C22×C8D33211yescharacteristic
252C22×C8D33211yescharacteristic
253#127—832no—
254#127—1661no—
255#127—1661no—
256#127—1661no—
257C4×Q8—1661no—
258C4×Q8—1661no—
259C4×Q8—1661no—
260C4×Q8—1661no—
261C2×C4⋊C4—831no—
262C2×C4⋊C4—1661no—
263C2×C4⋊C4—1661no—
264C2×C4⋊C4—1661no—
265C2×C4⋊C4—1661no—
266C2×C42—1661no—
267C2×C42C22×D8311yescharacteristic
268C4⋊C8—21no—
269C4⋊C8—21no—
270C4⋊C8—21no—
271C4⋊C8—1661no—
272C4⋊C8—1661no—
273C4⋊C8C22⋊D8312yes—
274#103—3322no—
275#102—3322no—
276#102—3322no—
277#102—1664no—
278C4×C8—21no—
279C4×C8—21no—
280C24—3321no—
281C2×Q8—832no—
282C2×Q8—1661no—
283C2×Q8—1661no—
284C2×Q8—1661no—
285C2×Q8—1661no—
286C2×Q8—1661no—
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—3321no—
304C22×C4—1661no—
305C22×C4—1661no—
306C22×C4D4×D8311yescharacteristic
307C22×C4C2×D33211yescharacteristic
308C22×C4#1305059111yescharacteristic
309SD16—3328no—
310C2×C8—21no—
311C2×C8—21no—
312C2×C8—22no—
313C2×C8—21no—
314C2×C8—21no—
315C2×C8—21no—
316C2×C8—41no—
317C2×C8—41no—
318C2×C8—41no—
319C2×C8—41no—
320C2×C8—41no—
321C2×C8—3321no—
322C2×C8—3321no—
323C2×C8—3321no—
324C2×C8#1305057912yescomplemented
325C4⋊C4—3321no—
326C4⋊C4—3321no—
327C4⋊C4—3321no—
328C4⋊C4—3321no—
329C4⋊C4—3321no—
330C4⋊C4—3321no—
331C4⋊C4—6641no—
332C4⋊C4—1661no—
333C4⋊C4—1661no—
334C42—3321no—
335C42—3321no—
336C42D4×D8311yescharacteristic
337C42C2×C22⋊D8311yescharacteristic
338C42#1305059011yescharacteristic
339C42#1305059111yescharacteristic
340C23—3322no—
341C23—6641no—
342C23—6641no—
343C23—6641no—
344C23—6641no—
345C23—6641no—
346C23—6641no—
347C23#1458176711yescharacteristic
348Q8—3321no—
349Q8—3321no—
350Q8—1662no—
351Q8—1662no—
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—3321no—
365C2×C4—3321no—
366C2×C4—3321no—
367C2×C4—3321no—
368C2×C4—3321no—
369C2×C4—6641no—
370C2×C4—6641no—
371C2×C4—6641no—
372C2×C4—1662no—
373C2×C4#1458176011yescharacteristic
374C2×C4#1458176611yescharacteristic
375C2×C4#1458176711yescharacteristic
376C2×C4#1458177711yescharacteristic
377C2×C4#1458177711yescharacteristic
378C2×C4#1458178111yescharacteristic
379C2×C4D8×D8311yescharacteristic
380C2×C4C2×#1305057911yescharacteristic
381C2×C4#1458181411yescharacteristic
382C2×C4#1458181511yescharacteristic
383C2×C4#1458181611yescharacteristic
384C2×C4#1458182011yescharacteristic
385C2×C4#1458182411yescharacteristic
386C2×C4#1458182511yescharacteristic
387C8—22no—
388C8—42no—
389C8—41no—
390C8—81no—
391C8—6641no—
392C22—13281no—
393C22—13281no—
394C22—13281no—
395C22—13281no—
396C22—6642no—
397C22—6642no—
398C22—6642no—
399C22#1912761811yescharacteristic
400C22#1912790211yescharacteristic
401C22#1912792411yescharacteristic
402C22#1912793811yescharacteristic
403C22#1912794811yescharacteristic
404C22#1912796511yescharacteristic
405C22#1912797211yescharacteristic
406C4—21no—
407C4—21no—
408C4—13281no—
409C4—3322no—
410C4—6641no—
411C4#1912783311yescharacteristic
412C4#1912786211yescharacteristic
413C4#1912790711yescharacteristic
414C4#1912791711yescharacteristic
415C4#1912793011yescharacteristic
416C4#1912794711yescharacteristic
417C4#1912801611yescharacteristic
418C4#1912802811yescharacteristic
419C2—13282no—
420C2#2490682411yescharacteristic
421C2#2490690811yescharacteristic
422C2#2490693611yescharacteristic
423C2#2490700411yescharacteristic
424C2#2490864711yescharacteristic
425C2#2490875711yescharacteristic
426C2#2490878111yescharacteristic
subgroupG/H
centerC23#14581767
commutator, cosocleC2×C42×C83C23
FrattiniC2×C42C22×D83
FittingC2×#335×C83C2
socleC23×C83#121
prime radicalC2×#14581789C22
hypercenterC2×#335D83
hypocenterC83#10474
LCSG ▷ C2×C42×C83 ▷ C22×C83 ▷ C83
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#335

Complements

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

NH ≅ G/Ntype
2. #24908399C2semidirect
4. #24907469C2semidirect
6. #24905347C2semidirect
7. C2×#335×C83C2semidirect
121. C2×C8×C83SD16semidirect
218. C2×#335D83semidirect
221. C83#10474semidirect
324. C2×C8#13050579semidirect

Maximal subgroups

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

subgroupindex
C2×#191279022
#249083992
#249082692
#249074692
C2×#191274442
#249053472
C2×#335×C832
#1047483