Finite Group Databasebeta

← #35559205

Subgroups — #35559205

Order 21248.

590 conjugacy classes of subgroups.

The subgroup lattice has 424 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#24917044C211yesmaximal, characteristic
2C2×#19127739C211yesmaximal, characteristic, complemented
3#24909672C212yesmaximal, complemented
4#24909666C211yesmaximal, characteristic, complemented
5#24909661C211yesmaximal, characteristic, complemented
6#24909655C212yesmaximal, complemented
7#24909649C211yesmaximal, characteristic, complemented
8#24909644C211yesmaximal, characteristic, complemented
9#24909247C212yesmaximal, complemented
10#24909171C212yesmaximal, complemented
11C2×C22⋊C16×C83C211yesmaximal, characteristic, complemented
12C2×#14581814C2211yescharacteristic
13#19128212—21no—
14C2×#14581751—21no—
15#19127942C2211yescharacteristic
16#19127942C2211yescharacteristic
17#19127937C2211yescharacteristic
18#19127937C2211yescharacteristic
19#19127930—21no—
20#19127930—21no—
21#19127739—21no—
22#19127739—21no—
23#19127739C2211yescharacteristic
24#19127739C2212yes—
25#19127739C2212yes—
26C2×#14581726C2211yescharacteristic
27C2×#14581726C2211yescharacteristic
28C2×#14581724—21no—
29#19127698—21no—
30#19127529C2212yescomplemented
31#19127529C2212yescomplemented
32#19127527C2212yescomplemented
33#19127527C2212yescomplemented
34#19127506C2212yescomplemented
35#19127506C2212yescomplemented
36#19127503C2212yescomplemented
37#19127503C2212yescomplemented
38C23×C8×C83C2211yescharacteristic
39C22×C16×C83C2212yes—
40C22⋊C16×C83C2211yescharacteristic
41C22⋊C16×C83C2211yescharacteristic
42C22⋊C16×C83C2212yes—
43C2×#13050590—21no—
44#14581814—24no—
45C2×Dic332C2311yescharacteristic
46C2×#13050579—21no—
47C2×D664C2311yescharacteristic
48#14581775—41no—
49#14581771—21no—
50#14581771—21no—
51#14581766—21no—
52#14581766—21no—
53#14581761—41no—
54C4×C22⋊D83—41no—
55C4×C22⋊D83—41no—
56#14581751—22no—
57#14581751—22no—
58#14581751—21no—
59#14581751—41no—
60#14581751—41no—
61C2×C4⋊Dic83—21no—
62C2×C4⋊Dic83—21no—
63C2×C22⋊Dic83—41no—
64C2×C4×Dic83—41no—
65#14581726—21no—
66#14581726—21no—
67#14581726C2311yescharacteristic
68#14581726C2311yescharacteristic
69#14581726C2312yes—
70#14581726C2312yes—
71#14581724—21no—
72#14581724—22no—
73#14581724—22no—
74#14581717—21no—
75#14581717—21no—
76#14581714—21no—
77#14581714—21no—
78C23×C4×C83D411yescharacteristic
79C22×C8×C83—24no—
80C22×C8×C83D411yescharacteristic
81C22×C8×C83D411yescharacteristic
82C22×C8×C83D411yescharacteristic
83C22×C8×C83C2311yescharacteristic
84C22×C8×C83C2311yescharacteristic
85C22×C8×C83C2311yescharacteristic
86C2×C16×C83—24no—
87C2×C16×C83C2312yes—
88C2×C16×C83C2312yes—
89#13050590—42no—
90#13050590—42no—
91C2×Dic166—21no—
92C2×D332—21no—
93C2×C22⋊D83—41no—
94C2×C22⋊D83—41no—
95C2×C22⋊D83—41no—
96C22×Dic83—41no—
97C22×Dic83—41no—
98C2×C4×D83—41no—
99Dic332—22no—
100#13050579—42no—
101D664—22no—
102C4⋊Dic83—21no—
103C4⋊Dic83—21no—
104C4⋊Dic83—22no—
105C4⋊Dic83—22no—
106C4⋊Dic83—41no—
107C4⋊Dic83—41no—
108Dic83⋊C4—41no—
109Dic83⋊C4—41no—
110C22⋊Dic83—41no—
111C22⋊Dic83—41no—
112C22⋊Dic83—41no—
113C22⋊Dic83—42no—
114C22⋊Dic83—42no—
115(C2×D83)⋊C4—41no—
116(C2×D83)⋊C4—41no—
117C4×Dic83—42no—
118C4×Dic83—42no—
119C4×Dic83—41no—
120C24×C83D811yescharacteristic
121C22×C4×C83—22no—
122C22×C4×C83—22no—
123C22×C4×C83D811yescharacteristic
124C22×C4×C83D811yescharacteristic
125C22×C4×C83D811yescharacteristic
126C22×C4×C83C2×D411yescharacteristic
127C22×C4×C83C2×D411yescharacteristic
128C22×C4×C83C2×D411yescharacteristic
129C2×C8×C83—22no—
130C2×C8×C83—22no—
131C2×C8×C83—22no—
132C2×C8×C83—22no—
133C2×C8×C83—22no—
134C2×C8×C83C2×D411yescharacteristic
135C2×C8×C83C2×D411yescharacteristic
136C2×C8×C83C2×D411yescharacteristic
137C2×C8×C83#4114yes—
138C2×C8×C83C2411yescharacteristic
139C16×C83—22no—
140C16×C83—22no—
141C23×C83—22no—
142C23×C83—22no—
143C23×C83—41no—
144C23×C83C2×D811yescharacteristic
145C23×C83C2×D811yescharacteristic
146C23×C83C2×D811yescharacteristic
147C22×D83—41no—
148C2×C4×C83—22no—
149C2×C4×C83—21no—
150C2×C4×C83—21no—
151C2×C4×C83—21no—
152C2×C4×C83—21no—
153C2×C4×C83—22no—
154C2×C4×C83—22no—
155C2×C4×C83#12312yes—
156C2×C4×C83C2×D811yescharacteristic
157C2×C4×C83C2×D811yescharacteristic
158C2×C4×C83C2×D811yescharacteristic
159C2×C4×C83#13712yes—
160C2×C4×C83C22×D411yescharacteristic
161C22⋊D83—82no—
162C22⋊D83—82no—
163C22⋊D83—82no—
164C2×Dic83—42no—
165C2×Dic83—42no—
166C2×Dic83—41no—
167C2×Dic83—41no—
168C2×Dic83—41no—
169C2×Dic83—82no—
170C2×Dic83—82no—
171D332—41no—
172D332—41no—
173C4×D83—82no—
174Dic166—41no—
175Dic166—41no—
176C8×C83—21no—
177C8×C83—21no—
178C8×C83#12312yes—
179C8×C83C2×#4112yes—
180C22×C83—21no—
181C22×C83—21no—
182C22×C83—21no—
183C22×C83—21no—
184C22×C83—21no—
185C22×C83—21no—
186C22×C83—21no—
187C22×C83—44no—
188C22×C83#48012yes—
189C22×C83#50812yes—
190C22×C83C22×D811yescharacteristic
191C2×D83—81no—
192C2×D83—81no—
193C2×D83—81no—
194C4×C83—21no—
195C4×C83—21no—
196C4×C83#48012yes—
197C4×C83C2×#12311yescharacteristic
198C4×C83C2×#13711yescharacteristic
199Dic83—81no—
200Dic83—82no—
201Dic83—81no—
202Dic83—82no—
203Dic83—81no—
204#33645—831nomaximal, Sylow
205C2×C83—22no—
206C2×C83—22no—
207C2×C83—41no—
208C2×C83C2×#48011yescharacteristic
209C2×C83C2×#50811yescharacteristic
210C2×C83#341411yescharacteristic
211D83—161no—
212#3050—831no—
213C2×#427—831no—
214#2237—832no—
215#2236—831no—
216#2235—831no—
217#2234—832no—
218#2233—831no—
219#2232—831no—
220#2158—832no—
221#2138—832no—
222C2×C22⋊C16D8311yescharacteristic, complemented
223C83#3364511yescharacteristic, complemented, Sylow
224C2×#135—831no—
225C23×C8C2×D8311yescharacteristic
226#525—1661no—
227C2×#117—1661no—
228C22×C16C2×D8312yes—
229#467—831no—
230#467—831no—
231#466—831no—
232#466—831no—
233#465—1661no—
234#465—1661no—
235#427—832no—
236#427—832no—
237#427—831no—
238#427—1661no—
239#427—1661no—
240C2×#107—831no—
241C2×#107—831no—
242C2×#106—1661no—
243#416—1661no—
244#367—832no—
245#367—832no—
246#366—832no—
247#366—832no—
248#358—832no—
249#358—832no—
250#357—832no—
251#357—832no—
252C22⋊C16C2×D8311yescharacteristic
253C22⋊C16C2×D8311yescharacteristic
254C22⋊C16C2×D8312yes—
255C2×#41—1661no—
256C23×C4D33211yescharacteristic
257#135—1664no—
258C2×Q16—831no—
259C2×SD16—1661no—
260C2×D8—831no—
261C22×C8—24no—
262C22×C8D33211yescharacteristic
263C22×C8D33211yescharacteristic
264C22×C8D33211yescharacteristic
265C22×C8C22×D8311yescharacteristic
266C22×C8C22×D8311yescharacteristic
267C22×C8C22×D8311yescharacteristic
268#123—3321no—
269C22⋊Q8—1661no—
270C22⋊Q8—1661no—
271#121—1661no—
272#121—1661no—
273#120—3321no—
274C4×D4—3321no—
275C4×D4—3321no—
276#117—3321no—
277#117—3321no—
278#117—1662no—
279#117—1662no—
280#117—1661no—
281C2×C4⋊C4—1661no—
282C2×C4⋊C4—1661no—
283C2×C22⋊C4—3321no—
284C2×C42—3321no—
285C2×C16—24no—
286C2×C16C22×D8312yes—
287C2×C16C22×D8312yes—
288#107—832no—
289#107—832no—
290#107—831no—
291#107—831no—
292#107—1661no—
293#107—1661no—
294#106—1662no—
295#106—1662no—
296#106—1661no—
297#103—1661no—
298#103—1661no—
299#102—1661no—
300#102—1661no—
301C24D66411yescharacteristic
302#41—3322no—
303#41—3322no—
304C2×Q8—1661no—
305C2×D4—3321no—
306C2×D4—3321no—
307C2×D4—3321no—
308C2×D4—1661no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—3321no—
312C22×C4—3321no—
313C22×C4—3321no—
314C22×C4D66411yescharacteristic
315C22×C4D66411yescharacteristic
316C22×C4D66411yescharacteristic
317C22×C4C2×D33211yescharacteristic
318C22×C4C2×D33211yescharacteristic
319C22×C4C2×D33211yescharacteristic
320Q16—1662no—
321SD16—3322no—
322D8—1662no—
323C2×C8—22no—
324C2×C8—22no—
325C2×C8—22no—
326C2×C8—22no—
327C2×C8—22no—
328C2×C8C2×D33211yescharacteristic
329C2×C8C2×D33211yescharacteristic
330C2×C8C2×D33211yescharacteristic
331C2×C8#1305058914yes—
332C2×C8C23×D8311yescharacteristic
333C4⋊C4—3321no—
334C4⋊C4—3321no—
335C4⋊C4—3321no—
336C4⋊C4—3321no—
337C4⋊C4—1661no—
338C4⋊C4—1661no—
339C4⋊C4—1662no—
340C4⋊C4—1662no—
341C22⋊C4—3322no—
342C22⋊C4—3322no—
343C22⋊C4—3321no—
344C22⋊C4—3321no—
345C22⋊C4—3321no—
346C22⋊C4—3321no—
347C22⋊C4—3321no—
348C42—3322no—
349C42—3322no—
350C42—3321no—
351C16—22no—
352C16—22no—
353C23—22no—
354C23—22no—
355C23—41no—
356C23—3321no—
357C23C2×D66411yescharacteristic
358C23C2×D66411yescharacteristic
359C23C2×D66411yescharacteristic
360Q8—3321no—
361Q8—3321no—
362D4—3321no—
363D4—3321no—
364D4—6642no—
365D4—6642no—
366D4—6642no—
367C2×C4—22no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—22no—
373C2×C4—22no—
374C2×C4—3322no—
375C2×C4—3322no—
376C2×C4—3321no—
377C2×C4—3321no—
378C2×C4—3321no—
379C2×C4—6642no—
380C2×C4—6642no—
381C2×C4—6642no—
382C2×C4#1458177612yes—
383C2×C4C2×D66411yescharacteristic
384C2×C4C2×D66411yescharacteristic
385C2×C4C2×D66411yescharacteristic
386C2×C4#1458182512yes—
387C2×C4C22×D33211yescharacteristic
388C8—21no—
389C8—21no—
390C8#1458177612yes—
391C8C2×#1305058912yes—
392C22—21no—
393C22—21no—
394C22—21no—
395C22—21no—
396C22—21no—
397C22—21no—
398C22—21no—
399C22—44no—
400C22—6641no—
401C22—6641no—
402C22—6641no—
403C22#1912801512yes—
404C22#1912813312yes—
405C22C22×D66411yescharacteristic
406C4—21no—
407C4—21no—
408C4—6642no—
409C4—6641no—
410C4—6642no—
411C4—6641no—
412C4—6641no—
413C4#1912801512yes—
414C4C2×#1458177611yescharacteristic
415C4C2×#1458182511yescharacteristic
416C2—22no—
417C2—22no—
418C2—41no—
419C2—13281no—
420C2C2×#1912801511yescharacteristic
421C2C2×#1912813311yescharacteristic
422C2#2492059911yescharacteristic
subgroupG/H
centerC2×C4C2×D664
commutator, cosocleC2×C8×C83C24
FrattiniC2×C8C23×D83
FittingC2×C22⋊C16×C83C2
socleC22×C83C22×D8
prime radical#24917044C2
hypercenterC2×C22⋊C16D83
hypocenterC83#33645
LCSG ▷ C2×C8×C83 ▷ C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ C23×C8 ◁ C2×C22⋊C16

Complements

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

NH ≅ G/Ntype
2. C2×#19127739C2semidirect
3. #24909672C2semidirect
4. #24909666C2semidirect
5. #24909661C2semidirect
6. #24909655C2semidirect
7. #24909649C2semidirect
8. #24909644C2semidirect
9. #24909247C2semidirect
10. #24909171C2semidirect
11. C2×C22⋊C16×C83C2semidirect
30. #19127529C22semidirect
31. #19127529C22semidirect
32. #19127527C22semidirect
33. #19127527C22semidirect
34. #19127506C22semidirect
35. #19127506C22semidirect
36. #19127503C22semidirect
37. #19127503C22semidirect
222. C2×C22⋊C16D83semidirect
223. C83#33645semidirect

Maximal subgroups

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

subgroupindex
#249170442
C2×#191277392
#249096722
#249096662
#249096612
#249096552
#249096492
#249096442
#249092472
#249091712
C2×C22⋊C16×C832
#3364583