Finite Group Databasebeta

← #35559479

Subgroups — #35559479

Order 21248.

592 conjugacy classes of subgroups.

The subgroup lattice has 416 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#24918042C211yesmaximal, characteristic, complemented
2#24909654C212yesmaximal, complemented
3#24909605C211yesmaximal, characteristic, complemented
4#24909600C211yesmaximal, characteristic, complemented
5#24909503C211yesmaximal, characteristic, complemented
6#24909498C211yesmaximal, characteristic, complemented
7#24909386C212yesmaximal, complemented
8#24909337C212yesmaximal, complemented
9#24909325C212yesmaximal, complemented
10#24909266C211yesmaximal, characteristic, complemented
11C4×#135×C83C211yesmaximal, characteristic, complemented
12#19128251—21no—
13C2×#14581735C2212yes—
14C2×#14581733C2212yes—
15#19128106C2211yescharacteristic
16#19128078C2211yescharacteristic
17#19128048—21no—
18#19127937—21no—
19#19127937C2212yes—
20C4×D664—21no—
21C4×D664C2212yes—
22#19127528C2212yes—
23#19127526C2212yes—
24C4⋊(C83⋊C16)C2211yescharacteristic
25C4⋊(C83⋊C16)C2211yescharacteristic
26#19127505C2212yes—
27#19127502C2212yes—
28#19127501C2212yes—
29#19127501C2212yes—
30C22⋊(C83⋊C16)C2212yes—
31#19127477C2212yes—
32C2×#135×C83C2211yescharacteristic
33C4×#41×C83—21no—
34C4×Q16×C83C2211yescharacteristic
35C4×Q16×C83C2211yescharacteristic
36C4×SD16×C83—21no—
37C4×SD16×C83—21no—
38C4×D8×C83C2211yescharacteristic
39C4×D8×C83C2211yescharacteristic
40#427×C83C2211yescharacteristic
41#416×C83—21no—
42C2×C4×C8×C83C2211yescharacteristic
43C2×#13050590—41no—
44C2×D664—21no—
45C2×D664C2312yes—
46#14581792—21no—
47#14581789—21no—
48#14581780—41no—
49#14581766—22no—
50#14581766—41no—
51C4×D332—22no—
52C4×D332—41no—
53#14581735—24no—
54#14581733—24no—
55C2×C83⋊C16C2312yes—
56C2×C83⋊C16C2312yes—
57#14581726—21no—
58#14581726C2312yes—
59#14581714—22no—
60#14581714—22no—
61C2×#41×C83—21no—
62#135×C83—22no—
63#135×C83—22no—
64C2×Q16×C83C2311yescharacteristic
65C2×SD16×C83—21no—
66C2×D8×C83C2311yescharacteristic
67C22×C8×C83D411yescharacteristic
68C22×C8×C83C2311yescharacteristic
69C4×Q8×C83—21no—
70C4×Q8×C83—21no—
71C4×D4×C83—21no—
72C4×D4×C83—21no—
73C4×D4×C83—41no—
74C4×D4×C83—41no—
75#117×C83—21no—
76#117×C83—41no—
77C2×C42×C83—41no—
78C2×C42×C83D411yescharacteristic
79#107×C83C2311yescharacteristic
80#107×C83C2311yescharacteristic
81#106×C83—21no—
82#103×C83—21no—
83#103×C83—21no—
84#102×C83—21no—
85#102×C83—21no—
86C4×C8×C83D411yescharacteristic
87C4×C8×C83D411yescharacteristic
88C4×C8×C83C2311yescharacteristic
89C4×C8×C83C2311yescharacteristic
90#13050590—84no—
91C2×Dic166—41no—
92C2×D332—22no—
93C2×D332—41no—
94C2×D332—41no—
95C2×C22⋊D83—42no—
96C2×C4×D83—42no—
97D664—22no—
98D664—22no—
99D664—42no—
100C4⋊Dic83—22no—
101C4⋊Dic83—41no—
102(C2×D83)⋊C4—42no—
103(C2×D83)⋊C4—42no—
104C83⋊C16—22no—
105C83⋊C16—22no—
106#41×C83—24no—
107#41×C83—42no—
108C2×Q8×C83—21no—
109C2×D4×C83—21no—
110C2×D4×C83—41no—
111C22×C4×C83—41no—
112C22×C4×C83—41no—
113C22×C4×C83—41no—
114C22×C4×C83D811yescharacteristic
115C22×C4×C83D811yescharacteristic
116C22×C4×C83C2×D411yescharacteristic
117Q16×C83—21no—
118Q16×C83#4112yescomplemented
119SD16×C83—24no—
120D8×C83—21no—
121D8×C83#4112yescomplemented
122C2×C8×C83—21no—
123C2×C8×C83—21no—
124C2×C8×C83—41no—
125C2×C8×C83C2×D411yescharacteristic
126C2×C8×C83C2×D411yescharacteristic
127C2×C8×C83C2×D411yescharacteristic
128C2×C8×C83C2411yescharacteristic
129C4⋊C4×C83—21no—
130C4⋊C4×C83—21no—
131C4⋊C4×C83—41no—
132C4⋊C4×C83—41no—
133C22⋊C4×C83—41no—
134C22⋊C4×C83—41no—
135C22⋊C4×C83—41no—
136C42×C83—41no—
137C42×C83—41no—
138C42×C83—41no—
139C42×C83D811yescharacteristic
140C42×C83D811yescharacteristic
141C42×C83C2×D411yescharacteristic
142C42×C83C2×D411yescharacteristic
143C23×C83—41no—
144C23×C83C2×D811yescharacteristic
145C22×D83—42no—
146Q8×C83—22no—
147Q8×C83—41no—
148D4×C83—22no—
149D4×C83—44no—
150D4×C83—41no—
151C2×C4×C83—21no—
152C2×C4×C83—21no—
153C2×C4×C83—22no—
154C2×C4×C83—41no—
155C2×C4×C83—44no—
156C2×C4×C83—41no—
157C2×C4×C83—41no—
158C2×C4×C83—41no—
159C2×C4×C83—82no—
160C2×C4×C83—82no—
161C2×C4×C83C2×D811yescharacteristic
162C2×C4×C83C2×D811yescharacteristic
163C2×C4×C83C2×D811yescharacteristic
164C2×C4×C83C2×D811yescharacteristic
165C2×C4×C83C2×D811yescharacteristic
166C2×C4×C83C22×D411yescharacteristic
167C22⋊D83—84no—
168C2×Dic83—42no—
169D332—42no—
170D332—42no—
171D332—82no—
172D332—82no—
173C4×D83—84no—
174Dic166—82no—
175C8×C83—21no—
176C8×C83—21no—
177C8×C83—21no—
178C8×C83C2×#4111yescharacteristic
179C8×C83C2×#4111yescharacteristic
180C22×C83—21no—
181C22×C83—21no—
182C22×C83—21no—
183C22×C83—42no—
184C22×C83—81no—
185C22×C83—81no—
186C22×C83C22×D811yescharacteristic
187C2×D83—82no—
188C2×D83—82no—
189C2×D83—82no—
190C4×C83—21no—
191C4×C83—21no—
192C4×C83—21no—
193C4×C83—21no—
194C4×C83—42no—
195C4×C83—81no—
196C4×C83—81no—
197C4×C83#50912yes—
198C4×C83#52511yescharacteristic
199C4×C83#57511yescharacteristic
200Dic83—82no—
201#33675—831nomaximal, Sylow
202C2×C83—41no—
203C2×C83—82no—
204C2×C83#310011yescharacteristic
205C2×C83C2×#50911yescharacteristic
206C2×C83#341411yescharacteristic
207D83—162no—
208#3169—831no—
209C4×#135D8311yescharacteristic, complemented
210#2234—832no—
211#2225—831no—
212#2224—831no—
213#2207—831no—
214#2206—831no—
215#2186—832no—
216#2177—832no—
217#2175—832no—
218#2163—831no—
219C83#3367511yescharacteristic, complemented, Sylow
220C2×#135C2×D8311yescharacteristic
221#532—1661no—
222C4×#41—21no—
223C2×SD32—832no—
224C2×D16—832no—
225#500—831no—
226#493—831no—
227#486—1661no—
228#466—832no—
229#466—1661no—
230C4×Q16C2×D8311yescharacteristic
231C4×Q16C2×D8311yescharacteristic
232C4×SD16—21no—
233C4×SD16—21no—
234C4×D8—832no—
235C4×D8—1661no—
236C4×D8C2×D8311yescharacteristic
237C4×D8C2×D8311yescharacteristic
238#427C2×D8311yescharacteristic
239#416—21no—
240C2×C4×C8C2×D8311yescharacteristic
241#367—832no—
242#366—832no—
243C4⋊C16—831no—
244C4⋊C16—831no—
245#358—832no—
246#357—832no—
247#357—832no—
248#357—832no—
249C22⋊C16—832no—
250#346—832no—
251C2×#41—21no—
252C2×#41—3321no—
253#135—22no—
254#135—22no—
255C2×Q16C22×D8311yescharacteristic
256C2×SD16—21no—
257C2×D8—832no—
258C2×D8—1661no—
259C2×D8C22×D8311yescharacteristic
260C22×C8C22⋊D8311yescharacteristic
261C22×C8C22×D8311yescharacteristic
262C4⋊Q8—1661no—
263#127—1661no—
264#124—3321no—
265#121—3321no—
266#121—1662no—
267C4×Q8—21no—
268C4×Q8—21no—
269C4×D4—21no—
270C4×D4—21no—
271C4×D4—41no—
272C4×D4—41no—
273C4×D4—3321no—
274C4×D4—1662no—
275#117—21no—
276#117—41no—
277C2×C42—41no—
278C2×C42C22⋊D8311yescharacteristic
279SD32—1664no—
280D16—1664no—
281C2×C16—832no—
282C2×C16—832no—
283#107—832no—
284#107—1661no—
285#107C22×D8311yescharacteristic
286#107C22×D8311yescharacteristic
287#106—21no—
288#103—21no—
289#103—21no—
290#102—21no—
291#102—21no—
292#102—1662no—
293#102—1662no—
294C4×C8C22⋊D8311yescharacteristic
295C4×C8C22⋊D8311yescharacteristic
296C4×C8C22×D8311yescharacteristic
297C4×C8C22×D8311yescharacteristic
298#41—24no—
299#41—42no—
300#41—6644no—
301C2×Q8—21no—
302C2×Q8—3321no—
303C2×D4—21no—
304C2×D4—41no—
305C2×D4—3322no—
306C2×D4—3321no—
307C2×D4—3321no—
308C2×D4—1662no—
309C22×C4—41no—
310C22×C4—41no—
311C22×C4—41no—
312C22×C4—3322no—
313C22×C4#1305057511yescharacteristic
314C22×C4#1305057511yescharacteristic
315C22×C4C2×C22⋊D8311yescharacteristic
316Q16—21no—
317Q16#1305059012yescomplemented
318SD16—24no—
319D8—21no—
320D8—3322no—
321D8—1662no—
322D8—1662no—
323D8#1305059012yescomplemented
324C2×C8—21no—
325C2×C8—21no—
326C2×C8—41no—
327C2×C8C2×C22⋊D8311yescharacteristic
328C2×C8C2×C22⋊D8311yescharacteristic
329C2×C8C2×C22⋊D8311yescharacteristic
330C2×C8C23×D8311yescharacteristic
331C4⋊C4—21no—
332C4⋊C4—21no—
333C4⋊C4—41no—
334C4⋊C4—41no—
335C4⋊C4—3321no—
336C4⋊C4—1662no—
337C22⋊C4—41no—
338C22⋊C4—41no—
339C22⋊C4—41no—
340C22⋊C4—3322no—
341C22⋊C4—3322no—
342C42—41no—
343C42—41no—
344C42—41no—
345C42#1305057511yescharacteristic
346C42#1305057511yescharacteristic
347C42C2×C22⋊D8311yescharacteristic
348C42C2×C22⋊D8311yescharacteristic
349C16—1662no—
350C16—1662no—
351C23—41no—
352C23—3322no—
353C23C2×#1305057511yescharacteristic
354Q8—22no—
355Q8—41no—
356Q8—6642no—
357D4—22no—
358D4—44no—
359D4—41no—
360D4—3322no—
361D4—3322no—
362D4—6642no—
363D4—6644no—
364D4—6642no—
365C2×C4—22no—
366C2×C4—21no—
367C2×C4—21no—
368C2×C4—41no—
369C2×C4—44no—
370C2×C4—41no—
371C2×C4—41no—
372C2×C4—41no—
373C2×C4—82no—
374C2×C4—82no—
375C2×C4—3322no—
376C2×C4—6644no—
377C2×C4C2×#1305057511yescharacteristic
378C2×C4C2×#1305057511yescharacteristic
379C2×C4C2×#1305057511yescharacteristic
380C2×C4C2×#1305057511yescharacteristic
381C2×C4C2×#1305057511yescharacteristic
382C2×C4C22×C22⋊D8311yescharacteristic
383C8—21no—
384C8—21no—
385C8—21no—
386C8C2×#1305059011yescharacteristic
387C8C2×#1305059011yescharacteristic
388C22—21no—
389C22—21no—
390C22—21no—
391C22—42no—
392C22—81no—
393C22—81no—
394C22—6642no—
395C22—6642no—
396C22—6642no—
397C22C22×#1305057511yescharacteristic
398C4—21no—
399C4—21no—
400C4—21no—
401C4—21no—
402C4—42no—
403C4—81no—
404C4—81no—
405C4—6642no—
406C4#1912813612yes—
407C4#1912821211yescharacteristic
408C4#1912849911yescharacteristic
409C2—41no—
410C2—82no—
411C2—13282no—
412C2#2491746311yescharacteristic
413C2C2×#1912813611yescharacteristic
414C2#2492059711yescharacteristic
subgroupG/H
centerC22C22×#13050575
commutator, cosocleC2×C8×C83C24
FrattiniC2×C8C23×D83
FittingC4×#135×C83C2
socleC22×C83C22×D8
hypercenterC4×#135D83
hypocenterC83#33675
LCSG ▷ C2×C8×C83 ▷ C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×C4×C8 ◁ C4×#135

Complements

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

NH ≅ G/Ntype
1. #24918042C2semidirect
2. #24909654C2semidirect
3. #24909605C2semidirect
4. #24909600C2semidirect
5. #24909503C2semidirect
6. #24909498C2semidirect
7. #24909386C2semidirect
8. #24909337C2semidirect
9. #24909325C2semidirect
10. #24909266C2semidirect
11. C4×#135×C83C2semidirect
118. Q16×C83#41semidirect
121. D8×C83#41semidirect
209. C4×#135D83semidirect
219. C83#33675semidirect
317. Q16#13050590semidirect
323. D8#13050590semidirect

Maximal subgroups

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

subgroupindex
#249180422
#249096542
#249096052
#249096002
#249095032
#249094982
#249093862
#249093372
#249093252
#249092662
C4×#135×C832
#3367583