Finite Group Databasebeta

← #35559462

Subgroups — #35559462

Order 21248.

612 conjugacy classes of subgroups.

The subgroup lattice has 408 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×D664C211yesmaximal, characteristic, complemented
2#24909658C211yesmaximal, characteristic, complemented
3#24909641C211yesmaximal, characteristic, complemented
4#24909617C212yesmaximal, complemented
5#24909515C212yesmaximal, complemented
6#24909391C211yesmaximal, characteristic, complemented
7#24909374C211yesmaximal, characteristic
8#24909337C212yesmaximal, complemented
9#24909325C212yesmaximal, complemented
10#24909266C211yesmaximal, characteristic, complemented
11C83×#3168C211yesmaximal, characteristic, complemented
12C22×D664C2211yescharacteristic
13C2×C4×D332—21no—
14C2×#14581735C2212yes—
15C2×#14581733C2212yes—
16C4×D664—21no—
17C4×D664—21no—
18C4×D664C2212yes—
19C4×D664C2212yes—
20C2×#14581726C2211yescharacteristic
21C2×#14581714—21no—
22#19127528C2212yescomplemented
23#19127526C2212yescomplemented
24C4⋊(C83⋊C16)C2212yescomplemented
25#19127505C2212yescomplemented
26#19127502C2212yescomplemented
27#19127501C2212yes—
28#19127501C2212yes—
29C22⋊(C83⋊C16)C2211yescharacteristic
30C22⋊(C83⋊C16)C2211yescharacteristic
31#19127477C2212yescomplemented
32#529×C83—21no—
33#499×C83C2211yescharacteristic, complemented
34#495×C83C2211yescharacteristic, complemented
35#488×C83—21no—
36#467×C83C2211yescharacteristic
37#466×C83C2211yescharacteristic
38#465×C83—21no—
39C4×Q16×C83C2211yescharacteristic, complemented
40C4×SD16×C83—21no—
41C4×D8×C83C2211yescharacteristic, complemented
42C2×C4×C8×C83C2211yescharacteristic
43C22×D332—21no—
44C22×C4×D83—41no—
45C2×D664—24no—
46C2×D664—21no—
47C2×D664C2312yes—
48C4×D332—22no—
49C4×D332—22no—
50C4×D332—41no—
51C4×D332—41no—
52C2×C4⋊Dic83—21no—
53C2×(C2×D83)⋊C4—41no—
54#14581735—24no—
55#14581733—24no—
56C2×C83⋊C16C2312yes—
57C2×C83⋊C16C2312yes—
58#14581726—21no—
59#14581726C2312yes—
60#14581714—22no—
61#14581714—22no—
62C2×Q16×C83C2311yescharacteristic
63C2×SD16×C83—21no—
64C2×D8×C83C2311yescharacteristic
65C22×C8×C83D411yescharacteristic
66C22×C8×C83C2311yescharacteristic
67#126×C83—41no—
68#125×C83—21no—
69#124×C83—21no—
70#123×C83—41no—
71C22⋊Q8×C83—21no—
72#121×C83—21no—
73C4×Q8×C83—21no—
74C4×D4×C83—21no—
75C4×D4×C83—41no—
76#117×C83—41no—
77C2×C42×C83D411yescharacteristic
78#107×C83C2311yescharacteristic
79#107×C83C2311yescharacteristic
80#106×C83—21no—
81#103×C83—21no—
82#103×C83—21no—
83#102×C83—21no—
84#102×C83—21no—
85C4×C8×C83D411yescharacteristic
86C4×C8×C83D411yescharacteristic
87C4×C8×C83C2311yescharacteristic
88C4×C8×C83C2311yescharacteristic
89C23×D83—41no—
90C2×D332—24no—
91C2×D332—22no—
92C2×D332—42no—
93C2×D332—41no—
94C22×Dic83—41no—
95C2×C4×D83—42no—
96C2×C4×D83—42no—
97C2×C4×D83—84no—
98D664—24no—
99D664—22no—
100D664#4114yes—
101C4⋊Dic83—22no—
102C4⋊Dic83—41no—
103(C2×D83)⋊C4—42no—
104(C2×D83)⋊C4—42no—
105C83⋊C16—22no—
106C83⋊C16—22no—
107C2×Q8×C83—21no—
108C2×D4×C83—21no—
109C2×D4×C83—41no—
110C22×C4×C83—41no—
111C22×C4×C83D811yescharacteristic
112C22×C4×C83D811yescharacteristic
113C22×C4×C83C2×D411yescharacteristic
114Q16×C83—22no—
115SD16×C83—42no—
116D8×C83—22no—
117C2×C8×C83—22no—
118C2×C8×C83—41no—
119C2×C8×C83C2×D411yescharacteristic
120C2×C8×C83C2×D411yescharacteristic
121C2×C8×C83C2×D411yescharacteristic
122C2×C8×C83C2411yescharacteristic
123C4⋊C4×C83—21no—
124C4⋊C4×C83—21no—
125C4⋊C4×C83—41no—
126C4⋊C4×C83—41no—
127C4⋊C4×C83—41no—
128C4⋊C4×C83—41no—
129C22⋊C4×C83—41no—
130C22⋊C4×C83—41no—
131C22⋊C4×C83—41no—
132C22⋊C4×C83—41no—
133C22⋊C4×C83—41no—
134C42×C83—41no—
135C42×C83D811yescharacteristic
136C42×C83D811yescharacteristic
137C42×C83C2×D411yescharacteristic
138C42×C83C2×D411yescharacteristic
139C23×C83—41no—
140C23×C83C2×D811yescharacteristic
141C22×D83—44no—
142C22×D83—42no—
143C22×D83—82no—
144C22×D83—82no—
145Q8×C83—41no—
146Q8×C83—41no—
147D4×C83—41no—
148D4×C83—41no—
149D4×C83—82no—
150C2×C4×C83—22no—
151C2×C4×C83—21no—
152C2×C4×C83—21no—
153C2×C4×C83—41no—
154C2×C4×C83—41no—
155C2×C4×C83—41no—
156C2×C4×C83—41no—
157C2×C4×C83—82no—
158C2×C4×C83C2×D811yescharacteristic
159C2×C4×C83C2×D811yescharacteristic
160C2×C4×C83C2×D811yescharacteristic
161C2×C4×C83C2×D811yescharacteristic
162C2×C4×C83C2×D811yescharacteristic
163C2×C4×C83C22×D411yescharacteristic
164C2×Dic83—42no—
165C2×Dic83—82no—
166D332—24no—
167D332—44no—
168D332—42no—
169C4×D83—84no—
170C4×D83—84no—
171C8×C83—21no—
172C8×C83—21no—
173C8×C83—21no—
174C8×C83C2×#4112yes—
175C22×C83—21no—
176C22×C83—21no—
177C22×C83—81no—
178C22×C83—81no—
179C22×C83—81no—
180C22×C83#50912yes—
181C22×C83C22×D811yescharacteristic
182C2×D83—44no—
183C2×D83—88no—
184C2×D83—82no—
185C2×D83—82no—
186C4×C83—21no—
187C4×C83—21no—
188C4×C83—21no—
189C4×C83—21no—
190C4×C83—21no—
191C4×C83—81no—
192C4×C83—81no—
193C4×C83—81no—
194C4×C83#52511yescharacteristic
195C4×C83#57511yescharacteristic
196Dic83—82no—
197#33674—831nomaximal, Sylow
198C2×C83—22no—
199C2×C83—161no—
200C2×C83#310011yescharacteristic
201C2×C83C2×#50911yescharacteristic
202C2×C83#341411yescharacteristic
203D83—84no—
204#3168D8311yescharacteristic, complemented
205C2×C4×D8—831no—
206#2235—831no—
207#2232—831no—
208#2228—832no—
209#2210—832no—
210#2187—831no—
211#2184—831no—
212#2177—832no—
213#2175—832no—
214#2163—831no—
215C83#3367411yescharacteristic, complemented, Sylow
216C22×D8—831no—
217#529—21no—
218C2×C4×D4—1661no—
219C2×SD32—832no—
220C2×D16—832no—
221#499C2×D8311yescharacteristic, complemented
222#495C2×D8311yescharacteristic, complemented
223#488—21no—
224#467C2×D8311yescharacteristic
225#466C2×D8311yescharacteristic
226#465—21no—
227C4×Q16C2×D8311yescharacteristic, complemented
228C4×SD16—21no—
229C4×D8—832no—
230C4×D8—832no—
231C4×D8—1661no—
232C4×D8—1661no—
233C4×D8C2×D8311yescharacteristic, complemented
234C2×#107—831no—
235C2×#102—1661no—
236C2×C4×C8C2×D8311yescharacteristic
237#367—832no—
238#366—832no—
239C4⋊C16—832no—
240#358—832no—
241#357—832no—
242#357—832no—
243#357—832no—
244C22⋊C16—831no—
245C22⋊C16—831no—
246#346—832no—
247C22×D4—1661no—
248C23×C4—3321no—
249C2×Q16C22×D8311yescharacteristic
250C2×SD16—21no—
251C2×D8—832no—
252C2×D8—1664no—
253C2×D8—1661no—
254C2×D8C22×D8311yescharacteristic
255C22×C8C22⋊D8311yescharacteristic
256C22×C8C22×D8311yescharacteristic
257#126—41no—
258#125—21no—
259#124—21no—
260#123—41no—
261C22⋊Q8—21no—
262#121—21no—
263C4×Q8—21no—
264C4×D4—21no—
265C4×D4—41no—
266C4×D4—3321no—
267C4×D4—3321no—
268C4×D4—1662no—
269C4×D4—1662no—
270#117—41no—
271C2×C4⋊C4—1661no—
272C2×C22⋊C4—3321no—
273C2×C42C22⋊D8311yescharacteristic
274SD32—1664no—
275D16—1664no—
276C2×C16—832no—
277C2×C16—832no—
278#107—832no—
279#107—1661no—
280#107C22×D8311yescharacteristic
281#107C22×D8311yescharacteristic
282#106—21no—
283#103—21no—
284#103—21no—
285#102—21no—
286#102—21no—
287#102—1662no—
288#102—1662no—
289C4×C8C22⋊D8311yescharacteristic
290C4×C8C22⋊D8311yescharacteristic
291C4×C8C22×D8311yescharacteristic
292C4×C8C22×D8311yescharacteristic
293C24—3321no—
294C2×Q8—21no—
295C2×D4—21no—
296C2×D4—41no—
297C2×D4—3322no—
298C2×D4—3321no—
299C2×D4—1664no—
300C2×D4—1662no—
301C22×C4—41no—
302C22×C4—3321no—
303C22×C4—3322no—
304C22×C4—3322no—
305C22×C4—6644no—
306C22×C4#1305057511yescharacteristic
307C22×C4#1305057511yescharacteristic
308C22×C4C2×C22⋊D8311yescharacteristic
309Q16—22no—
310SD16—42no—
311D8—22no—
312D8—834no—
313D8—1664no—
314D8—1662no—
315C2×C8—22no—
316C2×C8—41no—
317C2×C8C2×C22⋊D8311yescharacteristic
318C2×C8C2×C22⋊D8311yescharacteristic
319C2×C8C2×C22⋊D8311yescharacteristic
320C2×C8C23×D8311yescharacteristic
321C4⋊C4—21no—
322C4⋊C4—21no—
323C4⋊C4—41no—
324C4⋊C4—41no—
325C4⋊C4—41no—
326C4⋊C4—41no—
327C4⋊C4—3321no—
328C4⋊C4—1662no—
329C22⋊C4—41no—
330C22⋊C4—41no—
331C22⋊C4—41no—
332C22⋊C4—41no—
333C22⋊C4—41no—
334C22⋊C4—3322no—
335C22⋊C4—3322no—
336C42—41no—
337C42#1305057511yescharacteristic
338C42#1305057511yescharacteristic
339C42C2×C22⋊D8311yescharacteristic
340C42C2×C22⋊D8311yescharacteristic
341C16—1662no—
342C16—1662no—
343C23—41no—
344C23—3324no—
345C23—3322no—
346C23—6642no—
347C23—6642no—
348C23C2×#1305057511yescharacteristic
349Q8—41no—
350Q8—41no—
351D4—41no—
352D4—41no—
353D4—82no—
354D4—3324no—
355D4—3322no—
356D4—1664no—
357C2×C4—22no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—41no—
361C2×C4—41no—
362C2×C4—41no—
363C2×C4—41no—
364C2×C4—82no—
365C2×C4—3322no—
366C2×C4—6644no—
367C2×C4—6644no—
368C2×C4—6642no—
369C2×C4C2×#1305057511yescharacteristic
370C2×C4C2×#1305057511yescharacteristic
371C2×C4C2×#1305057511yescharacteristic
372C2×C4C2×#1305057511yescharacteristic
373C2×C4C2×#1305057511yescharacteristic
374C2×C4C22×C22⋊D8311yescharacteristic
375C8—21no—
376C8—21no—
377C8—21no—
378C8#41×D8312yes—
379C22—21no—
380C22—21no—
381C22—81no—
382C22—81no—
383C22—81no—
384C22—3324no—
385C22—6648no—
386C22—6642no—
387C22—6642no—
388C22#1912813612yes—
389C22C22×#1305057511yescharacteristic
390C4—21no—
391C4—21no—
392C4—21no—
393C4—21no—
394C4—21no—
395C4—81no—
396C4—81no—
397C4—81no—
398C4—6642no—
399C4#1912820811yescharacteristic
400C4#1912849911yescharacteristic
401C2—22no—
402C2—161no—
403C2—6644no—
404C2#2491745911yescharacteristic
405C2C2×#1912813611yescharacteristic
406C2#2492059711yescharacteristic
subgroupG/H
centerC22C22×#13050575
commutator, cosocleC2×C8×C83C24
FrattiniC2×C8C23×D83
FittingC83×#3168C2
socleC22×C83C22×D8
prime radical#24909374C2
hypercenter#3168D83
hypocenterC83#33674
LCSG ▷ C2×C8×C83 ▷ C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×C4×C8 ◁ #3168

Complements

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

NH ≅ G/Ntype
1. C2×C4×D664C2semidirect
2. #24909658C2semidirect
3. #24909641C2semidirect
4. #24909617C2semidirect
5. #24909515C2semidirect
6. #24909391C2semidirect
8. #24909337C2semidirect
9. #24909325C2semidirect
10. #24909266C2semidirect
11. C83×#3168C2semidirect
22. #19127528C22semidirect
23. #19127526C22semidirect
24. C4⋊(C83⋊C16)C22semidirect
25. #19127505C22semidirect
26. #19127502C22semidirect
31. #19127477C22semidirect
33. #499×C83C22semidirect
34. #495×C83C22semidirect
39. C4×Q16×C83C22semidirect
41. C4×D8×C83C22semidirect
204. #3168D83semidirect
215. C83#33674semidirect
221. #499C2×D83semidirect
222. #495C2×D83semidirect
227. C4×Q16C2×D83semidirect
233. C4×D8C2×D83semidirect

Maximal subgroups

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

subgroupindex
C2×C4×D6642
#249096582
#249096412
#249096172
#249095152
#249093912
#249093742
#249093372
#249093252
#249092662
C83×#31682
#3367483