Finite Group Databasebeta

← #35314554

Subgroups — #35314554

Order 21248.

448 conjugacy classes of subgroups.

The subgroup lattice has 386 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19127483C211yesmaximal, characteristic, complemented
2C2×#19127458C211yesmaximal, characteristic, complemented
3#24905451C212yesmaximal, complemented
4#24905448C212yesmaximal, complemented
5C83×#3054C211yesmaximal, characteristic
6C2×#14581751—21no—
7C2×#14581732—21no—
8#19127483—21no—
9#19127483C2212yes—
10#19127458—21no—
11#19127458C2212yes—
12C2×#135×C83C411yescharacteristic, complemented
13C22×M16×C83C2211yescharacteristic
14#525×C83—21no—
15#470×C83C411yescharacteristic, complemented
16#470×C83C2211yescharacteristic
17#469×C83C411yescharacteristic, complemented
18#469×C83C2211yescharacteristic
19#468×C83—21no—
20#468×C83—21no—
21#427×C83C411yescharacteristic, complemented
22#416×C83—21no—
23#14581751—22no—
24#14581751—22no—
25#14581751—41no—
26#14581751—41no—
27C2×C4⋊Dic83—41no—
28C2×C22⋊Dic83—41no—
29C2×C4×Dic83—22no—
30#14581732—22no—
31#14581732—41no—
32C2×C83⋊C16—41no—
33C2×#41×C83—21no—
34C23×C4×C83D411yescharacteristic
35#135×C83—41no—
36#135×C83—41no—
37C2×Q16×C83C2×C411yescharacteristic, complemented
38C2×SD16×C83—21no—
39C2×D8×C83C2×C411yescharacteristic, complemented
40C2×M16×C83—41no—
41C2×M16×C83—41no—
42C2×M16×C83C2×C411yescharacteristic
43C2×M16×C83D411yescharacteristic
44C2×M16×C83D411yescharacteristic
45C2×M16×C83C2311yescharacteristic
46C22×C8×C83C2×C411yescharacteristic
47C22×C8×C83D411yescharacteristic
48#123×C83—41no—
49C22⋊Q8×C83—21no—
50C22⋊Q8×C83—21no—
51#121×C83—21no—
52#121×C83—21no—
53#120×C83—41no—
54C4×D4×C83—41no—
55C4×D4×C83—41no—
56#117×C83—21no—
57#107×C83—21no—
58#106×C83C2×C411yescharacteristic, complemented
59#106×C83C2×C411yescharacteristic, complemented
60#103×C83—21no—
61#103×C83—21no—
62#102×C83—21no—
63#102×C83—21no—
64C22×Dic83—42no—
65C4⋊Dic83—42no—
66C4⋊Dic83—42no—
67C22⋊Dic83—42no—
68C22⋊Dic83—42no—
69C4×Dic83—24no—
70C4×Dic83—42no—
71C83⋊C16—42no—
72C24×C83D811yescharacteristic
73#41×C83—41no—
74#41×C83—41no—
75#41×C83—41no—
76#41×C83—41no—
77C2×Q8×C83—21no—
78C2×D4×C83—21no—
79C2×D4×C83—41no—
80C2×D4×C83—41no—
81C2×D4×C83—41no—
82C22×C4×C83—21no—
83C22×C4×C83—21no—
84C22×C4×C83—21no—
85C22×C4×C83—21no—
86C22×C4×C83—41no—
87C22×C4×C83C22⋊C411yescharacteristic
88C22×C4×C83C22⋊C411yescharacteristic
89C22×C4×C83D811yescharacteristic
90C22×C4×C83SD1611yescharacteristic
91C22×C4×C83SD1611yescharacteristic
92C22×C4×C83C2×D411yescharacteristic
93Q16×C83—41no—
94SD16×C83—42no—
95D8×C83—41no—
96M16×C83—41no—
97M16×C83—41no—
98M16×C83—41no—
99M16×C83—41no—
100C2×C8×C83—21no—
101C2×C8×C83—21no—
102C2×C8×C83—21no—
103C2×C8×C83—21no—
104C2×C8×C83C22⋊C411yescharacteristic
105C2×C8×C83C22⋊C411yescharacteristic
106C2×C8×C83C22×C411yescharacteristic
107C2×C8×C83C2×D411yescharacteristic
108C4⋊C4×C83—21no—
109C4⋊C4×C83—21no—
110C4⋊C4×C83—41no—
111C4⋊C4×C83—41no—
112C22⋊C4×C83—41no—
113C22⋊C4×C83—41no—
114C22⋊C4×C83—41no—
115C22⋊C4×C83—41no—
116C22⋊C4×C83—41no—
117C42×C83—41no—
118C23×C83—21no—
119C23×C83—21no—
120C23×C83—41no—
121C23×C83—81no—
122C23×C83#10211yescharacteristic
123C23×C83#10211yescharacteristic
124C23×C83C2×D811yescharacteristic
125Q8×C83—41no—
126Q8×C83—41no—
127D4×C83—41no—
128D4×C83—41no—
129D4×C83—82no—
130D4×C83—82no—
131D4×C83—81no—
132D4×C83—81no—
133C2×C4×C83—21no—
134C2×C4×C83—21no—
135C2×C4×C83—21no—
136C2×C4×C83—21no—
137C2×C4×C83—21no—
138C2×C4×C83—21no—
139C2×C4×C83—21no—
140C2×C4×C83—21no—
141C2×C4×C83—21no—
142C2×C4×C83—21no—
143C2×C4×C83—41no—
144C2×C4×C83—41no—
145C2×C4×C83—41no—
146C2×C4×C83—81no—
147C2×C4×C83—81no—
148C2×C4×C83#9911yescharacteristic
149C2×C4×C83#9911yescharacteristic
150C2×C4×C83#10211yescharacteristic
151C2×C4×C83#10211yescharacteristic
152C2×C4×C83#10411yescharacteristic
153C2×C4×C83#10411yescharacteristic
154C2×C4×C83C2×C22⋊C411yescharacteristic
155C2×C4×C83C2×SD1611yescharacteristic
156C2×Dic83—44no—
157C2×Dic83—84no—
158C8×C83—21no—
159C8×C83—21no—
160C8×C83—21no—
161C8×C83—21no—
162C22×C83—21no—
163C22×C83—21no—
164C22×C83—41no—
165C22×C83—41no—
166C22×C83—41no—
167C22×C83—42no—
168C22×C83—81no—
169C22×C83—81no—
170C22×C83—81no—
171C22×C83—81no—
172C22×C83#32711yescharacteristic
173C22×C83#32711yescharacteristic
174C22×C83C2×#10211yescharacteristic
175C4×C83—21no—
176C4×C83—21no—
177C4×C83—81no—
178C4×C83—81no—
179C4×C83—81no—
180C4×C83#32711yescharacteristic
181C4×C83#32711yescharacteristic
182C4×C83C2×#9911yescharacteristic
183C4×C83C2×#10411yescharacteristic
184Dic83—84no—
185#11685—831nomaximal, Sylow
186C2×C83—21no—
187C2×C83—21no—
188C2×C83—21no—
189C2×C83—41no—
190C2×C83—41no—
191C2×C83—161no—
192C2×C83C2×#32711yescharacteristic
193#3054D8311yescharacteristic
194C2×#349—831no—
195C2×H(3,ℤ/4)—831no—
196#1340—832no—
197#1339—832no—
198C83#1168511yescharacteristic, complemented, Sylow
199C2×#135Dic8311yescharacteristic, complemented
200C22×M16C2×D8311yescharacteristic
201#525—21no—
202C2×#117—1661no—
203C2×M32—1661no—
204#470Dic8311yescharacteristic, complemented
205#470C2×D8311yescharacteristic
206#469Dic8311yescharacteristic, complemented
207#469C2×D8311yescharacteristic
208#468—21no—
209#468—21no—
210#427Dic8311yescharacteristic, complemented
211#416—21no—
212#349—832no—
213#349—1661no—
214H(3,ℤ/4)—832no—
215H(3,ℤ/4)—1661no—
216C2×#41—21no—
217C23×C4C22⋊D8311yescharacteristic
218#135—41no—
219#135—41no—
220C2×Q16C2×Dic8311yescharacteristic, complemented
221C2×SD16—21no—
222C2×D8C2×Dic8311yescharacteristic, complemented
223C2×M16—41no—
224C2×M16—41no—
225C2×M16C2×Dic8311yescharacteristic
226C2×M16C22⋊D8311yescharacteristic
227C2×M16C22⋊D8311yescharacteristic
228C2×M16C22×D8311yescharacteristic
229C22×C8C2×Dic8311yescharacteristic
230C22×C8C22⋊D8311yescharacteristic
231#123—41no—
232C22⋊Q8—21no—
233C22⋊Q8—21no—
234#121—21no—
235#121—21no—
236#120—41no—
237C4×D4—41no—
238C4×D4—41no—
239#117—21no—
240#117—3321no—
241#117—3321no—
242#117—1662no—
243#117—1662no—
244C2×C4⋊C4—3321no—
245C2×C22⋊C4—3321no—
246C2×C42—1662no—
247M32—3321no—
248M32—1662no—
249C2×C16—3321no—
250#107—21no—
251#106C2×Dic8311yescharacteristic, complemented
252#106C2×Dic8311yescharacteristic, complemented
253#103—21no—
254#103—21no—
255#102—21no—
256#102—21no—
257C24#1305057511yescharacteristic
258#41—41no—
259#41—41no—
260#41—41no—
261#41—41no—
262C2×Q8—21no—
263C2×D4—21no—
264C2×D4—41no—
265C2×D4—41no—
266C2×D4—41no—
267C22×C4—21no—
268C22×C4—21no—
269C22×C4—21no—
270C22×C4—21no—
271C22×C4—41no—
272C22×C4—3322no—
273C22×C4C22⋊Dic8311yescharacteristic
274C22×C4C22⋊Dic8311yescharacteristic
275C22×C4#1305057511yescharacteristic
276C22×C4#1305057811yescharacteristic
277C22×C4#1305057811yescharacteristic
278C22×C4C2×C22⋊D8311yescharacteristic
279Q16—41no—
280SD16—42no—
281D8—41no—
282M16—41no—
283M16—41no—
284M16—41no—
285M16—41no—
286C2×C8—21no—
287C2×C8—21no—
288C2×C8—21no—
289C2×C8—21no—
290C2×C8C22⋊Dic8311yescharacteristic
291C2×C8C22⋊Dic8311yescharacteristic
292C2×C8C22×Dic8311yescharacteristic
293C2×C8C2×C22⋊D8311yescharacteristic
294C4⋊C4—21no—
295C4⋊C4—21no—
296C4⋊C4—41no—
297C4⋊C4—41no—
298C4⋊C4—3322no—
299C4⋊C4—3322no—
300C22⋊C4—41no—
301C22⋊C4—41no—
302C22⋊C4—41no—
303C22⋊C4—41no—
304C22⋊C4—41no—
305C22⋊C4—3322no—
306C22⋊C4—3322no—
307C42—41no—
308C42—3322no—
309C42—1664no—
310C16—3322no—
311C23—21no—
312C23—21no—
313C23—41no—
314C23—81no—
315C23#1458171311yescharacteristic
316C23#1458171311yescharacteristic
317C23C2×#1305057511yescharacteristic
318Q8—41no—
319Q8—41no—
320D4—41no—
321D4—41no—
322D4—82no—
323D4—82no—
324D4—81no—
325D4—81no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—41no—
337C2×C4—41no—
338C2×C4—41no—
339C2×C4—81no—
340C2×C4—81no—
341C2×C4—3324no—
342C2×C4—6644no—
343C2×C4#1458170711yescharacteristic
344C2×C4#1458170711yescharacteristic
345C2×C4#1458171311yescharacteristic
346C2×C4#1458171311yescharacteristic
347C2×C4#1458171911yescharacteristic
348C2×C4#1458171911yescharacteristic
349C2×C4C2×C22⋊Dic8311yescharacteristic
350C2×C4C2×#1305057811yescharacteristic
351C8—21no—
352C8—21no—
353C8—21no—
354C8—21no—
355C22—21no—
356C22—21no—
357C22—41no—
358C22—41no—
359C22—41no—
360C22—42no—
361C22—81no—
362C22—81no—
363C22—81no—
364C22—81no—
365C22#1912743511yescharacteristic
366C22#1912743511yescharacteristic
367C22C2×#1458171311yescharacteristic
368C4—21no—
369C4—21no—
370C4—81no—
371C4—81no—
372C4—81no—
373C4—6644no—
374C4#1912743511yescharacteristic
375C4#1912743511yescharacteristic
376C4C2×#1458170711yescharacteristic
377C4C2×#1458171911yescharacteristic
378C2—21no—
379C2—21no—
380C2—21no—
381C2—41no—
382C2—41no—
383C2—161no—
384C2C2×#1912743511yescharacteristic
subgroupG/H
centerC4#19127435
commutatorC2×C8×C83C22×C4
FrattiniC2×M16C22×D83
Fitting, prime radicalC83×#3054C2
socleC2×C83C2×#327
cosocleC2×M16×C83C23
hypercenter#3054D83
hypocenterC83#11685
LCSG ▷ C2×C8×C83 ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C4 ◁ C22×C4 ◁ C22×M16 ◁ #3054

Complements

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

NH ≅ G/Ntype
1. C2×#19127483C2semidirect
2. C2×#19127458C2semidirect
3. #24905451C2semidirect
4. #24905448C2semidirect
12. C2×#135×C83C4semidirect
15. #470×C83C4semidirect
17. #469×C83C4semidirect
21. #427×C83C4semidirect
37. C2×Q16×C83C2×C4semidirect
39. C2×D8×C83C2×C4semidirect
58. #106×C83C2×C4semidirect
59. #106×C83C2×C4semidirect
198. C83#11685semidirect
199. C2×#135Dic83semidirect
204. #470Dic83semidirect
206. #469Dic83semidirect
210. #427Dic83semidirect
220. C2×Q16C2×Dic83semidirect
222. C2×D8C2×Dic83semidirect
251. #106C2×Dic83semidirect
252. #106C2×Dic83semidirect

Maximal subgroups

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

subgroupindex
C2×#191274832
C2×#191274582
#249054512
#249054482
C83×#30542
#1168583