Finite Group Databasebeta

← #35559124

Subgroups — #35559124

Order 21248.

534 conjugacy classes of subgroups.

The subgroup lattice has 348 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#24918002C211yesmaximal, characteristic, complemented
2#24909627C212yesmaximal, complemented
3#24909620C212yesmaximal, complemented
4#24909610C212yesmaximal, complemented
5#24909605C212yesmaximal, complemented
6#24909249C211yesmaximal, characteristic, complemented
7C2×#19127528C211yesmaximal, characteristic, complemented
8#24909222C211yesmaximal, characteristic, complemented
9#24909175C211yesmaximal, characteristic
10#24909170C211yesmaximal, characteristic
11C83×#2951C211yesmaximal, characteristic
12#19128329—21no—
13C2×#14581732C2211yescharacteristic
14C22×C83⋊C16C2211yescharacteristic
15#19128110—22no—
16#19128106C2212yes—
17#19128103C2212yes—
18#19127743—21no—
19#19127739C2211yescharacteristic
20C2×#14581726C2211yescharacteristic
21#19127528C2212yes—
22#19127528C2212yes—
23#19127528C2212yes—
24#19127526C2212yes—
25C4⋊(C83⋊C16)C2212yes—
26C4⋊(C83⋊C16)C2212yes—
27#19127505C2211yescharacteristic
28#19127505C2212yes—
29#19127505C2211yescharacteristic
30#19127502C2211yescharacteristic
31#19127502C2212yes—
32#19127502C2211yescharacteristic
33C2×#135×C83C2211yescharacteristic
34#520×C83—21no—
35#440×C83—22no—
36C4×Q16×C83C2212yes—
37C4×D8×C83C2212yes—
38C2×#107×C83C2211yescharacteristic
39#417×C83—21no—
40#405×C83C2211yescharacteristic
41#14581792—22no—
42#14581791—41no—
43#14581784—22no—
44#14581783—41no—
45#14581769—42no—
46#14581751—21no—
47C2×C4⋊Dic83—21no—
48#14581732—22no—
49C2×C83⋊C16—22no—
50C2×C83⋊C16C2311yescharacteristic
51C2×C83⋊C16C2311yescharacteristic
52C2×C83⋊C16C2312yes—
53#14581726—22no—
54#14581726—21no—
55#14581726C2311yescharacteristic
56#14581726C2312yes—
57#14581726C2311yescharacteristic
58#14581724—22no—
59#14581724—21no—
60C2×#41×C83—21no—
61#135×C83—22no—
62#135×C83Q814yescomplemented
63C2×Q16×C83C2311yescharacteristic
64C2×SD16×C83—21no—
65C2×D8×C83C2311yescharacteristic
66C2×M16×C83D411yescharacteristic
67C22×C8×C83C2311yescharacteristic
68C4×Q8×C83—22no—
69C4×D4×C83—22no—
70C4×D4×C83—42no—
71#117×C83—41no—
72#117×C83D411yescharacteristic
73C2×C4⋊C4×C83—21no—
74C2×C4⋊C4×C83—41no—
75#107×C83—21no—
76#107×C83C2312yes—
77#103×C83—22no—
78#102×C83—22no—
79#97×C83D412yes—
80C4×C8×C83C2312yes—
81C2×Dic166—42no—
82C22×Dic83—41no—
83C4⋊Dic83—21no—
84C4⋊Dic83—22no—
85C4⋊Dic83—21no—
86C4⋊Dic83—42no—
87C4⋊Dic83—41no—
88Dic83⋊C4—42no—
89Dic83⋊C4—42no—
90C22⋊Dic83—41no—
91C4×Dic83—41no—
92C83⋊C16—21no—
93C83⋊C16—21no—
94C83⋊C16—22no—
95#41×C83—24no—
96#41×C83—42no—
97C2×Q8×C83—21no—
98C2×D4×C83—21no—
99C2×D4×C83—41no—
100C22×C4×C83—41no—
101C22×C4×C83—41no—
102C22×C4×C83—42no—
103C22×C4×C83C2×D411yescharacteristic
104Q16×C83—21no—
105Q16×C83C2×Q812yes—
106SD16×C83—24no—
107D8×C83—21no—
108D8×C83C2×Q812yes—
109M16×C83—22no—
110C2×C8×C83C2×D412yes—
111C2×C8×C83C2×D411yescharacteristic
112C2×C8×C83C2×Q812yes—
113C2×C8×C83#4112yes—
114C2×C8×C83C2411yescharacteristic
115C4⋊C4×C83—22no—
116C4⋊C4×C83—41no—
117C4⋊C4×C83—42no—
118C4⋊C4×C83D811yescharacteristic
119C4⋊C4×C83D811yescharacteristic
120C22⋊C4×C83—42no—
121C22⋊C4×C83D812yes—
122C42×C83—42no—
123C42×C83C2×D412yes—
124C23×C83—41no—
125C23×C83C2×D811yescharacteristic
126Q8×C83—22no—
127Q8×C83—41no—
128D4×C83—22no—
129D4×C83—44no—
130D4×C83—41no—
131C2×C4×C83—41no—
132C2×C4×C83—44no—
133C2×C4×C83—42no—
134C2×C4×C83—162no—
135C2×C4×C83—161no—
136C2×C4×C83C22⋊Q812yes—
137C2×C4×C83C2×D812yes—
138C2×C4×C83C2×D812yes—
139C2×C4×C83C2×D811yescharacteristic
140C2×C4×C83#13712yes—
141C2×C4×C83C22×D411yescharacteristic
142C2×Dic83—41no—
143C2×Dic83—41no—
144C2×Dic83—42no—
145C2×Dic83—161no—
146Dic166—162no—
147C8×C83—22no—
148C8×C83C22⋊Q812yes—
149C8×C83C22×Q811yescharacteristic
150C8×C83C2×#4111yescharacteristic
151C22×C83—21no—
152C22×C83—21no—
153C22×C83—42no—
154C22×C83—81no—
155C22×C83—81no—
156C22×C83#47412yes—
157C22×C83C22×D811yescharacteristic
158C4×C83—22no—
159C4×C83—22no—
160C4×C83—42no—
161C4×C83—82no—
162C4×C83#47412yes—
163C4×C83C2×C22⋊Q811yescharacteristic
164C4×C83C2×#13711yescharacteristic
165Dic83—81no—
166Dic83—82no—
167Dic83—81no—
168#33634—831nomaximal, Sylow
169C2×C83—22no—
170C2×C83—82no—
171C2×C83C2×#47411yescharacteristic
172C2×C83#341611yescharacteristic
173C2×C83#341611yescharacteristic
174#3164—831no—
175#2951D8311yescharacteristic
176#2229—832no—
177#2228—832no—
178#2226—832no—
179#2225—832no—
180#2159—831no—
181C2×#367—831no—
182#2151—831no—
183#2139—831no—
184#2138—831no—
185C83#3363411yescharacteristic, complemented, Sylow
186C2×#135C2×D8311yescharacteristic
187#544—1661no—
188#520—21no—
189C2×M32—831no—
190C22×C16—831no—
191#501—1662no—
192#500—832no—
193#499—832no—
194#440—22no—
195C4×Q16C2×D8312yes—
196C4×D8C2×D8312yes—
197#428—1661no—
198#427—831no—
199C2×#107—831no—
200C2×#107C2×D8311yescharacteristic
201#417—21no—
202#405C2×D8311yescharacteristic
203#367—832no—
204#367—832no—
205#367—832no—
206#366—832no—
207C4⋊C16—832no—
208C4⋊C16—832no—
209#358—831no—
210#358—831no—
211#358—832no—
212#357—831no—
213#357—831no—
214#357—832no—
215C2×#41—21no—
216#135—22no—
217#135Dic16614yescomplemented
218C2×Q16C22×D8311yescharacteristic
219C2×SD16—21no—
220C2×D8C22×D8311yescharacteristic
221C2×M16C22⋊D8311yescharacteristic
222C22×C8C22×D8311yescharacteristic
223C4⋊Q8—3321no—
224C4⋊Q8—1662no—
225#125—3321no—
226#125—1662no—
227C22⋊Q8—3322no—
228C4×Q8—22no—
229C4×D4—22no—
230C4×D4—42no—
231#117—41no—
232#117—1661no—
233#117C22⋊D8311yescharacteristic
234C2×C4⋊C4—21no—
235C2×C4⋊C4—41no—
236C2×C4⋊C4—1661no—
237M32—1662no—
238C2×C16—832no—
239C2×C16—831no—
240C2×C16—831no—
241C2×C16—1662no—
242#107—21no—
243#107—831no—
244#107—832no—
245#107—831no—
246#107—1661no—
247#107—1662no—
248#107C22×D8312yes—
249#106—1661no—
250#106—1662no—
251#103—22no—
252#102—22no—
253#97C22⋊D8312yes—
254C4×C8C22×D8312yes—
255#41—24no—
256#41—42no—
257C2×Q8—21no—
258C2×Q8—3322no—
259C2×D4—21no—
260C2×D4—41no—
261C22×C4—41no—
262C22×C4—41no—
263C22×C4—42no—
264C22×C4—3321no—
265C22×C4C2×C22⋊D8311yescharacteristic
266Q16—21no—
267Q16C2×Dic16612yes—
268SD16—24no—
269D8—21no—
270D8C2×Dic16612yes—
271M16—22no—
272C2×C8C2×C22⋊D8312yes—
273C2×C8C2×C22⋊D8311yescharacteristic
274C2×C8C2×Dic16612yes—
275C2×C8#1305059012yes—
276C2×C8C23×D8311yescharacteristic
277C4⋊C4—22no—
278C4⋊C4—41no—
279C4⋊C4—42no—
280C4⋊C4—3321no—
281C4⋊C4—3322no—
282C4⋊C4—3322no—
283C4⋊C4—3322no—
284C4⋊C4—1661no—
285C4⋊C4—1662no—
286C4⋊C4—1661no—
287C4⋊C4#1305057511yescharacteristic
288C4⋊C4#1305057511yescharacteristic
289C22⋊C4—42no—
290C22⋊C4—3321no—
291C22⋊C4#1305057512yes—
292C42—42no—
293C42—3321no—
294C42C2×C22⋊D8312yes—
295C16—1661no—
296C16—1661no—
297C16—1662no—
298C23—41no—
299C23C2×#1305057511yescharacteristic
300Q8—22no—
301Q8—41no—
302Q8—13282no—
303D4—22no—
304D4—44no—
305D4—41no—
306C2×C4—41no—
307C2×C4—44no—
308C2×C4—42no—
309C2×C4—162no—
310C2×C4—161no—
311C2×C4—13281no—
312C2×C4—3321no—
313C2×C4—3321no—
314C2×C4—3322no—
315C2×C4#1458177112yes—
316C2×C4C2×#1305057512yes—
317C2×C4C2×#1305057512yes—
318C2×C4C2×#1305057511yescharacteristic
319C2×C4#1458182212yes—
320C2×C4C22×C22⋊D8311yescharacteristic
321C8—22no—
322C8#1458177112yes—
323C8C22×Dic16611yescharacteristic
324C8C2×#1305059011yescharacteristic
325C22—21no—
326C22—21no—
327C22—42no—
328C22—81no—
329C22—81no—
330C22#1912798112yes—
331C22C22×#1305057511yescharacteristic
332C4—22no—
333C4—22no—
334C4—42no—
335C4—82no—
336C4—6642no—
337C4—6641no—
338C4—6641no—
339C4#1912798112yes—
340C4C2×#1458177111yescharacteristic
341C4C2×#1458182211yescharacteristic
342C2—22no—
343C2—82no—
344C2C2×#1912798111yescharacteristic
345C2#2492060711yescharacteristic
346C2#2492061011yescharacteristic
subgroupG/H
centerC22C22×#13050575
commutator, cosocleC2×C8×C83C24
FrattiniC2×C8C23×D83
FittingC83×#2951C2
socleC22×C83C22×D8
prime radicalC2×#135×C83C22
hypercenter#2951D83
hypocenterC83#33634
LCSG ▷ C2×C8×C83 ▷ C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ #117 ◁ #405 ◁ #2951

Complements

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

NH ≅ G/Ntype
1. #24918002C2semidirect
2. #24909627C2semidirect
3. #24909620C2semidirect
4. #24909610C2semidirect
5. #24909605C2semidirect
6. #24909249C2semidirect
7. C2×#19127528C2semidirect
8. #24909222C2semidirect
62. #135×C83Q8semidirect
185. C83#33634semidirect
217. #135Dic166semidirect

Maximal subgroups

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

subgroupindex
#249180022
#249096272
#249096202
#249096102
#249096052
#249092492
C2×#191275282
#249092222
#249091752
#249091702
C83×#29512
#3363483