Finite Group Databasebeta

← #35560719

Subgroups — #35560719

Order 21248.

496 conjugacy classes of subgroups.

The subgroup lattice has 390 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#24918049C211yesmaximal, characteristic, complemented
2#24909624C212yesmaximal, complemented
3#24909605C211yesmaximal, characteristic, complemented
4#24909600C211yesmaximal, characteristic, complemented
5#24909557C212yesmaximal
6#24909552C212yesmaximal, complemented
7#24909522C212yesmaximal, complemented
8#24909513C211yesmaximal, characteristic, complemented
9#24909508C211yesmaximal, characteristic, complemented
10C2×C4⋊(C83⋊C16)C211yesmaximal, characteristic, complemented
11C83×#3168C211yesmaximal, characteristic
12#19128258—21no—
13C2×#14581738C2212yes—
14C2×#14581736C2212yes—
15C22×C83⋊C16C2212yes—
16#19128106C2211yescharacteristic
17#19128081C2211yescharacteristic
18#19128052—21no—
19#19127942—21no—
20#19127942C2212yes—
21C4×Dic332—21no—
22C4×Dic332C2212yes—
23#19127528C2212yes—
24#19127526C2212yes—
25C4⋊(C83⋊C16)C2211yescharacteristic
26C4⋊(C83⋊C16)C2211yescharacteristic
27C4⋊(C83⋊C16)C2212yes—
28#19127505C2212yes—
29#19127504C2212yes—
30#19127504C2212yes—
31#19127502C2212yes—
32#529×C83—21no—
33#499×C83C2211yescharacteristic
34#495×C83C2211yescharacteristic
35#488×C83—21no—
36#467×C83C2211yescharacteristic
37#466×C83C2211yescharacteristic
38#465×C83—21no—
39C4×Q16×C83C2211yescharacteristic
40C4×SD16×C83—21no—
41C4×D8×C83C2211yescharacteristic
42C2×C4×C8×C83C2211yescharacteristic
43C2×Dic332—21no—
44C2×Dic332C2312yes—
45#14581792—21no—
46#14581792—21no—
47#14581784—41no—
48#14581771—22no—
49#14581771—41no—
50C4×Dic166—22no—
51C4×Dic166—41no—
52#14581751—41no—
53#14581738—42no—
54#14581736—42no—
55C2×C83⋊C16—42no—
56C2×C83⋊C16C2312yes—
57C2×C83⋊C16C2312yes—
58#14581726—21no—
59#14581726C2312yes—
60#14581717—22no—
61#14581717—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
89C2×Dic166—22no—
90C2×Dic166—41no—
91Dic332—22no—
92Dic332—22no—
93Dic332—42no—
94C4⋊Dic83—22no—
95C4⋊Dic83—41no—
96C4⋊Dic83—41no—
97C4⋊Dic83—41no—
98Dic83⋊C4—42no—
99Dic83⋊C4—42no—
100C22⋊Dic83—42no—
101C4×Dic83—42no—
102C83⋊C16—22no—
103C83⋊C16—22no—
104C2×Q8×C83—21no—
105C2×D4×C83—21no—
106C2×D4×C83—41no—
107C22×C4×C83—41no—
108C22×C4×C83D811yescharacteristic
109C22×C4×C83D811yescharacteristic
110C22×C4×C83C2×D411yescharacteristic
111Q16×C83—21no—
112Q16×C83—21no—
113SD16×C83—42no—
114D8×C83—21no—
115D8×C83—21no—
116C2×C8×C83—21no—
117C2×C8×C83—21no—
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
141Q8×C83—41no—
142Q8×C83—41no—
143D4×C83—41no—
144D4×C83—41no—
145D4×C83—82no—
146C2×C4×C83—22no—
147C2×C4×C83—21no—
148C2×C4×C83—21no—
149C2×C4×C83—22no—
150C2×C4×C83—41no—
151C2×C4×C83—41no—
152C2×C4×C83—41no—
153C2×C4×C83—82no—
154C2×C4×C83C2×D811yescharacteristic
155C2×C4×C83C2×D811yescharacteristic
156C2×C4×C83C2×D811yescharacteristic
157C2×C4×C83C2×D811yescharacteristic
158C2×C4×C83C2×D811yescharacteristic
159C2×C4×C83C22×D411yescharacteristic
160C2×Dic83—42no—
161C2×Dic83—42no—
162Dic166—42no—
163Dic166—42no—
164Dic166—82no—
165C8×C83—21no—
166C8×C83—21no—
167C8×C83—21no—
168C8×C83H(5,𝔽2)11yescharacteristic
169C8×C832−1+411yescharacteristic
170C22×C83—21no—
171C22×C83—21no—
172C22×C83—81no—
173C22×C83—81no—
174C22×C83—81no—
175C22×C83#51012yes—
176C22×C83C22×D811yescharacteristic
177C4×C83—21no—
178C4×C83—21no—
179C4×C83—21no—
180C4×C83—21no—
181C4×C83—81no—
182C4×C83—81no—
183C4×C83—81no—
184C4×C83#50812yes—
185C4×C83#53711yescharacteristic
186C4×C83#57511yescharacteristic
187Dic83—82no—
188Dic83—82no—
189#33814—831nomaximal, Sylow
190C2×C83—22no—
191C2×C83—161no—
192C2×C83#324511yescharacteristic
193C2×C83C2×#50811yescharacteristic
194C2×C83C2×#51011yescharacteristic
195#3170—831no—
196#3168D8311yescharacteristic
197#2229—832no—
198#2225—831no—
199#2224—831no—
200#2217—832no—
201#2216—832no—
202#2211—832no—
203#2209—831no—
204#2208—831no—
205C2×C4⋊C16—831no—
206C83#3381411yescharacteristic, complemented, Sylow
207#533—1661no—
208#529—21no—
209C2×Q32—832no—
210C2×SD32—832no—
211C22×C16—832no—
212#500—831no—
213#499C2×D8311yescharacteristic
214#495C2×D8311yescharacteristic
215#494—831no—
216#488—21no—
217#487—1661no—
218#467—832no—
219#467—1661no—
220#467C2×D8311yescharacteristic
221#466C2×D8311yescharacteristic
222#465—21no—
223C4×Q16—832no—
224C4×Q16—1661no—
225C4×Q16C2×D8311yescharacteristic
226C4×SD16—21no—
227C4×D8C2×D8311yescharacteristic
228C2×C4×C8C2×D8311yescharacteristic
229#367—832no—
230#366—832no—
231C4⋊C16—831no—
232C4⋊C16—831no—
233C4⋊C16—832no—
234#358—832no—
235#358—832no—
236#358—832no—
237#357—832no—
238C2×Q16—832no—
239C2×Q16—1661no—
240C2×Q16C22×D8311yescharacteristic
241C2×SD16—21no—
242C2×D8C22×D8311yescharacteristic
243C22×C8C22⋊D8311yescharacteristic
244C22×C8C22×D8311yescharacteristic
245C4⋊Q8—1661no—
246C4⋊Q8—1661no—
247#126—41no—
248#125—21no—
249#125—3321no—
250#124—21no—
251#123—41no—
252C22⋊Q8—21no—
253C22⋊Q8—3321no—
254C22⋊Q8—1662no—
255#121—21no—
256C4×Q8—21no—
257C4×Q8—3321no—
258C4×Q8—1662no—
259C4×D4—21no—
260C4×D4—41no—
261#117—41no—
262#117—3321no—
263C2×C42C22⋊D8311yescharacteristic
264Q32—3322no—
265SD32—3322no—
266C2×C16—3322no—
267C2×C16—832no—
268C2×C16—832no—
269#107—832no—
270#107—1661no—
271#107C22×D8311yescharacteristic
272#107C22×D8311yescharacteristic
273#106—21no—
274#103—21no—
275#103—21no—
276#103—1662no—
277#103—1662no—
278#102—21no—
279#102—21no—
280C4×C8C22⋊D8311yescharacteristic
281C4×C8C22⋊D8311yescharacteristic
282C4×C8C22×D8311yescharacteristic
283C4×C8C22×D8311yescharacteristic
284C2×Q8—21no—
285C2×Q8—3321no—
286C2×Q8—1662no—
287C2×D4—21no—
288C2×D4—41no—
289C22×C4—41no—
290C22×C4#1305057511yescharacteristic
291C22×C4#1305057511yescharacteristic
292C22×C4C2×C22⋊D8311yescharacteristic
293Q16—21no—
294Q16—21no—
295Q16—3322no—
296Q16—1662no—
297Q16—1662no—
298SD16—42no—
299D8—21no—
300D8—21no—
301C2×C8—21no—
302C2×C8—21no—
303C2×C8—41no—
304C2×C8C2×C22⋊D8311yescharacteristic
305C2×C8C2×C22⋊D8311yescharacteristic
306C2×C8C2×C22⋊D8311yescharacteristic
307C2×C8C23×D8311yescharacteristic
308C4⋊C4—21no—
309C4⋊C4—21no—
310C4⋊C4—41no—
311C4⋊C4—41no—
312C4⋊C4—41no—
313C4⋊C4—41no—
314C4⋊C4—3321no—
315C4⋊C4—3321no—
316C4⋊C4—3321no—
317C4⋊C4—3322no—
318C4⋊C4—3322no—
319C4⋊C4—1662no—
320C22⋊C4—41no—
321C22⋊C4—41no—
322C22⋊C4—41no—
323C22⋊C4—41no—
324C22⋊C4—41no—
325C22⋊C4—3322no—
326C42—41no—
327C42—3322no—
328C42#1305057511yescharacteristic
329C42#1305057511yescharacteristic
330C42C2×C22⋊D8311yescharacteristic
331C42C2×C22⋊D8311yescharacteristic
332C16—1662no—
333C16—1662no—
334C23—41no—
335C23C2×#1305057511yescharacteristic
336Q8—41no—
337Q8—41no—
338Q8—3322no—
339Q8—3322no—
340Q8—6642no—
341D4—41no—
342D4—41no—
343D4—82no—
344C2×C4—22no—
345C2×C4—22no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—41no—
351C2×C4—82no—
352C2×C4—3322no—
353C2×C4—3322no—
354C2×C4C2×#1305057511yescharacteristic
355C2×C4C2×#1305057511yescharacteristic
356C2×C4C2×#1305057511yescharacteristic
357C2×C4C2×#1305057511yescharacteristic
358C2×C4C2×#1305057511yescharacteristic
359C2×C4C22×C22⋊D8311yescharacteristic
360C8—21no—
361C8—21no—
362C8—21no—
363C8#1458183811yescharacteristic
364C8#1458183911yescharacteristic
365C22—21no—
366C22—21no—
367C22—81no—
368C22—81no—
369C22—81no—
370C22#1912814112yes—
371C22C22×#1305057511yescharacteristic
372C4—21no—
373C4—21no—
374C4—21no—
375C4—21no—
376C4—81no—
377C4—81no—
378C4—81no—
379C4—6642no—
380C4—6642no—
381C4#1912813112yes—
382C4#1912827911yescharacteristic
383C4#1912849911yescharacteristic
384C2—22no—
385C2—161no—
386C2#2491877011yescharacteristic
387C2C2×#1912813111yescharacteristic
388C2C2×#1912814111yescharacteristic
subgroupG/H
centerC22C22×#13050575
commutator, cosocleC2×C8×C83C24
FrattiniC2×C8C23×D83
FittingC83×#3168C2
socleC22×C83C22×D8
prime radical#466×C83C22
hypercenter#3168D83
hypocenterC83#33814
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. #24918049C2semidirect
2. #24909624C2semidirect
3. #24909605C2semidirect
4. #24909600C2semidirect
6. #24909552C2semidirect
7. #24909522C2semidirect
8. #24909513C2semidirect
9. #24909508C2semidirect
10. C2×C4⋊(C83⋊C16)C2semidirect
206. C83#33814semidirect

Maximal subgroups

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

subgroupindex
#249180492
#249096242
#249096052
#249096002
#249095572
#249095522
#249095222
#249095132
#249095082
C2×C4⋊(C83⋊C16)2
C83×#31682
#3381483