Finite Group Databasebeta

← #35559499

Subgroups — #35559499

Order 21248.

548 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#24918033C211yesmaximal, characteristic
2#24909665C211yesmaximal, characteristic, complemented
3#24909648C211yesmaximal, characteristic
4#24909627C212yesmaximal, complemented
5#24909525C212yesmaximal, complemented
6#24909397C211yesmaximal, characteristic, complemented
7#24909380C211yesmaximal, characteristic
8#24909332C212yesmaximal, complemented
9#24909325C212yesmaximal, complemented
10#24909266C211yesmaximal, characteristic, complemented
11C2×C4×Q16×C83C211yesmaximal, characteristic, complemented
12#19128235—21no—
13C2×#14581738C2212yes—
14C2×#14581735C2212yes—
15#19128103C2211yescharacteristic
16#19128085C2211yescharacteristic
17#19128056—21no—
18#19127942C2211yescharacteristic
19#19127937C2211yescharacteristic
20#19127930—21no—
21C4×Dic332C2211yescharacteristic
22C4×#13050579—21no—
23C4×D664C2211yescharacteristic
24#19127528C2212yes—
25#19127526C2212yes—
26C4⋊(C83⋊C16)C2212yes—
27#19127505C2212yes—
28#19127505C2212yes—
29#19127504C2212yes—
30#19127501C2212yes—
31C22⋊(C83⋊C16)C2211yescharacteristic
32C22⋊(C83⋊C16)C2211yescharacteristic
33#19127477C2212yes—
34C22×Q16×C83C2211yescharacteristic
35C2×C4×Q8×C83—21no—
36C4×Q16×C83—21no—
37C4×Q16×C83—21no—
38C4×Q16×C83C2212yes—
39C4×Q16×C83C2212yes—
40C2×#107×C83C2211yescharacteristic
41C2×#103×C83—21no—
42C2×C4×C8×C83C2211yescharacteristic
43C2×Dic332C2311yescharacteristic
44C2×#13050579—21no—
45C2×D664C2311yescharacteristic
46#14581786—41no—
47#14581784—21no—
48#14581780—21no—
49#14581775—41no—
50#14581771—21no—
51#14581766—21no—
52C4×Dic166—21no—
53C4×D332—21no—
54C4×C22⋊D83—41no—
55#14581749—41no—
56#14581738—24no—
57#14581735—24no—
58C2×C83⋊C16C2312yes—
59C2×C83⋊C16C2312yes—
60#14581726C2311yescharacteristic
61#14581726C2311yescharacteristic
62#14581724—21no—
63#14581717—21no—
64#14581717—21no—
65#14581714—21no—
66#14581714—21no—
67C22×Q8×C83—21no—
68C2×Q16×C83—24no—
69C2×Q16×C83—21no—
70C2×Q16×C83C2312yes—
71C22×C8×C83D411yescharacteristic
72C22×C8×C83C2311yescharacteristic
73C4×Q8×C83—22no—
74C4×Q8×C83—22no—
75C4×Q8×C83—41no—
76C4×Q8×C83—41no—
77C2×C4⋊C4×C83—21no—
78C2×C4⋊C4×C83—41no—
79C2×C42×C83—41no—
80C2×C42×C83D411yescharacteristic
81#107×C83—21no—
82#107×C83C2312yes—
83#103×C83—22no—
84#103×C83—22no—
85C4×C8×C83D411yescharacteristic
86C4×C8×C83D411yescharacteristic
87C4×C8×C83C2311yescharacteristic
88C4×C8×C83C2311yescharacteristic
89C2×Dic166—21no—
90C2×D332—21no—
91C2×C22⋊D83—41no—
92C2×C4×D83—41no—
93Dic332—22no—
94#13050579—42no—
95D664—22no—
96C4⋊Dic83—21no—
97C4⋊Dic83—21no—
98Dic83⋊C4—41no—
99Dic83⋊C4—41no—
100Dic83⋊C4—41no—
101Dic83⋊C4—41no—
102C22⋊Dic83—41no—
103(C2×D83)⋊C4—41no—
104(C2×D83)⋊C4—41no—
105(C2×D83)⋊C4—41no—
106(C2×D83)⋊C4—41no—
107C4×Dic83—41no—
108C83⋊C16—22no—
109C83⋊C16—22no—
110C2×Q8×C83—24no—
111C2×Q8×C83—22no—
112C2×Q8×C83—42no—
113C2×Q8×C83—41no—
114C22×C4×C83—41no—
115C22×C4×C83—41no—
116C22×C4×C83D811yescharacteristic
117C22×C4×C83D811yescharacteristic
118C22×C4×C83C2×D411yescharacteristic
119Q16×C83—24no—
120Q16×C83—22no—
121Q16×C83#4114yes—
122C2×C8×C83—22no—
123C2×C8×C83—41no—
124C2×C8×C83C2×D411yescharacteristic
125C2×C8×C83C2×D411yescharacteristic
126C2×C8×C83C2×D411yescharacteristic
127C2×C8×C83C2411yescharacteristic
128C4⋊C4×C83—22no—
129C4⋊C4×C83—41no—
130C4⋊C4×C83—42no—
131C4⋊C4×C83—42no—
132C42×C83—42no—
133C42×C83—42no—
134C42×C83D811yescharacteristic
135C42×C83D811yescharacteristic
136C42×C83C2×D411yescharacteristic
137C42×C83C2×D411yescharacteristic
138C23×C83C2×D811yescharacteristic
139C22×D83—41no—
140Q8×C83—24no—
141Q8×C83—44no—
142Q8×C83—42no—
143C2×C4×C83—22no—
144C2×C4×C83—21no—
145C2×C4×C83—21no—
146C2×C4×C83—41no—
147C2×C4×C83—44no—
148C2×C4×C83—42no—
149C2×C4×C83—42no—
150C2×C4×C83—82no—
151C2×C4×C83C2×D811yescharacteristic
152C2×C4×C83C2×D811yescharacteristic
153C2×C4×C83C2×D811yescharacteristic
154C2×C4×C83C2×D811yescharacteristic
155C2×C4×C83C2×D811yescharacteristic
156C2×C4×C83C22×D411yescharacteristic
157C22⋊D83—82no—
158C2×Dic83—41no—
159C2×Dic83—41no—
160C2×Dic83—41no—
161D332—41no—
162D332—41no—
163C4×D83—82no—
164Dic166—41no—
165Dic166—41no—
166C8×C83—21no—
167C8×C83—21no—
168C8×C83—21no—
169C8×C83C2×#4112yes—
170C22×C83—21no—
171C22×C83—21no—
172C22×C83#51012yes—
173C22×C83C22×D811yescharacteristic
174C2×D83—81no—
175C2×D83—81no—
176C2×D83—81no—
177C4×C83—21no—
178C4×C83—21no—
179C4×C83—21no—
180C4×C83—21no—
181C4×C83—21no—
182C4×C83—44no—
183C4×C83—82no—
184C4×C83#52511yescharacteristic
185C4×C83#57511yescharacteristic
186Dic83—81no—
187Dic83—81no—
188Dic83—81no—
189#33677—831nomaximal, Sylow
190C2×C83—22no—
191C2×C83#310011yescharacteristic
192C2×C83C2×#51011yescharacteristic
193C2×C83#341411yescharacteristic
194D83—161no—
195#3168—831no—
196C2×C4×Q16D8311yescharacteristic, complemented
197#2236—831no—
198#2233—831no—
199#2229—832no—
200#2211—832no—
201#2188—831no—
202#2185—831no—
203#2176—832no—
204#2175—832no—
205#2163—831no—
206C83#3367711yescharacteristic, complemented, Sylow
207C22×Q16C2×D8311yescharacteristic
208#529—1661no—
209C2×C4×Q8—21no—
210C2×Q32—832no—
211C2×SD32—832no—
212#499—831no—
213#495—831no—
214#488—1661no—
215#467—831no—
216#466—831no—
217#465—1661no—
218C4×Q16—21no—
219C4×Q16—21no—
220C4×Q16—831no—
221C4×Q16C2×D8312yes—
222C4×Q16C2×D8312yes—
223C4×SD16—1661no—
224C4×D8—831no—
225C2×#107C2×D8311yescharacteristic
226C2×#103—21no—
227C2×C4×C8C2×D8311yescharacteristic
228#367—832no—
229#366—832no—
230C4⋊C16—832no—
231#358—832no—
232#358—832no—
233#358—832no—
234#357—832no—
235C22⋊C16—831no—
236C22⋊C16—831no—
237#346—832no—
238C22×Q8—21no—
239C2×Q16—24no—
240C2×Q16—21no—
241C2×Q16—831no—
242C2×Q16C22×D8312yes—
243C2×SD16—1661no—
244C2×D8—831no—
245C22×C8C22⋊D8311yescharacteristic
246C22×C8C22×D8311yescharacteristic
247#126—3321no—
248#125—1661no—
249#124—1661no—
250#123—3321no—
251C22⋊Q8—1661no—
252#121—1661no—
253C4×Q8—22no—
254C4×Q8—22no—
255C4×Q8—41no—
256C4×Q8—41no—
257C4×Q8—1661no—
258C4×D4—3321no—
259C4×D4—1661no—
260#117—3321no—
261C2×C4⋊C4—21no—
262C2×C4⋊C4—41no—
263C2×C42—41no—
264C2×C42C22⋊D8311yescharacteristic
265Q32—1664no—
266SD32—1664no—
267C2×C16—832no—
268C2×C16—832no—
269#107—21no—
270#107—831no—
271#107—831no—
272#107C22×D8312yes—
273#106—1661no—
274#103—22no—
275#103—22no—
276#103—1661no—
277#103—1661no—
278#102—1661no—
279#102—1661no—
280C4×C8C22⋊D8311yescharacteristic
281C4×C8C22⋊D8311yescharacteristic
282C4×C8C22×D8311yescharacteristic
283C4×C8C22×D8311yescharacteristic
284C2×Q8—24no—
285C2×Q8—22no—
286C2×Q8—42no—
287C2×Q8—41no—
288C2×Q8—1661no—
289C2×D4—3321no—
290C2×D4—1661no—
291C22×C4—41no—
292C22×C4—41no—
293C22×C4—3321no—
294C22×C4#1305057511yescharacteristic
295C22×C4#1305057511yescharacteristic
296C22×C4C2×C22⋊D8311yescharacteristic
297Q16—24no—
298Q16—22no—
299Q16—1662no—
300Q16#1305059014yes—
301SD16—3322no—
302D8—1662no—
303C2×C8—22no—
304C2×C8—41no—
305C2×C8C2×C22⋊D8311yescharacteristic
306C2×C8C2×C22⋊D8311yescharacteristic
307C2×C8C2×C22⋊D8311yescharacteristic
308C2×C8C23×D8311yescharacteristic
309C4⋊C4—22no—
310C4⋊C4—41no—
311C4⋊C4—42no—
312C4⋊C4—42no—
313C4⋊C4—3321no—
314C4⋊C4—3321no—
315C4⋊C4—3321no—
316C4⋊C4—3321no—
317C4⋊C4—1661no—
318C4⋊C4—1661no—
319C22⋊C4—3321no—
320C22⋊C4—3321no—
321C22⋊C4—3321no—
322C22⋊C4—3321no—
323C22⋊C4—3321no—
324C42—42no—
325C42—42no—
326C42—3321no—
327C42#1305057511yescharacteristic
328C42#1305057511yescharacteristic
329C42C2×C22⋊D8311yescharacteristic
330C42C2×C22⋊D8311yescharacteristic
331C16—1662no—
332C16—1662no—
333C23—3321no—
334C23C2×#1305057511yescharacteristic
335Q8—24no—
336Q8—44no—
337Q8—42no—
338Q8—3321no—
339Q8—3321no—
340D4—3321no—
341D4—3321no—
342D4—6642no—
343C2×C4—22no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—41no—
347C2×C4—44no—
348C2×C4—42no—
349C2×C4—42no—
350C2×C4—82no—
351C2×C4—3321no—
352C2×C4—3321no—
353C2×C4—3321no—
354C2×C4—6642no—
355C2×C4C2×#1305057511yescharacteristic
356C2×C4C2×#1305057511yescharacteristic
357C2×C4C2×#1305057511yescharacteristic
358C2×C4C2×#1305057511yescharacteristic
359C2×C4C2×#1305057511yescharacteristic
360C2×C4C22×C22⋊D8311yescharacteristic
361C8—21no—
362C8—21no—
363C8—21no—
364C8C2×#1305059012yes—
365C22—21no—
366C22—21no—
367C22—6641no—
368C22—6641no—
369C22—6641no—
370C22#1912814212yes—
371C22C22×#1305057511yescharacteristic
372C4—21no—
373C4—21no—
374C4—21no—
375C4—21no—
376C4—21no—
377C4—44no—
378C4—82no—
379C4—6641no—
380C4—6641no—
381C4—6641no—
382C4#1912821211yescharacteristic
383C4#1912849911yescharacteristic
384C2—22no—
385C2—13281no—
386C2#2491746311yescharacteristic
387C2C2×#1912814211yescharacteristic
388C2#2492059711yescharacteristic
subgroupG/H
centerC22C22×#13050575
commutator, cosocleC2×C8×C83C24
FrattiniC2×C8C23×D83
FittingC2×C4×Q16×C83C2
socleC22×C83C22×D8
prime radical#19127937C22
hypercenterC2×C4×Q16D83
hypocenterC83#33677
LCSG ▷ C2×C8×C83 ▷ C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×C4×C8 ◁ C2×C4×Q16

Complements

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

NH ≅ G/Ntype
2. #24909665C2semidirect
4. #24909627C2semidirect
5. #24909525C2semidirect
6. #24909397C2semidirect
8. #24909332C2semidirect
9. #24909325C2semidirect
10. #24909266C2semidirect
11. C2×C4×Q16×C83C2semidirect
196. C2×C4×Q16D83semidirect
206. C83#33677semidirect

Maximal subgroups

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

subgroupindex
#249180332
#249096652
#249096482
#249096272
#249095252
#249093972
#249093802
#249093322
#249093252
#249092662
C2×C4×Q16×C832
#3367783