Finite Group Databasebeta

← #35301152

Subgroups — #35301152

Order 21248.

480 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#24918591C211yesmaximal, characteristic
2#24908295C211yesmaximal, characteristic
3#24906148C211yesmaximal, characteristic, complemented
4#24906022C211yesmaximal, characteristic
5#24905321C211yesmaximal, characteristic, complemented
6#24905292C211yesmaximal, characteristic, complemented
7C83×#2331C211yesmaximal, characteristic, complemented
8#19128251C2211yescharacteristic
9#19128097C412yescomplemented
10#19128049C412yescomplemented
11#19128019—22no—
12#19127988—22no—
13#19127754C2211yescharacteristic
14#19127754C2211yescharacteristic
15#19127725C2211yescharacteristic
16C2×#14581720—21no—
17C2×#14581720—21no—
18#19127447—22no—
19#19127434—22no—
20C2×C4⋊Q8×C83C2211yescharacteristic
21#391×C83—22no—
22#389×C83—21no—
23#387×C83—22no—
24#382×C83C2211yescharacteristic
25C4×C4⋊C4×C83—21no—
26#376×C83C2211yescharacteristic
27C2×#13050590—21no—
28C2×#13050574—21no—
29#14581792C2×C411yescharacteristic
30#14581789C2×C411yescharacteristic
31#14581780—21no—
32#14581766—21no—
33#14581766—21no—
34C4×D332—21no—
35C4×D332—21no—
36#14581723—24no—
37C4⋊(C83⋊C8)—21no—
38C4⋊(C83⋊C8)—21no—
39C4⋊(C83⋊C8)—21no—
40C4⋊(C83⋊C8)—21no—
41#14581720—42no—
42#14581720—42no—
43#14581712—24no—
44C22⋊(C83⋊C8)—21no—
45C22⋊(C83⋊C8)—21no—
46C4×C83⋊C8—21no—
47C4×C83⋊C8C2×C412yes—
48C22×Q8×C83—21no—
49C4⋊Q8×C83—21no—
50C4⋊Q8×C83—21no—
51C4⋊Q8×C83C2×C412yes—
52C4⋊Q8×C83D412yes—
53C2×C4⋊C4×C83—21no—
54C2×C4⋊C4×C83—21no—
55C2×C4⋊C4×C83—22no—
56C2×C4⋊C4×C83D412yes—
57C2×C42×C83—22no—
58C2×C42×C83D411yescharacteristic
59C2×C42×C83D411yescharacteristic
60C2×C42×C83C2311yescharacteristic
61#95×C83—22no—
62#95×C83—22no—
63#95×C83—21no—
64#95×C83—21no—
65#95×C83—21no—
66#95×C83D412yes—
67#13050590—41no—
68#13050590—41no—
69#13050590—41no—
70#13050590—41no—
71C2×Dic166—21no—
72C2×D332—21no—
73C2×D332—21no—
74C2×D332—21no—
75C2×C22⋊D83—41no—
76C2×C4×D83—41no—
77#13050574—41no—
78#13050574—41no—
79C2×C83⋊C8—22no—
80C2×C83⋊C8—41no—
81C2×C83⋊C8—41no—
82C4⋊Dic83—21no—
83C4⋊Dic83—21no—
84(C2×D83)⋊C4—41no—
85(C2×D83)⋊C4—41no—
86C2×Q8×C83—22no—
87C2×Q8×C83—22no—
88C2×Q8×C83—82no—
89C22×C4×C83—21no—
90C22×C4×C83—22no—
91C22×C4×C83—21no—
92C22×C4×C83—22no—
93C22×C4×C83C2×D411yescharacteristic
94C22×C4×C83C2×D411yescharacteristic
95C22×C4×C83C2×D411yescharacteristic
96C4⋊C4×C83—22no—
97C4⋊C4×C83—22no—
98C4⋊C4×C83—22no—
99C4⋊C4×C83—41no—
100C4⋊C4×C83—41no—
101C4⋊C4×C83—82no—
102C4⋊C4×C83C22⋊C414yes—
103C42×C83—21no—
104C42×C83—21no—
105C42×C83—21no—
106C42×C83—82no—
107C42×C83SD1612yes—
108C42×C83C22×C411yescharacteristic
109C42×C83C2×D411yescharacteristic
110C42×C83#4111yescharacteristic
111C42×C83#4111yescharacteristic
112C23×C83#12011yescharacteristic
113C22×D83—41no—
114Q8×C83—82no—
115Q8×C83—82no—
116C2×C4×C83—24no—
117C2×C4×C83—22no—
118C2×C4×C83—41no—
119C2×C4×C83—41no—
120C2×C4×C83—41no—
121C2×C4×C83—42no—
122C2×C4×C83—42no—
123C2×C4×C83—41no—
124C2×C4×C83—42no—
125C2×C4×C83—41no—
126C2×C4×C83—41no—
127C2×C4×C83—42no—
128C2×C4×C83—41no—
129C2×C4×C83—42no—
130C2×C4×C83C2×C22⋊C411yescharacteristic
131C2×C4×C83C4×D411yescharacteristic
132C2×C4×C83C4×D411yescharacteristic
133C2×C4×C83#12111yescharacteristic
134C2×C4×C83#12111yescharacteristic
135C2×C4×C83#12311yescharacteristic
136C2×C4×C83C2×SD1611yescharacteristic
137C2×C4×C83#13511yescharacteristic
138C2×C4×C83AGL(1,ℤ/8)11yescharacteristic
139C2×C4×C83#13711yescharacteristic
140C22⋊D83—81no—
141C22⋊D83—81no—
142C2×Dic83—41no—
143D332—42no—
144D332—41no—
145D332—41no—
146D332—81no—
147C4×D83—81no—
148C4×D83—81no—
149Dic166—41no—
150Dic166—41no—
151C83⋊C8—42no—
152C83⋊C8—81no—
153C83⋊C8—81no—
154C22×C83—21no—
155C22×C83—21no—
156C22×C83#38611yescharacteristic
157C22×C83#44811yescharacteristic
158C22×C83#44911yescharacteristic
159C2×D83—81no—
160C2×D83—81no—
161C2×D83—81no—
162C4×C83—21no—
163C4×C83—21no—
164C4×C83—21no—
165C4×C83—21no—
166C4×C83—42no—
167C4×C83—44no—
168C4×C83—82no—
169C4×C83—81no—
170C4×C83C4×SD1611yescharacteristic
171C4×C83#44011yescharacteristic
172C4×C83#46011yescharacteristic
173C4×C83#46411yescharacteristic
174Dic83—81no—
175#9302—831nomaximal, Sylow
176C2×C83—21no—
177C2×C83—21no—
178C2×C83#188011yescharacteristic
179C2×C83#190411yescharacteristic
180C2×C83#191211yescharacteristic
181D83—161no—
182#3227—831no—
183#2331D8311yescharacteristic, complemented
184#1963—831no—
185#1549—831no—
186#1517—831no—
187#1286—831no—
188#1274—831no—
189C83#930211yescharacteristic, complemented, Sylow
190#532—831no—
191C2×C4⋊Q8C2×D8311yescharacteristic
192#498—832no—
193#486—832no—
194#481—1662no—
195#476—1662no—
196#431—831no—
197#431—831no—
198#423—831no—
199C2×#104—1661no—
200C2×#104—1661no—
201#391—22no—
202#389—21no—
203#387—22no—
204#382C2×D8311yescharacteristic
205C4×C4⋊C4—21no—
206#376C2×D8311yescharacteristic
207#332—1662no—
208#327—1662no—
209C2×#41—1661no—
210C22×Q8—21no—
211C2×M16—1661no—
212C4⋊Q8—21no—
213C4⋊Q8—21no—
214C4⋊Q8—831no—
215C4⋊Q8C4×D8312yes—
216C4⋊Q8D33212yes—
217#127—831no—
218#124—1661no—
219#121—1661no—
220#121—1661no—
221C4×D4—1661no—
222C4×D4—1661no—
223C2×C4⋊C4—21no—
224C2×C4⋊C4—21no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4C22⋊D8312yes—
227C2×C42—22no—
228C2×C42C22⋊D8311yescharacteristic
229C2×C42C22⋊D8311yescharacteristic
230C2×C42C22×D8311yescharacteristic
231#106—1664no—
232C4⋊C8—1661no—
233C4⋊C8—1661no—
234C4⋊C8—1661no—
235C4⋊C8—1661no—
236#104—3322no—
237#104—3322no—
238#102—1664no—
239C22⋊C8—1661no—
240C22⋊C8—1661no—
241C4×C8—832no—
242C4×C8—1661no—
243#95—22no—
244#95—22no—
245#95—21no—
246#95—21no—
247#95—21no—
248#95C22⋊D8312yes—
249#41—3321no—
250#41—3321no—
251#41—3321no—
252#41—3321no—
253C2×Q8—22no—
254C2×Q8—22no—
255C2×Q8—82no—
256C2×Q8—1661no—
257C2×D4—3321no—
258C2×D4—1661no—
259C2×D4—1661no—
260C2×D4—1661no—
261C22×C4—21no—
262C22×C4—22no—
263C22×C4—21no—
264C22×C4—22no—
265C22×C4—3321no—
266C22×C4C2×C22⋊D8311yescharacteristic
267C22×C4C2×C22⋊D8311yescharacteristic
268C22×C4C2×C22⋊D8311yescharacteristic
269M16—3321no—
270M16—3321no—
271C2×C8—3321no—
272C2×C8—3321no—
273C2×C8—1662no—
274C4⋊C4—22no—
275C4⋊C4—22no—
276C4⋊C4—22no—
277C4⋊C4—41no—
278C4⋊C4—41no—
279C4⋊C4—82no—
280C4⋊C4—1661no—
281C4⋊C4—1661no—
282C4⋊C4(C2×D83)⋊C414yes—
283C22⋊C4—3321no—
284C22⋊C4—3321no—
285C42—21no—
286C42—21no—
287C42—21no—
288C42—82no—
289C42#1305057712yes—
290C42C2×C4×D8311yescharacteristic
291C42C2×D33211yescharacteristic
292C42#1305059011yescharacteristic
293C42#1305059011yescharacteristic
294C23—3321no—
295C23#1458176111yescharacteristic
296Q8—82no—
297Q8—82no—
298Q8—3321no—
299Q8—3321no—
300D4—3322no—
301D4—3321no—
302D4—3321no—
303D4—6641no—
304D4—6641no—
305D4—6641no—
306C2×C4—24no—
307C2×C4—22no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—42no—
312C2×C4—42no—
313C2×C4—41no—
314C2×C4—42no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—42no—
318C2×C4—41no—
319C2×C4—42no—
320C2×C4—3321no—
321C2×C4—6641no—
322C2×C4—6641no—
323C2×C4C2×(C2×D83)⋊C411yescharacteristic
324C2×C4C4×C22⋊D8311yescharacteristic
325C2×C4C4×C22⋊D8311yescharacteristic
326C2×C4#1458176611yescharacteristic
327C2×C4#1458176611yescharacteristic
328C2×C4#1458177511yescharacteristic
329C2×C4C2×#1305057711yescharacteristic
330C2×C4#1458181211yescharacteristic
331C2×C4#1458181711yescharacteristic
332C2×C4#1458182311yescharacteristic
333C8—3322no—
334C8—6641no—
335C8—6641no—
336C22—21no—
337C22—21no—
338C22—6641no—
339C22—6641no—
340C22—6641no—
341C22#1912758011yescharacteristic
342C22#1912784011yescharacteristic
343C22#1912784611yescharacteristic
344C4—21no—
345C4—21no—
346C4—21no—
347C4—21no—
348C4—42no—
349C4—44no—
350C4—82no—
351C4—81no—
352C4—6641no—
353C4C4×#1305057711yescharacteristic
354C4#1912779911yescharacteristic
355C4#1912790111yescharacteristic
356C4#1912792311yescharacteristic
357C2—21no—
358C2—21no—
359C2—13281no—
360C2#2490781811yescharacteristic
361C2#2490795011yescharacteristic
362C2#2490800011yescharacteristic
subgroupG/H
centerC22#19127580
commutatorC42×C83C22×C4
FrattiniC2×C42C22×D83
FittingC83×#2331C2
socleC22×C83#386
prime radical#19128251C22
cosocleC2×C42×C83C23
hypercenter#2331D83
hypocenterC83#9302
LCSG ▷ C42×C83 ▷ C22×C83 ▷ C83
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2331

Complements

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

NH ≅ G/Ntype
3. #24906148C2semidirect
5. #24905321C2semidirect
6. #24905292C2semidirect
7. C83×#2331C2semidirect
9. #19128097C4semidirect
10. #19128049C4semidirect
183. #2331D83semidirect
189. C83#9302semidirect

Maximal subgroups

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

subgroupindex
#249185912
#249082952
#249061482
#249060222
#249053212
#249052922
C83×#23312
#930283