Finite Group Databasebeta

← #35308232

Subgroups — #35308232

Order 21248.

506 conjugacy classes of subgroups.

The subgroup lattice has 420 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19127907C211yesmaximal, characteristic
2#24908423C211yesmaximal, characteristic, complemented
3#24908256C211yesmaximal, characteristic
4#24907460C211yesmaximal, characteristic
5C2×#19127451C211yesmaximal, characteristic, complemented
6C2×#19127446C211yesmaximal, characteristic, complemented
7C83×#1296C211yesmaximal, characteristic, complemented
8C22×#13050579—21no—
9C2×#14581792C2211yescharacteristic
10C2×C4×D332C2211yescharacteristic
11#19127907—22no—
12#19127907—22no—
13C2×C4⋊(C83⋊C8)C2211yescharacteristic
14C2×#14581717—21no—
15C2×#14581712—21no—
16C2×(C2×D83)⋊C8—21no—
17C2×(C2×D83)⋊C8—21no—
18C2×C4×C83⋊C8C2211yescharacteristic
19#19127463—21no—
20#19127463—21no—
21#19127463—21no—
22#19127451—22no—
23#19127446—22no—
24C2×C4⋊C8×C83C2211yescharacteristic
25C2×C4⋊C8×C83C2211yescharacteristic
26#384×C83C2211yescharacteristic
27C22×Dic166—21no—
28C22×D332D411yescharacteristic
29C22×C4×D83—21no—
30C2×#13050579—24no—
31C2×#13050579—42no—
32C2×#13050579—42no—
33C22×C83⋊C8—21no—
34C22×C83⋊C8D411yescharacteristic
35C22×C83⋊C8D411yescharacteristic
36#14581792—21no—
37#14581792—21no—
38#14581792—21no—
39#14581792D412yes—
40C4×D332—21no—
41C4×D332—21no—
42C4×D332—21no—
43C4×D332—21no—
44C2×C4⋊Dic83—21no—
45C2×C4⋊Dic83—21no—
46C2×C4⋊Dic83D411yescharacteristic
47C2×(C2×D83)⋊C4—21no—
48C4⋊(C83⋊C8)—21no—
49C4⋊(C83⋊C8)—21no—
50#14581717—24no—
51#14581712—42no—
52(C2×D83)⋊C8—42no—
53(C2×D83)⋊C8—42no—
54C4×C83⋊C8—21no—
55C4×C83⋊C8—21no—
56C22×C8×C83—21no—
57C22×C8×C83—21no—
58C2×C4⋊C4×C83—21no—
59C2×C4⋊C4×C83—21no—
60C2×C4⋊C4×C83—21no—
61C2×C42×C83C2311yescharacteristic
62C4⋊C8×C83—21no—
63C4⋊C8×C83—21no—
64C4⋊C8×C83—21no—
65C4⋊C8×C83D412yes—
66C23×D83—21no—
67C2×Dic166—22no—
68C2×Dic166—41no—
69C2×Dic166—41no—
70C2×Dic166—41no—
71C2×Dic166—41no—
72C2×Dic166—41no—
73C2×D332—21no—
74C2×D332—21no—
75C2×D332—21no—
76C2×D332—21no—
77C2×D332—21no—
78C2×D332SD1612yes—
79C22×Dic83—21no—
80C22×Dic83—41no—
81C2×C4×D83—41no—
82C2×C4×D83—41no—
83C2×C4×D83—41no—
84C2×C4×D83—41no—
85C2×C4×D83—41no—
86C2×C4×D83—41no—
87#13050579—48no—
88C2×C83⋊C8—21no—
89C2×C83⋊C8—21no—
90C2×C83⋊C8—21no—
91C2×C83⋊C8—21no—
92C2×C83⋊C8—21no—
93C2×C83⋊C8—41no—
94C2×C83⋊C8—41no—
95C2×C83⋊C8—41no—
96C2×C83⋊C8SD1612yescomplemented
97C4⋊Dic83—21no—
98C4⋊Dic83—22no—
99C4⋊Dic83—22no—
100C4⋊Dic83—21no—
101C4⋊Dic83—41no—
102C4⋊Dic83—41no—
103(C2×D83)⋊C4—41no—
104(C2×D83)⋊C4—41no—
105C22×C4×C83—41no—
106C22×C4×C83C2×D411yescharacteristic
107C22×C4×C83C2×D411yescharacteristic
108C22×C4×C83#4111yescharacteristic
109C2×C8×C83—22no—
110C2×C8×C83—41no—
111C2×C8×C83—41no—
112C2×C8×C83—41no—
113C2×C8×C83—41no—
114C2×C8×C83—41no—
115C4⋊C4×C83—41no—
116C4⋊C4×C83—41no—
117C4⋊C4×C83—41no—
118C4⋊C4×C83—41no—
119C4⋊C4×C83—81no—
120C42×C83C2×D411yescharacteristic
121C42×C83C2×D411yescharacteristic
122C42×C83#4111yescharacteristic
123C42×C83#4111yescharacteristic
124C23×C83#12111yescharacteristic
125C22×D83—22no—
126C22×D83—41no—
127C22×D83—41no—
128C22×D83—41no—
129C22×D83—41no—
130C22×D83—41no—
131C22×D83—41no—
132C2×C4×C83—21no—
133C2×C4×C83—21no—
134C2×C4×C83—81no—
135C2×C4×C83—81no—
136C2×C4×C83—81no—
137C2×C4×C83#12011yescharacteristic
138C2×C4×C83#12111yescharacteristic
139C2×C4×C83#12111yescharacteristic
140C2×C4×C83#12311yescharacteristic
141C2×C4×C83#12311yescharacteristic
142C2×C4×C83#12411yescharacteristic
143C2×C4×C83C2×SD1611yescharacteristic
144C2×C4×C83C2×SD1611yescharacteristic
145C2×C4×C83#13511yescharacteristic
146C2×C4×C83#13511yescharacteristic
147C2×C4×C83AGL(1,ℤ/8)11yescharacteristic
148C2×C4×C83AGL(1,ℤ/8)11yescharacteristic
149C2×C4×C83AGL(1,ℤ/8)11yescharacteristic
150C2×C4×C83#13711yescharacteristic
151C2×Dic83—41no—
152C2×Dic83—42no—
153C2×Dic83—41no—
154C2×Dic83—41no—
155C2×Dic83—81no—
156C2×Dic83—81no—
157D332—22no—
158D332—22no—
159D332—41no—
160D332—41no—
161C4×D83—81no—
162C4×D83—81no—
163C4×D83—81no—
164C4×D83—81no—
165Dic166—42no—
166Dic166—42no—
167Dic166—81no—
168Dic166—81no—
169C8×C83—42no—
170C8×C83—81no—
171C83⋊C8—22no—
172C83⋊C8—41no—
173C83⋊C8—81no—
174C22×C83#39411yescharacteristic
175C22×C83#46111yescharacteristic
176C22×C83#46311yescharacteristic
177C22×C83#46511yescharacteristic
178C22×C83#46911yescharacteristic
179C22×C83#47111yescharacteristic
180C22×C83#47311yescharacteristic
181C2×D83—42no—
182C2×D83—42no—
183C2×D83—42no—
184C2×D83—81no—
185C2×D83—81no—
186C2×D83—81no—
187C2×D83—81no—
188C4×C83—21no—
189C4×C83—21no—
190C4×C83—161no—
191C4×C83#44911yescharacteristic
192C4×C83#45011yescharacteristic
193C4×C83#46011yescharacteristic
194C4×C83#46411yescharacteristic
195C4×C83#46511yescharacteristic
196C4×C83#46811yescharacteristic
197C4×C83#48111yescharacteristic
198C4×C83#48211yescharacteristic
199Dic83—81no—
200Dic83—82no—
201#10477—831nomaximal, Sylow
202C2×C83#167111yescharacteristic
203C2×C83#167911yescharacteristic
204C2×C83#169311yescharacteristic
205C2×C83#170111yescharacteristic
206C2×C83#203411yescharacteristic
207C2×C83#205211yescharacteristic
208C2×C83#205511yescharacteristic
209D83—82no—
210C2×#461—831no—
211#1990—831no—
212#1954—831no—
213#1802—831no—
214C2×#334—831no—
215C2×#332—831no—
216#1296D8311yescharacteristic, complemented
217C83#1047711yescharacteristic, complemented, Sylow
218C22×SD16—1661no—
219C2×C4⋊Q8—831no—
220C2×C4×D4—831no—
221#461—1662no—
222#461—1662no—
223C2×C4⋊C8—831no—
224C2×C4⋊C8C2×D8311yescharacteristic
225C2×C4⋊C8C2×D8311yescharacteristic
226C2×#103—1661no—
227C2×#102—1661no—
228C2×C22⋊C8—1661no—
229C2×C22⋊C8—1661no—
230C2×C4×C8—831no—
231#384C2×D8311yescharacteristic
232#340—1661no—
233#340—1661no—
234#340—1661no—
235#334—1662no—
236#332—1662no—
237C22×Q8—1661no—
238C22×D4—831no—
239C23×C4—1661no—
240C2×SD16—3322no—
241C2×SD16—3322no—
242C2×SD16—1664no—
243C22×C8—21no—
244C22×C8—21no—
245C22×C8—831no—
246C22×C8—831no—
247C22×C8—1661no—
248C4⋊Q8—832no—
249C4⋊Q8—1661no—
250C4⋊Q8—1661no—
251C4⋊Q8—1661no—
252C4×D4—1661no—
253C4×D4—1661no—
254C4×D4—1661no—
255C4×D4—1661no—
256C2×C4⋊C4—21no—
257C2×C4⋊C4—21no—
258C2×C4⋊C4—21no—
259C2×C4⋊C4—831no—
260C2×C4⋊C4—1661no—
261C2×C4⋊C4—1661no—
262C2×C22⋊C4—1661no—
263C2×C42C22×D8311yescharacteristic
264C4⋊C8—21no—
265C4⋊C8—21no—
266C4⋊C8—21no—
267C4⋊C8—1661no—
268C4⋊C8—1661no—
269C4⋊C8D33212yes—
270#103—1664no—
271#102—3322no—
272C22⋊C8—3322no—
273C22⋊C8—3322no—
274C4×C8—1661no—
275C4×C8—1661no—
276C24—1661no—
277C2×Q8—3321no—
278C2×Q8—3321no—
279C2×Q8—3321no—
280C2×Q8—3321no—
281C2×Q8—3321no—
282C2×Q8—1662no—
283C2×D4—832no—
284C2×D4—1661no—
285C2×D4—1661no—
286C2×D4—1661no—
287C2×D4—1661no—
288C2×D4—1661no—
289C22×C4—41no—
290C22×C4—3321no—
291C22×C4—3321no—
292C22×C4—3321no—
293C22×C4—3321no—
294C22×C4—3321no—
295C22×C4—3321no—
296C22×C4—3321no—
297C22×C4—1661no—
298C22×C4D4×D8311yescharacteristic
299C22×C4D4×D8311yescharacteristic
300C22×C4#1305059011yescharacteristic
301SD16—3328no—
302C2×C8—22no—
303C2×C8—41no—
304C2×C8—41no—
305C2×C8—41no—
306C2×C8—41no—
307C2×C8—41no—
308C2×C8—3321no—
309C2×C8—3321no—
310C2×C8—3321no—
311C2×C8—832no—
312C2×C8—1661no—
313C2×C8—1661no—
314C2×C8—1661no—
315C2×C8—1661no—
316C2×C8—1661no—
317C4⋊C4—41no—
318C4⋊C4—41no—
319C4⋊C4—41no—
320C4⋊C4—41no—
321C4⋊C4—81no—
322C4⋊C4—3321no—
323C4⋊C4—3321no—
324C4⋊C4—1661no—
325C4⋊C4—1662no—
326C4⋊C4—1661no—
327C4⋊C4—1662no—
328C22⋊C4—3321no—
329C22⋊C4—3321no—
330C42D4×D8311yescharacteristic
331C42C2×D33211yescharacteristic
332C42#1305059011yescharacteristic
333C42#1305059111yescharacteristic
334C23—3321no—
335C23—3321no—
336C23—3321no—
337C23—3321no—
338C23—3321no—
339C23—3321no—
340C23—1662no—
341C23#1458176511yescharacteristic
342Q8—3322no—
343Q8—3322no—
344Q8—6641no—
345Q8—6641no—
346D4—3321no—
347D4—3321no—
348D4—1662no—
349D4—1662no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—81no—
353C2×C4—81no—
354C2×C4—81no—
355C2×C4—3321no—
356C2×C4—3322no—
357C2×C4—3321no—
358C2×C4—3321no—
359C2×C4—6641no—
360C2×C4—6641no—
361C2×C4—6641no—
362C2×C4—6641no—
363C2×C4—6641no—
364C2×C4—6641no—
365C2×C4#1458176211yescharacteristic
366C2×C4#1458176511yescharacteristic
367C2×C4#1458176711yescharacteristic
368C2×C4#1458177711yescharacteristic
369C2×C4#1458177711yescharacteristic
370C2×C4#1458178011yescharacteristic
371C2×C4SD16×D8311yescharacteristic
372C2×C4SD16×D8311yescharacteristic
373C2×C4#1458181311yescharacteristic
374C2×C4#1458181511yescharacteristic
375C2×C4#1458181611yescharacteristic
376C2×C4#1458181611yescharacteristic
377C2×C4#1458181911yescharacteristic
378C2×C4#1458182411yescharacteristic
379C8—42no—
380C8—81no—
381C8—3321no—
382C8—6641no—
383C8—1662no—
384C22—3322no—
385C22—3322no—
386C22—3322no—
387C22—6641no—
388C22—6641no—
389C22—6641no—
390C22—6641no—
391C22#1912761911yescharacteristic
392C22#1912790511yescharacteristic
393C22#1912791511yescharacteristic
394C22#1912793311yescharacteristic
395C22#1912795311yescharacteristic
396C22#1912796311yescharacteristic
397C22#1912797511yescharacteristic
398C4—21no—
399C4—21no—
400C4—161no—
401C4—6642no—
402C4—6641no—
403C4#1912784911yescharacteristic
404C4#1912785311yescharacteristic
405C4#1912790011yescharacteristic
406C4#1912792611yescharacteristic
407C4#1912793111yescharacteristic
408C4#1912794811yescharacteristic
409C4#1912802111yescharacteristic
410C4#1912802811yescharacteristic
411C2—6642no—
412C2#2490684111yescharacteristic
413C2#2490688511yescharacteristic
414C2#2490697511yescharacteristic
415C2#2490702311yescharacteristic
416C2#2490865511yescharacteristic
417C2#2490876511yescharacteristic
418C2#2490878211yescharacteristic
subgroupG/H
centerC23#14581765
commutator, cosocleC2×C42×C83C23
FrattiniC2×C42C22×D83
FittingC83×#1296C2
socleC23×C83#121
prime radicalC22×D332D4
hypercenter#1296D83
hypocenterC83#10477
LCSG ▷ C2×C42×C83 ▷ C22×C83 ▷ C83
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1296

Complements

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

NH ≅ G/Ntype
2. #24908423C2semidirect
5. C2×#19127451C2semidirect
6. C2×#19127446C2semidirect
7. C83×#1296C2semidirect
96. C2×C83⋊C8SD16semidirect
216. #1296D83semidirect
217. C83#10477semidirect

Maximal subgroups

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

subgroupindex
C2×#191279072
#249084232
#249082562
#249074602
C2×#191274512
C2×#191274462
C83×#12962
#1047783