Finite Group Databasebeta

← #35307937

Subgroups — #35307937

Order 21248.

546 conjugacy classes of subgroups.

The subgroup lattice has 446 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#24912381C211yesmaximal, characteristic
2#24907619C211yesmaximal, characteristic, complemented
3#24907460C211yesmaximal, characteristic
4C2×Q8⋊(C83⋊C8)C211yesmaximal, characteristic, complemented
5C2×#19127428C211yesmaximal, characteristic
6#24905347C211yesmaximal, characteristic, complemented
7C83×#1803C211yesmaximal, characteristic, complemented
8C2×#14581768—21no—
9C2×C4×D332C2211yescharacteristic
10C2×#14581723—22no—
11C2×C4⋊(C83⋊C8)C2211yescharacteristic
12C2×#14581712—21no—
13C2×C4×C83⋊C8C2211yescharacteristic
14#19127623—22no—
15#19127603—21no—
16#19127599—21no—
17#19127571C2211yescharacteristic
18#19127558—21no—
19#19127463—21no—
20Q8⋊(C83⋊C8)—22no—
21#19127428—22no—
22C2×C4×Q8×C83C2211yescharacteristic
23C2×C4⋊C8×C83C2211yescharacteristic
24C2×#103×C83—21no—
25#384×C83C2211yescharacteristic
26#340×C83—21no—
27C22×D332D411yescharacteristic
28C22×C4×D83—21no—
29C22×C83⋊C8—21no—
30C22×C83⋊C8—21no—
31#14581768—42no—
32#14581768—42no—
33C4×D332—21no—
34C4×D332—21no—
35C4×D332—21no—
36C4×D332—21no—
37C2×C4⋊Dic83—21no—
38C2×C4⋊Dic83—21no—
39C2×C4⋊Dic83D411yescharacteristic
40C2×C4⋊Dic83D412yes—
41C2×Dic83⋊C4—22no—
42C2×(C2×D83)⋊C4—22no—
43C2×(C2×D83)⋊C4—22no—
44C2×(C2×D83)⋊C4—21no—
45#14581723—44no—
46C4⋊(C83⋊C8)—21no—
47C4⋊(C83⋊C8)—21no—
48#14581712—42no—
49C4×C83⋊C8—21no—
50C4×C83⋊C8D412yes—
51C83⋊#95—22no—
52C22×Q8×C83D411yescharacteristic
53C22×C8×C83—21no—
54C4×Q8×C83—21no—
55C4×Q8×C83—21no—
56C4×Q8×C83—21no—
57C4×Q8×C83—21no—
58C2×C4⋊C4×C83—21no—
59C2×C4⋊C4×C83—21no—
60C2×C4⋊C4×C83—21no—
61C2×C4⋊C4×C83—21no—
62C2×C4⋊C4×C83D411yescharacteristic
63C2×C42×C83—21no—
64C2×C42×C83C2311yescharacteristic
65C4⋊C8×C83—21no—
66C4⋊C8×C83—21no—
67#103×C83—42no—
68C23×D83—21no—
69C2×D332—21no—
70C2×D332—21no—
71C2×D332—21no—
72C2×D332—21no—
73C2×D332—21no—
74C2×D332SD1612yes—
75C22×Dic83—21no—
76C22×Dic83—22no—
77C2×C4×D83—41no—
78C2×C4×D83—41no—
79C2×C4×D83—41no—
80C2×C4×D83—41no—
81C2×C4×D83—41no—
82C2×C4×D83—41no—
83C2×C83⋊C8—22no—
84C2×C83⋊C8—22no—
85C2×C83⋊C8—22no—
86C2×C83⋊C8—41no—
87C2×C83⋊C8—41no—
88C2×C83⋊C8—41no—
89C4⋊Dic83—21no—
90C4⋊Dic83—21no—
91C4⋊Dic83—41no—
92C4⋊Dic83—41no—
93C4⋊Dic83—41no—
94C4⋊Dic83—41no—
95C4⋊Dic83—42no—
96Dic83⋊C4—44no—
97(C2×D83)⋊C4—44no—
98(C2×D83)⋊C4—41no—
99(C2×D83)⋊C4—41no—
100(C2×D83)⋊C4—82no—
101C2×Q8×C83—21no—
102C2×Q8×C83—21no—
103C2×Q8×C83—21no—
104C2×Q8×C83—21no—
105C2×Q8×C83—21no—
106C2×Q8×C83SD1612yes—
107C22×C4×C83—21no—
108C22×C4×C83—21no—
109C22×C4×C83—41no—
110C22×C4×C83C2×D411yescharacteristic
111C22×C4×C83C2×D411yescharacteristic
112C22×C4×C83C2×D411yescharacteristic
113C2×C8×C83—41no—
114C2×C8×C83—41no—
115C2×C8×C83—41no—
116C4⋊C4×C83—21no—
117C4⋊C4×C83—21no—
118C4⋊C4×C83—42no—
119C4⋊C4×C83—41no—
120C4⋊C4×C83—41no—
121C4⋊C4×C83—41no—
122C4⋊C4×C83—41no—
123C4⋊C4×C83—41no—
124C4⋊C4×C83—41no—
125C42×C83—41no—
126C42×C83—41no—
127C42×C83C2×D411yescharacteristic
128C42×C83#4111yescharacteristic
129C42×C83#4111yescharacteristic
130C42×C83#4111yescharacteristic
131C23×C83#12011yescharacteristic
132C22×D83—22no—
133C22×D83—41no—
134C22×D83—41no—
135C22×D83—41no—
136C22×D83—41no—
137C22×D83—41no—
138C22×D83—41no—
139Q8×C83—22no—
140Q8×C83—22no—
141Q8×C83—41no—
142Q8×C83—41no—
143C2×C4×C83—21no—
144C2×C4×C83—21no—
145C2×C4×C83—22no—
146C2×C4×C83—41no—
147C2×C4×C83—41no—
148C2×C4×C83—41no—
149C2×C4×C83—41no—
150C2×C4×C83—41no—
151C2×C4×C83—81no—
152C2×C4×C83—81no—
153C2×C4×C83—81no—
154C2×C4×C83#12111yescharacteristic
155C2×C4×C83#12111yescharacteristic
156C2×C4×C83#12111yescharacteristic
157C2×C4×C83#12311yescharacteristic
158C2×C4×C83#12311yescharacteristic
159C2×C4×C83#12411yescharacteristic
160C2×C4×C83C2×SD1611yescharacteristic
161C2×C4×C83C2×SD1611yescharacteristic
162C2×C4×C83#13511yescharacteristic
163C2×C4×C83#13511yescharacteristic
164C2×C4×C83AGL(1,ℤ/8)11yescharacteristic
165C2×C4×C83AGL(1,ℤ/8)11yescharacteristic
166C2×C4×C83#13711yescharacteristic
167C2×C4×C83#13711yescharacteristic
168C2×Dic83—41no—
169C2×Dic83—41no—
170C2×Dic83—42no—
171C2×Dic83—41no—
172C2×Dic83—42no—
173C2×Dic83—42no—
174D332—22no—
175D332—22no—
176D332—41no—
177D332—41no—
178C4×D83—81no—
179C4×D83—81no—
180C4×D83—81no—
181C4×D83—81no—
182C8×C83—81no—
183C83⋊C8—24no—
184C83⋊C8—81no—
185C22×C83#39411yescharacteristic
186C22×C83#44811yescharacteristic
187C22×C83#44911yescharacteristic
188C22×C83#45011yescharacteristic
189C22×C83#45211yescharacteristic
190C22×C83#45311yescharacteristic
191C22×C83#45611yescharacteristic
192C2×D83—42no—
193C2×D83—42no—
194C2×D83—42no—
195C2×D83—81no—
196C2×D83—81no—
197C2×D83—81no—
198C2×D83—81no—
199C4×C83—21no—
200C4×C83—21no—
201C4×C83—42no—
202C4×C83—81no—
203C4×C83—161no—
204C4×C83#46511yescharacteristic
205C4×C83#46511yescharacteristic
206C4×C83#46911yescharacteristic
207C4×C83#47011yescharacteristic
208C4×C83#48111yescharacteristic
209C4×C83#48211yescharacteristic
210C4×C83#48311yescharacteristic
211C4×C83#48511yescharacteristic
212Dic83—82no—
213Dic83—81no—
214#10435—831nomaximal, Sylow
215C2×C83#162211yescharacteristic
216C2×C83#163111yescharacteristic
217C2×C83#165111yescharacteristic
218C2×C83#165211yescharacteristic
219C2×C83#203411yescharacteristic
220C2×C83#203511yescharacteristic
221C2×C83#203911yescharacteristic
222D83—82no—
223#2538—831no—
224#1844—831no—
225#1803D8311yescharacteristic, complemented
226#1802—831no—
227C2×Q8⋊C8—831no—
228C2×D4⋊C8—831no—
229#1296—831no—
230C83#1043511yescharacteristic, complemented, Sylow
231C2×C22⋊Q8—1661no—
232C2×C4×Q8C2×D8311yescharacteristic
233C2×C4×D4—831no—
234C2×#106—1662no—
235C2×C4⋊C8—831no—
236C2×C4⋊C8C2×D8311yescharacteristic
237C2×#103—21no—
238C2×#102—1661no—
239C2×C4×C8—831no—
240#396—1662no—
241#390—1661no—
242#389—1661no—
243#384—831no—
244#384C2×D8311yescharacteristic
245#380—1661no—
246#340—21no—
247#340—1661no—
248Q8⋊C8—1662no—
249D4⋊C8—1662no—
250C22×Q8C22⋊D8311yescharacteristic
251C22×D4—831no—
252C23×C4—1661no—
253C22×C8—21no—
254C22×C8—1661no—
255C22×C8—1661no—
256C22⋊Q8—3322no—
257C22⋊Q8—3322no—
258C4×Q8—21no—
259C4×Q8—21no—
260C4×Q8—21no—
261C4×Q8—21no—
262C4×D4—1661no—
263C4×D4—1661no—
264C4×D4—1661no—
265C4×D4—1661no—
266C2×C4⋊C4—21no—
267C2×C4⋊C4—21no—
268C2×C4⋊C4—21no—
269C2×C4⋊C4—21no—
270C2×C4⋊C4—832no—
271C2×C4⋊C4—831no—
272C2×C4⋊C4—1662no—
273C2×C4⋊C4—1661no—
274C2×C4⋊C4—1661no—
275C2×C4⋊C4C22⋊D8311yescharacteristic
276C2×C22⋊C4—1662no—
277C2×C22⋊C4—1661no—
278C2×C22⋊C4—1662no—
279C2×C42—21no—
280C2×C42C22×D8311yescharacteristic
281#106—3324no—
282C4⋊C8—21no—
283C4⋊C8—21no—
284C4⋊C8—1661no—
285C4⋊C8—1661no—
286#103—42no—
287#102—3322no—
288C4×C8—832no—
289C4×C8—1661no—
290#95—1662no—
291C24—1661no—
292C2×Q8—21no—
293C2×Q8—21no—
294C2×Q8—21no—
295C2×Q8—21no—
296C2×Q8—21no—
297C2×Q8#1305057712yes—
298C2×D4—832no—
299C2×D4—1661no—
300C2×D4—1661no—
301C2×D4—1661no—
302C2×D4—1661no—
303C2×D4—1661no—
304C22×C4—21no—
305C22×C4—21no—
306C22×C4—41no—
307C22×C4—3321no—
308C22×C4—3321no—
309C22×C4—3321no—
310C22×C4—3321no—
311C22×C4—3321no—
312C22×C4—3321no—
313C22×C4—1661no—
314C22×C4—1662no—
315C22×C4D4×D8311yescharacteristic
316C22×C4D4×D8311yescharacteristic
317C22×C4C2×C22⋊D8311yescharacteristic
318C2×C8—41no—
319C2×C8—41no—
320C2×C8—41no—
321C2×C8—3321no—
322C2×C8—3321no—
323C2×C8—3321no—
324C2×C8—1662no—
325C2×C8—1662no—
326C2×C8—1662no—
327C4⋊C4—21no—
328C4⋊C4—21no—
329C4⋊C4—42no—
330C4⋊C4—41no—
331C4⋊C4—41no—
332C4⋊C4—41no—
333C4⋊C4—41no—
334C4⋊C4—41no—
335C4⋊C4—41no—
336C4⋊C4—3322no—
337C4⋊C4—3324no—
338C4⋊C4—3321no—
339C4⋊C4—3321no—
340C4⋊C4—3321no—
341C4⋊C4—3321no—
342C4⋊C4—1661no—
343C4⋊C4—1661no—
344C22⋊C4—3324no—
345C22⋊C4—3321no—
346C22⋊C4—3321no—
347C22⋊C4—6642no—
348C42—41no—
349C42—41no—
350C42D4×D8311yescharacteristic
351C42#1305058911yescharacteristic
352C42#1305059011yescharacteristic
353C42#1305059011yescharacteristic
354C23—3321no—
355C23—3321no—
356C23—3321no—
357C23—3321no—
358C23—3321no—
359C23—3321no—
360C23—1662no—
361C23#1458176011yescharacteristic
362Q8—22no—
363Q8—22no—
364Q8—41no—
365Q8—41no—
366D4—3321no—
367D4—3321no—
368D4—1662no—
369D4—1662no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—22no—
373C2×C4—41no—
374C2×C4—41no—
375C2×C4—41no—
376C2×C4—41no—
377C2×C4—41no—
378C2×C4—81no—
379C2×C4—81no—
380C2×C4—81no—
381C2×C4—3321no—
382C2×C4—3322no—
383C2×C4—3321no—
384C2×C4—3322no—
385C2×C4—3322no—
386C2×C4—3321no—
387C2×C4—6641no—
388C2×C4—6641no—
389C2×C4—6641no—
390C2×C4—6641no—
391C2×C4#1458176411yescharacteristic
392C2×C4#1458176511yescharacteristic
393C2×C4#1458176511yescharacteristic
394C2×C4#1458177411yescharacteristic
395C2×C4#1458177511yescharacteristic
396C2×C4#1458177811yescharacteristic
397C2×C4SD16×D8311yescharacteristic
398C2×C4C2×#1305057711yescharacteristic
399C2×C4#1458181211yescharacteristic
400C2×C4#1458181511yescharacteristic
401C2×C4#1458181711yescharacteristic
402C2×C4#1458181811yescharacteristic
403C2×C4#1458182111yescharacteristic
404C2×C4#1458182411yescharacteristic
405C8—81no—
406C8—6641no—
407C8—1664no—
408C22—3322no—
409C22—3322no—
410C22—3322no—
411C22—6641no—
412C22—6641no—
413C22—6641no—
414C22—6641no—
415C22#1912761611yescharacteristic
416C22#1912783811yescharacteristic
417C22#1912784411yescharacteristic
418C22#1912785111yescharacteristic
419C22#1912786211yescharacteristic
420C22#1912786711yescharacteristic
421C22#1912788211yescharacteristic
422C4—21no—
423C4—21no—
424C4—42no—
425C4—81no—
426C4—161no—
427C4—6642no—
428C4—6641no—
429C4#1912793211yescharacteristic
430C4#1912793311yescharacteristic
431C4#1912795211yescharacteristic
432C4#1912795811yescharacteristic
433C4#1912801911yescharacteristic
434C4#1912802511yescharacteristic
435C4#1912803011yescharacteristic
436C4#1912804111yescharacteristic
437C2—6642no—
438C2#2490653411yescharacteristic
439C2#2490658811yescharacteristic
440C2#2490672511yescharacteristic
441C2#2490673111yescharacteristic
442C2#2490865211yescharacteristic
443C2#2490865711yescharacteristic
444C2#2490868511yescharacteristic
subgroupG/H
centerC23#14581760
commutator, cosocleC2×C42×C83C23
FrattiniC2×C42C22×D83
FittingC83×#1803C2
socleC23×C83#120
prime radicalC22×D332D4
hypercenter#1803D83
hypocenterC83#10435
LCSG ▷ C2×C42×C83 ▷ C22×C83 ▷ C83
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1803

Complements

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

NH ≅ G/Ntype
2. #24907619C2semidirect
4. C2×Q8⋊(C83⋊C8)C2semidirect
6. #24905347C2semidirect
7. C83×#1803C2semidirect
225. #1803D83semidirect
230. C83#10435semidirect

Maximal subgroups

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

subgroupindex
#249123812
#249076192
#249074602
C2×Q8⋊(C83⋊C8)2
C2×#191274282
#249053472
C83×#18032
#1043583