Finite Group Databasebeta

← #35307910

Subgroups — #35307910

Order 21248.

614 conjugacy classes of subgroups.

The subgroup lattice has 492 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#24907590C211yesmaximal, characteristic, complemented
2#24907463C211yesmaximal, characteristic, complemented
3#24907463C211yesmaximal, characteristic, complemented
4C2×D4⋊(C83⋊C8)C211yesmaximal, characteristic, complemented
5C2×D4⋊(C83⋊C8)C211yesmaximal, characteristic, complemented
6#24905346C211yesmaximal, characteristic, complemented
7C83×#2537C211yesmaximal, characteristic
8C2×C4⋊(C83⋊C8)C2211yescharacteristic
9C2×C4⋊(C83⋊C8)C2211yescharacteristic
10C2×#14581713—21no—
11C2×#14581713—21no—
12C2×C4×C83⋊C8C2211yescharacteristic
13#19127571C2211yescharacteristic
14#19127463—22no—
15#19127463—21no—
16#19127463—21no—
17D4⋊(C83⋊C8)—22no—
18D4⋊(C83⋊C8)—22no—
19C2×#121×C83—21no—
20C2×#121×C83—21no—
21C2×C4×D4×C83C2211yescharacteristic
22C2×C4×D4×C83C2211yescharacteristic
23#394×C83—22no—
24#384×C83C2211yescharacteristic
25#380×C83—21no—
26#380×C83—21no—
27C22×C83⋊C8—21no—
28C22×C83⋊C8—21no—
29C22×C83⋊C8—21no—
30C2×C4⋊Dic83—21no—
31C2×C4⋊Dic83—21no—
32C2×C4⋊Dic83—21no—
33C4⋊(C83⋊C8)—21no—
34C4⋊(C83⋊C8)—21no—
35C4⋊(C83⋊C8)—21no—
36C4⋊(C83⋊C8)—21no—
37#14581713—42no—
38#14581713—42no—
39C4×C83⋊C8—21no—
40C4×C83⋊C8D412yescomplemented
41C22×D4×C83—22no—
42C22×D4×C83D411yescharacteristic
43C22×D4×C83D411yescharacteristic
44C23×C4×C83—21no—
45C23×C4×C83—21no—
46#121×C83—42no—
47#121×C83—42no—
48#121×C83—42no—
49#121×C83—42no—
50C4×D4×C83—21no—
51C4×D4×C83—21no—
52C4×D4×C83—21no—
53C4×D4×C83—21no—
54C4×D4×C83—21no—
55C4×D4×C83—21no—
56C4×D4×C83—21no—
57C4×D4×C83—21no—
58C2×C4⋊C4×C83—21no—
59C2×C4⋊C4×C83—21no—
60C2×C4⋊C4×C83D411yescharacteristic
61C2×C4⋊C4×C83D411yescharacteristic
62C2×C4⋊C4×C83D412yes—
63C2×C22⋊C4×C83—22no—
64C2×C22⋊C4×C83—22no—
65C2×C22⋊C4×C83—21no—
66C2×C22⋊C4×C83—21no—
67C2×C42×C83C2311yescharacteristic
68#95×C83—22no—
69C22×Dic83—41no—
70C2×C83⋊C8—22no—
71C2×C83⋊C8—41no—
72C2×C83⋊C8—41no—
73C2×C83⋊C8—41no—
74C2×C83⋊C8—41no—
75C2×C83⋊C8—41no—
76C2×C83⋊C8—41no—
77C2×C83⋊C8—41no—
78C2×C83⋊C8—41no—
79C4⋊Dic83—42no—
80C4⋊Dic83—41no—
81C4⋊Dic83—41no—
82C4⋊Dic83—41no—
83C4⋊Dic83—41no—
84C24×C83—21no—
85C24×C83—21no—
86C2×D4×C83—21no—
87C2×D4×C83—21no—
88C2×D4×C83—21no—
89C2×D4×C83—21no—
90C2×D4×C83—21no—
91C2×D4×C83—21no—
92C2×D4×C83—21no—
93C2×D4×C83—21no—
94C2×D4×C83—21no—
95C2×D4×C83—21no—
96C2×D4×C83—44no—
97C2×D4×C83—44no—
98C2×D4×C83—82no—
99C2×D4×C83D812yes—
100C2×D4×C83SD1612yes—
101C22×C4×C83—21no—
102C22×C4×C83—21no—
103C22×C4×C83—22no—
104C22×C4×C83—41no—
105C22×C4×C83—41no—
106C22×C4×C83—41no—
107C22×C4×C83—41no—
108C22×C4×C83—41no—
109C22×C4×C83—41no—
110C22×C4×C83—41no—
111C22×C4×C83—41no—
112C22×C4×C83—41no—
113C22×C4×C83—41no—
114C22×C4×C83—41no—
115C22×C4×C83—41no—
116C22×C4×C83C2×D411yescharacteristic
117C22×C4×C83C2×D411yescharacteristic
118C22×C4×C83C2×D411yescharacteristic
119C4⋊C4×C83—21no—
120C4⋊C4×C83—21no—
121C4⋊C4×C83—21no—
122C4⋊C4×C83—21no—
123C4⋊C4×C83—42no—
124C4⋊C4×C83—41no—
125C4⋊C4×C83—41no—
126C4⋊C4×C83—41no—
127C4⋊C4×C83—41no—
128C22⋊C4×C83—44no—
129C22⋊C4×C83—44no—
130C22⋊C4×C83—41no—
131C22⋊C4×C83—41no—
132C22⋊C4×C83—41no—
133C22⋊C4×C83—41no—
134C42×C83C2×D411yescharacteristic
135C42×C83#4111yescharacteristic
136C42×C83#4111yescharacteristic
137C42×C83#4111yescharacteristic
138C23×C83—22no—
139C23×C83—22no—
140C23×C83—41no—
141C23×C83—41no—
142C23×C83—41no—
143C23×C83—41no—
144C23×C83—41no—
145C23×C83—41no—
146C23×C83—41no—
147C23×C83—41no—
148C23×C83—41no—
149C23×C83—41no—
150C23×C83—41no—
151C23×C83—41no—
152C23×C83#12011yescharacteristic
153D4×C83—22no—
154D4×C83—22no—
155D4×C83—22no—
156D4×C83—22no—
157D4×C83—41no—
158D4×C83—41no—
159D4×C83—41no—
160D4×C83—41no—
161D4×C83—88no—
162C2×C4×C83—21no—
163C2×C4×C83—21no—
164C2×C4×C83—41no—
165C2×C4×C83—41no—
166C2×C4×C83—41no—
167C2×C4×C83—42no—
168C2×C4×C83—41no—
169C2×C4×C83—41no—
170C2×C4×C83—42no—
171C2×C4×C83—41no—
172C2×C4×C83—42no—
173C2×C4×C83—81no—
174C2×C4×C83—81no—
175C2×C4×C83—81no—
176C2×C4×C83—81no—
177C2×C4×C83—81no—
178C2×C4×C83—81no—
179C2×C4×C83—81no—
180C2×C4×C83—81no—
181C2×C4×C83#12111yescharacteristic
182C2×C4×C83#12111yescharacteristic
183C2×C4×C83#12111yescharacteristic
184C2×C4×C83#12311yescharacteristic
185C2×C4×C83#12311yescharacteristic
186C2×C4×C83#12411yescharacteristic
187C2×C4×C83C2×D811yescharacteristic
188C2×C4×C83C2×SD1611yescharacteristic
189C2×C4×C83#13511yescharacteristic
190C2×C4×C83#13511yescharacteristic
191C2×C4×C83AGL(1,ℤ/8)11yescharacteristic
192C2×C4×C83AGL(1,ℤ/8)11yescharacteristic
193C2×C4×C83AGL(1,ℤ/8)11yescharacteristic
194C2×C4×C83#13711yescharacteristic
195C2×Dic83—81no—
196C2×Dic83—81no—
197C2×Dic83—81no—
198C83⋊C8—42no—
199C83⋊C8—81no—
200C83⋊C8—81no—
201C22×C83—42no—
202C22×C83—42no—
203C22×C83—42no—
204C22×C83—42no—
205C22×C83—42no—
206C22×C83—42no—
207C22×C83—81no—
208C22×C83—81no—
209C22×C83—81no—
210C22×C83—81no—
211C22×C83—81no—
212C22×C83—81no—
213C22×C83—81no—
214C22×C83—81no—
215C22×C83#39411yescharacteristic
216C22×C83#44711yescharacteristic
217C22×C83#44911yescharacteristic
218C22×C83#45011yescharacteristic
219C22×C83#45211yescharacteristic
220C22×C83#45411yescharacteristic
221C22×C83#45511yescharacteristic
222C4×C83—21no—
223C4×C83—21no—
224C4×C83—82no—
225C4×C83—81no—
226C4×C83—81no—
227C4×C83#46511yescharacteristic
228C4×C83#46611yescharacteristic
229C4×C83#46811yescharacteristic
230C4×C83#46911yescharacteristic
231C4×C83#48011yescharacteristic
232C4×C83#48111yescharacteristic
233C4×C83#48211yescharacteristic
234C4×C83#48211yescharacteristic
235Dic83—161no—
236#10431—831nomaximal, Sylow
237C2×C83—82no—
238C2×C83—82no—
239C2×C83#162511yescharacteristic
240C2×C83#163511yescharacteristic
241C2×C83#163911yescharacteristic
242C2×C83#164611yescharacteristic
243C2×C83#203311yescharacteristic
244C2×C83#203411yescharacteristic
245C2×C83#204011yescharacteristic
246#2537D8311yescharacteristic
247#1834—831no—
248#1802—831no—
249#1802—831no—
250C2×D4⋊C8—831no—
251C2×D4⋊C8—831no—
252#1296—831no—
253C83#1043111yescharacteristic, complemented, Sylow
254C2×#121—21no—
255C2×#121—21no—
256C2×C4×D4C2×D8311yescharacteristic
257C2×C4×D4C2×D8311yescharacteristic
258C2×C4⋊C8—831no—
259C2×C4⋊C8—831no—
260C2×#102—1661no—
261C2×#102—1661no—
262C2×C4×C8—831no—
263#394—22no—
264#384—831no—
265#384C2×D8311yescharacteristic
266#380—21no—
267#380—21no—
268#340—1661no—
269#340—1661no—
270#340—1662no—
271D4⋊C8—1662no—
272D4⋊C8—1662no—
273C22×D4—22no—
274C22×D4C22⋊D8311yescharacteristic
275C22×D4C22⋊D8311yescharacteristic
276C23×C4—21no—
277C23×C4—21no—
278C22×C8—1661no—
279C22×C8—1661no—
280C22×C8—1661no—
281#121—42no—
282#121—42no—
283#121—42no—
284#121—42no—
285C4×D4—21no—
286C4×D4—21no—
287C4×D4—21no—
288C4×D4—21no—
289C4×D4—21no—
290C4×D4—21no—
291C4×D4—21no—
292C4×D4—21no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—1661no—
296C2×C4⋊C4—1661no—
297C2×C4⋊C4—1661no—
298C2×C4⋊C4C22⋊D8311yescharacteristic
299C2×C4⋊C4C22⋊D8311yescharacteristic
300C2×C4⋊C4C22⋊D8312yes—
301C2×C22⋊C4—22no—
302C2×C22⋊C4—22no—
303C2×C22⋊C4—21no—
304C2×C22⋊C4—21no—
305C2×C42C22×D8311yescharacteristic
306C4⋊C8—1661no—
307C4⋊C8—1661no—
308C4⋊C8—1661no—
309C4⋊C8—1661no—
310#102—3322no—
311#102—3322no—
312C4×C8—832no—
313C4×C8—1661no—
314#95—22no—
315C24—21no—
316C24—21no—
317C2×D4—21no—
318C2×D4—21no—
319C2×D4—21no—
320C2×D4—21no—
321C2×D4—21no—
322C2×D4—21no—
323C2×D4—21no—
324C2×D4—21no—
325C2×D4—21no—
326C2×D4—21no—
327C2×D4—44no—
328C2×D4—44no—
329C2×D4—82no—
330C2×D4#1305057512yes—
331C2×D4#1305057812yes—
332C22×C4—21no—
333C22×C4—21no—
334C22×C4—22no—
335C22×C4—41no—
336C22×C4—41no—
337C22×C4—41no—
338C22×C4—41no—
339C22×C4—41no—
340C22×C4—41no—
341C22×C4—41no—
342C22×C4—41no—
343C22×C4—41no—
344C22×C4—41no—
345C22×C4—41no—
346C22×C4—41no—
347C22×C4—3321no—
348C22×C4C2×C22⋊D8311yescharacteristic
349C22×C4C2×C22⋊D8311yescharacteristic
350C22×C4C2×C22⋊D8311yescharacteristic
351C2×C8—3321no—
352C2×C8—3321no—
353C2×C8—3321no—
354C2×C8—3321no—
355C2×C8—3321no—
356C2×C8—3321no—
357C2×C8—3321no—
358C2×C8—3321no—
359C2×C8—1662no—
360C4⋊C4—21no—
361C4⋊C4—21no—
362C4⋊C4—21no—
363C4⋊C4—21no—
364C4⋊C4—42no—
365C4⋊C4—41no—
366C4⋊C4—41no—
367C4⋊C4—41no—
368C4⋊C4—41no—
369C4⋊C4—3322no—
370C4⋊C4—3321no—
371C4⋊C4—3321no—
372C4⋊C4—3321no—
373C4⋊C4—3321no—
374C22⋊C4—44no—
375C22⋊C4—44no—
376C22⋊C4—41no—
377C22⋊C4—41no—
378C22⋊C4—41no—
379C22⋊C4—41no—
380C42D4×D8311yescharacteristic
381C42#1305058911yescharacteristic
382C42#1305058911yescharacteristic
383C42#1305058911yescharacteristic
384C23—22no—
385C23—22no—
386C23—41no—
387C23—41no—
388C23—41no—
389C23—41no—
390C23—41no—
391C23—41no—
392C23—41no—
393C23—41no—
394C23—41no—
395C23—41no—
396C23—41no—
397C23—41no—
398C23#1458176111yescharacteristic
399D4—22no—
400D4—22no—
401D4—22no—
402D4—22no—
403D4—41no—
404D4—41no—
405D4—41no—
406D4—41no—
407D4—88no—
408C2×C4—21no—
409C2×C4—21no—
410C2×C4—41no—
411C2×C4—42no—
412C2×C4—41no—
413C2×C4—41no—
414C2×C4—42no—
415C2×C4—41no—
416C2×C4—41no—
417C2×C4—41no—
418C2×C4—42no—
419C2×C4—81no—
420C2×C4—81no—
421C2×C4—81no—
422C2×C4—81no—
423C2×C4—81no—
424C2×C4—81no—
425C2×C4—81no—
426C2×C4—81no—
427C2×C4—6641no—
428C2×C4—6641no—
429C2×C4—6641no—
430C2×C4#1458176311yescharacteristic
431C2×C4#1458176411yescharacteristic
432C2×C4#1458176411yescharacteristic
433C2×C4#1458177311yescharacteristic
434C2×C4#1458177311yescharacteristic
435C2×C4#1458177911yescharacteristic
436C2×C4C2×#1305057511yescharacteristic
437C2×C4C2×#1305057811yescharacteristic
438C2×C4#1458181211yescharacteristic
439C2×C4#1458181211yescharacteristic
440C2×C4#1458181711yescharacteristic
441C2×C4#1458181811yescharacteristic
442C2×C4#1458181811yescharacteristic
443C2×C4#1458182211yescharacteristic
444C8—3322no—
445C8—6641no—
446C8—6641no—
447C22—42no—
448C22—42no—
449C22—42no—
450C22—42no—
451C22—42no—
452C22—42no—
453C22—81no—
454C22—81no—
455C22—81no—
456C22—81no—
457C22—81no—
458C22—81no—
459C22—81no—
460C22—81no—
461C22#1912761511yescharacteristic
462C22#1912783511yescharacteristic
463C22#1912784611yescharacteristic
464C22#1912785211yescharacteristic
465C22#1912786311yescharacteristic
466C22#1912787411yescharacteristic
467C22#1912787911yescharacteristic
468C4—21no—
469C4—21no—
470C4—81no—
471C4—81no—
472C4—82no—
473C4—13281no—
474C4#1912792811yescharacteristic
475C4#1912793511yescharacteristic
476C4#1912794511yescharacteristic
477C4#1912795211yescharacteristic
478C4#1912801211yescharacteristic
479C4#1912801711yescharacteristic
480C4#1912802211yescharacteristic
481C4#1912802211yescharacteristic
482C2—82no—
483C2—82no—
484C2#2490655211yescharacteristic
485C2#2490661411yescharacteristic
486C2#2490664211yescharacteristic
487C2#2490669111yescharacteristic
488C2#2490864211yescharacteristic
489C2#2490864911yescharacteristic
490C2#2490868911yescharacteristic
subgroupG/H
centerC23#14581761
commutator, cosocleC2×C42×C83C23
FrattiniC2×C42C22×D83
Fitting, prime radicalC83×#2537C2
socleC23×C83#120
hypercenter#2537D83
hypocenterC83#10431
LCSG ▷ C2×C42×C83 ▷ C22×C83 ▷ C83
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2537

Complements

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

NH ≅ G/Ntype
1. #24907590C2semidirect
2. #24907463C2semidirect
3. #24907463C2semidirect
4. C2×D4⋊(C83⋊C8)C2semidirect
5. C2×D4⋊(C83⋊C8)C2semidirect
6. #24905346C2semidirect
40. C4×C83⋊C8D4semidirect
253. C83#10431semidirect

Maximal subgroups

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

subgroupindex
#249075902
#249074632
C2×D4⋊(C83⋊C8)2
#249053462
C83×#25372
#1043183