Finite Group Databasebeta

← C83×#8978 #35236802

Subgroups — C83×#8978 #35236802

Order 21248.

616 conjugacy classes of subgroups.

The subgroup lattice has 456 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#460×C83C211yesmaximal, characteristic
2C83×#2355C211yesmaximal, characteristic, complemented
3C83×#1982C211yesmaximal, characteristic
4C83×#1957C211yesmaximal, characteristic
5C2×Q8⋊C8×C83C211yesmaximal, characteristic, complemented
6C83×#1296C211yesmaximal, characteristic, complemented
7C83×#1279C211yesmaximal, characteristic, complemented
8C22×SD16×C83—21no—
9C2×#127×C83C2211yescharacteristic
10C2×C4×Q8×C83C2211yescharacteristic
11#460×C83—22no—
12#460×C83C414yescomplemented
13C2×C4⋊C8×C83C2211yescharacteristic
14C2×C4⋊C8×C83C2211yescharacteristic
15C2×#102×C83—21no—
16C2×#102×C83—22no—
17C2×#102×C83—21no—
18C2×#102×C83—21no—
19C2×C4×C8×C83C2211yescharacteristic
20#391×C83—21no—
21#387×C83—22no—
22#384×C83C2211yescharacteristic
23#382×C83—21no—
24C4×C4⋊C4×C83—21no—
25C4×C4⋊C4×C83—21no—
26C4×C4⋊C4×C83C2211yescharacteristic
27Q8⋊C8×C83—22no—
28C22×Q8×C83D411yescharacteristic
29C22×D4×C83—21no—
30C22×D4×C83—21no—
31C22×D4×C83—21no—
32C2×SD16×C83—24no—
33C2×SD16×C83—42no—
34C2×SD16×C83—42no—
35C22×C8×C83—21no—
36C22×C8×C83—21no—
37C22×C8×C83—21no—
38#127×C83—21no—
39#127×C83—21no—
40#127×C83—21no—
41#127×C83C2×C412yes—
42C4×Q8×C83—21no—
43C4×Q8×C83—21no—
44C4×Q8×C83C2×C412yes—
45C4×Q8×C83D412yes—
46C2×C4⋊C4×C83—21no—
47C2×C4⋊C4×C83—21no—
48C2×C4⋊C4×C83—21no—
49C2×C4⋊C4×C83D411yescharacteristic
50C2×C4⋊C4×C83D411yescharacteristic
51C2×C4⋊C4×C83D411yescharacteristic
52C2×C4⋊C4×C83D412yes—
53C2×C42×C83—21no—
54C2×C42×C83—21no—
55C2×C42×C83—22no—
56C2×C42×C83C2311yescharacteristic
57C4⋊C8×C83—21no—
58C4⋊C8×C83—21no—
59C4⋊C8×C83—21no—
60C4⋊C8×C83C2×C412yes—
61#102×C83—24no—
62#102×C83—42no—
63#102×C83—44no—
64#102×C83—42no—
65C4×C8×C83—21no—
66C4×C8×C83—21no—
67#95×C83—22no—
68#95×C83—22no—
69#95×C83—22no—
70#95×C83—21no—
71C24×C83—41no—
72C2×Q8×C83—21no—
73C2×Q8×C83—21no—
74C2×Q8×C83—21no—
75C2×Q8×C83—21no—
76C2×Q8×C83—21no—
77C2×Q8×C83C22⋊C412yes—
78C2×D4×C83—22no—
79C2×D4×C83—22no—
80C2×D4×C83—22no—
81C2×D4×C83—42no—
82C2×D4×C83—42no—
83C2×D4×C83—41no—
84C2×D4×C83—41no—
85C2×D4×C83—41no—
86C2×D4×C83—41no—
87C2×D4×C83—41no—
88C2×D4×C83—41no—
89C2×D4×C83—41no—
90C2×D4×C83—41no—
91C2×D4×C83—41no—
92C2×D4×C83—41no—
93C2×D4×C83—41no—
94C22×C4×C83—21no—
95C22×C4×C83—21no—
96C22×C4×C83—21no—
97C22×C4×C83—21no—
98C22×C4×C83—22no—
99C22×C4×C83C2×D411yescharacteristic
100C22×C4×C83C2×D411yescharacteristic
101C22×C4×C83C2×D411yescharacteristic
102SD16×C83—48no—
103C2×C8×C83—22no—
104C2×C8×C83—22no—
105C2×C8×C83—41no—
106C2×C8×C83—41no—
107C2×C8×C83—41no—
108C2×C8×C83—41no—
109C2×C8×C83—41no—
110C2×C8×C83—41no—
111C2×C8×C83—41no—
112C4⋊C4×C83—22no—
113C4⋊C4×C83—22no—
114C4⋊C4×C83—21no—
115C4⋊C4×C83—21no—
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×C83C22⋊C412yes—
124C4⋊C4×C83SD1612yescomplemented
125C42×C83—22no—
126C42×C83—22no—
127C42×C83—41no—
128C42×C83—41no—
129C42×C83—82no—
130C42×C83C22×C411yescharacteristic
131C42×C83C2×D411yescharacteristic
132C42×C83#4111yescharacteristic
133C42×C83#4111yescharacteristic
134C23×C83—42no—
135C23×C83—81no—
136C23×C83—81no—
137C23×C83—81no—
138C23×C83—81no—
139C23×C83—81no—
140C23×C83—81no—
141C23×C83#12011yescharacteristic
142Q8×C83—22no—
143Q8×C83—22no—
144Q8×C83—41no—
145Q8×C83—41no—
146D4×C83—44no—
147D4×C83—42no—
148D4×C83—42no—
149D4×C83—42no—
150D4×C83—42no—
151D4×C83—82no—
152D4×C83—81no—
153D4×C83—81no—
154D4×C83—81no—
155D4×C83—81no—
156C2×C4×C83—21no—
157C2×C4×C83—21no—
158C2×C4×C83—22no—
159C2×C4×C83—22no—
160C2×C4×C83—22no—
161C2×C4×C83—41no—
162C2×C4×C83—41no—
163C2×C4×C83—41no—
164C2×C4×C83—41no—
165C2×C4×C83—41no—
166C2×C4×C83—41no—
167C2×C4×C83—41no—
168C2×C4×C83—41no—
169C2×C4×C83—42no—
170C2×C4×C83—42no—
171C2×C4×C83—42no—
172C2×C4×C83—41no—
173C2×C4×C83C2×C22⋊C411yescharacteristic
174C2×C4×C83C4×D411yescharacteristic
175C2×C4×C83C4×D411yescharacteristic
176C2×C4×C83#12111yescharacteristic
177C2×C4×C83#12111yescharacteristic
178C2×C4×C83#12311yescharacteristic
179C2×C4×C83C2×SD1611yescharacteristic
180C2×C4×C83#13511yescharacteristic
181C2×C4×C83#13511yescharacteristic
182C2×C4×C83#13511yescharacteristic
183C2×C4×C83AGL(1,ℤ/8)11yescharacteristic
184C2×C4×C83AGL(1,ℤ/8)11yescharacteristic
185C2×C4×C83AGL(1,ℤ/8)11yescharacteristic
186C2×C4×C83#13711yescharacteristic
187C8×C83—42no—
188C8×C83—42no—
189C8×C83—81no—
190C22×C83—82no—
191C22×C83—82no—
192C22×C83—82no—
193C22×C83—161no—
194C22×C83—161no—
195C22×C83—161no—
196C22×C83—161no—
197C22×C83#38611yescharacteristic
198C22×C83#44811yescharacteristic
199C22×C83#44911yescharacteristic
200C22×C83#44911yescharacteristic
201C22×C83#44911yescharacteristic
202C22×C83#45311yescharacteristic
203C22×C83#45411yescharacteristic
204C4×C83—21no—
205C4×C83—21no—
206C4×C83—42no—
207C4×C83—42no—
208C4×C83—42no—
209C4×C83—82no—
210C4×C83—81no—
211C4×C83#41611yescharacteristic
212C4×C83#41811yescharacteristic
213C4×C83C4×SD1611yescharacteristic
214C4×C83#44011yescharacteristic
215C4×C83#46011yescharacteristic
216C4×C83#46411yescharacteristic
217C4×C83#48211yescharacteristic
218C4×C83#48211yescharacteristic
219#8978C8311yesmaximal, characteristic, direct product factor, Sylow
220C2×C83—162no—
221C2×C83#162011yescharacteristic
222C2×C83#162211yescharacteristic
223C2×C83#162611yescharacteristic
224C2×C83#162811yescharacteristic
225C2×C83#187511yescharacteristic
226C2×C83#188011yescharacteristic
227C2×C83#188311yescharacteristic
228C2×#460C2×C8311yescharacteristic
229#2355C2×C8311yescharacteristic, complemented
230#1982C2×C8311yescharacteristic
231#1957C2×C8311yescharacteristic
232C2×Q8⋊C8C2×C8311yescharacteristic, complemented
233#1296C2×C8311yescharacteristic, complemented
234#1279C2×C8311yescharacteristic, complemented
235C83#897811yescharacteristic, direct product factor, Sylow
236C22×SD16—21no—
237C2×#127C22×C8311yescharacteristic
238C2×C4×Q8C22×C8311yescharacteristic
239#460—22no—
240#460C4×C8314yescomplemented
241C2×C4⋊C8C22×C8311yescharacteristic
242C2×C4⋊C8C22×C8311yescharacteristic
243C2×#102—21no—
244C2×#102—22no—
245C2×#102—21no—
246C2×#102—21no—
247C2×C4×C8C22×C8311yescharacteristic
248#391—21no—
249#387—22no—
250#384C22×C8311yescharacteristic
251#382—21no—
252C4×C4⋊C4—21no—
253C4×C4⋊C4—21no—
254C4×C4⋊C4C22×C8311yescharacteristic
255Q8⋊C8—22no—
256C22×Q8D4×C8311yescharacteristic
257C22×D4—21no—
258C22×D4—21no—
259C22×D4—21no—
260C2×SD16—24no—
261C2×SD16—42no—
262C2×SD16—42no—
263C22×C8—21no—
264C22×C8—21no—
265C22×C8—21no—
266#127—21no—
267#127—21no—
268#127—21no—
269#127C2×C4×C8312yes—
270C4×Q8—21no—
271C4×Q8—21no—
272C4×Q8C2×C4×C8312yes—
273C4×Q8D4×C8312yes—
274C2×C4⋊C4—21no—
275C2×C4⋊C4—21no—
276C2×C4⋊C4—21no—
277C2×C4⋊C4D4×C8311yescharacteristic
278C2×C4⋊C4D4×C8311yescharacteristic
279C2×C4⋊C4D4×C8311yescharacteristic
280C2×C4⋊C4D4×C8312yes—
281C2×C42—21no—
282C2×C42—21no—
283C2×C42—22no—
284C2×C42C23×C8311yescharacteristic
285C4⋊C8—21no—
286C4⋊C8—21no—
287C4⋊C8—21no—
288C4⋊C8C2×C4×C8312yes—
289#102—24no—
290#102—42no—
291#102—44no—
292#102—42no—
293C4×C8—21no—
294C4×C8—21no—
295#95—22no—
296#95—22no—
297#95—22no—
298#95—21no—
299C24—41no—
300C2×Q8—21no—
301C2×Q8—21no—
302C2×Q8—21no—
303C2×Q8—21no—
304C2×Q8—21no—
305C2×Q8C22⋊C4×C8312yes—
306C2×D4—22no—
307C2×D4—22no—
308C2×D4—22no—
309C2×D4—42no—
310C2×D4—42no—
311C2×D4—41no—
312C2×D4—41no—
313C2×D4—41no—
314C2×D4—41no—
315C2×D4—41no—
316C2×D4—41no—
317C2×D4—41no—
318C2×D4—41no—
319C2×D4—41no—
320C2×D4—41no—
321C2×D4—41no—
322C22×C4—21no—
323C22×C4—21no—
324C22×C4—21no—
325C22×C4—21no—
326C22×C4—22no—
327C22×C4C2×D4×C8311yescharacteristic
328C22×C4C2×D4×C8311yescharacteristic
329C22×C4C2×D4×C8311yescharacteristic
330SD16—48no—
331C2×C8—22no—
332C2×C8—22no—
333C2×C8—41no—
334C2×C8—41no—
335C2×C8—41no—
336C2×C8—41no—
337C2×C8—41no—
338C2×C8—41no—
339C2×C8—41no—
340C4⋊C4—22no—
341C4⋊C4—22no—
342C4⋊C4—21no—
343C4⋊C4—21no—
344C4⋊C4—21no—
345C4⋊C4—21no—
346C4⋊C4—42no—
347C4⋊C4—41no—
348C4⋊C4—41no—
349C4⋊C4—41no—
350C4⋊C4—41no—
351C4⋊C4C22⋊C4×C8312yes—
352C4⋊C4SD16×C8312yescomplemented
353C42—22no—
354C42—22no—
355C42—41no—
356C42—41no—
357C42—82no—
358C42C22×C4×C8311yescharacteristic
359C42C2×D4×C8311yescharacteristic
360C42#41×C8311yescharacteristic
361C42#41×C8311yescharacteristic
362C23—42no—
363C23—81no—
364C23—81no—
365C23—81no—
366C23—81no—
367C23—81no—
368C23—81no—
369C23#120×C8311yescharacteristic
370Q8—22no—
371Q8—22no—
372Q8—41no—
373Q8—41no—
374D4—44no—
375D4—42no—
376D4—42no—
377D4—42no—
378D4—42no—
379D4—82no—
380D4—81no—
381D4—81no—
382D4—81no—
383D4—81no—
384C2×C4—21no—
385C2×C4—21no—
386C2×C4—22no—
387C2×C4—22no—
388C2×C4—22no—
389C2×C4—41no—
390C2×C4—41no—
391C2×C4—41no—
392C2×C4—41no—
393C2×C4—41no—
394C2×C4—41no—
395C2×C4—41no—
396C2×C4—41no—
397C2×C4—42no—
398C2×C4—42no—
399C2×C4—42no—
400C2×C4—41no—
401C2×C4C2×C22⋊C4×C8311yescharacteristic
402C2×C4C4×D4×C8311yescharacteristic
403C2×C4C4×D4×C8311yescharacteristic
404C2×C4#121×C8311yescharacteristic
405C2×C4#121×C8311yescharacteristic
406C2×C4#123×C8311yescharacteristic
407C2×C4C2×SD16×C8311yescharacteristic
408C2×C4#135×C8311yescharacteristic
409C2×C4#135×C8311yescharacteristic
410C2×C4#135×C8311yescharacteristic
411C2×C4AGL(1,ℤ/8)×C8311yescharacteristic
412C2×C4AGL(1,ℤ/8)×C8311yescharacteristic
413C2×C4AGL(1,ℤ/8)×C8311yescharacteristic
414C2×C4#137×C8311yescharacteristic
415C8—42no—
416C8—42no—
417C8—81no—
418C22—82no—
419C22—82no—
420C22—82no—
421C22—161no—
422C22—161no—
423C22—161no—
424C22—161no—
425C22#386×C8311yescharacteristic
426C22#448×C8311yescharacteristic
427C22#449×C8311yescharacteristic
428C22#449×C8311yescharacteristic
429C22#449×C8311yescharacteristic
430C22#453×C8311yescharacteristic
431C22#454×C8311yescharacteristic
432C4—21no—
433C4—21no—
434C4—42no—
435C4—42no—
436C4—42no—
437C4—82no—
438C4—81no—
439C4#416×C8311yescharacteristic
440C4#418×C8311yescharacteristic
441C4C4×SD16×C8311yescharacteristic
442C4#440×C8311yescharacteristic
443C4#460×C8311yescharacteristic
444C4#464×C8311yescharacteristic
445C4#482×C8311yescharacteristic
446C4#482×C8311yescharacteristic
447C2—162no—
448C2C83×#162011yescharacteristic
449C2C83×#162211yescharacteristic
450C2C83×#162611yescharacteristic
451C2C83×#162811yescharacteristic
452C2C83×#187511yescharacteristic
453C2C83×#188011yescharacteristic
454C2C83×#188311yescharacteristic
subgroupG/H
center, socleC23×C83#120
commutatorC42C22×C4×C83
Frattini, cosocleC2×C42C23×C83
prime radicalC2×#127×C83C22
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C23×C83 ◁ C2×C42×C83 ◁ G

Complements

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

NH ≅ G/Ntype
2. C83×#2355C2semidirect
5. C2×Q8⋊C8×C83C2semidirect
6. C83×#1296C2semidirect
7. C83×#1279C2semidirect
12. #460×C83C4semidirect
124. C4⋊C4×C83SD16semidirect
219. #8978C83direct (N × H)
229. #2355C2×C83semidirect
232. C2×Q8⋊C8C2×C83semidirect
233. #1296C2×C83semidirect
234. #1279C2×C83semidirect
235. C83#8978direct (N × H)
240. #460C4×C83semidirect
352. C4⋊C4SD16×C83semidirect

Maximal subgroups

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

subgroupindex
C2×#460×C832
C83×#23552
C83×#19822
C83×#19572
C2×Q8⋊C8×C832
C83×#12962
C83×#12792
#897883