Finite Group Databasebeta

← C83×#9652 #35237476

Subgroups — C83×#9652 #35237476

Order 21248.

704 conjugacy classes of subgroups.

The subgroup lattice has 472 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#388×C83C211yesmaximal, characteristic
2C83×#1797C211yesmaximal, characteristic, complemented
3C83×#1749C211yesmaximal, characteristic, complemented
4C2×#324×C83C212yesmaximal, complemented
5C2×#323×C83C212yesmaximal
6C22×C4⋊C4×C83C2211yescharacteristic
7C22×C22⋊C4×C83C411yescharacteristic
8C22×C22⋊C4×C83C411yescharacteristic
9C22×C22⋊C4×C83C2211yescharacteristic
10C22×C42×C83C2211yescharacteristic
11C2×C4⋊C8×C83—21no—
12C2×C22⋊C8×C83C2212yes—
13C2×C22⋊C8×C83C2212yes—
14C2×#97×C83—21no—
15#388×C83—21no—
16#388×C83—21no—
17#388×C83—21no—
18#388×C83—21no—
19C2×#95×C83C411yescharacteristic
20C2×#95×C83C411yescharacteristic
21#336×C83—21no—
22#336×C83—21no—
23#324×C83—24no—
24#323×C83—24no—
25C25×C83C2×C411yescharacteristic
26C23×C4×C83C2×C411yescharacteristic
27C23×C4×C83C2×C411yescharacteristic
28C23×C4×C83C2×C411yescharacteristic
29C23×C4×C83C2×C411yescharacteristic
30C23×C4×C83C2×C411yescharacteristic
31C23×C4×C83C2311yescharacteristic
32C22×C8×C83—22no—
33C22×C8×C83—22no—
34C2×C4⋊C4×C83—22no—
35C2×C4⋊C4×C83—21no—
36C2×C4⋊C4×C83—21no—
37C2×C4⋊C4×C83—22no—
38C2×C22⋊C4×C83—22no—
39C2×C22⋊C4×C83—21no—
40C2×C22⋊C4×C83—21no—
41C2×C22⋊C4×C83—22no—
42C2×C22⋊C4×C83—21no—
43C2×C22⋊C4×C83—21no—
44C2×C22⋊C4×C83—22no—
45C2×C22⋊C4×C83—21no—
46C2×C22⋊C4×C83—21no—
47C2×C22⋊C4×C83—22no—
48C2×C22⋊C4×C83—42no—
49C2×C42×C83—42no—
50C2×C42×C83D411yescharacteristic
51C2×C42×C83D411yescharacteristic
52C2×C42×C83D411yescharacteristic
53C2×C42×C83D411yescharacteristic
54C4⋊C8×C83—42no—
55C22⋊C8×C83—22no—
56C22⋊C8×C83—22no—
57C22⋊C8×C83—22no—
58C22⋊C8×C83D414yes—
59#97×C83—42no—
60#95×C83—21no—
61#95×C83—21no—
62#95×C83—21no—
63#95×C83—21no—
64C24×C83—21no—
65C24×C83—21no—
66C24×C83—22no—
67C24×C83—22no—
68C24×C83—41no—
69C24×C83—41no—
70C24×C83—41no—
71C24×C83—41no—
72C24×C83M1612yes—
73C24×C83C22×C411yescharacteristic
74C22×C4×C83—21no—
75C22×C4×C83—22no—
76C22×C4×C83—21no—
77C22×C4×C83—21no—
78C22×C4×C83—21no—
79C22×C4×C83—22no—
80C22×C4×C83—22no—
81C22×C4×C83—22no—
82C22×C4×C83—21no—
83C22×C4×C83—21no—
84C22×C4×C83—22no—
85C22×C4×C83—22no—
86C22×C4×C83—41no—
87C22×C4×C83—41no—
88C22×C4×C83—41no—
89C22×C4×C83—41no—
90C22×C4×C83—41no—
91C22×C4×C83—41no—
92C22×C4×C83—41no—
93C22×C4×C83—41no—
94C22×C4×C83—41no—
95C22×C4×C83C22⋊C411yescharacteristic
96C22×C4×C83C22⋊C411yescharacteristic
97C22×C4×C83C22⋊C411yescharacteristic
98C22×C4×C83C22⋊C411yescharacteristic
99C22×C4×C83M1612yes—
100C22×C4×C83C2×D411yescharacteristic
101C22×C4×C83C2×D411yescharacteristic
102C22×C4×C83C2×D412yes—
103C22×C4×C83#4112yes—
104C2×C8×C83—24no—
105C2×C8×C83—42no—
106C2×C8×C83—42no—
107C2×C8×C83—42no—
108C2×C8×C83—42no—
109C2×C8×C83—42no—
110C4⋊C4×C83—42no—
111C4⋊C4×C83—41no—
112C4⋊C4×C83—41no—
113C22⋊C4×C83—24no—
114C22⋊C4×C83—24no—
115C22⋊C4×C83—42no—
116C22⋊C4×C83—42no—
117C22⋊C4×C83—42no—
118C22⋊C4×C83—41no—
119C22⋊C4×C83—41no—
120C42×C83—22no—
121C42×C83—22no—
122C42×C83—41no—
123C42×C83—41no—
124C23×C83—21no—
125C23×C83—21no—
126C23×C83—21no—
127C23×C83—21no—
128C23×C83—22no—
129C23×C83—24no—
130C23×C83—42no—
131C23×C83—42no—
132C23×C83—42no—
133C23×C83—42no—
134C23×C83—41no—
135C23×C83—41no—
136C23×C83—41no—
137C23×C83—41no—
138C23×C83—41no—
139C23×C83—41no—
140C23×C83—42no—
141C23×C83—82no—
142C23×C83—82no—
143C23×C83—82no—
144C23×C83—82no—
145C23×C83C2×C22⋊C411yescharacteristic
146C23×C83C4×D412yes—
147C23×C83C2×M1611yescharacteristic
148C23×C83C2×M1611yescharacteristic
149C23×C83#13112yes—
150C2×C4×C83—21no—
151C2×C4×C83—21no—
152C2×C4×C83—21no—
153C2×C4×C83—21no—
154C2×C4×C83—21no—
155C2×C4×C83—21no—
156C2×C4×C83—21no—
157C2×C4×C83—21no—
158C2×C4×C83—22no—
159C2×C4×C83—21no—
160C2×C4×C83—21no—
161C2×C4×C83—21no—
162C2×C4×C83—21no—
163C2×C4×C83—21no—
164C2×C4×C83—21no—
165C2×C4×C83—21no—
166C2×C4×C83—24no—
167C2×C4×C83—42no—
168C2×C4×C83—42no—
169C2×C4×C83—44no—
170C2×C4×C83—42no—
171C2×C4×C83—42no—
172C2×C4×C83—41no—
173C2×C4×C83—41no—
174C2×C4×C83—42no—
175C2×C4×C83—41no—
176C2×C4×C83—41no—
177C2×C4×C83—41no—
178C2×C4×C83—41no—
179C2×C4×C83—41no—
180C2×C4×C83—41no—
181C2×C4×C83—82no—
182C2×C4×C83—82no—
183C2×C4×C83#9912yes—
184C2×C4×C83#12112yes—
185C2×C4×C83#12411yescharacteristic
186C2×C4×C83#12711yescharacteristic
187C8×C83—44no—
188C8×C83—82no—
189C22×C83—22no—
190C22×C83—22no—
191C22×C83—21no—
192C22×C83—21no—
193C22×C83—21no—
194C22×C83—21no—
195C22×C83—22no—
196C22×C83—22no—
197C22×C83—42no—
198C22×C83—42no—
199C22×C83—42no—
200C22×C83—84no—
201C22×C83—84no—
202C22×C83—81no—
203C22×C83—81no—
204C22×C83—81no—
205C22×C83—81no—
206C22×C83#39011yescharacteristic
207C22×C83#40711yescharacteristic
208C22×C83#40811yescharacteristic
209C22×C83C2×#9911yescharacteristic
210C22×C83#41011yescharacteristic
211C22×C83#41311yescharacteristic
212C22×C83#41311yescharacteristic
213C22×C83#43512yes—
214C22×C83#43512yes—
215C4×C83—22no—
216C4×C83—21no—
217C4×C83—21no—
218C4×C83—22no—
219C4×C83—42no—
220C4×C83—41no—
221C4×C83—41no—
222C4×C83—41no—
223C4×C83—81no—
224C4×C83—81no—
225#9652C8311yesmaximal, characteristic, direct product factor, Sylow
226C2×C83—22no—
227C2×C83—22no—
228C2×C83—82no—
229C2×C83#146411yescharacteristic
230C2×C83#146511yescharacteristic
231C2×C83#146811yescharacteristic
232C2×C83#147311yescharacteristic
233C2×C83#195511yescharacteristic
234C2×C83#196411yescharacteristic
235C2×C83#196711yescharacteristic
236C2×#388C2×C8311yescharacteristic
237#1797C2×C8311yescharacteristic, complemented
238#1749C2×C8311yescharacteristic, complemented
239C2×#324C2×C8312yescomplemented
240C2×#323C2×C8312yes—
241C83#965211yescharacteristic, direct product factor, Sylow
242C22×C4⋊C4C22×C8311yescharacteristic
243C22×C22⋊C4C4×C8311yescharacteristic
244C22×C22⋊C4C4×C8311yescharacteristic
245C22×C22⋊C4C22×C8311yescharacteristic
246C22×C42C22×C8311yescharacteristic
247C2×C4⋊C8—21no—
248C2×C22⋊C8C22×C8312yes—
249C2×C22⋊C8C22×C8312yes—
250C2×#97—21no—
251#388—21no—
252#388—21no—
253#388—21no—
254#388—21no—
255C2×#95C4×C8311yescharacteristic
256C2×#95C4×C8311yescharacteristic
257#336—21no—
258#336—21no—
259#324—24no—
260#323—24no—
261C25C2×C4×C8311yescharacteristic
262C23×C4C2×C4×C8311yescharacteristic
263C23×C4C2×C4×C8311yescharacteristic
264C23×C4C2×C4×C8311yescharacteristic
265C23×C4C2×C4×C8311yescharacteristic
266C23×C4C2×C4×C8311yescharacteristic
267C23×C4C23×C8311yescharacteristic
268C22×C8—22no—
269C22×C8—22no—
270C2×C4⋊C4—22no—
271C2×C4⋊C4—21no—
272C2×C4⋊C4—21no—
273C2×C4⋊C4—22no—
274C2×C22⋊C4—22no—
275C2×C22⋊C4—21no—
276C2×C22⋊C4—21no—
277C2×C22⋊C4—22no—
278C2×C22⋊C4—21no—
279C2×C22⋊C4—21no—
280C2×C22⋊C4—22no—
281C2×C22⋊C4—21no—
282C2×C22⋊C4—21no—
283C2×C22⋊C4—22no—
284C2×C22⋊C4—42no—
285C2×C42—42no—
286C2×C42D4×C8311yescharacteristic
287C2×C42D4×C8311yescharacteristic
288C2×C42D4×C8311yescharacteristic
289C2×C42D4×C8311yescharacteristic
290C4⋊C8—42no—
291C22⋊C8—22no—
292C22⋊C8—22no—
293C22⋊C8—22no—
294C22⋊C8D4×C8314yes—
295#97—42no—
296#95—21no—
297#95—21no—
298#95—21no—
299#95—21no—
300C24—21no—
301C24—21no—
302C24—22no—
303C24—22no—
304C24—41no—
305C24—41no—
306C24—41no—
307C24—41no—
308C24M16×C8312yes—
309C24C22×C4×C8311yescharacteristic
310C22×C4—21no—
311C22×C4—22no—
312C22×C4—21no—
313C22×C4—21no—
314C22×C4—21no—
315C22×C4—22no—
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—21no—
319C22×C4—21no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—41no—
323C22×C4—41no—
324C22×C4—41no—
325C22×C4—41no—
326C22×C4—41no—
327C22×C4—41no—
328C22×C4—41no—
329C22×C4—41no—
330C22×C4—41no—
331C22×C4C22⋊C4×C8311yescharacteristic
332C22×C4C22⋊C4×C8311yescharacteristic
333C22×C4C22⋊C4×C8311yescharacteristic
334C22×C4C22⋊C4×C8311yescharacteristic
335C22×C4M16×C8312yes—
336C22×C4C2×D4×C8311yescharacteristic
337C22×C4C2×D4×C8311yescharacteristic
338C22×C4C2×D4×C8312yes—
339C22×C4#41×C8312yes—
340C2×C8—24no—
341C2×C8—42no—
342C2×C8—42no—
343C2×C8—42no—
344C2×C8—42no—
345C2×C8—42no—
346C4⋊C4—42no—
347C4⋊C4—41no—
348C4⋊C4—41no—
349C22⋊C4—24no—
350C22⋊C4—24no—
351C22⋊C4—42no—
352C22⋊C4—42no—
353C22⋊C4—42no—
354C22⋊C4—41no—
355C22⋊C4—41no—
356C42—22no—
357C42—22no—
358C42—41no—
359C42—41no—
360C23—21no—
361C23—21no—
362C23—21no—
363C23—21no—
364C23—22no—
365C23—24no—
366C23—42no—
367C23—42no—
368C23—42no—
369C23—42no—
370C23—41no—
371C23—41no—
372C23—41no—
373C23—41no—
374C23—41no—
375C23—41no—
376C23—42no—
377C23—82no—
378C23—82no—
379C23—82no—
380C23—82no—
381C23C2×C22⋊C4×C8311yescharacteristic
382C23C4×D4×C8312yes—
383C23C2×M16×C8311yescharacteristic
384C23C2×M16×C8311yescharacteristic
385C23#131×C8312yes—
386C2×C4—21no—
387C2×C4—21no—
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—21no—
391C2×C4—21no—
392C2×C4—21no—
393C2×C4—21no—
394C2×C4—22no—
395C2×C4—21no—
396C2×C4—21no—
397C2×C4—21no—
398C2×C4—21no—
399C2×C4—21no—
400C2×C4—21no—
401C2×C4—21no—
402C2×C4—24no—
403C2×C4—42no—
404C2×C4—42no—
405C2×C4—44no—
406C2×C4—42no—
407C2×C4—42no—
408C2×C4—41no—
409C2×C4—41no—
410C2×C4—42no—
411C2×C4—41no—
412C2×C4—41no—
413C2×C4—41no—
414C2×C4—41no—
415C2×C4—41no—
416C2×C4—41no—
417C2×C4—82no—
418C2×C4—82no—
419C2×C4#99×C8312yes—
420C2×C4#121×C8312yes—
421C2×C4#124×C8311yescharacteristic
422C2×C4#127×C8311yescharacteristic
423C8—44no—
424C8—82no—
425C22—22no—
426C22—22no—
427C22—21no—
428C22—21no—
429C22—21no—
430C22—21no—
431C22—22no—
432C22—22no—
433C22—42no—
434C22—42no—
435C22—42no—
436C22—84no—
437C22—84no—
438C22—81no—
439C22—81no—
440C22—81no—
441C22—81no—
442C22#390×C8311yescharacteristic
443C22#407×C8311yescharacteristic
444C22#408×C8311yescharacteristic
445C22C2×#99×C8311yescharacteristic
446C22#410×C8311yescharacteristic
447C22#413×C8311yescharacteristic
448C22#413×C8311yescharacteristic
449C22#435×C8312yes—
450C22#435×C8312yes—
451C4—22no—
452C4—21no—
453C4—21no—
454C4—22no—
455C4—42no—
456C4—41no—
457C4—41no—
458C4—41no—
459C4—81no—
460C4—81no—
461C2—22no—
462C2—22no—
463C2—82no—
464C2C83×#146411yescharacteristic
465C2C83×#146511yescharacteristic
466C2C83×#146811yescharacteristic
467C2C83×#147311yescharacteristic
468C2C83×#195511yescharacteristic
469C2C83×#196411yescharacteristic
470C2C83×#196711yescharacteristic
subgroupG/H
center, socleC23×C83C2×C22⋊C4
commutatorC24C22×C4×C83
Frattini, cosocleC23×C4C23×C83
prime radicalC25×C83C2×C4
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C23×C83 ◁ C22×C42×C83 ◁ G

Complements

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

NH ≅ G/Ntype
2. C83×#1797C2semidirect
3. C83×#1749C2semidirect
4. C2×#324×C83C2semidirect
225. #9652C83direct (N × H)
237. #1797C2×C83semidirect
238. #1749C2×C83semidirect
239. C2×#324C2×C83semidirect
241. C83#9652direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#388×C832
C83×#17972
C83×#17492
C2×#324×C832
C2×#323×C832
#965283