Finite Group Databasebeta

← C83×#10004 #35237828

Subgroups — C83×#10004 #35237828

Order 21248.

692 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 692 conjugacy classes of subgroups — too many to draw. Only its 102 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C83×#3213C211yesmaximal, characteristic, complemented
2C83×#3109C211yesmaximal, characteristic, complemented
3C83×#2540C211yesmaximal, characteristic, complemented
4C83×#1980C211yesmaximal, characteristic, complemented
5C83×#1527C211yesmaximal, characteristic, complemented
6C83×#1490C211yesmaximal, characteristic, complemented
7C83×#1285C211yesmaximal, characteristic, complemented
8#532×C83C2211yescharacteristic
9C2×#124×C83C2211yescharacteristic
10C2×C22⋊Q8×C83—21no—
11C2×#121×C83—21no—
12C2×C4×D4×C83C2211yescharacteristic
13#496×C83—21no—
14#489×C83—21no—
15#481×C83—21no—
16#476×C83—21no—
17#463×C83—22no—
18#459×C83—21no—
19#448×C83—21no—
20#447×C83—21no—
21#447×C83—21no—
22#442×C83—22no—
23#432×C83C2211yescharacteristic
24#432×C83C2211yescharacteristic
25C2×#104×C83—22no—
26C4×M16×C83C2211yescharacteristic
27#399×C83—21no—
28#392×C83—21no—
29#390×C83—22no—
30C4×C4⋊C4×C83C2211yescharacteristic
31#329×C83—22no—
32D4⋊C8×C83—22no—
33C2×#41×C83—21no—
34C22×Q8×C83D411yescharacteristic
35C22×D4×C83—22no—
36C22×D4×C83D411yescharacteristic
37C22×D4×C83D411yescharacteristic
38C23×C4×C83—21no—
39C2×SD16×C83—22no—
40C2×D8×C83—22no—
41C2×D8×C83—22no—
42C2×M16×C83—21no—
43C4⋊Q8×C83D411yescharacteristic, complemented
44#127×C83D411yescharacteristic, complemented
45#124×C83—21no—
46#124×C83—21no—
47#124×C83—21no—
48#124×C83—21no—
49#124×C83—21no—
50C22⋊Q8×C83—44no—
51#121×C83—21no—
52#121×C83—21no—
53#121×C83—44no—
54C4×D4×C83—21no—
55C4×D4×C83—21no—
56C4×D4×C83—21no—
57C4×D4×C83—21no—
58C4×D4×C83—21no—
59C4×D4×C83—21no—
60C2×C4⋊C4×C83—21no—
61C2×C4⋊C4×C83—22no—
62C2×C4⋊C4×C83D411yescharacteristic
63C2×C22⋊C4×C83—22no—
64C2×C22⋊C4×C83—22no—
65C2×C22⋊C4×C83—21no—
66C2×C22⋊C4×C83—21no—
67C2×C22⋊C4×C83—21no—
68C2×C42×C83D412yes—
69C2×C42×C83C2311yescharacteristic
70#106×C83—22no—
71C4⋊C8×C83—21no—
72C4⋊C8×C83D412yescomplemented
73#104×C83—44no—
74#103×C83—22no—
75#102×C83—22no—
76#102×C83—22no—
77#102×C83—22no—
78C22⋊C8×C83—21no—
79C22⋊C8×C83—21no—
80#97×C83—21no—
81#97×C83—21no—
82#97×C83D412yescomplemented
83C4×C8×C83—21no—
84#95×C83—21no—
85C24×C83—21no—
86C24×C83—21no—
87#41×C83—42no—
88#41×C83—42no—
89C2×Q8×C83—21no—
90C2×Q8×C83—21no—
91C2×Q8×C83—21no—
92C2×Q8×C83—42no—
93C2×D4×C83—21no—
94C2×D4×C83—21no—
95C2×D4×C83—21no—
96C2×D4×C83—21no—
97C2×D4×C83—21no—
98C2×D4×C83—21no—
99C2×D4×C83—21no—
100C2×D4×C83—42no—
101C2×D4×C83—44no—
102C2×D4×C83—42no—
103C2×D4×C83—44no—
104C2×D4×C83—41no—
105C2×D4×C83—82no—
106C22×C4×C83—22no—
107C22×C4×C83—21no—
108C22×C4×C83—21no—
109C22×C4×C83—42no—
110C22×C4×C83—42no—
111C22×C4×C83—41no—
112C22×C4×C83—41no—
113C22×C4×C83—41no—
114C22×C4×C83C2×D411yescharacteristic
115C22×C4×C83C2×D411yescharacteristic
116C22×C4×C83C2×D411yescharacteristic
117SD16×C83—82no—
118D8×C83—44no—
119D8×C83—82no—
120M16×C83—22no—
121M16×C83—41no—
122C2×C8×C83—22no—
123C2×C8×C83—22no—
124C2×C8×C83—41no—
125C4⋊C4×C83—24no—
126C4⋊C4×C83—21no—
127C4⋊C4×C83—21no—
128C4⋊C4×C83—21no—
129C4⋊C4×C83—21no—
130C4⋊C4×C83—44no—
131C22⋊C4×C83—44no—
132C22⋊C4×C83—44no—
133C22⋊C4×C83—41no—
134C22⋊C4×C83—41no—
135C22⋊C4×C83—41no—
136C22⋊C4×C83—41no—
137C22⋊C4×C83—41no—
138C22⋊C4×C83—41no—
139C22⋊C4×C83—41no—
140C22⋊C4×C83—41no—
141C42×C83—42no—
142C42×C83C2×D411yescharacteristic
143C42×C83C2×D411yescharacteristic
144C42×C83C2×D411yescharacteristic
145C42×C83#4111yescharacteristic
146C23×C83—22no—
147C23×C83—41no—
148C23×C83—41no—
149C23×C83—41no—
150C23×C83—42no—
151C23×C83—42no—
152C23×C83—41no—
153C23×C83—41no—
154C23×C83—41no—
155C23×C83—41no—
156C23×C83—41no—
157C23×C83—41no—
158C23×C83—41no—
159C23×C83#12011yescharacteristic
160Q8×C83—42no—
161Q8×C83—42no—
162Q8×C83—81no—
163D4×C83—22no—
164D4×C83—22no—
165D4×C83—41no—
166D4×C83—41no—
167D4×C83—42no—
168D4×C83—42no—
169D4×C83—42no—
170D4×C83—42no—
171D4×C83—81no—
172D4×C83—88no—
173D4×C83—81no—
174D4×C83—81no—
175C2×C4×C83—21no—
176C2×C4×C83—21no—
177C2×C4×C83—22no—
178C2×C4×C83—24no—
179C2×C4×C83—41no—
180C2×C4×C83—42no—
181C2×C4×C83—42no—
182C2×C4×C83—41no—
183C2×C4×C83—42no—
184C2×C4×C83—41no—
185C2×C4×C83—41no—
186C2×C4×C83—41no—
187C2×C4×C83—41no—
188C2×C4×C83—82no—
189C2×C4×C83—82no—
190C2×C4×C83—82no—
191C2×C4×C83#12011yescharacteristic
192C2×C4×C83#12011yescharacteristic
193C2×C4×C83#12111yescharacteristic
194C2×C4×C83#12111yescharacteristic
195C2×C4×C83#12111yescharacteristic
196C2×C4×C83#12711yescharacteristic
197C8×C83—42no—
198C8×C83—42no—
199C8×C83—42no—
200C22×C83—21no—
201C22×C83—21no—
202C22×C83—42no—
203C22×C83—42no—
204C22×C83—42no—
205C22×C83—81no—
206C22×C83—81no—
207C22×C83—82no—
208C22×C83—82no—
209C22×C83—81no—
210C22×C83—81no—
211C22×C83—81no—
212C22×C83—81no—
213C22×C83—81no—
214C22×C83—81no—
215C22×C83—81no—
216C22×C83—81no—
217C22×C83#39211yescharacteristic
218C22×C83#45311yescharacteristic
219C22×C83#45411yescharacteristic
220C4×C83—21no—
221C4×C83—21no—
222C4×C83—21no—
223C4×C83—21no—
224C4×C83—21no—
225C4×C83—44no—
226C4×C83—81no—
227C4×C83—81no—
228C4×C83—81no—
229C4×C83#45312yes—
230#10004C8311yesmaximal, characteristic, direct product factor, Sylow
231C2×C83—21no—
232C2×C83—21no—
233C2×C83—82no—
234C2×C83—161no—
235C2×C83—161no—
236C2×C83#200211yescharacteristic
237C2×C83#200611yescharacteristic
238C2×C83#201011yescharacteristic
239#3213C2×C8311yescharacteristic, complemented
240#3109C2×C8311yescharacteristic, complemented
241#2540C2×C8311yescharacteristic, complemented
242#1980C2×C8311yescharacteristic, complemented
243#1527C2×C8311yescharacteristic, complemented
244#1490C2×C8311yescharacteristic, complemented
245#1285C2×C8311yescharacteristic, complemented
246C83#1000411yescharacteristic, direct product factor, Sylow
247#532C22×C8311yescharacteristic
248C2×#124C22×C8311yescharacteristic
249C2×C22⋊Q8—21no—
250C2×#121—21no—
251C2×C4×D4C22×C8311yescharacteristic
252#496—21no—
253#489—21no—
254#481—21no—
255#476—21no—
256#463—22no—
257#459—21no—
258#448—21no—
259#447—21no—
260#447—21no—
261#442—22no—
262#432C22×C8311yescharacteristic
263#432C22×C8311yescharacteristic
264C2×#104—22no—
265C4×M16C22×C8311yescharacteristic
266#399—21no—
267#392—21no—
268#390—22no—
269C4×C4⋊C4C22×C8311yescharacteristic
270#329—22no—
271D4⋊C8—22no—
272C2×#41—21no—
273C22×Q8D4×C8311yescharacteristic
274C22×D4—22no—
275C22×D4D4×C8311yescharacteristic
276C22×D4D4×C8311yescharacteristic
277C23×C4—21no—
278C2×SD16—22no—
279C2×D8—22no—
280C2×D8—22no—
281C2×M16—21no—
282C4⋊Q8D4×C8311yescharacteristic, complemented
283#127D4×C8311yescharacteristic, complemented
284#124—21no—
285#124—21no—
286#124—21no—
287#124—21no—
288#124—21no—
289C22⋊Q8—44no—
290#121—21no—
291#121—21no—
292#121—44no—
293C4×D4—21no—
294C4×D4—21no—
295C4×D4—21no—
296C4×D4—21no—
297C4×D4—21no—
298C4×D4—21no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4D4×C8311yescharacteristic
302C2×C22⋊C4—22no—
303C2×C22⋊C4—22no—
304C2×C22⋊C4—21no—
305C2×C22⋊C4—21no—
306C2×C22⋊C4—21no—
307C2×C42D4×C8312yes—
308C2×C42C23×C8311yescharacteristic
309#106—22no—
310C4⋊C8—21no—
311C4⋊C8D4×C8312yescomplemented
312#104—44no—
313#103—22no—
314#102—22no—
315#102—22no—
316#102—22no—
317C22⋊C8—21no—
318C22⋊C8—21no—
319#97—21no—
320#97—21no—
321#97D4×C8312yescomplemented
322C4×C8—21no—
323#95—21no—
324C24—21no—
325C24—21no—
326#41—42no—
327#41—42no—
328C2×Q8—21no—
329C2×Q8—21no—
330C2×Q8—21no—
331C2×Q8—42no—
332C2×D4—21no—
333C2×D4—21no—
334C2×D4—21no—
335C2×D4—21no—
336C2×D4—21no—
337C2×D4—21no—
338C2×D4—21no—
339C2×D4—42no—
340C2×D4—44no—
341C2×D4—42no—
342C2×D4—44no—
343C2×D4—41no—
344C2×D4—82no—
345C22×C4—22no—
346C22×C4—21no—
347C22×C4—21no—
348C22×C4—42no—
349C22×C4—42no—
350C22×C4—41no—
351C22×C4—41no—
352C22×C4—41no—
353C22×C4C2×D4×C8311yescharacteristic
354C22×C4C2×D4×C8311yescharacteristic
355C22×C4C2×D4×C8311yescharacteristic
356SD16—82no—
357D8—44no—
358D8—82no—
359M16—22no—
360M16—41no—
361C2×C8—22no—
362C2×C8—22no—
363C2×C8—41no—
364C4⋊C4—24no—
365C4⋊C4—21no—
366C4⋊C4—21no—
367C4⋊C4—21no—
368C4⋊C4—21no—
369C4⋊C4—44no—
370C22⋊C4—44no—
371C22⋊C4—44no—
372C22⋊C4—41no—
373C22⋊C4—41no—
374C22⋊C4—41no—
375C22⋊C4—41no—
376C22⋊C4—41no—
377C22⋊C4—41no—
378C22⋊C4—41no—
379C22⋊C4—41no—
380C42—42no—
381C42C2×D4×C8311yescharacteristic
382C42C2×D4×C8311yescharacteristic
383C42C2×D4×C8311yescharacteristic
384C42#41×C8311yescharacteristic
385C23—22no—
386C23—41no—
387C23—41no—
388C23—41no—
389C23—42no—
390C23—42no—
391C23—41no—
392C23—41no—
393C23—41no—
394C23—41no—
395C23—41no—
396C23—41no—
397C23—41no—
398C23#120×C8311yescharacteristic
399Q8—42no—
400Q8—42no—
401Q8—81no—
402D4—22no—
403D4—22no—
404D4—41no—
405D4—41no—
406D4—42no—
407D4—42no—
408D4—42no—
409D4—42no—
410D4—81no—
411D4—88no—
412D4—81no—
413D4—81no—
414C2×C4—21no—
415C2×C4—21no—
416C2×C4—22no—
417C2×C4—24no—
418C2×C4—41no—
419C2×C4—42no—
420C2×C4—42no—
421C2×C4—41no—
422C2×C4—42no—
423C2×C4—41no—
424C2×C4—41no—
425C2×C4—41no—
426C2×C4—41no—
427C2×C4—82no—
428C2×C4—82no—
429C2×C4—82no—
430C2×C4#120×C8311yescharacteristic
431C2×C4#120×C8311yescharacteristic
432C2×C4#121×C8311yescharacteristic
433C2×C4#121×C8311yescharacteristic
434C2×C4#121×C8311yescharacteristic
435C2×C4#127×C8311yescharacteristic
436C8—42no—
437C8—42no—
438C8—42no—
439C22—21no—
440C22—21no—
441C22—42no—
442C22—42no—
443C22—42no—
444C22—81no—
445C22—81no—
446C22—82no—
447C22—82no—
448C22—81no—
449C22—81no—
450C22—81no—
451C22—81no—
452C22—81no—
453C22—81no—
454C22—81no—
455C22—81no—
456C22#392×C8311yescharacteristic
457C22#453×C8311yescharacteristic
458C22#454×C8311yescharacteristic
459C4—21no—
460C4—21no—
461C4—21no—
462C4—21no—
463C4—21no—
464C4—44no—
465C4—81no—
466C4—81no—
467C4—81no—
468C4#453×C8312yes—
469C2—21no—
470C2—21no—
471C2—82no—
472C2—161no—
473C2—161no—
474C2C83×#200211yescharacteristic
475C2C83×#200611yescharacteristic
476C2C83×#201011yescharacteristic
subgroupG/H
center, socleC22×C83#392
commutator, Frattini, cosocleC2×C42C23×C83
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C83 ◁ C2×C42×C83 ◁ G

Complements

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

NH ≅ G/Ntype
1. C83×#3213C2semidirect
2. C83×#3109C2semidirect
3. C83×#2540C2semidirect
4. C83×#1980C2semidirect
5. C83×#1527C2semidirect
6. C83×#1490C2semidirect
7. C83×#1285C2semidirect
43. C4⋊Q8×C83D4semidirect
44. #127×C83D4semidirect
72. C4⋊C8×C83D4semidirect
82. #97×C83D4semidirect
230. #10004C83direct (N × H)
239. #3213C2×C83semidirect
240. #3109C2×C83semidirect
241. #2540C2×C83semidirect
242. #1980C2×C83semidirect
243. #1527C2×C83semidirect
244. #1490C2×C83semidirect
245. #1285C2×C83semidirect
246. C83#10004direct (N × H)
282. C4⋊Q8D4×C83semidirect
283. #127D4×C83semidirect
311. C4⋊C8D4×C83semidirect
321. #97D4×C83semidirect

Maximal subgroups

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

subgroupindex
C83×#32132
C83×#31092
C83×#25402
C83×#19802
C83×#15272
C83×#14902
C83×#12852
#1000483