Finite Group Databasebeta

← C83×#9318 #35237142

Subgroups — C83×#9318 #35237142

Order 21248.

580 conjugacy classes of subgroups.

The subgroup lattice has 442 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C83×#3038C211yesmaximal, characteristic
2C83×#2988C211yesmaximal, characteristic, complemented
3C83×#1982C211yesmaximal, characteristic, complemented
4C83×#1806C211yesmaximal, characteristic, complemented
5C83×#1545C211yesmaximal, characteristic, complemented
6C83×#1303C211yesmaximal, characteristic, complemented
7C83×#1292C211yesmaximal, characteristic, complemented
8C2×AGL(1,ℤ/8)×C83—21no—
9C2×#127×C83C2211yescharacteristic
10C4×#41×C83C2211yescharacteristic
11#460×C83—21no—
12#460×C83C412yescomplemented
13#459×C83—21no—
14#459×C83C412yescomplemented
15C8×Q8×C83—21no—
16C8×D4×C83—22no—
17C8×D4×C83—21no—
18#431×C83—21no—
19#431×C83C2211yescharacteristic
20#423×C83C2211yescharacteristic
21#423×C83C2211yescharacteristic
22C2×C4⋊C8×C83—21no—
23C2×C4⋊C8×C83C2211yescharacteristic
24C2×#104×C83—21no—
25#418×C83—21no—
26C2×#102×C83—22no—
27C4×C4⋊C4×C83C2211yescharacteristic
28#339×C83—21no—
29#331×C83—21no—
30#331×C83—21no—
31C2×#41×C83D411yescharacteristic
32C22×D4×C83—21no—
33C22×D4×C83—21no—
34C22×D4×C83—21no—
35AGL(1,ℤ/8)×C83—42no—
36AGL(1,ℤ/8)×C83—42no—
37C2×SD16×C83—22no—
38C2×D8×C83—22no—
39C2×M16×C83—21no—
40C2×M16×C83—21no—
41C22×C8×C83—22no—
42C22×C8×C83D412yes—
43#127×C83—21no—
44#127×C83—21no—
45#127×C83—21no—
46#127×C83C2×C412yes—
47C4×Q8×C83C2×C411yescharacteristic
48C4×Q8×C83D411yescharacteristic
49C4×D4×C83—21no—
50C4×D4×C83—21no—
51C4×D4×C83C2×C411yescharacteristic
52C4×D4×C83D411yescharacteristic
53#117×C83—21no—
54#117×C83D411yescharacteristic
55C2×C4⋊C4×C83—21no—
56C2×C42×C83—21no—
57C2×C42×C83—21no—
58C2×C42×C83C2311yescharacteristic
59C4⋊C8×C83—22no—
60C4⋊C8×C83—21no—
61C4⋊C8×C83—21no—
62C4⋊C8×C83—21no—
63C4⋊C8×C83—21no—
64C4⋊C8×C83—21no—
65C4⋊C8×C83C2×C412yes—
66C4⋊C8×C83D411yescharacteristic
67C4⋊C8×C83D411yescharacteristic
68#104×C83—42no—
69#102×C83—22no—
70#102×C83—22no—
71#102×C83—44no—
72C22⋊C8×C83—22no—
73C22⋊C8×C83—21no—
74C4×C8×C83—22no—
75C4×C8×C83—21no—
76#95×C83—21no—
77C24×C83—41no—
78#41×C83—21no—
79#41×C83—21no—
80#41×C83D812yes—
81#41×C83SD1612yes—
82C2×Q8×C83C22⋊C411yescharacteristic
83C2×D4×C83—21no—
84C2×D4×C83—22no—
85C2×D4×C83—22no—
86C2×D4×C83—22no—
87C2×D4×C83—42no—
88C2×D4×C83—42no—
89C2×D4×C83—41no—
90C2×D4×C83—41no—
91C2×D4×C83—41no—
92C2×D4×C83—41no—
93C2×D4×C83—41no—
94C2×D4×C83—41no—
95C2×D4×C83—41no—
96C2×D4×C83—82no—
97C2×D4×C83C22⋊C411yescharacteristic
98C22×C4×C83—21no—
99C22×C4×C83—21no—
100C22×C4×C83—21no—
101C22×C4×C83—41no—
102C22×C4×C83C2×D411yescharacteristic
103C22×C4×C83C2×D411yescharacteristic
104C22×C4×C83#4111yescharacteristic
105SD16×C83—44no—
106D8×C83—44no—
107M16×C83—41no—
108M16×C83—41no—
109M16×C83—41no—
110M16×C83—41no—
111C2×C8×C83—22no—
112C2×C8×C83—21no—
113C2×C8×C83—22no—
114C2×C8×C83—21no—
115C2×C8×C83—42no—
116C2×C8×C83—41no—
117C2×C8×C83—82no—
118C4⋊C4×C83—21no—
119C4⋊C4×C83—21no—
120C4⋊C4×C83—22no—
121C4⋊C4×C83—41no—
122C4⋊C4×C83C22⋊C411yescharacteristic
123C4⋊C4×C83C22⋊C411yescharacteristic
124C22⋊C4×C83—21no—
125C22⋊C4×C83—21no—
126C22⋊C4×C83—21no—
127C42×C83—21no—
128C42×C83—21no—
129C42×C83—21no—
130C42×C83—41no—
131C42×C83—41no—
132C42×C83C22×C411yescharacteristic
133C42×C83C2×D411yescharacteristic
134C42×C83C2×D411yescharacteristic
135C42×C83#4111yescharacteristic
136C23×C83—21no—
137C23×C83—42no—
138C23×C83—82no—
139C23×C83—82no—
140C23×C83—81no—
141C23×C83—81no—
142C23×C83#12111yescharacteristic
143Q8×C83—21no—
144Q8×C83#10212yescomplemented
145D4×C83—22no—
146D4×C83—22no—
147D4×C83—21no—
148D4×C83—44no—
149D4×C83—42no—
150D4×C83—42no—
151D4×C83—82no—
152D4×C83—82no—
153D4×C83—81no—
154D4×C83—81no—
155D4×C83—81no—
156D4×C83—81no—
157D4×C83#10212yescomplemented
158C2×C4×C83—21no—
159C2×C4×C83—21no—
160C2×C4×C83—21no—
161C2×C4×C83—22no—
162C2×C4×C83—22no—
163C2×C4×C83—21no—
164C2×C4×C83—21no—
165C2×C4×C83—41no—
166C2×C4×C83—41no—
167C2×C4×C83—41no—
168C2×C4×C83—42no—
169C2×C4×C83—81no—
170C2×C4×C83—81no—
171C2×C4×C83—81no—
172C2×C4×C83C2×C22⋊C411yescharacteristic
173C2×C4×C83C4×D411yescharacteristic
174C2×C4×C83C4×D411yescharacteristic
175C2×C4×C83#12011yescharacteristic
176C2×C4×C83#12111yescharacteristic
177C2×C4×C83#12311yescharacteristic
178C2×C4×C83C2×D811yescharacteristic
179C2×C4×C83C2×SD1611yescharacteristic
180C2×C4×C83#13511yescharacteristic
181C2×C4×C83#13511yescharacteristic
182C8×C83—42no—
183C8×C83—42no—
184C8×C83—41no—
185C8×C83—41no—
186C8×C83—81no—
187C22×C83—21no—
188C22×C83—21no—
189C22×C83—22no—
190C22×C83—41no—
191C22×C83—41no—
192C22×C83—82no—
193C22×C83—82no—
194C22×C83—82no—
195C22×C83—162no—
196C22×C83—162no—
197C22×C83#38611yescharacteristic
198C22×C83#46511yescharacteristic
199C22×C83#46611yescharacteristic
200C4×C83—21no—
201C4×C83—22no—
202C4×C83—21no—
203C4×C83—21no—
204C4×C83—41no—
205C4×C83—41no—
206C4×C83—82no—
207C4×C83C2×#10211yescharacteristic
208C4×C83#41611yescharacteristic
209C4×C83#45311yescharacteristic
210C4×C83#45411yescharacteristic
211C4×C83#46511yescharacteristic
212C4×C83#46611yescharacteristic
213#9318C8311yesmaximal, characteristic, direct product factor, Sylow
214C2×C83—21no—
215C2×C83—21no—
216C2×C83—42no—
217C2×C83—162no—
218C2×C83#188311yescharacteristic
219C2×C83#189311yescharacteristic
220C2×C83#190311yescharacteristic
221#3038C2×C8311yescharacteristic
222#2988C2×C8311yescharacteristic, complemented
223#1982C2×C8311yescharacteristic, complemented
224#1806C2×C8311yescharacteristic, complemented
225#1545C2×C8311yescharacteristic, complemented
226#1303C2×C8311yescharacteristic, complemented
227#1292C2×C8311yescharacteristic, complemented
228C83#931811yescharacteristic, direct product factor, Sylow
229C2×AGL(1,ℤ/8)—21no—
230C2×#127C22×C8311yescharacteristic
231C4×#41C22×C8311yescharacteristic
232#460—21no—
233#460C4×C8312yescomplemented
234#459—21no—
235#459C4×C8312yescomplemented
236C8×Q8—21no—
237C8×D4—22no—
238C8×D4—21no—
239#431—21no—
240#431C22×C8311yescharacteristic
241#423C22×C8311yescharacteristic
242#423C22×C8311yescharacteristic
243C2×C4⋊C8—21no—
244C2×C4⋊C8C22×C8311yescharacteristic
245C2×#104—21no—
246#418—21no—
247C2×#102—22no—
248C4×C4⋊C4C22×C8311yescharacteristic
249#339—21no—
250#331—21no—
251#331—21no—
252C2×#41D4×C8311yescharacteristic
253C22×D4—21no—
254C22×D4—21no—
255C22×D4—21no—
256AGL(1,ℤ/8)—42no—
257AGL(1,ℤ/8)—42no—
258C2×SD16—22no—
259C2×D8—22no—
260C2×M16—21no—
261C2×M16—21no—
262C22×C8—22no—
263C22×C8D4×C8312yes—
264#127—21no—
265#127—21no—
266#127—21no—
267#127C2×C4×C8312yes—
268C4×Q8C2×C4×C8311yescharacteristic
269C4×Q8D4×C8311yescharacteristic
270C4×D4—21no—
271C4×D4—21no—
272C4×D4C2×C4×C8311yescharacteristic
273C4×D4D4×C8311yescharacteristic
274#117—21no—
275#117D4×C8311yescharacteristic
276C2×C4⋊C4—21no—
277C2×C42—21no—
278C2×C42—21no—
279C2×C42C23×C8311yescharacteristic
280C4⋊C8—22no—
281C4⋊C8—21no—
282C4⋊C8—21no—
283C4⋊C8—21no—
284C4⋊C8—21no—
285C4⋊C8—21no—
286C4⋊C8C2×C4×C8312yes—
287C4⋊C8D4×C8311yescharacteristic
288C4⋊C8D4×C8311yescharacteristic
289#104—42no—
290#102—22no—
291#102—22no—
292#102—44no—
293C22⋊C8—22no—
294C22⋊C8—21no—
295C4×C8—22no—
296C4×C8—21no—
297#95—21no—
298C24—41no—
299#41—21no—
300#41—21no—
301#41D8×C8312yes—
302#41SD16×C8312yes—
303C2×Q8C22⋊C4×C8311yescharacteristic
304C2×D4—21no—
305C2×D4—22no—
306C2×D4—22no—
307C2×D4—22no—
308C2×D4—42no—
309C2×D4—42no—
310C2×D4—41no—
311C2×D4—41no—
312C2×D4—41no—
313C2×D4—41no—
314C2×D4—41no—
315C2×D4—41no—
316C2×D4—41no—
317C2×D4—82no—
318C2×D4C22⋊C4×C8311yescharacteristic
319C22×C4—21no—
320C22×C4—21no—
321C22×C4—21no—
322C22×C4—41no—
323C22×C4C2×D4×C8311yescharacteristic
324C22×C4C2×D4×C8311yescharacteristic
325C22×C4#41×C8311yescharacteristic
326SD16—44no—
327D8—44no—
328M16—41no—
329M16—41no—
330M16—41no—
331M16—41no—
332C2×C8—22no—
333C2×C8—21no—
334C2×C8—22no—
335C2×C8—21no—
336C2×C8—42no—
337C2×C8—41no—
338C2×C8—82no—
339C4⋊C4—21no—
340C4⋊C4—21no—
341C4⋊C4—22no—
342C4⋊C4—41no—
343C4⋊C4C22⋊C4×C8311yescharacteristic
344C4⋊C4C22⋊C4×C8311yescharacteristic
345C22⋊C4—21no—
346C22⋊C4—21no—
347C22⋊C4—21no—
348C42—21no—
349C42—21no—
350C42—21no—
351C42—41no—
352C42—41no—
353C42C22×C4×C8311yescharacteristic
354C42C2×D4×C8311yescharacteristic
355C42C2×D4×C8311yescharacteristic
356C42#41×C8311yescharacteristic
357C23—21no—
358C23—42no—
359C23—82no—
360C23—82no—
361C23—81no—
362C23—81no—
363C23#121×C8311yescharacteristic
364Q8—21no—
365Q8#102×C8312yescomplemented
366D4—22no—
367D4—22no—
368D4—21no—
369D4—44no—
370D4—42no—
371D4—42no—
372D4—82no—
373D4—82no—
374D4—81no—
375D4—81no—
376D4—81no—
377D4—81no—
378D4#102×C8312yescomplemented
379C2×C4—21no—
380C2×C4—22no—
381C2×C4—22no—
382C2×C4—21no—
383C2×C4—21no—
384C2×C4—21no—
385C2×C4—21no—
386C2×C4—41no—
387C2×C4—41no—
388C2×C4—41no—
389C2×C4—42no—
390C2×C4—81no—
391C2×C4—81no—
392C2×C4—81no—
393C2×C4C2×C22⋊C4×C8311yescharacteristic
394C2×C4C4×D4×C8311yescharacteristic
395C2×C4C4×D4×C8311yescharacteristic
396C2×C4#120×C8311yescharacteristic
397C2×C4#121×C8311yescharacteristic
398C2×C4#123×C8311yescharacteristic
399C2×C4C2×D8×C8311yescharacteristic
400C2×C4C2×SD16×C8311yescharacteristic
401C2×C4#135×C8311yescharacteristic
402C2×C4#135×C8311yescharacteristic
403C8—42no—
404C8—42no—
405C8—41no—
406C8—41no—
407C8—81no—
408C22—21no—
409C22—21no—
410C22—22no—
411C22—41no—
412C22—41no—
413C22—82no—
414C22—82no—
415C22—82no—
416C22—162no—
417C22—162no—
418C22#386×C8311yescharacteristic
419C22#465×C8311yescharacteristic
420C22#466×C8311yescharacteristic
421C4—21no—
422C4—22no—
423C4—21no—
424C4—21no—
425C4—41no—
426C4—41no—
427C4—82no—
428C4C2×#102×C8311yescharacteristic
429C4#416×C8311yescharacteristic
430C4#453×C8311yescharacteristic
431C4#454×C8311yescharacteristic
432C4#465×C8311yescharacteristic
433C4#466×C8311yescharacteristic
434C2—21no—
435C2—21no—
436C2—42no—
437C2—162no—
438C2C83×#188311yescharacteristic
439C2C83×#189311yescharacteristic
440C2C83×#190311yescharacteristic
subgroupG/H
center, socleC22×C83#386
commutatorC42C22×C4×C83
Frattini, cosocleC2×C42C23×C83
prime radicalC83×#3038C2
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C83 ◁ C2×C42×C83 ◁ G

Complements

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

NH ≅ G/Ntype
2. C83×#2988C2semidirect
3. C83×#1982C2semidirect
4. C83×#1806C2semidirect
5. C83×#1545C2semidirect
6. C83×#1303C2semidirect
7. C83×#1292C2semidirect
12. #460×C83C4semidirect
14. #459×C83C4semidirect
144. Q8×C83#102semidirect
157. D4×C83#102semidirect
213. #9318C83direct (N × H)
222. #2988C2×C83semidirect
223. #1982C2×C83semidirect
224. #1806C2×C83semidirect
225. #1545C2×C83semidirect
226. #1303C2×C83semidirect
227. #1292C2×C83semidirect
228. C83#9318direct (N × H)
233. #460C4×C83semidirect
235. #459C4×C83semidirect
365. Q8#102×C83semidirect
378. D4#102×C83semidirect

Maximal subgroups

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

subgroupindex
C83×#30382
C83×#29882
C83×#19822
C83×#18062
C83×#15452
C83×#13032
C83×#12922
#931883