Finite Group Databasebeta

← C83×#9361 #35237185

Subgroups — C83×#9361 #35237185

Order 21248.

760 conjugacy classes of subgroups.

The subgroup lattice has 454 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C83×#3007C211yesmaximal, characteristic
2C83×#2339C211yesmaximal, characteristic
3C83×#1996C211yesmaximal, characteristic, complemented
4C83×#1789C211yesmaximal, characteristic
5C83×#1508C211yesmaximal, characteristic, complemented
6C83×#1305C211yesmaximal, characteristic, complemented
7C83×#1297C211yesmaximal, characteristic, complemented
8C22×#41×C83C2211yescharacteristic
9C2×#137×C83C412yescomplemented
10C2×C22⋊Q8×C83C2211yescharacteristic
11C2×C4×D4×C83—21no—
12#452×C83—22no—
13#451×C83—21no—
14#450×C83—21no—
15C2×#104×C83—22no—
16#417×C83C412yescomplemented
17#407×C83C2211yescharacteristic
18#407×C83C2211yescharacteristic
19#391×C83—21no—
20#386×C83—22no—
21#381×C83—21no—
22#380×C83C2211yescharacteristic
23C4×C22⋊C4×C83C2211yescharacteristic
24C4×C22⋊C4×C83C2211yescharacteristic
25C2×#95×C83—21no—
26#342×C83—22no—
27#328×C83—21no—
28#328×C83—21no—
29C2×#41×C83—21no—
30C2×#41×C83—21no—
31C2×#41×C83—44no—
32C2×#41×C83C2×C412yes—
33C2×#41×C83D412yes—
34C22×Q8×C83C2×C411yescharacteristic, complemented
35C22×Q8×C83D411yescharacteristic
36C22×D4×C83—21no—
37C22×D4×C83D411yescharacteristic
38C23×C4×C83—21no—
39C23×C4×C83—22no—
40C23×C4×C83C2311yescharacteristic
41#137×C83—44no—
42C2×Q16×C83—22no—
43C2×SD16×C83—22no—
44C2×M16×C83—21no—
45C2×M16×C83C2×C412yescomplemented
46C22⋊Q8×C83—21no—
47C22⋊Q8×C83—21no—
48C22⋊Q8×C83—21no—
49C22⋊Q8×C83—21no—
50C4×D4×C83—44no—
51C2×C4⋊C4×C83—21no—
52C2×C4⋊C4×C83—21no—
53C2×C4⋊C4×C83C2×C411yescharacteristic, complemented
54C2×C22⋊C4×C83—21no—
55C2×C22⋊C4×C83—22no—
56C2×C22⋊C4×C83—21no—
57C2×C22⋊C4×C83—21no—
58C2×C22⋊C4×C83D412yes—
59C2×C42×C83—21no—
60C2×C42×C83—21no—
61#104×C83—44no—
62#103×C83—22no—
63#102×C83—22no—
64C22⋊C8×C83—21no—
65C22⋊C8×C83—21no—
66C22⋊C8×C83—21no—
67C22⋊C8×C83—21no—
68#95×C83—22no—
69#95×C83—22no—
70#95×C83—21no—
71#95×C83—21no—
72#95×C83D411yescharacteristic
73#95×C83D411yescharacteristic
74C24×C83—21no—
75C24×C83C2×D411yescharacteristic
76#41×C83—24no—
77#41×C83—24no—
78#41×C83—42no—
79#41×C83—42no—
80#41×C83—44no—
81#41×C83—44no—
82C2×Q8×C83—21no—
83C2×Q8×C83—21no—
84C2×Q8×C83—21no—
85C2×Q8×C83—42no—
86C2×Q8×C83—42no—
87C2×Q8×C83C22⋊C412yes—
88C2×D4×C83—24no—
89C2×D4×C83—22no—
90C2×D4×C83—22no—
91C2×D4×C83—21no—
92C2×D4×C83—44no—
93C2×D4×C83C22⋊C412yes—
94C22×C4×C83—21no—
95C22×C4×C83—21no—
96C22×C4×C83—22no—
97C22×C4×C83—22no—
98C22×C4×C83—22no—
99C22×C4×C83—21no—
100C22×C4×C83—21no—
101C22×C4×C83—21no—
102C22×C4×C83—42no—
103C22×C4×C83—44no—
104C22×C4×C83—44no—
105C22×C4×C83—44no—
106C22×C4×C83—41no—
107C22×C4×C83—82no—
108C22×C4×C83C22×C411yescharacteristic
109C22×C4×C83C2×D411yescharacteristic
110C22×C4×C83C2×D411yescharacteristic
111C22×C4×C83C2×D411yescharacteristic
112C22×C4×C83#4111yescharacteristic
113C22×C4×C83#4111yescharacteristic
114Q16×C83—44no—
115SD16×C83—44no—
116M16×C83—42no—
117M16×C83—42no—
118C2×C8×C83—22no—
119C2×C8×C83—41no—
120C4⋊C4×C83—21no—
121C4⋊C4×C83—21no—
122C4⋊C4×C83—42no—
123C4⋊C4×C83—41no—
124C4⋊C4×C83—41no—
125C22⋊C4×C83—22no—
126C22⋊C4×C83—42no—
127C22⋊C4×C83—41no—
128C22⋊C4×C83—44no—
129C22⋊C4×C83—41no—
130C22⋊C4×C83—41no—
131C22⋊C4×C83—42no—
132C42×C83—42no—
133C42×C83—42no—
134C23×C83—24no—
135C23×C83—22no—
136C23×C83—21no—
137C23×C83—21no—
138C23×C83—41no—
139C23×C83—41no—
140C23×C83—42no—
141C23×C83—41no—
142C23×C83—41no—
143C23×C83C4×D411yescharacteristic
144C23×C83#12011yescharacteristic
145C23×C83#12111yescharacteristic
146Q8×C83—22no—
147Q8×C83—22no—
148Q8×C83—42no—
149Q8×C83—42no—
150Q8×C83—81no—
151D4×C83—24no—
152D4×C83—22no—
153D4×C83—44no—
154D4×C83—44no—
155D4×C83#9914yes—
156C2×C4×C83—22no—
157C2×C4×C83—22no—
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—44no—
165C2×C4×C83—44no—
166C2×C4×C83—41no—
167C2×C4×C83—42no—
168C2×C4×C83—41no—
169C2×C4×C83—41no—
170C2×C4×C83—42no—
171C2×C4×C83—42no—
172C2×C4×C83—41no—
173C2×C4×C83—42no—
174C2×C4×C83—41no—
175C2×C4×C83—41no—
176C2×C4×C83—42no—
177C2×C4×C83—41no—
178C2×C4×C83—88no—
179C2×C4×C83—81no—
180C2×C4×C83—81no—
181C2×C4×C83—81no—
182C2×C4×C83—81no—
183C2×C4×C83C2×C22⋊C411yescharacteristic
184C2×C4×C83C4×D411yescharacteristic
185C2×C4×C83#12111yescharacteristic
186C2×C4×C83#12311yescharacteristic
187C8×C83—42no—
188C8×C83—81no—
189C22×C83—21no—
190C22×C83—21no—
191C22×C83—21no—
192C22×C83—21no—
193C22×C83—21no—
194C22×C83—21no—
195C22×C83—24no—
196C22×C83—21no—
197C22×C83—21no—
198C22×C83—44no—
199C22×C83—44no—
200C22×C83—42no—
201C22×C83—42no—
202C22×C83—81no—
203C22×C83—81no—
204C22×C83#38611yescharacteristic
205C22×C83#45511yescharacteristic
206C22×C83#45611yescharacteristic
207C4×C83—21no—
208C4×C83—21no—
209C4×C83—21no—
210C4×C83—42no—
211C4×C83—42no—
212C4×C83—81no—
213C4×C83—82no—
214C4×C83—82no—
215C4×C83—81no—
216C4×C83—81no—
217C4×C83C2×#9912yes—
218#9361C8311yesmaximal, characteristic, direct product factor, Sylow
219C2×C83—21no—
220C2×C83—21no—
221C2×C83—41no—
222C2×C83—41no—
223C2×C83—44no—
224C2×C83#188211yescharacteristic
225C2×C83#189811yescharacteristic
226C2×C83#190611yescharacteristic
227#3007C2×C8311yescharacteristic
228#2339C2×C8311yescharacteristic
229#1996C2×C8311yescharacteristic, complemented
230#1789C2×C8311yescharacteristic
231#1508C2×C8311yescharacteristic, complemented
232#1305C2×C8311yescharacteristic, complemented
233#1297C2×C8311yescharacteristic, complemented
234C83#936111yescharacteristic, direct product factor, Sylow
235C22×#41C22×C8311yescharacteristic
236C2×#137C4×C8312yescomplemented
237C2×C22⋊Q8C22×C8311yescharacteristic
238C2×C4×D4—21no—
239#452—22no—
240#451—21no—
241#450—21no—
242C2×#104—22no—
243#417C4×C8312yescomplemented
244#407C22×C8311yescharacteristic
245#407C22×C8311yescharacteristic
246#391—21no—
247#386—22no—
248#381—21no—
249#380C22×C8311yescharacteristic
250C4×C22⋊C4C22×C8311yescharacteristic
251C4×C22⋊C4C22×C8311yescharacteristic
252C2×#95—21no—
253#342—22no—
254#328—21no—
255#328—21no—
256C2×#41—21no—
257C2×#41—21no—
258C2×#41—44no—
259C2×#41C2×C4×C8312yes—
260C2×#41D4×C8312yes—
261C22×Q8C2×C4×C8311yescharacteristic, complemented
262C22×Q8D4×C8311yescharacteristic
263C22×D4—21no—
264C22×D4D4×C8311yescharacteristic
265C23×C4—21no—
266C23×C4—22no—
267C23×C4C23×C8311yescharacteristic
268#137—44no—
269C2×Q16—22no—
270C2×SD16—22no—
271C2×M16—21no—
272C2×M16C2×C4×C8312yescomplemented
273C22⋊Q8—21no—
274C22⋊Q8—21no—
275C22⋊Q8—21no—
276C22⋊Q8—21no—
277C4×D4—44no—
278C2×C4⋊C4—21no—
279C2×C4⋊C4—21no—
280C2×C4⋊C4C2×C4×C8311yescharacteristic, complemented
281C2×C22⋊C4—21no—
282C2×C22⋊C4—22no—
283C2×C22⋊C4—21no—
284C2×C22⋊C4—21no—
285C2×C22⋊C4D4×C8312yes—
286C2×C42—21no—
287C2×C42—21no—
288#104—44no—
289#103—22no—
290#102—22no—
291C22⋊C8—21no—
292C22⋊C8—21no—
293C22⋊C8—21no—
294C22⋊C8—21no—
295#95—22no—
296#95—22no—
297#95—21no—
298#95—21no—
299#95D4×C8311yescharacteristic
300#95D4×C8311yescharacteristic
301C24—21no—
302C24C2×D4×C8311yescharacteristic
303#41—24no—
304#41—24no—
305#41—42no—
306#41—42no—
307#41—44no—
308#41—44no—
309C2×Q8—21no—
310C2×Q8—21no—
311C2×Q8—21no—
312C2×Q8—42no—
313C2×Q8—42no—
314C2×Q8C22⋊C4×C8312yes—
315C2×D4—24no—
316C2×D4—22no—
317C2×D4—22no—
318C2×D4—21no—
319C2×D4—44no—
320C2×D4C22⋊C4×C8312yes—
321C22×C4—21no—
322C22×C4—21no—
323C22×C4—22no—
324C22×C4—22no—
325C22×C4—22no—
326C22×C4—21no—
327C22×C4—21no—
328C22×C4—21no—
329C22×C4—42no—
330C22×C4—44no—
331C22×C4—44no—
332C22×C4—44no—
333C22×C4—41no—
334C22×C4—82no—
335C22×C4C22×C4×C8311yescharacteristic
336C22×C4C2×D4×C8311yescharacteristic
337C22×C4C2×D4×C8311yescharacteristic
338C22×C4C2×D4×C8311yescharacteristic
339C22×C4#41×C8311yescharacteristic
340C22×C4#41×C8311yescharacteristic
341Q16—44no—
342SD16—44no—
343M16—42no—
344M16—42no—
345C2×C8—22no—
346C2×C8—41no—
347C4⋊C4—21no—
348C4⋊C4—21no—
349C4⋊C4—42no—
350C4⋊C4—41no—
351C4⋊C4—41no—
352C22⋊C4—22no—
353C22⋊C4—42no—
354C22⋊C4—44no—
355C22⋊C4—41no—
356C22⋊C4—41no—
357C22⋊C4—41no—
358C22⋊C4—42no—
359C42—42no—
360C42—42no—
361C23—24no—
362C23—22no—
363C23—21no—
364C23—21no—
365C23—41no—
366C23—41no—
367C23—42no—
368C23—41no—
369C23—41no—
370C23C4×D4×C8311yescharacteristic
371C23#120×C8311yescharacteristic
372C23#121×C8311yescharacteristic
373Q8—22no—
374Q8—22no—
375Q8—42no—
376Q8—42no—
377Q8—81no—
378D4—24no—
379D4—22no—
380D4—44no—
381D4—44no—
382D4#99×C8314yes—
383C2×C4—22no—
384C2×C4—22no—
385C2×C4—22no—
386C2×C4—22no—
387C2×C4—22no—
388C2×C4—41no—
389C2×C4—41no—
390C2×C4—41no—
391C2×C4—44no—
392C2×C4—44no—
393C2×C4—41no—
394C2×C4—42no—
395C2×C4—41no—
396C2×C4—41no—
397C2×C4—42no—
398C2×C4—42no—
399C2×C4—41no—
400C2×C4—42no—
401C2×C4—41no—
402C2×C4—41no—
403C2×C4—42no—
404C2×C4—41no—
405C2×C4—88no—
406C2×C4—81no—
407C2×C4—81no—
408C2×C4—81no—
409C2×C4—81no—
410C2×C4C2×C22⋊C4×C8311yescharacteristic
411C2×C4C4×D4×C8311yescharacteristic
412C2×C4#121×C8311yescharacteristic
413C2×C4#123×C8311yescharacteristic
414C8—42no—
415C8—81no—
416C22—21no—
417C22—21no—
418C22—21no—
419C22—21no—
420C22—21no—
421C22—21no—
422C22—24no—
423C22—21no—
424C22—21no—
425C22—44no—
426C22—44no—
427C22—42no—
428C22—42no—
429C22—81no—
430C22—81no—
431C22#386×C8311yescharacteristic
432C22#455×C8311yescharacteristic
433C22#456×C8311yescharacteristic
434C4—21no—
435C4—21no—
436C4—21no—
437C4—42no—
438C4—42no—
439C4—82no—
440C4—81no—
441C4—82no—
442C4—81no—
443C4—81no—
444C4C2×#99×C8312yes—
445C2—21no—
446C2—21no—
447C2—41no—
448C2—41no—
449C2—44no—
450C2C83×#188211yescharacteristic
451C2C83×#189811yescharacteristic
452C2C83×#190611yescharacteristic
subgroupG/H
center, socleC22×C83#386
commutatorC22×C4C22×C4×C83
Frattini, cosocleC23×C4C23×C83
prime radicalC22×#41×C83C22
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C83 ◁ C23×C4×C83 ◁ G

Complements

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

NH ≅ G/Ntype
3. C83×#1996C2semidirect
5. C83×#1508C2semidirect
6. C83×#1305C2semidirect
7. C83×#1297C2semidirect
9. C2×#137×C83C4semidirect
16. #417×C83C4semidirect
34. C22×Q8×C83C2×C4semidirect
45. C2×M16×C83C2×C4semidirect
53. C2×C4⋊C4×C83C2×C4semidirect
218. #9361C83direct (N × H)
229. #1996C2×C83semidirect
231. #1508C2×C83semidirect
232. #1305C2×C83semidirect
233. #1297C2×C83semidirect
234. C83#9361direct (N × H)
236. C2×#137C4×C83semidirect
243. #417C4×C83semidirect
261. C22×Q8C2×C4×C83semidirect
272. C2×M16C2×C4×C83semidirect
280. C2×C4⋊C4C2×C4×C83semidirect

Maximal subgroups

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

subgroupindex
C83×#30072
C83×#23392
C83×#19962
C83×#17892
C83×#15082
C83×#13052
C83×#12972
#936183