Finite Group Databasebeta

← C83×#9381 #35237205

Subgroups — C83×#9381 #35237205

Order 21248.

530 conjugacy classes of subgroups.

The subgroup lattice has 444 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C83×#3213C211yesmaximal, characteristic
2C83×#2337C211yesmaximal, characteristic, complemented
3C83×#1980C211yesmaximal, characteristic, complemented
4C83×#1527C211yesmaximal, characteristic, complemented
5C83×#1514C211yesmaximal, characteristic, complemented
6C83×#1288C211yesmaximal, characteristic, complemented
7C83×#1285C211yesmaximal, characteristic, complemented
8#532×C83C2211yescharacteristic
9C2×#124×C83C2211yescharacteristic
10#496×C83C412yescomplemented
11#489×C83C412yescomplemented
12#463×C83—22no—
13#448×C83—21no—
14#447×C83—21no—
15#432×C83C2211yescharacteristic
16#432×C83C2211yescharacteristic
17C2×#104×C83—21no—
18C2×#104×C83—21no—
19C4×M16×C83C2211yescharacteristic
20#391×C83—21no—
21#388×C83—21no—
22#388×C83—21no—
23#386×C83—21no—
24#385×C83—21no—
25#383×C83C2211yescharacteristic
26C4×C4⋊C4×C83C2211yescharacteristic
27C4×C22⋊C4×C83—21no—
28#329×C83—22no—
29#327×C83—21no—
30#327×C83—21no—
31C2×#41×C83—21no—
32C22×Q8×C83D411yescharacteristic
33C22×D4×C83D411yescharacteristic
34C23×C4×C83—21no—
35C2×SD16×C83—22no—
36C2×D8×C83—22no—
37C2×M16×C83—21no—
38C4⋊Q8×C83C2×C411yescharacteristic, complemented
39#127×C83C2×C411yescharacteristic, complemented
40#124×C83—21no—
41#124×C83—21no—
42#124×C83—21no—
43#124×C83C2×C412yes—
44#124×C83D412yes—
45#121×C83—21no—
46#121×C83—21no—
47C4×D4×C83—21no—
48C4×D4×C83—21no—
49C2×C4⋊C4×C83—21no—
50C2×C4⋊C4×C83—21no—
51C2×C22⋊C4×C83—21no—
52C2×C22⋊C4×C83—21no—
53C2×C22⋊C4×C83—21no—
54C2×C22⋊C4×C83—21no—
55C2×C22⋊C4×C83—21no—
56C2×C42×C83—21no—
57C2×C42×C83D411yescharacteristic
58C2×C42×C83D411yescharacteristic
59C2×C42×C83C2311yescharacteristic
60C4⋊C8×C83—21no—
61C4⋊C8×C83—21no—
62#104×C83—42no—
63#104×C83—42no—
64#103×C83—22no—
65#102×C83—22no—
66C22⋊C8×C83—21no—
67C22⋊C8×C83—21no—
68#97×C83—21no—
69#97×C83—21no—
70#97×C83C2×C412yescomplemented
71C4×C8×C83—21no—
72#95×C83—21no—
73#95×C83—21no—
74#95×C83—21no—
75#95×C83—21no—
76#95×C83—21no—
77#95×C83D411yescharacteristic
78#95×C83D411yescharacteristic
79C24×C83—21no—
80#41×C83—41no—
81#41×C83—41no—
82#41×C83—41no—
83#41×C83—41no—
84C2×Q8×C83—21no—
85C2×Q8×C83—21no—
86C2×Q8×C83—81no—
87C2×Q8×C83C22⋊C412yes—
88C2×D4×C83—21no—
89C2×D4×C83—21no—
90C2×D4×C83—21no—
91C2×D4×C83—21no—
92C2×D4×C83—41no—
93C2×D4×C83—81no—
94C2×D4×C83C22⋊C412yes—
95C22×C4×C83—21no—
96C22×C4×C83—21no—
97C22×C4×C83—21no—
98C22×C4×C83—21no—
99C22×C4×C83—21no—
100C22×C4×C83—42no—
101C22×C4×C83—41no—
102C22×C4×C83—81no—
103C22×C4×C83—81no—
104C22×C4×C83C2×D411yescharacteristic
105C22×C4×C83C2×D411yescharacteristic
106C22×C4×C83C2×D411yescharacteristic
107SD16×C83—82no—
108D8×C83—82no—
109M16×C83—22no—
110M16×C83—41no—
111C2×C8×C83—22no—
112C2×C8×C83—41no—
113C2×C8×C83—41no—
114C4⋊C4×C83—21no—
115C4⋊C4×C83—21no—
116C4⋊C4×C83—42no—
117C4⋊C4×C83—81no—
118C22⋊C4×C83—22no—
119C22⋊C4×C83—22no—
120C22⋊C4×C83—22no—
121C22⋊C4×C83—22no—
122C22⋊C4×C83—42no—
123C22⋊C4×C83—41no—
124C22⋊C4×C83—41no—
125C22⋊C4×C83—81no—
126C22⋊C4×C83—81no—
127C42×C83—41no—
128C42×C83—41no—
129C42×C83—81no—
130C42×C83C22×C411yescharacteristic
131C42×C83C2×D411yescharacteristic
132C42×C83#4111yescharacteristic
133C42×C83#4111yescharacteristic
134C23×C83—22no—
135C23×C83—41no—
136C23×C83—41no—
137C23×C83—41no—
138C23×C83—41no—
139C23×C83—41no—
140C23×C83—41no—
141C23×C83—41no—
142C23×C83#12011yescharacteristic
143Q8×C83—42no—
144Q8×C83—41no—
145Q8×C83—41no—
146Q8×C83—81no—
147D4×C83—42no—
148D4×C83—42no—
149D4×C83—41no—
150D4×C83—41no—
151D4×C83—81no—
152D4×C83—81no—
153D4×C83—81no—
154D4×C83—81no—
155C2×C4×C83—21no—
156C2×C4×C83—21no—
157C2×C4×C83—22no—
158C2×C4×C83—22no—
159C2×C4×C83—41no—
160C2×C4×C83—41no—
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—41no—
170C2×C4×C83—41no—
171C2×C4×C83—41no—
172C2×C4×C83—41no—
173C2×C4×C83—41no—
174C2×C4×C83—41no—
175C2×C4×C83—81no—
176C2×C4×C83—81no—
177C2×C4×C83—84no—
178C2×C4×C83C2×C22⋊C411yescharacteristic
179C2×C4×C83C4×D411yescharacteristic
180C2×C4×C83C4×D411yescharacteristic
181C2×C4×C83#12111yescharacteristic
182C2×C4×C83#12111yescharacteristic
183C2×C4×C83#12311yescharacteristic
184C8×C83—42no—
185C8×C83—42no—
186C8×C83—81no—
187C22×C83—21no—
188C22×C83—21no—
189C22×C83—42no—
190C22×C83—42no—
191C22×C83—42no—
192C22×C83—81no—
193C22×C83—81no—
194C22×C83—81no—
195C22×C83—81no—
196C22×C83—81no—
197C22×C83—81no—
198C22×C83—81no—
199C22×C83#38611yescharacteristic
200C22×C83#45311yescharacteristic
201C22×C83#45411yescharacteristic
202C4×C83—21no—
203C4×C83—21no—
204C4×C83—21no—
205C4×C83—21no—
206C4×C83—21no—
207C4×C83—21no—
208C4×C83—42no—
209C4×C83—42no—
210C4×C83—81no—
211C4×C83—81no—
212C4×C83—81no—
213C4×C83—81no—
214#9381C8311yesmaximal, characteristic, direct product factor, Sylow
215C2×C83—21no—
216C2×C83—21no—
217C2×C83—82no—
218C2×C83—161no—
219C2×C83#188311yescharacteristic
220C2×C83#190611yescharacteristic
221C2×C83#191111yescharacteristic
222#3213C2×C8311yescharacteristic
223#2337C2×C8311yescharacteristic, complemented
224#1980C2×C8311yescharacteristic, complemented
225#1527C2×C8311yescharacteristic, complemented
226#1514C2×C8311yescharacteristic, complemented
227#1288C2×C8311yescharacteristic, complemented
228#1285C2×C8311yescharacteristic, complemented
229C83#938111yescharacteristic, direct product factor, Sylow
230#532C22×C8311yescharacteristic
231C2×#124C22×C8311yescharacteristic
232#496C4×C8312yescomplemented
233#489C4×C8312yescomplemented
234#463—22no—
235#448—21no—
236#447—21no—
237#432C22×C8311yescharacteristic
238#432C22×C8311yescharacteristic
239C2×#104—21no—
240C2×#104—21no—
241C4×M16C22×C8311yescharacteristic
242#391—21no—
243#388—21no—
244#388—21no—
245#386—21no—
246#385—21no—
247#383C22×C8311yescharacteristic
248C4×C4⋊C4C22×C8311yescharacteristic
249C4×C22⋊C4—21no—
250#329—22no—
251#327—21no—
252#327—21no—
253C2×#41—21no—
254C22×Q8D4×C8311yescharacteristic
255C22×D4D4×C8311yescharacteristic
256C23×C4—21no—
257C2×SD16—22no—
258C2×D8—22no—
259C2×M16—21no—
260C4⋊Q8C2×C4×C8311yescharacteristic, complemented
261#127C2×C4×C8311yescharacteristic, complemented
262#124—21no—
263#124—21no—
264#124—21no—
265#124C2×C4×C8312yes—
266#124D4×C8312yes—
267#121—21no—
268#121—21no—
269C4×D4—21no—
270C4×D4—21no—
271C2×C4⋊C4—21no—
272C2×C4⋊C4—21no—
273C2×C22⋊C4—21no—
274C2×C22⋊C4—21no—
275C2×C22⋊C4—21no—
276C2×C22⋊C4—21no—
277C2×C22⋊C4—21no—
278C2×C42—21no—
279C2×C42D4×C8311yescharacteristic
280C2×C42D4×C8311yescharacteristic
281C2×C42C23×C8311yescharacteristic
282C4⋊C8—21no—
283C4⋊C8—21no—
284#104—42no—
285#104—42no—
286#103—22no—
287#102—22no—
288C22⋊C8—21no—
289C22⋊C8—21no—
290#97—21no—
291#97—21no—
292#97C2×C4×C8312yescomplemented
293C4×C8—21no—
294#95—21no—
295#95—21no—
296#95—21no—
297#95—21no—
298#95—21no—
299#95D4×C8311yescharacteristic
300#95D4×C8311yescharacteristic
301C24—21no—
302#41—41no—
303#41—41no—
304#41—41no—
305#41—41no—
306C2×Q8—21no—
307C2×Q8—21no—
308C2×Q8—81no—
309C2×Q8C22⋊C4×C8312yes—
310C2×D4—21no—
311C2×D4—21no—
312C2×D4—21no—
313C2×D4—21no—
314C2×D4—41no—
315C2×D4—81no—
316C2×D4C22⋊C4×C8312yes—
317C22×C4—21no—
318C22×C4—21no—
319C22×C4—21no—
320C22×C4—21no—
321C22×C4—21no—
322C22×C4—42no—
323C22×C4—41no—
324C22×C4—81no—
325C22×C4—81no—
326C22×C4C2×D4×C8311yescharacteristic
327C22×C4C2×D4×C8311yescharacteristic
328C22×C4C2×D4×C8311yescharacteristic
329SD16—82no—
330D8—82no—
331M16—22no—
332M16—41no—
333C2×C8—22no—
334C2×C8—41no—
335C2×C8—41no—
336C4⋊C4—21no—
337C4⋊C4—21no—
338C4⋊C4—42no—
339C4⋊C4—81no—
340C22⋊C4—22no—
341C22⋊C4—22no—
342C22⋊C4—22no—
343C22⋊C4—22no—
344C22⋊C4—42no—
345C22⋊C4—41no—
346C22⋊C4—41no—
347C22⋊C4—81no—
348C22⋊C4—81no—
349C42—41no—
350C42—41no—
351C42—81no—
352C42C22×C4×C8311yescharacteristic
353C42C2×D4×C8311yescharacteristic
354C42#41×C8311yescharacteristic
355C42#41×C8311yescharacteristic
356C23—22no—
357C23—41no—
358C23—41no—
359C23—41no—
360C23—41no—
361C23—41no—
362C23—41no—
363C23—41no—
364C23#120×C8311yescharacteristic
365Q8—42no—
366Q8—41no—
367Q8—41no—
368Q8—81no—
369D4—42no—
370D4—42no—
371D4—41no—
372D4—41no—
373D4—81no—
374D4—81no—
375D4—81no—
376D4—81no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—22no—
380C2×C4—22no—
381C2×C4—41no—
382C2×C4—41no—
383C2×C4—41no—
384C2×C4—41no—
385C2×C4—41no—
386C2×C4—41no—
387C2×C4—41no—
388C2×C4—41no—
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—81no—
398C2×C4—81no—
399C2×C4—84no—
400C2×C4C2×C22⋊C4×C8311yescharacteristic
401C2×C4C4×D4×C8311yescharacteristic
402C2×C4C4×D4×C8311yescharacteristic
403C2×C4#121×C8311yescharacteristic
404C2×C4#121×C8311yescharacteristic
405C2×C4#123×C8311yescharacteristic
406C8—42no—
407C8—42no—
408C8—81no—
409C22—21no—
410C22—21no—
411C22—42no—
412C22—42no—
413C22—42no—
414C22—81no—
415C22—81no—
416C22—81no—
417C22—81no—
418C22—81no—
419C22—81no—
420C22—81no—
421C22#386×C8311yescharacteristic
422C22#453×C8311yescharacteristic
423C22#454×C8311yescharacteristic
424C4—21no—
425C4—21no—
426C4—21no—
427C4—21no—
428C4—21no—
429C4—21no—
430C4—42no—
431C4—42no—
432C4—81no—
433C4—81no—
434C4—81no—
435C4—81no—
436C2—21no—
437C2—21no—
438C2—82no—
439C2—161no—
440C2C83×#188311yescharacteristic
441C2C83×#190611yescharacteristic
442C2C83×#191111yescharacteristic
subgroupG/H
center, socleC22×C83#386
commutatorC42C22×C4×C83
Frattini, cosocleC2×C42C23×C83
prime radicalC83×#3213C2
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×#2337C2semidirect
3. C83×#1980C2semidirect
4. C83×#1527C2semidirect
5. C83×#1514C2semidirect
6. C83×#1288C2semidirect
7. C83×#1285C2semidirect
10. #496×C83C4semidirect
11. #489×C83C4semidirect
38. C4⋊Q8×C83C2×C4semidirect
39. #127×C83C2×C4semidirect
70. #97×C83C2×C4semidirect
214. #9381C83direct (N × H)
223. #2337C2×C83semidirect
224. #1980C2×C83semidirect
225. #1527C2×C83semidirect
226. #1514C2×C83semidirect
227. #1288C2×C83semidirect
228. #1285C2×C83semidirect
229. C83#9381direct (N × H)
232. #496C4×C83semidirect
233. #489C4×C83semidirect
260. C4⋊Q8C2×C4×C83semidirect
261. #127C2×C4×C83semidirect
292. #97C2×C4×C83semidirect

Maximal subgroups

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

subgroupindex
C83×#32132
C83×#23372
C83×#19802
C83×#15272
C83×#15142
C83×#12882
C83×#12852
#938183