Finite Group Databasebeta

← C83×#9316 #35237140

Subgroups — C83×#9316 #35237140

Order 21248.

512 conjugacy classes of subgroups.

The subgroup lattice has 410 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C83×#3039C211yesmaximal, characteristic
2C83×#2988C211yesmaximal, characteristic
3C83×#1988C211yesmaximal, characteristic, complemented
4C83×#1806C211yesmaximal, characteristic
5C83×#1546C211yesmaximal, characteristic, complemented
6C83×#1303C211yesmaximal, characteristic, complemented
7C83×#1292C211yesmaximal, characteristic, complemented
8C2×#137×C83—21no—
9C2×C4⋊Q8×C83C2211yescharacteristic
10C4×#41×C83C2211yescharacteristic
11#462×C83—21no—
12#462×C83C412yescomplemented
13#461×C83—21no—
14#461×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#419×C83—21no—
26C4×C4⋊C4×C83C2211yescharacteristic
27#340×C83—22no—
28#339×C83—21no—
29#332×C83—21no—
30#332×C83—21no—
31C2×#41×C83D411yescharacteristic
32C22×Q8×C83—21no—
33#137×C83—42no—
34#137×C83—42no—
35C2×Q16×C83—22no—
36C2×SD16×C83—22no—
37C2×M16×C83—21no—
38C2×M16×C83—21no—
39C22×C8×C83—22no—
40C22×C8×C83D412yes—
41C4⋊Q8×C83—21no—
42C4⋊Q8×C83—21no—
43C4⋊Q8×C83—21no—
44C4⋊Q8×C83C2×C412yes—
45C4×Q8×C83C2×C411yescharacteristic
46C4×Q8×C83D411yescharacteristic
47C4×D4×C83—21no—
48C4×D4×C83—21no—
49C4×D4×C83C2×C411yescharacteristic
50C4×D4×C83D411yescharacteristic
51#117×C83—21no—
52#117×C83D411yescharacteristic
53C2×C4⋊C4×C83—21no—
54C2×C4⋊C4×C83—21no—
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#103×C83—22no—
70#103×C83—22no—
71C22⋊C8×C83—22no—
72C22⋊C8×C83—21no—
73C4×C8×C83—22no—
74C4×C8×C83—21no—
75#95×C83—21no—
76#41×C83—21no—
77#41×C83—21no—
78#41×C83D812yes—
79#41×C83SD1612yes—
80C2×Q8×C83—22no—
81C2×Q8×C83—41no—
82C2×Q8×C83—41no—
83C2×Q8×C83—41no—
84C2×Q8×C83—41no—
85C2×Q8×C83—41no—
86C2×Q8×C83C22⋊C411yescharacteristic
87C2×D4×C83—21no—
88C2×D4×C83C22⋊C411yescharacteristic
89C22×C4×C83—21no—
90C22×C4×C83—21no—
91C22×C4×C83—21no—
92C22×C4×C83—41no—
93C22×C4×C83—41no—
94C22×C4×C83C2×D411yescharacteristic
95C22×C4×C83C2×D411yescharacteristic
96C22×C4×C83#4111yescharacteristic
97Q16×C83—44no—
98SD16×C83—44no—
99M16×C83—41no—
100M16×C83—41no—
101M16×C83—41no—
102M16×C83—41no—
103C2×C8×C83—22no—
104C2×C8×C83—21no—
105C2×C8×C83—22no—
106C2×C8×C83—21no—
107C2×C8×C83—42no—
108C2×C8×C83—41no—
109C2×C8×C83—82no—
110C4⋊C4×C83—21no—
111C4⋊C4×C83—21no—
112C4⋊C4×C83—22no—
113C4⋊C4×C83—22no—
114C4⋊C4×C83—22no—
115C4⋊C4×C83—41no—
116C4⋊C4×C83—41no—
117C4⋊C4×C83—41no—
118C4⋊C4×C83C22⋊C411yescharacteristic
119C4⋊C4×C83C22⋊C411yescharacteristic
120C22⋊C4×C83—21no—
121C22⋊C4×C83—21no—
122C22⋊C4×C83—21no—
123C42×C83—21no—
124C42×C83—21no—
125C42×C83—21no—
126C42×C83—41no—
127C42×C83—41no—
128C42×C83C22×C411yescharacteristic
129C42×C83C2×D411yescharacteristic
130C42×C83C2×D411yescharacteristic
131C42×C83#4111yescharacteristic
132C23×C83—21no—
133C23×C83#12111yescharacteristic
134Q8×C83—21no—
135Q8×C83—42no—
136Q8×C83—42no—
137Q8×C83—81no—
138Q8×C83—81no—
139Q8×C83#10212yes—
140D4×C83—22no—
141D4×C83—22no—
142D4×C83—21no—
143D4×C83#10212yes—
144C2×C4×C83—21no—
145C2×C4×C83—21no—
146C2×C4×C83—21no—
147C2×C4×C83—22no—
148C2×C4×C83—22no—
149C2×C4×C83—21no—
150C2×C4×C83—21no—
151C2×C4×C83—41no—
152C2×C4×C83—41no—
153C2×C4×C83—41no—
154C2×C4×C83—42no—
155C2×C4×C83—42no—
156C2×C4×C83—81no—
157C2×C4×C83—81no—
158C2×C4×C83—81no—
159C2×C4×C83—81no—
160C2×C4×C83—81no—
161C2×C4×C83C2×C22⋊C411yescharacteristic
162C2×C4×C83C4×D411yescharacteristic
163C2×C4×C83C4×D411yescharacteristic
164C2×C4×C83#12011yescharacteristic
165C2×C4×C83#12111yescharacteristic
166C2×C4×C83#12311yescharacteristic
167C2×C4×C83C2×D811yescharacteristic
168C2×C4×C83C2×SD1611yescharacteristic
169C2×C4×C83#13511yescharacteristic
170C2×C4×C83#13511yescharacteristic
171C8×C83—42no—
172C8×C83—42no—
173C8×C83—41no—
174C8×C83—41no—
175C8×C83—81no—
176C22×C83—21no—
177C22×C83—21no—
178C22×C83—22no—
179C22×C83—41no—
180C22×C83—41no—
181C22×C83#38611yescharacteristic
182C22×C83#46511yescharacteristic
183C22×C83#46611yescharacteristic
184C4×C83—21no—
185C4×C83—22no—
186C4×C83—21no—
187C4×C83—21no—
188C4×C83—41no—
189C4×C83—41no—
190C4×C83—82no—
191C4×C83—82no—
192C4×C83C2×#10211yescharacteristic
193C4×C83#41611yescharacteristic
194C4×C83#45511yescharacteristic
195C4×C83#45611yescharacteristic
196C4×C83#46511yescharacteristic
197C4×C83#46611yescharacteristic
198#9316C8311yesmaximal, characteristic, direct product factor, Sylow
199C2×C83—21no—
200C2×C83—21no—
201C2×C83—42no—
202C2×C83#188211yescharacteristic
203C2×C83#189311yescharacteristic
204C2×C83#190211yescharacteristic
205#3039C2×C8311yescharacteristic
206#2988C2×C8311yescharacteristic
207#1988C2×C8311yescharacteristic, complemented
208#1806C2×C8311yescharacteristic
209#1546C2×C8311yescharacteristic, complemented
210#1303C2×C8311yescharacteristic, complemented
211#1292C2×C8311yescharacteristic, complemented
212C83#931611yescharacteristic, direct product factor, Sylow
213C2×#137—21no—
214C2×C4⋊Q8C22×C8311yescharacteristic
215C4×#41C22×C8311yescharacteristic
216#462—21no—
217#462C4×C8312yescomplemented
218#461—21no—
219#461C4×C8312yescomplemented
220C8×Q8—21no—
221C8×D4—22no—
222C8×D4—21no—
223#431—21no—
224#431C22×C8311yescharacteristic
225#423C22×C8311yescharacteristic
226#423C22×C8311yescharacteristic
227C2×C4⋊C8—21no—
228C2×C4⋊C8C22×C8311yescharacteristic
229C2×#104—21no—
230#419—21no—
231C4×C4⋊C4C22×C8311yescharacteristic
232#340—22no—
233#339—21no—
234#332—21no—
235#332—21no—
236C2×#41D4×C8311yescharacteristic
237C22×Q8—21no—
238#137—42no—
239#137—42no—
240C2×Q16—22no—
241C2×SD16—22no—
242C2×M16—21no—
243C2×M16—21no—
244C22×C8—22no—
245C22×C8D4×C8312yes—
246C4⋊Q8—21no—
247C4⋊Q8—21no—
248C4⋊Q8—21no—
249C4⋊Q8C2×C4×C8312yes—
250C4×Q8C2×C4×C8311yescharacteristic
251C4×Q8D4×C8311yescharacteristic
252C4×D4—21no—
253C4×D4—21no—
254C4×D4C2×C4×C8311yescharacteristic
255C4×D4D4×C8311yescharacteristic
256#117—21no—
257#117D4×C8311yescharacteristic
258C2×C4⋊C4—21no—
259C2×C4⋊C4—21no—
260C2×C4⋊C4—21no—
261C2×C42—21no—
262C2×C42—21no—
263C2×C42C23×C8311yescharacteristic
264C4⋊C8—22no—
265C4⋊C8—21no—
266C4⋊C8—21no—
267C4⋊C8—21no—
268C4⋊C8—21no—
269C4⋊C8—21no—
270C4⋊C8C2×C4×C8312yes—
271C4⋊C8D4×C8311yescharacteristic
272C4⋊C8D4×C8311yescharacteristic
273#104—42no—
274#103—22no—
275#103—22no—
276C22⋊C8—22no—
277C22⋊C8—21no—
278C4×C8—22no—
279C4×C8—21no—
280#95—21no—
281#41—21no—
282#41—21no—
283#41D8×C8312yes—
284#41SD16×C8312yes—
285C2×Q8—22no—
286C2×Q8—41no—
287C2×Q8—41no—
288C2×Q8—41no—
289C2×Q8—41no—
290C2×Q8—41no—
291C2×Q8C22⋊C4×C8311yescharacteristic
292C2×D4—21no—
293C2×D4C22⋊C4×C8311yescharacteristic
294C22×C4—21no—
295C22×C4—21no—
296C22×C4—21no—
297C22×C4—41no—
298C22×C4—41no—
299C22×C4C2×D4×C8311yescharacteristic
300C22×C4C2×D4×C8311yescharacteristic
301C22×C4#41×C8311yescharacteristic
302Q16—44no—
303SD16—44no—
304M16—41no—
305M16—41no—
306M16—41no—
307M16—41no—
308C2×C8—22no—
309C2×C8—21no—
310C2×C8—22no—
311C2×C8—21no—
312C2×C8—42no—
313C2×C8—41no—
314C2×C8—82no—
315C4⋊C4—21no—
316C4⋊C4—21no—
317C4⋊C4—22no—
318C4⋊C4—22no—
319C4⋊C4—22no—
320C4⋊C4—41no—
321C4⋊C4—41no—
322C4⋊C4—41no—
323C4⋊C4C22⋊C4×C8311yescharacteristic
324C4⋊C4C22⋊C4×C8311yescharacteristic
325C22⋊C4—21no—
326C22⋊C4—21no—
327C22⋊C4—21no—
328C42—21no—
329C42—21no—
330C42—21no—
331C42—41no—
332C42—41no—
333C42C22×C4×C8311yescharacteristic
334C42C2×D4×C8311yescharacteristic
335C42C2×D4×C8311yescharacteristic
336C42#41×C8311yescharacteristic
337C23—21no—
338C23#121×C8311yescharacteristic
339Q8—21no—
340Q8—42no—
341Q8—42no—
342Q8—81no—
343Q8—81no—
344Q8#102×C8312yes—
345D4—22no—
346D4—22no—
347D4—21no—
348D4#102×C8312yes—
349C2×C4—21no—
350C2×C4—22no—
351C2×C4—22no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—41no—
357C2×C4—41no—
358C2×C4—41no—
359C2×C4—42no—
360C2×C4—42no—
361C2×C4—81no—
362C2×C4—81no—
363C2×C4—81no—
364C2×C4—81no—
365C2×C4—81no—
366C2×C4C2×C22⋊C4×C8311yescharacteristic
367C2×C4C4×D4×C8311yescharacteristic
368C2×C4C4×D4×C8311yescharacteristic
369C2×C4#120×C8311yescharacteristic
370C2×C4#121×C8311yescharacteristic
371C2×C4#123×C8311yescharacteristic
372C2×C4C2×D8×C8311yescharacteristic
373C2×C4C2×SD16×C8311yescharacteristic
374C2×C4#135×C8311yescharacteristic
375C2×C4#135×C8311yescharacteristic
376C8—42no—
377C8—42no—
378C8—41no—
379C8—41no—
380C8—81no—
381C22—21no—
382C22—21no—
383C22—22no—
384C22—41no—
385C22—41no—
386C22#386×C8311yescharacteristic
387C22#465×C8311yescharacteristic
388C22#466×C8311yescharacteristic
389C4—21no—
390C4—22no—
391C4—21no—
392C4—21no—
393C4—41no—
394C4—41no—
395C4—82no—
396C4—82no—
397C4C2×#102×C8311yescharacteristic
398C4#416×C8311yescharacteristic
399C4#455×C8311yescharacteristic
400C4#456×C8311yescharacteristic
401C4#465×C8311yescharacteristic
402C4#466×C8311yescharacteristic
403C2—21no—
404C2—21no—
405C2—42no—
406C2C83×#188211yescharacteristic
407C2C83×#189311yescharacteristic
408C2C83×#190211yescharacteristic
subgroupG/H
center, socleC22×C83#386
commutatorC42C22×C4×C83
Frattini, cosocleC2×C42C23×C83
prime radicalC2×#41×C83D4
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
3. C83×#1988C2semidirect
5. C83×#1546C2semidirect
6. C83×#1303C2semidirect
7. C83×#1292C2semidirect
12. #462×C83C4semidirect
14. #461×C83C4semidirect
198. #9316C83direct (N × H)
207. #1988C2×C83semidirect
209. #1546C2×C83semidirect
210. #1303C2×C83semidirect
211. #1292C2×C83semidirect
212. C83#9316direct (N × H)
217. #462C4×C83semidirect
219. #461C4×C83semidirect

Maximal subgroups

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

subgroupindex
C83×#30392
C83×#29882
C83×#19882
C83×#18062
C83×#15462
C83×#13032
C83×#12922
#931683