Finite Group Databasebeta

← C83×#9667 #35237491

Subgroups — C83×#9667 #35237491

Order 21248.

478 conjugacy classes of subgroups.

The subgroup lattice has 394 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×SD16×C83C211yesmaximal, characteristic
2C83×#1564C211yesmaximal, characteristic, complemented
3C83×#1479C211yesmaximal, characteristic, complemented
4C83×#1478C211yesmaximal, characteristic
5C83×#1461C211yesmaximal, characteristic, complemented
6C83×#1460C211yesmaximal, characteristic
7C83×#1450C211yesmaximal, characteristic, complemented
8C22×SD16×C83C411yescharacteristic
9C2×C4×Q8×C83C2211yescharacteristic
10C2×C4×D4×C83C2211yescharacteristic
11C4×SD16×C83—21no—
12C4×SD16×C83—21no—
13C4×SD16×C83—21no—
14C4×SD16×C83—21no—
15#431×C83C2211yescharacteristic
16#431×C83C2211yescharacteristic
17#431×C83C2211yescharacteristic
18#431×C83C2211yescharacteristic
19C2×#106×C83C411yescharacteristic
20C2×#103×C83C411yescharacteristic
21C2×#102×C83C411yescharacteristic
22C2×C4×C8×C83C2211yescharacteristic
23#335×C83—21no—
24#334×C83—21no—
25Q8⋊C8×C83—22no—
26D4⋊C8×C83—22no—
27#324×C83—22no—
28#323×C83—22no—
29#322×C83—21no—
30C82×C83—21no—
31C22×Q8×C83C2×C411yescharacteristic
32C22×D4×C83C2×C411yescharacteristic
33C23×C4×C83—21no—
34C2×SD16×C83—21no—
35C2×SD16×C83—21no—
36C2×SD16×C83—22no—
37C2×SD16×C83C814yescomplemented
38C22×C8×C83C2×C411yescharacteristic
39C22×C8×C83C2×C411yescharacteristic
40C4×Q8×C83—21no—
41C4×Q8×C83—21no—
42C4×Q8×C83—21no—
43C4×Q8×C83—21no—
44C4×D4×C83—21no—
45C4×D4×C83—21no—
46C4×D4×C83—21no—
47C4×D4×C83—21no—
48C2×C4⋊C4×C83—21no—
49C2×C4⋊C4×C83C2×C411yescharacteristic
50C2×C4⋊C4×C83C2×C411yescharacteristic
51C2×C22⋊C4×C83—21no—
52C2×C42×C83—21no—
53C2×C42×C83C2311yescharacteristic
54#106×C83—21no—
55#106×C83—21no—
56C4⋊C8×C83—21no—
57C4⋊C8×C83—21no—
58C4⋊C8×C83—21no—
59C4⋊C8×C83D412yes—
60#103×C83—21no—
61#103×C83—21no—
62#102×C83—21no—
63#102×C83—21no—
64C22⋊C8×C83—21no—
65C22⋊C8×C83—21no—
66C22⋊C8×C83—21no—
67C22⋊C8×C83D412yes—
68C4×C8×C83—21no—
69C4×C8×C83—21no—
70C4×C8×C83—21no—
71C4×C8×C83—21no—
72C4×C8×C83D411yescharacteristic
73C4×C8×C83D411yescharacteristic
74C4×C8×C83D411yescharacteristic
75C4×C8×C83D411yescharacteristic
76C24×C83—21no—
77C2×Q8×C83—21no—
78C2×Q8×C83—21no—
79C2×Q8×C83—21no—
80C2×Q8×C83—21no—
81C2×Q8×C83—21no—
82C2×Q8×C83C2×C812yes—
83C2×D4×C83—21no—
84C2×D4×C83—21no—
85C2×D4×C83—21no—
86C2×D4×C83—21no—
87C2×D4×C83—21no—
88C2×D4×C83C2×C812yes—
89C22×C4×C83—21no—
90C22×C4×C83—21no—
91C22×C4×C83—21no—
92C22×C4×C83—41no—
93C22×C4×C83—41no—
94C22×C4×C83—82no—
95C22×C4×C83C22×C411yescharacteristic
96C22×C4×C83C2×D411yescharacteristic
97C22×C4×C83#4111yescharacteristic
98SD16×C83—24no—
99SD16×C83—24no—
100C2×C8×C83—21no—
101C2×C8×C83—21no—
102C2×C8×C83—22no—
103C2×C8×C83—22no—
104C2×C8×C83—41no—
105C2×C8×C83—41no—
106C2×C8×C83C22⋊C411yescharacteristic
107C2×C8×C83C22⋊C411yescharacteristic
108C2×C8×C83C22⋊C411yescharacteristic
109C2×C8×C83C22⋊C411yescharacteristic
110C2×C8×C83C2×C812yes—
111C2×C8×C83M1612yes—
112C4⋊C4×C83—21no—
113C4⋊C4×C83—21no—
114C4⋊C4×C83—21no—
115C4⋊C4×C83—21no—
116C4⋊C4×C83—41no—
117C4⋊C4×C83—41no—
118C22⋊C4×C83—41no—
119C22⋊C4×C83—41no—
120C42×C83—41no—
121C42×C83—41no—
122C42×C83C2×D411yescharacteristic
123C42×C83C2×D411yescharacteristic
124C42×C83C2×D411yescharacteristic
125C42×C83#4111yescharacteristic
126C23×C83—22no—
127C23×C83—41no—
128C23×C83—41no—
129C23×C83—81no—
130C23×C83—81no—
131C23×C83C4×D411yescharacteristic
132Q8×C83—22no—
133Q8×C83—22no—
134Q8×C83—41no—
135Q8×C83—41no—
136D4×C83—22no—
137D4×C83—22no—
138D4×C83—41no—
139D4×C83—41no—
140C2×C4×C83—22no—
141C2×C4×C83—22no—
142C2×C4×C83—41no—
143C2×C4×C83—41no—
144C2×C4×C83—41no—
145C2×C4×C83—41no—
146C2×C4×C83—41no—
147C2×C4×C83—82no—
148C2×C4×C83—82no—
149C2×C4×C83—81no—
150C2×C4×C83—81no—
151C2×C4×C83C2×C22⋊C411yescharacteristic
152C2×C4×C83C4×D411yescharacteristic
153C2×C4×C83#12111yescharacteristic
154C2×C4×C83#12111yescharacteristic
155C2×C4×C83#12411yescharacteristic
156C2×C4×C83#12711yescharacteristic
157C2×C4×C83C22×C811yescharacteristic
158C2×C4×C83C2×M1611yescharacteristic
159C2×C4×C83#13111yescharacteristic
160C2×C4×C83#13111yescharacteristic
161C8×C83—21no—
162C8×C83—21no—
163C8×C83—42no—
164C8×C83—42no—
165C8×C83—81no—
166C8×C83—81no—
167C8×C83C22⋊C812yes—
168C8×C83C22⋊C812yes—
169C22×C83—21no—
170C22×C83—21no—
171C22×C83—42no—
172C22×C83—82no—
173C22×C83—82no—
174C22×C83—81no—
175C22×C83—81no—
176C22×C83#39011yescharacteristic
177C22×C83C8×D411yescharacteristic
178C22×C83#43611yescharacteristic
179C4×C83—21no—
180C4×C83—21no—
181C4×C83—21no—
182C4×C83—42no—
183C4×C83—81no—
184C4×C83—81no—
185C4×C83C2×C22⋊C811yescharacteristic
186C4×C83#40811yescharacteristic
187C4×C83C8×D411yescharacteristic
188C4×C83#43611yescharacteristic
189C4×C83#47112yes—
190#9667C8311yesmaximal, characteristic, direct product factor, Sylow
191C2×C83—21no—
192C2×C83—21no—
193C2×C83—82no—
194C2×C83#195411yescharacteristic
195C2×C83#197611yescharacteristic
196C2×C83#197711yescharacteristic
197C2×C4×SD16C2×C8311yescharacteristic
198#1564C2×C8311yescharacteristic, complemented
199#1479C2×C8311yescharacteristic, complemented
200#1478C2×C8311yescharacteristic
201#1461C2×C8311yescharacteristic, complemented
202#1460C2×C8311yescharacteristic
203#1450C2×C8311yescharacteristic, complemented
204C83#966711yescharacteristic, direct product factor, Sylow
205C22×SD16C4×C8311yescharacteristic
206C2×C4×Q8C22×C8311yescharacteristic
207C2×C4×D4C22×C8311yescharacteristic
208C4×SD16—21no—
209C4×SD16—21no—
210C4×SD16—21no—
211C4×SD16—21no—
212#431C22×C8311yescharacteristic
213#431C22×C8311yescharacteristic
214#431C22×C8311yescharacteristic
215#431C22×C8311yescharacteristic
216C2×#106C4×C8311yescharacteristic
217C2×#103C4×C8311yescharacteristic
218C2×#102C4×C8311yescharacteristic
219C2×C4×C8C22×C8311yescharacteristic
220#335—21no—
221#334—21no—
222Q8⋊C8—22no—
223D4⋊C8—22no—
224#324—22no—
225#323—22no—
226#322—21no—
227C82—21no—
228C22×Q8C2×C4×C8311yescharacteristic
229C22×D4C2×C4×C8311yescharacteristic
230C23×C4—21no—
231C2×SD16—21no—
232C2×SD16—21no—
233C2×SD16—22no—
234C2×SD16C8×C8314yescomplemented
235C22×C8C2×C4×C8311yescharacteristic
236C22×C8C2×C4×C8311yescharacteristic
237C4×Q8—21no—
238C4×Q8—21no—
239C4×Q8—21no—
240C4×Q8—21no—
241C4×D4—21no—
242C4×D4—21no—
243C4×D4—21no—
244C4×D4—21no—
245C2×C4⋊C4—21no—
246C2×C4⋊C4C2×C4×C8311yescharacteristic
247C2×C4⋊C4C2×C4×C8311yescharacteristic
248C2×C22⋊C4—21no—
249C2×C42—21no—
250C2×C42C23×C8311yescharacteristic
251#106—21no—
252#106—21no—
253C4⋊C8—21no—
254C4⋊C8—21no—
255C4⋊C8—21no—
256C4⋊C8D4×C8312yes—
257#103—21no—
258#103—21no—
259#102—21no—
260#102—21no—
261C22⋊C8—21no—
262C22⋊C8—21no—
263C22⋊C8—21no—
264C22⋊C8D4×C8312yes—
265C4×C8—21no—
266C4×C8—21no—
267C4×C8—21no—
268C4×C8—21no—
269C4×C8D4×C8311yescharacteristic
270C4×C8D4×C8311yescharacteristic
271C4×C8D4×C8311yescharacteristic
272C4×C8D4×C8311yescharacteristic
273C24—21no—
274C2×Q8—21no—
275C2×Q8—21no—
276C2×Q8—21no—
277C2×Q8—21no—
278C2×Q8—21no—
279C2×Q8C2×C8×C8312yes—
280C2×D4—21no—
281C2×D4—21no—
282C2×D4—21no—
283C2×D4—21no—
284C2×D4—21no—
285C2×D4C2×C8×C8312yes—
286C22×C4—21no—
287C22×C4—21no—
288C22×C4—21no—
289C22×C4—41no—
290C22×C4—41no—
291C22×C4—82no—
292C22×C4C22×C4×C8311yescharacteristic
293C22×C4C2×D4×C8311yescharacteristic
294C22×C4#41×C8311yescharacteristic
295SD16—24no—
296SD16—24no—
297C2×C8—21no—
298C2×C8—21no—
299C2×C8—22no—
300C2×C8—22no—
301C2×C8—41no—
302C2×C8—41no—
303C2×C8C22⋊C4×C8311yescharacteristic
304C2×C8C22⋊C4×C8311yescharacteristic
305C2×C8C22⋊C4×C8311yescharacteristic
306C2×C8C22⋊C4×C8311yescharacteristic
307C2×C8C2×C8×C8312yes—
308C2×C8M16×C8312yes—
309C4⋊C4—21no—
310C4⋊C4—21no—
311C4⋊C4—21no—
312C4⋊C4—21no—
313C4⋊C4—41no—
314C4⋊C4—41no—
315C22⋊C4—41no—
316C22⋊C4—41no—
317C42—41no—
318C42—41no—
319C42C2×D4×C8311yescharacteristic
320C42C2×D4×C8311yescharacteristic
321C42C2×D4×C8311yescharacteristic
322C42#41×C8311yescharacteristic
323C23—22no—
324C23—41no—
325C23—41no—
326C23—81no—
327C23—81no—
328C23C4×D4×C8311yescharacteristic
329Q8—22no—
330Q8—22no—
331Q8—41no—
332Q8—41no—
333D4—22no—
334D4—22no—
335D4—41no—
336D4—41no—
337C2×C4—22no—
338C2×C4—22no—
339C2×C4—41no—
340C2×C4—41no—
341C2×C4—41no—
342C2×C4—41no—
343C2×C4—41no—
344C2×C4—82no—
345C2×C4—82no—
346C2×C4—81no—
347C2×C4—81no—
348C2×C4C2×C22⋊C4×C8311yescharacteristic
349C2×C4C4×D4×C8311yescharacteristic
350C2×C4#121×C8311yescharacteristic
351C2×C4#121×C8311yescharacteristic
352C2×C4#124×C8311yescharacteristic
353C2×C4#127×C8311yescharacteristic
354C2×C4C22×C8×C8311yescharacteristic
355C2×C4C2×M16×C8311yescharacteristic
356C2×C4#131×C8311yescharacteristic
357C2×C4#131×C8311yescharacteristic
358C8—21no—
359C8—21no—
360C8—42no—
361C8—42no—
362C8—81no—
363C8—81no—
364C8C22⋊C8×C8312yes—
365C8C22⋊C8×C8312yes—
366C22—21no—
367C22—21no—
368C22—42no—
369C22—82no—
370C22—82no—
371C22—81no—
372C22—81no—
373C22#390×C8311yescharacteristic
374C22C8×D4×C8311yescharacteristic
375C22#436×C8311yescharacteristic
376C4—21no—
377C4—21no—
378C4—21no—
379C4—42no—
380C4—81no—
381C4—81no—
382C4C2×C22⋊C8×C8311yescharacteristic
383C4#408×C8311yescharacteristic
384C4C8×D4×C8311yescharacteristic
385C4#436×C8311yescharacteristic
386C4#471×C8312yes—
387C2—21no—
388C2—21no—
389C2—82no—
390C2C83×#195411yescharacteristic
391C2C83×#197611yescharacteristic
392C2C83×#197711yescharacteristic
subgroupG/H
centerC2×C4×C83#121
commutatorC2×C4C22×C8×C83
Frattini, cosocleC2×C42C23×C83
socleC22×C83#390
prime radicalC22×D4×C83C2×C4
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4×C83 ◁ C2×C42×C83 ◁ G

Complements

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

NH ≅ G/Ntype
2. C83×#1564C2semidirect
3. C83×#1479C2semidirect
5. C83×#1461C2semidirect
7. C83×#1450C2semidirect
37. C2×SD16×C83C8semidirect
190. #9667C83direct (N × H)
198. #1564C2×C83semidirect
199. #1479C2×C83semidirect
201. #1461C2×C83semidirect
203. #1450C2×C83semidirect
204. C83#9667direct (N × H)
234. C2×SD16C8×C83semidirect

Maximal subgroups

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

subgroupindex
C2×C4×SD16×C832
C83×#15642
C83×#14792
C83×#14782
C83×#14612
C83×#14602
C83×#14502
#966783