Finite Group Databasebeta

← C83×#7658 #35235482

Subgroups — C83×#7658 #35235482

Order 21248.

544 conjugacy classes of subgroups.

The subgroup lattice has 396 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C83×#2854C211yesmaximal, characteristic
2C83×#1797C211yesmaximal, characteristic, complemented
3C4×C22⋊C8×C83C211yesmaximal, characteristic, complemented
4C83×#1271C212yesmaximal, complemented
5C83×#1270C212yesmaximal, complemented
6C2×#123×C83—21no—
7C2×#121×C83C411yescharacteristic
8C22×C42×C83C2211yescharacteristic
9C2×C4⋊C8×C83—21no—
10C2×C22⋊C8×C83—21no—
11C2×C22⋊C8×C83—21no—
12C2×C4×C8×C83—21no—
13#390×C83C411yescharacteristic
14#390×C83C2211yescharacteristic
15#389×C83C411yescharacteristic
16#389×C83C2211yescharacteristic
17#388×C83—21no—
18#388×C83—21no—
19#386×C83—21no—
20#380×C83C411yescharacteristic
21#336×C83C2212yes—
22#336×C83C2212yes—
23C22×D4×C83—21no—
24C22×D4×C83C2×C411yescharacteristic
25C23×C4×C83C2×C411yescharacteristic
26C23×C4×C83D411yescharacteristic
27C23×C4×C83D411yescharacteristic
28C22×C8×C83—22no—
29C22×C8×C83—22no—
30#123×C83—42no—
31#123×C83—42no—
32#121×C83—42no—
33C2×C4⋊C4×C83—21no—
34C2×C4⋊C4×C83—21no—
35C2×C4⋊C4×C83C2×C411yescharacteristic
36C2×C4⋊C4×C83C2×C411yescharacteristic
37C2×C4⋊C4×C83C2×C411yescharacteristic
38C2×C22⋊C4×C83—21no—
39C2×C22⋊C4×C83—21no—
40C2×C22⋊C4×C83—21no—
41C2×C22⋊C4×C83—21no—
42C2×C22⋊C4×C83—21no—
43C2×C22⋊C4×C83—21no—
44C2×C42×C83—42no—
45C2×C42×C83C2×C411yescharacteristic
46C2×C42×C83D411yescharacteristic
47C2×C42×C83D411yescharacteristic
48C2×C42×C83C2311yescharacteristic
49C4⋊C8×C83—42no—
50C22⋊C8×C83—24no—
51C22⋊C8×C83—42no—
52C4×C8×C83—42no—
53#95×C83—21no—
54#95×C83—21no—
55#95×C83—21no—
56C24×C83—21no—
57C24×C83C22⋊C411yescharacteristic
58C2×D4×C83—22no—
59C2×D4×C83—41no—
60C2×D4×C83—41no—
61C2×D4×C83—42no—
62C2×D4×C83—42no—
63C2×D4×C83—42no—
64C22×C4×C83—22no—
65C22×C4×C83—22no—
66C22×C4×C83—22no—
67C22×C4×C83—22no—
68C22×C4×C83—21no—
69C22×C4×C83—21no—
70C22×C4×C83—21no—
71C22×C4×C83—41no—
72C22×C4×C83—41no—
73C22×C4×C83—41no—
74C22×C4×C83—41no—
75C22×C4×C83C22⋊C411yescharacteristic
76C22×C4×C83C22⋊C411yescharacteristic
77C22×C4×C83C22⋊C411yescharacteristic
78C22×C4×C83D812yes—
79C22×C4×C83SD1612yes—
80C22×C4×C83C22×C411yescharacteristic
81C22×C4×C83C2×D411yescharacteristic
82C22×C4×C83C2×D411yescharacteristic
83C2×C8×C83—24no—
84C2×C8×C83—42no—
85C2×C8×C83—42no—
86C2×C8×C83—42no—
87C2×C8×C83—42no—
88C2×C8×C83—42no—
89C4⋊C4×C83—22no—
90C4⋊C4×C83—41no—
91C4⋊C4×C83—41no—
92C4⋊C4×C83—42no—
93C4⋊C4×C83—41no—
94C4⋊C4×C83—41no—
95C22⋊C4×C83—42no—
96C22⋊C4×C83—42no—
97C22⋊C4×C83—42no—
98C22⋊C4×C83—42no—
99C22⋊C4×C83—41no—
100C22⋊C4×C83—41no—
101C22⋊C4×C83—42no—
102C42×C83—22no—
103C42×C83—22no—
104C42×C83—41no—
105C42×C83—41no—
106C23×C83—21no—
107C23×C83—21no—
108C23×C83—21no—
109C23×C83—21no—
110C23×C83—21no—
111C23×C83—21no—
112C23×C83—41no—
113C23×C83—41no—
114C23×C83—41no—
115C23×C83—41no—
116C23×C83—41no—
117C23×C83—41no—
118C23×C83—41no—
119C23×C83#10212yes—
120C23×C83C2×C22⋊C411yescharacteristic
121D4×C83—82no—
122D4×C83—84no—
123C2×C4×C83—21no—
124C2×C4×C83—21no—
125C2×C4×C83—21no—
126C2×C4×C83—21no—
127C2×C4×C83—21no—
128C2×C4×C83—21no—
129C2×C4×C83—21no—
130C2×C4×C83—21no—
131C2×C4×C83—21no—
132C2×C4×C83—21no—
133C2×C4×C83—21no—
134C2×C4×C83—21no—
135C2×C4×C83—42no—
136C2×C4×C83—42no—
137C2×C4×C83—44no—
138C2×C4×C83—44no—
139C2×C4×C83—41no—
140C2×C4×C83—41no—
141C2×C4×C83—41no—
142C2×C4×C83—41no—
143C2×C4×C83—41no—
144C2×C4×C83—41no—
145C2×C4×C83—41no—
146C2×C4×C83—41no—
147C2×C4×C83—41no—
148C2×C4×C83#9912yes—
149C2×C4×C83#10212yes—
150C2×C4×C83#10412yes—
151C2×C4×C83#13112yes—
152C2×C4×C83C2×D811yescharacteristic
153C2×C4×C83C2×SD1611yescharacteristic
154C2×C4×C83#13512yes—
155C8×C83—44no—
156C8×C83—82no—
157C22×C83—22no—
158C22×C83—22no—
159C22×C83—22no—
160C22×C83—41no—
161C22×C83—41no—
162C22×C83—41no—
163C22×C83—41no—
164C22×C83—81no—
165C22×C83—81no—
166C22×C83—81no—
167C22×C83—81no—
168C22×C83—81no—
169C22×C83—81no—
170C22×C83—81no—
171C22×C83#40811yescharacteristic
172C22×C83C2×#9911yescharacteristic
173C22×C83#41311yescharacteristic
174C22×C83C2×#10211yescharacteristic
175C22×C83#41611yescharacteristic
176C22×C83C2×#10411yescharacteristic
177C22×C83#42111yescharacteristic
178C4×C83—22no—
179C4×C83—22no—
180C4×C83—22no—
181C4×C83—22no—
182C4×C83—21no—
183C4×C83—22no—
184C4×C83—21no—
185C4×C83—81no—
186C4×C83—81no—
187C4×C83—81no—
188#7658C8311yesmaximal, characteristic, direct product factor, Sylow
189C2×C83—42no—
190C2×C83—161no—
191C2×C83#146811yescharacteristic
192C2×C83#148211yescharacteristic
193C2×C83#149211yescharacteristic
194C2×C83#150111yescharacteristic
195C2×C83#151911yescharacteristic
196C2×C83#152911yescharacteristic
197C2×C83#154011yescharacteristic
198#2854C2×C8311yescharacteristic
199#1797C2×C8311yescharacteristic, complemented
200C4×C22⋊C8C2×C8311yescharacteristic, complemented
201#1271C2×C8312yescomplemented
202#1270C2×C8312yescomplemented
203C83#765811yescharacteristic, direct product factor, Sylow
204C2×#123—21no—
205C2×#121C4×C8311yescharacteristic
206C22×C42C22×C8311yescharacteristic
207C2×C4⋊C8—21no—
208C2×C22⋊C8—21no—
209C2×C22⋊C8—21no—
210C2×C4×C8—21no—
211#390C4×C8311yescharacteristic
212#390C22×C8311yescharacteristic
213#389C4×C8311yescharacteristic
214#389C22×C8311yescharacteristic
215#388—21no—
216#388—21no—
217#386—21no—
218#380C4×C8311yescharacteristic
219#336C22×C8312yes—
220#336C22×C8312yes—
221C22×D4—21no—
222C22×D4C2×C4×C8311yescharacteristic
223C23×C4C2×C4×C8311yescharacteristic
224C23×C4D4×C8311yescharacteristic
225C23×C4D4×C8311yescharacteristic
226C22×C8—22no—
227C22×C8—22no—
228#123—42no—
229#123—42no—
230#121—42no—
231C2×C4⋊C4—21no—
232C2×C4⋊C4—21no—
233C2×C4⋊C4C2×C4×C8311yescharacteristic
234C2×C4⋊C4C2×C4×C8311yescharacteristic
235C2×C4⋊C4C2×C4×C8311yescharacteristic
236C2×C22⋊C4—21no—
237C2×C22⋊C4—21no—
238C2×C22⋊C4—21no—
239C2×C22⋊C4—21no—
240C2×C22⋊C4—21no—
241C2×C22⋊C4—21no—
242C2×C42—42no—
243C2×C42C2×C4×C8311yescharacteristic
244C2×C42D4×C8311yescharacteristic
245C2×C42D4×C8311yescharacteristic
246C2×C42C23×C8311yescharacteristic
247C4⋊C8—42no—
248C22⋊C8—24no—
249C22⋊C8—42no—
250C4×C8—42no—
251#95—21no—
252#95—21no—
253#95—21no—
254C24—21no—
255C24C22⋊C4×C8311yescharacteristic
256C2×D4—22no—
257C2×D4—41no—
258C2×D4—41no—
259C2×D4—42no—
260C2×D4—42no—
261C2×D4—42no—
262C22×C4—22no—
263C22×C4—22no—
264C22×C4—22no—
265C22×C4—22no—
266C22×C4—21no—
267C22×C4—21no—
268C22×C4—21no—
269C22×C4—41no—
270C22×C4—41no—
271C22×C4—41no—
272C22×C4—41no—
273C22×C4C22⋊C4×C8311yescharacteristic
274C22×C4C22⋊C4×C8311yescharacteristic
275C22×C4C22⋊C4×C8311yescharacteristic
276C22×C4D8×C8312yes—
277C22×C4SD16×C8312yes—
278C22×C4C22×C4×C8311yescharacteristic
279C22×C4C2×D4×C8311yescharacteristic
280C22×C4C2×D4×C8311yescharacteristic
281C2×C8—24no—
282C2×C8—42no—
283C2×C8—42no—
284C2×C8—42no—
285C2×C8—42no—
286C2×C8—42no—
287C4⋊C4—22no—
288C4⋊C4—41no—
289C4⋊C4—41no—
290C4⋊C4—42no—
291C4⋊C4—41no—
292C4⋊C4—41no—
293C22⋊C4—42no—
294C22⋊C4—42no—
295C22⋊C4—42no—
296C22⋊C4—42no—
297C22⋊C4—41no—
298C22⋊C4—41no—
299C22⋊C4—42no—
300C42—22no—
301C42—22no—
302C42—41no—
303C42—41no—
304C23—21no—
305C23—21no—
306C23—21no—
307C23—21no—
308C23—21no—
309C23—21no—
310C23—41no—
311C23—41no—
312C23—41no—
313C23—41no—
314C23—41no—
315C23—41no—
316C23—41no—
317C23#102×C8312yes—
318C23C2×C22⋊C4×C8311yescharacteristic
319D4—82no—
320D4—84no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—42no—
334C2×C4—42no—
335C2×C4—44no—
336C2×C4—44no—
337C2×C4—41no—
338C2×C4—41no—
339C2×C4—41no—
340C2×C4—41no—
341C2×C4—41no—
342C2×C4—41no—
343C2×C4—41no—
344C2×C4—41no—
345C2×C4—41no—
346C2×C4#99×C8312yes—
347C2×C4#102×C8312yes—
348C2×C4#104×C8312yes—
349C2×C4#131×C8312yes—
350C2×C4C2×D8×C8311yescharacteristic
351C2×C4C2×SD16×C8311yescharacteristic
352C2×C4#135×C8312yes—
353C8—44no—
354C8—82no—
355C22—22no—
356C22—22no—
357C22—22no—
358C22—41no—
359C22—41no—
360C22—41no—
361C22—41no—
362C22—81no—
363C22—81no—
364C22—81no—
365C22—81no—
366C22—81no—
367C22—81no—
368C22—81no—
369C22#408×C8311yescharacteristic
370C22C2×#99×C8311yescharacteristic
371C22#413×C8311yescharacteristic
372C22C2×#102×C8311yescharacteristic
373C22#416×C8311yescharacteristic
374C22C2×#104×C8311yescharacteristic
375C22#421×C8311yescharacteristic
376C4—22no—
377C4—22no—
378C4—21no—
379C4—22no—
380C4—21no—
381C4—22no—
382C4—22no—
383C4—81no—
384C4—81no—
385C4—81no—
386C2—42no—
387C2—161no—
388C2C83×#146811yescharacteristic
389C2C83×#148211yescharacteristic
390C2C83×#149211yescharacteristic
391C2C83×#150111yescharacteristic
392C2C83×#151911yescharacteristic
393C2C83×#152911yescharacteristic
394C2C83×#154011yescharacteristic
subgroupG/H
center, socleC23×C83C2×C22⋊C4
commutatorC22×C4C22×C4×C83
Frattini, cosocleC2×C42C23×C83
prime radicalC2×#121×C83C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23×C83 ◁ C22×C42×C83 ◁ G

Complements

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

NH ≅ G/Ntype
2. C83×#1797C2semidirect
3. C4×C22⋊C8×C83C2semidirect
4. C83×#1271C2semidirect
5. C83×#1270C2semidirect
188. #7658C83direct (N × H)
199. #1797C2×C83semidirect
200. C4×C22⋊C8C2×C83semidirect
201. #1271C2×C83semidirect
202. #1270C2×C83semidirect
203. C83#7658direct (N × H)

Maximal subgroups

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

subgroupindex
C83×#28542
C83×#17972
C4×C22⋊C8×C832
C83×#12712
C83×#12702
#765883