Finite Group Databasebeta

← C83×#11762 #35239586

Subgroups — C83×#11762 #35239586

Order 21248.

622 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 622 conjugacy classes of subgroups — too many to draw. Only its 106 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C83×#2406C211yesmaximal, characteristic
2C2×#343×C83C211yesmaximal, characteristic, complemented
3C83×#1463C211yesmaximal, characteristic, complemented
4C83×#1354C212yesmaximal, complemented
5C83×#1345C212yesmaximal, complemented
6C22×#41×C83C411yescharacteristic
7C22×M16×C83—21no—
8C2×#123×C83—21no—
9C22×C4⋊C4×C83C2211yescharacteristic
10#407×C83—21no—
11#395×C83C411yescharacteristic
12#395×C83C2211yescharacteristic
13#394×C83C411yescharacteristic
14#394×C83C2211yescharacteristic
15#386×C83—21no—
16#385×C83—21no—
17#385×C83—21no—
18#381×C83C411yescharacteristic
19#343×C83—21no—
20#343×C83C2212yes—
21#324×C83—21no—
22#324×C83C2212yes—
23C2×#41×C83—41no—
24C2×#41×C83—41no—
25C2×#41×C83—44no—
26C22×Q8×C83C2×C411yescharacteristic
27C22×D4×C83—21no—
28C22×D4×C83C2×C411yescharacteristic
29C23×C4×C83—21no—
30C23×C4×C83C2×C411yescharacteristic
31C23×C4×C83D411yescharacteristic
32C23×C4×C83D411yescharacteristic
33C2×M16×C83—24no—
34C2×M16×C83—22no—
35C2×M16×C83—41no—
36C2×M16×C83—42no—
37C2×M16×C83—41no—
38C22×C8×C83—41no—
39#123×C83—44no—
40C2×C4⋊C4×C83—21no—
41C2×C4⋊C4×C83—21no—
42C2×C4⋊C4×C83—41no—
43C2×C4⋊C4×C83—41no—
44C2×C4⋊C4×C83C2×C411yescharacteristic
45C2×C4⋊C4×C83D411yescharacteristic
46C2×C4⋊C4×C83D411yescharacteristic
47C2×C4⋊C4×C83C2311yescharacteristic
48C2×C22⋊C4×C83—21no—
49C2×C22⋊C4×C83—21no—
50C2×C22⋊C4×C83—21no—
51C2×C22⋊C4×C83—21no—
52C2×C22⋊C4×C83—21no—
53C22⋊C8×C83—22no—
54C22⋊C8×C83—41no—
55#95×C83—21no—
56#95×C83—21no—
57#95×C83—21no—
58#95×C83C2×C411yescharacteristic
59#95×C83C2×C411yescharacteristic
60C24×C83—21no—
61C24×C83C22⋊C411yescharacteristic
62#41×C83—44no—
63#41×C83—44no—
64#41×C83—44no—
65#41×C83—44no—
66C2×Q8×C83—42no—
67C2×Q8×C83—41no—
68C2×D4×C83—24no—
69C2×D4×C83—41no—
70C2×D4×C83—42no—
71C2×D4×C83—41no—
72C2×D4×C83—41no—
73C2×D4×C83—42no—
74C22×C4×C83—21no—
75C22×C4×C83—21no—
76C22×C4×C83—22no—
77C22×C4×C83—21no—
78C22×C4×C83—21no—
79C22×C4×C83—21no—
80C22×C4×C83—21no—
81C22×C4×C83—21no—
82C22×C4×C83—21no—
83C22×C4×C83—21no—
84C22×C4×C83—21no—
85C22×C4×C83—21no—
86C22×C4×C83—42no—
87C22×C4×C83—42no—
88C22×C4×C83—42no—
89C22×C4×C83—42no—
90C22×C4×C83—42no—
91C22×C4×C83C22⋊C411yescharacteristic
92C22×C4×C83C22⋊C411yescharacteristic
93C22×C4×C83C22⋊C411yescharacteristic
94C22×C4×C83C22×C411yescharacteristic
95C22×C4×C83C2×D411yescharacteristic
96C22×C4×C83C2×D411yescharacteristic
97M16×C83—24no—
98M16×C83—44no—
99M16×C83—42no—
100M16×C83—82no—
101C2×C8×C83—44no—
102C2×C8×C83—42no—
103C2×C8×C83—42no—
104C4⋊C4×C83—41no—
105C4⋊C4×C83—41no—
106C4⋊C4×C83—41no—
107C4⋊C4×C83—41no—
108C4⋊C4×C83—42no—
109C4⋊C4×C83—42no—
110C22⋊C4×C83—42no—
111C22⋊C4×C83—42no—
112C22⋊C4×C83—42no—
113C22⋊C4×C83—42no—
114C22⋊C4×C83—41no—
115C22⋊C4×C83—41no—
116C23×C83—22no—
117C23×C83—21no—
118C23×C83—21no—
119C23×C83—22no—
120C23×C83—41no—
121C23×C83—41no—
122C23×C83—41no—
123C23×C83—41no—
124C23×C83—41no—
125C23×C83—41no—
126C23×C83—41no—
127C23×C83#9911yescharacteristic
128C23×C83#9911yescharacteristic
129C23×C83C2×C22⋊C411yescharacteristic
130Q8×C83—42no—
131Q8×C83—42no—
132D4×C83—44no—
133D4×C83—42no—
134D4×C83—42no—
135D4×C83—44no—
136C2×C4×C83—22no—
137C2×C4×C83—22no—
138C2×C4×C83—22no—
139C2×C4×C83—22no—
140C2×C4×C83—21no—
141C2×C4×C83—21no—
142C2×C4×C83—21no—
143C2×C4×C83—21no—
144C2×C4×C83—22no—
145C2×C4×C83—21no—
146C2×C4×C83—21no—
147C2×C4×C83—21no—
148C2×C4×C83—21no—
149C2×C4×C83—41no—
150C2×C4×C83—42no—
151C2×C4×C83—48no—
152C2×C4×C83—42no—
153C2×C4×C83—42no—
154C2×C4×C83—41no—
155C2×C4×C83—41no—
156C2×C4×C83—41no—
157C2×C4×C83—41no—
158C2×C4×C83—41no—
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—81no—
166C2×C4×C83#10412yes—
167C2×C4×C83AGL(1,ℤ/8)11yescharacteristic
168C2×C4×C83#13711yescharacteristic
169C8×C83—44no—
170C8×C83—82no—
171C22×C83—21no—
172C22×C83—21no—
173C22×C83—21no—
174C22×C83—21no—
175C22×C83—21no—
176C22×C83—21no—
177C22×C83—21no—
178C22×C83—42no—
179C22×C83—42no—
180C22×C83—42no—
181C22×C83—42no—
182C22×C83—42no—
183C22×C83—81no—
184C22×C83—81no—
185C22×C83—81no—
186C22×C83—81no—
187C22×C83AΣL(1,𝔽16)12yes—
188C22×C83C2×#9911yescharacteristic
189C22×C83#41711yescharacteristic
190C22×C83C2×#10411yescharacteristic
191C4×C83—21no—
192C4×C83—22no—
193C4×C83—21no—
194C4×C83—41no—
195C4×C83—41no—
196C4×C83—42no—
197C4×C83—41no—
198C4×C83—41no—
199C4×C83—81no—
200C4×C83—81no—
201#11762C8311yesmaximal, characteristic, direct product factor, Sylow
202C2×C83—21no—
203C2×C83—21no—
204C2×C83—22no—
205C2×C83—41no—
206C2×C83—82no—
207C2×C83#150411yescharacteristic
208C2×C83C2×AΣL(1,𝔽16)11yescharacteristic
209C2×C83#213311yescharacteristic
210#2406C2×C8311yescharacteristic
211C2×#343C2×C8311yescharacteristic, complemented
212#1463C2×C8311yescharacteristic, complemented
213#1354C2×C8312yescomplemented
214#1345C2×C8312yescomplemented
215C83#1176211yescharacteristic, direct product factor, Sylow
216C22×#41C4×C8311yescharacteristic
217C22×M16—21no—
218C2×#123—21no—
219C22×C4⋊C4C22×C8311yescharacteristic
220#407—21no—
221#395C4×C8311yescharacteristic
222#395C22×C8311yescharacteristic
223#394C4×C8311yescharacteristic
224#394C22×C8311yescharacteristic
225#386—21no—
226#385—21no—
227#385—21no—
228#381C4×C8311yescharacteristic
229#343—21no—
230#343C22×C8312yes—
231#324—21no—
232#324C22×C8312yes—
233C2×#41—41no—
234C2×#41—41no—
235C2×#41—44no—
236C22×Q8C2×C4×C8311yescharacteristic
237C22×D4—21no—
238C22×D4C2×C4×C8311yescharacteristic
239C23×C4—21no—
240C23×C4C2×C4×C8311yescharacteristic
241C23×C4D4×C8311yescharacteristic
242C23×C4D4×C8311yescharacteristic
243C2×M16—24no—
244C2×M16—22no—
245C2×M16—41no—
246C2×M16—42no—
247C2×M16—41no—
248C22×C8—41no—
249#123—44no—
250C2×C4⋊C4—21no—
251C2×C4⋊C4—21no—
252C2×C4⋊C4—41no—
253C2×C4⋊C4—41no—
254C2×C4⋊C4C2×C4×C8311yescharacteristic
255C2×C4⋊C4D4×C8311yescharacteristic
256C2×C4⋊C4D4×C8311yescharacteristic
257C2×C4⋊C4C23×C8311yescharacteristic
258C2×C22⋊C4—21no—
259C2×C22⋊C4—21no—
260C2×C22⋊C4—21no—
261C2×C22⋊C4—21no—
262C2×C22⋊C4—21no—
263C22⋊C8—22no—
264C22⋊C8—41no—
265#95—21no—
266#95—21no—
267#95—21no—
268#95C2×C4×C8311yescharacteristic
269#95C2×C4×C8311yescharacteristic
270C24—21no—
271C24C22⋊C4×C8311yescharacteristic
272#41—44no—
273#41—44no—
274#41—44no—
275#41—44no—
276C2×Q8—42no—
277C2×Q8—41no—
278C2×D4—24no—
279C2×D4—41no—
280C2×D4—42no—
281C2×D4—41no—
282C2×D4—41no—
283C2×D4—42no—
284C22×C4—21no—
285C22×C4—21no—
286C22×C4—22no—
287C22×C4—21no—
288C22×C4—21no—
289C22×C4—21no—
290C22×C4—21no—
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—21no—
294C22×C4—21no—
295C22×C4—21no—
296C22×C4—42no—
297C22×C4—42no—
298C22×C4—42no—
299C22×C4—42no—
300C22×C4—42no—
301C22×C4C22⋊C4×C8311yescharacteristic
302C22×C4C22⋊C4×C8311yescharacteristic
303C22×C4C22⋊C4×C8311yescharacteristic
304C22×C4C22×C4×C8311yescharacteristic
305C22×C4C2×D4×C8311yescharacteristic
306C22×C4C2×D4×C8311yescharacteristic
307M16—24no—
308M16—44no—
309M16—42no—
310M16—82no—
311C2×C8—44no—
312C2×C8—42no—
313C2×C8—42no—
314C4⋊C4—41no—
315C4⋊C4—41no—
316C4⋊C4—41no—
317C4⋊C4—41no—
318C4⋊C4—42no—
319C4⋊C4—42no—
320C22⋊C4—42no—
321C22⋊C4—42no—
322C22⋊C4—42no—
323C22⋊C4—42no—
324C22⋊C4—41no—
325C22⋊C4—41no—
326C23—22no—
327C23—21no—
328C23—21no—
329C23—22no—
330C23—41no—
331C23—41no—
332C23—41no—
333C23—41no—
334C23—41no—
335C23—41no—
336C23—41no—
337C23#99×C8311yescharacteristic
338C23#99×C8311yescharacteristic
339C23C2×C22⋊C4×C8311yescharacteristic
340Q8—42no—
341Q8—42no—
342D4—44no—
343D4—42no—
344D4—42no—
345D4—44no—
346C2×C4—22no—
347C2×C4—22no—
348C2×C4—22no—
349C2×C4—22no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—22no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—41no—
360C2×C4—42no—
361C2×C4—48no—
362C2×C4—42no—
363C2×C4—42no—
364C2×C4—41no—
365C2×C4—41no—
366C2×C4—41no—
367C2×C4—41no—
368C2×C4—41no—
369C2×C4—41no—
370C2×C4—41no—
371C2×C4—41no—
372C2×C4—41no—
373C2×C4—41no—
374C2×C4—41no—
375C2×C4—81no—
376C2×C4#104×C8312yes—
377C2×C4AGL(1,ℤ/8)×C8311yescharacteristic
378C2×C4#137×C8311yescharacteristic
379C8—44no—
380C8—82no—
381C22—21no—
382C22—21no—
383C22—21no—
384C22—21no—
385C22—21no—
386C22—21no—
387C22—21no—
388C22—42no—
389C22—42no—
390C22—42no—
391C22—42no—
392C22—42no—
393C22—81no—
394C22—81no—
395C22—81no—
396C22—81no—
397C22AΣL(1,𝔽16)×C8312yes—
398C22C2×#99×C8311yescharacteristic
399C22#417×C8311yescharacteristic
400C22C2×#104×C8311yescharacteristic
401C4—21no—
402C4—22no—
403C4—21no—
404C4—41no—
405C4—41no—
406C4—42no—
407C4—41no—
408C4—41no—
409C4—81no—
410C4—81no—
411C2—21no—
412C2—21no—
413C2—22no—
414C2—41no—
415C2—82no—
416C2C83×#150411yescharacteristic
417C2C2×AΣL(1,𝔽16)×C8311yescharacteristic
418C2C83×#213311yescharacteristic
subgroupG/H
center, socleC22×C83C2×#99
commutatorC22×C4C22×C4×C83
Frattini, cosocleC2×C4⋊C4C23×C83
prime radicalC22×#41×C83C4
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22×C83 ◁ C24×C83 ◁ C22×C4⋊C4×C83 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×#343×C83C2semidirect
3. C83×#1463C2semidirect
4. C83×#1354C2semidirect
5. C83×#1345C2semidirect
201. #11762C83direct (N × H)
211. C2×#343C2×C83semidirect
212. #1463C2×C83semidirect
213. #1354C2×C83semidirect
214. #1345C2×C83semidirect
215. C83#11762direct (N × H)

Maximal subgroups

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

subgroupindex
C83×#24062
C2×#343×C832
C83×#14632
C83×#13542
C83×#13452
#1176283