Finite Group Databasebeta

← C83×#8300 #35236124

Subgroups — C83×#8300 #35236124

Order 21248.

1158 conjugacy classes of subgroups.

The subgroup lattice has 422 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#117×C83C211yesmaximal, characteristic
2C83×#1817C212yesmaximal, complemented
3C2×#324×C83C212yesmaximal, complemented
4C2×#323×C83C212yesmaximal
5C24×C4×C83C411yescharacteristic
6C2×#117×C83—24no—
7C2×#117×C83—24no—
8C2×#117×C83—24no—
9C22×C4⋊C4×C83C411yescharacteristic
10C22×C4⋊C4×C83C2211yescharacteristic
11C22×C22⋊C4×C83C411yescharacteristic
12C22×C22⋊C4×C83C2211yescharacteristic
13C22×C42×C83C411yescharacteristic
14C22×C42×C83C2211yescharacteristic
15C2×C22⋊C8×C83C2214yes—
16#336×C83—24no—
17#324×C83—24no—
18#323×C83—24no—
19C25×C83C2×C411yescharacteristic
20C23×C4×C83—22no—
21C23×C4×C83—22no—
22C23×C4×C83—22no—
23C23×C4×C83—22no—
24C23×C4×C83—22no—
25C23×C4×C83—22no—
26C23×C4×C83C814yescomplemented
27C23×C4×C83C2×C411yescharacteristic
28C23×C4×C83C2×C411yescharacteristic
29C23×C4×C83C2×C411yescharacteristic
30C23×C4×C83C2×C411yescharacteristic
31C23×C4×C83C2×C411yescharacteristic
32C23×C4×C83C2311yescharacteristic
33C22×C8×C83—24no—
34#117×C83—232no—
35C2×C4⋊C4×C83—22no—
36C2×C4⋊C4×C83—22no—
37C2×C4⋊C4×C83—22no—
38C2×C4⋊C4×C83—22no—
39C2×C4⋊C4×C83—22no—
40C2×C4⋊C4×C83D414yes—
41C2×C22⋊C4×C83—22no—
42C2×C22⋊C4×C83—22no—
43C2×C22⋊C4×C83—22no—
44C2×C22⋊C4×C83—22no—
45C2×C22⋊C4×C83—22no—
46C2×C22⋊C4×C83D414yes—
47C2×C42×C83—22no—
48C2×C42×C83—22no—
49C2×C42×C83—22no—
50C2×C42×C83—22no—
51C2×C42×C83—22no—
52C2×C42×C83D414yes—
53C22⋊C8×C83—24no—
54C22⋊C8×C83—24no—
55C24×C83—21no—
56C24×C83—21no—
57C24×C83—21no—
58C24×C83—21no—
59C24×C83—42no—
60C24×C83—42no—
61C24×C83C22⋊C412yes—
62C24×C83C2×C812yes—
63C24×C83M1612yes—
64C24×C83C22×C411yescharacteristic
65C22×C4×C83—28no—
66C22×C4×C83—28no—
67C22×C4×C83—28no—
68C22×C4×C83—28no—
69C22×C4×C83—28no—
70C22×C4×C83—21no—
71C22×C4×C83—21no—
72C22×C4×C83—21no—
73C22×C4×C83—21no—
74C22×C4×C83—21no—
75C22×C4×C83—21no—
76C22×C4×C83—21no—
77C22×C4×C83—21no—
78C22×C4×C83—28no—
79C22×C4×C83—28no—
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×C83C22⋊C412yes—
91C22×C4×C83C22⋊C412yes—
92C22×C4×C83C22⋊C412yes—
93C22×C4×C83C22⋊C412yes—
94C22×C4×C83C22⋊C412yes—
95C22×C4×C83C2×C812yes—
96C22×C4×C83C2×C812yes—
97C22×C4×C83M1612yes—
98C22×C4×C83M1612yes—
99C22×C4×C83C2×D412yes—
100C22×C4×C83C2×D412yes—
101C22×C4×C83C2×D412yes—
102C2×C8×C83—48no—
103C2×C8×C83—44no—
104C4⋊C4×C83—28no—
105C4⋊C4×C83—28no—
106C22⋊C4×C83—28no—
107C22⋊C4×C83—28no—
108C42×C83—28no—
109C42×C83—28no—
110C23×C83—22no—
111C23×C83—22no—
112C23×C83—22no—
113C23×C83—22no—
114C23×C83—22no—
115C23×C83—22no—
116C23×C83—22no—
117C23×C83—22no—
118C23×C83—42no—
119C23×C83—42no—
120C23×C83—44no—
121C23×C83—44no—
122C23×C83—44no—
123C23×C83—44no—
124C23×C83—42no—
125C23×C83—42no—
126C23×C83—42no—
127C23×C83—42no—
128C23×C83C22⋊C814yes—
129C23×C83C2×C22⋊C411yescharacteristic
130C23×C83C2×C22⋊C411yescharacteristic
131C23×C83C2×C22⋊C411yescharacteristic
132C23×C83C22×C811yescharacteristic
133C23×C83C2×M1611yescharacteristic
134C23×C83C2×M1611yescharacteristic
135C23×C83C2×M1611yescharacteristic
136C2×C4×C83—232no—
137C2×C4×C83—22no—
138C2×C4×C83—22no—
139C2×C4×C83—22no—
140C2×C4×C83—22no—
141C2×C4×C83—22no—
142C2×C4×C83—22no—
143C2×C4×C83—22no—
144C2×C4×C83—22no—
145C2×C4×C83—22no—
146C2×C4×C83—22no—
147C2×C4×C83—22no—
148C2×C4×C83—22no—
149C2×C4×C83—24no—
150C2×C4×C83—22no—
151C2×C4×C83—22no—
152C2×C4×C83—22no—
153C2×C4×C83—24no—
154C2×C4×C83—42no—
155C2×C4×C83—42no—
156C2×C4×C83—42no—
157C2×C4×C83—42no—
158C2×C4×C83—42no—
159C2×C4×C83—42no—
160C2×C4×C83—42no—
161C2×C4×C83—42no—
162C2×C4×C83—42no—
163C2×C4×C83—42no—
164C2×C4×C83C22⋊C814yes—
165C2×C4×C83C22⋊C814yes—
166C2×C4×C83#12014yes—
167C8×C83—84no—
168C22×C83—22no—
169C22×C83—22no—
170C22×C83—22no—
171C22×C83—22no—
172C22×C83—21no—
173C22×C83—21no—
174C22×C83—21no—
175C22×C83—21no—
176C22×C83—24no—
177C22×C83—44no—
178C22×C83—44no—
179C22×C83—44no—
180C22×C83—44no—
181C22×C83—44no—
182C22×C83—44no—
183C22×C83—42no—
184C22×C83—42no—
185C22×C83#37911yescharacteristic
186C22×C83C2×C22⋊C811yescharacteristic
187C22×C83C2×C22⋊C811yescharacteristic
188C22×C83C2×C22⋊C811yescharacteristic
189C22×C83#40711yescharacteristic
190C22×C83#40711yescharacteristic
191C22×C83#40711yescharacteristic
192C22×C83#41012yes—
193C22×C83#41312yes—
194C4×C83—24no—
195C4×C83—24no—
196C4×C83—24no—
197C4×C83—42no—
198C4×C83—42no—
199C4×C83—42no—
200#8300C8311yesmaximal, characteristic, direct product factor, Sylow
201C2×C83—21no—
202C2×C83—21no—
203C2×C83—21no—
204C2×C83—21no—
205C2×C83—44no—
206C2×C83#145812yes—
207C2×C83#146412yes—
208C2×C83#177911yescharacteristic
209C2×C83#178211yescharacteristic
210C2×C83#178511yescharacteristic
211C22×#117C2×C8311yescharacteristic
212#1817C2×C8312yescomplemented
213C2×#324C2×C8312yescomplemented
214C2×#323C2×C8312yes—
215C83#830011yescharacteristic, direct product factor, Sylow
216C24×C4C4×C8311yescharacteristic
217C2×#117—24no—
218C2×#117—24no—
219C2×#117—24no—
220C22×C4⋊C4C4×C8311yescharacteristic
221C22×C4⋊C4C22×C8311yescharacteristic
222C22×C22⋊C4C4×C8311yescharacteristic
223C22×C22⋊C4C22×C8311yescharacteristic
224C22×C42C4×C8311yescharacteristic
225C22×C42C22×C8311yescharacteristic
226C2×C22⋊C8C22×C8314yes—
227#336—24no—
228#324—24no—
229#323—24no—
230C25C2×C4×C8311yescharacteristic
231C23×C4—22no—
232C23×C4—22no—
233C23×C4—22no—
234C23×C4—22no—
235C23×C4—22no—
236C23×C4—22no—
237C23×C4C8×C8314yescomplemented
238C23×C4C2×C4×C8311yescharacteristic
239C23×C4C2×C4×C8311yescharacteristic
240C23×C4C2×C4×C8311yescharacteristic
241C23×C4C2×C4×C8311yescharacteristic
242C23×C4C2×C4×C8311yescharacteristic
243C23×C4C23×C8311yescharacteristic
244C22×C8—24no—
245#117—232no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—22no—
251C2×C4⋊C4D4×C8314yes—
252C2×C22⋊C4—22no—
253C2×C22⋊C4—22no—
254C2×C22⋊C4—22no—
255C2×C22⋊C4—22no—
256C2×C22⋊C4—22no—
257C2×C22⋊C4D4×C8314yes—
258C2×C42—22no—
259C2×C42—22no—
260C2×C42—22no—
261C2×C42—22no—
262C2×C42—22no—
263C2×C42D4×C8314yes—
264C22⋊C8—24no—
265C22⋊C8—24no—
266C24—21no—
267C24—21no—
268C24—21no—
269C24—21no—
270C24—42no—
271C24—42no—
272C24C22⋊C4×C8312yes—
273C24C2×C8×C8312yes—
274C24M16×C8312yes—
275C24C22×C4×C8311yescharacteristic
276C22×C4—28no—
277C22×C4—28no—
278C22×C4—28no—
279C22×C4—28no—
280C22×C4—28no—
281C22×C4—21no—
282C22×C4—21no—
283C22×C4—21no—
284C22×C4—21no—
285C22×C4—21no—
286C22×C4—21no—
287C22×C4—21no—
288C22×C4—21no—
289C22×C4—28no—
290C22×C4—28no—
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—21no—
294C22×C4—21no—
295C22×C4—21no—
296C22×C4—21no—
297C22×C4—42no—
298C22×C4—42no—
299C22×C4—42no—
300C22×C4—42no—
301C22×C4C22⋊C4×C8312yes—
302C22×C4C22⋊C4×C8312yes—
303C22×C4C22⋊C4×C8312yes—
304C22×C4C22⋊C4×C8312yes—
305C22×C4C22⋊C4×C8312yes—
306C22×C4C2×C8×C8312yes—
307C22×C4C2×C8×C8312yes—
308C22×C4M16×C8312yes—
309C22×C4M16×C8312yes—
310C22×C4C2×D4×C8312yes—
311C22×C4C2×D4×C8312yes—
312C22×C4C2×D4×C8312yes—
313C2×C8—48no—
314C2×C8—44no—
315C4⋊C4—28no—
316C4⋊C4—28no—
317C22⋊C4—28no—
318C22⋊C4—28no—
319C42—28no—
320C42—28no—
321C23—22no—
322C23—22no—
323C23—22no—
324C23—22no—
325C23—22no—
326C23—22no—
327C23—22no—
328C23—22no—
329C23—42no—
330C23—42no—
331C23—44no—
332C23—44no—
333C23—42no—
334C23—42no—
335C23—44no—
336C23—44no—
337C23—42no—
338C23—42no—
339C23C22⋊C8×C8314yes—
340C23C2×C22⋊C4×C8311yescharacteristic
341C23C2×C22⋊C4×C8311yescharacteristic
342C23C2×C22⋊C4×C8311yescharacteristic
343C23C22×C8×C8311yescharacteristic
344C23C2×M16×C8311yescharacteristic
345C23C2×M16×C8311yescharacteristic
346C23C2×M16×C8311yescharacteristic
347C2×C4—232no—
348C2×C4—22no—
349C2×C4—22no—
350C2×C4—22no—
351C2×C4—22no—
352C2×C4—22no—
353C2×C4—22no—
354C2×C4—22no—
355C2×C4—22no—
356C2×C4—22no—
357C2×C4—22no—
358C2×C4—22no—
359C2×C4—22no—
360C2×C4—24no—
361C2×C4—22no—
362C2×C4—22no—
363C2×C4—22no—
364C2×C4—24no—
365C2×C4—42no—
366C2×C4—42no—
367C2×C4—42no—
368C2×C4—42no—
369C2×C4—42no—
370C2×C4—42no—
371C2×C4—42no—
372C2×C4—42no—
373C2×C4—42no—
374C2×C4—42no—
375C2×C4C22⋊C8×C8314yes—
376C2×C4C22⋊C8×C8314yes—
377C2×C4#120×C8314yes—
378C8—84no—
379C22—21no—
380C22—21no—
381C22—21no—
382C22—21no—
383C22—22no—
384C22—22no—
385C22—22no—
386C22—22no—
387C22—24no—
388C22—44no—
389C22—44no—
390C22—44no—
391C22—44no—
392C22—42no—
393C22—42no—
394C22—44no—
395C22—44no—
396C22#379×C8311yescharacteristic
397C22C2×C22⋊C8×C8311yescharacteristic
398C22C2×C22⋊C8×C8311yescharacteristic
399C22C2×C22⋊C8×C8311yescharacteristic
400C22#407×C8311yescharacteristic
401C22#407×C8311yescharacteristic
402C22#407×C8311yescharacteristic
403C22#410×C8312yes—
404C22#413×C8312yes—
405C4—24no—
406C4—24no—
407C4—24no—
408C4—42no—
409C4—42no—
410C4—42no—
411C2—21no—
412C2—21no—
413C2—21no—
414C2—21no—
415C2—44no—
416C2C83×#145812yes—
417C2C83×#146412yes—
418C2C83×#177911yescharacteristic
419C2C83×#178211yescharacteristic
420C2C83×#178511yescharacteristic
subgroupG/H
center, socleC23×C83C2×C22⋊C4
commutatorC23C22×C8×C83
Frattini, cosocleC23×C4C23×C83
prime radicalC25×C83C2×C4
LCSG ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C23×C83 ◁ C22×C42×C83 ◁ G

Complements

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

NH ≅ G/Ntype
2. C83×#1817C2semidirect
3. C2×#324×C83C2semidirect
26. C23×C4×C83C8semidirect
200. #8300C83direct (N × H)
212. #1817C2×C83semidirect
213. C2×#324C2×C83semidirect
215. C83#8300direct (N × H)
237. C23×C4C8×C83semidirect

Maximal subgroups

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

subgroupindex
C22×#117×C832
C83×#18172
C2×#324×C832
C2×#323×C832
#830083