Finite Group Databasebeta

← C83×#8627 #35236451

Subgroups — C83×#8627 #35236451

Order 21248.

456 conjugacy classes of subgroups.

The subgroup lattice has 380 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C83×#2929C211yesmaximal, characteristic
2C83×#1810C211yesmaximal, characteristic, complemented
3C83×#1808C211yesmaximal, characteristic, complemented
4C83×#1513C212yesmaximal
5C83×#1305C212yesmaximal, complemented
6C2×#131×C83C411yescharacteristic, complemented
7C23×C8×C83C2211yescharacteristic
8#525×C83C2211yescharacteristic
9#435×C83—21no—
10#435×C83—21no—
11C8×D4×C83—21no—
12C8×D4×C83—21no—
13C2×#108×C83—21no—
14C2×#107×C83—21no—
15#424×C83C411yescharacteristic, complemented
16#408×C83C411yescharacteristic, complemented
17#407×C83C2212yes—
18#407×C83C2211yescharacteristic
19#405×C83C411yescharacteristic, complemented
20#380×C83C2212yes—
21#341×C83—21no—
22#340×C83—21no—
23#328×C83—22no—
24#327×C83—22no—
25C2×#41×C83C2×C411yescharacteristic
26C23×C4×C83C2311yescharacteristic
27#131×C83—42no—
28C2×M16×C83—21no—
29C2×M16×C83—22no—
30C2×M16×C83C2×C411yescharacteristic
31C2×M16×C83C2×C411yescharacteristic
32C22×C8×C83—21no—
33C22×C8×C83—42no—
34C22×C8×C83C2×C411yescharacteristic
35C22×C8×C83C2×C411yescharacteristic
36C22×C8×C83D411yescharacteristic
37C22×C8×C83D411yescharacteristic
38C22×C8×C83D411yescharacteristic
39C22×C8×C83D411yescharacteristic
40#123×C83—21no—
41C22⋊Q8×C83D411yescharacteristic
42C22⋊Q8×C83Q811yescharacteristic
43#121×C83D411yescharacteristic
44#121×C83Q811yescharacteristic
45#120×C83—21no—
46C4×D4×C83—21no—
47C4×D4×C83—21no—
48#117×C83C2×C411yescharacteristic
49C2×C4⋊C4×C83—22no—
50C2×C22⋊C4×C83—22no—
51#108×C83—42no—
52#107×C83—42no—
53C4⋊C8×C83—21no—
54C4⋊C8×C83—21no—
55C22⋊C8×C83—21no—
56C22⋊C8×C83—21no—
57C22⋊C8×C83—21no—
58C22⋊C8×C83—21no—
59C22⋊C8×C83—22no—
60C22⋊C8×C83—22no—
61#97×C83—21no—
62C4×C8×C83—21no—
63#95×C83—22no—
64C24×C83#4111yescharacteristic
65#41×C83—41no—
66#41×C83—41no—
67C2×Q8×C83C4⋊C411yescharacteristic
68C2×D4×C83—21no—
69C2×D4×C83—21no—
70C2×D4×C83—21no—
71C2×D4×C83C4⋊C411yescharacteristic
72C22×C4×C83—21no—
73C22×C4×C83—41no—
74C22×C4×C83—41no—
75C22×C4×C83—42no—
76C22×C4×C83C22×C411yescharacteristic
77C22×C4×C83C2×D411yescharacteristic
78C22×C4×C83C2×D411yescharacteristic
79C22×C4×C83C2×D411yescharacteristic
80C22×C4×C83C2×Q811yescharacteristic
81C22×C4×C83#4111yescharacteristic
82M16×C83—42no—
83M16×C83—41no—
84M16×C83—41no—
85M16×C83—42no—
86M16×C83—42no—
87C2×C8×C83—22no—
88C2×C8×C83—22no—
89C2×C8×C83—22no—
90C2×C8×C83—21no—
91C2×C8×C83—21no—
92C2×C8×C83—41no—
93C2×C8×C83—41no—
94C2×C8×C83—41no—
95C2×C8×C83—42no—
96C2×C8×C83—42no—
97C2×C8×C83C22⋊C411yescharacteristic
98C2×C8×C83C22⋊C411yescharacteristic
99C2×C8×C83C22⋊C411yescharacteristic
100C2×C8×C83C22⋊C411yescharacteristic
101C4⋊C4×C83—21no—
102C4⋊C4×C83—21no—
103C4⋊C4×C83—42no—
104C4⋊C4×C83—42no—
105C4⋊C4×C83C4⋊C411yescharacteristic
106C4⋊C4×C83C4⋊C411yescharacteristic
107C22⋊C4×C83—21no—
108C22⋊C4×C83—21no—
109C22⋊C4×C83—21no—
110C22⋊C4×C83—21no—
111C22⋊C4×C83—21no—
112C22⋊C4×C83—82no—
113C42×C83—21no—
114C23×C83—21no—
115C23×C83—21no—
116C23×C83—21no—
117C23×C83—41no—
118C23×C83—41no—
119C23×C83#11711yescharacteristic
120C23×C83#12111yescharacteristic
121C23×C83C22⋊Q811yescharacteristic
122Q8×C83—41no—
123D4×C83—41no—
124D4×C83—42no—
125D4×C83—42no—
126D4×C83—41no—
127D4×C83—41no—
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—21no—
136C2×C4×C83—21no—
137C2×C4×C83—41no—
138C2×C4×C83—41no—
139C2×C4×C83—41no—
140C2×C4×C83—41no—
141C2×C4×C83—41no—
142C2×C4×C83—82no—
143C2×C4×C83—82no—
144C2×C4×C83—82no—
145C2×C4×C83C2×C22⋊C411yescharacteristic
146C2×C4×C83C2×C4⋊C411yescharacteristic
147C2×C4×C83#12111yescharacteristic
148C2×C4×C83C22⋊Q811yescharacteristic
149C8×C83—21no—
150C8×C83—21no—
151C8×C83—21no—
152C8×C83—41no—
153C8×C83—41no—
154C8×C83—82no—
155C8×C83#9912yes—
156C22×C83—21no—
157C22×C83—21no—
158C22×C83—21no—
159C22×C83—21no—
160C22×C83—21no—
161C22×C83—21no—
162C22×C83—21no—
163C22×C83—21no—
164C22×C83—42no—
165C22×C83—42no—
166C22×C83—41no—
167C22×C83—41no—
168C22×C83—41no—
169C22×C83#38011yescharacteristic
170C22×C83#47211yescharacteristic
171C22×C83#47311yescharacteristic
172C4×C83—21no—
173C4×C83—21no—
174C4×C83—21no—
175C4×C83—41no—
176C4×C83—41no—
177C4×C83—41no—
178C4×C83—162no—
179C4×C83C2×#9911yescharacteristic
180C4×C83#41011yescharacteristic
181#8627C8311yesmaximal, characteristic, direct product factor, Sylow
182C2×C83—21no—
183C2×C83—21no—
184C2×C83—41no—
185C2×C83—41no—
186C2×C83—81no—
187C2×C83#179911yescharacteristic
188C2×C83#181411yescharacteristic
189C2×C83#181611yescharacteristic
190#2929C2×C8311yescharacteristic
191#1810C2×C8311yescharacteristic, complemented
192#1808C2×C8311yescharacteristic, complemented
193#1513C2×C8312yes—
194#1305C2×C8312yescomplemented
195C83#862711yescharacteristic, direct product factor, Sylow
196C2×#131C4×C8311yescharacteristic, complemented
197C23×C8C22×C8311yescharacteristic
198#525C22×C8311yescharacteristic
199#435—21no—
200#435—21no—
201C8×D4—21no—
202C8×D4—21no—
203C2×#108—21no—
204C2×#107—21no—
205#424C4×C8311yescharacteristic, complemented
206#408C4×C8311yescharacteristic, complemented
207#407C22×C8312yes—
208#407C22×C8311yescharacteristic
209#405C4×C8311yescharacteristic, complemented
210#380C22×C8312yes—
211#341—21no—
212#340—21no—
213#328—22no—
214#327—22no—
215C2×#41C2×C4×C8311yescharacteristic
216C23×C4C23×C8311yescharacteristic
217#131—42no—
218C2×M16—21no—
219C2×M16—22no—
220C2×M16C2×C4×C8311yescharacteristic
221C2×M16C2×C4×C8311yescharacteristic
222C22×C8—21no—
223C22×C8—42no—
224C22×C8C2×C4×C8311yescharacteristic
225C22×C8C2×C4×C8311yescharacteristic
226C22×C8D4×C8311yescharacteristic
227C22×C8D4×C8311yescharacteristic
228C22×C8D4×C8311yescharacteristic
229C22×C8D4×C8311yescharacteristic
230#123—21no—
231C22⋊Q8D4×C8311yescharacteristic
232C22⋊Q8Q8×C8311yescharacteristic
233#121D4×C8311yescharacteristic
234#121Q8×C8311yescharacteristic
235#120—21no—
236C4×D4—21no—
237C4×D4—21no—
238#117C2×C4×C8311yescharacteristic
239C2×C4⋊C4—22no—
240C2×C22⋊C4—22no—
241#108—42no—
242#107—42no—
243C4⋊C8—21no—
244C4⋊C8—21no—
245C22⋊C8—21no—
246C22⋊C8—21no—
247C22⋊C8—21no—
248C22⋊C8—21no—
249C22⋊C8—22no—
250C22⋊C8—22no—
251#97—21no—
252C4×C8—21no—
253#95—22no—
254C24#41×C8311yescharacteristic
255#41—41no—
256#41—41no—
257C2×Q8C4⋊C4×C8311yescharacteristic
258C2×D4—21no—
259C2×D4—21no—
260C2×D4—21no—
261C2×D4C4⋊C4×C8311yescharacteristic
262C22×C4—21no—
263C22×C4—41no—
264C22×C4—41no—
265C22×C4—42no—
266C22×C4C22×C4×C8311yescharacteristic
267C22×C4C2×D4×C8311yescharacteristic
268C22×C4C2×D4×C8311yescharacteristic
269C22×C4C2×D4×C8311yescharacteristic
270C22×C4C2×Q8×C8311yescharacteristic
271C22×C4#41×C8311yescharacteristic
272M16—42no—
273M16—41no—
274M16—41no—
275M16—42no—
276M16—42no—
277C2×C8—22no—
278C2×C8—22no—
279C2×C8—22no—
280C2×C8—21no—
281C2×C8—21no—
282C2×C8—41no—
283C2×C8—41no—
284C2×C8—41no—
285C2×C8—42no—
286C2×C8—42no—
287C2×C8C22⋊C4×C8311yescharacteristic
288C2×C8C22⋊C4×C8311yescharacteristic
289C2×C8C22⋊C4×C8311yescharacteristic
290C2×C8C22⋊C4×C8311yescharacteristic
291C4⋊C4—21no—
292C4⋊C4—21no—
293C4⋊C4—42no—
294C4⋊C4—42no—
295C4⋊C4C4⋊C4×C8311yescharacteristic
296C4⋊C4C4⋊C4×C8311yescharacteristic
297C22⋊C4—21no—
298C22⋊C4—21no—
299C22⋊C4—21no—
300C22⋊C4—21no—
301C22⋊C4—21no—
302C22⋊C4—82no—
303C42—21no—
304C23—21no—
305C23—21no—
306C23—21no—
307C23—41no—
308C23—41no—
309C23#117×C8311yescharacteristic
310C23#121×C8311yescharacteristic
311C23C22⋊Q8×C8311yescharacteristic
312Q8—41no—
313D4—41no—
314D4—42no—
315D4—42no—
316D4—41no—
317D4—41no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—41no—
328C2×C4—41no—
329C2×C4—41no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—82no—
333C2×C4—82no—
334C2×C4—82no—
335C2×C4C2×C22⋊C4×C8311yescharacteristic
336C2×C4C2×C4⋊C4×C8311yescharacteristic
337C2×C4#121×C8311yescharacteristic
338C2×C4C22⋊Q8×C8311yescharacteristic
339C8—21no—
340C8—21no—
341C8—21no—
342C8—41no—
343C8—41no—
344C8—82no—
345C8#99×C8312yes—
346C22—21no—
347C22—21no—
348C22—21no—
349C22—21no—
350C22—21no—
351C22—21no—
352C22—21no—
353C22—21no—
354C22—42no—
355C22—42no—
356C22—41no—
357C22—41no—
358C22—41no—
359C22#380×C8311yescharacteristic
360C22#472×C8311yescharacteristic
361C22#473×C8311yescharacteristic
362C4—21no—
363C4—21no—
364C4—21no—
365C4—41no—
366C4—41no—
367C4—41no—
368C4—162no—
369C4C2×#99×C8311yescharacteristic
370C4#410×C8311yescharacteristic
371C2—21no—
372C2—21no—
373C2—41no—
374C2—41no—
375C2—81no—
376C2C83×#179911yescharacteristic
377C2C83×#181411yescharacteristic
378C2C83×#181611yescharacteristic
subgroupG/H
center, socleC22×C83#380
commutatorC22×C4C22×C4×C83
Frattini, cosocleC23×C4C23×C83
prime radical#525×C83C22
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C83 ◁ C23×C4×C83 ◁ G

Complements

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

NH ≅ G/Ntype
2. C83×#1810C2semidirect
3. C83×#1808C2semidirect
5. C83×#1305C2semidirect
6. C2×#131×C83C4semidirect
15. #424×C83C4semidirect
16. #408×C83C4semidirect
19. #405×C83C4semidirect
181. #8627C83direct (N × H)
191. #1810C2×C83semidirect
192. #1808C2×C83semidirect
194. #1305C2×C83semidirect
195. C83#8627direct (N × H)
196. C2×#131C4×C83semidirect
205. #424C4×C83semidirect
206. #408C4×C83semidirect
209. #405C4×C83semidirect

Maximal subgroups

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

subgroupindex
C83×#29292
C83×#18102
C83×#18082
C83×#15132
C83×#13052
#862783