Finite Group Databasebeta

← C8×AΣL(1,𝔽16) #128449

Subgroups — C8×AΣL(1,𝔽16) #128449

Order 512.

627 conjugacy classes of subgroups; only the 122 normal subgroups are shown.

normal subgroups only

This group has 627 conjugacy classes of subgroups — too many to draw. Only its 122 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
1C8×#120C211yesmaximal, characteristic
2C4×AΣL(1,𝔽16)C211yesmaximal, characteristic
3#11471C211yesmaximal, characteristic
4C8×#100C211yesmaximal, characteristic, complemented
5C8×#99C211yesmaximal, characteristic, complemented
6#7204C211yesmaximal, characteristic, complemented
7#7203C211yesmaximal, characteristic, complemented
8C24×C8C411yescharacteristic, complemented
9C2×C8×D4—21no—
10C2×C8×D4C2211yescharacteristic
11C4×#120C2211yescharacteristic
12C2×AΣL(1,𝔽16)C412yes—
13#1869—21no—
14#1869C2211yescharacteristic
15#1850—21no—
16#1850C411yescharacteristic, complemented
17#1779C411yescharacteristic, complemented
18C4×#100C2211yescharacteristic
19C4×#99C2211yescharacteristic
20C8×C22⋊C4—21no—
21C8×C22⋊C4—21no—
22C8×C22⋊C4C411yescharacteristic, complemented
23#1460C2211yescharacteristic
24#1459C2211yescharacteristic
25C8×M16—21no—
26#1390C412yes—
27#1326C412yescomplemented
28#1316C412yescomplemented
29#1283—21no—
30#1282—21no—
31C24×C4C2×C411yescharacteristic
32C23×C8—21no—
33C23×C8—21no—
34C23×C8—44no—
35C23×C8—41no—
36C23×C8—41no—
37C23×C8C2×C411yescharacteristic
38C23×C8C2×C411yescharacteristic
39C23×C8D411yescharacteristic
40C2×#120C2×C411yescharacteristic
41C2×C4×D4—21no—
42C2×C4×D4C2311yescharacteristic
43C8×D4—41no—
44C8×D4—41no—
45C8×D4—44no—
46#431—21no—
47C2×C4⋊C8—21no—
48C2×C4⋊C8D411yescharacteristic
49C2×#100C2×C412yes—
50C2×#99C2×C412yes—
51C2×C22⋊C8—21no—
52C2×C22⋊C8—21no—
53C2×C22⋊C8—21no—
54C2×C22⋊C8—21no—
55C2×C22⋊C8C2×C411yescharacteristic
56C2×C22⋊C8C2×C411yescharacteristic
57C2×C22⋊C8D411yescharacteristic
58C4×M16—21no—
59C2×C4×C8—21no—
60C2×C4×C8—41no—
61C2×C4×C8D411yescharacteristic
62#386—21no—
63#386C2×C411yescharacteristic
64#385—21no—
65#385C2×C411yescharacteristic
66#379C2×C411yescharacteristic
67C4×C22⋊C4—21no—
68C4×C22⋊C4—21no—
69C4×C22⋊C4C2×C411yescharacteristic
70AΣL(1,𝔽16)C818yesdirect product factor
71#343—21no—
72#342—21no—
73#336—21no—
74#336—21no—
75#336—21no—
76#336—41no—
77#324—21no—
78#323—21no—
79#323C2×C412yes—
80#323C2×C412yes—
81#322—41no—
82C82—41no—
83C25C4211yescharacteristic
84C22×D4—21no—
85C22×D4C22×C411yescharacteristic
86C23×C4—21no—
87C23×C4—21no—
88C23×C4—42no—
89C23×C4—41no—
90C23×C4—42no—
91C23×C4—41no—
92C23×C4C4211yescharacteristic
93C23×C4C22×C411yescharacteristic
94C23×C4C2×D411yescharacteristic
95C2×M16—22no—
96C22×C8—24no—
97C22×C8—21no—
98C22×C8—21no—
99C22×C8—21no—
100C22×C8—21no—
101C22×C8—21no—
102C22×C8—21no—
103C22×C8—42no—
104C22×C8—42no—
105C22×C8—42no—
106C22×C8—42no—
107C22×C8—42no—
108C22×C8—42no—
109C22×C8—42no—
110C22×C8—44no—
111C22×C8—44no—
112C22×C8—44no—
113C22×C8—41no—
114C22×C8—41no—
115C22×C8—41no—
116C22×C8—44no—
117C22×C8—41no—
118C22×C8C22⋊C411yescharacteristic
119C22×C8C22⋊C411yescharacteristic
120C22×C8C22⋊C411yescharacteristic
121C22×C8C22⋊C411yescharacteristic
122#120—22no—
123#120C2×C814yes—
124C4×D4—41no—
125C4×D4—41no—
126C4×D4—42no—
127C4×D4—42no—
128C2×C4⋊C4—21no—
129C2×C4⋊C4#4111yescharacteristic
130C2×C22⋊C4—21no—
131C2×C22⋊C4—22no—
132C2×C22⋊C4—21no—
133C2×C22⋊C4—21no—
134C2×C22⋊C4—21no—
135C2×C22⋊C4C4211yescharacteristic
136C2×C22⋊C4C4211yescharacteristic
137C2×C22⋊C4C22×C411yescharacteristic
138C2×C22⋊C4#4111yescharacteristic
139C2×C42—21no—
140C2×C42—41no—
141C2×C42C2×D411yescharacteristic
142C4⋊C8—41no—
143C4⋊C8—41no—
144C4⋊C8—42no—
145#100—22no—
146#100C2×C814yescomplemented
147#99—22no—
148#99C2×C814yescomplemented
149C22⋊C8—22no—
150C22⋊C8—22no—
151C22⋊C8—41no—
152C22⋊C8—41no—
153C22⋊C8—42no—
154C22⋊C8—42no—
155C22⋊C8—41no—
156C22⋊C8—42no—
157C22⋊C8—41no—
158#97—41no—
159C4×C8—21no—
160C4×C8—21no—
161C4×C8—41no—
162C4×C8—41no—
163C4×C8—42no—
164C4×C8—82no—
165#95—21no—
166#95—21no—
167#95—21no—
168#95—41no—
169C24—21no—
170C24—21no—
171C24—42no—
172C24—41no—
173C24—42no—
174C24—41no—
175C24C4×C812yescomplemented
176C24C2×C4211yescharacteristic
177C24C4×D411yescharacteristic
178C2×D4—24no—
179C2×D4—41no—
180C2×D4—41no—
181C2×D4—42no—
182C2×D4—42no—
183C2×D4C22×C812yes—
184C2×D4#13112yes—
185C22×C4—22no—
186C22×C4—22no—
187C22×C4—21no—
188C22×C4—21no—
189C22×C4—21no—
190C22×C4—21no—
191C22×C4—21no—
192C22×C4—21no—
193C22×C4—21no—
194C22×C4—41no—
195C22×C4—41no—
196C22×C4—42no—
197C22×C4—42no—
198C22×C4—41no—
199C22×C4—42no—
200C22×C4—41no—
201C22×C4—41no—
202C22×C4—41no—
203C22×C4—42no—
204C22×C4—42no—
205C22×C4—44no—
206C22×C4—44no—
207C22×C4—42no—
208C22×C4—41no—
209C22×C4—41no—
210C22×C4—41no—
211C22×C4—42no—
212C22×C4—42no—
213C22×C4—42no—
214C22×C4C2×C22⋊C411yescharacteristic
215C22×C4#11711yescharacteristic
216C22×C4C4×D411yescharacteristic
217C22×C4C4×D411yescharacteristic
218C22×C4C4×D411yescharacteristic
219M16—24no—
220M16—42no—
221C2×C8—22no—
222C2×C8—22no—
223C2×C8—21no—
224C2×C8—21no—
225C2×C8—21no—
226C2×C8—44no—
227C2×C8—42no—
228C2×C8—42no—
229C2×C8—41no—
230C2×C8—48no—
231C2×C8—42no—
232C2×C8—41no—
233C2×C8—42no—
234C2×C8—44no—
235C2×C8—42no—
236C2×C8—41no—
237C2×C8—41no—
238C2×C8—41no—
239C2×C8—42no—
240C2×C8—41no—
241C2×C8—42no—
242C2×C8—42no—
243C2×C8—41no—
244C2×C8—41no—
245C2×C8—81no—
246C2×C8—81no—
247C2×C8—81no—
248C2×C8#9911yescharacteristic
249C2×C8#9911yescharacteristic
250C4⋊C4—41no—
251C4⋊C4—42no—
252C22⋊C4—21no—
253C22⋊C4—24no—
254C22⋊C4—24no—
255C22⋊C4—41no—
256C22⋊C4—41no—
257C22⋊C4—42no—
258C22⋊C4—42no—
259C22⋊C4—42no—
260C22⋊C4—42no—
261C22⋊C4—41no—
262C22⋊C4C4×C812yescomplemented
263C42—21no—
264C42—21no—
265C42—41no—
266C42—41no—
267C42—81no—
268C42—81no—
269C23—22no—
270C23—22no—
271C23—22no—
272C23—21no—
273C23—21no—
274C23—21no—
275C23—21no—
276C23—41no—
277C23—44no—
278C23—42no—
279C23—41no—
280C23—41no—
281C23—41no—
282C23—44no—
283C23—42no—
284C23—41no—
285C23—41no—
286C23—42no—
287C23—42no—
288C23—42no—
289C23—48no—
290C23—42no—
291C23—41no—
292C23—41no—
293C23C4×C22⋊C411yescharacteristic
294C23C2×C4×C811yescharacteristic
295C23#40511yescharacteristic
296C23C8×D412yes—
297D4—42no—
298D4—42no—
299D4—48no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—22no—
305C2×C4—22no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—42no—
310C2×C4—42no—
311C2×C4—42no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—44no—
316C2×C4—44no—
317C2×C4—42no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—42no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—42no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—41no—
327C2×C4—41no—
328C2×C4—42no—
329C2×C4—41no—
330C2×C4—44no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—81no—
334C2×C4—82no—
335C2×C4—82no—
336C2×C4C2×#9911yescharacteristic
337C2×C4#41011yescharacteristic
338C2×C4C8×D412yes—
339C8—21no—
340C8—44no—
341C8—42no—
342C8—41no—
343C8—41no—
344C8—41no—
345C8—81no—
346C8—82no—
347C8—82no—
348C8AΣL(1,𝔽16)12yesdirect product factor
349C22—21no—
350C22—21no—
351C22—24no—
352C22—22no—
353C22—21no—
354C22—21no—
355C22—42no—
356C22—41no—
357C22—41no—
358C22—42no—
359C22—42no—
360C22—42no—
361C22—42no—
362C22—44no—
363C22—44no—
364C22—44no—
365C22—44no—
366C22—42no—
367C22—41no—
368C22—41no—
369C22—41no—
370C22—42no—
371C22—41no—
372C22—42no—
373C22—81no—
374C22C8×C22⋊C411yescharacteristic
375C22C4×#9911yescharacteristic
376C22#175711yescharacteristic
377C4—21no—
378C4—22no—
379C4—42no—
380C4—41no—
381C4—42no—
382C4—41no—
383C4—41no—
384C4—81no—
385C4—84no—
386C4C2×AΣL(1,𝔽16)11yescharacteristic
387C4#212011yescharacteristic
388C2—21no—
389C2—21no—
390C2—42no—
391C2—44no—
392C2—41no—
393C2—41no—
394C2C8×#9911yescharacteristic
395C2C4×AΣL(1,𝔽16)11yescharacteristic
396C2#1262311yescharacteristic
subgroupG/H
centerC2×C8#99
commutatorC23C2×C4×C8
Frattini, cosocleC2×C4×D4C23
socleC22C4×#99
prime radicalC2×#120C2×C4
LCSG ▷ C23 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2×C8 ◁ C22×C8 ◁ C2×C8×D4 ◁ G
LECSG ▷ C2×C4×D4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. C8×#100C2semidirect
5. C8×#99C2semidirect
6. #7204C2semidirect
7. #7203C2semidirect
8. C24×C8C4semidirect
16. #1850C4semidirect
17. #1779C4semidirect
22. C8×C22⋊C4C4semidirect
27. #1326C4semidirect
28. #1316C4semidirect
70. AΣL(1,𝔽16)C8direct (N × H)
146. #100C2×C8semidirect
148. #99C2×C8semidirect
175. C24C4×C8semidirect
262. C22⋊C4C4×C8semidirect
348. C8AΣL(1,𝔽16)direct (N × H)

Maximal subgroups

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

subgroupindex
C8×#1202
C4×AΣL(1,𝔽16)2
#114712
C8×#1002
C8×#992
#72042
#72032