Finite Group Databasebeta

← #25094

Subgroups — #25094

Order 256.

586 conjugacy classes of subgroups.

The subgroup lattice has 404 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#468C211yesmaximal, characteristic, complemented
2C2×#461C211yesmaximal, characteristic, complemented
3C2×#449C211yesmaximal, characteristic
4C2×#417C211yesmaximal, characteristic, complemented
5#2432C211yesmaximal, characteristic, complemented
6#2361C211yesmaximal, characteristic, complemented
7#2067C211yesmaximal, characteristic, complemented
8#2055C211yesmaximal, characteristic, complemented
9#2043C211yesmaximal, characteristic, complemented
10#2012C211yesmaximal, characteristic, complemented
11#1983C211yesmaximal, characteristic, complemented
12#1928C211yesmaximal, characteristic, complemented
13#1879C211yesmaximal, characteristic, complemented
14#1872C211yesmaximal, characteristic, complemented
15#1851C211yesmaximal, characteristic, complemented
16C22×#41C2211yescharacteristic
17C22×SD16C2211yescharacteristic
18C22×SD16C2211yescharacteristic
19C22×D8C2211yescharacteristic
20C22×M16C2211yescharacteristic
21C2×C4⋊Q8C2211yescharacteristic
22C2×#124C2211yescharacteristic
23C2×C22⋊Q8C2211yescharacteristic
24C2×#121C2211yescharacteristic
25C2×C4×D4—21no—
26C2×C4×D4C2211yescharacteristic
27C22×C4⋊C4C2211yescharacteristic
28#468—24no—
29#461—24no—
30#449—24no—
31C2×#107C2211yescharacteristic
32C2×C4⋊C8C2211yescharacteristic
33C2×C4⋊C8C2211yescharacteristic
34#417—24no—
35C2×#103C2211yescharacteristic
36C2×#103C2211yescharacteristic
37C2×#103C2211yescharacteristic
38C2×#103C2211yescharacteristic
39C2×#103C2211yescharacteristic
40C2×#103C2211yescharacteristic
41C2×#102C2211yescharacteristic
42C2×#102C2211yescharacteristic
43C2×#102C2211yescharacteristic
44C2×#102C2211yescharacteristic
45C2×C22⋊C8C2211yescharacteristic
46#394—21no—
47#394—21no—
48#391C2211yescharacteristic
49#390C2211yescharacteristic
50#390C2211yescharacteristic
51#389C2211yescharacteristic
52#388—21no—
53#386—21no—
54#385—21no—
55#385—21no—
56#385—21no—
57C4×C4⋊C4C2211yescharacteristic
58C4×C22⋊C4C2211yescharacteristic
59#340C2211yescharacteristic
60#340C2211yescharacteristic
61#336C2211yescharacteristic
62#336C2211yescharacteristic
63C2×#41—22no—
64C2×#41—22no—
65C2×#41—22no—
66C2×#41—22no—
67C2×#41—41no—
68C2×#41—41no—
69C22×Q8C2311yescharacteristic
70C22×Q8C2311yescharacteristic
71C22×D4—21no—
72C22×D4—21no—
73C22×D4C2311yescharacteristic
74C22×D4C2311yescharacteristic
75C23×C4—21no—
76C23×C4—21no—
77C23×C4—21no—
78C23×C4C2311yescharacteristic
79C2×SD16—22no—
80C2×SD16—22no—
81C2×SD16—22no—
82C2×SD16—22no—
83C2×SD16—41no—
84C2×SD16—41no—
85C2×D8—22no—
86C2×D8—22no—
87C2×D8—41no—
88C2×M16—22no—
89C2×M16—22no—
90C2×M16—41no—
91C22×C8C2311yescharacteristic
92C22×C8C2311yescharacteristic
93C22×C8C2311yescharacteristic
94C22×C8C2311yescharacteristic
95C4⋊Q8—22no—
96C4⋊Q8—41no—
97#124—22no—
98#124—41no—
99C22⋊Q8—22no—
100C22⋊Q8—22no—
101#121—22no—
102#121D414yes—
103C4×D4—22no—
104C4×D4—41no—
105C4×D4—44no—
106C2×C4⋊C4—22no—
107C2×C4⋊C4—21no—
108C2×C4⋊C4—21no—
109C2×C4⋊C4—21no—
110C2×C4⋊C4—21no—
111C2×C4⋊C4—21no—
112C2×C4⋊C4—41no—
113C2×C4⋊C4C2311yescharacteristic
114C2×C4⋊C4C2311yescharacteristic
115C2×C4⋊C4C2311yescharacteristic
116C2×C4⋊C4C2311yescharacteristic
117C2×C22⋊C4—21no—
118C2×C22⋊C4—21no—
119C2×C22⋊C4—21no—
120C2×C22⋊C4—21no—
121C2×C22⋊C4—21no—
122C2×C22⋊C4—21no—
123C2×C22⋊C4—21no—
124C2×C22⋊C4—21no—
125C2×C22⋊C4—21no—
126C2×C22⋊C4D411yescharacteristic
127C2×C22⋊C4D411yescharacteristic
128C2×C42—21no—
129C2×C42C2311yescharacteristic
130C2×C42C2311yescharacteristic
131#107—22no—
132C4⋊C8—22no—
133C4⋊C8—41no—
134#103—22no—
135#103—22no—
136#103—22no—
137#103—22no—
138#103—41no—
139#103D414yescomplemented
140#102—22no—
141#102—22no—
142#102—22no—
143#102—22no—
144C22⋊C8—22no—
145#95—21no—
146#95—21no—
147#95—21no—
148#95—21no—
149#95D411yescharacteristic
150#95D411yescharacteristic
151C24—21no—
152C24—21no—
153C24C2×D411yescharacteristic
154#41—28no—
155#41—44no—
156#41—44no—
157#41—44no—
158C2×Q8—21no—
159C2×Q8—21no—
160C2×Q8—21no—
161C2×Q8—21no—
162C2×Q8—21no—
163C2×Q8—21no—
164C2×Q8—21no—
165C2×Q8—21no—
166C2×Q8—21no—
167C2×Q8—21no—
168C2×Q8—21no—
169C2×Q8C2×D412yes—
170C2×D4—24no—
171C2×D4—21no—
172C2×D4—21no—
173C2×D4—21no—
174C2×D4—21no—
175C2×D4—21no—
176C2×D4—21no—
177C2×D4—21no—
178C2×D4—21no—
179C2×D4—21no—
180C2×D4—21no—
181C2×D4—42no—
182C2×D4—42no—
183C2×D4—42no—
184C2×D4—42no—
185C2×D4—42no—
186C2×D4C2×D412yes—
187C2×D4#4112yes—
188C22×C4—21no—
189C22×C4—21no—
190C22×C4—21no—
191C22×C4—21no—
192C22×C4—21no—
193C22×C4—21no—
194C22×C4—21no—
195C22×C4—21no—
196C22×C4—21no—
197C22×C4—21no—
198C22×C4—42no—
199C22×C4—42no—
200C22×C4—42no—
201C22×C4—42no—
202C22×C4—42no—
203C22×C4—42no—
204C22×C4—42no—
205C22×C4—81no—
206C22×C4—81no—
207C22×C4C2×D411yescharacteristic
208C22×C4C2×D411yescharacteristic
209C22×C4C2×D411yescharacteristic
210C22×C4C2×D411yescharacteristic
211C22×C4C2×D412yes—
212C22×C4C2×D411yescharacteristic
213C22×C4#4112yes—
214C22×C4C2411yescharacteristic
215SD16—44no—
216SD16—44no—
217D8—44no—
218M16—44no—
219C2×C8—21no—
220C2×C8—21no—
221C2×C8—21no—
222C2×C8—21no—
223C2×C8—21no—
224C2×C8—21no—
225C2×C8—21no—
226C2×C8—21no—
227C2×C8—21no—
228C2×C8—21no—
229C2×C8—21no—
230C2×C8C2×D412yes—
231C4⋊C4—21no—
232C4⋊C4—21no—
233C4⋊C4—21no—
234C4⋊C4—21no—
235C4⋊C4—21no—
236C4⋊C4—21no—
237C4⋊C4—42no—
238C4⋊C4—42no—
239C4⋊C4—42no—
240C4⋊C4—42no—
241C4⋊C4—42no—
242C4⋊C4C2×D412yes—
243C4⋊C4C2×D412yes—
244C22⋊C4—24no—
245C22⋊C4—22no—
246C22⋊C4—41no—
247C22⋊C4—42no—
248C22⋊C4—42no—
249C22⋊C4—42no—
250C22⋊C4—42no—
251C22⋊C4—42no—
252C22⋊C4—42no—
253C22⋊C4—42no—
254C22⋊C4—81no—
255C42—21no—
256C42—21no—
257C42—21no—
258C42—21no—
259C42—42no—
260C23—21no—
261C23—21no—
262C23—21no—
263C23—21no—
264C23—21no—
265C23—21no—
266C23—22no—
267C23—22no—
268C23—41no—
269C23—41no—
270C23—41no—
271C23—41no—
272C23—41no—
273C23—41no—
274C23—41no—
275C23—41no—
276C23—41no—
277C23—41no—
278C23—41no—
279C23—41no—
280C23#12112yes—
281C23C22×D411yescharacteristic
282Q8—22no—
283Q8—22no—
284Q8—41no—
285Q8—41no—
286Q8—41no—
287Q8—41no—
288Q8—41no—
289Q8—41no—
290D4—22no—
291D4—22no—
292D4—22no—
293D4—22no—
294D4—48no—
295D4—48no—
296D4—41no—
297D4—41no—
298D4—41no—
299D4—41no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—22no—
305C2×C4—22no—
306C2×C4—22no—
307C2×C4—22no—
308C2×C4—22no—
309C2×C4—21no—
310C2×C4—22no—
311C2×C4—22no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—48no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—41no—
327C2×C4—41no—
328C2×C4—41no—
329C2×C4—41no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—41no—
334C2×C4—41no—
335C2×C4—41no—
336C2×C4—84no—
337C2×C4—84no—
338C2×C4#12112yes—
339C2×C4AGL(1,ℤ/8)12yes—
340C2×C4C22×D411yescharacteristic
341C2×C4C22×D411yescharacteristic
342C2×C4C2×#4111yescharacteristic
343C2×C4C2×#4111yescharacteristic
344C2×C4H(5,𝔽2)11yescharacteristic
345C2×C4H(5,𝔽2)11yescharacteristic
346C8—22no—
347C8—41no—
348C8—41no—
349C8—41no—
350C22—22no—
351C22—22no—
352C22—22no—
353C22—41no—
354C22—41no—
355C22—41no—
356C22—41no—
357C22—42no—
358C22—42no—
359C22—42no—
360C22—42no—
361C22—42no—
362C22—42no—
363C22—81no—
364C22—81no—
365C22—81no—
366C22—81no—
367C22—81no—
368C22—81no—
369C22—81no—
370C22—81no—
371C22C2×#12111yescharacteristic
372C22D4211yescharacteristic
373C22#54611yescharacteristic
374C22C2×AGL(1,ℤ/8)11yescharacteristic
375C22#57511yescharacteristic
376C22#57711yescharacteristic
377C22#57811yescharacteristic
378C4—22no—
379C4—22no—
380C4—41no—
381C4—42no—
382C4—42no—
383C4—41no—
384C4—42no—
385C4—41no—
386C4—81no—
387C4—81no—
388C4—81no—
389C4#52511yescharacteristic
390C4#53811yescharacteristic
391C4D4211yescharacteristic
392C4#54611yescharacteristic
393C2—42no—
394C2—82no—
395C2—82no—
396C2#242411yescharacteristic
397C2#303611yescharacteristic
398C2#304611yescharacteristic
399C2#326511yescharacteristic
400C2#327711yescharacteristic
401C2#330711yescharacteristic
402C2#333011yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC2×#449C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#468C2semidirect
2. C2×#461C2semidirect
4. C2×#417C2semidirect
5. #2432C2semidirect
6. #2361C2semidirect
7. #2067C2semidirect
8. #2055C2semidirect
9. #2043C2semidirect
10. #2012C2semidirect
11. #1983C2semidirect
12. #1928C2semidirect
13. #1879C2semidirect
14. #1872C2semidirect
15. #1851C2semidirect
139. #103D4semidirect

Maximal subgroups

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

subgroupindex
C2×#4682
C2×#4612
C2×#4492
C2×#4172
#24322
#23612
#20672
#20552
#20432
#20122
#19832
#19282
#18792
#18722
#18512