Finite Group Databasebeta

← #30607

Subgroups — #30607

Order 256.

468 conjugacy classes of subgroups.

The subgroup lattice has 375 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#469C211yesmaximal, characteristic
2C2×#463C211yesmaximal, characteristic, complemented
3#2427C211yesmaximal, characteristic, complemented
4#2422C211yesmaximal, characteristic, complemented
5#2096C211yesmaximal, characteristic, complemented
6#2088C211yesmaximal, characteristic, complemented
7#2055C211yesmaximal, characteristic, complemented
8#2042C211yesmaximal, characteristic, complemented
9#2034C211yesmaximal, characteristic, complemented
10#2000C211yesmaximal, characteristic, complemented
11#1894C211yesmaximal, characteristic, complemented
12#1879C211yesmaximal, characteristic, complemented
13#1864C211yesmaximal, characteristic, complemented
14#1851C211yesmaximal, characteristic, complemented
15#1802C211yesmaximal, characteristic, complemented
16C22×SD16C2211yescharacteristic
17C22×D8C2211yescharacteristic
18C22×M16C2211yescharacteristic
19C2×#124C2211yescharacteristic
20C2×#123—21no—
21C2×C22⋊Q8C2211yescharacteristic
22C2×#121C2211yescharacteristic
23C2×#121C2211yescharacteristic
24C2×C4×D4C2211yescharacteristic
25C2×C4×D4C2211yescharacteristic
26#469—24no—
27#463—24no—
28C2×#106C2211yescharacteristic
29C2×#106C2211yescharacteristic
30C2×C4⋊C8C2211yescharacteristic
31C2×C4⋊C8C2211yescharacteristic
32C2×#103C2211yescharacteristic
33C2×#103C2211yescharacteristic
34C2×#103C2211yescharacteristic
35C2×#102C2211yescharacteristic
36C2×#102C2211yescharacteristic
37C2×#102C2211yescharacteristic
38C2×#102C2211yescharacteristic
39C2×#102C2211yescharacteristic
40C2×C22⋊C8C2211yescharacteristic
41#397—21no—
42#397—21no—
43#394—21no—
44#394—21no—
45#391C2211yescharacteristic
46#390C2211yescharacteristic
47#389C2211yescharacteristic
48#389C2211yescharacteristic
49#386—21no—
50#385—21no—
51#384C2211yescharacteristic
52#380—21no—
53#380C2211yescharacteristic
54C4×C22⋊C4C2211yescharacteristic
55#340C2211yescharacteristic
56#340C2211yescharacteristic
57#340C2211yescharacteristic
58#340C2211yescharacteristic
59#336C2211yescharacteristic
60#336C2211yescharacteristic
61C22×Q8C2311yescharacteristic
62C22×D4—21no—
63C22×D4—21no—
64C22×D4C2311yescharacteristic
65C22×D4C2311yescharacteristic
66C23×C4—21no—
67C23×C4—21no—
68C23×C4C2311yescharacteristic
69C2×SD16—22no—
70C2×SD16—41no—
71C2×SD16—41no—
72C2×D8—41no—
73C2×D8—41no—
74C2×D8D414yes—
75C2×M16—22no—
76C2×M16—41no—
77C2×M16—41no—
78C22×C8C2311yescharacteristic
79C22×C8C2311yescharacteristic
80C22×C8C2311yescharacteristic
81C22×C8C2311yescharacteristic
82#124—22no—
83#124—41no—
84#123—44no—
85C22⋊Q8—22no—
86C22⋊Q8—41no—
87#121—22no—
88#121—22no—
89#121—41no—
90#121—41no—
91C4×D4—22no—
92C4×D4—22no—
93C4×D4—41no—
94C4×D4—41no—
95C2×C4⋊C4—21no—
96C2×C4⋊C4—21no—
97C2×C4⋊C4—21no—
98C2×C4⋊C4—21no—
99C2×C4⋊C4—21no—
100C2×C4⋊C4C2311yescharacteristic
101C2×C4⋊C4C2311yescharacteristic
102C2×C4⋊C4C2311yescharacteristic
103C2×C4⋊C4C2311yescharacteristic
104C2×C4⋊C4C2311yescharacteristic
105C2×C22⋊C4—21no—
106C2×C22⋊C4—21no—
107C2×C22⋊C4—21no—
108C2×C22⋊C4—21no—
109C2×C22⋊C4—21no—
110C2×C22⋊C4—21no—
111C2×C22⋊C4—21no—
112C2×C22⋊C4—21no—
113C2×C22⋊C4—21no—
114C2×C22⋊C4—21no—
115C2×C22⋊C4D411yescharacteristic
116C2×C22⋊C4D411yescharacteristic
117C2×C42C2311yescharacteristic
118C2×C42C2311yescharacteristic
119#106—22no—
120#106—41no—
121C4⋊C8—22no—
122C4⋊C8—41no—
123#103—22no—
124#103—41no—
125#103—41no—
126#102—22no—
127#102—22no—
128#102—22no—
129#102—22no—
130#102—41no—
131C22⋊C8—41no—
132#95—21no—
133#95—21no—
134#95—21no—
135#95—21no—
136#95—21no—
137#95D411yescharacteristic
138#95D411yescharacteristic
139C24—21no—
140C24—21no—
141C24C2×D411yescharacteristic
142C2×Q8—21no—
143C2×Q8—21no—
144C2×Q8—21no—
145C2×Q8—21no—
146C2×Q8—21no—
147C2×Q8—21no—
148C2×D4—21no—
149C2×D4—21no—
150C2×D4—21no—
151C2×D4—21no—
152C2×D4—21no—
153C2×D4—21no—
154C2×D4—21no—
155C2×D4—21no—
156C2×D4—21no—
157C2×D4—21no—
158C2×D4—42no—
159C2×D4—42no—
160C2×D4—42no—
161C2×D4—42no—
162C2×D4—81no—
163C2×D4—81no—
164C2×D4C2×D412yes—
165C2×D4C2×D412yes—
166C22×C4—21no—
167C22×C4—21no—
168C22×C4—21no—
169C22×C4—21no—
170C22×C4—21no—
171C22×C4—21no—
172C22×C4—21no—
173C22×C4—21no—
174C22×C4—21no—
175C22×C4—21no—
176C22×C4—21no—
177C22×C4—21no—
178C22×C4—42no—
179C22×C4—42no—
180C22×C4—42no—
181C22×C4—42no—
182C22×C4—81no—
183C22×C4—81no—
184C22×C4C2×D411yescharacteristic
185C22×C4C2×D411yescharacteristic
186C22×C4C2×D411yescharacteristic
187C22×C4C2×D411yescharacteristic
188C22×C4C2×D411yescharacteristic
189C22×C4C2411yescharacteristic
190SD16—44no—
191D8—44no—
192M16—44no—
193C2×C8—21no—
194C2×C8—21no—
195C2×C8—21no—
196C2×C8—21no—
197C2×C8—21no—
198C2×C8—21no—
199C2×C8—21no—
200C2×C8—21no—
201C2×C8—21no—
202C2×C8—21no—
203C2×C8—21no—
204C2×C8C2×D412yes—
205C4⋊C4—21no—
206C4⋊C4—21no—
207C4⋊C4—21no—
208C4⋊C4—21no—
209C4⋊C4—21no—
210C4⋊C4—21no—
211C4⋊C4—21no—
212C4⋊C4—21no—
213C4⋊C4—21no—
214C4⋊C4—42no—
215C4⋊C4—42no—
216C4⋊C4—42no—
217C4⋊C4—81no—
218C4⋊C4—81no—
219C4⋊C4#4112yes—
220C22⋊C4—22no—
221C22⋊C4—41no—
222C22⋊C4—42no—
223C22⋊C4—42no—
224C22⋊C4—42no—
225C22⋊C4—42no—
226C22⋊C4—42no—
227C22⋊C4—42no—
228C22⋊C4—42no—
229C22⋊C4—42no—
230C22⋊C4—42no—
231C22⋊C4—42no—
232C42—21no—
233C42—21no—
234C42—21no—
235C42—21no—
236C23—21no—
237C23—22no—
238C23—22no—
239C23—21no—
240C23—21no—
241C23—21no—
242C23—21no—
243C23—21no—
244C23—21no—
245C23—41no—
246C23—41no—
247C23—41no—
248C23—41no—
249C23—41no—
250C23—41no—
251C23—41no—
252C23—41no—
253C23—41no—
254C23—41no—
255C23—41no—
256C23—41no—
257C23C22×D411yescharacteristic
258Q8—41no—
259Q8—41no—
260Q8—41no—
261Q8—41no—
262D4—22no—
263D4—22no—
264D4—22no—
265D4—22no—
266D4—41no—
267D4—41no—
268D4—41no—
269D4—41no—
270D4—84no—
271D4—84no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—22no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—41no—
288C2×C4—41no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—41no—
295C2×C4—41no—
296C2×C4—41no—
297C2×C4—41no—
298C2×C4—41no—
299C2×C4—41no—
300C2×C4—41no—
301C2×C4—41no—
302C2×C4—41no—
303C2×C4—41no—
304C2×C4—41no—
305C2×C4—41no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—84no—
309C2×C4—84no—
310C2×C4AGL(1,ℤ/8)12yes—
311C2×C4C22×D411yescharacteristic
312C2×C4C2×#4111yescharacteristic
313C2×C4H(5,𝔽2)11yescharacteristic
314C2×C4H(5,𝔽2)11yescharacteristic
315C2×C4H(5,𝔽2)11yescharacteristic
316C2×C42−1+411yescharacteristic
317C8—22no—
318C8—41no—
319C8—41no—
320C8—41no—
321C22—41no—
322C22—42no—
323C22—42no—
324C22—41no—
325C22—41no—
326C22—41no—
327C22—42no—
328C22—42no—
329C22—42no—
330C22—41no—
331C22—42no—
332C22—41no—
333C22—41no—
334C22—81no—
335C22—81no—
336C22—81no—
337C22—81no—
338C22—81no—
339C22—81no—
340C22—81no—
341C22—81no—
342C22#53711yescharacteristic
343C22D4211yescharacteristic
344C22#54611yescharacteristic
345C22C2×AGL(1,ℤ/8)11yescharacteristic
346C22#57511yescharacteristic
347C22#57711yescharacteristic
348C22#57711yescharacteristic
349C4—22no—
350C4—41no—
351C4—41no—
352C4—41no—
353C4—42no—
354C4—41no—
355C4—81no—
356C4—81no—
357C4—81no—
358C4—81no—
359C4—81no—
360C4#53411yescharacteristic
361C4#54011yescharacteristic
362C4#54711yescharacteristic
363C4#56211yescharacteristic
364C2—81no—
365C2—82no—
366C2—82no—
367C2#267111yescharacteristic
368C2#325111yescharacteristic
369C2#326311yescharacteristic
370C2#326711yescharacteristic
371C2#327211yescharacteristic
372C2#330611yescharacteristic
373C2#332611yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC2×#469C2
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
2. C2×#463C2semidirect
3. #2427C2semidirect
4. #2422C2semidirect
5. #2096C2semidirect
6. #2088C2semidirect
7. #2055C2semidirect
8. #2042C2semidirect
9. #2034C2semidirect
10. #2000C2semidirect
11. #1894C2semidirect
12. #1879C2semidirect
13. #1864C2semidirect
14. #1851C2semidirect
15. #1802C2semidirect

Maximal subgroups

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

subgroupindex
C2×#4692
C2×#4632
#24272
#24222
#20962
#20882
#20552
#20422
#20342
#20002
#18942
#18792
#18642
#18512
#18022