Finite Group Databasebeta

← #28155

Subgroups — #28155

Order 256.

478 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×#440C211yesmaximal, characteristic, complemented
2C2×#435C211yesmaximal, characteristic, complemented
3#2538C211yesmaximal, characteristic, complemented
4#2535C211yesmaximal, characteristic
5#2076C211yesmaximal, characteristic, complemented
6#2075C211yesmaximal, characteristic, complemented
7#2000C211yesmaximal, characteristic, complemented
8#2000C211yesmaximal, characteristic, complemented
9#1895C211yesmaximal, characteristic, complemented
10#1894C211yesmaximal, characteristic, complemented
11#1893C211yesmaximal, characteristic, complemented
12#1892C211yesmaximal, characteristic, complemented
13#1836C211yesmaximal, characteristic, complemented
14#1805C211yesmaximal, characteristic, complemented
15#1802C211yesmaximal, characteristic, complemented
16C22×SD16C2211yescharacteristic
17C22×M16C2211yescharacteristic
18C23×C8C2211yescharacteristic
19C2×C22⋊Q8C2211yescharacteristic
20C2×C22⋊Q8C2211yescharacteristic
21C2×#121C2211yescharacteristic
22C2×#121C2211yescharacteristic
23C2×C4×Q8C2211yescharacteristic
24C2×C4×D4C2211yescharacteristic
25C2×C4×D4C2211yescharacteristic
26#440—24no—
27#435—24no—
28C2×#107C2211yescharacteristic
29C2×C4⋊C8C2211yescharacteristic
30C2×#103C2211yescharacteristic
31C2×#103C2211yescharacteristic
32C2×#103C2211yescharacteristic
33C2×#102C2211yescharacteristic
34C2×#102C2211yescharacteristic
35C2×#102C2211yescharacteristic
36C2×C22⋊C8C2211yescharacteristic
37C2×C22⋊C8C2211yescharacteristic
38C2×#97C2211yescharacteristic
39#397—21no—
40#396—21no—
41#396—21no—
42#392—21no—
43#390—21no—
44#389—21no—
45#386—21no—
46#386—21no—
47#384C2211yescharacteristic
48#382C2211yescharacteristic
49#380C2211yescharacteristic
50#380C2211yescharacteristic
51#380C2211yescharacteristic
52#380C2211yescharacteristic
53#340C2211yescharacteristic
54#340C2211yescharacteristic
55#340C2211yescharacteristic
56#340C2211yescharacteristic
57#340C2211yescharacteristic
58#340C2211yescharacteristic
59#340C2211yescharacteristic
60#340C2211yescharacteristic
61C22×Q8C2311yescharacteristic
62C22×D4—21no—
63C22×D4—21no—
64C22×D4D411yescharacteristic
65C22×D4C2311yescharacteristic
66C23×C4—21no—
67C23×C4C2311yescharacteristic
68C23×C4C2311yescharacteristic
69C2×SD16—22no—
70C2×SD16—22no—
71C2×SD16—22no—
72C2×M16—22no—
73C2×M16—41no—
74C2×M16—41no—
75C22×C8—22no—
76C22×C8—41no—
77C22×C8—41no—
78C22×C8C2311yescharacteristic
79C22×C8C2311yescharacteristic
80C22×C8C2311yescharacteristic
81C22×C8C2311yescharacteristic
82C22⋊Q8—22no—
83C22⋊Q8—22no—
84C22⋊Q8—41no—
85C22⋊Q8—41no—
86#121—22no—
87#121—22no—
88#121—41no—
89#121—41no—
90C4×Q8—22no—
91C4×Q8—22no—
92C4×D4—22no—
93C4×D4—22no—
94C4×D4—22no—
95C4×D4—22no—
96C2×C4⋊C4—21no—
97C2×C4⋊C4—21no—
98C2×C4⋊C4—21no—
99C2×C4⋊C4—21no—
100C2×C4⋊C4—21no—
101C2×C4⋊C4D411yescharacteristic
102C2×C4⋊C4C2311yescharacteristic
103C2×C4⋊C4C2311yescharacteristic
104C2×C4⋊C4C2311yescharacteristic
105C2×C4⋊C4C2311yescharacteristic
106C2×C4⋊C4C2311yescharacteristic
107C2×C4⋊C4C2311yescharacteristic
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⋊C4—21no—
116C2×C22⋊C4—21no—
117C2×C22⋊C4D411yescharacteristic
118C2×C22⋊C4D411yescharacteristic
119C2×C42—21no—
120C2×C42C2311yescharacteristic
121#107—22no—
122C4⋊C8—22no—
123#103—22no—
124#103—41no—
125#103—41no—
126#102—22no—
127#102—41no—
128#102—41no—
129C22⋊C8—22no—
130C22⋊C8—22no—
131#97D414yescomplemented
132#95—21no—
133#95—21no—
134#95—21no—
135#95—21no—
136#95—21no—
137#95—21no—
138C24—21no—
139C24C2×D411yescharacteristic
140C24C2×D411yescharacteristic
141C2×Q8—21no—
142C2×Q8—21no—
143C2×Q8—21no—
144C2×Q8—21no—
145C2×Q8—21no—
146C2×Q8#4112yes—
147C2×D4—22no—
148C2×D4—22no—
149C2×D4—21no—
150C2×D4—21no—
151C2×D4—21no—
152C2×D4—21no—
153C2×D4—21no—
154C2×D4—41no—
155C2×D4—42no—
156C2×D4—42no—
157C2×D4—42no—
158C2×D4—42no—
159C2×D4—81no—
160C2×D4—81no—
161C2×D4#4112yes—
162C22×C4—21no—
163C22×C4—21no—
164C22×C4—21no—
165C22×C4—21no—
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—21no—
179C22×C4—21no—
180C22×C4—21no—
181C22×C4—42no—
182C22×C4—81no—
183C22×C4—81no—
184C22×C4C2×D411yescharacteristic
185C22×C4C2×D411yescharacteristic
186C22×C4C2×D411yescharacteristic
187C22×C4C2×D411yescharacteristic
188C22×C4C2411yescharacteristic
189SD16—28no—
190M16—44no—
191C2×C8—21no—
192C2×C8—21no—
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—44no—
202C2×C8C2×D412yes—
203C2×C8C2×D412yes—
204C4⋊C4—22no—
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—21no—
215C4⋊C4—21no—
216C4⋊C4—21no—
217C4⋊C4—42no—
218C4⋊C4—42no—
219C4⋊C4—42no—
220C4⋊C4—42no—
221C4⋊C4—81no—
222C22⋊C4—22no—
223C22⋊C4—22no—
224C22⋊C4—42no—
225C22⋊C4—42no—
226C22⋊C4—42no—
227C22⋊C4—42no—
228C22⋊C4—42no—
229C22⋊C4—81no—
230C22⋊C4—81no—
231C22⋊C4—81no—
232C22⋊C4—81no—
233C42—42no—
234C42C2×D412yes—
235C42#4112yes—
236C23—21no—
237C23—21no—
238C23—21no—
239C23—21no—
240C23—21no—
241C23—21no—
242C23—21no—
243C23—21no—
244C23—22no—
245C23—21no—
246C23—21no—
247C23—21no—
248C23—21no—
249C23—21no—
250C23—41no—
251C23—41no—
252C23—41no—
253C23—41no—
254C23—41no—
255C23—41no—
256C23#13712yes—
257C23C22×D411yescharacteristic
258Q8—22no—
259Q8—22no—
260Q8—41no—
261Q8—41no—
262D4—22no—
263D4—22no—
264D4—44no—
265D4—41no—
266D4—41no—
267D4—84no—
268D4—84no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—22no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—41no—
281C2×C4—41no—
282C2×C4—41no—
283C2×C4—41no—
284C2×C4—41no—
285C2×C4—41no—
286C2×C4—41no—
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#12112yes—
310C2×C4#12112yes—
311C2×C4C22×D411yescharacteristic
312C2×C4C2×#4111yescharacteristic
313C2×C4C2×#4111yescharacteristic
314C2×C4C2×#4111yescharacteristic
315C2×C4H(5,𝔽2)11yescharacteristic
316C2×C4H(5,𝔽2)11yescharacteristic
317C8—22no—
318C8—22no—
319C8—41no—
320C8—41no—
321C22—22no—
322C22—22no—
323C22—22no—
324C22—41no—
325C22—41no—
326C22—41no—
327C22—41no—
328C22—41no—
329C22—42no—
330C22—41no—
331C22—41no—
332C22—42no—
333C22—41no—
334C22—42no—
335C22—41no—
336C22—41no—
337C22—41no—
338C22—81no—
339C22—81no—
340C22—81no—
341C22—81no—
342C22C2×#12111yescharacteristic
343C22#54611yescharacteristic
344C22#54611yescharacteristic
345C22C2×#13711yescharacteristic
346C22#57511yescharacteristic
347C22#57611yescharacteristic
348C22#57711yescharacteristic
349C4—22no—
350C4—22no—
351C4—41no—
352C4—41no—
353C4—42no—
354C4—81no—
355C4—81no—
356C4—81no—
357C4—81no—
358C4—81no—
359C4—81no—
360C4#52511yescharacteristic
361C4#53511yescharacteristic
362C4#55311yescharacteristic
363C4#55311yescharacteristic
364C2—42no—
365C2—81no—
366C2—82no—
367C2#253511yescharacteristic
368C2#305611yescharacteristic
369C2#306211yescharacteristic
370C2#330911yescharacteristic
371C2#331111yescharacteristic
372C2#331411yescharacteristic
373C2#332011yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radical#2535C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#440C2semidirect
2. C2×#435C2semidirect
3. #2538C2semidirect
5. #2076C2semidirect
6. #2075C2semidirect
7. #2000C2semidirect
8. #2000C2semidirect
9. #1895C2semidirect
10. #1894C2semidirect
11. #1893C2semidirect
12. #1892C2semidirect
13. #1836C2semidirect
14. #1805C2semidirect
15. #1802C2semidirect
131. #97D4semidirect

Maximal subgroups

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

subgroupindex
C2×#4402
C2×#4352
#25382
#25352
#20762
#20752
#20002
#18952
#18942
#18932
#18922
#18362
#18052
#18022