Finite Group Databasebeta

← #28364

Subgroups — #28364

Order 256.

605 conjugacy classes of subgroups.

The subgroup lattice has 361 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#481C211yesmaximal, characteristic, complemented
2C2×#469C211yesmaximal, characteristic, complemented
3C2×#449C211yesmaximal, characteristic, complemented
4C2×#417C211yesmaximal, characteristic, complemented
5#2501C211yesmaximal, characteristic, complemented
6#2431C211yesmaximal, characteristic, complemented
7#2096C211yesmaximal, characteristic, complemented
8#2094C211yesmaximal, characteristic, complemented
9#2012C211yesmaximal, characteristic, complemented
10#2011C211yesmaximal, characteristic, complemented
11#1972C211yesmaximal, characteristic, complemented
12#1968C211yesmaximal, characteristic, complemented
13#1959C211yesmaximal, characteristic, complemented
14#1957C211yesmaximal, characteristic, complemented
15#1753C211yesmaximal, characteristic, complemented
16C22×#41C2211yescharacteristic
17C22×SD16C2211yescharacteristic
18C22×D8C2211yescharacteristic
19C22×D8C2211yescharacteristic
20C22×M16C2211yescharacteristic
21C2×#127C2211yescharacteristic
22C2×#124C2211yescharacteristic
23C2×#123—22no—
24C2×#121C2211yescharacteristic
25C2×#121C2211yescharacteristic
26C22×C4⋊C4C2211yescharacteristic
27#481—24no—
28#469—24no—
29#449—24no—
30C2×#106C2211yescharacteristic
31C2×#106C2211yescharacteristic
32C2×#106C2211yescharacteristic
33#417—24no—
34C2×#103C2211yescharacteristic
35C2×#103C2211yescharacteristic
36C2×#103C2211yescharacteristic
37C2×#102C2211yescharacteristic
38C2×#102C2211yescharacteristic
39C2×#102C2211yescharacteristic
40C2×#102C2211yescharacteristic
41C2×#102C2211yescharacteristic
42C2×#102C2211yescharacteristic
43C2×#102C2211yescharacteristic
44C2×C22⋊C8C2211yescharacteristic
45C2×#97C2211yescharacteristic
46C2×C4×C8C2211yescharacteristic
47#399—22no—
48#392—22no—
49#391C2211yescharacteristic
50#390C2211yescharacteristic
51#390C2211yescharacteristic
52#390C2211yescharacteristic
53#386—22no—
54#384C2211yescharacteristic
55#384C2211yescharacteristic
56C4×C22⋊C4C2211yescharacteristic
57#336C2211yescharacteristic
58#336C2211yescharacteristic
59C2×#41—22no—
60C2×#41—22no—
61C2×#41—22no—
62C2×#41—22no—
63C2×#41—41no—
64C2×#41—41no—
65C22×Q8C2311yescharacteristic
66C22×D4—21no—
67C22×D4—21no—
68C22×D4—21no—
69C22×D4—22no—
70C22×D4C2311yescharacteristic
71C22×D4C2311yescharacteristic
72C22×D4C2311yescharacteristic
73C23×C4—21no—
74C23×C4—21no—
75C23×C4C2311yescharacteristic
76C2×SD16—22no—
77C2×SD16—22no—
78C2×SD16—41no—
79C2×D8—22no—
80C2×D8—22no—
81C2×D8—22no—
82C2×D8—41no—
83C2×D8—41no—
84C2×D8—41no—
85C2×M16—41no—
86C2×M16—41no—
87C2×M16D414yescomplemented
88C22×C8C2311yescharacteristic
89C22×C8C2311yescharacteristic
90C22×C8C2311yescharacteristic
91C22×C8C2311yescharacteristic
92#127—22no—
93#127—41no—
94#124—22no—
95#124—41no—
96#123—48no—
97#121—22no—
98#121—22no—
99#121—22no—
100#121—22no—
101C2×C4⋊C4—22no—
102C2×C4⋊C4—22no—
103C2×C4⋊C4—21no—
104C2×C4⋊C4—22no—
105C2×C4⋊C4—22no—
106C2×C4⋊C4C2311yescharacteristic
107C2×C4⋊C4C2311yescharacteristic
108C2×C4⋊C4C2311yescharacteristic
109C2×C4⋊C4C2311yescharacteristic
110C2×C22⋊C4—21no—
111C2×C22⋊C4—21no—
112C2×C22⋊C4—22no—
113C2×C22⋊C4—22no—
114C2×C22⋊C4—22no—
115C2×C22⋊C4—22no—
116C2×C22⋊C4D412yes—
117C2×C42C2311yescharacteristic
118C2×C42C2311yescharacteristic
119#106—22no—
120#106—22no—
121#106—22no—
122#103—22no—
123#103—22no—
124#103—22no—
125#102—22no—
126#102—22no—
127#102—22no—
128#102—22no—
129#102—22no—
130#102—22no—
131#102—22no—
132C22⋊C8D414yescomplemented
133#97—41no—
134C4×C8—41no—
135#95—22no—
136#95D412yes—
137C24—21no—
138C24—21no—
139C24—21no—
140C24C2×D411yescharacteristic
141#41—28no—
142#41—44no—
143#41—44no—
144#41—44no—
145C2×Q8—21no—
146C2×Q8—21no—
147C2×Q8—21no—
148C2×Q8—21no—
149C2×Q8—21no—
150C2×Q8#4112yes—
151C2×D4—21no—
152C2×D4—21no—
153C2×D4—21no—
154C2×D4—21no—
155C2×D4—21no—
156C2×D4—21no—
157C2×D4—21no—
158C2×D4—21no—
159C2×D4—21no—
160C2×D4—21no—
161C2×D4—21no—
162C2×D4—21no—
163C2×D4—21no—
164C2×D4—21no—
165C2×D4—21no—
166C2×D4—21no—
167C2×D4—21no—
168C2×D4—42no—
169C2×D4—44no—
170C2×D4—42no—
171C2×D4—42no—
172C2×D4—44no—
173C2×D4—42no—
174C2×D4—42no—
175C2×D4—44no—
176C2×D4—82no—
177C2×D4—82no—
178C2×D4#4112yes—
179C22×C4—21no—
180C22×C4—21no—
181C22×C4—21no—
182C22×C4—21no—
183C22×C4—21no—
184C22×C4—21no—
185C22×C4—21no—
186C22×C4—21no—
187C22×C4—21no—
188C22×C4—42no—
189C22×C4—42no—
190C22×C4—42no—
191C22×C4—44no—
192C22×C4—42no—
193C22×C4C2×D412yes—
194C22×C4C2×D411yescharacteristic
195C22×C4C2×D412yes—
196C22×C4C2×D412yes—
197C22×C4C2×D412yes—
198C22×C4C2411yescharacteristic
199SD16—44no—
200D8—44no—
201D8—44no—
202M16—44no—
203C2×C8—21no—
204C2×C8—21no—
205C2×C8—21no—
206C2×C8—21no—
207C2×C8—21no—
208C2×C8—21no—
209C2×C8—21no—
210C2×C8—21no—
211C2×C8C2×D412yes—
212C2×C8C2×D412yes—
213C2×C8C2×D412yes—
214C2×C8C2×D412yes—
215C4⋊C4—21no—
216C4⋊C4—21no—
217C4⋊C4—21no—
218C4⋊C4—21no—
219C4⋊C4—21no—
220C4⋊C4—21no—
221C4⋊C4—21no—
222C4⋊C4—21no—
223C4⋊C4—44no—
224C4⋊C4—42no—
225C4⋊C4—44no—
226C22⋊C4—24no—
227C22⋊C4—44no—
228C22⋊C4—44no—
229C22⋊C4—42no—
230C22⋊C4—42no—
231C22⋊C4—44no—
232C22⋊C4—82no—
233C42—21no—
234C42—21no—
235C42—21no—
236C42—21no—
237C23—21no—
238C23—21no—
239C23—22no—
240C23—22no—
241C23—22no—
242C23—41no—
243C23—41no—
244C23—42no—
245C23—42no—
246C23—41no—
247C23—41no—
248C23—41no—
249C23—41no—
250C23—41no—
251C23—41no—
252C23—42no—
253C23—42no—
254C23—42no—
255C23—42no—
256C23#12712yes—
257C23C22×D411yescharacteristic
258Q8—22no—
259Q8—22no—
260Q8—41no—
261Q8—41no—
262D4—22no—
263D4—22no—
264D4—48no—
265D4—41no—
266D4—41no—
267D4—41no—
268D4—41no—
269D4—41no—
270D4—41no—
271D4—41no—
272D4—41no—
273D4—41no—
274D4—41no—
275D4—88no—
276D4—84no—
277D4—84no—
278C2×C4—22no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—22no—
282C2×C4—22no—
283C2×C4—22no—
284C2×C4—22no—
285C2×C4—22no—
286C2×C4—22no—
287C2×C4—22no—
288C2×C4—22no—
289C2×C4—22no—
290C2×C4—48no—
291C2×C4—41no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—41no—
295C2×C4—41no—
296C2×C4—41no—
297C2×C4—42no—
298C2×C4—42no—
299C2×C4—42no—
300C2×C4—42no—
301C2×C4—84no—
302C2×C4#12712yes—
303C2×C4AGL(1,ℤ/8)14yes—
304C2×C4C22×D411yescharacteristic
305C2×C4C22×D411yescharacteristic
306C2×C4C2×#4111yescharacteristic
307C2×C4C2×#4111yescharacteristic
308C2×C4H(5,𝔽2)11yescharacteristic
309C2×C4H(5,𝔽2)11yescharacteristic
310C8—22no—
311C8—22no—
312C8—22no—
313C8—22no—
314C22—22no—
315C22—22no—
316C22—22no—
317C22—42no—
318C22—42no—
319C22—44no—
320C22—42no—
321C22—82no—
322C22—81no—
323C22—81no—
324C22—82no—
325C22—82no—
326C22—81no—
327C22—81no—
328C22—81no—
329C22—81no—
330C22—82no—
331C22—82no—
332C22—81no—
333C22—81no—
334C22C2×#12711yescharacteristic
335C22#54611yescharacteristic
336C22#54611yescharacteristic
337C22C2×AGL(1,ℤ/8)12yes—
338C22#57712yes—
339C4—24no—
340C4—22no—
341C4—42no—
342C4—42no—
343C4—81no—
344C4—81no—
345C4—81no—
346C4—81no—
347C4#52511yescharacteristic
348C4#53411yescharacteristic
349C4#54611yescharacteristic
350C4#54611yescharacteristic
351C2—42no—
352C2—82no—
353C2—161no—
354C2—161no—
355C2#253911yescharacteristic
356C2#315111yescharacteristic
357C2#315411yescharacteristic
358C2#330612yes—
359C2#333212yes—
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
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×#481C2semidirect
2. C2×#469C2semidirect
3. C2×#449C2semidirect
4. C2×#417C2semidirect
5. #2501C2semidirect
6. #2431C2semidirect
7. #2096C2semidirect
8. #2094C2semidirect
9. #2012C2semidirect
10. #2011C2semidirect
11. #1972C2semidirect
12. #1968C2semidirect
13. #1959C2semidirect
14. #1957C2semidirect
15. #1753C2semidirect
87. C2×M16D4semidirect
132. C22⋊C8D4semidirect

Maximal subgroups

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

subgroupindex
C2×#4812
C2×#4692
C2×#4492
C2×#4172
#25012
#24312
#20962
#20942
#20122
#20112
#19722
#19682
#19592
#19572
#17532