Finite Group Databasebeta

← #24787

Subgroups — #24787

Order 256.

527 conjugacy classes of subgroups.

The subgroup lattice has 385 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#468C211yesmaximal, characteristic, complemented
2C2×#465C211yesmaximal, characteristic, complemented
3C2×#435C211yesmaximal, characteristic, complemented
4C22×#107C211yesmaximal, characteristic, complemented
5#2417C211yesmaximal, characteristic, complemented
6#2416C211yesmaximal, characteristic
7#2068C211yesmaximal, characteristic, complemented
8#2067C211yesmaximal, characteristic, complemented
9#2036C211yesmaximal, characteristic, complemented
10#2033C211yesmaximal, characteristic, complemented
11#1973C211yesmaximal, characteristic, complemented
12#1943C211yesmaximal, characteristic, complemented
13#1874C211yesmaximal, characteristic, complemented
14#1872C211yesmaximal, characteristic, complemented
15#1809C211yesmaximal, characteristic, complemented
16C22×SD16C2211yescharacteristic
17C22×M16C2211yescharacteristic
18C23×C8C2211yescharacteristic
19C2×C22⋊Q8C2211yescharacteristic
20C2×C22⋊Q8C2211yescharacteristic
21C2×#121—21no—
22C2×#121C2211yescharacteristic
23C2×#121C2211yescharacteristic
24C2×C4×D4C2211yescharacteristic
25C22×C4⋊C4C2211yescharacteristic
26C22×C4⋊C4C2211yescharacteristic
27#468—24no—
28#465—24no—
29#435—24no—
30C2×#107—24no—
31C2×#107C2211yescharacteristic
32C2×#107C2211yescharacteristic
33C2×#107C2211yescharacteristic
34C2×#107C2211yescharacteristic
35C2×#106C2211yescharacteristic
36C2×C4⋊C8C2211yescharacteristic
37C2×#103C2211yescharacteristic
38C2×#103C2211yescharacteristic
39C2×#103C2211yescharacteristic
40C2×#102C2211yescharacteristic
41C2×#102C2211yescharacteristic
42C2×#102C2211yescharacteristic
43C2×C22⋊C8C2211yescharacteristic
44C2×C22⋊C8C2211yescharacteristic
45C2×#97C2211yescharacteristic
46#398—21no—
47#396—21no—
48#396—21no—
49#394—21no—
50#391C2211yescharacteristic
51#390C2211yescharacteristic
52#389C2211yescharacteristic
53#389C2211yescharacteristic
54#386—21no—
55#386—21no—
56#380—21no—
57#380C2211yescharacteristic
58#380C2211yescharacteristic
59#340C2211yescharacteristic
60#340C2211yescharacteristic
61#340C2211yescharacteristic
62#340C2211yescharacteristic
63C22×Q8C2311yescharacteristic
64C22×D4—21no—
65C22×D4—21no—
66C22×D4D411yescharacteristic
67C22×D4C2311yescharacteristic
68C23×C4—21no—
69C23×C4—21no—
70C23×C4C2311yescharacteristic
71C23×C4C2311yescharacteristic
72C2×SD16—22no—
73C2×SD16—22no—
74C2×SD16—22no—
75C2×M16—22no—
76C2×M16—22no—
77C2×M16—22no—
78C22×C8—22no—
79C22×C8—22no—
80C22×C8D414yescomplemented
81C22×C8C2311yescharacteristic
82C22×C8C2311yescharacteristic
83C22×C8C2311yescharacteristic
84C22×C8C2311yescharacteristic
85C22⋊Q8—22no—
86C22⋊Q8—22no—
87C22⋊Q8—22no—
88C22⋊Q8—41no—
89#121—22no—
90#121—22no—
91#121—22no—
92#121—41no—
93#121—44no—
94C4×D4—22no—
95C4×D4—22no—
96C2×C4⋊C4—22no—
97C2×C4⋊C4—22no—
98C2×C4⋊C4—21no—
99C2×C4⋊C4—21no—
100C2×C4⋊C4—21no—
101C2×C4⋊C4—21no—
102C2×C4⋊C4—21no—
103C2×C4⋊C4—21no—
104C2×C4⋊C4—41no—
105C2×C4⋊C4—41no—
106C2×C4⋊C4D411yescharacteristic
107C2×C4⋊C4C2311yescharacteristic
108C2×C4⋊C4C2311yescharacteristic
109C2×C4⋊C4C2311yescharacteristic
110C2×C4⋊C4C2311yescharacteristic
111C2×C4⋊C4C2311yescharacteristic
112C2×C4⋊C4C2311yescharacteristic
113C2×C22⋊C4—21no—
114C2×C22⋊C4—21no—
115C2×C22⋊C4—21no—
116C2×C22⋊C4—21no—
117C2×C22⋊C4—21no—
118C2×C22⋊C4—21no—
119C2×C22⋊C4—21no—
120C2×C22⋊C4—21no—
121C2×C22⋊C4D411yescharacteristic
122C2×C22⋊C4D411yescharacteristic
123C2×C42C2311yescharacteristic
124#107—22no—
125#107—22no—
126#107—22no—
127#107D414yescomplemented
128#106—22no—
129C4⋊C8—22no—
130#103—22no—
131#103—22no—
132#103—41no—
133#102—22no—
134#102—22no—
135#102—41no—
136C22⋊C8—22no—
137C22⋊C8—22no—
138#97—22no—
139#95—21no—
140#95—21no—
141#95—21no—
142#95—21no—
143#95—21no—
144#95—21no—
145C24—21no—
146C24C2×D411yescharacteristic
147C24C2×D411yescharacteristic
148C2×Q8—21no—
149C2×Q8—21no—
150C2×Q8—21no—
151C2×Q8—21no—
152C2×Q8—21no—
153C2×Q8—21no—
154C2×D4—22no—
155C2×D4—22no—
156C2×D4—22no—
157C2×D4—21no—
158C2×D4—21no—
159C2×D4—21no—
160C2×D4—21no—
161C2×D4—21no—
162C2×D4—21no—
163C2×D4—42no—
164C2×D4—42no—
165C2×D4—42no—
166C2×D4—42no—
167C2×D4—42no—
168C2×D4—81no—
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—21no—
182C22×C4—21no—
183C22×C4—21no—
184C22×C4—21no—
185C22×C4—21no—
186C22×C4—42no—
187C22×C4—42no—
188C22×C4—42no—
189C22×C4—42no—
190C22×C4—81no—
191C22×C4—81no—
192C22×C4C2×D411yescharacteristic
193C22×C4C2×D411yescharacteristic
194C22×C4C2×D411yescharacteristic
195C22×C4C2×D411yescharacteristic
196C22×C4C2×D412yes—
197C22×C4#4112yes—
198C22×C4C2411yescharacteristic
199SD16—44no—
200M16—44no—
201C2×C8—21no—
202C2×C8—21no—
203C2×C8—28no—
204C2×C8—21no—
205C2×C8—21no—
206C2×C8—21no—
207C2×C8—21no—
208C2×C8—21no—
209C2×C8—21no—
210C2×C8C2×D412yes—
211C2×C8C2×D412yes—
212C2×C8C2×D412yes—
213C2×C8#4112yes—
214C4⋊C4—22no—
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—21no—
224C4⋊C4—21no—
225C4⋊C4—42no—
226C4⋊C4—42no—
227C4⋊C4—42no—
228C4⋊C4—42no—
229C4⋊C4—42no—
230C4⋊C4—81no—
231C4⋊C4C2×D412yes—
232C4⋊C4C2×D412yes—
233C22⋊C4—22no—
234C22⋊C4—22no—
235C22⋊C4—42no—
236C22⋊C4—42no—
237C22⋊C4—42no—
238C22⋊C4—42no—
239C22⋊C4—42no—
240C22⋊C4—42no—
241C22⋊C4—81no—
242C22⋊C4—81no—
243C42—21no—
244C42—21no—
245C23—21no—
246C23—21no—
247C23—21no—
248C23—21no—
249C23—21no—
250C23—21no—
251C23—21no—
252C23—21no—
253C23—21no—
254C23—21no—
255C23—21no—
256C23—41no—
257C23—41no—
258C23—41no—
259C23—41no—
260C23—41no—
261C23—41no—
262C23—41no—
263C23#12112yes—
264C23AGL(1,ℤ/8)12yes—
265C23#13712yes—
266C23C22×D411yescharacteristic
267Q8—41no—
268Q8—41no—
269Q8—41no—
270Q8—41no—
271D4—44no—
272D4—41no—
273D4—41no—
274D4—41no—
275D4—41no—
276D4—84no—
277D4—84no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—22no—
281C2×C4—22no—
282C2×C4—22no—
283C2×C4—22no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—22no—
289C2×C4—22no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
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—41no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—84no—
319C2×C4—84no—
320C2×C4#12112yes—
321C2×C4C22×D411yescharacteristic
322C2×C4C22×D411yescharacteristic
323C2×C4C2×#4111yescharacteristic
324C2×C4C2×#4111yescharacteristic
325C2×C4H(5,𝔽2)11yescharacteristic
326C2×C42−1+411yescharacteristic
327C8—22no—
328C8—41no—
329C8—41no—
330C8#12114yescomplemented
331C22—22no—
332C22—22no—
333C22—22no—
334C22—22no—
335C22—22no—
336C22—22no—
337C22—41no—
338C22—41no—
339C22—41no—
340C22—41no—
341C22—41no—
342C22—41no—
343C22—41no—
344C22—81no—
345C22—81no—
346C22—81no—
347C22—81no—
348C22—81no—
349C22—81no—
350C22—81no—
351C22#46814yescomplemented
352C22C2×#12111yescharacteristic
353C22D4211yescharacteristic
354C22#54711yescharacteristic
355C22C2×AGL(1,ℤ/8)11yescharacteristic
356C22C2×#13711yescharacteristic
357C22#57611yescharacteristic
358C22#57811yescharacteristic
359C4—22no—
360C4—41no—
361C4—42no—
362C4—42no—
363C4—41no—
364C4—41no—
365C4—81no—
366C4—81no—
367C4—81no—
368C4—81no—
369C4—81no—
370C4C2×#12111yescharacteristic
371C4#53711yescharacteristic
372C4#54811yescharacteristic
373C4#55311yescharacteristic
374C2—24no—
375C2—81no—
376C2—161no—
377C2#241611yescharacteristic
378C2C2×#46811yescharacteristic
379C2#306111yescharacteristic
380C2#326411yescharacteristic
381C2#327011yescharacteristic
382C2#334311yescharacteristic
383C2#334611yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radical#2416C2
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×#468C2semidirect
2. C2×#465C2semidirect
3. C2×#435C2semidirect
4. C22×#107C2semidirect
5. #2417C2semidirect
7. #2068C2semidirect
8. #2067C2semidirect
9. #2036C2semidirect
10. #2033C2semidirect
11. #1973C2semidirect
12. #1943C2semidirect
13. #1874C2semidirect
14. #1872C2semidirect
15. #1809C2semidirect
80. C22×C8D4semidirect
127. #107D4semidirect
330. C8#121semidirect
351. C22#468semidirect

Maximal subgroups

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

subgroupindex
C2×#4682
C2×#4652
C2×#4352
C22×#1072
#24172
#24162
#20682
#20672
#20362
#20332
#19732
#19432
#18742
#18722
#18092