Finite Group Databasebeta

← #28916

Subgroups — #28916

Order 256.

498 conjugacy classes of subgroups.

The subgroup lattice has 394 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#490C211yesmaximal, characteristic, complemented
2C2×#469C211yesmaximal, characteristic, complemented
3#2490C211yesmaximal, characteristic, complemented
4#2428C211yesmaximal, characteristic, complemented
5#2059C211yesmaximal, characteristic, complemented
6#2033C211yesmaximal, characteristic, complemented
7#2011C211yesmaximal, characteristic, complemented
8#1999C211yesmaximal, characteristic, complemented
9#1970C211yesmaximal, characteristic, complemented
10#1936C211yesmaximal, characteristic, complemented
11#1926C211yesmaximal, characteristic, complemented
12#1895C211yesmaximal, characteristic, complemented
13#1879C211yesmaximal, characteristic, complemented
14C4×#102C211yesmaximal, characteristic, complemented
15#1753C211yesmaximal, characteristic, complemented
16C22×D8C2211yescharacteristic
17C22×D8C2211yescharacteristic
18C22×M16C2211yescharacteristic
19C2×#127C2211yescharacteristic
20C2×#125C2211yescharacteristic
21C2×#121—21no—
22C2×#121C2211yescharacteristic
23C2×#121C2211yescharacteristic
24C2×#121C2211yescharacteristic
25C2×C4×D4C2211yescharacteristic
26#490—24no—
27#469—24no—
28C2×#107C2211yescharacteristic
29C2×#106C2211yescharacteristic
30C2×#102C2211yescharacteristic
31C2×#102C2211yescharacteristic
32C2×#102C2211yescharacteristic
33C2×#102C2211yescharacteristic
34C2×#102C2211yescharacteristic
35C2×#102C2211yescharacteristic
36C2×#102C2211yescharacteristic
37C2×#102C2211yescharacteristic
38C2×C22⋊C8C2211yescharacteristic
39C2×#97C2211yescharacteristic
40C2×C4×C8C2211yescharacteristic
41#397—21no—
42#394—21no—
43#394—21no—
44#392—21no—
45#390—21no—
46#390C2211yescharacteristic
47#390C2211yescharacteristic
48#390C2211yescharacteristic
49#389C2211yescharacteristic
50#388—21no—
51#386—21no—
52#380C2211yescharacteristic
53C4×C4⋊C4C2211yescharacteristic
54C4×C22⋊C4C2211yescharacteristic
55#340C2211yescharacteristic
56#340C2211yescharacteristic
57#340C2211yescharacteristic
58#340C2211yescharacteristic
59#336C2211yescharacteristic
60#336C2211yescharacteristic
61C22×D4—21no—
62C22×D4—21no—
63C22×D4—21no—
64C22×D4—21no—
65C22×D4—21no—
66C22×D4C2311yescharacteristic
67C22×D4C2311yescharacteristic
68C22×D4C2311yescharacteristic
69C23×C4—21no—
70C23×C4C2311yescharacteristic
71C2×D8—22no—
72C2×D8—22no—
73C2×D8—22no—
74C2×D8—41no—
75C2×D8—41no—
76C2×D8—41no—
77C2×M16—22no—
78C2×M16—22no—
79C2×M16—41no—
80C22×C8C2311yescharacteristic
81C22×C8C2311yescharacteristic
82C22×C8C2311yescharacteristic
83C22×C8C2311yescharacteristic
84#127—22no—
85#127—41no—
86#125—22no—
87#125—41no—
88#121—22no—
89#121—22no—
90#121—22no—
91#121—44no—
92#121—41no—
93#121—41no—
94#121—41no—
95C4×D4—22no—
96C4×D4—41no—
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⋊C4—21no—
116C2×C22⋊C4D411yescharacteristic
117C2×C22⋊C4D411yescharacteristic
118C2×C42—21no—
119C2×C42C2311yescharacteristic
120C2×C42C2311yescharacteristic
121#107—41no—
122#106—22no—
123#102—22no—
124#102—22no—
125#102—22no—
126#102—22no—
127#102—22no—
128#102—22no—
129#102—22no—
130#102D414yescomplemented
131C22⋊C8—22no—
132#97—22no—
133C4×C8—41no—
134#95—21no—
135#95—21no—
136#95—21no—
137#95D411yescharacteristic
138#95D411yescharacteristic
139C24—21no—
140C24—21no—
141C24—21no—
142C24C2×D411yescharacteristic
143C2×D4—21no—
144C2×D4—21no—
145C2×D4—21no—
146C2×D4—21no—
147C2×D4—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—21no—
159C2×D4—21no—
160C2×D4—42no—
161C2×D4—42no—
162C2×D4—42no—
163C2×D4—42no—
164C2×D4—42no—
165C2×D4—42no—
166C2×D4—42no—
167C2×D4—42no—
168C2×D4—42no—
169C2×D4—42no—
170C2×D4—81no—
171C2×D4—81no—
172C2×D4—81no—
173C2×D4—81no—
174C2×D4—81no—
175C2×D4C2×D412yes—
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—21no—
187C22×C4—42no—
188C22×C4—42no—
189C22×C4—81no—
190C22×C4C2×D411yescharacteristic
191C22×C4C2×D411yescharacteristic
192C22×C4C2×D411yescharacteristic
193C22×C4C2×D411yescharacteristic
194C22×C4C2×D411yescharacteristic
195C22×C4C2411yescharacteristic
196D8—44no—
197D8—44no—
198M16—44no—
199C2×C8—21no—
200C2×C8—21no—
201C2×C8—21no—
202C2×C8—21no—
203C2×C8—21no—
204C2×C8—21no—
205C2×C8—21no—
206C2×C8—21no—
207C2×C8—21no—
208C2×C8C2×D412yes—
209C2×C8#4112yes—
210C2×C8#4112yes—
211C4⋊C4—21no—
212C4⋊C4—21no—
213C4⋊C4—21no—
214C4⋊C4—21no—
215C4⋊C4—21no—
216C4⋊C4—21no—
217C4⋊C4—21no—
218C4⋊C4—21no—
219C4⋊C4—42no—
220C4⋊C4—42no—
221C4⋊C4—42no—
222C4⋊C4C2×D412yes—
223C4⋊C4#4112yes—
224C22⋊C4—22no—
225C22⋊C4—41no—
226C22⋊C4—42no—
227C22⋊C4—42no—
228C22⋊C4—42no—
229C22⋊C4—42no—
230C22⋊C4—42no—
231C22⋊C4—42no—
232C22⋊C4—81no—
233C22⋊C4—81no—
234C22⋊C4—81no—
235C22⋊C4—81no—
236C22⋊C4—81no—
237C42—21no—
238C42—21no—
239C42—21no—
240C42—21no—
241C42—81no—
242C23—21no—
243C23—22no—
244C23—21no—
245C23—21no—
246C23—21no—
247C23—21no—
248C23—21no—
249C23—21no—
250C23—41no—
251C23—41no—
252C23—41no—
253C23—41no—
254C23—41no—
255C23—41no—
256C23—41no—
257C23—41no—
258C23—41no—
259C23—41no—
260C23—41no—
261C23—41no—
262C23—41no—
263C23—41no—
264C23—41no—
265C23—41no—
266C23—41no—
267C23—41no—
268C23—41no—
269C23—41no—
270C23C22×D411yescharacteristic
271D4—22no—
272D4—22no—
273D4—41no—
274D4—41no—
275D4—41no—
276D4—41no—
277D4—41no—
278D4—41no—
279D4—41no—
280D4—41no—
281D4—41no—
282D4—41no—
283D4—84no—
284D4—84no—
285D4—84no—
286D4—84no—
287D4—84no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—22no—
295C2×C4—21no—
296C2×C4—22no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
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—41no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—84no—
322C2×C4AGL(1,ℤ/8)12yes—
323C2×C4C22×D411yescharacteristic
324C2×C4C2×#4111yescharacteristic
325C2×C4C2×#4111yescharacteristic
326C2×C4C2×#4111yescharacteristic
327C2×C4H(5,𝔽2)11yescharacteristic
328C2×C4H(5,𝔽2)11yescharacteristic
329C8—22no—
330C8—22no—
331C8—22no—
332C8—41no—
333C22—41no—
334C22—41no—
335C22—41no—
336C22—42no—
337C22—41no—
338C22—42no—
339C22—41no—
340C22—41no—
341C22—41no—
342C22—42no—
343C22—81no—
344C22—81no—
345C22—81no—
346C22—81no—
347C22—81no—
348C22—81no—
349C22—81no—
350C22—81no—
351C22—81no—
352C22—81no—
353C22—81no—
354C22—81no—
355C22—81no—
356C22—81no—
357C22—81no—
358C22—81no—
359C22—81no—
360C22—81no—
361C22#53211yescharacteristic
362C22D4211yescharacteristic
363C22#54611yescharacteristic
364C22C2×AGL(1,ℤ/8)11yescharacteristic
365C22#57511yescharacteristic
366C22#57611yescharacteristic
367C22#57611yescharacteristic
368C4—22no—
369C4—41no—
370C4—42no—
371C4—41no—
372C4—41no—
373C4—42no—
374C4—41no—
375C4—81no—
376C4—81no—
377C4—81no—
378C4#53211yescharacteristic
379C4#53811yescharacteristic
380C4#54611yescharacteristic
381C4#55511yescharacteristic
382C2—81no—
383C2—82no—
384C2—161no—
385C2—161no—
386C2#255511yescharacteristic
387C2#317111yescharacteristic
388C2#317611yescharacteristic
389C2#327311yescharacteristic
390C2#327411yescharacteristic
391C2#330911yescharacteristic
392C2#331811yescharacteristic
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×#490C2semidirect
2. C2×#469C2semidirect
3. #2490C2semidirect
4. #2428C2semidirect
5. #2059C2semidirect
6. #2033C2semidirect
7. #2011C2semidirect
8. #1999C2semidirect
9. #1970C2semidirect
10. #1936C2semidirect
11. #1926C2semidirect
12. #1895C2semidirect
13. #1879C2semidirect
14. C4×#102C2semidirect
15. #1753C2semidirect
130. #102D4semidirect

Maximal subgroups

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

subgroupindex
C2×#4902
C2×#4692
#24902
#24282
#20592
#20332
#20112
#19992
#19702
#19362
#19262
#18952
#18792
C4×#1022
#17532