Finite Group Databasebeta

← #26943

Subgroups — #26943

Order 256.

538 conjugacy classes of subgroups.

The subgroup lattice has 400 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#481C211yesmaximal, characteristic, complemented
2C2×#459C211yesmaximal, characteristic, complemented
3#2428C211yesmaximal, characteristic, complemented
4#2361C211yesmaximal, characteristic, complemented
5#2088C211yesmaximal, characteristic, complemented
6#2041C211yesmaximal, characteristic, complemented
7#1999C211yesmaximal, characteristic, complemented
8#1999C211yesmaximal, characteristic, complemented
9#1972C211yesmaximal, characteristic, complemented
10#1926C211yesmaximal, characteristic, complemented
11#1924C211yesmaximal, characteristic, complemented
12#1879C211yesmaximal, characteristic, complemented
13#1874C211yesmaximal, characteristic, complemented
14#1864C211yesmaximal, characteristic, complemented
15#1762C211yesmaximal, characteristic, complemented
16C22×D8C2211yescharacteristic
17C22×D8C2211yescharacteristic
18C2×#127C2211yescharacteristic
19C2×#121—21no—
20C2×#121C2211yescharacteristic
21C2×#121C2211yescharacteristic
22C2×#121C2211yescharacteristic
23C2×C4×D4—21no—
24C2×C4×D4C2211yescharacteristic
25C2×C4×D4C2211yescharacteristic
26C22×C4⋊C4C2211yescharacteristic
27#481—24no—
28#459—24no—
29C2×#106C2211yescharacteristic
30C2×#106C2211yescharacteristic
31C2×C4⋊C8C2211yescharacteristic
32C2×#102C2211yescharacteristic
33C2×#102C2211yescharacteristic
34C2×#102C2211yescharacteristic
35C2×#102C2211yescharacteristic
36C2×#102C2211yescharacteristic
37C2×#102C2211yescharacteristic
38C2×#102C2211yescharacteristic
39C2×#102C2211yescharacteristic
40C2×C22⋊C8C2211yescharacteristic
41C2×C22⋊C8C2211yescharacteristic
42C2×#97C2211yescharacteristic
43#394—21no—
44#392—21no—
45#390C2211yescharacteristic
46#390C2211yescharacteristic
47#389C2211yescharacteristic
48#389C2211yescharacteristic
49#388—21no—
50#386—21no—
51#386—21no—
52#385—21no—
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×C4—21no—
71C23×C4—21no—
72C23×C4C2311yescharacteristic
73C2×D8—22no—
74C2×D8—22no—
75C2×D8—22no—
76C2×D8—22no—
77C2×D8—41no—
78C2×D8—41no—
79C22×C8C2311yescharacteristic
80C22×C8C2311yescharacteristic
81C22×C8C2311yescharacteristic
82C22×C8C2311yescharacteristic
83#127—22no—
84#127—41no—
85#121—22no—
86#121—22no—
87#121—22no—
88#121—22no—
89#121—44no—
90#121—41no—
91#121—41no—
92C4×D4—22no—
93C4×D4—22no—
94C4×D4—41no—
95C4×D4—41no—
96C4×D4—44no—
97C2×C4⋊C4—22no—
98C2×C4⋊C4—21no—
99C2×C4⋊C4—21no—
100C2×C4⋊C4—21no—
101C2×C4⋊C4—41no—
102C2×C4⋊C4C2311yescharacteristic
103C2×C4⋊C4C2311yescharacteristic
104C2×C4⋊C4C2311yescharacteristic
105C2×C4⋊C4C2311yescharacteristic
106C2×C4⋊C4C2311yescharacteristic
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#106—22no—
122#106—22no—
123C4⋊C8—22no—
124#102—22no—
125#102—22no—
126#102—22no—
127#102—22no—
128#102—22no—
129#102—41no—
130#102—41no—
131#102D414yescomplemented
132C22⋊C8—22no—
133C22⋊C8—41no—
134#97—41no—
135#95—21no—
136#95—21no—
137#95—21no—
138#95D411yescharacteristic
139#95D411yescharacteristic
140C24—21no—
141C24—21no—
142C24—21no—
143C24C2×D411yescharacteristic
144C2×D4—24no—
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—21no—
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—42no—
171C2×D4—81no—
172C2×D4—81no—
173C2×D4—81no—
174C2×D4—81no—
175C2×D4C2×D412yes—
176C2×D4#4112yes—
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—21no—
188C22×C4—42no—
189C22×C4—42no—
190C22×C4—42no—
191C22×C4—42no—
192C22×C4—42no—
193C22×C4—42no—
194C22×C4—81no—
195C22×C4—81no—
196C22×C4—81no—
197C22×C4C2×D411yescharacteristic
198C22×C4C2×D411yescharacteristic
199C22×C4C2×D411yescharacteristic
200C22×C4C2×D411yescharacteristic
201C22×C4C2×D411yescharacteristic
202C22×C4C2411yescharacteristic
203D8—44no—
204D8—44no—
205C2×C8—21no—
206C2×C8—21no—
207C2×C8—21no—
208C2×C8—21no—
209C2×C8—21no—
210C2×C8—21no—
211C2×C8—21no—
212C2×C8—21no—
213C2×C8—21no—
214C2×C8—21no—
215C2×C8—21no—
216C2×C8C2×D412yes—
217C4⋊C4—21no—
218C4⋊C4—21no—
219C4⋊C4—21no—
220C4⋊C4—21no—
221C4⋊C4—21no—
222C4⋊C4—21no—
223C4⋊C4—21no—
224C4⋊C4—42no—
225C4⋊C4—42no—
226C4⋊C4—42no—
227C4⋊C4C2×D412yes—
228C4⋊C4#4112yes—
229C4⋊C4#4112yes—
230C22⋊C4—24no—
231C22⋊C4—22no—
232C22⋊C4—22no—
233C22⋊C4—42no—
234C22⋊C4—42no—
235C22⋊C4—42no—
236C22⋊C4—42no—
237C22⋊C4—42no—
238C22⋊C4—42no—
239C22⋊C4—81no—
240C22⋊C4—81no—
241C42—21no—
242C42—21no—
243C42—21no—
244C42—21no—
245C42—42no—
246C23—21no—
247C23—21no—
248C23—21no—
249C23—21no—
250C23—21no—
251C23—21no—
252C23—22no—
253C23—22no—
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—
270C23—41no—
271C23—41no—
272C23—41no—
273C23AGL(1,ℤ/8)12yes—
274C23C22×D411yescharacteristic
275D4—22no—
276D4—22no—
277D4—22no—
278D4—22no—
279D4—48no—
280D4—41no—
281D4—41no—
282D4—41no—
283D4—41no—
284D4—41no—
285D4—41no—
286D4—41no—
287D4—41no—
288D4—84no—
289D4—84no—
290D4—84no—
291D4—84no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—22no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—22no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—22no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
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—41no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—84no—
326C2×C4—84no—
327C2×C4—84no—
328C2×C4AGL(1,ℤ/8)12yes—
329C2×C4C22×D411yescharacteristic
330C2×C4C2×#4111yescharacteristic
331C2×C4C2×#4111yescharacteristic
332C2×C4C2×#4111yescharacteristic
333C2×C4H(5,𝔽2)11yescharacteristic
334C2×C4H(5,𝔽2)11yescharacteristic
335C8—22no—
336C8—41no—
337C8—41no—
338C8—41no—
339C22—22no—
340C22—22no—
341C22—22no—
342C22—41no—
343C22—41no—
344C22—41no—
345C22—41no—
346C22—42no—
347C22—42no—
348C22—42no—
349C22—42no—
350C22—42no—
351C22—42no—
352C22—81no—
353C22—81no—
354C22—81no—
355C22—81no—
356C22—81no—
357C22—81no—
358C22—81no—
359C22—81no—
360C22—81no—
361C22—81no—
362C22—81no—
363C22—81no—
364C22—81no—
365C22—81no—
366C22—81no—
367C22#52511yescharacteristic
368C22D4211yescharacteristic
369C22#54611yescharacteristic
370C22C2×AGL(1,ℤ/8)11yescharacteristic
371C22C2×AGL(1,ℤ/8)11yescharacteristic
372C22#57611yescharacteristic
373C22#57711yescharacteristic
374C4—22no—
375C4—41no—
376C4—42no—
377C4—41no—
378C4—42no—
379C4—41no—
380C4—41no—
381C4—42no—
382C4—81no—
383C4—81no—
384C4#53211yescharacteristic
385C4#54011yescharacteristic
386C4#54611yescharacteristic
387C4#55311yescharacteristic
388C2—42no—
389C2—82no—
390C2—82no—
391C2—161no—
392C2#246411yescharacteristic
393C2#310911yescharacteristic
394C2#311911yescharacteristic
395C2#326411yescharacteristic
396C2#326511yescharacteristic
397C2#331811yescharacteristic
398C2#333211yescharacteristic
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×#459C2semidirect
3. #2428C2semidirect
4. #2361C2semidirect
5. #2088C2semidirect
6. #2041C2semidirect
7. #1999C2semidirect
8. #1999C2semidirect
9. #1972C2semidirect
10. #1926C2semidirect
11. #1924C2semidirect
12. #1879C2semidirect
13. #1874C2semidirect
14. #1864C2semidirect
15. #1762C2semidirect
131. #102D4semidirect

Maximal subgroups

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

subgroupindex
C2×#4812
C2×#4592
#24282
#23612
#20882
#20412
#19992
#19722
#19262
#19242
#18792
#18742
#18642
#17622