Finite Group Databasebeta

← #32180

Subgroups — #32180

Order 256.

454 conjugacy classes of subgroups.

The subgroup lattice has 374 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#2541C211yesmaximal, characteristic
2#2514C211yesmaximal, characteristic, complemented
3#2073C211yesmaximal, characteristic, complemented
4#2055C211yesmaximal, characteristic, complemented
5#2041C211yesmaximal, characteristic, complemented
6#2014C211yesmaximal, characteristic, complemented
7#2014C211yesmaximal, characteristic, complemented
8#2014C211yesmaximal, characteristic, complemented
9#1982C211yesmaximal, characteristic, complemented
10#1973C211yesmaximal, characteristic, complemented
11#1961C211yesmaximal, characteristic, complemented
12#1925C211yesmaximal, characteristic, complemented
13#1917C211yesmaximal, characteristic, complemented
14#1877C211yesmaximal, characteristic, complemented
15#1862C211yesmaximal, characteristic, complemented
16C22×SD16C2211yescharacteristic
17C22×SD16C2211yescharacteristic
18C22×SD16C2211yescharacteristic
19C2×#127C2211yescharacteristic
20C2×#127C2211yescharacteristic
21C2×#127C2211yescharacteristic
22C2×#121—21no—
23C2×#121—21no—
24C2×C4×D4C2211yescharacteristic
25C2×#107C2211yescharacteristic
26C2×C4⋊C8C2211yescharacteristic
27C2×#103C2211yescharacteristic
28C2×#102C2211yescharacteristic
29C2×#102C2211yescharacteristic
30C2×#102C2211yescharacteristic
31C2×#102C2211yescharacteristic
32C2×#102C2211yescharacteristic
33C2×#102C2211yescharacteristic
34C2×#102C2211yescharacteristic
35C2×#102C2211yescharacteristic
36C2×#102C2211yescharacteristic
37C2×#97C2211yescharacteristic
38#401—21no—
39#400—21no—
40#392—21no—
41#392—21no—
42#391C2211yescharacteristic
43#391C2211yescharacteristic
44#391C2211yescharacteristic
45#390C2211yescharacteristic
46#390C2211yescharacteristic
47#389C2211yescharacteristic
48#387—21no—
49#383—21no—
50#382C2211yescharacteristic
51C4×C4⋊C4C2211yescharacteristic
52C4×C4⋊C4C2211yescharacteristic
53#340C2211yescharacteristic
54#340C2211yescharacteristic
55#336C2211yescharacteristic
56#336C2211yescharacteristic
57#336C2211yescharacteristic
58#336C2211yescharacteristic
59C22×Q8C2311yescharacteristic
60C22×D4—21no—
61C22×D4—21no—
62C22×D4—21no—
63C22×D4—21no—
64C22×D4—21no—
65C22×D4—21no—
66C22×D4C2311yescharacteristic
67C22×D4C2311yescharacteristic
68C22×D4C2311yescharacteristic
69C23×C4—21no—
70C2×SD16—22no—
71C2×SD16—22no—
72C2×SD16—41no—
73C2×SD16—41no—
74C2×SD16—41no—
75C2×SD16—41no—
76C2×SD16—41no—
77C2×SD16—41no—
78C2×SD16—41no—
79C22×C8C2311yescharacteristic
80C22×C8C2311yescharacteristic
81C22×C8C2311yescharacteristic
82C22×C8C2311yescharacteristic
83#127—22no—
84#127—22no—
85#127—41no—
86#127—41no—
87#127—41no—
88#127—41no—
89#121—44no—
90#121—44no—
91C4×D4—41no—
92C4×D4—41no—
93C2×C4⋊C4—21no—
94C2×C4⋊C4D411yescharacteristic
95C2×C4⋊C4D411yescharacteristic
96C2×C4⋊C4C2311yescharacteristic
97C2×C4⋊C4C2311yescharacteristic
98C2×C4⋊C4C2311yescharacteristic
99C2×C4⋊C4C2311yescharacteristic
100C2×C22⋊C4—21no—
101C2×C22⋊C4—21no—
102C2×C22⋊C4—21no—
103C2×C22⋊C4—21no—
104C2×C22⋊C4—21no—
105C2×C42—21no—
106C2×C42C2311yescharacteristic
107C2×C42C2311yescharacteristic
108C2×C42C2311yescharacteristic
109#107—41no—
110C4⋊C8—41no—
111#103—41no—
112#102—22no—
113#102—22no—
114#102—41no—
115#102—41no—
116#102—41no—
117#102—41no—
118#102—41no—
119#102—41no—
120#102—41no—
121#97—41no—
122#95—21no—
123#95—21no—
124#95—21no—
125#95—21no—
126#95—21no—
127#95—21no—
128#95—21no—
129#95—21no—
130#95—21no—
131#95—21no—
132#95D411yescharacteristic
133#95D411yescharacteristic
134C24—21no—
135C24—21no—
136C24—21no—
137C2×Q8—21no—
138C2×Q8—21no—
139C2×Q8—21no—
140C2×Q8—21no—
141C2×Q8—21no—
142C2×Q8—21no—
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—42no—
171C2×D4—42no—
172C2×D4—81no—
173C2×D4—81no—
174C2×D4—81no—
175C2×D4—81no—
176C2×D4—81no—
177C2×D4—81no—
178C2×D4#4112yes—
179C22×C4—21no—
180C22×C4—21no—
181C22×C4—21no—
182C22×C4—21no—
183C22×C4—21no—
184C22×C4—42no—
185C22×C4—42no—
186C22×C4—42no—
187C22×C4C2×D411yescharacteristic
188C22×C4C2×D411yescharacteristic
189C22×C4C2×D411yescharacteristic
190C22×C4C2×D411yescharacteristic
191C22×C4C2×D411yescharacteristic
192C22×C4C2×D411yescharacteristic
193C22×C4C2411yescharacteristic
194SD16—44no—
195SD16—44no—
196SD16—44no—
197C2×C8—21no—
198C2×C8—21no—
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×C8#4112yes—
209C4⋊C4—22no—
210C4⋊C4—22no—
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—
220C22⋊C4—42no—
221C22⋊C4—42no—
222C22⋊C4—42no—
223C22⋊C4—42no—
224C22⋊C4—42no—
225C42—21no—
226C42—21no—
227C42—21no—
228C42—21no—
229C42—21no—
230C42—21no—
231C42—42no—
232C23—22no—
233C23—41no—
234C23—41no—
235C23—41no—
236C23—41no—
237C23—41no—
238C23—41no—
239C23—41no—
240C23—41no—
241C23—41no—
242C23—41no—
243C23—41no—
244C23—41no—
245C23—41no—
246C23—41no—
247C23—41no—
248C23—41no—
249C23—41no—
250C23—41no—
251C23—41no—
252C23—41no—
253C23C22×D411yescharacteristic
254Q8—41no—
255Q8—41no—
256Q8—41no—
257Q8—41no—
258D4—22no—
259D4—22no—
260D4—41no—
261D4—41no—
262D4—41no—
263D4—41no—
264D4—41no—
265D4—41no—
266D4—41no—
267D4—41no—
268D4—41no—
269D4—41no—
270D4—84no—
271D4—84no—
272D4—84no—
273D4—84no—
274D4—84no—
275D4—84no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
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—84no—
308C2×C4AGL(1,ℤ/8)12yes—
309C2×C4AGL(1,ℤ/8)12yes—
310C2×C4C2×#4111yescharacteristic
311C2×C4C2×#4111yescharacteristic
312C2×C4H(5,𝔽2)11yescharacteristic
313C2×C4H(5,𝔽2)11yescharacteristic
314C2×C4H(5,𝔽2)11yescharacteristic
315C2×C4H(5,𝔽2)11yescharacteristic
316C8—22no—
317C8—41no—
318C8—41no—
319C8—41no—
320C22—42no—
321C22—42no—
322C22—42no—
323C22—81no—
324C22—81no—
325C22—81no—
326C22—81no—
327C22—81no—
328C22—81no—
329C22—81no—
330C22—81no—
331C22—81no—
332C22—81no—
333C22—81no—
334C22—81no—
335C22—81no—
336C22—81no—
337C22—81no—
338C22—81no—
339C22—81no—
340C22—81no—
341C22#53511yescharacteristic
342C22#54611yescharacteristic
343C22#55011yescharacteristic
344C22C2×AGL(1,ℤ/8)11yescharacteristic
345C22C2×AGL(1,ℤ/8)11yescharacteristic
346C22#57711yescharacteristic
347C22#57711yescharacteristic
348C4—22no—
349C4—22no—
350C4—41no—
351C4—41no—
352C4—41no—
353C4—41no—
354C4—81no—
355C4—81no—
356C4—81no—
357C4—81no—
358C4—81no—
359C4#53811yescharacteristic
360C4#53811yescharacteristic
361C4#55311yescharacteristic
362C4#56011yescharacteristic
363C2—82no—
364C2—161no—
365C2—161no—
366C2#271411yescharacteristic
367C2#321711yescharacteristic
368C2#322211yescharacteristic
369C2#331011yescharacteristic
370C2#332411yescharacteristic
371C2#339711yescharacteristic
372C2#339711yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radical#2541C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #2514C2semidirect
3. #2073C2semidirect
4. #2055C2semidirect
5. #2041C2semidirect
6. #2014C2semidirect
7. #2014C2semidirect
8. #2014C2semidirect
9. #1982C2semidirect
10. #1973C2semidirect
11. #1961C2semidirect
12. #1925C2semidirect
13. #1917C2semidirect
14. #1877C2semidirect
15. #1862C2semidirect

Maximal subgroups

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

subgroupindex
#25412
#25142
#20732
#20552
#20412
#20142
#19822
#19732
#19612
#19252
#19172
#18772
#18622