Finite Group Databasebeta

← #32188

Subgroups — #32188

Order 256.

424 conjugacy classes of subgroups.

The subgroup lattice has 367 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#2544C211yesmaximal, characteristic, complemented
2#2512C211yesmaximal, characteristic, complemented
3#2071C211yesmaximal, characteristic, complemented
4#2055C211yesmaximal, characteristic, complemented
5#2042C211yesmaximal, characteristic, complemented
6#2012C211yesmaximal, characteristic, complemented
7#2012C211yesmaximal, characteristic, complemented
8#2011C211yesmaximal, characteristic
9#1982C211yesmaximal, characteristic, complemented
10#1972C211yesmaximal, characteristic, complemented
11#1959C211yesmaximal, characteristic, complemented
12#1927C211yesmaximal, characteristic, complemented
13#1917C211yesmaximal, characteristic, complemented
14#1879C211yesmaximal, characteristic, complemented
15#1864C211yesmaximal, characteristic, complemented
16C22×SD16C2211yescharacteristic
17C22×D8C2211yescharacteristic
18C22×D8C2211yescharacteristic
19C2×#127C2211yescharacteristic
20C2×#126—21no—
21C2×#124C2211yescharacteristic
22C2×#124C2211yescharacteristic
23C2×C4×D4C2211yescharacteristic
24C2×#106C2211yescharacteristic
25C2×C4⋊C8C2211yescharacteristic
26C2×#103C2211yescharacteristic
27C2×#103C2211yescharacteristic
28C2×#103C2211yescharacteristic
29C2×#102C2211yescharacteristic
30C2×#102C2211yescharacteristic
31C2×#102C2211yescharacteristic
32C2×#102C2211yescharacteristic
33C2×#102C2211yescharacteristic
34C2×#102C2211yescharacteristic
35C2×#102C2211yescharacteristic
36C2×#97C2211yescharacteristic
37#397—21no—
38#396—21no—
39#394—21no—
40#394—21no—
41#391C2211yescharacteristic
42#390C2211yescharacteristic
43#390C2211yescharacteristic
44#390C2211yescharacteristic
45#390C2211yescharacteristic
46#389C2211yescharacteristic
47#388—21no—
48#386—21no—
49#386—21no—
50#384C2211yescharacteristic
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×D4C2311yescharacteristic
63C22×D4C2311yescharacteristic
64C22×D4C2311yescharacteristic
65C23×C4—21no—
66C2×SD16—41no—
67C2×SD16—41no—
68C2×SD16—41no—
69C2×D8—22no—
70C2×D8—22no—
71C2×D8—41no—
72C2×D8—41no—
73C2×D8—41no—
74C2×D8—41no—
75C22×C8C2311yescharacteristic
76C22×C8C2311yescharacteristic
77C22×C8C2311yescharacteristic
78C22×C8C2311yescharacteristic
79#127—22no—
80#127—41no—
81#126—44no—
82#124—22no—
83#124—41no—
84#124—41no—
85#124—41no—
86C4×D4—41no—
87C4×D4—41no—
88C2×C4⋊C4—21no—
89C2×C4⋊C4—21no—
90C2×C4⋊C4—21no—
91C2×C4⋊C4D411yescharacteristic
92C2×C4⋊C4D411yescharacteristic
93C2×C4⋊C4C2311yescharacteristic
94C2×C4⋊C4C2311yescharacteristic
95C2×C4⋊C4C2311yescharacteristic
96C2×C4⋊C4C2311yescharacteristic
97C2×C22⋊C4—21no—
98C2×C22⋊C4—21no—
99C2×C22⋊C4—21no—
100C2×C22⋊C4—21no—
101C2×C22⋊C4—21no—
102C2×C22⋊C4—21no—
103C2×C22⋊C4—21no—
104C2×C22⋊C4—21no—
105C2×C22⋊C4—21no—
106C2×C22⋊C4—21no—
107C2×C22⋊C4—21no—
108C2×C22⋊C4—21no—
109C2×C22⋊C4—21no—
110C2×C42—21no—
111C2×C42C2311yescharacteristic
112C2×C42C2311yescharacteristic
113C2×C42C2311yescharacteristic
114#106—41no—
115C4⋊C8—41no—
116#103—22no—
117#103—41no—
118#103—41no—
119#102—22no—
120#102—41no—
121#102—41no—
122#102—41no—
123#102—41no—
124#102—41no—
125#102—41no—
126#97—41no—
127#95—21no—
128#95—21no—
129#95—21no—
130#95—21no—
131#95D411yescharacteristic
132#95D411yescharacteristic
133C24—21no—
134C24—21no—
135C24—21no—
136C2×Q8—21no—
137C2×Q8—21no—
138C2×Q8—21no—
139C2×Q8—21no—
140C2×Q8—21no—
141C2×Q8—21no—
142C2×D4—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—42no—
160C2×D4—42no—
161C2×D4—42no—
162C2×D4—42no—
163C2×D4—81no—
164C2×D4—81no—
165C2×D4#4112yes—
166C22×C4—21no—
167C22×C4—21no—
168C22×C4—21no—
169C22×C4—21no—
170C22×C4—21no—
171C22×C4—42no—
172C22×C4—81no—
173C22×C4—81no—
174C22×C4C2×D411yescharacteristic
175C22×C4C2×D411yescharacteristic
176C22×C4C2×D411yescharacteristic
177C22×C4C2×D411yescharacteristic
178C22×C4C2×D411yescharacteristic
179C22×C4C2×D411yescharacteristic
180C22×C4C2411yescharacteristic
181SD16—44no—
182D8—44no—
183D8—44no—
184C2×C8—21no—
185C2×C8—21no—
186C2×C8—21no—
187C2×C8—21no—
188C2×C8—21no—
189C2×C8—21no—
190C2×C8—21no—
191C2×C8—21no—
192C2×C8—21no—
193C2×C8—21no—
194C2×C8—21no—
195C2×C8#4112yes—
196C4⋊C4—22no—
197C4⋊C4—21no—
198C4⋊C4—21no—
199C4⋊C4—21no—
200C4⋊C4—21no—
201C4⋊C4—21no—
202C4⋊C4—21no—
203C4⋊C4—21no—
204C4⋊C4—21no—
205C4⋊C4—41no—
206C4⋊C4—42no—
207C4⋊C4—42no—
208C4⋊C4—81no—
209C22⋊C4—42no—
210C22⋊C4—42no—
211C22⋊C4—42no—
212C22⋊C4—42no—
213C22⋊C4—42no—
214C22⋊C4—42no—
215C22⋊C4—42no—
216C22⋊C4—42no—
217C22⋊C4—42no—
218C22⋊C4—81no—
219C22⋊C4—81no—
220C22⋊C4—81no—
221C22⋊C4—81no—
222C42—21no—
223C42—21no—
224C42—21no—
225C42—21no—
226C42—21no—
227C42—21no—
228C42—42no—
229C23—22no—
230C23—41no—
231C23—41no—
232C23—41no—
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—
250C23C22×D411yescharacteristic
251Q8—41no—
252Q8—41no—
253Q8—41no—
254Q8—41no—
255D4—22no—
256D4—22no—
257D4—41no—
258D4—41no—
259D4—41no—
260D4—41no—
261D4—41no—
262D4—41no—
263D4—41no—
264D4—41no—
265D4—41no—
266D4—41no—
267D4—84no—
268D4—84no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—21no—
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—41no—
287C2×C4—41no—
288C2×C4—41no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—41no—
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—84no—
302C2×C4AGL(1,ℤ/8)12yes—
303C2×C4C2×#4111yescharacteristic
304C2×C4C2×#4111yescharacteristic
305C2×C4H(5,𝔽2)11yescharacteristic
306C2×C4H(5,𝔽2)11yescharacteristic
307C2×C4H(5,𝔽2)11yescharacteristic
308C2×C4H(5,𝔽2)11yescharacteristic
309C8—22no—
310C8—41no—
311C8—41no—
312C8—41no—
313C22—42no—
314C22—42no—
315C22—42no—
316C22—81no—
317C22—81no—
318C22—81no—
319C22—81no—
320C22—81no—
321C22—81no—
322C22—81no—
323C22—81no—
324C22—81no—
325C22—81no—
326C22—81no—
327C22—81no—
328C22—81no—
329C22—81no—
330C22—81no—
331C22—81no—
332C22—81no—
333C22—81no—
334C22#53511yescharacteristic
335C22#54611yescharacteristic
336C22#55011yescharacteristic
337C22C2×AGL(1,ℤ/8)11yescharacteristic
338C22#57511yescharacteristic
339C22#57611yescharacteristic
340C22#57711yescharacteristic
341C4—22no—
342C4—41no—
343C4—41no—
344C4—41no—
345C4—41no—
346C4—41no—
347C4—81no—
348C4—81no—
349C4—81no—
350C4—81no—
351C4—81no—
352C4#53811yescharacteristic
353C4#53811yescharacteristic
354C4#55311yescharacteristic
355C4#56011yescharacteristic
356C2—82no—
357C2—161no—
358C2—161no—
359C2#271411yescharacteristic
360C2#321411yescharacteristic
361C2#322411yescharacteristic
362C2#330611yescharacteristic
363C2#332211yescharacteristic
364C2#339911yescharacteristic
365C2#340111yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radical#2011C2
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
1. #2544C2semidirect
2. #2512C2semidirect
3. #2071C2semidirect
4. #2055C2semidirect
5. #2042C2semidirect
6. #2012C2semidirect
7. #2012C2semidirect
9. #1982C2semidirect
10. #1972C2semidirect
11. #1959C2semidirect
12. #1927C2semidirect
13. #1917C2semidirect
14. #1879C2semidirect
15. #1864C2semidirect

Maximal subgroups

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

subgroupindex
#25442
#25122
#20712
#20552
#20422
#20122
#20112
#19822
#19722
#19592
#19272
#19172
#18792
#18642