Finite Group Databasebeta

← #30044

Subgroups — #30044

Order 256.

407 conjugacy classes of subgroups.

The subgroup lattice has 348 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#2861C211yesmaximal, characteristic, complemented
2C4×#124C211yesmaximal, characteristic, complemented
3#2074C211yesmaximal, characteristic, complemented
4#2073C211yesmaximal, characteristic, complemented
5#2026C211yesmaximal, characteristic, complemented
6#2025C211yesmaximal, characteristic, complemented
7#2015C211yesmaximal, characteristic, complemented
8#2015C211yesmaximal, characteristic, complemented
9#2012C211yesmaximal, characteristic
10#2012C211yesmaximal, characteristic
11#1880C211yesmaximal, characteristic, complemented
12#1879C211yesmaximal, characteristic
13#1878C211yesmaximal, characteristic, complemented
14#1877C211yesmaximal, characteristic, complemented
15#1839C211yesmaximal, characteristic, complemented
16C22×Q16C2211yescharacteristic
17C22×SD16C2211yescharacteristic
18C22×SD16C2211yescharacteristic
19C22×D8C2211yescharacteristic
20C2×#125C2211yescharacteristic
21C2×#124C2211yescharacteristic
22C2×#124C2211yescharacteristic
23C2×C4×Q8C2211yescharacteristic
24C2×C4×D4C2211yescharacteristic
25C2×#103C2211yescharacteristic
26C2×#103C2211yescharacteristic
27C2×#103C2211yescharacteristic
28C2×#103C2211yescharacteristic
29C2×#102C2211yescharacteristic
30C2×#102C2211yescharacteristic
31C2×#102C2211yescharacteristic
32C2×#102C2211yescharacteristic
33#391C2211yescharacteristic
34#391C2211yescharacteristic
35#391C2211yescharacteristic
36#390C2211yescharacteristic
37#390C2211yescharacteristic
38#390C2211yescharacteristic
39#388—21no—
40#388—21no—
41#388—21no—
42#388—21no—
43#388—21no—
44#388—21no—
45#382C2211yescharacteristic
46C4×C22⋊C4—21no—
47C4×C22⋊C4—21no—
48C43C2211yescharacteristic
49#340C2211yescharacteristic
50#340C2211yescharacteristic
51#340C2211yescharacteristic
52#340C2211yescharacteristic
53#336C2211yescharacteristic
54#336C2211yescharacteristic
55#336C2211yescharacteristic
56#336C2211yescharacteristic
57#336C2211yescharacteristic
58#336C2211yescharacteristic
59C22×Q8C2311yescharacteristic
60C22×Q8C2311yescharacteristic
61C22×D4C2311yescharacteristic
62C22×D4C2311yescharacteristic
63C23×C4—21no—
64C2×Q16—22no—
65C2×Q16—41no—
66C2×Q16—41no—
67C2×SD16—22no—
68C2×SD16—22no—
69C2×SD16—41no—
70C2×SD16—41no—
71C2×SD16—41no—
72C2×SD16—41no—
73C2×D8—22no—
74C2×D8—41no—
75C2×D8—41no—
76C22×C8C2311yescharacteristic
77C22×C8C2311yescharacteristic
78C22×C8C2311yescharacteristic
79C22×C8C2311yescharacteristic
80#125—41no—
81#125—41no—
82#124—22no—
83#124—22no—
84#124—41no—
85#124—41no—
86C4×Q8—41no—
87C4×Q8—41no—
88C4×D4—41no—
89C4×D4—41no—
90C2×C4⋊C4—21no—
91C2×C4⋊C4—21no—
92C2×C4⋊C4—21no—
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×C42—21no—
109C2×C42D411yescharacteristic
110C2×C42D411yescharacteristic
111C2×C42D411yescharacteristic
112C2×C42D411yescharacteristic
113C2×C42C2311yescharacteristic
114C2×C42C2311yescharacteristic
115C2×C42C2311yescharacteristic
116#103—41no—
117#103—41no—
118#103—41no—
119#103—41no—
120#102—41no—
121#102—41no—
122#102—41no—
123#102—41no—
124#95—21no—
125#95—21no—
126#95—21no—
127#95—21no—
128#95—21no—
129#95—21no—
130#95—21no—
131#95—21no—
132C24—21no—
133C24—21no—
134C2×Q8—21no—
135C2×Q8—21no—
136C2×Q8—21no—
137C2×Q8—21no—
138C2×Q8—21no—
139C2×Q8—21no—
140C2×Q8—21no—
141C2×Q8—21no—
142C2×Q8—21no—
143C2×Q8—21no—
144C2×Q8—21no—
145C2×Q8#4112yes—
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#4112yes—
158C22×C4—21no—
159C22×C4—21no—
160C22×C4—21no—
161C22×C4—21no—
162C22×C4—21no—
163C22×C4—21no—
164C22×C4—42no—
165C22×C4—42no—
166C22×C4—42no—
167C22×C4C2×D411yescharacteristic
168C22×C4C2×D411yescharacteristic
169C22×C4C2×D411yescharacteristic
170C22×C4C2×D411yescharacteristic
171C22×C4C2×D411yescharacteristic
172C22×C4C2×D411yescharacteristic
173C22×C4C2411yescharacteristic
174Q16—44no—
175SD16—44no—
176SD16—44no—
177D8—44no—
178C2×C8—21no—
179C2×C8—21no—
180C2×C8—21no—
181C2×C8—21no—
182C2×C8—21no—
183C2×C8—21no—
184C2×C8—21no—
185C2×C8—21no—
186C2×C8—21no—
187C2×C8—21no—
188C2×C8—21no—
189C2×C8—21no—
190C4⋊C4—21no—
191C4⋊C4—21no—
192C4⋊C4—21no—
193C4⋊C4—21no—
194C4⋊C4—21no—
195C4⋊C4—21no—
196C4⋊C4—21no—
197C4⋊C4—21no—
198C4⋊C4—42no—
199C4⋊C4—42no—
200C4⋊C4—42no—
201C22⋊C4—24no—
202C22⋊C4—24no—
203C22⋊C4—42no—
204C22⋊C4—42no—
205C22⋊C4—42no—
206C22⋊C4—42no—
207C22⋊C4—42no—
208C22⋊C4—42no—
209C22⋊C4—42no—
210C22⋊C4—42no—
211C22⋊C4—42no—
212C42—22no—
213C42—21no—
214C42—21no—
215C42—21no—
216C42—21no—
217C42—21no—
218C42—21no—
219C42—41no—
220C42—41no—
221C42—41no—
222C42—42no—
223C23—22no—
224C23—41no—
225C23—41no—
226C23—41no—
227C23—41no—
228C23—41no—
229C23—41no—
230C23—41no—
231C23—41no—
232C23—41no—
233C23—41no—
234C23—41no—
235C23—41no—
236C23—41no—
237C23C22×D411yescharacteristic
238Q8—22no—
239Q8—22no—
240Q8—41no—
241Q8—41no—
242Q8—41no—
243Q8—41no—
244Q8—41no—
245Q8—41no—
246D4—22no—
247D4—22no—
248D4—41no—
249D4—41no—
250D4—41no—
251D4—41no—
252D4—41no—
253D4—41no—
254C2×C4—21no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—21no—
258C2×C4—22no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—41no—
273C2×C4—41no—
274C2×C4—41no—
275C2×C4—41no—
276C2×C4—41no—
277C2×C4—41no—
278C2×C4—41no—
279C2×C4—41no—
280C2×C4—41no—
281C2×C4—41no—
282C2×C4—41no—
283C2×C4—41no—
284C2×C4—41no—
285C2×C4—41no—
286C2×C4—41no—
287C2×C4—41no—
288C2×C4—41no—
289C2×C4—84no—
290C2×C4#13512yes—
291C2×C4C2×#4111yescharacteristic
292C2×C4C2×#4111yescharacteristic
293C2×C4H(5,𝔽2)11yescharacteristic
294C2×C4H(5,𝔽2)11yescharacteristic
295C2×C4H(5,𝔽2)11yescharacteristic
296C2×C4H(5,𝔽2)11yescharacteristic
297C8—41no—
298C8—41no—
299C8—41no—
300C8—41no—
301C22—42no—
302C22—42no—
303C22—42no—
304C22—81no—
305C22—81no—
306C22—81no—
307C22—81no—
308C22—81no—
309C22—81no—
310C22—81no—
311C22—81no—
312C22—81no—
313C22—81no—
314C22—81no—
315C22#52511yescharacteristic
316C22#53511yescharacteristic
317C22#53511yescharacteristic
318C22C2×#13511yescharacteristic
319C22#57511yescharacteristic
320C22#57511yescharacteristic
321C22#57511yescharacteristic
322C4—22no—
323C4—41no—
324C4—41no—
325C4—42no—
326C4—41no—
327C4—41no—
328C4—41no—
329C4—81no—
330C4—81no—
331C4—81no—
332C4—81no—
333C4—81no—
334C4#53811yescharacteristic
335C4#53811yescharacteristic
336C4#53811yescharacteristic
337C4#53811yescharacteristic
338C2—82no—
339C2—161no—
340C2#263111yescharacteristic
341C2#310211yescharacteristic
342C2#311111yescharacteristic
343C2#321111yescharacteristic
344C2#321211yescharacteristic
345C2#321911yescharacteristic
346C2#321911yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC22×D8C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C43 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #2861C2semidirect
2. C4×#124C2semidirect
3. #2074C2semidirect
4. #2073C2semidirect
5. #2026C2semidirect
6. #2025C2semidirect
7. #2015C2semidirect
8. #2015C2semidirect
11. #1880C2semidirect
13. #1878C2semidirect
14. #1877C2semidirect
15. #1839C2semidirect

Maximal subgroups

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

subgroupindex
#28612
C4×#1242
#20742
#20732
#20262
#20252
#20152
#20122
#18802
#18792
#18782
#18772
#18392