Finite Group Databasebeta

← #30614

Subgroups — #30614

Order 256.

423 conjugacy classes of subgroups.

The subgroup lattice has 354 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#470C211yesmaximal, characteristic, complemented
2C2×#464C211yesmaximal, characteristic, complemented
3#2426C211yesmaximal, characteristic
4#2423C211yesmaximal, characteristic, complemented
5#2094C211yesmaximal, characteristic, complemented
6#2088C211yesmaximal, characteristic, complemented
7#2055C211yesmaximal, characteristic, complemented
8#2042C211yesmaximal, characteristic, complemented
9#2035C211yesmaximal, characteristic, complemented
10#2000C211yesmaximal, characteristic
11#1895C211yesmaximal, characteristic
12#1878C211yesmaximal, characteristic, complemented
13#1865C211yesmaximal, characteristic, complemented
14#1851C211yesmaximal, characteristic, complemented
15#1803C211yesmaximal, characteristic, complemented
16C22×Q16C2211yescharacteristic
17C22×SD16C2211yescharacteristic
18C22×M16C2211yescharacteristic
19C2×#124C2211yescharacteristic
20C2×C22⋊Q8C2211yescharacteristic
21C2×C22⋊Q8C2211yescharacteristic
22C2×#121C2211yescharacteristic
23C2×C4×Q8C2211yescharacteristic
24C2×C4×Q8C2211yescharacteristic
25#470—24no—
26#464—24no—
27C2×#106C2211yescharacteristic
28C2×#106C2211yescharacteristic
29C2×C4⋊C8C2211yescharacteristic
30C2×C4⋊C8C2211yescharacteristic
31C2×#103C2211yescharacteristic
32C2×#103C2211yescharacteristic
33C2×#103C2211yescharacteristic
34C2×#103C2211yescharacteristic
35C2×#103C2211yescharacteristic
36C2×#102C2211yescharacteristic
37C2×#102C2211yescharacteristic
38C2×#102C2211yescharacteristic
39C2×C22⋊C8C2211yescharacteristic
40#399—21no—
41#398—21no—
42#396—21no—
43#394—21no—
44#391C2211yescharacteristic
45#390—21no—
46#390C2211yescharacteristic
47#389C2211yescharacteristic
48#389C2211yescharacteristic
49#388—21no—
50#388—21no—
51#387—21no—
52#384C2211yescharacteristic
53#380C2211yescharacteristic
54C4×C22⋊C4C2211yescharacteristic
55#340C2211yescharacteristic
56#340C2211yescharacteristic
57#340C2211yescharacteristic
58#340C2211yescharacteristic
59#336C2211yescharacteristic
60#336C2211yescharacteristic
61C22×Q8C2311yescharacteristic
62C22×Q8C2311yescharacteristic
63C22×D4—21no—
64C22×D4C2311yescharacteristic
65C23×C4C2311yescharacteristic
66C2×Q16—41no—
67C2×Q16—41no—
68C2×Q16D414yescomplemented
69C2×SD16—22no—
70C2×SD16—41no—
71C2×SD16—41no—
72C2×M16—22no—
73C2×M16—41no—
74C2×M16—41no—
75C22×C8C2311yescharacteristic
76C22×C8C2311yescharacteristic
77C22×C8C2311yescharacteristic
78C22×C8C2311yescharacteristic
79#124—22no—
80#124—41no—
81C22⋊Q8—22no—
82C22⋊Q8—22no—
83C22⋊Q8—41no—
84C22⋊Q8—41no—
85#121—22no—
86#121—41no—
87C4×Q8—22no—
88C4×Q8—22no—
89C4×Q8—41no—
90C4×Q8—41no—
91C2×C4⋊C4—21no—
92C2×C4⋊C4—21no—
93C2×C4⋊C4—21no—
94C2×C4⋊C4—21no—
95C2×C4⋊C4—21no—
96C2×C4⋊C4—21no—
97C2×C4⋊C4—21no—
98C2×C4⋊C4—21no—
99C2×C4⋊C4C2311yescharacteristic
100C2×C4⋊C4C2311yescharacteristic
101C2×C4⋊C4C2311yescharacteristic
102C2×C4⋊C4C2311yescharacteristic
103C2×C4⋊C4C2311yescharacteristic
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×C22⋊C4—21no—
111C2×C22⋊C4—21no—
112C2×C22⋊C4D411yescharacteristic
113C2×C22⋊C4D411yescharacteristic
114C2×C42—21no—
115C2×C42—21no—
116C2×C42C2311yescharacteristic
117C2×C42C2311yescharacteristic
118#106—22no—
119#106—41no—
120C4⋊C8—22no—
121C4⋊C8—41no—
122#103—22no—
123#103—22no—
124#103—22no—
125#103—22no—
126#103—41no—
127#102—22no—
128#102—41no—
129#102—41no—
130C22⋊C8—41no—
131#95—21no—
132#95—21no—
133#95—21no—
134#95—21no—
135#95—21no—
136#95D411yescharacteristic
137#95D411yescharacteristic
138C24—21no—
139C24C2×D411yescharacteristic
140C2×Q8—21no—
141C2×Q8—21no—
142C2×Q8—21no—
143C2×Q8—21no—
144C2×Q8—21no—
145C2×Q8—21no—
146C2×Q8—21no—
147C2×Q8—21no—
148C2×Q8—21no—
149C2×Q8—21no—
150C2×Q8C2×D412yes—
151C2×Q8C2×D412yes—
152C2×D4—21no—
153C2×D4—21no—
154C2×D4—21no—
155C2×D4—21no—
156C2×D4—21no—
157C2×D4—21no—
158C2×D4—42no—
159C2×D4—42no—
160C2×D4—81no—
161C22×C4—21no—
162C22×C4—21no—
163C22×C4—21no—
164C22×C4—21no—
165C22×C4—21no—
166C22×C4—21no—
167C22×C4—21no—
168C22×C4—21no—
169C22×C4—21no—
170C22×C4—21no—
171C22×C4—21no—
172C22×C4—21no—
173C22×C4—21no—
174C22×C4C2×D411yescharacteristic
175C22×C4C2×D411yescharacteristic
176C22×C4C2×D411yescharacteristic
177C22×C4C2×D411yescharacteristic
178C22×C4C2×D411yescharacteristic
179C22×C4C2411yescharacteristic
180Q16—44no—
181SD16—44no—
182M16—44no—
183C2×C8—21no—
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×C8C2×D412yes—
195C4⋊C4—21no—
196C4⋊C4—21no—
197C4⋊C4—21no—
198C4⋊C4—21no—
199C4⋊C4—21no—
200C4⋊C4—21no—
201C4⋊C4—21no—
202C4⋊C4—21no—
203C4⋊C4—21no—
204C4⋊C4—42no—
205C4⋊C4—42no—
206C4⋊C4—42no—
207C4⋊C4—42no—
208C4⋊C4—42no—
209C4⋊C4—81no—
210C4⋊C4—81no—
211C4⋊C4—81no—
212C4⋊C4#4112yes—
213C22⋊C4—22no—
214C22⋊C4—41no—
215C22⋊C4—42no—
216C22⋊C4—42no—
217C22⋊C4—42no—
218C22⋊C4—42no—
219C22⋊C4—42no—
220C22⋊C4—42no—
221C22⋊C4—81no—
222C22⋊C4—81no—
223C42—21no—
224C42—21no—
225C42—21no—
226C42—21no—
227C42—42no—
228C42—42no—
229C23—21no—
230C23—21no—
231C23—21no—
232C23—21no—
233C23—21no—
234C23—21no—
235C23—21no—
236C23—41no—
237C23—41no—
238C23—41no—
239C23—41no—
240C23—41no—
241C23—41no—
242C23—41no—
243C23C22×D411yescharacteristic
244Q8—22no—
245Q8—22no—
246Q8—22no—
247Q8—22no—
248Q8—41no—
249Q8—41no—
250Q8—41no—
251Q8—41no—
252D4—41no—
253D4—41no—
254D4—41no—
255D4—41no—
256D4—84no—
257C2×C4—21no—
258C2×C4—21no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—21no—
262C2×C4—22no—
263C2×C4—22no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—22no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—21no—
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—41no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—41no—
295C2×C4—41no—
296C2×C4#13712yes—
297C2×C4C22×D411yescharacteristic
298C2×C4C2×#4111yescharacteristic
299C2×C4H(5,𝔽2)11yescharacteristic
300C2×C4H(5,𝔽2)11yescharacteristic
301C2×C4H(5,𝔽2)11yescharacteristic
302C2×C42−1+411yescharacteristic
303C8—22no—
304C8—41no—
305C8—41no—
306C8—41no—
307C22—41no—
308C22—41no—
309C22—41no—
310C22—41no—
311C22—41no—
312C22—41no—
313C22—41no—
314C22—81no—
315C22—81no—
316C22—81no—
317C22—81no—
318C22—81no—
319C22—81no—
320C22—81no—
321C22#53711yescharacteristic
322C22D4211yescharacteristic
323C22#54611yescharacteristic
324C22C2×#13711yescharacteristic
325C22#57511yescharacteristic
326C22#57711yescharacteristic
327C22#57711yescharacteristic
328C4—22no—
329C4—41no—
330C4—42no—
331C4—42no—
332C4—41no—
333C4—41no—
334C4—41no—
335C4—42no—
336C4—81no—
337C4—81no—
338C4—81no—
339C4—81no—
340C4#53411yescharacteristic
341C4#54011yescharacteristic
342C4#54711yescharacteristic
343C4#56211yescharacteristic
344C2—81no—
345C2—161no—
346C2#267111yescharacteristic
347C2#325211yescharacteristic
348C2#326311yescharacteristic
349C2#327111yescharacteristic
350C2#327811yescharacteristic
351C2#331611yescharacteristic
352C2#332311yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC2×#121C22
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×#470C2semidirect
2. C2×#464C2semidirect
4. #2423C2semidirect
5. #2094C2semidirect
6. #2088C2semidirect
7. #2055C2semidirect
8. #2042C2semidirect
9. #2035C2semidirect
12. #1878C2semidirect
13. #1865C2semidirect
14. #1851C2semidirect
15. #1803C2semidirect
68. C2×Q16D4semidirect

Maximal subgroups

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

subgroupindex
C2×#4702
C2×#4642
#24262
#24232
#20942
#20882
#20552
#20422
#20352
#20002
#18952
#18782
#18652
#18512
#18032