Finite Group Databasebeta

← #26176

Subgroups — #26176

Order 256.

451 conjugacy classes of subgroups.

The subgroup lattice has 344 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#463C211yesmaximal, characteristic
2C2×#442C211yesmaximal, characteristic
3#2447C211yesmaximal, characteristic, complemented
4#2349C211yesmaximal, characteristic, complemented
5#2097C211yesmaximal, characteristic, complemented
6#2088C211yesmaximal, characteristic, complemented
7#2073C211yesmaximal, characteristic, complemented
8#2042C211yesmaximal, characteristic, complemented
9#1935C211yesmaximal, characteristic, complemented
10#1917C211yesmaximal, characteristic, complemented
11#1879C211yesmaximal, characteristic
12#1877C211yesmaximal, characteristic, complemented
13#1877C211yesmaximal, characteristic, complemented
14#1862C211yesmaximal, characteristic, complemented
15#1862C211yesmaximal, characteristic, complemented
16C22×SD16C2211yescharacteristic
17C22×D8C2211yescharacteristic
18C2×#124C2211yescharacteristic
19C2×C4×D4C2211yescharacteristic
20C2×C4×D4C2211yescharacteristic
21C2×C4×D4C2211yescharacteristic
22C2×C4×D4C2211yescharacteristic
23#463—24no—
24#442—24no—
25C2×#107C2211yescharacteristic
26C2×#106C2211yescharacteristic
27C2×C4⋊C8C2211yescharacteristic
28C2×#103C2211yescharacteristic
29C2×#103C2211yescharacteristic
30C2×#103C2211yescharacteristic
31C2×#102C2211yescharacteristic
32C2×#102C2211yescharacteristic
33C2×#102C2211yescharacteristic
34C2×#102C2211yescharacteristic
35C2×#102C2211yescharacteristic
36C2×#97C2211yescharacteristic
37#397—21no—
38#397—21no—
39#391C2211yescharacteristic
40#391C2211yescharacteristic
41#390C2211yescharacteristic
42#389C2211yescharacteristic
43#389C2211yescharacteristic
44#389C2211yescharacteristic
45#385—21no—
46#385—21no—
47#385—21no—
48#382C2211yescharacteristic
49#381—21no—
50#380—21no—
51C4×C4⋊C4C2211yescharacteristic
52C2×#95—21no—
53#340C2211yescharacteristic
54#340C2211yescharacteristic
55#340C2211yescharacteristic
56#340C2211yescharacteristic
57#336C2211yescharacteristic
58#336C2211yescharacteristic
59#336C2211yescharacteristic
60#336C2211yescharacteristic
61C22×Q8C2311yescharacteristic
62C22×D4C2311yescharacteristic
63C22×D4C2311yescharacteristic
64C23×C4—21no—
65C23×C4—21no—
66C23×C4—21no—
67C23×C4—21no—
68C2×SD16—22no—
69C2×SD16—22no—
70C2×SD16—22no—
71C2×D8—22no—
72C2×D8—22no—
73C2×D8D414yes—
74C22×C8C2311yescharacteristic
75C22×C8C2311yescharacteristic
76C22×C8C2311yescharacteristic
77C22×C8C2311yescharacteristic
78#124—22no—
79#124—41no—
80C4×D4—22no—
81C4×D4—22no—
82C4×D4—22no—
83C4×D4—22no—
84C4×D4—22no—
85C4×D4—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×C4⋊C4C2311yescharacteristic
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×C42C2311yescharacteristic
107C2×C42C2311yescharacteristic
108C2×C42C2311yescharacteristic
109#107—41no—
110#106—22no—
111C4⋊C8—22no—
112#103—22no—
113#103—22no—
114#103—41no—
115#102—22no—
116#102—22no—
117#102—22no—
118#102—22no—
119#102—22no—
120#97—22no—
121#95—21no—
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#95D411yescharacteristic
131#95D411yescharacteristic
132C24—21no—
133C24—21no—
134C2×Q8—21no—
135C2×Q8—21no—
136C2×Q8—21no—
137C2×Q8—21no—
138C2×Q8—21no—
139C2×Q8—21no—
140C2×D4—21no—
141C2×D4—21no—
142C2×D4—21no—
143C2×D4—21no—
144C2×D4—21no—
145C2×D4—21no—
146C2×D4—21no—
147C2×D4—21no—
148C2×D4C2×D412yes—
149C2×D4C2×D412yes—
150C2×D4#4112yes—
151C2×D4#4112yes—
152C22×C4—21no—
153C22×C4—24no—
154C22×C4—21no—
155C22×C4—21no—
156C22×C4—21no—
157C22×C4—21no—
158C22×C4—21no—
159C22×C4—42no—
160C22×C4—42no—
161C22×C4—42no—
162C22×C4—42no—
163C22×C4—42no—
164C22×C4—42no—
165C22×C4—81no—
166C22×C4—81no—
167C22×C4—81no—
168C22×C4—81no—
169C22×C4—81no—
170C22×C4C2×D411yescharacteristic
171C22×C4C2×D411yescharacteristic
172C22×C4C2×D411yescharacteristic
173C22×C4C2×D411yescharacteristic
174C22×C4C2×D411yescharacteristic
175C22×C4C2×D411yescharacteristic
176C22×C4C2411yescharacteristic
177SD16—44no—
178D8—28no—
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—
190C2×C8C2×D412yes—
191C4⋊C4—21no—
192C4⋊C4—21no—
193C4⋊C4—21no—
194C4⋊C4—21no—
195C4⋊C4—21no—
196C4⋊C4—21no—
197C4⋊C4—21no—
198C4⋊C4—21no—
199C4⋊C4—21no—
200C4⋊C4—41no—
201C4⋊C4—41no—
202C4⋊C4—81no—
203C4⋊C4—81no—
204C4⋊C4—81no—
205C4⋊C4#4112yes—
206C22⋊C4—24no—
207C22⋊C4—42no—
208C22⋊C4—42no—
209C22⋊C4—42no—
210C22⋊C4—42no—
211C22⋊C4—42no—
212C22⋊C4—42no—
213C22⋊C4—81no—
214C42—21no—
215C42—21no—
216C42—21no—
217C42—21no—
218C42—21no—
219C42—21no—
220C23—22no—
221C23—22no—
222C23—22no—
223C23—22no—
224C23—41no—
225C23—41no—
226C23—41no—
227C23—41no—
228C23—41no—
229C23—41no—
230C23—41no—
231C23—41no—
232C23—41no—
233C23—41no—
234C23C22×D411yescharacteristic
235Q8—41no—
236Q8—41no—
237Q8—41no—
238Q8—41no—
239D4—22no—
240D4—22no—
241D4—22no—
242D4—22no—
243D4—22no—
244D4—41no—
245D4—41no—
246D4#12314yes—
247C2×C4—21no—
248C2×C4—21no—
249C2×C4—22no—
250C2×C4—21no—
251C2×C4—21no—
252C2×C4—21no—
253C2×C4—21no—
254C2×C4—21no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—21no—
258C2×C4—21no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—41no—
267C2×C4—41no—
268C2×C4—41no—
269C2×C4—41no—
270C2×C4—41no—
271C2×C4—41no—
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—84no—
284C2×C4—84no—
285C2×C4—84no—
286C2×C4—84no—
287C2×C4C22×D411yescharacteristic
288C2×C4C2×#4111yescharacteristic
289C2×C4C2×#4111yescharacteristic
290C2×C4C2×#4111yescharacteristic
291C2×C4H(5,𝔽2)11yescharacteristic
292C2×C42−1+411yescharacteristic
293C8—22no—
294C8—41no—
295C8—41no—
296C8—41no—
297C22—24no—
298C22—42no—
299C22—42no—
300C22—42no—
301C22—42no—
302C22—42no—
303C22—42no—
304C22—42no—
305C22—42no—
306C22—42no—
307C22—81no—
308C22—81no—
309C22—81no—
310C22—81no—
311C22#52511yescharacteristic
312C22D4211yescharacteristic
313C22#54811yescharacteristic
314C22#57511yescharacteristic
315C22#57511yescharacteristic
316C22#57611yescharacteristic
317C22#57711yescharacteristic
318C4—41no—
319C4—41no—
320C4—42no—
321C4—41no—
322C4—41no—
323C4—41no—
324C4—41no—
325C4—81no—
326C4—81no—
327C4—81no—
328C4—81no—
329C4—81no—
330C4C2×#12311yescharacteristic
331C4#53911yescharacteristic
332C4#54711yescharacteristic
333C4#55311yescharacteristic
334C2—44no—
335C2—82no—
336C2#244311yescharacteristic
337C2#311111yescharacteristic
338C2#311811yescharacteristic
339C2#326711yescharacteristic
340C2#327311yescharacteristic
341C2#337011yescharacteristic
342C2#337611yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC22×D8C22
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
3. #2447C2semidirect
4. #2349C2semidirect
5. #2097C2semidirect
6. #2088C2semidirect
7. #2073C2semidirect
8. #2042C2semidirect
9. #1935C2semidirect
10. #1917C2semidirect
12. #1877C2semidirect
13. #1877C2semidirect
14. #1862C2semidirect
15. #1862C2semidirect

Maximal subgroups

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

subgroupindex
C2×#4632
C2×#4422
#24472
#23492
#20972
#20882
#20732
#20422
#19352
#19172
#18792
#18772
#18622