Finite Group Databasebeta

← #25238

Subgroups — #25238

Order 256.

484 conjugacy classes of subgroups.

The subgroup lattice has 363 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#481C211yesmaximal, characteristic, complemented
2C2×C8×D4C211yesmaximal, characteristic, complemented
3#2487C211yesmaximal, characteristic, complemented
4#2416C211yesmaximal, characteristic
5#2088C211yesmaximal, characteristic, complemented
6#2077C211yesmaximal, characteristic, complemented
7#2065C211yesmaximal, characteristic, complemented
8#2034C211yesmaximal, characteristic, complemented
9#1941C211yesmaximal, characteristic, complemented
10#1934C211yesmaximal, characteristic, complemented
11#1893C211yesmaximal, characteristic, complemented
12#1893C211yesmaximal, characteristic, complemented
13#1853C211yesmaximal, characteristic, complemented
14#1807C211yesmaximal, characteristic, complemented
15C2×#340C211yesmaximal, characteristic, complemented
16C23×C8C2211yescharacteristic
17C23×C8C2211yescharacteristic
18C2×#121—21no—
19C2×#121C2211yescharacteristic
20C2×#121C2211yescharacteristic
21C2×C4×D4C2211yescharacteristic
22C22×C4⋊C4C2211yescharacteristic
23C22×C4⋊C4C2211yescharacteristic
24#481—24no—
25C8×D4—24no—
26C2×#106C2211yescharacteristic
27C2×#106C2211yescharacteristic
28C2×#106C2211yescharacteristic
29C2×#106C2211yescharacteristic
30C2×C4⋊C8C2211yescharacteristic
31C2×#102C2211yescharacteristic
32C2×#102C2211yescharacteristic
33C2×#102C2211yescharacteristic
34C2×#102C2211yescharacteristic
35C2×C22⋊C8C2211yescharacteristic
36C2×C22⋊C8C2211yescharacteristic
37C2×C4×C8C2211yescharacteristic
38#400—21no—
39#399—21no—
40#396—21no—
41#394—21no—
42#390C2211yescharacteristic
43#389C2211yescharacteristic
44#389C2211yescharacteristic
45#389C2211yescharacteristic
46#386—21no—
47#386—21no—
48#385—21no—
49#380C2211yescharacteristic
50#380C2211yescharacteristic
51#380C2211yescharacteristic
52#380C2211yescharacteristic
53#340C2211yescharacteristic
54#340C2211yescharacteristic
55#340C2211yescharacteristic
56#340C2211yescharacteristic
57#340C2211yescharacteristic
58#340C2211yescharacteristic
59#340C2211yescharacteristic
60#340C2211yescharacteristic
61C22×D4—21no—
62C22×D4—21no—
63C22×D4D411yescharacteristic
64C22×D4C2311yescharacteristic
65C23×C4—21no—
66C23×C4—21no—
67C23×C4C2311yescharacteristic
68C23×C4C2311yescharacteristic
69C22×C8—22no—
70C22×C8—22no—
71C22×C8—22no—
72C22×C8—22no—
73C22×C8—41no—
74C22×C8—41no—
75C22×C8C2311yescharacteristic
76C22×C8C2311yescharacteristic
77C22×C8C2311yescharacteristic
78C22×C8C2311yescharacteristic
79#121—22no—
80#121—22no—
81#121—22no—
82#121—41no—
83#121—44no—
84C4×D4—22no—
85C4×D4—22no—
86C2×C4⋊C4—22no—
87C2×C4⋊C4—22no—
88C2×C4⋊C4—22no—
89C2×C4⋊C4—21no—
90C2×C4⋊C4—21no—
91C2×C4⋊C4—21no—
92C2×C4⋊C4—21no—
93C2×C4⋊C4—21no—
94C2×C4⋊C4—41no—
95C2×C4⋊C4D411yescharacteristic
96C2×C4⋊C4C2311yescharacteristic
97C2×C4⋊C4C2311yescharacteristic
98C2×C4⋊C4C2311yescharacteristic
99C2×C4⋊C4C2311yescharacteristic
100C2×C4⋊C4C2311yescharacteristic
101C2×C4⋊C4C2311yescharacteristic
102C2×C4⋊C4C2311yescharacteristic
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×C22⋊C4—21no—
111C2×C22⋊C4D411yescharacteristic
112C2×C22⋊C4D411yescharacteristic
113C2×C42C2311yescharacteristic
114#106—22no—
115#106—22no—
116#106—22no—
117#106—41no—
118C4⋊C8—22no—
119#102—22no—
120#102—22no—
121#102—22no—
122#102—41no—
123C22⋊C8—22no—
124C22⋊C8D414yescomplemented
125C4×C8—22no—
126#95—21no—
127#95—21no—
128#95—21no—
129#95—21no—
130#95—21no—
131#95—21no—
132#95—21no—
133C24—21no—
134C24C2×D411yescharacteristic
135C24C2×D411yescharacteristic
136C2×D4—22no—
137C2×D4—22no—
138C2×D4—21no—
139C2×D4—21no—
140C2×D4—21no—
141C2×D4—21no—
142C2×D4—21no—
143C2×D4—21no—
144C2×D4—42no—
145C2×D4—42no—
146C2×D4—42no—
147C2×D4—42no—
148C2×D4—42no—
149C2×D4—81no—
150C2×D4SD1614yes—
151C22×C4—21no—
152C22×C4—21no—
153C22×C4—21no—
154C22×C4—21no—
155C22×C4—21no—
156C22×C4—21no—
157C22×C4—21no—
158C22×C4—21no—
159C22×C4—21no—
160C22×C4—21no—
161C22×C4—21no—
162C22×C4—21no—
163C22×C4—21no—
164C22×C4—21no—
165C22×C4—21no—
166C22×C4—21no—
167C22×C4—21no—
168C22×C4—42no—
169C22×C4—42no—
170C22×C4—42no—
171C22×C4—81no—
172C22×C4—81no—
173C22×C4—81no—
174C22×C4C2×D411yescharacteristic
175C22×C4C2×D411yescharacteristic
176C22×C4C2×D411yescharacteristic
177C22×C4C2×D412yes—
178C22×C4C2×D411yescharacteristic
179C22×C4#4112yes—
180C22×C4C2411yescharacteristic
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×C8—44no—
191C2×C8—44no—
192C2×C8C2×D412yes—
193C2×C8C2×D412yes—
194C2×C8#4112yes—
195C4⋊C4—22no—
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—21no—
205C4⋊C4—21no—
206C4⋊C4—21no—
207C4⋊C4—21no—
208C4⋊C4—21no—
209C4⋊C4—42no—
210C4⋊C4—42no—
211C4⋊C4—42no—
212C4⋊C4—81no—
213C4⋊C4—81no—
214C4⋊C4#4112yes—
215C22⋊C4—22no—
216C22⋊C4—22no—
217C22⋊C4—42no—
218C22⋊C4—42no—
219C22⋊C4—42no—
220C22⋊C4—42no—
221C22⋊C4—42no—
222C22⋊C4—81no—
223C22⋊C4—81no—
224C22⋊C4—81no—
225C42—21no—
226C42—21no—
227C23—21no—
228C23—21no—
229C23—21no—
230C23—21no—
231C23—21no—
232C23—21no—
233C23—21no—
234C23—21no—
235C23—21no—
236C23—21no—
237C23—41no—
238C23—41no—
239C23—41no—
240C23—41no—
241C23—41no—
242C23—41no—
243C23—41no—
244C23#12112yes—
245C23C2×SD1612yes—
246C23C2×SD1612yes—
247C23#13512yes—
248C23C22×D411yescharacteristic
249D4—28no—
250D4—41no—
251D4—41no—
252D4—41no—
253D4—41no—
254D4—84no—
255D4—84no—
256C2×C4—21no—
257C2×C4—21no—
258C2×C4—21no—
259C2×C4—21no—
260C2×C4—22no—
261C2×C4—22no—
262C2×C4—22no—
263C2×C4—22no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—22no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—21no—
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—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—84no—
296C2×C4—84no—
297C2×C4#12112yes—
298C2×C4C2×SD1612yes—
299C2×C4C22×D411yescharacteristic
300C2×C4C2×#4111yescharacteristic
301C2×C4C2×#4111yescharacteristic
302C2×C4C2×#4111yescharacteristic
303C2×C4H(5,𝔽2)11yescharacteristic
304C2×C42−1+411yescharacteristic
305C8—22no—
306C8—22no—
307C8—22no—
308C8—41no—
309C22—22no—
310C22—22no—
311C22—22no—
312C22—22no—
313C22—22no—
314C22—22no—
315C22—22no—
316C22—22no—
317C22—22no—
318C22—41no—
319C22—41no—
320C22—41no—
321C22—41no—
322C22—81no—
323C22—81no—
324C22—81no—
325C22—81no—
326C22—81no—
327C22—81no—
328C22—81no—
329C22#46514yes—
330C22C2×#12111yescharacteristic
331C22#54611yescharacteristic
332C22#54711yescharacteristic
333C22C22×SD1611yescharacteristic
334C22C2×#13511yescharacteristic
335C22#57611yescharacteristic
336C22#57711yescharacteristic
337C4—22no—
338C4—22no—
339C4—41no—
340C4—42no—
341C4—41no—
342C4—81no—
343C4—81no—
344C4—81no—
345C4—81no—
346C4—81no—
347C4—81no—
348C4#52911yescharacteristic
349C4#53911yescharacteristic
350C4#54611yescharacteristic
351C4#54711yescharacteristic
352C2—24no—
353C2—42no—
354C2—161no—
355C2#242511yescharacteristic
356C2C2×#46511yescharacteristic
357C2#306211yescharacteristic
358C2#329511yescharacteristic
359C2#330111yescharacteristic
360C2#333511yescharacteristic
361C2#333911yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radical#2416C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#481C2semidirect
2. C2×C8×D4C2semidirect
3. #2487C2semidirect
5. #2088C2semidirect
6. #2077C2semidirect
7. #2065C2semidirect
8. #2034C2semidirect
9. #1941C2semidirect
10. #1934C2semidirect
11. #1893C2semidirect
12. #1893C2semidirect
13. #1853C2semidirect
14. #1807C2semidirect
15. C2×#340C2semidirect
124. C22⋊C8D4semidirect

Maximal subgroups

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

subgroupindex
C2×#4812
C2×C8×D42
#24872
#24162
#20882
#20772
#20652
#20342
#19412
#19342
#18932
#18532
#18072
C2×#3402