Finite Group Databasebeta

← #31396

Subgroups — #31396

Order 256.

469 conjugacy classes of subgroups.

The subgroup lattice has 389 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#2535C211yesmaximal, characteristic, complemented
2#2416C211yesmaximal, characteristic, complemented
3#2101C211yesmaximal, characteristic, complemented
4#2077C211yesmaximal, characteristic, complemented
5#2073C211yesmaximal, characteristic, complemented
6#2034C211yesmaximal, characteristic, complemented
7#1999C211yesmaximal, characteristic, complemented
8#1999C211yesmaximal, characteristic, complemented
9#1893C211yesmaximal, characteristic, complemented
10#1893C211yesmaximal, characteristic, complemented
11#1879C211yesmaximal, characteristic, complemented
12#1874C211yesmaximal, characteristic, complemented
13#1869C211yesmaximal, characteristic, complemented
14#1864C211yesmaximal, characteristic, complemented
15#1824C211yesmaximal, characteristic, complemented
16C22×D8C2211yescharacteristic
17C23×C8C2211yescharacteristic
18C2×#121—21no—
19C2×#121C2211yescharacteristic
20C2×#121C2211yescharacteristic
21C2×#121C2211yescharacteristic
22C2×#121C2211yescharacteristic
23C2×C4×D4C2211yescharacteristic
24C2×C4×D4C2211yescharacteristic
25C22×C4⋊C4C2211yescharacteristic
26C2×#106C2211yescharacteristic
27C2×#102C2211yescharacteristic
28C2×#102C2211yescharacteristic
29C2×#102C2211yescharacteristic
30C2×#102C2211yescharacteristic
31C2×#102C2211yescharacteristic
32C2×#102C2211yescharacteristic
33C2×C22⋊C8C2211yescharacteristic
34C2×C22⋊C8C2211yescharacteristic
35C2×C22⋊C8C2211yescharacteristic
36#397—21no—
37#396—21no—
38#394—21no—
39#392—21no—
40#390C2211yescharacteristic
41#389C2211yescharacteristic
42#386—21no—
43#386—21no—
44#386—21no—
45#382C2211yescharacteristic
46#380C2211yescharacteristic
47#380C2211yescharacteristic
48#380C2211yescharacteristic
49#340C2211yescharacteristic
50#340C2211yescharacteristic
51#340C2211yescharacteristic
52#340C2211yescharacteristic
53#340C2211yescharacteristic
54#340C2211yescharacteristic
55#340C2211yescharacteristic
56#340C2211yescharacteristic
57#336C2211yescharacteristic
58#336C2211yescharacteristic
59C22×D4—21no—
60C22×D4—21no—
61C22×D4—21no—
62C22×D4—21no—
63C22×D4D411yescharacteristic
64C22×D4C2311yescharacteristic
65C22×D4C2311yescharacteristic
66C23×C4—21no—
67C23×C4—21no—
68C23×C4C2311yescharacteristic
69C23×C4C2311yescharacteristic
70C2×D8—22no—
71C2×D8—41no—
72C2×D8—41no—
73C22×C8—41no—
74C22×C8—41no—
75C22×C8—41no—
76C22×C8C2311yescharacteristic
77C22×C8C2311yescharacteristic
78C22×C8C2311yescharacteristic
79C22×C8C2311yescharacteristic
80#121—22no—
81#121—22no—
82#121—22no—
83#121—41no—
84#121—41no—
85#121—41no—
86#121—41no—
87#121—41no—
88#121—44no—
89C4×D4—41no—
90C4×D4—41no—
91C4×D4—41no—
92C4×D4—41no—
93C2×C4⋊C4—22no—
94C2×C4⋊C4—21no—
95C2×C4⋊C4—21no—
96C2×C4⋊C4—41no—
97C2×C4⋊C4D411yescharacteristic
98C2×C4⋊C4C2311yescharacteristic
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⋊C4—21no—
113C2×C22⋊C4—21no—
114C2×C22⋊C4D411yescharacteristic
115C2×C22⋊C4D411yescharacteristic
116C2×C42C2311yescharacteristic
117#106—41no—
118#102—22no—
119#102—41no—
120#102—41no—
121#102—41no—
122#102—41no—
123#102—41no—
124C22⋊C8—41no—
125C22⋊C8—41no—
126C22⋊C8—41no—
127#95—21no—
128#95—21no—
129#95—21no—
130#95—21no—
131#95—21no—
132#95—21no—
133C24—21no—
134C24—21no—
135C24C2×D411yescharacteristic
136C24C2×D411yescharacteristic
137C2×D4—22no—
138C2×D4—22no—
139C2×D4—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×D4—21no—
149C2×D4—21no—
150C2×D4—41no—
151C2×D4—42no—
152C2×D4—42no—
153C2×D4—42no—
154C2×D4—42no—
155C2×D4—42no—
156C2×D4—42no—
157C2×D4—42no—
158C2×D4—42no—
159C2×D4—42no—
160C2×D4—81no—
161C2×D4—81no—
162C2×D4—81no—
163C2×D4#4112yes—
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×C4—21no—
175C22×C4—21no—
176C22×C4—21no—
177C22×C4—21no—
178C22×C4—21no—
179C22×C4—21no—
180C22×C4—21no—
181C22×C4—21no—
182C22×C4—42no—
183C22×C4—42no—
184C22×C4—42no—
185C22×C4—81no—
186C22×C4—81no—
187C22×C4—81no—
188C22×C4C2×D411yescharacteristic
189C22×C4C2×D411yescharacteristic
190C22×C4C2×D411yescharacteristic
191C22×C4C2×D411yescharacteristic
192C22×C4C2411yescharacteristic
193D8—44no—
194C2×C8—21no—
195C2×C8—21no—
196C2×C8—21no—
197C2×C8—21no—
198C2×C8—21no—
199C2×C8—21no—
200C2×C8—21no—
201C2×C8—21no—
202C2×C8—21no—
203C2×C8—21no—
204C2×C8—21no—
205C2×C8—21no—
206C2×C8—44no—
207C4⋊C4—21no—
208C4⋊C4—21no—
209C4⋊C4—21no—
210C4⋊C4—21no—
211C4⋊C4—21no—
212C4⋊C4—21no—
213C4⋊C4—21no—
214C4⋊C4—21no—
215C4⋊C4—21no—
216C4⋊C4—21no—
217C4⋊C4—21no—
218C4⋊C4—41no—
219C4⋊C4—42no—
220C4⋊C4—42no—
221C4⋊C4#4112yes—
222C22⋊C4—41no—
223C22⋊C4—41no—
224C22⋊C4—42no—
225C22⋊C4—42no—
226C22⋊C4—42no—
227C22⋊C4—42no—
228C22⋊C4—42no—
229C22⋊C4—42no—
230C22⋊C4—42no—
231C22⋊C4—81no—
232C22⋊C4—81no—
233C22⋊C4—81no—
234C42—21no—
235C42—21no—
236C23—21no—
237C23—21no—
238C23—21no—
239C23—21no—
240C23—21no—
241C23—21no—
242C23—21no—
243C23—21no—
244C23—21no—
245C23—21no—
246C23—21no—
247C23—21no—
248C23—22no—
249C23—41no—
250C23—41no—
251C23—41no—
252C23—41no—
253C23—41no—
254C23—41no—
255C23—41no—
256C23—41no—
257C23—41no—
258C23—41no—
259C23—41no—
260C23—41no—
261C23—41no—
262C23#13512yes—
263C23AGL(1,ℤ/8)12yes—
264C23C22×D411yescharacteristic
265D4—22no—
266D4—22no—
267D4—44no—
268D4—41no—
269D4—41no—
270D4—41no—
271D4—41no—
272D4—41no—
273D4—41no—
274D4—84no—
275D4—84no—
276D4—84no—
277D4—84no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—22no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
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—41no—
302C2×C4—41no—
303C2×C4—41no—
304C2×C4—41no—
305C2×C4—41no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—84no—
317C2×C4—84no—
318C2×C4C2×#4111yescharacteristic
319C2×C4C2×#4111yescharacteristic
320C2×C4H(5,𝔽2)11yescharacteristic
321C2×C4H(5,𝔽2)11yescharacteristic
322C2×C4H(5,𝔽2)11yescharacteristic
323C2×C4H(5,𝔽2)11yescharacteristic
324C8—41no—
325C8—41no—
326C8—41no—
327C8—41no—
328C22—22no—
329C22—22no—
330C22—22no—
331C22—22no—
332C22—22no—
333C22—22no—
334C22—41no—
335C22—41no—
336C22—41no—
337C22—41no—
338C22—41no—
339C22—41no—
340C22—41no—
341C22—41no—
342C22—42no—
343C22—42no—
344C22—42no—
345C22—81no—
346C22—81no—
347C22—81no—
348C22—81no—
349C22—81no—
350C22—81no—
351C22—81no—
352C22—81no—
353C22—81no—
354C22—81no—
355C22—81no—
356C22#53411yescharacteristic
357C22#54611yescharacteristic
358C22#54611yescharacteristic
359C22C2×#13511yescharacteristic
360C22C2×AGL(1,ℤ/8)11yescharacteristic
361C22#57611yescharacteristic
362C22#57711yescharacteristic
363C4—41no—
364C4—41no—
365C4—41no—
366C4—41no—
367C4—42no—
368C4—81no—
369C4—81no—
370C4—81no—
371C4—81no—
372C4—81no—
373C4#53811yescharacteristic
374C4#54011yescharacteristic
375C4#55311yescharacteristic
376C4#56011yescharacteristic
377C2—42no—
378C2—42no—
379C2—82no—
380C2—161no—
381C2#268811yescharacteristic
382C2#318811yescharacteristic
383C2#319811yescharacteristic
384C2#329711yescharacteristic
385C2#330111yescharacteristic
386C2#330611yescharacteristic
387C2#331811yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
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. #2535C2semidirect
2. #2416C2semidirect
3. #2101C2semidirect
4. #2077C2semidirect
5. #2073C2semidirect
6. #2034C2semidirect
7. #1999C2semidirect
8. #1999C2semidirect
9. #1893C2semidirect
10. #1893C2semidirect
11. #1879C2semidirect
12. #1874C2semidirect
13. #1869C2semidirect
14. #1864C2semidirect
15. #1824C2semidirect

Maximal subgroups

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

subgroupindex
#25352
#24162
#21012
#20772
#20732
#20342
#19992
#18932
#18792
#18742
#18692
#18642
#18242