Finite Group Databasebeta

← #31382

Subgroups — #31382

Order 256.

483 conjugacy classes of subgroups.

The subgroup lattice has 392 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#2537C211yesmaximal, characteristic, complemented
2#2416C211yesmaximal, characteristic, complemented
3#2099C211yesmaximal, characteristic, complemented
4#2075C211yesmaximal, characteristic, complemented
5#2071C211yesmaximal, characteristic, complemented
6#2033C211yesmaximal, 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#1862C211yesmaximal, characteristic, complemented
15#1823C211yesmaximal, characteristic, complemented
16C22×D8C2211yescharacteristic
17C23×C8C2211yescharacteristic
18C2×#121—21no—
19C2×#121—21no—
20C2×#121C2211yescharacteristic
21C2×#121C2211yescharacteristic
22C2×#121C2211yescharacteristic
23C2×#121C2211yescharacteristic
24C2×C4×D4C2211yescharacteristic
25C2×C4×D4C2211yescharacteristic
26C22×C4⋊C4C2211yescharacteristic
27C2×#107C2211yescharacteristic
28C2×#102C2211yescharacteristic
29C2×#102C2211yescharacteristic
30C2×#102C2211yescharacteristic
31C2×#102C2211yescharacteristic
32C2×#102C2211yescharacteristic
33C2×#102C2211yescharacteristic
34C2×C22⋊C8C2211yescharacteristic
35C2×C22⋊C8C2211yescharacteristic
36C2×C22⋊C8C2211yescharacteristic
37#396—21no—
38#394—21no—
39#394—21no—
40#394—21no—
41#390C2211yescharacteristic
42#389C2211yescharacteristic
43#386—21no—
44#384C2211yescharacteristic
45#380—21no—
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—
89#121—44no—
90C4×D4—41no—
91C4×D4—41no—
92C4×D4—41no—
93C4×D4—41no—
94C2×C4⋊C4—22no—
95C2×C4⋊C4—21no—
96C2×C4⋊C4—21no—
97C2×C4⋊C4—21no—
98C2×C4⋊C4—21no—
99C2×C4⋊C4—41no—
100C2×C4⋊C4D411yescharacteristic
101C2×C4⋊C4C2311yescharacteristic
102C2×C4⋊C4C2311yescharacteristic
103C2×C4⋊C4C2311yescharacteristic
104C2×C4⋊C4C2311yescharacteristic
105C2×C4⋊C4C2311yescharacteristic
106C2×C4⋊C4C2311yescharacteristic
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⋊C4—21no—
115C2×C22⋊C4—21no—
116C2×C22⋊C4—21no—
117C2×C22⋊C4D411yescharacteristic
118C2×C22⋊C4D411yescharacteristic
119C2×C42C2311yescharacteristic
120#107—41no—
121#102—22no—
122#102—41no—
123#102—41no—
124#102—41no—
125#102—41no—
126#102—41no—
127C22⋊C8—41no—
128C22⋊C8—41no—
129C22⋊C8—41no—
130#95—21no—
131#95—21no—
132#95—21no—
133#95—21no—
134C24—21no—
135C24—21no—
136C24C2×D411yescharacteristic
137C24C2×D411yescharacteristic
138C2×D4—21no—
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—41no—
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×D4D814yescomplemented
164C2×D4#4112yes—
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—21no—
183C22×C4—42no—
184C22×C4—42no—
185C22×C4—42no—
186C22×C4—42no—
187C22×C4—42no—
188C22×C4—81no—
189C22×C4C2×D411yescharacteristic
190C22×C4C2×D411yescharacteristic
191C22×C4C2×D411yescharacteristic
192C22×C4C2×D411yescharacteristic
193C22×C4C2411yescharacteristic
194D8—44no—
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—21no—
207C2×C8—44no—
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—21no—
219C4⋊C4—41no—
220C4⋊C4—42no—
221C4⋊C4—42no—
222C4⋊C4—42no—
223C4⋊C4—81no—
224C4⋊C4#4112yes—
225C22⋊C4—41no—
226C22⋊C4—41no—
227C22⋊C4—42no—
228C22⋊C4—42no—
229C22⋊C4—42no—
230C22⋊C4—42no—
231C22⋊C4—42no—
232C22⋊C4—42no—
233C22⋊C4—42no—
234C22⋊C4—42no—
235C22⋊C4—42no—
236C22⋊C4—81no—
237C42—21no—
238C42—21no—
239C23—21no—
240C23—21no—
241C23—21no—
242C23—21no—
243C23—21no—
244C23—21no—
245C23—21no—
246C23—21no—
247C23—21no—
248C23—21no—
249C23—21no—
250C23—21no—
251C23—22no—
252C23—41no—
253C23—41no—
254C23—41no—
255C23—41no—
256C23—41no—
257C23—41no—
258C23—41no—
259C23—41no—
260C23—41no—
261C23—41no—
262C23—41no—
263C23—41no—
264C23—41no—
265C23C2×D812yes—
266C23C2×D812yes—
267C23C22×D411yescharacteristic
268D4—22no—
269D4—22no—
270D4—44no—
271D4—41no—
272D4—41no—
273D4—41no—
274D4—41no—
275D4—41no—
276D4—41no—
277D4—84no—
278D4—84no—
279D4—84no—
280D4—84no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—22no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
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—41no—
317C2×C4—41no—
318C2×C4—84no—
319C2×C4—84no—
320C2×C4C2×D812yes—
321C2×C4C2×#4111yescharacteristic
322C2×C4C2×#4111yescharacteristic
323C2×C4H(5,𝔽2)11yescharacteristic
324C2×C4H(5,𝔽2)11yescharacteristic
325C2×C4H(5,𝔽2)11yescharacteristic
326C2×C4H(5,𝔽2)11yescharacteristic
327C8—41no—
328C8—41no—
329C8—41no—
330C8—41no—
331C22—22no—
332C22—22no—
333C22—22no—
334C22—22no—
335C22—22no—
336C22—22no—
337C22—41no—
338C22—41no—
339C22—41no—
340C22—41no—
341C22—41no—
342C22—41no—
343C22—41no—
344C22—41no—
345C22—42no—
346C22—42no—
347C22—42no—
348C22—81no—
349C22—81no—
350C22—81no—
351C22—81no—
352C22—81no—
353C22—81no—
354C22—81no—
355C22—81no—
356C22—81no—
357C22—81no—
358C22—81no—
359C22#53411yescharacteristic
360C22#54611yescharacteristic
361C22#54611yescharacteristic
362C22C22×D811yescharacteristic
363C22#57511yescharacteristic
364C22#57711yescharacteristic
365C22#57711yescharacteristic
366C4—22no—
367C4—41no—
368C4—41no—
369C4—41no—
370C4—42no—
371C4—81no—
372C4—81no—
373C4—81no—
374C4—81no—
375C4—81no—
376C4#53811yescharacteristic
377C4#54011yescharacteristic
378C4#55311yescharacteristic
379C4#56011yescharacteristic
380C2—42no—
381C2—42no—
382C2—82no—
383C2—161no—
384C2#268811yescharacteristic
385C2#318611yescharacteristic
386C2#320411yescharacteristic
387C2#329411yescharacteristic
388C2#329411yescharacteristic
389C2#330711yescharacteristic
390C2#332111yescharacteristic
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. #2537C2semidirect
2. #2416C2semidirect
3. #2099C2semidirect
4. #2075C2semidirect
5. #2071C2semidirect
6. #2033C2semidirect
7. #1999C2semidirect
8. #1999C2semidirect
9. #1893C2semidirect
10. #1893C2semidirect
11. #1879C2semidirect
12. #1874C2semidirect
13. #1869C2semidirect
14. #1862C2semidirect
15. #1823C2semidirect
163. C2×D4D8semidirect

Maximal subgroups

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

subgroupindex
#25372
#24162
#20992
#20752
#20712
#20332
#19992
#18932
#18792
#18742
#18692
#18622
#18232