Finite Group Databasebeta

← #23971

Subgroups — #23971

Order 256.

637 conjugacy classes of subgroups.

The subgroup lattice has 409 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#552C211yesmaximal, characteristic, complemented
2C2×#452C211yesmaximal, characteristic, complemented
3C2×#416C211yesmaximal, characteristic, complemented
4#2407C211yesmaximal, characteristic
5#2026C211yesmaximal, characteristic, complemented
6#2025C211yesmaximal, characteristic, complemented
7#2001C211yesmaximal, characteristic, complemented
8#2000C211yesmaximal, characteristic, complemented
9#1892C211yesmaximal, characteristic, complemented
10#1880C211yesmaximal, characteristic, complemented
11#1878C211yesmaximal, characteristic, complemented
12#1874C211yesmaximal, characteristic, complemented
13#1872C211yesmaximal, characteristic, complemented
14#1869C211yesmaximal, characteristic, complemented
15#1825C211yesmaximal, characteristic, complemented
16C22×#41C2211yescharacteristic
17C22×Q16C2211yescharacteristic
18C22×SD16C2211yescharacteristic
19C23×C8C2211yescharacteristic
20#552—24no—
21#552—24no—
22C2×#126—21no—
23C2×#125—21no—
24C2×#125C2211yescharacteristic
25C2×#123—21no—
26C2×C22⋊Q8C2211yescharacteristic
27C2×C22⋊Q8C2211yescharacteristic
28C2×C22⋊Q8C2211yescharacteristic
29C2×#121C2211yescharacteristic
30C2×C4×Q8C2211yescharacteristic
31C2×C4×D4C2211yescharacteristic
32C2×#117—21no—
33C2×#117C2211yescharacteristic
34C22×C4⋊C4C2211yescharacteristic
35#452—24no—
36#416—24no—
37C2×#103C2211yescharacteristic
38C2×#103C2211yescharacteristic
39C2×#103C2211yescharacteristic
40C2×#103C2211yescharacteristic
41C2×#103C2211yescharacteristic
42C2×#102C2211yescharacteristic
43C2×#102C2211yescharacteristic
44C2×#102C2211yescharacteristic
45C2×C22⋊C8C2211yescharacteristic
46C2×C22⋊C8C2211yescharacteristic
47C2×C22⋊C8C2211yescharacteristic
48#394—21no—
49#392—21no—
50#391C2211yescharacteristic
51#390C2211yescharacteristic
52#386—21no—
53#385—21no—
54#380C2211yescharacteristic
55#340C2211yescharacteristic
56#340C2211yescharacteristic
57#340C2211yescharacteristic
58#340C2211yescharacteristic
59#340C2211yescharacteristic
60#340C2211yescharacteristic
61#336C2211yescharacteristic
62#336C2211yescharacteristic
63C2×#41—22no—
64C2×#41—22no—
65C2×#41—22no—
66C2×#41—22no—
67C2×#41—41no—
68C2×#41—41no—
69C22×Q8C2311yescharacteristic
70C22×Q8C2311yescharacteristic
71C22×D4—21no—
72C22×D4—21no—
73C22×D4D411yescharacteristic
74C22×D4C2311yescharacteristic
75C23×C4—21no—
76C23×C4—21no—
77C23×C4C2311yescharacteristic
78C23×C4C2311yescharacteristic
79C2×Q16—22no—
80C2×Q16—22no—
81C2×Q16—41no—
82C2×SD16—22no—
83C2×SD16—22no—
84C2×SD16—41no—
85C22×C8—22no—
86C22×C8—41no—
87C22×C8—41no—
88C22×C8C2311yescharacteristic
89C22×C8C2311yescharacteristic
90C22×C8C2311yescharacteristic
91C22×C8C2311yescharacteristic
92#126—28no—
93#125—28no—
94#125—22no—
95#125—22no—
96#123—28no—
97C22⋊Q8—22no—
98C22⋊Q8—22no—
99C22⋊Q8—22no—
100C22⋊Q8—22no—
101C22⋊Q8—41no—
102C22⋊Q8D414yescomplemented
103#121—22no—
104#121—41no—
105C4×Q8—22no—
106C4×Q8—22no—
107C4×D4—22no—
108C4×D4—22no—
109#117—28no—
110#117—22no—
111#117D414yescomplemented
112C2×C4⋊C4—21no—
113C2×C4⋊C4—21no—
114C2×C4⋊C4—21no—
115C2×C4⋊C4—21no—
116C2×C4⋊C4—21no—
117C2×C4⋊C4—21no—
118C2×C4⋊C4—22no—
119C2×C4⋊C4—22no—
120C2×C4⋊C4—21no—
121C2×C4⋊C4D411yescharacteristic
122C2×C4⋊C4C2311yescharacteristic
123C2×C4⋊C4C2311yescharacteristic
124C2×C4⋊C4C2311yescharacteristic
125C2×C4⋊C4C2311yescharacteristic
126C2×C4⋊C4C2311yescharacteristic
127C2×C22⋊C4—21no—
128C2×C22⋊C4—21no—
129C2×C22⋊C4—21no—
130C2×C22⋊C4—21no—
131C2×C22⋊C4—21no—
132C2×C22⋊C4—21no—
133C2×C22⋊C4—21no—
134C2×C22⋊C4—21no—
135C2×C22⋊C4D411yescharacteristic
136C2×C22⋊C4D411yescharacteristic
137C2×C42—21no—
138C2×C42—21no—
139C2×C42C2311yescharacteristic
140#103—22no—
141#103—22no—
142#103—22no—
143#103—41no—
144#103—41no—
145#102—22no—
146#102—22no—
147#102—22no—
148C22⋊C8—22no—
149C22⋊C8—41no—
150C22⋊C8—41no—
151#95—21no—
152#95—21no—
153#95—21no—
154C24—21no—
155C24C2×D411yescharacteristic
156C24C2×D411yescharacteristic
157#41—28no—
158#41—44no—
159#41—44no—
160#41—44no—
161C2×Q8—21no—
162C2×Q8—21no—
163C2×Q8—21no—
164C2×Q8—21no—
165C2×Q8—21no—
166C2×Q8—21no—
167C2×Q8—21no—
168C2×Q8—21no—
169C2×Q8—21no—
170C2×Q8—21no—
171C2×Q8C2×D412yes—
172C2×Q8#4112yes—
173C2×D4—22no—
174C2×D4—22no—
175C2×D4—21no—
176C2×D4—21no—
177C2×D4—21no—
178C2×D4—21no—
179C2×D4—21no—
180C2×D4—41no—
181C2×D4—42no—
182C2×D4—42no—
183C2×D4—42no—
184C2×D4—42no—
185C2×D4—42no—
186C2×D4—81no—
187C2×D4#4112yes—
188C22×C4—21no—
189C22×C4—21no—
190C22×C4—21no—
191C22×C4—21no—
192C22×C4—24no—
193C22×C4—21no—
194C22×C4—21no—
195C22×C4—21no—
196C22×C4—21no—
197C22×C4—21no—
198C22×C4—21no—
199C22×C4—21no—
200C22×C4—21no—
201C22×C4—21no—
202C22×C4—21no—
203C22×C4—21no—
204C22×C4—21no—
205C22×C4—21no—
206C22×C4—42no—
207C22×C4—42no—
208C22×C4—42no—
209C22×C4—42no—
210C22×C4—42no—
211C22×C4C2×D411yescharacteristic
212C22×C4C2×D411yescharacteristic
213C22×C4C2×D411yescharacteristic
214C22×C4C2×D411yescharacteristic
215C22×C4C2×D412yes—
216C22×C4C2×D412yes—
217C22×C4C2411yescharacteristic
218Q16—44no—
219SD16—44no—
220C2×C8—21no—
221C2×C8—21no—
222C2×C8—21no—
223C2×C8—21no—
224C2×C8—21no—
225C2×C8—21no—
226C2×C8—21no—
227C2×C8—21no—
228C2×C8—21no—
229C2×C8—21no—
230C2×C8—21no—
231C2×C8—21no—
232C2×C8—44no—
233C4⋊C4—24no—
234C4⋊C4—24no—
235C4⋊C4—24no—
236C4⋊C4—24no—
237C4⋊C4—24no—
238C4⋊C4—24no—
239C4⋊C4—22no—
240C4⋊C4—21no—
241C4⋊C4—21no—
242C4⋊C4—21no—
243C4⋊C4—21no—
244C4⋊C4—21no—
245C4⋊C4—21no—
246C4⋊C4—21no—
247C4⋊C4—42no—
248C4⋊C4C2×D412yes—
249C4⋊C4C2×D412yes—
250C4⋊C4C2×D412yes—
251C22⋊C4—24no—
252C22⋊C4—24no—
253C22⋊C4—24no—
254C22⋊C4—22no—
255C22⋊C4—22no—
256C22⋊C4—42no—
257C22⋊C4—42no—
258C22⋊C4—42no—
259C22⋊C4—81no—
260C22⋊C4—81no—
261C42—24no—
262C42—24no—
263C42—21no—
264C42—21no—
265C23—21no—
266C23—21no—
267C23—21no—
268C23—21no—
269C23—21no—
270C23—21no—
271C23—21no—
272C23—21no—
273C23—21no—
274C23—21no—
275C23—21no—
276C23—21no—
277C23—22no—
278C23—41no—
279C23—41no—
280C23—41no—
281C23—41no—
282C23—41no—
283C23—41no—
284C23#12012yes—
285C23#13512yes—
286C23C22×D411yescharacteristic
287Q8—22no—
288Q8—22no—
289Q8—22no—
290Q8—22no—
291Q8—41no—
292Q8—41no—
293Q8—41no—
294Q8—41no—
295D4—22no—
296D4—22no—
297D4—44no—
298D4—48no—
299D4—41no—
300D4—41no—
301D4—84no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—22no—
305C2×C4—22no—
306C2×C4—22no—
307C2×C4—22no—
308C2×C4—21no—
309C2×C4—22no—
310C2×C4—22no—
311C2×C4—22no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—22no—
315C2×C4—21no—
316C2×C4—22no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—48no—
323C2×C4—48no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—41no—
327C2×C4—41no—
328C2×C4—41no—
329C2×C4—41no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—41no—
334C2×C4—41no—
335C2×C4—41no—
336C2×C4—41no—
337C2×C4—41no—
338C2×C4—41no—
339C2×C4—41no—
340C2×C4—41no—
341C2×C4—41no—
342C2×C4—41no—
343C2×C4—41no—
344C2×C4#12012yes—
345C2×C4C22×D411yescharacteristic
346C2×C4C22×D411yescharacteristic
347C2×C4C2×#4111yescharacteristic
348C2×C4C2×#4111yescharacteristic
349C2×C4H(5,𝔽2)11yescharacteristic
350C2×C4H(5,𝔽2)11yescharacteristic
351C8—41no—
352C8—41no—
353C8—41no—
354C8—41no—
355C22—22no—
356C22—22no—
357C22—22no—
358C22—22no—
359C22—22no—
360C22—22no—
361C22—41no—
362C22—41no—
363C22—41no—
364C22—41no—
365C22—41no—
366C22—41no—
367C22—41no—
368C22—41no—
369C22—42no—
370C22—42no—
371C22—42no—
372C22—81no—
373C22—81no—
374C22—81no—
375C22—81no—
376C22C2×#12011yescharacteristic
377C22#54611yescharacteristic
378C22#54611yescharacteristic
379C22C2×#13511yescharacteristic
380C22#57511yescharacteristic
381C22#57711yescharacteristic
382C22#57811yescharacteristic
383C4—22no—
384C4—41no—
385C4—42no—
386C4—42no—
387C4—41no—
388C4—42no—
389C4—42no—
390C4—41no—
391C4—42no—
392C4—81no—
393C4—81no—
394C4#52511yescharacteristic
395C4#53511yescharacteristic
396C4#54611yescharacteristic
397C4#54611yescharacteristic
398C2—42no—
399C2—42no—
400C2—82no—
401C2#240411yescharacteristic
402C2#300211yescharacteristic
403C2#301311yescharacteristic
404C2#329611yescharacteristic
405C2#330011yescharacteristic
406C2#331511yescharacteristic
407C2#331711yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radical#2407C2
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×#552C2semidirect
2. C2×#452C2semidirect
3. C2×#416C2semidirect
5. #2026C2semidirect
6. #2025C2semidirect
7. #2001C2semidirect
8. #2000C2semidirect
9. #1892C2semidirect
10. #1880C2semidirect
11. #1878C2semidirect
12. #1874C2semidirect
13. #1872C2semidirect
14. #1869C2semidirect
15. #1825C2semidirect
102. C22⋊Q8D4semidirect
111. #117D4semidirect

Maximal subgroups

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

subgroupindex
C2×#5522
C2×#4522
C2×#4162
#24072
#20262
#20252
#20012
#20002
#18922
#18802
#18782
#18742
#18722
#18692
#18252