Finite Group Databasebeta

← #24124

Subgroups — #24124

Order 256.

608 conjugacy classes of subgroups.

The subgroup lattice has 408 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#553C211yesmaximal, characteristic, complemented
2C2×#478C211yesmaximal, characteristic, complemented
3C2×C4×D8C211yesmaximal, characteristic, complemented
4#2535C211yesmaximal, characteristic, complemented
5#2034C211yesmaximal, characteristic, complemented
6#2033C211yesmaximal, characteristic, complemented
7#1999C211yesmaximal, characteristic, complemented
8#1999C211yesmaximal, characteristic, complemented
9#1991C211yesmaximal, characteristic, complemented
10#1954C211yesmaximal, characteristic, complemented
11#1875C211yesmaximal, characteristic, complemented
12#1874C211yesmaximal, characteristic, complemented
13#1854C211yesmaximal, characteristic, complemented
14#1852C211yesmaximal, characteristic, complemented
15#1804C211yesmaximal, characteristic, complemented
16C22×D8C2211yescharacteristic
17#553—24no—
18#553—24no—
19C2×#126—21no—
20C2×#125C2211yescharacteristic
21C2×#123—21no—
22C2×#121—21no—
23C2×#121C2211yescharacteristic
24C2×#121C2211yescharacteristic
25C2×#121C2211yescharacteristic
26C2×#121C2211yescharacteristic
27C2×C4×D4—21no—
28C2×C4×D4C2211yescharacteristic
29C2×C4×D4C2211yescharacteristic
30C2×C4×D4C2211yescharacteristic
31C2×#117C2211yescharacteristic
32C22×C4⋊C4C2211yescharacteristic
33#478—24no—
34C4×D8—24no—
35C2×#107C2211yescharacteristic
36C2×#107C2211yescharacteristic
37C2×#106C2211yescharacteristic
38C2×C4⋊C8C2211yescharacteristic
39C2×#102C2211yescharacteristic
40C2×#102C2211yescharacteristic
41C2×#102C2211yescharacteristic
42C2×#102C2211yescharacteristic
43C2×#102C2211yescharacteristic
44C2×#102C2211yescharacteristic
45C2×C22⋊C8C2211yescharacteristic
46C2×C22⋊C8C2211yescharacteristic
47C2×C22⋊C8C2211yescharacteristic
48C2×C22⋊C8C2211yescharacteristic
49C2×C4×C8C2211yescharacteristic
50#397—21no—
51#392—21no—
52#386—21no—
53#386—21no—
54#382C2211yescharacteristic
55#380C2211yescharacteristic
56#380C2211yescharacteristic
57#340C2211yescharacteristic
58#340C2211yescharacteristic
59#340C2211yescharacteristic
60#340C2211yescharacteristic
61#340C2211yescharacteristic
62#340C2211yescharacteristic
63C22×D4—21no—
64C22×D4—21no—
65C22×D4—21no—
66C22×D4—21no—
67C22×D4D411yescharacteristic
68C22×D4C2311yescharacteristic
69C22×D4C2311yescharacteristic
70C23×C4—21no—
71C23×C4—21no—
72C23×C4—21no—
73C23×C4C2311yescharacteristic
74C23×C4C2311yescharacteristic
75C2×D8—22no—
76C2×D8—22no—
77C2×D8—22no—
78C22×C8C2311yescharacteristic
79C22×C8C2311yescharacteristic
80C22×C8C2311yescharacteristic
81C22×C8C2311yescharacteristic
82#126—28no—
83#125—22no—
84#125D414yescomplemented
85#123—28no—
86#121—28no—
87#121—22no—
88#121—22no—
89#121—22no—
90#121—22no—
91#121—22no—
92#121—22no—
93#121—41no—
94#121—41no—
95C4×D4—28no—
96C4×D4—22no—
97C4×D4—22no—
98C4×D4—22no—
99C4×D4—22no—
100C4×D4—22no—
101C4×D4D414yescomplemented
102#117—22no—
103#117—22no—
104C2×C4⋊C4—21no—
105C2×C4⋊C4—21no—
106C2×C4⋊C4—21no—
107C2×C4⋊C4—22no—
108C2×C4⋊C4—22no—
109C2×C4⋊C4D411yescharacteristic
110C2×C4⋊C4C2311yescharacteristic
111C2×C4⋊C4C2311yescharacteristic
112C2×C4⋊C4C2311yescharacteristic
113C2×C4⋊C4C2311yescharacteristic
114C2×C4⋊C4C2311yescharacteristic
115C2×C4⋊C4C2311yescharacteristic
116C2×C22⋊C4—21no—
117C2×C22⋊C4—21no—
118C2×C22⋊C4—21no—
119C2×C22⋊C4—21no—
120C2×C22⋊C4—21no—
121C2×C22⋊C4—21no—
122C2×C22⋊C4—21no—
123C2×C22⋊C4—21no—
124C2×C22⋊C4—21no—
125C2×C22⋊C4D411yescharacteristic
126C2×C22⋊C4D411yescharacteristic
127C2×C42—21no—
128C2×C42C2311yescharacteristic
129#107—22no—
130#107—22no—
131#106—22no—
132C4⋊C8—22no—
133#102—22no—
134#102—22no—
135#102—22no—
136#102—22no—
137#102—41no—
138#102—41no—
139C22⋊C8—22no—
140C22⋊C8—22no—
141C22⋊C8—41no—
142C22⋊C8—41no—
143C4×C8—22no—
144#95—21no—
145#95—21no—
146#95—21no—
147#95—21no—
148C24—21no—
149C24—21no—
150C24C2×D411yescharacteristic
151C24C2×D411yescharacteristic
152C2×D4—24no—
153C2×D4—22no—
154C2×D4—24no—
155C2×D4—22no—
156C2×D4—21no—
157C2×D4—21no—
158C2×D4—21no—
159C2×D4—21no—
160C2×D4—21no—
161C2×D4—21no—
162C2×D4—21no—
163C2×D4—21no—
164C2×D4—21no—
165C2×D4—21no—
166C2×D4—41no—
167C2×D4—42no—
168C2×D4—42no—
169C2×D4—42no—
170C2×D4—42no—
171C2×D4—42no—
172C2×D4—42no—
173C2×D4—81no—
174C2×D4—81no—
175C2×D4—81no—
176C2×D4—81no—
177C2×D4C2×D412yes—
178C2×D4#4112yes—
179C22×C4—21no—
180C22×C4—21no—
181C22×C4—21no—
182C22×C4—24no—
183C22×C4—21no—
184C22×C4—21no—
185C22×C4—21no—
186C22×C4—21no—
187C22×C4—21no—
188C22×C4—21no—
189C22×C4—21no—
190C22×C4—21no—
191C22×C4—21no—
192C22×C4—21no—
193C22×C4—21no—
194C22×C4—21no—
195C22×C4—24no—
196C22×C4—21no—
197C22×C4—21no—
198C22×C4—21no—
199C22×C4—42no—
200C22×C4—81no—
201C22×C4—81no—
202C22×C4—81no—
203C22×C4—81no—
204C22×C4—81no—
205C22×C4—81no—
206C22×C4C2×D411yescharacteristic
207C22×C4C2×D411yescharacteristic
208C22×C4C2×D411yescharacteristic
209C22×C4C2×D411yescharacteristic
210C22×C4C2411yescharacteristic
211D8—28no—
212C2×C8—21no—
213C2×C8—21no—
214C2×C8—21no—
215C2×C8—21no—
216C2×C8—21no—
217C2×C8—21no—
218C2×C8—21no—
219C2×C8—21no—
220C2×C8—21no—
221C2×C8—21no—
222C2×C8—21no—
223C2×C8#4112yes—
224C4⋊C4—24no—
225C4⋊C4—24no—
226C4⋊C4—24no—
227C4⋊C4—22no—
228C4⋊C4—21no—
229C4⋊C4—21no—
230C4⋊C4—21no—
231C4⋊C4—21no—
232C4⋊C4—21no—
233C4⋊C4—21no—
234C4⋊C4—21no—
235C4⋊C4—21no—
236C4⋊C4—21no—
237C4⋊C4C2×D412yes—
238C4⋊C4C2×D412yes—
239C4⋊C4C2×D412yes—
240C22⋊C4—24no—
241C22⋊C4—24no—
242C22⋊C4—24no—
243C22⋊C4—24no—
244C22⋊C4—22no—
245C22⋊C4—22no—
246C22⋊C4—42no—
247C22⋊C4—42no—
248C22⋊C4—42no—
249C22⋊C4—81no—
250C22⋊C4—81no—
251C42—24no—
252C42C2×D412yes—
253C42C2×D412yes—
254C23—21no—
255C23—21no—
256C23—21no—
257C23—21no—
258C23—21no—
259C23—21no—
260C23—21no—
261C23—22no—
262C23—22no—
263C23—21no—
264C23—21no—
265C23—21no—
266C23—21no—
267C23—21no—
268C23—21no—
269C23—41no—
270C23—41no—
271C23—41no—
272C23—41no—
273C23—41no—
274C23—41no—
275C23—41no—
276C23—41no—
277C23—41no—
278C23—41no—
279C23—41no—
280C23—41no—
281C23#13512yes—
282C23C22×D411yescharacteristic
283D4—22no—
284D4—22no—
285D4—22no—
286D4—22no—
287D4—44no—
288D4—48no—
289D4—41no—
290D4—41no—
291D4—41no—
292D4—41no—
293D4—84no—
294D4—84no—
295D4—84no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—22no—
302C2×C4—22no—
303C2×C4—21no—
304C2×C4—22no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
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—41no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—41no—
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—84no—
333C2×C4—84no—
334C2×C4—84no—
335C2×C4#12012yes—
336C2×C4#12012yes—
337C2×C4C22×D411yescharacteristic
338C2×C4C22×D411yescharacteristic
339C2×C4C2×#4111yescharacteristic
340C2×C4C2×#4111yescharacteristic
341C2×C4H(5,𝔽2)11yescharacteristic
342C2×C42−1+411yescharacteristic
343C8—22no—
344C8—41no—
345C8—41no—
346C8—41no—
347C22—22no—
348C22—22no—
349C22—22no—
350C22—41no—
351C22—41no—
352C22—41no—
353C22—41no—
354C22—41no—
355C22—41no—
356C22—41no—
357C22—42no—
358C22—42no—
359C22—41no—
360C22—42no—
361C22—42no—
362C22—41no—
363C22—42no—
364C22—41no—
365C22—41no—
366C22—42no—
367C22—81no—
368C22—81no—
369C22—81no—
370C22—81no—
371C22—81no—
372C22—81no—
373C22—81no—
374C22—81no—
375C22C2×#12011yescharacteristic
376C22#54611yescharacteristic
377C22#54711yescharacteristic
378C22C2×#13511yescharacteristic
379C22#57511yescharacteristic
380C22#57611yescharacteristic
381C22#57611yescharacteristic
382C4—22no—
383C4—22no—
384C4—41no—
385C4—41no—
386C4—42no—
387C4—42no—
388C4—42no—
389C4—81no—
390C4—81no—
391C4—81no—
392C4#52511yescharacteristic
393C4#53711yescharacteristic
394C4#54811yescharacteristic
395C4#55011yescharacteristic
396C2—42no—
397C2—81no—
398C2—82no—
399C2—82no—
400C2#240711yescharacteristic
401C2#300211yescharacteristic
402C2#300811yescharacteristic
403C2#329711yescharacteristic
404C2#332211yescharacteristic
405C2#333911yescharacteristic
406C2#335511yescharacteristic
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. C2×#553C2semidirect
2. C2×#478C2semidirect
3. C2×C4×D8C2semidirect
4. #2535C2semidirect
5. #2034C2semidirect
6. #2033C2semidirect
7. #1999C2semidirect
8. #1999C2semidirect
9. #1991C2semidirect
10. #1954C2semidirect
11. #1875C2semidirect
12. #1874C2semidirect
13. #1854C2semidirect
14. #1852C2semidirect
15. #1804C2semidirect
84. #125D4semidirect
101. C4×D4D4semidirect

Maximal subgroups

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

subgroupindex
C2×#5532
C2×#4782
C2×C4×D82
#25352
#20342
#20332
#19992
#19912
#19542
#18752
#18742
#18542
#18522
#18042