Finite Group Databasebeta

← #25554

Subgroups — #25554

Order 256.

618 conjugacy classes of subgroups.

The subgroup lattice has 412 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#555C211yesmaximal, characteristic, complemented
2C2×#482C211yesmaximal, characteristic, complemented
3C2×#469C211yesmaximal, characteristic, complemented
4#2416C211yesmaximal, characteristic, complemented
5#2058C211yesmaximal, characteristic, complemented
6#2034C211yesmaximal, characteristic, complemented
7#2023C211yesmaximal, characteristic, complemented
8#1999C211yesmaximal, characteristic, complemented
9#1879C211yesmaximal, characteristic, complemented
10#1875C211yesmaximal, characteristic, complemented
11#1874C211yesmaximal, characteristic, complemented
12#1870C211yesmaximal, characteristic, complemented
13#1862C211yesmaximal, characteristic, complemented
14#1854C211yesmaximal, characteristic, complemented
15#1736C211yesmaximal, characteristic, complemented
16C22×D8C2211yescharacteristic
17C22×M16C2211yescharacteristic
18#555—24no—
19#555—24no—
20C2×C4⋊Q8C2211yescharacteristic
21C2×#124—21no—
22C2×#124—21no—
23C2×#121—21no—
24C2×#121—21no—
25C2×#121C2211yescharacteristic
26C2×#121C2211yescharacteristic
27C2×#121C2211yescharacteristic
28C2×#121C2211yescharacteristic
29C2×C4×D4C2211yescharacteristic
30C2×C4×D4C2211yescharacteristic
31C2×#117—21no—
32C2×#117C2211yescharacteristic
33C2×#117C2211yescharacteristic
34C22×C4⋊C4C2211yescharacteristic
35#482—24no—
36#469—24no—
37C2×#107C2211yescharacteristic
38C2×#106C2211yescharacteristic
39C2×#106C2211yescharacteristic
40C2×#102C2211yescharacteristic
41C2×#102C2211yescharacteristic
42C2×#102C2211yescharacteristic
43C2×#102C2211yescharacteristic
44C2×#102C2211yescharacteristic
45C2×#102C2211yescharacteristic
46C2×C22⋊C8C2211yescharacteristic
47C2×C22⋊C8C2211yescharacteristic
48C2×C22⋊C8C2211yescharacteristic
49#396—21no—
50#394—21no—
51#390C2211yescharacteristic
52#389C2211yescharacteristic
53#386—21no—
54#380C2211yescharacteristic
55#340C2211yescharacteristic
56#340C2211yescharacteristic
57#340C2211yescharacteristic
58#340C2211yescharacteristic
59#340C2211yescharacteristic
60#340C2211yescharacteristic
61#336C2211yescharacteristic
62#336C2211yescharacteristic
63C22×Q8—21no—
64C22×D4—21no—
65C22×D4—21no—
66C22×D4—21no—
67C22×D4—21no—
68C22×D4D411yescharacteristic
69C22×D4C2311yescharacteristic
70C22×D4C2311yescharacteristic
71C23×C4—21no—
72C23×C4—21no—
73C23×C4C2311yescharacteristic
74C23×C4C2311yescharacteristic
75C2×D8—22no—
76C2×D8—22no—
77C2×D8—41no—
78C2×M16—22no—
79C2×M16—22no—
80C2×M16—41no—
81C22×C8C2311yescharacteristic
82C22×C8C2311yescharacteristic
83C22×C8C2311yescharacteristic
84C22×C8C2311yescharacteristic
85C4⋊Q8—22no—
86C4⋊Q8—22no—
87#124—28no—
88#124—28no—
89#121—28no—
90#121—22no—
91#121—22no—
92#121—22no—
93#121—22no—
94#121—22no—
95#121—22no—
96#121—41no—
97#121—44no—
98#121D414yescomplemented
99C4×D4—22no—
100C4×D4—22no—
101C4×D4—22no—
102C4×D4—22no—
103#117—28no—
104#117—22no—
105#117—22no—
106#117—22no—
107#117D414yescomplemented
108C2×C4⋊C4—21no—
109C2×C4⋊C4—22no—
110C2×C4⋊C4—21no—
111C2×C4⋊C4—21no—
112C2×C4⋊C4—41no—
113C2×C4⋊C4D411yescharacteristic
114C2×C4⋊C4C2311yescharacteristic
115C2×C4⋊C4C2311yescharacteristic
116C2×C4⋊C4C2311yescharacteristic
117C2×C4⋊C4C2311yescharacteristic
118C2×C4⋊C4C2311yescharacteristic
119C2×C4⋊C4C2311yescharacteristic
120C2×C22⋊C4—21no—
121C2×C22⋊C4—21no—
122C2×C22⋊C4—21no—
123C2×C22⋊C4—21no—
124C2×C22⋊C4—21no—
125C2×C22⋊C4—21no—
126C2×C22⋊C4—21no—
127C2×C22⋊C4—21no—
128C2×C22⋊C4—21no—
129C2×C22⋊C4—21no—
130C2×C22⋊C4D411yescharacteristic
131C2×C22⋊C4D411yescharacteristic
132C2×C42—21no—
133C2×C42—21no—
134C2×C42C2311yescharacteristic
135#107—22no—
136#106—22no—
137#106—22no—
138#102—22no—
139#102—22no—
140#102—22no—
141#102—22no—
142#102—22no—
143#102—41no—
144C22⋊C8—22no—
145C22⋊C8—22no—
146C22⋊C8—41no—
147#95—21no—
148#95—21no—
149C24—21no—
150C24—21no—
151C24C2×D411yescharacteristic
152C24C2×D411yescharacteristic
153C2×Q8—24no—
154C2×Q8—81no—
155C2×Q8—81no—
156C2×D4—22no—
157C2×D4—24no—
158C2×D4—24no—
159C2×D4—22no—
160C2×D4—21no—
161C2×D4—21no—
162C2×D4—21no—
163C2×D4—21no—
164C2×D4—21no—
165C2×D4—21no—
166C2×D4—21no—
167C2×D4—21no—
168C2×D4—21no—
169C2×D4—21no—
170C2×D4—21no—
171C2×D4—41no—
172C2×D4—42no—
173C2×D4—42no—
174C2×D4—42no—
175C2×D4—42no—
176C2×D4—42no—
177C2×D4—81no—
178C2×D4—81no—
179C2×D4—81no—
180C2×D4—81no—
181C2×D4—81no—
182C2×D4C2×D412yes—
183C22×C4—21no—
184C22×C4—21no—
185C22×C4—21no—
186C22×C4—21no—
187C22×C4—24no—
188C22×C4—21no—
189C22×C4—21no—
190C22×C4—21no—
191C22×C4—21no—
192C22×C4—21no—
193C22×C4—21no—
194C22×C4—21no—
195C22×C4—21no—
196C22×C4—21no—
197C22×C4—21no—
198C22×C4—21no—
199C22×C4—21no—
200C22×C4—42no—
201C22×C4—42no—
202C22×C4—42no—
203C22×C4—42no—
204C22×C4—81no—
205C22×C4C2×D411yescharacteristic
206C22×C4C2×D411yescharacteristic
207C22×C4C2×D411yescharacteristic
208C22×C4C2×D412yes—
209C22×C4C2×D412yes—
210C22×C4C2×D411yescharacteristic
211C22×C4C2411yescharacteristic
212D8—44no—
213M16—44no—
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—21no—
224C2×C8—21no—
225C2×C8#4112yes—
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—42no—
237C4⋊C4—42no—
238C4⋊C4C2×D412yes—
239C4⋊C4C2×D412yes—
240C4⋊C4C2×D412yes—
241C4⋊C4#4112yes—
242C22⋊C4—24no—
243C22⋊C4—24no—
244C22⋊C4—24no—
245C22⋊C4—24no—
246C22⋊C4—22no—
247C22⋊C4—24no—
248C22⋊C4—22no—
249C22⋊C4—42no—
250C22⋊C4—42no—
251C22⋊C4—42no—
252C22⋊C4—42no—
253C22⋊C4—81no—
254C42—24no—
255C42—24no—
256C42—21no—
257C42—21no—
258C23—21no—
259C23—21no—
260C23—21no—
261C23—21no—
262C23—21no—
263C23—21no—
264C23—21no—
265C23—22no—
266C23—21no—
267C23—21no—
268C23—21no—
269C23—21no—
270C23—21no—
271C23—21no—
272C23—41no—
273C23—41no—
274C23—41no—
275C23—41no—
276C23—41no—
277C23—41no—
278C23—41no—
279C23—41no—
280C23—41no—
281C23—41no—
282C23—41no—
283C23—41no—
284C23—41no—
285C23#12012yes—
286C23C22×D411yescharacteristic
287Q8—84no—
288D4—22no—
289D4—22no—
290D4—44no—
291D4—41no—
292D4—41no—
293D4—41no—
294D4—41no—
295D4—41no—
296D4—41no—
297D4—84no—
298D4—84no—
299D4—84no—
300D4—84no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—22no—
306C2×C4—22no—
307C2×C4—22no—
308C2×C4—22no—
309C2×C4—21no—
310C2×C4—22no—
311C2×C4—21no—
312C2×C4—22no—
313C2×C4—22no—
314C2×C4—21no—
315C2×C4—22no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—48no—
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—41no—
333C2×C4—41no—
334C2×C4—41no—
335C2×C4—41no—
336C2×C4—41no—
337C2×C4—41no—
338C2×C4—84no—
339C2×C4#12012yes—
340C2×C4AGL(1,ℤ/8)12yes—
341C2×C4C22×D411yescharacteristic
342C2×C4C22×D411yescharacteristic
343C2×C4C2×#4111yescharacteristic
344C2×C4C2×#4111yescharacteristic
345C2×C4H(5,𝔽2)11yescharacteristic
346C2×C42−1+411yescharacteristic
347C8—22no—
348C8—41no—
349C8—41no—
350C8—41no—
351C22—22no—
352C22—22no—
353C22—22no—
354C22—41no—
355C22—41no—
356C22—41no—
357C22—41no—
358C22—41no—
359C22—41no—
360C22—41no—
361C22—42no—
362C22—42no—
363C22—41no—
364C22—41no—
365C22—42no—
366C22—41no—
367C22—41no—
368C22—81no—
369C22—81no—
370C22—81no—
371C22—81no—
372C22—81no—
373C22—81no—
374C22—81no—
375C22—81no—
376C22—81no—
377C22—81no—
378C22—81no—
379C22C2×#12011yescharacteristic
380C22#54611yescharacteristic
381C22#54711yescharacteristic
382C22C2×AGL(1,ℤ/8)11yescharacteristic
383C22#57511yescharacteristic
384C22#57611yescharacteristic
385C22#57711yescharacteristic
386C4—22no—
387C4—22no—
388C4—41no—
389C4—42no—
390C4—41no—
391C4—42no—
392C4—42no—
393C4—42no—
394C4—81no—
395C4—81no—
396C4#53211yescharacteristic
397C4#53711yescharacteristic
398C4#54611yescharacteristic
399C4#54711yescharacteristic
400C2—42no—
401C2—81no—
402C2—82no—
403C2—161no—
404C2#243011yescharacteristic
405C2#300611yescharacteristic
406C2#301211yescharacteristic
407C2#332111yescharacteristic
408C2#332511yescharacteristic
409C2#334711yescharacteristic
410C2#335411yescharacteristic
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×#555C2semidirect
2. C2×#482C2semidirect
3. C2×#469C2semidirect
4. #2416C2semidirect
5. #2058C2semidirect
6. #2034C2semidirect
7. #2023C2semidirect
8. #1999C2semidirect
9. #1879C2semidirect
10. #1875C2semidirect
11. #1874C2semidirect
12. #1870C2semidirect
13. #1862C2semidirect
14. #1854C2semidirect
15. #1736C2semidirect
98. #121D4semidirect
107. #117D4semidirect

Maximal subgroups

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

subgroupindex
C2×#5552
C2×#4822
C2×#4692
#24162
#20582
#20342
#20232
#19992
#18792
#18752
#18742
#18702
#18622
#18542
#17362