Finite Group Databasebeta

← #21456

Subgroups — #21456

Order 256.

776 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×C4×#41C211yesmaximal, characteristic, complemented
2C2×#449C211yesmaximal, characteristic
3C2×#442C211yesmaximal, characteristic, complemented
4C2×#440C211yesmaximal, characteristic, complemented
5C4×#121C211yesmaximal, characteristic, complemented
6#1880C211yesmaximal, characteristic, complemented
7#1879C211yesmaximal, characteristic, complemented
8#1875C211yesmaximal, characteristic, complemented
9#1865C211yesmaximal, characteristic, complemented
10#1862C211yesmaximal, characteristic, complemented
11#1854C211yesmaximal, characteristic, complemented
12#1788C211yesmaximal, characteristic, complemented
13#1763C211yesmaximal, characteristic, complemented
14#1762C211yesmaximal, characteristic, complemented
15#1749C211yesmaximal, characteristic, complemented
16C22×#41C2211yescharacteristic
17C22×SD16C2211yescharacteristic
18C22×D8C2211yescharacteristic
19C2×#121C2211yescharacteristic
20C4×#41—24no—
21C4×#41—24no—
22C2×C4×Q8C2211yescharacteristic
23C2×C4×Q8C2211yescharacteristic
24C2×C4×D4—21no—
25C2×C4×D4—21no—
26C2×C4×D4—21no—
27C2×C4×D4C2211yescharacteristic
28C2×C4×D4C2211yescharacteristic
29C2×C4×D4C2211yescharacteristic
30C2×#117—21no—
31C2×#117C2211yescharacteristic
32C22×C42—21no—
33C22×C42C2211yescharacteristic
34#449C418yescomplemented
35#442—24no—
36#440—24no—
37C2×#107C2211yescharacteristic
38C2×#106C2211yescharacteristic
39C2×#103C2211yescharacteristic
40C2×#103C2211yescharacteristic
41C2×#103C2211yescharacteristic
42C2×#102C2211yescharacteristic
43C2×#102C2211yescharacteristic
44C2×#102C2211yescharacteristic
45C2×#102C2211yescharacteristic
46C2×#102C2211yescharacteristic
47C2×C22⋊C8C2211yescharacteristic
48C2×C22⋊C8C2211yescharacteristic
49C2×#97C2211yescharacteristic
50C2×#97C2211yescharacteristic
51#390C2211yescharacteristic
52#389C2211yescharacteristic
53#388—21no—
54#386—21no—
55#380C2211yescharacteristic
56C4×C4⋊C4C2211yescharacteristic
57C4×C22⋊C4—21no—
58#340C2211yescharacteristic
59#340C2211yescharacteristic
60#340C2211yescharacteristic
61#340C2211yescharacteristic
62#336C2211yescharacteristic
63#336C2211yescharacteristic
64C2×#41—22no—
65C2×#41—22no—
66C2×#41—22no—
67C2×#41—22no—
68C2×#41—22no—
69C2×#41C2×C414yescomplemented
70C22×Q8C2311yescharacteristic
71C22×D4—21no—
72C22×D4—21no—
73C22×D4C2311yescharacteristic
74C22×D4C2311yescharacteristic
75C23×C4—21no—
76C23×C4—21no—
77C23×C4—21no—
78C23×C4—21no—
79C23×C4D411yescharacteristic
80C23×C4D411yescharacteristic
81C23×C4C2311yescharacteristic
82C2×SD16—22no—
83C2×SD16—22no—
84C2×SD16C2×C414yescomplemented
85C2×D8—22no—
86C2×D8—22no—
87C2×D8C2×C414yescomplemented
88C22×C8C2311yescharacteristic
89C22×C8C2311yescharacteristic
90C22×C8C2311yescharacteristic
91C22×C8C2311yescharacteristic
92#121—22no—
93#121C2×C414yescomplemented
94C4×Q8—22no—
95C4×Q8—22no—
96C4×Q8—22no—
97C4×Q8D414yescomplemented
98C4×D4—28no—
99C4×D4—28no—
100C4×D4—22no—
101C4×D4—22no—
102C4×D4—22no—
103C4×D4—22no—
104C4×D4—22no—
105C4×D4—44no—
106C4×D4D414yescomplemented
107#117—28no—
108#117—22no—
109#117—22no—
110C2×C4⋊C4—21no—
111C2×C4⋊C4—21no—
112C2×C4⋊C4—21no—
113C2×C4⋊C4C2311yescharacteristic
114C2×C4⋊C4C2311yescharacteristic
115C2×C4⋊C4C2311yescharacteristic
116C2×C4⋊C4C2311yescharacteristic
117C2×C4⋊C4C2311yescharacteristic
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⋊C4—21no—
126C2×C42—28no—
127C2×C42—21no—
128C2×C42—21no—
129C2×C42—21no—
130C2×C42—22no—
131C2×C42—22no—
132C2×C42—21no—
133C2×C42D411yescharacteristic
134C2×C42D411yescharacteristic
135C2×C42C2311yescharacteristic
136C2×C42C2311yescharacteristic
137#107—22no—
138#106—22no—
139#103—22no—
140#103—22no—
141#103C2×C414yescomplemented
142#102—22no—
143#102—22no—
144#102—22no—
145#102—22no—
146#102C2×C414yescomplemented
147C22⋊C8—22no—
148C22⋊C8C2×C414yescomplemented
149#97—22no—
150#97—22no—
151#95—21no—
152#95—21no—
153#95—21no—
154C24—21no—
155C24—21no—
156C24C2×D411yescharacteristic
157#41—28no—
158#41—28no—
159#41—28no—
160#41—28no—
161C2×Q8—21no—
162C2×Q8—21no—
163C2×Q8—21no—
164C2×Q8C22×C412yes—
165C2×Q8C2×D412yes—
166C2×Q8#4112yes—
167C2×D4—24no—
168C2×D4—24no—
169C2×D4—24no—
170C2×D4—24no—
171C2×D4—21no—
172C2×D4—21no—
173C2×D4—21no—
174C2×D4—21no—
175C2×D4—21no—
176C2×D4—21no—
177C2×D4—21no—
178C2×D4—21no—
179C2×D4—42no—
180C2×D4—42no—
181C2×D4C22×C412yes—
182C2×D4C22×C412yes—
183C2×D4C2×D412yes—
184C2×D4#4112yes—
185C22×C4—24no—
186C22×C4—24no—
187C22×C4—21no—
188C22×C4—24no—
189C22×C4—21no—
190C22×C4—21no—
191C22×C4—21no—
192C22×C4—21no—
193C22×C4—24no—
194C22×C4—21no—
195C22×C4—21no—
196C22×C4—21no—
197C22×C4—22no—
198C22×C4—22no—
199C22×C4—24no—
200C22×C4—22no—
201C22×C4—21no—
202C22×C4—21no—
203C22×C4—41no—
204C22×C4—41no—
205C22×C4—41no—
206C22×C4—42no—
207C22×C4—42no—
208C22×C4—42no—
209C22×C4—42no—
210C22×C4—42no—
211C22×C4—42no—
212C22×C4—42no—
213C22×C4C22×C412yes—
214C22×C4C2×D411yescharacteristic
215C22×C4C2×D411yescharacteristic
216C22×C4C2×D411yescharacteristic
217C22×C4C2×D411yescharacteristic
218C22×C4C2×D411yescharacteristic
219C22×C4#4112yes—
220C22×C4C2411yescharacteristic
221SD16—28no—
222D8—28no—
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—21no—
233C2×C8C22×C412yes—
234C2×C8C22×C412yes—
235C4⋊C4—24no—
236C4⋊C4—24no—
237C4⋊C4—21no—
238C4⋊C4—21no—
239C4⋊C4—21no—
240C4⋊C4—21no—
241C4⋊C4—21no—
242C4⋊C4—21no—
243C4⋊C4—21no—
244C4⋊C4—42no—
245C4⋊C4C22×C412yes—
246C4⋊C4C2×D412yes—
247C4⋊C4C2×D412yes—
248C22⋊C4—24no—
249C22⋊C4—24no—
250C22⋊C4—24no—
251C22⋊C4—24no—
252C22⋊C4—42no—
253C22⋊C4—42no—
254C22⋊C4—42no—
255C22⋊C4—42no—
256C42—24no—
257C42—24no—
258C42—22no—
259C42—24no—
260C42—22no—
261C42—21no—
262C42—21no—
263C42—42no—
264C42C2×D412yes—
265C42C2×D412yes—
266C23—22no—
267C23—21no—
268C23—22no—
269C23—21no—
270C23—22no—
271C23—21no—
272C23—21no—
273C23—21no—
274C23—21no—
275C23—22no—
276C23—41no—
277C23—41no—
278C23—41no—
279C23—41no—
280C23—41no—
281C23—41no—
282C23—41no—
283C23—41no—
284C23—41no—
285C23—41no—
286C23C4×D412yes—
287C23C22×D411yescharacteristic
288Q8—22no—
289Q8—22no—
290Q8—22no—
291Q8C4×D414yescomplemented
292D4—216no—
293D4—22no—
294D4—22no—
295D4—22no—
296D4—22no—
297D4—22no—
298D4—48no—
299D4—41no—
300D4—41no—
301D4C4×D414yescomplemented
302C2×C4—216no—
303C2×C4—22no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—22no—
307C2×C4—21no—
308C2×C4—22no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—22no—
315C2×C4—22no—
316C2×C4—22no—
317C2×C4—22no—
318C2×C4—22no—
319C2×C4—21no—
320C2×C4—22no—
321C2×C4—22no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—44no—
325C2×C4—44no—
326C2×C4—48no—
327C2×C4—48no—
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—84no—
341C2×C4C4×D412yes—
342C2×C4#12012yes—
343C2×C4#12012yes—
344C2×C4AGL(1,ℤ/8)12yes—
345C2×C4C23×C411yescharacteristic
346C2×C4C22×D411yescharacteristic
347C2×C4C22×D411yescharacteristic
348C2×C4C2×#4111yescharacteristic
349C2×C4C2×#4111yescharacteristic
350C2×C4C2×#4111yescharacteristic
351C8—22no—
352C8—22no—
353C8—41no—
354C8—41no—
355C22—22no—
356C22—22no—
357C22—22no—
358C22—24no—
359C22—41no—
360C22—42no—
361C22—41no—
362C22—42no—
363C22—42no—
364C22—42no—
365C22—41no—
366C22—42no—
367C22—41no—
368C22—42no—
369C22—42no—
370C22—42no—
371C22—42no—
372C22—81no—
373C22—81no—
374C22—81no—
375C22—81no—
376C22C2×C4×D411yescharacteristic
377C22C2×#12011yescharacteristic
378C22#52511yescharacteristic
379C22C2×AGL(1,ℤ/8)11yescharacteristic
380C22#57511yescharacteristic
381C22#57511yescharacteristic
382C22#57511yescharacteristic
383C4—22no—
384C4—22no—
385C4—22no—
386C4—22no—
387C4—24no—
388C4—42no—
389C4—41no—
390C4—42no—
391C4—42no—
392C4—81no—
393C4—81no—
394C4C2×C4×D411yescharacteristic
395C4C2×C4×D411yescharacteristic
396C4#52511yescharacteristic
397C4#52511yescharacteristic
398C2—42no—
399C2—44no—
400C2—82no—
401C2C4×#12011yescharacteristic
402C2#294311yescharacteristic
403C2C4×AGL(1,ℤ/8)11yescharacteristic
404C2#300511yescharacteristic
405C2#300611yescharacteristic
406C2#311111yescharacteristic
407C2#311511yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutatorC2×C4C23×C4
Frattini, cosocleC22×C4C24
prime radicalC2×#449C2
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. C2×C4×#41C2semidirect
3. C2×#442C2semidirect
4. C2×#440C2semidirect
5. C4×#121C2semidirect
6. #1880C2semidirect
7. #1879C2semidirect
8. #1875C2semidirect
9. #1865C2semidirect
10. #1862C2semidirect
11. #1854C2semidirect
12. #1788C2semidirect
13. #1763C2semidirect
14. #1762C2semidirect
15. #1749C2semidirect
34. #449C4semidirect
69. C2×#41C2×C4semidirect
84. C2×SD16C2×C4semidirect
87. C2×D8C2×C4semidirect
93. #121C2×C4semidirect
97. C4×Q8D4semidirect
106. C4×D4D4semidirect
141. #103C2×C4semidirect
146. #102C2×C4semidirect
148. C22⋊C8C2×C4semidirect
291. Q8C4×D4semidirect
301. D4C4×D4semidirect

Maximal subgroups

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

subgroupindex
C2×C4×#412
C2×#4492
C2×#4422
C2×#4402
C4×#1212
#18802
#18792
#18752
#18652
#18622
#18542
#17882
#17632
#17622
#17492