Finite Group Databasebeta

← #21459

Subgroups — #21459

Order 256.

780 conjugacy classes of subgroups.

The subgroup lattice has 411 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#518C211yesmaximal, characteristic
2C2×#450C211yesmaximal, characteristic
3C2×#440C211yesmaximal, characteristic, complemented
4C2×C4×SD16C211yesmaximal, characteristic, complemented
5#2323C211yesmaximal, characteristic, complemented
6#1877C211yesmaximal, characteristic, complemented
7#1877C211yesmaximal, characteristic, complemented
8#1873C211yesmaximal, characteristic, complemented
9#1864C211yesmaximal, characteristic, complemented
10#1862C211yesmaximal, characteristic, complemented
11#1854C211yesmaximal, characteristic, complemented
12#1786C211yesmaximal, characteristic
13#1762C211yesmaximal, characteristic, complemented
14C4×#102C211yesmaximal, characteristic, complemented
15#1750C211yesmaximal, characteristic, complemented
16C23×D4C2211yescharacteristic
17C22×SD16C2211yescharacteristic
18C22×SD16C2211yescharacteristic
19C2×C22⋊Q8C2211yescharacteristic
20#518—24no—
21#518—24no—
22C2×C4×Q8C2211yescharacteristic, complemented
23C2×C4×D4—21no—
24C2×C4×D4—21no—
25C2×C4×D4C2211yescharacteristic
26C2×C4×D4C2211yescharacteristic
27C2×C4×D4C2211yescharacteristic
28C2×C4×D4C2211yescharacteristic
29C2×#117C2211yescharacteristic
30C2×#117C2211yescharacteristic
31C22×C22⋊C4—21no—
32C22×C22⋊C4—21no—
33#450C418yescomplemented
34#440—24no—
35C4×SD16—24no—
36C2×#107C2211yescharacteristic, complemented
37C2×#106C2211yescharacteristic, complemented
38C2×#103C2211yescharacteristic, complemented
39C2×#103C2211yescharacteristic, complemented
40C2×#102C2211yescharacteristic
41C2×#102C2211yescharacteristic
42C2×#102C2211yescharacteristic
43C2×#102C2211yescharacteristic
44C2×#102C2211yescharacteristic
45C2×#102C2211yescharacteristic
46C2×C22⋊C8C2211yescharacteristic
47C2×C22⋊C8C2211yescharacteristic
48C2×#97C2211yescharacteristic, complemented
49C2×C4×C8C2211yescharacteristic, complemented
50#391C2211yescharacteristic, complemented
51#389—21no—
52#389C2211yescharacteristic, complemented
53#388—21no—
54#388—21no—
55#387—21no—
56#380C2211yescharacteristic
57C4×C4⋊C4C2211yescharacteristic, complemented
58#340C2211yescharacteristic, complemented
59#340C2211yescharacteristic, complemented
60#340C2211yescharacteristic, complemented
61#340C2211yescharacteristic, complemented
62#336C2211yescharacteristic, complemented
63#336C2211yescharacteristic, complemented
64C25—21no—
65C22×Q8C2311yescharacteristic
66C22×D4—22no—
67C22×D4—22no—
68C22×D4—22no—
69C22×D4—22no—
70C22×D4—21no—
71C22×D4—22no—
72C22×D4C2×C414yes—
73C22×D4C2311yescharacteristic
74C22×D4C2311yescharacteristic
75C23×C4—21no—
76C23×C4—21no—
77C23×C4—21no—
78C23×C4—21no—
79C23×C4C2311yescharacteristic
80C2×SD16—22no—
81C2×SD16—22no—
82C2×SD16—22no—
83C2×SD16—22no—
84C2×SD16C2×C414yescomplemented
85C2×SD16C2×C414yescomplemented
86C22×C8C2311yescharacteristic
87C22×C8C2311yescharacteristic
88C22×C8C2311yescharacteristic
89C22×C8C2311yescharacteristic
90C22⋊Q8—22no—
91C22⋊Q8C2×C414yescomplemented
92C4×Q8—22no—
93C4×Q8—22no—
94C4×D4—28no—
95C4×D4—28no—
96C4×D4—22no—
97C4×D4—22no—
98C4×D4—22no—
99C4×D4—22no—
100C4×D4—22no—
101C4×D4—22no—
102C4×D4D414yes—
103C4×D4D414yes—
104#117—22no—
105#117—22no—
106#117—22no—
107#117—22no—
108C2×C4⋊C4—21no—
109C2×C4⋊C4—21no—
110C2×C4⋊C4—21no—
111C2×C4⋊C4D411yescharacteristic
112C2×C4⋊C4D411yescharacteristic
113C2×C4⋊C4C2311yescharacteristic
114C2×C4⋊C4C2311yescharacteristic
115C2×C4⋊C4C2311yescharacteristic
116C2×C4⋊C4C2311yescharacteristic
117C2×C4⋊C4C2311yescharacteristic
118C2×C22⋊C4—28no—
119C2×C22⋊C4—28no—
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⋊C4D411yescharacteristic
128C2×C22⋊C4D411yescharacteristic
129C2×C42—21no—
130C2×C42—21no—
131C2×C42—21no—
132C2×C42C2311yescharacteristic
133C2×C42C2311yescharacteristic
134#107—22no—
135#106—22no—
136#103—22no—
137#103—22no—
138#102—22no—
139#102—22no—
140#102—22no—
141#102—22no—
142#102C2×C414yescomplemented
143#102C2×C414yescomplemented
144C22⋊C8—22no—
145C22⋊C8C2×C414yescomplemented
146#97—22no—
147C4×C8—22no—
148#95—21no—
149#95—21no—
150#95—21no—
151#95—21no—
152#95—21no—
153C24—24no—
154C24—24no—
155C24—21no—
156C24—21no—
157C24—24no—
158C24—81no—
159C24—81no—
160C24—81no—
161C24—81no—
162C24C2×D411yescharacteristic
163C2×Q8—21no—
164C2×Q8—21no—
165C2×Q8—21no—
166C2×Q8—21no—
167C2×Q8—21no—
168C2×Q8C22×C412yes—
169C2×D4—24no—
170C2×D4—24no—
171C2×D4—28no—
172C2×D4—28no—
173C2×D4—28no—
174C2×D4—28no—
175C2×D4—24no—
176C2×D4—21no—
177C2×D4—21no—
178C2×D4—21no—
179C2×D4—21no—
180C2×D4—21no—
181C2×D4—21no—
182C2×D4C22×C412yes—
183C2×D4C22×C412yes—
184C2×D4C2×D412yes—
185C2×D4C2×D412yes—
186C2×D4#4112yes—
187C2×D4#4112yes—
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—24no—
197C22×C4—24no—
198C22×C4—24no—
199C22×C4—24no—
200C22×C4—21no—
201C22×C4—21no—
202C22×C4—81no—
203C22×C4—81no—
204C22×C4—81no—
205C22×C4—81no—
206C22×C4—81no—
207C22×C4—81no—
208C22×C4—81no—
209C22×C4—81no—
210C22×C4C22×C412yes—
211C22×C4C2×D411yescharacteristic
212C22×C4C2×D411yescharacteristic
213C22×C4C2×D411yescharacteristic
214C22×C4C2×D411yescharacteristic
215C22×C4C2×D411yescharacteristic
216C22×C4#4112yes—
217C22×C4C2411yescharacteristic
218SD16—28no—
219SD16—28no—
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×C8C22×C412yes—
231C2×C8C22×C412yes—
232C4⋊C4—24no—
233C4⋊C4—22no—
234C4⋊C4—22no—
235C4⋊C4—21no—
236C4⋊C4—21no—
237C4⋊C4—21no—
238C4⋊C4—21no—
239C4⋊C4—21no—
240C4⋊C4—21no—
241C4⋊C4—21no—
242C4⋊C4—42no—
243C4⋊C4—81no—
244C4⋊C4C22×C412yes—
245C4⋊C4C2×D412yes—
246C4⋊C4C2×D412yes—
247C22⋊C4—24no—
248C22⋊C4—24no—
249C22⋊C4—24no—
250C22⋊C4—24no—
251C22⋊C4—22no—
252C22⋊C4—22no—
253C22⋊C4—24no—
254C22⋊C4—24no—
255C22⋊C4—81no—
256C42—24no—
257C42—21no—
258C42—21no—
259C42—42no—
260C42—81no—
261C42C2×D412yes—
262C42C2×D412yes—
263C23—216no—
264C23—22no—
265C23—22no—
266C23—21no—
267C23—22no—
268C23—22no—
269C23—21no—
270C23—22no—
271C23—22no—
272C23—21no—
273C23—21no—
274C23—21no—
275C23—21no—
276C23—41no—
277C23—41no—
278C23—41no—
279C23—41no—
280C23—41no—
281C23—41no—
282C23—41no—
283C23—41no—
284C23—84no—
285C23—84no—
286C23—84no—
287C23—84no—
288C23—84no—
289C23—84no—
290C23C4×D412yes—
291C23C22×D411yescharacteristic
292Q8—22no—
293Q8—22no—
294Q8—41no—
295Q8—41no—
296D4—216no—
297D4—22no—
298D4—22no—
299D4—22no—
300D4—22no—
301D4—22no—
302D4—22no—
303D4C4×D414yes—
304D4C4×D414yes—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—22no—
312C2×C4—22no—
313C2×C4—22no—
314C2×C4—22no—
315C2×C4—22no—
316C2×C4—22no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—22no—
320C2×C4—22no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
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—84no—
340C2×C4—84no—
341C2×C4—84no—
342C2×C4—84no—
343C2×C4C4×D412yes—
344C2×C4#12012yes—
345C2×C4#12012yes—
346C2×C4C23×C411yescharacteristic
347C2×C4C22×D411yescharacteristic
348C2×C4C22×D411yescharacteristic
349C2×C4C2×#4111yescharacteristic
350C2×C4C2×#4111yescharacteristic
351C2×C4C2×#4111yescharacteristic
352C8—22no—
353C8—22no—
354C8—41no—
355C8—41no—
356C22—22no—
357C22—22no—
358C22—22no—
359C22—24no—
360C22—24no—
361C22—41no—
362C22—42no—
363C22—42no—
364C22—41no—
365C22—42no—
366C22—42no—
367C22—42no—
368C22—42no—
369C22—42no—
370C22—42no—
371C22—42no—
372C22—42no—
373C22—41no—
374C22—42no—
375C22—42no—
376C22—41no—
377C22—816no—
378C22C2×C4×D411yescharacteristic
379C22C2×#12011yescharacteristic
380C22#52511yescharacteristic
381C22#57611yescharacteristic
382C22#57711yescharacteristic
383C22#57711yescharacteristic
384C22#57711yescharacteristic
385C4—22no—
386C4—22no—
387C4—22no—
388C4—42no—
389C4—41no—
390C4—42no—
391C4—41no—
392C4—42no—
393C4—42no—
394C4—81no—
395C4—81no—
396C4C2×C4×D411yescharacteristic
397C4C2×C4×D411yescharacteristic
398C4#52511yescharacteristic
399C4#52511yescharacteristic
400C2—42no—
401C2—44no—
402C2—44no—
403C2C4×#12011yescharacteristic
404C2#294611yescharacteristic
405C2#294911yescharacteristic
406C2#300911yescharacteristic
407C2#301211yescharacteristic
408C2#311811yescharacteristic
409C2#312311yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutatorC2×C4C23×C4
Frattini, cosocleC22×C4C24
prime radicalC23×D4C22
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. C2×#440C2semidirect
4. C2×C4×SD16C2semidirect
5. #2323C2semidirect
6. #1877C2semidirect
7. #1877C2semidirect
8. #1873C2semidirect
9. #1864C2semidirect
10. #1862C2semidirect
11. #1854C2semidirect
13. #1762C2semidirect
14. C4×#102C2semidirect
15. #1750C2semidirect
22. C2×C4×Q8C22semidirect
33. #450C4semidirect
36. C2×#107C22semidirect
37. C2×#106C22semidirect
38. C2×#103C22semidirect
39. C2×#103C22semidirect
48. C2×#97C22semidirect
49. C2×C4×C8C22semidirect
50. #391C22semidirect
52. #389C22semidirect
57. C4×C4⋊C4C22semidirect
58. #340C22semidirect
59. #340C22semidirect
60. #340C22semidirect
61. #340C22semidirect
62. #336C22semidirect
63. #336C22semidirect
84. C2×SD16C2×C4semidirect
85. C2×SD16C2×C4semidirect
91. C22⋊Q8C2×C4semidirect
142. #102C2×C4semidirect
143. #102C2×C4semidirect
145. C22⋊C8C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#5182
C2×#4502
C2×#4402
C2×C4×SD162
#23232
#18772
#18732
#18642
#18622
#18542
#17862
#17622
C4×#1022
#17502