Finite Group Databasebeta

← #24259

Subgroups — #24259

Order 256.

783 conjugacy classes of subgroups.

The subgroup lattice has 462 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#546C211yesmaximal, characteristic
2C2×#464C211yesmaximal, characteristic, complemented
3C2×#452C211yesmaximal, characteristic, complemented
4C2×#450C211yesmaximal, characteristic, complemented
5C2×#440C211yesmaximal, characteristic, complemented
6#2538C211yesmaximal, characteristic, complemented
7#2036C211yesmaximal, characteristic, complemented
8#2033C211yesmaximal, characteristic, complemented
9#2001C211yesmaximal, characteristic, complemented
10#2000C211yesmaximal, characteristic, complemented
11#1959C211yesmaximal, characteristic, complemented
12#1955C211yesmaximal, characteristic, complemented
13#1802C211yesmaximal, characteristic, complemented
14#1788C211yesmaximal, characteristic, complemented
15#1786C211yesmaximal, characteristic, complemented
16C22×#41C2211yescharacteristic
17C23×D4C2211yescharacteristic
18C22×Q16C2211yescharacteristic, complemented
19C22×SD16C2211yescharacteristic
20C22×SD16C2211yescharacteristic
21#546—24no—
22#546—24no—
23C2×#124C2211yescharacteristic
24C2×#123—21no—
25C2×C22⋊Q8C2211yescharacteristic
26C2×C22⋊Q8C2211yescharacteristic
27C2×C22⋊Q8C2211yescharacteristic, complemented
28C2×#121—21no—
29C2×#121C2211yescharacteristic
30C2×#120—21no—
31C2×C4×Q8C2211yescharacteristic, complemented
32C2×C4×D4C2211yescharacteristic
33C2×C4×D4C2211yescharacteristic
34C22×C22⋊C4—21no—
35#464—24no—
36#452—24no—
37#450—24no—
38#440—24no—
39C2×#107C2211yescharacteristic, complemented
40C2×C4⋊C8C2211yescharacteristic, complemented
41C2×#103C2211yescharacteristic, complemented
42C2×#103C2211yescharacteristic, complemented
43C2×#103C2211yescharacteristic, complemented
44C2×#103C2211yescharacteristic, complemented
45C2×#102C2211yescharacteristic
46C2×#102C2211yescharacteristic
47C2×#102C2211yescharacteristic
48C2×#102C2211yescharacteristic
49C2×#102C2211yescharacteristic
50C2×#102C2211yescharacteristic
51C2×C22⋊C8C2211yescharacteristic
52C2×C22⋊C8C2211yescharacteristic
53C2×C22⋊C8C2211yescharacteristic, complemented
54C2×C22⋊C8C2211yescharacteristic, complemented
55C2×#97C2211yescharacteristic, complemented
56#396—21no—
57#396—21no—
58#390—21no—
59#389—21no—
60#384C2211yescharacteristic, complemented
61#380C2211yescharacteristic
62#380C2211yescharacteristic, complemented
63#340C2211yescharacteristic, complemented
64#340C2211yescharacteristic, complemented
65C25—21no—
66C2×#41—22no—
67C2×#41—22no—
68C2×#41—22no—
69C2×#41—22no—
70C2×#41—41no—
71C2×#41—41no—
72C22×Q8C2311yescharacteristic
73C22×Q8C2311yescharacteristic
74C22×D4—21no—
75C22×D4—21no—
76C22×D4—22no—
77C22×D4—22no—
78C22×D4—22no—
79C22×D4—22no—
80C22×D4—21no—
81C22×D4—41no—
82C22×D4—41no—
83C22×D4D411yescharacteristic
84C22×D4C2311yescharacteristic
85C22×D4C2311yescharacteristic
86C23×C4—21no—
87C23×C4—21no—
88C23×C4C2311yescharacteristic
89C23×C4C2311yescharacteristic
90C2×Q16—22no—
91C2×Q16—22no—
92C2×Q16—41no—
93C2×SD16—22no—
94C2×SD16—22no—
95C2×SD16—22no—
96C2×SD16—22no—
97C2×SD16—41no—
98C2×SD16D414yescomplemented
99C22×C8C2311yescharacteristic
100C22×C8C2311yescharacteristic
101C22×C8C2311yescharacteristic
102C22×C8C2311yescharacteristic
103#124—22no—
104#124D414yes—
105#123—28no—
106C22⋊Q8—22no—
107C22⋊Q8—22no—
108C22⋊Q8—22no—
109C22⋊Q8—22no—
110C22⋊Q8—41no—
111C22⋊Q8—41no—
112#121—28no—
113#121—22no—
114#121—22no—
115#120—28no—
116C4×Q8—22no—
117C4×Q8—22no—
118C4×D4—22no—
119C4×D4—22no—
120C4×D4—22no—
121C4×D4D414yes—
122C2×C4⋊C4—21no—
123C2×C4⋊C4—21no—
124C2×C4⋊C4—21no—
125C2×C4⋊C4—21no—
126C2×C4⋊C4—21no—
127C2×C4⋊C4—21no—
128C2×C4⋊C4D411yescharacteristic
129C2×C4⋊C4C2311yescharacteristic
130C2×C4⋊C4C2311yescharacteristic
131C2×C4⋊C4C2311yescharacteristic
132C2×C4⋊C4C2311yescharacteristic
133C2×C22⋊C4—28no—
134C2×C22⋊C4—21no—
135C2×C22⋊C4—21no—
136C2×C22⋊C4—21no—
137C2×C22⋊C4—21no—
138C2×C22⋊C4—21no—
139C2×C22⋊C4—21no—
140C2×C22⋊C4—21no—
141C2×C22⋊C4—21no—
142C2×C22⋊C4—21no—
143C2×C22⋊C4D411yescharacteristic
144C2×C22⋊C4D411yescharacteristic
145C2×C42—21no—
146C2×C42C2311yescharacteristic
147#107—22no—
148C4⋊C8—22no—
149#103—22no—
150#103—22no—
151#103—22no—
152#103—41no—
153#102—22no—
154#102—22no—
155#102—22no—
156#102—22no—
157#102—22no—
158#102—41no—
159C22⋊C8—22no—
160C22⋊C8—22no—
161C22⋊C8—41no—
162C22⋊C8—41no—
163#97—22no—
164#95—21no—
165#95—21no—
166C24—21no—
167C24—24no—
168C24—21no—
169C24—42no—
170C24—42no—
171C24—42no—
172C24—42no—
173C24—81no—
174C24—81no—
175C24C2×D411yescharacteristic
176C24C2×D411yescharacteristic
177#41—44no—
178#41—44no—
179#41—44no—
180#41—44no—
181C2×Q8—21no—
182C2×Q8—21no—
183C2×Q8—21no—
184C2×Q8—21no—
185C2×Q8—21no—
186C2×Q8—21no—
187C2×Q8—21no—
188C2×Q8—21no—
189C2×Q8—21no—
190C2×Q8—21no—
191C2×Q8C2×D412yes—
192C2×Q8C2×D412yes—
193C2×D4—24no—
194C2×D4—24no—
195C2×D4—22no—
196C2×D4—24no—
197C2×D4—22no—
198C2×D4—21no—
199C2×D4—21no—
200C2×D4—21no—
201C2×D4—21no—
202C2×D4—21no—
203C2×D4—21no—
204C2×D4—21no—
205C2×D4—21no—
206C2×D4—41no—
207C2×D4—42no—
208C2×D4—42no—
209C2×D4—44no—
210C2×D4—44no—
211C2×D4—44no—
212C2×D4—42no—
213C2×D4—42no—
214C2×D4—44no—
215C2×D4—81no—
216C2×D4—81no—
217C2×D4C2×D412yes—
218C2×D4C2×D412yes—
219C2×D4C2×D412yes—
220C2×D4#4112yes—
221C22×C4—24no—
222C22×C4—21no—
223C22×C4—21no—
224C22×C4—21no—
225C22×C4—21no—
226C22×C4—21no—
227C22×C4—21no—
228C22×C4—21no—
229C22×C4—21no—
230C22×C4—21no—
231C22×C4—21no—
232C22×C4—21no—
233C22×C4—21no—
234C22×C4—21no—
235C22×C4—21no—
236C22×C4—21no—
237C22×C4—24no—
238C22×C4—21no—
239C22×C4—21no—
240C22×C4—21no—
241C22×C4—42no—
242C22×C4—42no—
243C22×C4—81no—
244C22×C4—81no—
245C22×C4C2×D411yescharacteristic
246C22×C4C2×D411yescharacteristic
247C22×C4C2×D411yescharacteristic
248C22×C4C2×D411yescharacteristic
249C22×C4C2411yescharacteristic
250Q16—44no—
251SD16—28no—
252SD16—44no—
253C2×C8—21no—
254C2×C8—21no—
255C2×C8—21no—
256C2×C8—21no—
257C2×C8—21no—
258C2×C8—21no—
259C2×C8—21no—
260C2×C8—21no—
261C2×C8—21no—
262C2×C8—21no—
263C2×C8—21no—
264C2×C8C2×D412yes—
265C4⋊C4—24no—
266C4⋊C4—22no—
267C4⋊C4—21no—
268C4⋊C4—21no—
269C4⋊C4—21no—
270C4⋊C4—21no—
271C4⋊C4—21no—
272C4⋊C4—21no—
273C4⋊C4—21no—
274C4⋊C4—42no—
275C4⋊C4—42no—
276C4⋊C4—42no—
277C4⋊C4—42no—
278C4⋊C4—81no—
279C4⋊C4C2×D412yes—
280C22⋊C4—24no—
281C22⋊C4—24no—
282C22⋊C4—22no—
283C22⋊C4—24no—
284C22⋊C4—24no—
285C22⋊C4—24no—
286C22⋊C4—22no—
287C22⋊C4—42no—
288C22⋊C4—42no—
289C22⋊C4—81no—
290C22⋊C4—81no—
291C42—42no—
292C42C2×D412yes—
293C42C2×D412yes—
294C23—21no—
295C23—22no—
296C23—21no—
297C23—22no—
298C23—21no—
299C23—22no—
300C23—21no—
301C23—21no—
302C23—21no—
303C23—22no—
304C23—21no—
305C23—21no—
306C23—21no—
307C23—21no—
308C23—21no—
309C23—21no—
310C23—21no—
311C23—48no—
312C23—48no—
313C23—41no—
314C23—41no—
315C23—41no—
316C23—41no—
317C23—41no—
318C23—41no—
319C23—41no—
320C23—41no—
321C23—41no—
322C23—41no—
323C23—84no—
324C23—84no—
325C23—84no—
326C23—84no—
327C23—84no—
328C23#13712yes—
329C23C22×D411yescharacteristic
330Q8—22no—
331Q8—22no—
332Q8—22no—
333Q8—22no—
334Q8—41no—
335Q8—41no—
336Q8—41no—
337Q8—41no—
338D4—22no—
339D4—22no—
340D4—22no—
341D4—22no—
342D4—22no—
343D4—44no—
344D4—48no—
345D4—48no—
346D4—41no—
347D4—41no—
348D4—84no—
349D4#12114yes—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—22no—
355C2×C4—21no—
356C2×C4—22no—
357C2×C4—21no—
358C2×C4—22no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—48no—
364C2×C4—41no—
365C2×C4—41no—
366C2×C4—41no—
367C2×C4—41no—
368C2×C4—41no—
369C2×C4—41no—
370C2×C4—41no—
371C2×C4—41no—
372C2×C4—41no—
373C2×C4—41no—
374C2×C4—41no—
375C2×C4—41no—
376C2×C4—41no—
377C2×C4—41no—
378C2×C4—41no—
379C2×C4—41no—
380C2×C4—41no—
381C2×C4—41no—
382C2×C4—41no—
383C2×C4—41no—
384C2×C4—41no—
385C2×C4—41no—
386C2×C4—41no—
387C2×C4—84no—
388C2×C4#12012yes—
389C2×C4#12012yes—
390C2×C4C22×D411yescharacteristic
391C2×C4C22×D411yescharacteristic
392C2×C4C22×D411yescharacteristic
393C2×C4C2×#4111yescharacteristic
394C2×C4H(5,𝔽2)11yescharacteristic
395C2×C4H(5,𝔽2)11yescharacteristic
396C8—22no—
397C8—41no—
398C8—41no—
399C8—41no—
400C22—24no—
401C22—22no—
402C22—22no—
403C22—22no—
404C22—41no—
405C22—42no—
406C22—42no—
407C22—42no—
408C22—41no—
409C22—41no—
410C22—42no—
411C22—41no—
412C22—42no—
413C22—42no—
414C22—41no—
415C22—42no—
416C22—41no—
417C22—42no—
418C22—41no—
419C22—41no—
420C22—42no—
421C22—41no—
422C22—41no—
423C22—41no—
424C22—816no—
425C22—81no—
426C22—81no—
427C22—81no—
428C22—81no—
429C22C2×#12011yescharacteristic
430C22D4211yescharacteristic
431C22#54611yescharacteristic
432C22C2×#13711yescharacteristic
433C22#57511yescharacteristic
434C22#57611yescharacteristic
435C22#57711yescharacteristic
436C4—22no—
437C4—22no—
438C4—41no—
439C4—41no—
440C4—42no—
441C4—42no—
442C4—42no—
443C4—81no—
444C4—81no—
445C4—81no—
446C4C2×#12111yescharacteristic
447C4#53511yescharacteristic
448C4D4211yescharacteristic
449C4#54611yescharacteristic
450C2—42no—
451C2—44no—
452C2—81no—
453C2—82no—
454C2#240811yescharacteristic
455C2#300711yescharacteristic
456C2#301211yescharacteristic
457C2#326611yescharacteristic
458C2#326811yescharacteristic
459C2#331111yescharacteristic
460C2#331411yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC2×#546C2
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
2. C2×#464C2semidirect
3. C2×#452C2semidirect
4. C2×#450C2semidirect
5. C2×#440C2semidirect
6. #2538C2semidirect
7. #2036C2semidirect
8. #2033C2semidirect
9. #2001C2semidirect
10. #2000C2semidirect
11. #1959C2semidirect
12. #1955C2semidirect
13. #1802C2semidirect
14. #1788C2semidirect
15. #1786C2semidirect
18. C22×Q16C22semidirect
27. C2×C22⋊Q8C22semidirect
31. C2×C4×Q8C22semidirect
39. C2×#107C22semidirect
40. C2×C4⋊C8C22semidirect
41. C2×#103C22semidirect
42. C2×#103C22semidirect
43. C2×#103C22semidirect
44. C2×#103C22semidirect
53. C2×C22⋊C8C22semidirect
54. C2×C22⋊C8C22semidirect
55. C2×#97C22semidirect
60. #384C22semidirect
62. #380C22semidirect
63. #340C22semidirect
64. #340C22semidirect
98. C2×SD16D4semidirect

Maximal subgroups

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

subgroupindex
C2×#5462
C2×#4642
C2×#4522
C2×#4502
C2×#4402
#25382
#20362
#20332
#20012
#20002
#19592
#19552
#18022
#17882
#17862