Finite Group Databasebeta

← #24394

Subgroups — #24394

Order 256.

789 conjugacy classes of subgroups.

The subgroup lattice has 482 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#535C211yesmaximal, characteristic
2C2×#468C211yesmaximal, characteristic, complemented
3C2×#465C211yesmaximal, characteristic, complemented
4C2×#450C211yesmaximal, characteristic, complemented
5C2×#408C211yesmaximal, characteristic, complemented
6#2616C211yesmaximal, characteristic, complemented
7#2034C211yesmaximal, characteristic, complemented
8#2033C211yesmaximal, characteristic, complemented
9#2000C211yesmaximal, characteristic, complemented
10#2000C211yesmaximal, characteristic, complemented
11#1895C211yesmaximal, characteristic, complemented
12#1893C211yesmaximal, characteristic, complemented
13#1873C211yesmaximal, characteristic, complemented
14#1854C211yesmaximal, characteristic, complemented
15#1786C211yesmaximal, characteristic, complemented
16C22×#41C2211yescharacteristic
17C23×D4C2211yescharacteristic
18C22×SD16C2211yescharacteristic
19C22×SD16C2211yescharacteristic
20C22×M16C2211yescharacteristic, complemented
21C23×C8C2211yescharacteristic, complemented
22#535—24no—
23#535—24no—
24C2×#127—21no—
25C2×#124—21no—
26C2×C22⋊Q8C2211yescharacteristic
27C2×C22⋊Q8C2211yescharacteristic, complemented
28C2×C22⋊Q8C2211yescharacteristic, complemented
29C2×#121C2211yescharacteristic
30C2×#121C2211yescharacteristic
31C2×#121C2211yescharacteristic
32C2×#121C2211yescharacteristic
33C2×#120—21no—
34C2×#120—21no—
35C2×#117C2211yescharacteristic
36#468—24no—
37#465—24no—
38#450—24no—
39C2×#107C2211yescharacteristic, complemented
40C2×#106C2211yescharacteristic, complemented
41C2×#103C2211yescharacteristic, complemented
42C2×#103C2211yescharacteristic, complemented
43C2×#102C2211yescharacteristic
44C2×#102C2211yescharacteristic
45C2×#102C2211yescharacteristic
46C2×#102C2211yescharacteristic
47C2×#102C2211yescharacteristic
48C2×#102C2211yescharacteristic
49#408—24no—
50C2×C22⋊C8C2211yescharacteristic
51C2×C22⋊C8C2211yescharacteristic
52C2×C22⋊C8C2211yescharacteristic, complemented
53C2×C22⋊C8C2211yescharacteristic, complemented
54#395—21no—
55#394—21no—
56#394—21no—
57#394—21no—
58#380C2211yescharacteristic
59#380C2211yescharacteristic, complemented
60#380C2211yescharacteristic, complemented
61#340C2211yescharacteristic, complemented
62#340C2211yescharacteristic, 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×Q8D411yescharacteristic
73C22×Q8C2311yescharacteristic
74C22×D4—21no—
75C22×D4—21no—
76C22×D4—21no—
77C22×D4—21no—
78C22×D4—22no—
79C22×D4—22no—
80C22×D4—21no—
81C22×D4—22no—
82C22×D4—22no—
83C22×D4—41no—
84C22×D4—41no—
85C22×D4D411yescharacteristic
86C22×D4D411yescharacteristic
87C22×D4D411yescharacteristic
88C22×D4C2311yescharacteristic
89C22×D4C2311yescharacteristic
90C23×C4C2311yescharacteristic
91C23×C4C2311yescharacteristic
92C23×C4C2311yescharacteristic
93C2×SD16—22no—
94C2×SD16—22no—
95C2×SD16—22no—
96C2×SD16—22no—
97C2×SD16—41no—
98C2×SD16—41no—
99C2×M16—22no—
100C2×M16—22no—
101C2×M16—41no—
102C22×C8—22no—
103C22×C8—22no—
104C22×C8—41no—
105C22×C8C2311yescharacteristic
106C22×C8C2311yescharacteristic
107C22×C8C2311yescharacteristic
108C22×C8C2311yescharacteristic
109#127—28no—
110#124—28no—
111C22⋊Q8—22no—
112C22⋊Q8—22no—
113C22⋊Q8—22no—
114C22⋊Q8—22no—
115C22⋊Q8—41no—
116C22⋊Q8—41no—
117#121—22no—
118#121—22no—
119#121—22no—
120#121—22no—
121#121—22no—
122#121—22no—
123#121D414yes—
124#121D414yes—
125#120—28no—
126#120—28no—
127#117—22no—
128#117—22no—
129C2×C4⋊C4—21no—
130C2×C4⋊C4—21no—
131C2×C4⋊C4—21no—
132C2×C4⋊C4C2311yescharacteristic
133C2×C4⋊C4C2311yescharacteristic
134C2×C4⋊C4C2311yescharacteristic
135C2×C4⋊C4C2311yescharacteristic
136C2×C4⋊C4C2311yescharacteristic
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⋊C4—21no—
144C2×C22⋊C4—21no—
145C2×C22⋊C4—21no—
146C2×C22⋊C4—21no—
147C2×C22⋊C4—21no—
148C2×C42—21no—
149#107—22no—
150#106—22no—
151#103—22no—
152#103—22no—
153#102—22no—
154#102—22no—
155#102—22no—
156#102—22no—
157#102—41no—
158#102—41no—
159C22⋊C8—22no—
160C22⋊C8—22no—
161C22⋊C8—22no—
162C22⋊C8D414yescomplemented
163#95—21no—
164#95—21no—
165#95—21no—
166C24—24no—
167C24—21no—
168C24—21no—
169C24—42no—
170C24—42no—
171C24—81no—
172C24—81no—
173C24—81no—
174C24—81no—
175C24C2×D411yescharacteristic
176C24C2×D411yescharacteristic
177C24C2×D411yescharacteristic
178#41—44no—
179#41—44no—
180#41—44no—
181#41—44no—
182C2×Q8—22no—
183C2×Q8—22no—
184C2×Q8—21no—
185C2×Q8—21no—
186C2×Q8—21no—
187C2×Q8—21no—
188C2×Q8—21no—
189C2×Q8—21no—
190C2×Q8—41no—
191C2×D4—24no—
192C2×D4—22no—
193C2×D4—22no—
194C2×D4—22no—
195C2×D4—22no—
196C2×D4—24no—
197C2×D4—24no—
198C2×D4—24no—
199C2×D4—24no—
200C2×D4—22no—
201C2×D4—22no—
202C2×D4—21no—
203C2×D4—21no—
204C2×D4—21no—
205C2×D4—21no—
206C2×D4—21no—
207C2×D4—21no—
208C2×D4—21no—
209C2×D4—21no—
210C2×D4—21no—
211C2×D4—21no—
212C2×D4—44no—
213C2×D4—44no—
214C2×D4—44no—
215C2×D4—44no—
216C2×D4—41no—
217C2×D4—41no—
218C2×D4—41no—
219C2×D4—42no—
220C2×D4—42no—
221C2×D4—81no—
222C2×D4—81no—
223C2×D4—81no—
224C2×D4—81no—
225C2×D4—81no—
226C2×D4—81no—
227C2×D4—81no—
228C2×D4—81no—
229C2×D4C2×D412yes—
230C2×D4C2×D412yes—
231C22×C4—21no—
232C22×C4—21no—
233C22×C4—21no—
234C22×C4—21no—
235C22×C4—21no—
236C22×C4—21no—
237C22×C4—21no—
238C22×C4—21no—
239C22×C4—21no—
240C22×C4—21no—
241C22×C4—21no—
242C22×C4—21no—
243C22×C4—21no—
244C22×C4—21no—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—21no—
248C22×C4—21no—
249C22×C4—21no—
250C22×C4—21no—
251C22×C4C2×D411yescharacteristic
252C22×C4C2×D411yescharacteristic
253C22×C4C2×D412yes—
254C22×C4C2×D412yes—
255C22×C4C2×D411yescharacteristic
256C22×C4C2×D412yes—
257C22×C4#4112yes—
258C22×C4C2411yescharacteristic
259SD16—44no—
260SD16—44no—
261M16—44no—
262C2×C8—21no—
263C2×C8—21no—
264C2×C8—21no—
265C2×C8—21no—
266C2×C8—21no—
267C2×C8—21no—
268C2×C8—21no—
269C2×C8—21no—
270C2×C8—21no—
271C2×C8—21no—
272C2×C8—44no—
273C2×C8C2×D412yes—
274C2×C8C2×D412yes—
275C4⋊C4—21no—
276C4⋊C4—21no—
277C4⋊C4—21no—
278C4⋊C4—21no—
279C4⋊C4—21no—
280C4⋊C4—21no—
281C4⋊C4—21no—
282C4⋊C4—21no—
283C4⋊C4—42no—
284C4⋊C4—42no—
285C4⋊C4—42no—
286C4⋊C4C2×D412yes—
287C4⋊C4C2×D412yes—
288C22⋊C4—24no—
289C22⋊C4—24no—
290C22⋊C4—24no—
291C22⋊C4—24no—
292C22⋊C4—24no—
293C22⋊C4—42no—
294C22⋊C4—42no—
295C22⋊C4—42no—
296C22⋊C4—81no—
297C22⋊C4—81no—
298C22⋊C4—81no—
299C42—24no—
300C23—21no—
301C23—21no—
302C23—21no—
303C23—21no—
304C23—21no—
305C23—21no—
306C23—21no—
307C23—21no—
308C23—21no—
309C23—21no—
310C23—21no—
311C23—21no—
312C23—22no—
313C23—22no—
314C23—21no—
315C23—21no—
316C23—21no—
317C23—21no—
318C23—21no—
319C23—21no—
320C23—21no—
321C23—48no—
322C23—41no—
323C23—41no—
324C23—41no—
325C23—41no—
326C23—41no—
327C23—41no—
328C23—41no—
329C23—41no—
330C23—41no—
331C23—41no—
332C23—41no—
333C23—41no—
334C23—84no—
335C23—84no—
336C23—84no—
337C23—84no—
338C23—84no—
339C23—84no—
340C23#12012yes—
341C23#12112yes—
342C23C22×D411yescharacteristic
343Q8—44no—
344Q8—41no—
345Q8—41no—
346Q8—41no—
347Q8—41no—
348D4—22no—
349D4—22no—
350D4—22no—
351D4—22no—
352D4—44no—
353D4—44no—
354D4—48no—
355D4—44no—
356D4—41no—
357D4—41no—
358D4—41no—
359D4—41no—
360D4—84no—
361D4—84no—
362D4—84no—
363D4—84no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—21no—
367C2×C4—21no—
368C2×C4—22no—
369C2×C4—22no—
370C2×C4—22no—
371C2×C4—22no—
372C2×C4—22no—
373C2×C4—22no—
374C2×C4—22no—
375C2×C4—22no—
376C2×C4—22no—
377C2×C4—22no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—41no—
382C2×C4—41no—
383C2×C4—41no—
384C2×C4—41no—
385C2×C4—41no—
386C2×C4—41no—
387C2×C4—41no—
388C2×C4—41no—
389C2×C4—41no—
390C2×C4—41no—
391C2×C4—41no—
392C2×C4—41no—
393C2×C4—41no—
394C2×C4—41no—
395C2×C4—41no—
396C2×C4—41no—
397C2×C4—41no—
398C2×C4—41no—
399C2×C4—41no—
400C2×C4—41no—
401C2×C4#12012yes—
402C2×C4#12112yes—
403C2×C4C22×D411yescharacteristic
404C2×C4C22×D411yescharacteristic
405C2×C4C22×D411yescharacteristic
406C2×C4C2×#4111yescharacteristic
407C2×C4H(5,𝔽2)11yescharacteristic
408C2×C4H(5,𝔽2)11yescharacteristic
409C8—22no—
410C8—22no—
411C8—41no—
412C8—41no—
413C22—22no—
414C22—22no—
415C22—22no—
416C22—22no—
417C22—22no—
418C22—22no—
419C22—41no—
420C22—41no—
421C22—41no—
422C22—41no—
423C22—41no—
424C22—41no—
425C22—41no—
426C22—41no—
427C22—41no—
428C22—42no—
429C22—42no—
430C22—41no—
431C22—41no—
432C22—41no—
433C22—42no—
434C22—42no—
435C22—41no—
436C22—42no—
437C22—42no—
438C22—41no—
439C22—41no—
440C22—816no—
441C22—81no—
442C22—81no—
443C22—81no—
444C22—81no—
445C22—81no—
446C22—81no—
447C22—81no—
448C22—81no—
449C22C2×#12011yescharacteristic
450C22C2×#12111yescharacteristic
451C22#53411yescharacteristic
452C22#57611yescharacteristic
453C22#57711yescharacteristic
454C22#57711yescharacteristic
455C22#57711yescharacteristic
456C4—22no—
457C4—22no—
458C4—41no—
459C4—42no—
460C4—42no—
461C4—81no—
462C4—81no—
463C4—81no—
464C4—81no—
465C4D4211yescharacteristic
466C4D4211yescharacteristic
467C4#54611yescharacteristic
468C4#54611yescharacteristic
469C2—42no—
470C2—42no—
471C2—81no—
472C2—82no—
473C2—82no—
474C2#241011yescharacteristic
475C2#300911yescharacteristic
476C2#301211yescharacteristic
477C2#305711yescharacteristic
478C2#306211yescharacteristic
479C2#319811yescharacteristic
480C2#320011yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC2×#535C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×#41 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#468C2semidirect
3. C2×#465C2semidirect
4. C2×#450C2semidirect
5. C2×#408C2semidirect
6. #2616C2semidirect
7. #2034C2semidirect
8. #2033C2semidirect
9. #2000C2semidirect
10. #2000C2semidirect
11. #1895C2semidirect
12. #1893C2semidirect
13. #1873C2semidirect
14. #1854C2semidirect
15. #1786C2semidirect
20. C22×M16C22semidirect
21. C23×C8C22semidirect
27. C2×C22⋊Q8C22semidirect
28. C2×C22⋊Q8C22semidirect
39. C2×#107C22semidirect
40. C2×#106C22semidirect
41. C2×#103C22semidirect
42. C2×#103C22semidirect
52. C2×C22⋊C8C22semidirect
53. C2×C22⋊C8C22semidirect
59. #380C22semidirect
60. #380C22semidirect
61. #340C22semidirect
62. #340C22semidirect
63. #340C22semidirect
64. #340C22semidirect
162. C22⋊C8D4semidirect

Maximal subgroups

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

subgroupindex
C2×#5352
C2×#4682
C2×#4652
C2×#4502
C2×#4082
#26162
#20342
#20332
#20002
#18952
#18932
#18732
#18542
#17862