Finite Group Databasebeta

← #22400

Subgroups — #22400

Order 256.

791 conjugacy classes of subgroups.

The subgroup lattice has 495 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#525C211yesmaximal, characteristic
2C2×#436C211yesmaximal, characteristic, complemented
3C2×#435C211yesmaximal, characteristic, complemented
4C2×#408C211yesmaximal, characteristic, complemented
5C2×#407C211yesmaximal, characteristic, complemented
6#1955C211yesmaximal, characteristic, complemented
7#1954C211yesmaximal, characteristic, complemented
8#1870C211yesmaximal, characteristic, complemented
9#1870C211yesmaximal, characteristic, complemented
10#1870C211yesmaximal, characteristic, complemented
11#1869C211yesmaximal, characteristic, complemented
12#1798C211yesmaximal, characteristic, complemented
13#1780C211yesmaximal, characteristic, complemented
14#1779C211yesmaximal, characteristic, complemented
15#1750C211yesmaximal, characteristic, complemented
16C22×#41C2211yescharacteristic
17C24×C4C2211yescharacteristic
18C22×M16C2211yescharacteristic
19C22×M16C2211yescharacteristic
20C22×M16C2211yescharacteristic, complemented
21C23×C8C2211yescharacteristic, complemented
22#525—24no—
23#525—24no—
24C2×#123C411yescharacteristic
25C2×#123C411yescharacteristic
26C2×C22⋊Q8C411yescharacteristic
27C2×C22⋊Q8C411yescharacteristic
28C2×#121C411yescharacteristic
29C2×#121C411yescharacteristic
30C2×#120C411yescharacteristic
31C2×#120C411yescharacteristic
32C2×C4×D4C2211yescharacteristic
33C2×C4×D4C2211yescharacteristic
34C2×C4×D4C2211yescharacteristic
35C2×C4×D4C2211yescharacteristic
36C2×#117C2211yescharacteristic
37#436—24no—
38#435—24no—
39C2×C4⋊C8C2211yescharacteristic, complemented
40C2×C4⋊C8C2211yescharacteristic, complemented
41#408—24no—
42#407—24no—
43C2×C22⋊C8C2211yescharacteristic
44C2×C22⋊C8C2211yescharacteristic
45C2×C22⋊C8C2211yescharacteristic
46C2×C22⋊C8C2211yescharacteristic
47C2×C22⋊C8C2211yescharacteristic
48C2×C22⋊C8C2211yescharacteristic
49C2×C22⋊C8C2211yescharacteristic
50C2×C22⋊C8C2211yescharacteristic
51C2×C22⋊C8C2211yescharacteristic
52C2×C22⋊C8C2211yescharacteristic
53C2×C22⋊C8C2211yescharacteristic, complemented
54C2×C22⋊C8C2211yescharacteristic, complemented
55C2×C22⋊C8C2211yescharacteristic, complemented
56C2×C22⋊C8C2211yescharacteristic, complemented
57C2×C22⋊C8C2211yescharacteristic, complemented
58C2×C22⋊C8C2211yescharacteristic, complemented
59C2×#97C2211yescharacteristic, complemented
60C2×C4×C8C2211yescharacteristic, complemented
61#336C2211yescharacteristic, complemented
62#336C2211yescharacteristic, complemented
63#336C2211yescharacteristic, complemented
64#336C2211yescharacteristic, complemented
65C25C2×C411yescharacteristic
66C2×#41—22no—
67C2×#41—22no—
68C2×#41—22no—
69C2×#41—22no—
70C2×#41—41no—
71C2×#41—41no—
72C22×Q8C2×C411yescharacteristic
73C22×D4C2×C411yescharacteristic
74C22×D4C2×C411yescharacteristic
75C22×D4C2×C411yescharacteristic
76C22×D4C2×C411yescharacteristic
77C22×D4C2×C411yescharacteristic
78C22×D4C2×C411yescharacteristic
79C22×D4C2×C411yescharacteristic
80C23×C4—22no—
81C23×C4—22no—
82C23×C4—22no—
83C23×C4—22no—
84C23×C4—41no—
85C23×C4—41no—
86C23×C4C2×C411yescharacteristic
87C23×C4C2×C411yescharacteristic
88C23×C4C2×C411yescharacteristic
89C23×C4C2311yescharacteristic
90C23×C4C2311yescharacteristic
91C23×C4C2311yescharacteristic
92C23×C4C2311yescharacteristic
93C23×C4C2311yescharacteristic
94C2×M16—22no—
95C2×M16—22no—
96C2×M16—22no—
97C2×M16—22no—
98C2×M16—22no—
99C2×M16—22no—
100C2×M16—41no—
101C2×M16—41no—
102C2×M16—41no—
103C22×C8—22no—
104C22×C8—22no—
105C22×C8—41no—
106C22×C8C2311yescharacteristic
107C22×C8C2311yescharacteristic
108C22×C8C2311yescharacteristic
109C22×C8C2311yescharacteristic
110C22×C8C2311yescharacteristic
111C22×C8C2311yescharacteristic
112C22×C8C2311yescharacteristic
113C22×C8C2311yescharacteristic
114#123—24no—
115#123—24no—
116C22⋊Q8—24no—
117C22⋊Q8—24no—
118#121—24no—
119#121—24no—
120#120—24no—
121#120—24no—
122C4×D4—22no—
123C4×D4—22no—
124C4×D4—22no—
125C4×D4—22no—
126C4×D4—22no—
127C4×D4—22no—
128C4×D4D414yes—
129C4×D4D414yes—
130#117—22no—
131#117—22no—
132C2×C4⋊C4C2×C411yescharacteristic
133C2×C4⋊C4C2×C411yescharacteristic
134C2×C4⋊C4C2×C411yescharacteristic
135C2×C4⋊C4C2×C411yescharacteristic
136C2×C4⋊C4C2×C411yescharacteristic
137C2×C4⋊C4C2×C411yescharacteristic
138C2×C22⋊C4C2×C411yescharacteristic
139C2×C22⋊C4C2×C411yescharacteristic
140C2×C22⋊C4C2×C411yescharacteristic
141C2×C22⋊C4C2×C411yescharacteristic
142C2×C22⋊C4C2×C411yescharacteristic
143C2×C22⋊C4C2×C411yescharacteristic
144C2×C22⋊C4C2×C411yescharacteristic
145C2×C22⋊C4C2×C411yescharacteristic
146C2×C22⋊C4C2×C411yescharacteristic
147C2×C22⋊C4C2×C411yescharacteristic
148C2×C42C2311yescharacteristic
149C2×C42C2311yescharacteristic
150C4⋊C8—22no—
151C4⋊C8—22no—
152C22⋊C8—22no—
153C22⋊C8—22no—
154C22⋊C8—22no—
155C22⋊C8—22no—
156C22⋊C8—22no—
157C22⋊C8—22no—
158C22⋊C8—22no—
159C22⋊C8—22no—
160C22⋊C8—22no—
161C22⋊C8—22no—
162C22⋊C8—22no—
163C22⋊C8—41no—
164C22⋊C8—41no—
165C22⋊C8—41no—
166C22⋊C8—41no—
167C22⋊C8D414yescomplemented
168#97—22no—
169C4×C8—22no—
170C24—22no—
171C24—22no—
172C24—22no—
173C24—41no—
174C24—41no—
175C24—41no—
176C24—41no—
177C24C22×C411yescharacteristic
178C24C22×C411yescharacteristic
179C24C22×C411yescharacteristic
180C24C22×C411yescharacteristic
181C24C22×C411yescharacteristic
182#41—44no—
183#41—44no—
184#41—44no—
185#41—44no—
186C2×Q8—22no—
187C2×Q8—22no—
188C2×Q8—41no—
189C2×D4—22no—
190C2×D4—22no—
191C2×D4—22no—
192C2×D4—22no—
193C2×D4—22no—
194C2×D4—22no—
195C2×D4—22no—
196C2×D4—22no—
197C2×D4—22no—
198C2×D4—22no—
199C2×D4—22no—
200C2×D4—41no—
201C2×D4—41no—
202C2×D4—41no—
203C2×D4—41no—
204C2×D4—41no—
205C2×D4—41no—
206C2×D4—41no—
207C2×D4—41no—
208C2×D4—41no—
209C2×D4C22⋊C414yes—
210C22×C4—21no—
211C22×C4—21no—
212C22×C4—21no—
213C22×C4—21no—
214C22×C4—21no—
215C22×C4—21no—
216C22×C4—21no—
217C22×C4—21no—
218C22×C4—21no—
219C22×C4—21no—
220C22×C4—21no—
221C22×C4—21no—
222C22×C4—22no—
223C22×C4—22no—
224C22×C4—22no—
225C22×C4—22no—
226C22×C4—22no—
227C22×C4—22no—
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—21no—
238C22×C4—21no—
239C22×C4—21no—
240C22×C4—41no—
241C22×C4—41no—
242C22×C4—41no—
243C22×C4—44no—
244C22×C4—44no—
245C22×C4—44no—
246C22×C4—44no—
247C22×C4C22×C411yescharacteristic
248C22×C4C22×C411yescharacteristic
249C22×C4C22×C411yescharacteristic
250C22×C4C22×C411yescharacteristic
251C22×C4C22×C411yescharacteristic
252C22×C4C22×C411yescharacteristic
253C22×C4C22×C411yescharacteristic
254C22×C4C22×C411yescharacteristic
255C22×C4C22×C411yescharacteristic
256C22×C4C2×D412yes—
257C22×C4C2×D412yes—
258C22×C4C2×D412yes—
259C22×C4C2×D412yes—
260C22×C4C2×D412yes—
261C22×C4#4112yes—
262C22×C4C2411yescharacteristic
263M16—44no—
264M16—44no—
265M16—44no—
266C2×C8—21no—
267C2×C8—21no—
268C2×C8—21no—
269C2×C8—21no—
270C2×C8—21no—
271C2×C8—21no—
272C2×C8—21no—
273C2×C8—21no—
274C2×C8—21no—
275C2×C8—21no—
276C2×C8—21no—
277C2×C8—21no—
278C2×C8—21no—
279C2×C8—21no—
280C2×C8—21no—
281C2×C8—21no—
282C2×C8—21no—
283C2×C8—21no—
284C2×C8—21no—
285C2×C8—21no—
286C2×C8—21no—
287C2×C8—21no—
288C2×C8—44no—
289C2×C8C2×D412yes—
290C2×C8C2×D412yes—
291C4⋊C4—22no—
292C4⋊C4—22no—
293C4⋊C4—22no—
294C4⋊C4—22no—
295C4⋊C4—22no—
296C4⋊C4C22⋊C414yes—
297C22⋊C4—22no—
298C22⋊C4—22no—
299C22⋊C4—22no—
300C22⋊C4—22no—
301C22⋊C4—22no—
302C22⋊C4—22no—
303C22⋊C4—22no—
304C22⋊C4—22no—
305C22⋊C4C22⋊C414yes—
306C22⋊C4C22⋊C414yes—
307C42—21no—
308C42—21no—
309C42C2×D412yes—
310C42C2×D412yes—
311C23—21no—
312C23—21no—
313C23—21no—
314C23—21no—
315C23—21no—
316C23—21no—
317C23—21no—
318C23—21no—
319C23—21no—
320C23—21no—
321C23—21no—
322C23—21no—
323C23—21no—
324C23—21no—
325C23—21no—
326C23—21no—
327C23—21no—
328C23—21no—
329C23—21no—
330C23—21no—
331C23—21no—
332C23—21no—
333C23—21no—
334C23—21no—
335C23—21no—
336C23—21no—
337C23—21no—
338C23—21no—
339C23—21no—
340C23—21no—
341C23—21no—
342C23—44no—
343C23—44no—
344C23—44no—
345C23—44no—
346C23—44no—
347C23—44no—
348C23—44no—
349C23C2×C22⋊C412yes—
350C23C2×C22⋊C412yes—
351C23C4×D412yes—
352C23#13112yes—
353C23C23×C411yescharacteristic
354Q8—44no—
355D4—44no—
356D4—44no—
357D4—44no—
358D4—44no—
359D4—44no—
360D4—44no—
361D4—44no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—21no—
367C2×C4—21no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—22no—
373C2×C4—22no—
374C2×C4—22no—
375C2×C4—22no—
376C2×C4—22no—
377C2×C4—22no—
378C2×C4—22no—
379C2×C4—22no—
380C2×C4—22no—
381C2×C4—22no—
382C2×C4—22no—
383C2×C4—22no—
384C2×C4—21no—
385C2×C4—21no—
386C2×C4—21no—
387C2×C4—21no—
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—21no—
391C2×C4—21no—
392C2×C4—21no—
393C2×C4—21no—
394C2×C4—21no—
395C2×C4—21no—
396C2×C4—44no—
397C2×C4—44no—
398C2×C4—44no—
399C2×C4—41no—
400C2×C4—41no—
401C2×C4—41no—
402C2×C4—41no—
403C2×C4—41no—
404C2×C4—41no—
405C2×C4—41no—
406C2×C4—41no—
407C2×C4C2×C22⋊C412yes—
408C2×C4C2×C22⋊C412yes—
409C2×C4C2×C22⋊C412yes—
410C2×C4C2×C22⋊C412yes—
411C2×C4C4×D412yes—
412C2×C4C22×D411yescharacteristic
413C2×C4C22×D411yescharacteristic
414C2×C4C22×D411yescharacteristic
415C2×C4C2×#4111yescharacteristic
416C2×C4H(5,𝔽2)11yescharacteristic
417C2×C4H(5,𝔽2)11yescharacteristic
418C8—22no—
419C8—22no—
420C8—41no—
421C8—41no—
422C8—41no—
423C8—41no—
424C8—41no—
425C8—41no—
426C22—22no—
427C22—22no—
428C22—22no—
429C22—22no—
430C22—22no—
431C22—22no—
432C22—22no—
433C22—22no—
434C22—22no—
435C22—22no—
436C22—22no—
437C22—22no—
438C22—416no—
439C22—41no—
440C22—41no—
441C22—41no—
442C22—41no—
443C22—41no—
444C22—41no—
445C22—41no—
446C22—41no—
447C22—41no—
448C22—41no—
449C22—41no—
450C22—41no—
451C22—41no—
452C22—41no—
453C22—41no—
454C22—41no—
455C22—41no—
456C22—41no—
457C22—41no—
458C22—41no—
459C22—41no—
460C22—41no—
461C22—41no—
462C22C22×C22⋊C411yescharacteristic
463C22C2×C4×D411yescharacteristic
464C22#51811yescharacteristic
465C22C2×#13111yescharacteristic
466C22#56811yescharacteristic
467C22#56811yescharacteristic
468C22#56811yescharacteristic
469C4—22no—
470C4—22no—
471C4—22no—
472C4—22no—
473C4—22no—
474C4—41no—
475C4—41no—
476C4—41no—
477C4—41no—
478C4D4211yescharacteristic
479C4D4211yescharacteristic
480C4#54611yescharacteristic
481C4#54611yescharacteristic
482C2—42no—
483C2—42no—
484C2—42no—
485C2—42no—
486C2—81no—
487C2D4×C22⋊C411yescharacteristic
488C2#287911yescharacteristic
489C2#288011yescharacteristic
490C2#293011yescharacteristic
491C2#293311yescharacteristic
492C2#297611yescharacteristic
493C2#297811yescharacteristic
subgroupG/H
center, Frattini, cosocleC22×C4C24
commutator, socleC23C23×C4
prime radicalC2×#525C2
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C22×C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#436C2semidirect
3. C2×#435C2semidirect
4. C2×#408C2semidirect
5. C2×#407C2semidirect
6. #1955C2semidirect
7. #1954C2semidirect
8. #1870C2semidirect
9. #1870C2semidirect
10. #1870C2semidirect
11. #1869C2semidirect
12. #1798C2semidirect
13. #1780C2semidirect
14. #1779C2semidirect
15. #1750C2semidirect
20. C22×M16C22semidirect
21. C23×C8C22semidirect
39. C2×C4⋊C8C22semidirect
40. C2×C4⋊C8C22semidirect
53. C2×C22⋊C8C22semidirect
54. C2×C22⋊C8C22semidirect
55. C2×C22⋊C8C22semidirect
56. C2×C22⋊C8C22semidirect
57. C2×C22⋊C8C22semidirect
58. C2×C22⋊C8C22semidirect
59. C2×#97C22semidirect
60. C2×C4×C8C22semidirect
61. #336C22semidirect
62. #336C22semidirect
63. #336C22semidirect
64. #336C22semidirect
167. C22⋊C8D4semidirect

Maximal subgroups

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

subgroupindex
C2×#5252
C2×#4362
C2×#4352
C2×#4082
C2×#4072
#19552
#19542
#18702
#18692
#17982
#17802
#17792
#17502