Finite Group Databasebeta

← #22506

Subgroups — #22506

Order 256.

574 conjugacy classes of subgroups.

The subgroup lattice has 524 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3475C211yesmaximal, characteristic
2D4×M16C211yesmaximal, characteristic, complemented
3#2290C211yesmaximal, characteristic, complemented
4#1995C211yesmaximal, characteristic, complemented
5#1986C211yesmaximal, characteristic, complemented
6#1897C211yesmaximal, characteristic, complemented
7#1896C211yesmaximal, characteristic, complemented
8#1871C211yesmaximal, characteristic, complemented
9#1871C211yesmaximal, characteristic, complemented
10#1826C211yesmaximal, characteristic, complemented
11#1826C211yesmaximal, characteristic, complemented
12#1806C211yesmaximal, characteristic, complemented
13#1759C211yesmaximal, characteristic, complemented
14#1733C211yesmaximal, characteristic, complemented
15C2×#339C211yesmaximal, characteristic, complemented
16C22×M16C2211yescharacteristic
17C22×M16C2211yescharacteristic
18#556C411yescharacteristic, complemented
19#555C411yescharacteristic, complemented
20#554C411yescharacteristic, complemented
21#553C411yescharacteristic, complemented
22#553C411yescharacteristic, complemented
23#552—21no—
24#552C411yescharacteristic, complemented
25#552C411yescharacteristic, complemented
26#551—21no—
27#551—21no—
28#547C411yescharacteristic, complemented
29#546—21no—
30#533C2211yescharacteristic
31#529—21no—
32#529—21no—
33#529—21no—
34#525C2211yescharacteristic
35#525C2211yescharacteristic
36C4×#41—21no—
37C4×#41C2211yescharacteristic
38C2×C4×D4C2211yescharacteristic
39C2×#117C2211yescharacteristic
40C2×#117C2211yescharacteristic
41C2×#117C2211yescharacteristic
42C22×C42C2211yescharacteristic
43#436—21no—
44#435—21no—
45#435—21no—
46C8×D4—21no—
47#423C2211yescharacteristic
48C2×#104C2211yescharacteristic
49C2×#104C2211yescharacteristic
50C2×#104C2211yescharacteristic
51C2×#104C2211yescharacteristic
52#407C2211yescharacteristic
53#407C2211yescharacteristic
54C4×M16C2211yescharacteristic
55#388—21no—
56#387—21no—
57#386—21no—
58#385—21no—
59C4×C4⋊C4C2211yescharacteristic
60C4×C22⋊C4C2211yescharacteristic
61C4×C22⋊C4C2211yescharacteristic
62#376C2211yescharacteristic
63#339C2211yescharacteristic
64#339C2211yescharacteristic
65#339C2211yescharacteristic
66#339C2211yescharacteristic
67#339C2211yescharacteristic
68#339C2211yescharacteristic
69#339C2211yescharacteristic
70#339C2211yescharacteristic
71#339C2211yescharacteristic
72#339C2211yescharacteristic
73#339C2211yescharacteristic
74#339C2211yescharacteristic
75C2×#41—21no—
76C2×#41C2311yescharacteristic
77C22×D4D411yescharacteristic
78C23×C4—21no—
79C23×C4C2311yescharacteristic
80C23×C4C2311yescharacteristic
81C2×M16—21no—
82C2×M16—21no—
83C2×M16—22no—
84C2×M16—22no—
85C2×M16—41no—
86C2×M16—41no—
87C2×M16C2311yescharacteristic
88C2×M16C2311yescharacteristic
89C2×M16C2311yescharacteristic
90C2×M16C2311yescharacteristic
91C22×C8—21no—
92C22×C8—21no—
93C4⋊Q8C2×C411yescharacteristic, complemented
94C4⋊Q8C2×C411yescharacteristic, complemented
95#126—21no—
96#126—21no—
97#126—21no—
98#126—21no—
99#126—21no—
100#126—21no—
101#125—21no—
102#125—21no—
103#125—21no—
104#125C2×C411yescharacteristic, complemented
105#125C2×C411yescharacteristic, complemented
106#124—21no—
107#124—21no—
108#124—21no—
109#123—21no—
110#123—21no—
111#123—21no—
112#123—21no—
113#123—21no—
114#123—21no—
115#123—21no—
116#123—21no—
117C22⋊Q8—21no—
118C22⋊Q8—21no—
119C22⋊Q8—21no—
120C22⋊Q8—21no—
121C22⋊Q8—21no—
122C22⋊Q8C2×C411yescharacteristic
123C22⋊Q8C2×C411yescharacteristic
124C22⋊Q8C2×C411yescharacteristic, complemented
125C22⋊Q8C2×C411yescharacteristic, complemented
126#121—21no—
127#121—21no—
128#121—21no—
129#121C2×C411yescharacteristic
130#121C2×C411yescharacteristic
131#121C2×C411yescharacteristic, complemented
132#121C2×C411yescharacteristic, complemented
133#120—21no—
134#120—21no—
135C4×Q8—21no—
136C4×Q8—21no—
137C4×Q8—21no—
138C4×Q8C2×C411yescharacteristic, complemented
139C4×Q8C2×C411yescharacteristic, complemented
140C4×D4—21no—
141C4×D4—21no—
142C4×D4—21no—
143C4×D4—21no—
144C4×D4—21no—
145C4×D4—21no—
146C4×D4—21no—
147C4×D4—21no—
148C4×D4—21no—
149C4×D4C2×C411yescharacteristic
150C4×D4C2×C411yescharacteristic
151C4×D4C2×C411yescharacteristic
152C4×D4C2×C411yescharacteristic
153C4×D4C2×C411yescharacteristic, complemented
154C4×D4C2×C411yescharacteristic, complemented
155C4×D4D411yescharacteristic
156C4×D4D411yescharacteristic
157C4×D4D411yescharacteristic
158C4×D4D411yescharacteristic
159#117—21no—
160#117—21no—
161#117—21no—
162#117—21no—
163#117—21no—
164#117—21no—
165#117—21no—
166#117—21no—
167#117—22no—
168#117—41no—
169#117C2×C411yescharacteristic
170#117C2×C411yescharacteristic
171#117C2×C411yescharacteristic, complemented
172#117C2×C411yescharacteristic, complemented
173#117C2311yescharacteristic
174#117C2311yescharacteristic
175#117C2311yescharacteristic
176C2×C4⋊C4—21no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4C2×C411yescharacteristic
179C2×C4⋊C4C2×C411yescharacteristic
180C2×C4⋊C4C2×C411yescharacteristic, complemented
181C2×C4⋊C4C2×C411yescharacteristic, complemented
182C2×C4⋊C4D411yescharacteristic
183C2×C22⋊C4—21no—
184C2×C22⋊C4—21no—
185C2×C22⋊C4—21no—
186C2×C22⋊C4—21no—
187C2×C22⋊C4—21no—
188C2×C22⋊C4D411yescharacteristic
189C2×C22⋊C4D411yescharacteristic
190C2×C42—21no—
191C2×C42—21no—
192C2×C42—21no—
193C2×C42—22no—
194C2×C42—21no—
195C2×C42—41no—
196C2×C42C2311yescharacteristic
197C2×C42C2311yescharacteristic
198C2×C42C2311yescharacteristic
199C2×C42C2311yescharacteristic
200C2×C42C2311yescharacteristic
201C4⋊C8—21no—
202C4⋊C8—21no—
203#104—22no—
204#104—22no—
205#104—41no—
206#104—41no—
207C22⋊C8—21no—
208C22⋊C8—21no—
209C22⋊C8—21no—
210C22⋊C8—21no—
211#97—21no—
212C4×C8—21no—
213#95—21no—
214#95—21no—
215#95—21no—
216#95—21no—
217#95—21no—
218C24C2×D411yescharacteristic
219C24C2×D411yescharacteristic
220#41—21no—
221#41—21no—
222#41—21no—
223#41—21no—
224#41—44no—
225C2×Q8—21no—
226C2×Q8—21no—
227C2×Q8C22×C411yescharacteristic
228C2×D4—21no—
229C2×D4—21no—
230C2×D4—21no—
231C2×D4—21no—
232C2×D4—21no—
233C2×D4—42no—
234C2×D4C22⋊C411yescharacteristic
235C2×D4C22⋊C411yescharacteristic
236C2×D4C22⋊C411yescharacteristic
237C2×D4C22⋊C411yescharacteristic
238C2×D4C22×C411yescharacteristic
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×C4—21no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—21no—
256C22×C4—21no—
257C22×C4—21no—
258C22×C4—42no—
259C22×C4—42no—
260C22×C4—42no—
261C22×C4C22×C411yescharacteristic
262C22×C4C22×C411yescharacteristic
263C22×C4C22×C411yescharacteristic
264C22×C4C22×C411yescharacteristic
265C22×C4C2×D411yescharacteristic
266C22×C4C2×D411yescharacteristic
267C22×C4C2×D411yescharacteristic
268C22×C4C2×D411yescharacteristic
269C22×C4C2×D411yescharacteristic
270C22×C4C2×D411yescharacteristic
271C22×C4C2×D411yescharacteristic
272C22×C4C2×D411yescharacteristic
273C22×C4C2411yescharacteristic
274M16—21no—
275M16—21no—
276M16—21no—
277M16—21no—
278M16—21no—
279M16—21no—
280M16—21no—
281M16—42no—
282M16—42no—
283M16#4112yescomplemented
284C2×C8—21no—
285C2×C8—21no—
286C2×C8—21no—
287C2×C8—21no—
288C2×C8—42no—
289C2×C8—42no—
290C4⋊C4—21no—
291C4⋊C4—21no—
292C4⋊C4—21no—
293C4⋊C4—21no—
294C4⋊C4—21no—
295C4⋊C4—21no—
296C4⋊C4—21no—
297C4⋊C4—21no—
298C4⋊C4—21no—
299C4⋊C4—21no—
300C4⋊C4—21no—
301C4⋊C4—21no—
302C4⋊C4—21no—
303C4⋊C4—21no—
304C4⋊C4—21no—
305C4⋊C4—21no—
306C4⋊C4—42no—
307C4⋊C4—42no—
308C4⋊C4C22⋊C411yescharacteristic
309C4⋊C4C22⋊C411yescharacteristic
310C4⋊C4C22⋊C411yescharacteristic
311C4⋊C4C22⋊C411yescharacteristic
312C4⋊C4C22×C411yescharacteristic
313C4⋊C4C22×C411yescharacteristic
314C4⋊C4C22×C411yescharacteristic
315C4⋊C4C22×C411yescharacteristic
316C4⋊C4C22×C411yescharacteristic
317C4⋊C4C22×C411yescharacteristic
318C22⋊C4—21no—
319C22⋊C4—21no—
320C22⋊C4—21no—
321C22⋊C4—21no—
322C22⋊C4—21no—
323C22⋊C4—21no—
324C22⋊C4—21no—
325C22⋊C4—21no—
326C22⋊C4—21no—
327C22⋊C4—21no—
328C22⋊C4—21no—
329C22⋊C4—21no—
330C22⋊C4—21no—
331C22⋊C4—21no—
332C22⋊C4—21no—
333C22⋊C4—21no—
334C22⋊C4—21no—
335C22⋊C4—42no—
336C22⋊C4—42no—
337C22⋊C4—42no—
338C22⋊C4C22⋊C411yescharacteristic
339C22⋊C4C22⋊C411yescharacteristic
340C22⋊C4C22⋊C411yescharacteristic
341C22⋊C4C22⋊C411yescharacteristic
342C22⋊C4C22⋊C411yescharacteristic
343C22⋊C4C22⋊C411yescharacteristic
344C22⋊C4C22⋊C411yescharacteristic
345C22⋊C4C22⋊C411yescharacteristic
346C42—21no—
347C42—21no—
348C42—21no—
349C42—21no—
350C42—21no—
351C42—21no—
352C42—21no—
353C42—21no—
354C42—21no—
355C42—21no—
356C42—21no—
357C42—21no—
358C42—21no—
359C42—21no—
360C42—21no—
361C42—21no—
362C42—42no—
363C42C22×C411yescharacteristic
364C42C22×C411yescharacteristic
365C42C2×D411yescharacteristic
366C42C2×D411yescharacteristic
367C42#4112yescomplemented
368C23—21no—
369C23—21no—
370C23—22no—
371C23—21no—
372C23—21no—
373C23—21no—
374C23—41no—
375C23—41no—
376C23—41no—
377C23C2×C22⋊C411yescharacteristic
378C23C2×C22⋊C411yescharacteristic
379C23C2×C22⋊C411yescharacteristic
380C23C2×C22⋊C411yescharacteristic
381C23C22×D411yescharacteristic
382Q8—21no—
383Q8—21no—
384Q8—42no—
385Q8—81no—
386D4—22no—
387D4—22no—
388D4—21no—
389D4—21no—
390D4—41no—
391D4—41no—
392D4—42no—
393D4—41no—
394D4—41no—
395D4—42no—
396D4—81no—
397D4—81no—
398C2×C4—21no—
399C2×C4—21no—
400C2×C4—21no—
401C2×C4—21no—
402C2×C4—21no—
403C2×C4—21no—
404C2×C4—21no—
405C2×C4—21no—
406C2×C4—21no—
407C2×C4—21no—
408C2×C4—21no—
409C2×C4—21no—
410C2×C4—21no—
411C2×C4—21no—
412C2×C4—21no—
413C2×C4—21no—
414C2×C4—21no—
415C2×C4—21no—
416C2×C4—21no—
417C2×C4—21no—
418C2×C4—22no—
419C2×C4—21no—
420C2×C4—21no—
421C2×C4—21no—
422C2×C4—21no—
423C2×C4—21no—
424C2×C4—21no—
425C2×C4—21no—
426C2×C4—22no—
427C2×C4—41no—
428C2×C4—41no—
429C2×C4—42no—
430C2×C4—41no—
431C2×C4—41no—
432C2×C4—41no—
433C2×C4—42no—
434C2×C4—48no—
435C2×C4—41no—
436C2×C4—42no—
437C2×C4—41no—
438C2×C4—41no—
439C2×C4—41no—
440C2×C4—41no—
441C2×C4—41no—
442C2×C4—41no—
443C2×C4—41no—
444C2×C4—41no—
445C2×C4—81no—
446C2×C4—81no—
447C2×C4—81no—
448C2×C4C2×C22⋊C411yescharacteristic
449C2×C4C2×C22⋊C411yescharacteristic
450C2×C4C2×C22⋊C411yescharacteristic
451C2×C4C2×C22⋊C411yescharacteristic
452C2×C4C2×C22⋊C411yescharacteristic
453C2×C4C2×C22⋊C411yescharacteristic
454C2×C4C2×C22⋊C411yescharacteristic
455C2×C4C2×C22⋊C411yescharacteristic
456C2×C4C23×C411yescharacteristic
457C2×C4C22×D411yescharacteristic
458C2×C4C2×#4111yescharacteristic
459C2×C4C2×#4111yescharacteristic
460C2×C4H(5,𝔽2)11yescharacteristic
461C2×C42−1+411yescharacteristic
462C8—22no—
463C8—41no—
464C8—41no—
465C8—41no—
466C22—21no—
467C22—21no—
468C22—21no—
469C22—21no—
470C22—21no—
471C22—21no—
472C22—21no—
473C22—21no—
474C22—21no—
475C22—21no—
476C22—21no—
477C22—21no—
478C22—21no—
479C22—42no—
480C22—42no—
481C22—41no—
482C22—41no—
483C22—41no—
484C22—81no—
485C22—81no—
486C22#42112yes—
487C22C22×C22⋊C411yescharacteristic
488C22#54611yescharacteristic
489C22#54711yescharacteristic
490C4—21no—
491C4—21no—
492C4—21no—
493C4—21no—
494C4—21no—
495C4—21no—
496C4—21no—
497C4—21no—
498C4—42no—
499C4—42no—
500C4—41no—
501C4—41no—
502C4—41no—
503C4—41no—
504C4—41no—
505C4—41no—
506C4—41no—
507C4—81no—
508C4—81no—
509C4C4×#4111yescharacteristic
510C4#52011yescharacteristic
511C4#54611yescharacteristic
512C4#54711yescharacteristic
513C2—21no—
514C2—21no—
515C2—22no—
516C2—41no—
517C2—41no—
518C2—41no—
519C2—81no—
520C2#233911yescharacteristic
521C2C2×#42111yescharacteristic
522C2#290111yescharacteristic
subgroupG/H
centerC2×C4C22×D4
commutatorC2×C4C23×C4
Frattini, cosocleC22×C4C24
socleC22C22×C22⋊C4
prime radical#3475C2
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C2×C4×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. D4×M16C2semidirect
3. #2290C2semidirect
4. #1995C2semidirect
5. #1986C2semidirect
6. #1897C2semidirect
7. #1896C2semidirect
8. #1871C2semidirect
9. #1871C2semidirect
10. #1826C2semidirect
11. #1826C2semidirect
12. #1806C2semidirect
13. #1759C2semidirect
14. #1733C2semidirect
15. C2×#339C2semidirect
18. #556C4semidirect
19. #555C4semidirect
20. #554C4semidirect
21. #553C4semidirect
22. #553C4semidirect
24. #552C4semidirect
25. #552C4semidirect
28. #547C4semidirect
93. C4⋊Q8C2×C4semidirect
94. C4⋊Q8C2×C4semidirect
104. #125C2×C4semidirect
105. #125C2×C4semidirect
124. C22⋊Q8C2×C4semidirect
125. C22⋊Q8C2×C4semidirect
131. #121C2×C4semidirect
132. #121C2×C4semidirect
138. C4×Q8C2×C4semidirect
139. C4×Q8C2×C4semidirect
153. C4×D4C2×C4semidirect
154. C4×D4C2×C4semidirect
171. #117C2×C4semidirect
172. #117C2×C4semidirect
180. C2×C4⋊C4C2×C4semidirect
181. C2×C4⋊C4C2×C4semidirect
283. M16#41semidirect
367. C42#41semidirect

Maximal subgroups

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

subgroupindex
#34752
D4×M162
#22902
#19952
#19862
#18972
#18962
#18712
#18262
#18062
#17592
#17332
C2×#3392