Finite Group Databasebeta

← #18658

Subgroups — #18658

Order 256.

486 conjugacy classes of subgroups.

The subgroup lattice has 391 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3514C211yesmaximal, characteristic, complemented
2#3212C211yesmaximal, characteristic, complemented
3#3169C211yesmaximal, characteristic, complemented
4#3115C211yesmaximal, characteristic, complemented
5#1650C211yesmaximal, characteristic, complemented
6#1649C211yesmaximal, characteristic, complemented
7#1640C211yesmaximal, characteristic, complemented
8#1639C211yesmaximal, characteristic, complemented
9#1628C211yesmaximal, characteristic, complemented
10#1627C211yesmaximal, characteristic, complemented
11#1620C211yesmaximal, characteristic, complemented
12#1619C211yesmaximal, characteristic, complemented
13#1553C211yesmaximal, characteristic, complemented
14#1494C211yesmaximal, characteristic, complemented
15#1476C211yesmaximal, characteristic, complemented
16#559C2211yescharacteristic
17#556C2211yescharacteristic
18#555—21no—
19#551—22no—
20D4×Q8C2211yescharacteristic
21#548—21no—
22#546—21no—
23#546—21no—
24D42C2211yescharacteristic
25#542—21no—
26#538—22no—
27#532C2211yescharacteristic
28#532C2211yescharacteristic, complemented
29#529—21no—
30#529C2211yescharacteristic
31C2×#124—21no—
32#518—21no—
33C4×#41C2211yescharacteristic
34#500C2211yescharacteristic, complemented
35#495—21no—
36#493C2211yescharacteristic
37#488—21no—
38#486C2211yescharacteristic
39#486C2211yescharacteristic, complemented
40#482—21no—
41#475C2211yescharacteristic, complemented
42#474C2211yescharacteristic
43#466—22no—
44#462C2211yescharacteristic, complemented
45#461C2211yescharacteristic
46#460C2211yescharacteristic, complemented
47#460C2211yescharacteristic, complemented
48#459C2211yescharacteristic
49#459C2211yescharacteristic
50#452—21no—
51#449—21no—
52#449—21no—
53#442—21no—
54#440—21no—
55C4×D8—22no—
56#432C2211yescharacteristic, complemented
57#431C2211yescharacteristic, complemented
58C2×C4×C8C2211yescharacteristic, complemented
59#332C2211yescharacteristic, complemented
60#332C2211yescharacteristic, complemented
61#331C2211yescharacteristic
62#331C2211yescharacteristic
63Q8⋊C8C2211yescharacteristic, complemented
64Q8⋊C8C2211yescharacteristic, complemented
65Q8⋊C8C2211yescharacteristic, complemented
66Q8⋊C8C2211yescharacteristic, complemented
67D4⋊C8C2211yescharacteristic
68D4⋊C8C2211yescharacteristic
69D4⋊C8C2211yescharacteristic
70D4⋊C8C2211yescharacteristic
71C2×#41—21no—
72C2×#41—21no—
73C2×#41D411yescharacteristic
74C22×Q8—21no—
75C22×D4—21no—
76C22×D4—21no—
77C2×Q16—21no—
78C2×SD16—21no—
79C2×SD16—21no—
80C2×SD16—21no—
81C2×D8—22no—
82C2×D8—21no—
83C2×D8—21no—
84C22×C8—21no—
85C4⋊Q8—21no—
86C4⋊Q8C2311yescharacteristic
87C4⋊Q8C2311yescharacteristic
88#127C2311yescharacteristic
89#127C2311yescharacteristic
90#126—22no—
91#126—22no—
92#126—21no—
93#125D411yescharacteristic
94#124—22no—
95#124—22no—
96#124—22no—
97#124—21no—
98#124—21no—
99#124—21no—
100#124—21no—
101#124—21no—
102#124D411yescharacteristic
103#123—22no—
104#123—22no—
105#123—22no—
106#123—21no—
107C22⋊Q8—22no—
108C22⋊Q8—21no—
109C22⋊Q8—21no—
110C22⋊Q8—22no—
111C22⋊Q8D411yescharacteristic
112#121—22no—
113#121—22no—
114#121—21no—
115#121—21no—
116#121—21no—
117#121—21no—
118#121—21no—
119#121D411yescharacteristic
120#121D412yes—
121#120—22no—
122#120—22no—
123C4×Q8—21no—
124C4×Q8D411yescharacteristic
125C4×Q8C2311yescharacteristic
126C4×Q8C2311yescharacteristic
127C4×D4—22no—
128C4×D4—22no—
129C4×D4—21no—
130C4×D4—21no—
131C4×D4—21no—
132C4×D4—21no—
133C4×D4—21no—
134C4×D4—21no—
135C4×D4—21no—
136C4×D4—21no—
137C4×D4—21no—
138C4×D4D411yescharacteristic
139C4×D4D412yes—
140C4×D4C2311yescharacteristic
141C4×D4C2311yescharacteristic
142#117—22no—
143#117—21no—
144#117—21no—
145#117—21no—
146#117D411yescharacteristic
147C2×C22⋊C4—22no—
148C2×C22⋊C4—22no—
149C2×C22⋊C4—22no—
150C2×C42—21no—
151C2×C42C2311yescharacteristic
152#107—22no—
153#107—21no—
154#106—21no—
155C4⋊C8C2311yescharacteristic
156C4⋊C8C2311yescharacteristic
157C4⋊C8C2311yescharacteristic
158C4⋊C8C2311yescharacteristic
159#103—21no—
160#103—21no—
161#103—21no—
162#102—22no—
163#102—21no—
164#102—22no—
165#102—21no—
166#102—21no—
167#102—21no—
168#102—21no—
169C22⋊C8—21no—
170C22⋊C8—21no—
171#97—21no—
172C4×C8—21no—
173C4×C8—21no—
174C4×C8C2311yescharacteristic
175C4×C8C2311yescharacteristic
176C24—22no—
177#41—22no—
178#41—41no—
179#41—81no—
180#41—81no—
181#41—81no—
182#41—81no—
183C2×Q8—22no—
184C2×Q8—21no—
185C2×Q8—21no—
186C2×Q8—81no—
187C2×Q8—81no—
188C2×Q8C2×D411yescharacteristic
189C2×Q8C2×D411yescharacteristic
190C2×D4—22no—
191C2×D4—21no—
192C2×D4—22no—
193C2×D4—21no—
194C2×D4—22no—
195C2×D4—21no—
196C2×D4—21no—
197C2×D4—21no—
198C2×D4—21no—
199C2×D4—21no—
200C2×D4—42no—
201C2×D4—42no—
202C2×D4—41no—
203C2×D4—81no—
204C2×D4—81no—
205C2×D4C2×D411yescharacteristic
206C2×D4C2×D411yescharacteristic
207C2×D4C2×D412yes—
208C22×C4—22no—
209C22×C4—22no—
210C22×C4—21no—
211C22×C4—21no—
212C22×C4—21no—
213C22×C4—21no—
214C22×C4—22no—
215C22×C4—41no—
216C22×C4C2×D411yescharacteristic
217C22×C4C2×D411yescharacteristic
218C22×C4C2×D411yescharacteristic
219Q16—81no—
220SD16—42no—
221SD16—81no—
222SD16—81no—
223D8—44no—
224D8—42no—
225D8—81no—
226C2×C8—21no—
227C2×C8—21no—
228C2×C8—21no—
229C2×C8—21no—
230C2×C8—21no—
231C2×C8—21no—
232C2×C8—42no—
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—21no—
243C4⋊C4—21no—
244C4⋊C4C2×D411yescharacteristic
245C4⋊C4C2×D411yescharacteristic
246C4⋊C4C2×D411yescharacteristic
247C4⋊C4C2×D411yescharacteristic
248C4⋊C4C2×D412yes—
249C22⋊C4—22no—
250C22⋊C4—22no—
251C22⋊C4—22no—
252C22⋊C4—22no—
253C22⋊C4—22no—
254C22⋊C4—22no—
255C22⋊C4—22no—
256C22⋊C4—21no—
257C22⋊C4—21no—
258C22⋊C4—21no—
259C22⋊C4—21no—
260C22⋊C4—21no—
261C22⋊C4—21no—
262C22⋊C4—22no—
263C22⋊C4—22no—
264C22⋊C4—21no—
265C22⋊C4—21no—
266C22⋊C4—41no—
267C22⋊C4—41no—
268C42—22no—
269C42—21no—
270C42—21no—
271C42—21no—
272C42—21no—
273C42C2×D411yescharacteristic
274C42C2×D411yescharacteristic
275C42C2×D411yescharacteristic
276C42C2411yescharacteristic
277C23—21no—
278C23—21no—
279C23—22no—
280C23—44no—
281C23—41no—
282C23—82no—
283C23—82no—
284C23#12011yescharacteristic
285Q8—21no—
286Q8—21no—
287Q8—21no—
288Q8—21no—
289Q8—41no—
290Q8—41no—
291Q8—41no—
292Q8—41no—
293Q8—82no—
294D4—21no—
295D4—21no—
296D4—21no—
297D4—42no—
298D4—42no—
299D4—42no—
300D4—44no—
301D4—41no—
302D4—41no—
303D4—41no—
304D4—41no—
305D4—41no—
306D4—41no—
307D4—41no—
308D4—41no—
309D4—81no—
310D4—82no—
311D4—82no—
312D4—82no—
313D4#13512yes—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—22no—
325C2×C4—41no—
326C2×C4—42no—
327C2×C4—41no—
328C2×C4—41no—
329C2×C4—41no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—81no—
333C2×C4—82no—
334C2×C4—82no—
335C2×C4—82no—
336C2×C4—82no—
337C2×C4#12011yescharacteristic
338C2×C4#12011yescharacteristic
339C2×C4#12011yescharacteristic
340C2×C4C22×D411yescharacteristic
341C2×C4C22×D411yescharacteristic
342C2×C4C22×D411yescharacteristic
343C8—22no—
344C8—41no—
345C8—41no—
346C8—41no—
347C8—41no—
348C8—41no—
349C22—21no—
350C22—22no—
351C22—21no—
352C22—21no—
353C22—41no—
354C22—41no—
355C22—41no—
356C22—42no—
357C22—42no—
358C22—41no—
359C22—42no—
360C22—41no—
361C22—81no—
362C22—81no—
363C22—81no—
364C22—164no—
365C22C2×#12011yescharacteristic
366C4—21no—
367C4—21no—
368C4—21no—
369C4—21no—
370C4—41no—
371C4—41no—
372C4—41no—
373C4—42no—
374C4—41no—
375C4—41no—
376C4—41no—
377C4—81no—
378C4C2×#13511yescharacteristic
379C4#57511yescharacteristic
380C4#57611yescharacteristic
381C4#57611yescharacteristic
382C2—41no—
383C2—42no—
384C2—82no—
385C2—81no—
386C2—161no—
387C2#300211yescharacteristic
388C2#300811yescharacteristic
389C2#301811yescharacteristic
subgroupG/H
center, socleC22C2×#120
commutator, Frattini, cosocleC42C24
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ G
LECSG ▷ C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #3514C2semidirect
2. #3212C2semidirect
3. #3169C2semidirect
4. #3115C2semidirect
5. #1650C2semidirect
6. #1649C2semidirect
7. #1640C2semidirect
8. #1639C2semidirect
9. #1628C2semidirect
10. #1627C2semidirect
11. #1620C2semidirect
12. #1619C2semidirect
13. #1553C2semidirect
14. #1494C2semidirect
15. #1476C2semidirect
28. #532C22semidirect
34. #500C22semidirect
39. #486C22semidirect
41. #475C22semidirect
44. #462C22semidirect
46. #460C22semidirect
47. #460C22semidirect
56. #432C22semidirect
57. #431C22semidirect
58. C2×C4×C8C22semidirect
59. #332C22semidirect
60. #332C22semidirect
63. Q8⋊C8C22semidirect
64. Q8⋊C8C22semidirect
65. Q8⋊C8C22semidirect
66. Q8⋊C8C22semidirect

Maximal subgroups

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

subgroupindex
#35142
#32122
#31692
#31152
#16502
#16492
#16402
#16392
#16282
#16272
#16202
#16192
#15532
#14942
#14762