Finite Group Databasebeta

← #18974

Subgroups — #18974

Order 256.

465 conjugacy classes of subgroups.

The subgroup lattice has 390 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3546C211yesmaximal, characteristic
2#3254C211yesmaximal, characteristic, complemented
3#3248C211yesmaximal, characteristic, complemented
4#3228C211yesmaximal, characteristic, complemented
5#1654C211yesmaximal, characteristic, complemented
6#1652C211yesmaximal, characteristic, complemented
7#1650C211yesmaximal, characteristic, complemented
8#1647C211yesmaximal, characteristic, complemented
9#1642C211yesmaximal, characteristic, complemented
10#1641C211yesmaximal, characteristic, complemented
11#1639C211yesmaximal, characteristic, complemented
12#1639C211yesmaximal, characteristic, complemented
13#1547C211yesmaximal, characteristic, complemented
14#1495C211yesmaximal, characteristic, complemented
15#1480C211yesmaximal, characteristic, complemented
16#560—22no—
17#556C2211yescharacteristic
18#553—21no—
19#550—21no—
20#550C2211yescharacteristic
21#547C2211yescharacteristic
22#546—21no—
23#546—21no—
24D42C2211yescharacteristic
25#542—21no—
26#538—22no—
27#537—21no—
28#535—21no—
29#535—21no—
30#533C2211yescharacteristic, complemented
31#532C2211yescharacteristic
32#532C2211yescharacteristic
33#529C2211yescharacteristic
34#500C2211yescharacteristic, complemented
35#498C2211yescharacteristic, complemented
36#486C2211yescharacteristic
37#486C2211yescharacteristic
38#482—22no—
39#479—21no—
40#478—22no—
41#477—21no—
42#475C2211yescharacteristic, complemented
43#474C2211yescharacteristic
44#474C2211yescharacteristic
45#474C2211yescharacteristic, complemented
46#468—21no—
47#468—21no—
48#467—21no—
49#465—21no—
50#464—21no—
51#462C2211yescharacteristic, complemented
52#461C2211yescharacteristic
53#461C2211yescharacteristic
54#461C2211yescharacteristic, complemented
55#460—21no—
56#431C2211yescharacteristic, complemented
57#423C2211yescharacteristic, complemented
58C2×C4⋊C8C2211yescharacteristic, complemented
59#332C2211yescharacteristic, complemented
60#332C2211yescharacteristic, complemented
61#331C2211yescharacteristic
62#331C2211yescharacteristic
63Q8⋊C8C2211yescharacteristic, complemented
64Q8⋊C8C2211yescharacteristic, complemented
65D4⋊C8C2211yescharacteristic
66D4⋊C8C2211yescharacteristic
67D4⋊C8C2211yescharacteristic
68D4⋊C8C2211yescharacteristic
69D4⋊C8C2211yescharacteristic, complemented
70D4⋊C8C2211yescharacteristic, complemented
71C2×#41—21no—
72C2×#41—21no—
73C2×#41—21no—
74C2×#41—21no—
75C22×D4—21no—
76C22×D4—21no—
77C2×Q16—21no—
78C2×SD16—21no—
79C2×SD16—21no—
80C2×SD16—21no—
81C2×M16—21no—
82C22×C8—21no—
83C4⋊Q8D411yescharacteristic
84C4⋊Q8C2311yescharacteristic
85C4⋊Q8C2311yescharacteristic
86C4⋊Q8C2311yescharacteristic
87#127—22no—
88#127—21no—
89#127D411yescharacteristic
90#127C2311yescharacteristic
91#126—22no—
92#126—22no—
93#126—21no—
94#125—21no—
95#125D411yescharacteristic
96#124—22no—
97#124—21no—
98#124—21no—
99#124—21no—
100#124D411yescharacteristic
101#123—22no—
102#123—22no—
103#123—21no—
104#123—22no—
105C22⋊Q8—22no—
106C22⋊Q8—21no—
107C22⋊Q8—21no—
108C22⋊Q8—21no—
109C22⋊Q8—21no—
110C22⋊Q8D411yescharacteristic
111#121—22no—
112#121—22no—
113#121—22no—
114#121—22no—
115#121—22no—
116#121—21no—
117#121—22no—
118#121—21no—
119#121—21no—
120#121—21no—
121#121—21no—
122#121—21no—
123#121D411yescharacteristic
124#121D411yescharacteristic
125#121D411yescharacteristic
126#121D412yes—
127#120—22no—
128#120—22no—
129#120—22no—
130#120—22no—
131C4×Q8—21no—
132C4×Q8—21no—
133C4×Q8—21no—
134C4×Q8C2311yescharacteristic
135C4×D4—22no—
136C4×D4—21no—
137C4×D4—21no—
138C4×D4—21no—
139C4×D4—21no—
140C4×D4D412yes—
141C4×D4C2311yescharacteristic
142C4×D4C2311yescharacteristic
143C4×D4C2311yescharacteristic
144#117—21no—
145#117—21no—
146#117—21no—
147C2×C4⋊C4—21no—
148C2×C22⋊C4—22no—
149C2×C42C2311yescharacteristic
150#107—21no—
151#107—21no—
152#107—21no—
153#107—22no—
154#106—21no—
155#106—22no—
156C4⋊C8—21no—
157C4⋊C8—21no—
158C4⋊C8—21no—
159C4⋊C8C2311yescharacteristic
160C4⋊C8C2311yescharacteristic
161C4⋊C8C2311yescharacteristic
162C4⋊C8C2311yescharacteristic
163#103—21no—
164#103—21no—
165#103—21no—
166#103—21no—
167#103—21no—
168#102—21no—
169#102—21no—
170#102—21no—
171#102—22no—
172#102—22no—
173C22⋊C8—21no—
174C4×C8C2311yescharacteristic
175C4×C8C2311yescharacteristic
176C24—22no—
177#41—81no—
178#41—81no—
179#41—81no—
180#41—81no—
181#41—81no—
182#41—81no—
183#41—81no—
184#41—81no—
185C2×Q8—21no—
186C2×Q8—21no—
187C2×Q8—21no—
188C2×Q8—21no—
189C2×Q8C2×D411yescharacteristic
190C2×D4—22no—
191C2×D4—22no—
192C2×D4—22no—
193C2×D4—22no—
194C2×D4—22no—
195C2×D4—21no—
196C2×D4—21no—
197C2×D4—22no—
198C2×D4—21no—
199C2×D4—21no—
200C2×D4—21no—
201C2×D4—21no—
202C2×D4—21no—
203C2×D4—81no—
204C2×D4—81no—
205C2×D4—81no—
206C2×D4—81no—
207C2×D4C2×D411yescharacteristic
208C2×D4C2×D411yescharacteristic
209C2×D4C2×D411yescharacteristic
210C2×D4C2×D412yes—
211C22×C4—22no—
212C22×C4—22no—
213C22×C4—21no—
214C22×C4—21no—
215C22×C4—21no—
216C22×C4—22no—
217C22×C4C2×D411yescharacteristic
218C22×C4C2×D411yescharacteristic
219C22×C4C2×D411yescharacteristic
220Q16—81no—
221SD16—81no—
222SD16—81no—
223SD16—81no—
224M16—81no—
225C2×C8—21no—
226C2×C8—21no—
227C2×C8—21no—
228C2×C8—21no—
229C2×C8—21no—
230C2×C8—21no—
231C2×C8—81no—
232C4⋊C4—22no—
233C4⋊C4—22no—
234C4⋊C4—21no—
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⋊C4—41no—
245C4⋊C4—41no—
246C4⋊C4C2×D411yescharacteristic
247C4⋊C4C2×D411yescharacteristic
248C4⋊C4C2×D411yescharacteristic
249C4⋊C4C2×D411yescharacteristic
250C4⋊C4C2×D412yes—
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—21no—
267C22⋊C4—41no—
268C42—22no—
269C42—21no—
270C42—41no—
271C42C2×D411yescharacteristic
272C42C2×D411yescharacteristic
273C42C2×D411yescharacteristic
274C42C2411yescharacteristic
275C23—21no—
276C23—21no—
277C23—22no—
278C23—21no—
279C23—82no—
280C23—82no—
281C23—82no—
282C23#12011yescharacteristic
283Q8—21no—
284Q8—21no—
285Q8—41no—
286Q8—41no—
287Q8—41no—
288Q8—41no—
289Q8—41no—
290Q8—41no—
291Q8—81no—
292D4—21no—
293D4—21no—
294D4—21no—
295D4—21no—
296D4—21no—
297D4—21no—
298D4—42no—
299D4—41no—
300D4—41no—
301D4—41no—
302D4—41no—
303D4—81no—
304D4—81no—
305D4—81no—
306D4—81no—
307D4—81no—
308D4—82no—
309D4—82no—
310D4—82no—
311D4—82no—
312D4—82no—
313D4—82no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—22no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—41no—
327C2×C4—41no—
328C2×C4—41no—
329C2×C4—41no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—82no—
334C2×C4—82no—
335C2×C4—82no—
336C2×C4#12011yescharacteristic
337C2×C4#12011yescharacteristic
338C2×C4#12011yescharacteristic
339C2×C4C22×D411yescharacteristic
340C2×C4C22×D411yescharacteristic
341C2×C4C22×D411yescharacteristic
342C8—41no—
343C8—41no—
344C8—41no—
345C8—41no—
346C8—41no—
347C8—41no—
348C22—21no—
349C22—21no—
350C22—21no—
351C22—41no—
352C22—41no—
353C22—41no—
354C22—41no—
355C22—42no—
356C22—42no—
357C22—41no—
358C22—41no—
359C22—42no—
360C22—41no—
361C22—41no—
362C22—41no—
363C22—164no—
364C22C2×#12011yescharacteristic
365C4—21no—
366C4—21no—
367C4—21no—
368C4—21no—
369C4—41no—
370C4—41no—
371C4—41no—
372C4—41no—
373C4—42no—
374C4—41no—
375C4—81no—
376C4—81no—
377C4#57611yescharacteristic
378C4#57611yescharacteristic
379C4#57711yescharacteristic
380C4#57711yescharacteristic
381C2—41no—
382C2—81no—
383C2—82no—
384C2—81no—
385C2—81no—
386C2#301211yescharacteristic
387C2#301211yescharacteristic
388C2#302211yescharacteristic
subgroupG/H
center, socleC22C2×#120
commutator, Frattini, cosocleC42C24
prime radical#3546C2
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
2. #3254C2semidirect
3. #3248C2semidirect
4. #3228C2semidirect
5. #1654C2semidirect
6. #1652C2semidirect
7. #1650C2semidirect
8. #1647C2semidirect
9. #1642C2semidirect
10. #1641C2semidirect
11. #1639C2semidirect
12. #1639C2semidirect
13. #1547C2semidirect
14. #1495C2semidirect
15. #1480C2semidirect
30. #533C22semidirect
34. #500C22semidirect
35. #498C22semidirect
42. #475C22semidirect
45. #474C22semidirect
51. #462C22semidirect
54. #461C22semidirect
56. #431C22semidirect
57. #423C22semidirect
58. C2×C4⋊C8C22semidirect
59. #332C22semidirect
60. #332C22semidirect
63. Q8⋊C8C22semidirect
64. Q8⋊C8C22semidirect
69. D4⋊C8C22semidirect
70. D4⋊C8C22semidirect

Maximal subgroups

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

subgroupindex
#35462
#32542
#32482
#32282
#16542
#16522
#16502
#16472
#16422
#16412
#16392
#15472
#14952
#14802