Finite Group Databasebeta

← #18798

Subgroups — #18798

Order 256.

613 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 613 conjugacy classes of subgroups — too many to draw. Only its 131 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1D4×#41C211yesmaximal, characteristic
2#3226C211yesmaximal, characteristic, complemented
3C2×#474C211yesmaximal, characteristic, complemented
4#3039C211yesmaximal, characteristic, complemented
5#1649C212yesmaximal, complemented
6#1647C212yesmaximal, complemented
7#1641C212yesmaximal, complemented
8#1639C212yesmaximal, complemented
9#1547C211yesmaximal, characteristic, complemented
10#1486C211yesmaximal, characteristic, complemented
11#1477C211yesmaximal, characteristic, complemented
12C22×#41—21no—
13C2×#137—21no—
14#550C2211yescharacteristic
15D4×Q8C2211yescharacteristic
16#548—21no—
17#547—21no—
18#547C2211yescharacteristic
19#546—22no—
20#546—21no—
21D42—21no—
22D42C2211yescharacteristic
23#532—21no—
24#532C2211yescharacteristic
25C2×C4⋊Q8C2211yescharacteristic, complemented
26#525—22no—
27#525—21no—
28C4×#41C2211yescharacteristic
29C2×C4×D4—21no—
30C2×C4×D4C2211yescharacteristic
31#500C2212yescomplemented
32#486C2212yescomplemented
33#480—22no—
34#474—22no—
35#474—21no—
36#474—21no—
37#474C2212yes—
38#474C2212yescomplemented
39#462—21no—
40#462C2212yescomplemented
41#461—21no—
42#461C2212yescomplemented
43#431C2211yescharacteristic, complemented
44C2×#107—21no—
45#423C2211yescharacteristic, complemented
46C2×C4⋊C8C2211yescharacteristic, complemented
47#419—21no—
48C2×#102—21no—
49#332C2212yescomplemented
50#331C2212yescomplemented
51Q8⋊C8C2212yescomplemented
52D4⋊C8C2212yes—
53D4⋊C8C2212yescomplemented
54D4⋊C8C2212yescomplemented
55C2×#41—22no—
56C2×#41—22no—
57C2×#41—21no—
58C2×#41—21no—
59C2×#41—21no—
60C2×#41—44no—
61C2×#41—44no—
62C2×#41D411yescharacteristic
63C22×Q8—21no—
64C22×Q8—21no—
65C22×D4—22no—
66C22×D4—21no—
67C22×D4—21no—
68C22×D4D411yescharacteristic
69C23×C4—22no—
70C23×C4—21no—
71#137—44no—
72C2×Q16—22no—
73C2×SD16—22no—
74C2×M16—21no—
75C22×C8—21no—
76C4⋊Q8—21no—
77C4⋊Q8—21no—
78C4⋊Q8—21no—
79C4⋊Q8—21no—
80C4⋊Q8C2312yes—
81C4⋊Q8C2311yescharacteristic
82#127—21no—
83#127C2311yescharacteristic
84#124—22no—
85#124—21no—
86#123—22no—
87#123—22no—
88#123—22no—
89C22⋊Q8—22no—
90C22⋊Q8—21no—
91C22⋊Q8—22no—
92C22⋊Q8—21no—
93#121—22no—
94#121—22no—
95#121—22no—
96#121—21no—
97#121—21no—
98#121—21no—
99#121—21no—
100#121—21no—
101#121D412yes—
102#120—22no—
103#120—22no—
104#120—22no—
105C4×Q8D411yescharacteristic
106C4×Q8C2311yescharacteristic
107C4×D4—22no—
108C4×D4—22no—
109C4×D4—22no—
110C4×D4—21no—
111C4×D4—21no—
112C4×D4—21no—
113C4×D4—21no—
114C4×D4—21no—
115C4×D4—21no—
116C4×D4—21no—
117C4×D4D411yescharacteristic
118C4×D4D411yescharacteristic
119C4×D4D411yescharacteristic
120C4×D4D412yes—
121C4×D4C2311yescharacteristic
122C4×D4C2311yescharacteristic
123C4×D4C2311yescharacteristic
124#117—21no—
125#117D411yescharacteristic
126C2×C4⋊C4—21no—
127C2×C4⋊C4—21no—
128C2×C4⋊C4—21no—
129C2×C4⋊C4D411yescharacteristic
130C2×C22⋊C4—22no—
131C2×C22⋊C4—21no—
132C2×C42—21no—
133C2×C42C2311yescharacteristic
134#107—22no—
135#107—22no—
136#107—24no—
137C4⋊C8—21no—
138C4⋊C8—21no—
139C4⋊C8—21no—
140C4⋊C8C2312yes—
141C4⋊C8C2312yes—
142#103—22no—
143#103—22no—
144#102—22no—
145#102—22no—
146#102—24no—
147C22⋊C8—21no—
148C4×C8C2312yes—
149C24—22no—
150C24—21no—
151#41—22no—
152#41—48no—
153#41—48no—
154#41—42no—
155#41—42no—
156#41—42no—
157#41—42no—
158#41—42no—
159#41—42no—
160#41D814yes—
161C2×Q8—22no—
162C2×Q8—21no—
163C2×Q8—22no—
164C2×Q8—42no—
165C2×Q8—42no—
166C2×Q8—41no—
167C2×Q8—42no—
168C2×Q8—42no—
169C2×Q8C2×D411yescharacteristic
170C2×D4—21no—
171C2×D4—22no—
172C2×D4—22no—
173C2×D4—22no—
174C2×D4—22no—
175C2×D4—22no—
176C2×D4—22no—
177C2×D4—22no—
178C2×D4—21no—
179C2×D4—21no—
180C2×D4—21no—
181C2×D4—41no—
182C2×D4—42no—
183C2×D4—48no—
184C2×D4—42no—
185C2×D4—42no—
186C2×D4—42no—
187C2×D4C2×D411yescharacteristic
188C2×D4C2×D411yescharacteristic
189C2×D4C2×D411yescharacteristic
190C2×D4C2×D412yes—
191C22×C4—21no—
192C22×C4—22no—
193C22×C4—22no—
194C22×C4—22no—
195C22×C4—21no—
196C22×C4—21no—
197C22×C4—22no—
198C22×C4—21no—
199C22×C4—21no—
200C22×C4—21no—
201C22×C4—21no—
202C22×C4—21no—
203C22×C4—48no—
204C22×C4—42no—
205C22×C4—42no—
206C22×C4—41no—
207C22×C4C2×D411yescharacteristic
208C22×C4C2×D411yescharacteristic
209C22×C4C2×D411yescharacteristic
210Q16—44no—
211SD16—44no—
212M16—42no—
213C2×C8—22no—
214C2×C8—22no—
215C2×C8—22no—
216C2×C8—42no—
217C4⋊C4—22no—
218C4⋊C4—22no—
219C4⋊C4—21no—
220C4⋊C4—21no—
221C4⋊C4—21no—
222C4⋊C4—21no—
223C4⋊C4—22no—
224C4⋊C4—22no—
225C4⋊C4—41no—
226C4⋊C4—41no—
227C4⋊C4C2×D411yescharacteristic
228C4⋊C4C2×D411yescharacteristic
229C4⋊C4C2×D411yescharacteristic
230C4⋊C4C2×D411yescharacteristic
231C4⋊C4C2×D412yes—
232C22⋊C4—22no—
233C22⋊C4—22no—
234C22⋊C4—22no—
235C22⋊C4—22no—
236C22⋊C4—21no—
237C22⋊C4—21no—
238C22⋊C4—22no—
239C22⋊C4—21no—
240C22⋊C4—21no—
241C22⋊C4—22no—
242C22⋊C4—21no—
243C22⋊C4—21no—
244C22⋊C4—21no—
245C42—21no—
246C42—21no—
247C42—21no—
248C42C2×D411yescharacteristic
249C42C2×D411yescharacteristic
250C42C2×D411yescharacteristic
251C42C2411yescharacteristic
252C23—21no—
253C23—21no—
254C23—21no—
255C23—22no—
256C23—42no—
257C23—44no—
258C23—44no—
259C23—42no—
260C23—81no—
261C23—82no—
262C23#12011yescharacteristic
263Q8—21no—
264Q8—44no—
265Q8—41no—
266Q8—41no—
267Q8—42no—
268Q8—42no—
269Q8—82no—
270Q8C2×D812yes—
271D4—24no—
272D4—21no—
273D4—21no—
274D4—21no—
275D4—21no—
276D4—42no—
277D4—42no—
278D4—44no—
279D4—44no—
280D4—44no—
281D4—44no—
282D4—44no—
283D4—41no—
284D4—41no—
285D4—41no—
286D4—41no—
287D4—82no—
288D4C2×D812yes—
289D4#13712yescomplemented
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—24no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—22no—
298C2×C4—21no—
299C2×C4—41no—
300C2×C4—44no—
301C2×C4—44no—
302C2×C4—44no—
303C2×C4—42no—
304C2×C4—41no—
305C2×C4—41no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—42no—
311C2×C4—44no—
312C2×C4—42no—
313C2×C4—42no—
314C2×C4—81no—
315C2×C4—82no—
316C2×C4#12011yescharacteristic
317C2×C4#12011yescharacteristic
318C2×C4#12011yescharacteristic
319C2×C4C2×D812yes—
320C2×C4AGL(1,ℤ/8)12yes—
321C2×C4C22×D411yescharacteristic
322C2×C4C22×D411yescharacteristic
323C2×C4C22×D411yescharacteristic
324C8—42no—
325C8—42no—
326C8—42no—
327C22—21no—
328C22—21no—
329C22—22no—
330C22—22no—
331C22—41no—
332C22—41no—
333C22—41no—
334C22—41no—
335C22—42no—
336C22—41no—
337C22—41no—
338C22—42no—
339C22—42no—
340C22—41no—
341C22—84no—
342C22—88no—
343C22#44712yes—
344C22C2×#12011yescharacteristic
345C4—22no—
346C4—21no—
347C4—21no—
348C4—21no—
349C4—41no—
350C4—42no—
351C4—41no—
352C4—41no—
353C4—41no—
354C4—82no—
355C4#44712yes—
356C4C22×D811yescharacteristic
357C4C2×AGL(1,ℤ/8)11yescharacteristic
358C4C2×#13711yescharacteristic
359C4#57511yescharacteristic
360C2—22no—
361C2—42no—
362C2—42no—
363C2—81no—
364C2—82no—
365C2C2×#44711yescharacteristic
366C2#300711yescharacteristic
367C2#302211yescharacteristic
subgroupG/H
center, socleC22C2×#120
commutator, Frattini, cosocleC42C24
prime radicalD4×#41C2
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. #3226C2semidirect
3. C2×#474C2semidirect
4. #3039C2semidirect
5. #1649C2semidirect
6. #1647C2semidirect
7. #1641C2semidirect
8. #1639C2semidirect
9. #1547C2semidirect
10. #1486C2semidirect
11. #1477C2semidirect
25. C2×C4⋊Q8C22semidirect
31. #500C22semidirect
32. #486C22semidirect
38. #474C22semidirect
40. #462C22semidirect
42. #461C22semidirect
43. #431C22semidirect
45. #423C22semidirect
46. C2×C4⋊C8C22semidirect
49. #332C22semidirect
50. #331C22semidirect
51. Q8⋊C8C22semidirect
53. D4⋊C8C22semidirect
54. D4⋊C8C22semidirect
289. D4#137semidirect

Maximal subgroups

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

subgroupindex
D4×#412
#32262
C2×#4742
#30392
#16492
#16472
#16412
#16392
#15472
#14862
#14772