Finite Group Databasebeta

← #18882

Subgroups — #18882

Order 256.

575 conjugacy classes of subgroups; only the 129 normal subgroups are shown.

normal subgroups only

This group has 575 conjugacy classes of subgroups — too many to draw. Only its 129 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
1#3499C211yesmaximal, characteristic, complemented
2#3245C211yesmaximal, characteristic, complemented
3#3226C211yesmaximal, characteristic, complemented
4#3035C211yesmaximal, characteristic, complemented
5#1649C212yesmaximal, complemented
6#1647C212yesmaximal, complemented
7#1627C212yesmaximal, complemented
8#1625C212yesmaximal, complemented
9C2×#332C211yesmaximal, characteristic, complemented
10#1480C211yesmaximal, characteristic, complemented
11#1477C211yesmaximal, characteristic, complemented
12C2×2−1+4—21no—
13C2×#135—21no—
14D4×Q8—21no—
15D4×Q8C2212yes—
16#547—22no—
17#547—21no—
18#547—22no—
19#547C2212yes—
20#536—22no—
21#536—21no—
22#532—21no—
23#532C2211yescharacteristic
24#532C2211yescharacteristic
25C2×C4⋊Q8—21no—
26C2×C4⋊Q8C2211yescharacteristic
27C4×#41C2211yescharacteristic
28#500C2212yes—
29#486C2212yes—
30#480—22no—
31#474—22no—
32#474—21no—
33#474C2212yes—
34#466—22no—
35#464—21no—
36#463—21no—
37#462C2211yescharacteristic
38#461C2211yescharacteristic
39#460C2211yescharacteristic
40#459—21no—
41#459C2211yescharacteristic
42#459C2212yes—
43#431C2211yescharacteristic
44C2×C4⋊C8C2211yescharacteristic
45C2×C4⋊C8C2211yescharacteristic
46#416—21no—
47#332C2212yes—
48#332C2212yes—
49Q8⋊C8C2212yes—
50D4⋊C8C2212yes—
51D4⋊C8C2212yes—
52D4⋊C8C2212yes—
532−1+4—44no—
542−1+4—44no—
55C2×#41—22no—
56C2×#41—24no—
57C2×#41—22no—
58C2×#41—21no—
59C2×#41—21no—
60C2×#41D411yescharacteristic
61C22×Q8—22no—
62C22×Q8—22no—
63C22×Q8—21no—
64#135—44no—
65C2×Q16—21no—
66C2×SD16—21no—
67C2×SD16—21no—
68C2×D8—21no—
69C2×D8—22no—
70C22×C8—21no—
71C22×C8—21no—
72C4⋊Q8—22no—
73C4⋊Q8—22no—
74C4⋊Q8—22no—
75C4⋊Q8—21no—
76C4⋊Q8—21no—
77C4⋊Q8D411yescharacteristic
78C4⋊Q8D412yes—
79C4⋊Q8C2311yescharacteristic
80C4⋊Q8C2312yes—
81#127D411yescharacteristic
82#127C2311yescharacteristic
83#124—22no—
84#124—21no—
85#124—21no—
86#123—24no—
87#123—24no—
88#123—24no—
89C22⋊Q8—24no—
90C22⋊Q8—22no—
91C22⋊Q8—24no—
92C22⋊Q8—22no—
93#121—22no—
94#121—22no—
95#121—21no—
96#121—21no—
97#121—21no—
98#121D412yes—
99C4×Q8D411yescharacteristic
100C4×Q8C2311yescharacteristic
101C4×D4—22no—
102C4×D4—22no—
103C4×D4—21no—
104C4×D4—21no—
105C4×D4—21no—
106C4×D4—21no—
107C4×D4—21no—
108C4×D4D411yescharacteristic
109C4×D4C2311yescharacteristic
110C4×D4C2312yes—
111#117—21no—
112#117D411yescharacteristic
113C2×C4⋊C4—22no—
114C2×C4⋊C4—21no—
115C2×C4⋊C4—22no—
116C2×C4⋊C4D412yes—
117C2×C42—21no—
118C2×C42C2311yescharacteristic
119#107—22no—
120#107—24no—
121C4⋊C8—21no—
122C4⋊C8—21no—
123C4⋊C8—21no—
124C4⋊C8C2311yescharacteristic
125C4⋊C8C2311yescharacteristic
126C4⋊C8C2312yes—
127#103—21no—
128#103—21no—
129#102—22no—
130#102—21no—
131#102—21no—
132#102—22no—
133#102—24no—
134C22⋊C8—21no—
135C4×C8C2312yes—
136#41—22no—
137#41—48no—
138#41—416no—
139#41—44no—
140#41—44no—
141#41—42no—
142#41—42no—
143#41—42no—
144#41—42no—
145#41D814yescomplemented
146C2×Q8—24no—
147C2×Q8—22no—
148C2×Q8—21no—
149C2×Q8—22no—
150C2×Q8—21no—
151C2×Q8—48no—
152C2×Q8—44no—
153C2×Q8—44no—
154C2×Q8—42no—
155C2×Q8—42no—
156C2×Q8C2×D411yescharacteristic
157C2×D4—21no—
158C2×D4—24no—
159C2×D4—22no—
160C2×D4—21no—
161C2×D4—21no—
162C2×D4—21no—
163C2×D4—21no—
164C2×D4—21no—
165C2×D4—41no—
166C2×D4C2×D412yes—
167C2×D4C2×D411yescharacteristic
168C22×C4—21no—
169C22×C4—24no—
170C22×C4—24no—
171C22×C4—22no—
172C22×C4—21no—
173C22×C4—21no—
174C22×C4—22no—
175C22×C4—41no—
176C22×C4C2×D411yescharacteristic
177C22×C4C2×D411yescharacteristic
178C22×C4C2×D411yescharacteristic
179Q16—42no—
180SD16—42no—
181SD16—42no—
182D8—42no—
183D8—82no—
184C2×C8—22no—
185C2×C8—22no—
186C2×C8—21no—
187C2×C8—21no—
188C2×C8—42no—
189C2×C8—81no—
190C4⋊C4—24no—
191C4⋊C4—24no—
192C4⋊C4—21no—
193C4⋊C4—22no—
194C4⋊C4—22no—
195C4⋊C4—21no—
196C4⋊C4—22no—
197C4⋊C4—21no—
198C4⋊C4—21no—
199C4⋊C4—21no—
200C4⋊C4C2×D411yescharacteristic
201C4⋊C4C2×D411yescharacteristic
202C4⋊C4C2×D412yes—
203C4⋊C4C2×D414yes—
204C22⋊C4—24no—
205C22⋊C4—24no—
206C22⋊C4—21no—
207C22⋊C4—21no—
208C22⋊C4—22no—
209C22⋊C4—21no—
210C22⋊C4—22no—
211C22⋊C4—41no—
212C22⋊C4—41no—
213C42—21no—
214C42—21no—
215C42—21no—
216C42C2×D411yescharacteristic
217C42C2×D411yescharacteristic
218C42C2×D411yescharacteristic
219C42C2411yescharacteristic
220C23—21no—
221C23—22no—
222C23—41no—
223C23#12011yescharacteristic
224Q8—21no—
225Q8—48no—
226Q8—42no—
227Q8—42no—
228Q8—42no—
229Q8—44no—
230Q8—41no—
231Q8—41no—
232Q8C2×D812yes—
233D4—24no—
234D4—21no—
235D4—22no—
236D4—22no—
237D4—42no—
238D4—42no—
239D4—48no—
240D4—44no—
241D4—41no—
242D4—41no—
243D4—41no—
244D4—41no—
245D4—42no—
246D4—82no—
247D4C2×D812yes—
248C2×C4—21no—
249C2×C4—21no—
250C2×C4—21no—
251C2×C4—24no—
252C2×C4—21no—
253C2×C4—21no—
254C2×C4—24no—
255C2×C4—22no—
256C2×C4—41no—
257C2×C4—42no—
258C2×C4—48no—
259C2×C4—48no—
260C2×C4—44no—
261C2×C4—42no—
262C2×C4—42no—
263C2×C4—41no—
264C2×C4—41no—
265C2×C4—41no—
266C2×C4—82no—
267C2×C4#12011yescharacteristic
268C2×C4#12011yescharacteristic
269C2×C4#12011yescharacteristic
270C2×C4C2×D812yes—
271C2×C4AGL(1,ℤ/8)12yes—
272C2×C4C22×D411yescharacteristic
273C2×C4C22×D411yescharacteristic
274C2×C4C22×D411yescharacteristic
275C8—42no—
276C8—41no—
277C8—41no—
278C8—42no—
279C22—21no—
280C22—21no—
281C22—22no—
282C22—41no—
283C22—41no—
284C22—42no—
285C22—42no—
286C22—42no—
287C22—81no—
288C22—81no—
289C22—81no—
290C22#44712yes—
291C22C2×#12011yescharacteristic
292C4—22no—
293C4—21no—
294C4—21no—
295C4—21no—
296C4—44no—
297C4—42no—
298C4—41no—
299C4—41no—
300C4—81no—
301C4#44712yes—
302C4C22×D811yescharacteristic
303C4C2×AGL(1,ℤ/8)11yescharacteristic
304C4#57711yescharacteristic
305C4#57811yescharacteristic
306C2—22no—
307C2—42no—
308C2—82no—
309C2—161no—
310C2C2×#44711yescharacteristic
311C2#301311yescharacteristic
312C2#302011yescharacteristic
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. #3499C2semidirect
2. #3245C2semidirect
3. #3226C2semidirect
4. #3035C2semidirect
5. #1649C2semidirect
6. #1647C2semidirect
7. #1627C2semidirect
8. #1625C2semidirect
9. C2×#332C2semidirect
10. #1480C2semidirect
11. #1477C2semidirect
145. #41D8semidirect

Maximal subgroups

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

subgroupindex
#34992
#32452
#32262
#30352
#16492
#16472
#16272
#16252
C2×#3322
#14802
#14772