Finite Group Databasebeta

← #24173

Subgroups — #24173

Order 256.

537 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 537 conjugacy classes of subgroups — too many to draw. Only its 125 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#3511C211yesmaximal, characteristic
2#3037C211yesmaximal, characteristic, complemented
3C4×#137C211yesmaximal, characteristic, complemented
4#2567C211yesmaximal, characteristic, complemented
5#2039C212yesmaximal, complemented
6#2007C212yesmaximal, complemented
7#1997C211yesmaximal, characteristic, complemented
8#1980C211yesmaximal, characteristic, complemented
9#1882C212yesmaximal, complemented
10#1859C212yesmaximal, complemented
11#1806C211yesmaximal, characteristic, complemented
12C2×#137C2211yescharacteristic
13C2×#137C2212yes—
14#556—21no—
15#555—21no—
16#554—21no—
17#553—22no—
18#548—22no—
19#547—21no—
20#542—22no—
21#537C2212yes—
22#533—21no—
23#532—21no—
24#532C2211yescharacteristic
25C2×C4⋊Q8C2211yescharacteristic
26#520C2212yes—
27C4×#41C2211yescharacteristic
28C2×C4×Q8C2211yescharacteristic
29#464—22no—
30#462—21no—
31#460—21no—
32#441—21no—
33#440—21no—
34C4×Q16—21no—
35C4×SD16—21no—
36#428C2211yescharacteristic
37#423C2211yescharacteristic
38C2×#104C2212yes—
39C2×#104C2212yes—
40C2×#104C2212yes—
41#419C2211yescharacteristic
42#417C2212yes—
43#417C2211yescharacteristic
44C4×M16C2211yescharacteristic
45#395—22no—
46#391C2212yes—
47#389—22no—
48#384C2212yes—
49C4×C4⋊C4C2211yescharacteristic
50#343C2212yes—
51#343C2212yes—
52#339C2212yes—
53C2×#41—22no—
54C2×#41C2311yescharacteristic
55C2×#41C2312yes—
56C22×Q8D412yes—
57C22×Q8C2311yescharacteristic
58#137—22no—
59#137—24no—
60#137—22no—
61#137—42no—
62C2×Q16—21no—
63C2×Q16—22no—
64C2×SD16—21no—
65C2×SD16—22no—
66C2×M16C2312yes—
67C2×M16C2311yescharacteristic
68C2×M16C2311yescharacteristic
69C4⋊Q8—21no—
70C4⋊Q8—21no—
71C4⋊Q8—21no—
72C4⋊Q8—21no—
73C4⋊Q8D411yescharacteristic
74#127D411yescharacteristic
75#126—24no—
76#125—22no—
77#124—22no—
78#124—24no—
79#124D412yes—
80#123—24no—
81C22⋊Q8—22no—
82C22⋊Q8—24no—
83C22⋊Q8—22no—
84C22⋊Q8—22no—
85#121—22no—
86#121—22no—
87#121—22no—
88#121—21no—
89#121—21no—
90#121—24no—
91C4×Q8—22no—
92C4×Q8—22no—
93C4×Q8—21no—
94C4×Q8—21no—
95C4×Q8—21no—
96C4×Q8—21no—
97C4×Q8D411yescharacteristic
98C4×Q8D411yescharacteristic
99C4×D4—22no—
100C4×D4—22no—
101C4×D4—21no—
102C4×D4—21no—
103C4×D4—24no—
104C4×D4—21no—
105C4×D4—21no—
106C4×D4D411yescharacteristic
107C4×D4D411yescharacteristic
108#117—22no—
109#117—22no—
110#117—21no—
111#117C2311yescharacteristic
112C2×C4⋊C4—21no—
113C2×C4⋊C4—22no—
114C2×C4⋊C4—21no—
115C2×C4⋊C4—24no—
116C2×C4⋊C4D412yes—
117C2×C4⋊C4C2311yescharacteristic
118C2×C4⋊C4C2312yes—
119C2×C42—21no—
120C2×C42—21no—
121C2×C42C2312yes—
122C2×C42C2311yescharacteristic
123#107—21no—
124#106—21no—
125C4⋊C8—21no—
126C4⋊C8—21no—
127#104—24no—
128#104—42no—
129#104—42no—
130#103—21no—
131#103—22no—
132#103—21no—
133#103—21no—
134#102—21no—
135#102—22no—
136#97—21no—
137C4×C8—21no—
138#95—21no—
139#95—24no—
140#41—22no—
141#41—22no—
142#41—21no—
143#41—21no—
144#41—21no—
145#41—22no—
146#41—22no—
147#41—21no—
148#41—84no—
149C2×Q8—22no—
150C2×Q8—22no—
151C2×Q8—21no—
152C2×Q8—21no—
153C2×Q8—21no—
154C2×Q8—21no—
155C2×Q8—21no—
156C2×Q8—82no—
157C2×Q8C2×D412yes—
158C2×Q8C2×D411yescharacteristic
159C2×Q8#4112yescomplemented
160C2×D4—21no—
161C2×D4—22no—
162C2×D4—24no—
163C2×D4—21no—
164C2×D4—21no—
165C2×D4C2×D412yes—
166C2×D4C2×D411yescharacteristic
167C22×C4—21no—
168C22×C4—22no—
169C22×C4—24no—
170C22×C4—21no—
171C22×C4—21no—
172C22×C4—21no—
173C22×C4—22no—
174C22×C4—21no—
175C22×C4C2×D411yescharacteristic
176C22×C4C2×D414yes—
177C22×C4C2×D411yescharacteristic
178C22×C4C2411yescharacteristic
179Q16—24no—
180Q16—44no—
181SD16—24no—
182SD16—44no—
183M16—21no—
184M16—22no—
185M16—21no—
186M16—22no—
187M16—21no—
188M16#4112yescomplemented
189C2×C8—21no—
190C2×C8—21no—
191C2×C8—22no—
192C4⋊C4—24no—
193C4⋊C4—24no—
194C4⋊C4—24no—
195C4⋊C4—22no—
196C4⋊C4—22no—
197C4⋊C4—21no—
198C4⋊C4—21no—
199C4⋊C4—21no—
200C4⋊C4—21no—
201C4⋊C4—22no—
202C4⋊C4—22no—
203C4⋊C4—21no—
204C4⋊C4—21no—
205C4⋊C4—41no—
206C4⋊C4—41no—
207C4⋊C4—84no—
208C4⋊C4C2×D411yescharacteristic
209C4⋊C4C2×D411yescharacteristic
210C22⋊C4—24no—
211C22⋊C4—24no—
212C22⋊C4—24no—
213C22⋊C4—21no—
214C22⋊C4—21no—
215C22⋊C4—21no—
216C22⋊C4—22no—
217C22⋊C4—22no—
218C42—24no—
219C42—21no—
220C42—21no—
221C42—21no—
222C42—22no—
223C42—22no—
224C42—41no—
225C42—41no—
226C42C2×D411yescharacteristic
227C42C2×D411yescharacteristic
228C42C2×D411yescharacteristic
229C42C2×D411yescharacteristic
230C23—21no—
231C23—22no—
232C23C22×D411yescharacteristic
233Q8—22no—
234Q8—21no—
235Q8—21no—
236Q8—22no—
237Q8—22no—
238Q8—22no—
239Q8—44no—
240Q8—41no—
241Q8—41no—
242Q8—82no—
243D4—21no—
244D4—21no—
245D4—22no—
246D4—22no—
247D4—42no—
248D4—42no—
249D4—41no—
250D4—42no—
251D4—41no—
252D4—42no—
253D4—84no—
254C2×C4—22no—
255C2×C4—24no—
256C2×C4—21no—
257C2×C4—22no—
258C2×C4—21no—
259C2×C4—21no—
260C2×C4—24no—
261C2×C4—22no—
262C2×C4—41no—
263C2×C4—42no—
264C2×C4—42no—
265C2×C4—41no—
266C2×C4—42no—
267C2×C4—41no—
268C2×C4—42no—
269C2×C4—42no—
270C2×C4—41no—
271C2×C4—41no—
272C2×C4—42no—
273C2×C4—44no—
274C2×C4—42no—
275C2×C4—41no—
276C2×C4—84no—
277C2×C4#12011yescharacteristic
278C2×C4#12011yescharacteristic
279C2×C4#12011yescharacteristic
280C2×C4#12011yescharacteristic
281C2×C4C22×D411yescharacteristic
282C2×C4C22×D411yescharacteristic
283C2×C4C2×#4111yescharacteristic
284C2×C4C2×#4111yescharacteristic
285C2×C4H(5,𝔽2)11yescharacteristic
286C2×C42−1+411yescharacteristic
287C8—22no—
288C8—41no—
289C8—42no—
290C22—21no—
291C22—21no—
292C22—42no—
293C22—41no—
294C22—41no—
295C22—42no—
296C22—41no—
297C22—42no—
298C22C2×#12011yescharacteristic
299C22#54811yescharacteristic
300C22#55011yescharacteristic
301C4—21no—
302C4—21no—
303C4—21no—
304C4—44no—
305C4—42no—
306C4—41no—
307C4—42no—
308C4—41no—
309C4—41no—
310C4—44no—
311C4—81no—
312C4#45612yes—
313C4#52511yescharacteristic
314C4#53711yescharacteristic
315C4#54611yescharacteristic
316C4#54711yescharacteristic
317C2—21no—
318C2—21no—
319C2—81no—
320C2—82no—
321C2#240711yescharacteristic
322C2C2×#45611yescharacteristic
323C2#301911yescharacteristic
subgroupG/H
center, socleC22C2×#120
commutator, Frattini, cosocleC22×C4C24
prime radical#3511C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #3037C2semidirect
3. C4×#137C2semidirect
4. #2567C2semidirect
5. #2039C2semidirect
6. #2007C2semidirect
7. #1997C2semidirect
8. #1980C2semidirect
9. #1882C2semidirect
10. #1859C2semidirect
11. #1806C2semidirect
159. C2×Q8#41semidirect
188. M16#41semidirect

Maximal subgroups

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

subgroupindex
#35112
#30372
C4×#1372
#25672
#20392
#20072
#19972
#19802
#18822
#18592
#18062