Finite Group Databasebeta

← #8437

Subgroups — #8437

Order 256.

391 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 391 conjugacy classes of subgroups — too many to draw. Only its 83 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#3480C211yesmaximal, characteristic
2#1806C211yesmaximal, characteristic, complemented
3C4×#103C211yesmaximal, characteristic, complemented
4#1502C211yesmaximal, characteristic, complemented
5#1492C211yesmaximal, characteristic, complemented
6#1487C211yesmaximal, characteristic, complemented
7#1476C211yesmaximal, characteristic, complemented
8#559—21no—
9#559C411yescharacteristic, complemented
10#557C411yescharacteristic, complemented
11#556—21no—
12#556C411yescharacteristic, complemented
13#553—22no—
14#553—21no—
15#550C411yescharacteristic, complemented
16#548—21no—
17#532—21no—
18#532C2211yescharacteristic
19#529—21no—
20#529—22no—
21C4×#41—21no—
22C4×#41—21no—
23C4×#41C2211yescharacteristic
24C2×C4×Q8C2211yescharacteristic
25#431C2211yescharacteristic
26#423C2211yescharacteristic
27C2×#104—21no—
28C2×#103—21no—
29C2×C4×C8C2211yescharacteristic
30C4×C4⋊C4C2211yescharacteristic
31#340—21no—
32#339—21no—
33#327—22no—
34Q8⋊C8—21no—
35Q8⋊C8—21no—
36Q8⋊C8—21no—
37D4⋊C8—21no—
38D4⋊C8—22no—
39C2×#41—21no—
40C2×#41—21no—
41C2×#41D411yescharacteristic
42C22×Q8D411yescharacteristic
43C2×M16—21no—
44C22×C8—21no—
45C4⋊Q8—21no—
46C4⋊Q8C2×C411yescharacteristic
47#127—21no—
48#127C2×C411yescharacteristic
49#126—22no—
50#126—22no—
51#126—22no—
52#125—22no—
53#125—21no—
54#124—22no—
55#124—21no—
56#124—21no—
57#124—21no—
58#124—21no—
59#123—22no—
60#123—22no—
61#123—22no—
62C22⋊Q8—22no—
63C22⋊Q8—21no—
64#121—22no—
65#121—22no—
66#121—22no—
67#121—21no—
68#121—21no—
69#121D412yes—
70C4×Q8—21no—
71C4×Q8—21no—
72C4×Q8—21no—
73C4×Q8—21no—
74C4×Q8—21no—
75C4×Q8C2×C411yescharacteristic
76C4×Q8C2×C411yescharacteristic
77C4×Q8C2×C411yescharacteristic
78C4×Q8D411yescharacteristic
79C4×Q8D411yescharacteristic
80C4×Q8D411yescharacteristic
81C4×D4—22no—
82C4×D4—22no—
83C4×D4—22no—
84C4×D4—22no—
85C4×D4—22no—
86C4×D4—21no—
87C4×D4—21no—
88C4×D4—21no—
89C4×D4—21no—
90C4×D4—21no—
91C4×D4—21no—
92C4×D4C2×C411yescharacteristic
93C4×D4D411yescharacteristic
94C4×D4D412yes—
95#117—22no—
96#117—22no—
97#117—21no—
98#117—21no—
99#117D411yescharacteristic
100C2×C4⋊C4—21no—
101C2×C4⋊C4—21no—
102C2×C4⋊C4D411yescharacteristic
103C2×C42—22no—
104C2×C42—21no—
105C2×C42—21no—
106C2×C42—21no—
107C2×C42C2311yescharacteristic
108C4⋊C8—21no—
109C4⋊C8—21no—
110C4⋊C8—21no—
111#104—42no—
112#103—24no—
113C22⋊C8—21no—
114C4×C8—21no—
115C4×C8—21no—
116C4×C8—21no—
117#95—21no—
118#41—21no—
119#41—21no—
120#41—21no—
121#41—21no—
122#41—42no—
123#41—42no—
124#41—41no—
125#41—41no—
126#41—41no—
127#41—41no—
128C2×Q8—21no—
129C2×Q8—21no—
130C2×Q8—22no—
131C2×Q8—41no—
132C2×Q8C22⋊C411yescharacteristic
133C2×Q8C22⋊C411yescharacteristic
134C2×Q8C22⋊C411yescharacteristic
135C2×D4—22no—
136C2×D4—21no—
137C2×D4—22no—
138C2×D4—22no—
139C2×D4—21no—
140C2×D4—21no—
141C2×D4C22⋊C411yescharacteristic
142C2×D4C22⋊C412yes—
143C22×C4—22no—
144C22×C4—21no—
145C22×C4—22no—
146C22×C4—21no—
147C22×C4—21no—
148C22×C4—21no—
149C22×C4—22no—
150C22×C4—21no—
151C22×C4—41no—
152C22×C4C2×D411yescharacteristic
153C22×C4C2×D411yescharacteristic
154C22×C4C2×D411yescharacteristic
155M16—41no—
156M16—41no—
157C2×C8—22no—
158C2×C8—41no—
159C2×C8—41no—
160C2×C8—41no—
161C2×C8—41no—
162C4⋊C4—22no—
163C4⋊C4—22no—
164C4⋊C4—22no—
165C4⋊C4—22no—
166C4⋊C4—21no—
167C4⋊C4—21no—
168C4⋊C4—21no—
169C4⋊C4—21no—
170C4⋊C4—21no—
171C4⋊C4—21no—
172C4⋊C4—21no—
173C4⋊C4—21no—
174C4⋊C4—22no—
175C4⋊C4—41no—
176C4⋊C4C22⋊C411yescharacteristic
177C4⋊C4C22⋊C411yescharacteristic
178C4⋊C4C22⋊C411yescharacteristic
179C4⋊C4C22⋊C411yescharacteristic
180C4⋊C4C22⋊C412yes—
181C22⋊C4—22no—
182C22⋊C4—22no—
183C22⋊C4—22no—
184C22⋊C4—22no—
185C22⋊C4—22no—
186C22⋊C4—22no—
187C22⋊C4—21no—
188C22⋊C4—21no—
189C22⋊C4—22no—
190C22⋊C4—21no—
191C22⋊C4—22no—
192C42—22no—
193C42—22no—
194C42—21no—
195C42—21no—
196C42—21no—
197C42—21no—
198C42—21no—
199C42—21no—
200C42—21no—
201C42—41no—
202C42—41no—
203C42C22×C411yescharacteristic
204C42C2×D411yescharacteristic
205C42C2×D411yescharacteristic
206C42C2×D411yescharacteristic
207C23—22no—
208C23—21no—
209C23#12011yescharacteristic
210Q8—21no—
211Q8—21no—
212Q8—21no—
213Q8—21no—
214Q8—21no—
215Q8—42no—
216Q8—41no—
217Q8—41no—
218Q8#10412yescomplemented
219D4—21no—
220D4—21no—
221D4—42no—
222D4—44no—
223D4—41no—
224D4—42no—
225D4—41no—
226D4—42no—
227D4—41no—
228D4—41no—
229D4—41no—
230D4—41no—
231D4—82no—
232C2×C4—21no—
233C2×C4—21no—
234C2×C4—21no—
235C2×C4—21no—
236C2×C4—21no—
237C2×C4—21no—
238C2×C4—21no—
239C2×C4—21no—
240C2×C4—22no—
241C2×C4—21no—
242C2×C4—41no—
243C2×C4—42no—
244C2×C4—44no—
245C2×C4—41no—
246C2×C4—41no—
247C2×C4—41no—
248C2×C4—42no—
249C2×C4—41no—
250C2×C4—42no—
251C2×C4—42no—
252C2×C4—81no—
253C2×C4—81no—
254C2×C4—82no—
255C2×C4—81no—
256C2×C4—81no—
257C2×C4C2×C22⋊C411yescharacteristic
258C2×C4C2×C22⋊C411yescharacteristic
259C2×C4C2×C22⋊C411yescharacteristic
260C2×C4#12011yescharacteristic
261C2×C4#12011yescharacteristic
262C2×C4#12011yescharacteristic
263C2×C4#13511yescharacteristic
264C2×C4#13511yescharacteristic
265C2×C4AGL(1,ℤ/8)11yescharacteristic
266C2×C4#13711yescharacteristic
267C8—42no—
268C8—81no—
269C8—81no—
270C22—21no—
271C22—21no—
272C22—42no—
273C22—41no—
274C22—41no—
275C22—42no—
276C22—42no—
277C22—41no—
278C22#37911yescharacteristic
279C22#44911yescharacteristic
280C22#45211yescharacteristic
281C4—21no—
282C4—21no—
283C4—21no—
284C4—22no—
285C4—41no—
286C4—41no—
287C4—41no—
288C4—41no—
289C4—42no—
290C4—41no—
291C4—41no—
292C4—82no—
293C4#41611yescharacteristic
294C4#41711yescharacteristic
295C4C2×#10411yescharacteristic
296C4#42111yescharacteristic
297C4#44911yescharacteristic
298C4#45211yescharacteristic
299C2—21no—
300C2—21no—
301C2—82no—
302C2—81no—
303C2#178811yescharacteristic
304C2#178911yescharacteristic
305C2#179411yescharacteristic
subgroupG/H
centerC2×C4#120
commutatorC42C22×C4
Frattini, cosocleC2×C42C23
socleC22#379
prime radical#3480C2
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #1806C2semidirect
3. C4×#103C2semidirect
4. #1502C2semidirect
5. #1492C2semidirect
6. #1487C2semidirect
7. #1476C2semidirect
9. #559C4semidirect
10. #557C4semidirect
12. #556C4semidirect
15. #550C4semidirect
218. Q8#104semidirect

Maximal subgroups

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

subgroupindex
#34802
#18062
C4×#1032
#15022
#14922
#14872
#14762