Finite Group Databasebeta

← #8438

Subgroups — #8438

Order 256.

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

normal subgroups only

This group has 396 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#3475C211yesmaximal, characteristic
2#1806C211yesmaximal, characteristic, complemented
3C4×#102C211yesmaximal, characteristic, complemented
4#1503C211yesmaximal, characteristic, complemented
5#1493C211yesmaximal, characteristic, complemented
6#1486C211yesmaximal, characteristic, complemented
7#1476C211yesmaximal, characteristic, complemented
8#556C411yescharacteristic, complemented
9#555C411yescharacteristic, complemented
10#554C411yescharacteristic, complemented
11#553—21no—
12#552—21no—
13#552—21no—
14#551—22no—
15#547C411yescharacteristic, complemented
16#546—21no—
17#533C2211yescharacteristic
18#529—22no—
19#529—21no—
20#525—21no—
21C4×#41—21no—
22C4×#41C2211yescharacteristic
23C2×C4×D4C2211yescharacteristic
24C2×#117—21no—
25#431C2211yescharacteristic
26#423C2211yescharacteristic
27C2×#104—21no—
28C2×#102—21no—
29C2×C4×C8C2211yescharacteristic
30C4×C4⋊C4C2211yescharacteristic
31#340—21no—
32#339—21no—
33#328—22no—
34Q8⋊C8—21no—
35Q8⋊C8—22no—
36D4⋊C8—21no—
37D4⋊C8—21no—
38D4⋊C8—21no—
39C2×#41—21no—
40C2×#41D411yescharacteristic
41C22×D4D411yescharacteristic
42C23×C4—21no—
43C2×M16—21no—
44C22×C8—21no—
45C4⋊Q8C2×C411yescharacteristic
46C4⋊Q8C2×C411yescharacteristic
47#126—22no—
48#126—22no—
49#126—22no—
50#125—22no—
51#125—21no—
52#125—21no—
53#124—22no—
54#124—21no—
55#123—22no—
56#123—22no—
57#123—22no—
58#123—22no—
59C22⋊Q8—21no—
60C22⋊Q8—21no—
61C22⋊Q8—22no—
62C22⋊Q8—21no—
63C22⋊Q8—21no—
64C22⋊Q8D412yes—
65#121—22no—
66#121—21no—
67#121—21no—
68#121—21no—
69#120—22no—
70C4×Q8—21no—
71C4×Q8—21no—
72C4×Q8C2×C411yescharacteristic
73C4×Q8D411yescharacteristic
74C4×Q8D412yes—
75C4×D4—22no—
76C4×D4—22no—
77C4×D4—22no—
78C4×D4—21no—
79C4×D4—21no—
80C4×D4—21no—
81C4×D4—21no—
82C4×D4—21no—
83C4×D4C2×C411yescharacteristic
84C4×D4C2×C411yescharacteristic
85C4×D4C2×C411yescharacteristic
86C4×D4D411yescharacteristic
87C4×D4D411yescharacteristic
88C4×D4D411yescharacteristic
89#117—22no—
90#117—22no—
91#117—22no—
92#117—21no—
93#117—21no—
94#117—21no—
95#117—21no—
96#117—21no—
97#117D411yescharacteristic
98C2×C4⋊C4—21no—
99C2×C4⋊C4—21no—
100C2×C4⋊C4—21no—
101C2×C4⋊C4D411yescharacteristic
102C2×C22⋊C4—22no—
103C2×C22⋊C4—21no—
104C2×C42—22no—
105C2×C42—21no—
106C2×C42—21no—
107C2×C42C2311yescharacteristic
108C4⋊C8—21no—
109C4⋊C8—21no—
110C4⋊C8—21no—
111#104—42no—
112#102—24no—
113C22⋊C8—21no—
114C4×C8—21no—
115C4×C8—21no—
116C4×C8—21no—
117#95—21no—
118C24—21no—
119#41—21no—
120#41—21no—
121#41—21no—
122#41—21no—
123#41—42no—
124#41—42no—
125C2×Q8—21no—
126C2×Q8C22⋊C411yescharacteristic
127C2×Q8C22⋊C412yes—
128C2×D4—22no—
129C2×D4—22no—
130C2×D4—21no—
131C2×D4—22no—
132C2×D4—21no—
133C2×D4—41no—
134C2×D4C22⋊C411yescharacteristic
135C2×D4C22⋊C411yescharacteristic
136C2×D4C22⋊C411yescharacteristic
137C22×C4—22no—
138C22×C4—21no—
139C22×C4—22no—
140C22×C4—21no—
141C22×C4—21no—
142C22×C4—21no—
143C22×C4—21no—
144C22×C4—21no—
145C22×C4—21no—
146C22×C4—21no—
147C22×C4—41no—
148C22×C4—41no—
149C22×C4—41no—
150C22×C4—41no—
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—22no—
172C4⋊C4—21no—
173C4⋊C4—22no—
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—22no—
189C22⋊C4—21no—
190C22⋊C4—21no—
191C22⋊C4—21no—
192C22⋊C4—21no—
193C22⋊C4—21no—
194C22⋊C4—21no—
195C42—22no—
196C42—22no—
197C42—21no—
198C42—21no—
199C42—21no—
200C42—22no—
201C42—41no—
202C42—41no—
203C42C22×C411yescharacteristic
204C42C2×D411yescharacteristic
205C42C2×D411yescharacteristic
206C42C2×D411yescharacteristic
207C23—21no—
208C23—21no—
209C23—21no—
210C23—42no—
211C23—41no—
212C23—41no—
213C23—81no—
214C23#12011yescharacteristic
215Q8—21no—
216Q8—21no—
217Q8—42no—
218Q8—41no—
219Q8—41no—
220D4—21no—
221D4—21no—
222D4—21no—
223D4—21no—
224D4—21no—
225D4—42no—
226D4—44no—
227D4—41no—
228D4—41no—
229D4—82no—
230D4#10412yescomplemented
231C2×C4—21no—
232C2×C4—21no—
233C2×C4—21no—
234C2×C4—21no—
235C2×C4—21no—
236C2×C4—22no—
237C2×C4—21no—
238C2×C4—22no—
239C2×C4—21no—
240C2×C4—41no—
241C2×C4—44no—
242C2×C4—41no—
243C2×C4—41no—
244C2×C4—41no—
245C2×C4—41no—
246C2×C4—41no—
247C2×C4—41no—
248C2×C4—41no—
249C2×C4—41no—
250C2×C4—41no—
251C2×C4—41no—
252C2×C4—41no—
253C2×C4—41no—
254C2×C4—41no—
255C2×C4—41no—
256C2×C4—42no—
257C2×C4—81no—
258C2×C4—82no—
259C2×C4—81no—
260C2×C4—81no—
261C2×C4C2×C22⋊C411yescharacteristic
262C2×C4C2×C22⋊C411yescharacteristic
263C2×C4C2×C22⋊C411yescharacteristic
264C2×C4#12011yescharacteristic
265C2×C4#12011yescharacteristic
266C2×C4#12011yescharacteristic
267C2×C4#13511yescharacteristic
268C2×C4#13511yescharacteristic
269C2×C4AGL(1,ℤ/8)11yescharacteristic
270C2×C4#13711yescharacteristic
271C8—42no—
272C8—81no—
273C8—81no—
274C22—21no—
275C22—21no—
276C22—22no—
277C22—41no—
278C22—41no—
279C22—41no—
280C22—41no—
281C22—41no—
282C22—41no—
283C22—41no—
284C22—41no—
285C22—82no—
286C22—82no—
287C22#37911yescharacteristic
288C22#44911yescharacteristic
289C22#45211yescharacteristic
290C4—21no—
291C4—21no—
292C4—21no—
293C4—41no—
294C4—42no—
295C4—41no—
296C4—41no—
297C4—41no—
298C4—41no—
299C4—42no—
300C4—82no—
301C4#41611yescharacteristic
302C4#41711yescharacteristic
303C4C2×#10411yescharacteristic
304C4#42111yescharacteristic
305C4#44911yescharacteristic
306C4#45211yescharacteristic
307C2—21no—
308C2—21no—
309C2—42no—
310C2—81no—
311C2—81no—
312C2#178811yescharacteristic
313C2#178911yescharacteristic
314C2#179411yescharacteristic
subgroupG/H
centerC2×C4#120
commutatorC42C22×C4
Frattini, cosocleC2×C42C23
socleC22#379
prime radical#3475C2
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×#102C2semidirect
4. #1503C2semidirect
5. #1493C2semidirect
6. #1486C2semidirect
7. #1476C2semidirect
8. #556C4semidirect
9. #555C4semidirect
10. #554C4semidirect
15. #547C4semidirect
230. D4#104semidirect

Maximal subgroups

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

subgroupindex
#34752
#18062
C4×#1022
#15032
#14932
#14862
#14762