Finite Group Databasebeta

← #8436

Subgroups — #8436

Order 256.

525 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 525 conjugacy classes of subgroups — too many to draw. Only its 91 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#1806C211yesmaximal, characteristic, complemented
3C4×#102C211yesmaximal, characteristic, complemented
4#1502C211yesmaximal, characteristic, complemented
5#1492C211yesmaximal, characteristic, complemented
6#1486C211yesmaximal, characteristic, complemented
7#1476C211yesmaximal, characteristic, complemented
8C22×#41—21no—
9#550C411yescharacteristic, complemented
10D4×Q8C411yescharacteristic, complemented
11#548—21no—
12#547—21no—
13#547C411yescharacteristic, complemented
14#546—22no—
15#546—21no—
16D42—21no—
17D42C411yescharacteristic, complemented
18#532—21no—
19#532C2211yescharacteristic
20#525—22no—
21#525—21no—
22C4×#41C2211yescharacteristic
23C2×C4×D4—21no—
24C2×C4×D4C2211yescharacteristic
25#431C2211yescharacteristic, complemented
26#423C2211yescharacteristic, complemented
27C2×#104—21no—
28C2×#102—21no—
29C2×C4×C8C2211yescharacteristic, complemented
30C4×C4⋊C4C2211yescharacteristic, complemented
31#340—21no—
32#339—21no—
33#327—22no—
34Q8⋊C8—21no—
35D4⋊C8—21no—
36D4⋊C8—21no—
37D4⋊C8—21no—
38D4⋊C8—22no—
39C2×#41—22no—
40C2×#41—22no—
41C2×#41—21no—
42C2×#41—21no—
43C2×#41—21no—
44C2×#41—42no—
45C2×#41—42no—
46C2×#41—42no—
47C2×#41—42no—
48C2×#41D411yescharacteristic
49C22×Q8—21no—
50C22×D4—22no—
51C22×D4—21no—
52C22×D4—21no—
53C22×D4D411yescharacteristic
54C23×C4—22no—
55C23×C4—21no—
56C2×M16—21no—
57C22×C8—21no—
58C4⋊Q8—21no—
59C4⋊Q8C2×C411yescharacteristic
60#127—21no—
61#127C2×C411yescharacteristic
62#124—22no—
63#124—21no—
64#123—22no—
65#123—22no—
66#123—22no—
67C22⋊Q8—22no—
68C22⋊Q8—21no—
69C22⋊Q8—22no—
70C22⋊Q8—21no—
71#121—22no—
72#121—22no—
73#121—22no—
74#121—21no—
75#121—21no—
76#121—21no—
77#121—21no—
78#121—21no—
79#121D412yes—
80#120—22no—
81#120—22no—
82#120—22no—
83C4×Q8C2×C411yescharacteristic
84C4×Q8D411yescharacteristic
85C4×D4—22no—
86C4×D4—22no—
87C4×D4—22no—
88C4×D4—21no—
89C4×D4—21no—
90C4×D4—21no—
91C4×D4—21no—
92C4×D4—21no—
93C4×D4—21no—
94C4×D4—21no—
95C4×D4C2×C411yescharacteristic
96C4×D4C2×C411yescharacteristic
97C4×D4C2×C411yescharacteristic
98C4×D4D411yescharacteristic
99C4×D4D411yescharacteristic
100C4×D4D411yescharacteristic
101C4×D4D412yes—
102#117—21no—
103#117D411yescharacteristic
104C2×C4⋊C4—21no—
105C2×C4⋊C4—21no—
106C2×C4⋊C4D411yescharacteristic
107C2×C22⋊C4—22no—
108C2×C22⋊C4—21no—
109C2×C42—21no—
110C2×C42—21no—
111C2×C42C2311yescharacteristic
112C4⋊C8—21no—
113C4⋊C8—21no—
114C4⋊C8—21no—
115#104—42no—
116#102—24no—
117C22⋊C8—21no—
118C4×C8—21no—
119C4×C8—21no—
120C4×C8—21no—
121#95—21no—
122C24—22no—
123C24—21no—
124#41—21no—
125#41—21no—
126#41—44no—
127#41—44no—
128#41—44no—
129#41—44no—
130#41—42no—
131#41—42no—
132#41—42no—
133#41—42no—
134#41—41no—
135#41—41no—
136#41—41no—
137#41—41no—
138#41D812yes—
139#41SD1612yes—
140C2×Q8—22no—
141C2×Q8—21no—
142C2×Q8—42no—
143C2×Q8—42no—
144C2×Q8C22⋊C411yescharacteristic
145C2×D4—21no—
146C2×D4—22no—
147C2×D4—22no—
148C2×D4—22no—
149C2×D4—22no—
150C2×D4—22no—
151C2×D4—22no—
152C2×D4—22no—
153C2×D4—21no—
154C2×D4—21no—
155C2×D4—21no—
156C2×D4—41no—
157C2×D4—42no—
158C2×D4—44no—
159C2×D4—44no—
160C2×D4—42no—
161C2×D4—42no—
162C2×D4—42no—
163C2×D4C22⋊C411yescharacteristic
164C2×D4C22⋊C411yescharacteristic
165C2×D4C22⋊C411yescharacteristic
166C2×D4C22⋊C412yes—
167C22×C4—21no—
168C22×C4—22no—
169C22×C4—22no—
170C22×C4—22no—
171C22×C4—21no—
172C22×C4—21no—
173C22×C4—21no—
174C22×C4—21no—
175C22×C4—21no—
176C22×C4—22no—
177C22×C4—21no—
178C22×C4—21no—
179C22×C4—44no—
180C22×C4—44no—
181C22×C4—41no—
182C22×C4—41no—
183C22×C4—41no—
184C22×C4—41no—
185C22×C4—41no—
186C22×C4C2×D411yescharacteristic
187C22×C4C2×D411yescharacteristic
188C22×C4C2×D411yescharacteristic
189M16—41no—
190M16—41no—
191C2×C8—22no—
192C2×C8—41no—
193C2×C8—41no—
194C2×C8—41no—
195C2×C8—41no—
196C4⋊C4—22no—
197C4⋊C4—22no—
198C4⋊C4—21no—
199C4⋊C4—21no—
200C4⋊C4—21no—
201C4⋊C4—21no—
202C4⋊C4—22no—
203C4⋊C4—41no—
204C4⋊C4C22⋊C411yescharacteristic
205C4⋊C4C22⋊C411yescharacteristic
206C4⋊C4C22⋊C411yescharacteristic
207C4⋊C4C22⋊C411yescharacteristic
208C4⋊C4C22⋊C412yes—
209C22⋊C4—22no—
210C22⋊C4—22no—
211C22⋊C4—22no—
212C22⋊C4—22no—
213C22⋊C4—21no—
214C22⋊C4—21no—
215C22⋊C4—21no—
216C22⋊C4—21no—
217C22⋊C4—21no—
218C22⋊C4—22no—
219C22⋊C4—21no—
220C22⋊C4—21no—
221C22⋊C4—22no—
222C42—21no—
223C42—21no—
224C42—21no—
225C42—41no—
226C42—41no—
227C42C22×C411yescharacteristic
228C42C2×D411yescharacteristic
229C42C2×D411yescharacteristic
230C42C2×D411yescharacteristic
231C23—21no—
232C23—21no—
233C23—22no—
234C23—21no—
235C23—42no—
236C23—44no—
237C23—44no—
238C23—41no—
239C23—41no—
240C23—81no—
241C23—82no—
242C23#12011yescharacteristic
243Q8—21no—
244Q8—44no—
245Q8—41no—
246Q8—41no—
247Q8#10212yescomplemented
248D4—22no—
249D4—22no—
250D4—21no—
251D4—21no—
252D4—21no—
253D4—21no—
254D4—42no—
255D4—42no—
256D4—44no—
257D4—44no—
258D4—44no—
259D4—44no—
260D4—42no—
261D4—42no—
262D4—41no—
263D4—41no—
264D4—41no—
265D4—41no—
266D4—82no—
267D4#10212yescomplemented
268D4#10412yescomplemented
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—22no—
273C2×C4—22no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—22no—
278C2×C4—21no—
279C2×C4—41no—
280C2×C4—44no—
281C2×C4—44no—
282C2×C4—44no—
283C2×C4—41no—
284C2×C4—41no—
285C2×C4—41no—
286C2×C4—41no—
287C2×C4—41no—
288C2×C4—41no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—42no—
293C2×C4—41no—
294C2×C4—41no—
295C2×C4—42no—
296C2×C4—41no—
297C2×C4—42no—
298C2×C4—81no—
299C2×C4—81no—
300C2×C4—81no—
301C2×C4C2×C22⋊C411yescharacteristic
302C2×C4C2×C22⋊C411yescharacteristic
303C2×C4C2×C22⋊C411yescharacteristic
304C2×C4#12011yescharacteristic
305C2×C4#12011yescharacteristic
306C2×C4#12011yescharacteristic
307C2×C4C2×D811yescharacteristic
308C2×C4C2×SD1611yescharacteristic
309C2×C4AGL(1,ℤ/8)11yescharacteristic
310C2×C4AGL(1,ℤ/8)11yescharacteristic
311C8—42no—
312C8—81no—
313C8—81no—
314C22—21no—
315C22—21no—
316C22—22no—
317C22—22no—
318C22—41no—
319C22—41no—
320C22—41no—
321C22—41no—
322C22—42no—
323C22—41no—
324C22—42no—
325C22—41no—
326C22—41no—
327C22—42no—
328C22—82no—
329C22—82no—
330C22—88no—
331C22#37911yescharacteristic
332C22#44711yescharacteristic
333C22#45011yescharacteristic
334C4—21no—
335C4—21no—
336C4—22no—
337C4—21no—
338C4—41no—
339C4—41no—
340C4—42no—
341C4—41no—
342C4—41no—
343C4—82no—
344C4C2×#10211yescharacteristic
345C4#41811yescharacteristic
346C4C2×#10411yescharacteristic
347C4#42111yescharacteristic
348C4#44711yescharacteristic
349C4#45011yescharacteristic
350C2—21no—
351C2—21no—
352C2—42no—
353C2—42no—
354C2—82no—
355C2—81no—
356C2#178611yescharacteristic
357C2#178911yescharacteristic
358C2#179411yescharacteristic
subgroupG/H
centerC2×C4#120
commutatorC42C22×C4
Frattini, cosocleC2×C42C23
socleC22#379
prime radicalD4×#41C2
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. #1502C2semidirect
5. #1492C2semidirect
6. #1486C2semidirect
7. #1476C2semidirect
9. #550C4semidirect
10. D4×Q8C4semidirect
13. #547C4semidirect
17. D42C4semidirect
25. #431C22semidirect
26. #423C22semidirect
29. C2×C4×C8C22semidirect
30. C4×C4⋊C4C22semidirect
247. Q8#102semidirect
267. D4#102semidirect
268. D4#104semidirect

Maximal subgroups

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

subgroupindex
D4×#412
#18062
C4×#1022
#15022
#14922
#14862
#14762