Finite Group Databasebeta

← #30423

Subgroups — #30423

Order 256.

484 conjugacy classes of subgroups.

The subgroup lattice has 378 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#470C211yesmaximal, characteristic, complemented
2C2×#451C211yesmaximal, characteristic, complemented
3#2538C211yesmaximal, characteristic
4#2409C211yesmaximal, characteristic
5#2096C211yesmaximal, characteristic, complemented
6#2077C211yesmaximal, characteristic, complemented
7#2072C211yesmaximal, characteristic, complemented
8#2035C211yesmaximal, characteristic, complemented
9#2035C211yesmaximal, characteristic, complemented
10#2001C211yesmaximal, characteristic, complemented
11#1894C211yesmaximal, characteristic, complemented
12#1878C211yesmaximal, characteristic, complemented
13#1878C211yesmaximal, characteristic, complemented
14#1870C211yesmaximal, characteristic
15#1787C211yesmaximal, characteristic, complemented
16C23×Q8C2211yescharacteristic
17C22×Q16C2211yescharacteristic
18C22×Q16C2211yescharacteristic
19C22×M16C2211yescharacteristic
20C2×C22⋊Q8C2211yescharacteristic
21C2×C22⋊Q8C2211yescharacteristic
22C2×C22⋊Q8C2211yescharacteristic
23C2×C22⋊Q8C2211yescharacteristic
24C2×C4×Q8C2211yescharacteristic
25C2×C4×D4C2211yescharacteristic
26#470—24no—
27#451—24no—
28C2×#106C2211yescharacteristic
29C2×#106C2211yescharacteristic
30C2×#103C2211yescharacteristic
31C2×#103C2211yescharacteristic
32C2×#103C2211yescharacteristic
33C2×#103C2211yescharacteristic
34C2×#103C2211yescharacteristic
35C2×#103C2211yescharacteristic
36C2×#103C2211yescharacteristic
37C2×#103C2211yescharacteristic
38C2×C22⋊C8C2211yescharacteristic
39C2×C22⋊C8C2211yescharacteristic
40C2×C22⋊C8C2211yescharacteristic
41#396—21no—
42#396—21no—
43#395—21no—
44#393—21no—
45#391C2211yescharacteristic
46#391C2211yescharacteristic
47#390—21no—
48#389—21no—
49#386—21no—
50#385—21no—
51#384C2211yescharacteristic
52#380C2211yescharacteristic
53#380C2211yescharacteristic
54#380C2211yescharacteristic
55#340C2211yescharacteristic
56#340C2211yescharacteristic
57#340C2211yescharacteristic
58#340C2211yescharacteristic
59#336C2211yescharacteristic
60#336C2211yescharacteristic
61C22×Q8—21no—
62C22×Q8—22no—
63C22×Q8—22no—
64C22×Q8—41no—
65C22×Q8—41no—
66C22×Q8—41no—
67C22×Q8—41no—
68C22×Q8C2311yescharacteristic
69C22×Q8C2311yescharacteristic
70C22×Q8C2311yescharacteristic
71C22×D4D411yescharacteristic
72C23×C4—21no—
73C23×C4C2311yescharacteristic
74C23×C4C2311yescharacteristic
75C2×Q16—22no—
76C2×Q16—22no—
77C2×Q16—41no—
78C2×Q16—41no—
79C2×Q16—41no—
80C2×Q16D414yescomplemented
81C2×M16—22no—
82C2×M16—41no—
83C2×M16—41no—
84C22×C8C2311yescharacteristic
85C22×C8C2311yescharacteristic
86C22×C8C2311yescharacteristic
87C22×C8C2311yescharacteristic
88C22⋊Q8—22no—
89C22⋊Q8—22no—
90C22⋊Q8—22no—
91C22⋊Q8—22no—
92C22⋊Q8—41no—
93C22⋊Q8—41no—
94C22⋊Q8—41no—
95C22⋊Q8—41no—
96C4×Q8—41no—
97C4×Q8—41no—
98C4×D4—41no—
99C4×D4—41no—
100C2×C4⋊C4—21no—
101C2×C4⋊C4—21no—
102C2×C4⋊C4—21no—
103C2×C4⋊C4—21no—
104C2×C4⋊C4—21no—
105C2×C4⋊C4—21no—
106C2×C4⋊C4—21no—
107C2×C4⋊C4D411yescharacteristic
108C2×C4⋊C4C2311yescharacteristic
109C2×C4⋊C4C2311yescharacteristic
110C2×C4⋊C4C2311yescharacteristic
111C2×C4⋊C4C2311yescharacteristic
112C2×C4⋊C4C2311yescharacteristic
113C2×C22⋊C4—21no—
114C2×C22⋊C4—21no—
115C2×C22⋊C4—21no—
116C2×C22⋊C4—21no—
117C2×C22⋊C4—21no—
118C2×C22⋊C4—21no—
119C2×C22⋊C4—21no—
120C2×C22⋊C4D411yescharacteristic
121C2×C22⋊C4D411yescharacteristic
122C2×C42—21no—
123C2×C42C2311yescharacteristic
124#106—22no—
125#106—41no—
126#103—22no—
127#103—22no—
128#103—22no—
129#103—22no—
130#103—41no—
131#103—41no—
132#103—41no—
133#103—41no—
134C22⋊C8—22no—
135C22⋊C8—41no—
136C22⋊C8—41no—
137#95—21no—
138#95—21no—
139#95—21no—
140#95—21no—
141#95—21no—
142#95—21no—
143#95—21no—
144C24C2×D411yescharacteristic
145C24C2×D411yescharacteristic
146C2×Q8—21no—
147C2×Q8—21no—
148C2×Q8—21no—
149C2×Q8—21no—
150C2×Q8—21no—
151C2×Q8—21no—
152C2×Q8—21no—
153C2×Q8—21no—
154C2×Q8—21no—
155C2×Q8—21no—
156C2×Q8—21no—
157C2×Q8—21no—
158C2×Q8—21no—
159C2×Q8—21no—
160C2×Q8—21no—
161C2×Q8—42no—
162C2×Q8—42no—
163C2×Q8—42no—
164C2×Q8—44no—
165C2×Q8—44no—
166C2×Q8—44no—
167C2×Q8—44no—
168C2×Q8C2×D412yes—
169C2×Q8C2×D412yes—
170C2×Q8#4112yes—
171C2×D4—22no—
172C2×D4—22no—
173C2×D4—41no—
174C22×C4—21no—
175C22×C4—21no—
176C22×C4—21no—
177C22×C4—21no—
178C22×C4—21no—
179C22×C4—21no—
180C22×C4—21no—
181C22×C4—21no—
182C22×C4—21no—
183C22×C4—21no—
184C22×C4—21no—
185C22×C4—21no—
186C22×C4—21no—
187C22×C4—21no—
188C22×C4—21no—
189C22×C4—21no—
190C22×C4—21no—
191C22×C4—21no—
192C22×C4—21no—
193C22×C4—21no—
194C22×C4—42no—
195C22×C4—42no—
196C22×C4—42no—
197C22×C4C2×D411yescharacteristic
198C22×C4C2×D411yescharacteristic
199C22×C4C2×D411yescharacteristic
200C22×C4C2×D411yescharacteristic
201C22×C4C2411yescharacteristic
202Q16—44no—
203Q16—44no—
204M16—44no—
205C2×C8—21no—
206C2×C8—21no—
207C2×C8—21no—
208C2×C8—21no—
209C2×C8—21no—
210C2×C8—21no—
211C2×C8—21no—
212C2×C8—21no—
213C2×C8—21no—
214C2×C8—21no—
215C2×C8—21no—
216C2×C8C2×D412yes—
217C4⋊C4—21no—
218C4⋊C4—21no—
219C4⋊C4—21no—
220C4⋊C4—21no—
221C4⋊C4—21no—
222C4⋊C4—21no—
223C4⋊C4—21no—
224C4⋊C4—21no—
225C4⋊C4—21no—
226C4⋊C4—21no—
227C4⋊C4—41no—
228C4⋊C4—42no—
229C4⋊C4—42no—
230C4⋊C4—42no—
231C4⋊C4—42no—
232C4⋊C4—42no—
233C4⋊C4—42no—
234C4⋊C4—81no—
235C22⋊C4—41no—
236C22⋊C4—41no—
237C22⋊C4—42no—
238C22⋊C4—42no—
239C22⋊C4—42no—
240C22⋊C4—42no—
241C22⋊C4—42no—
242C22⋊C4—81no—
243C22⋊C4—81no—
244C42—21no—
245C42—21no—
246C42—42no—
247C23—21no—
248C23—21no—
249C23—21no—
250C23—21no—
251C23—21no—
252C23—21no—
253C23—21no—
254C23—21no—
255C23—21no—
256C23—21no—
257C23—21no—
258C23—21no—
259C23—21no—
260C23#13712yes—
261C23C22×D411yescharacteristic
262Q8—22no—
263Q8—22no—
264Q8—22no—
265Q8—22no—
266Q8—22no—
267Q8—22no—
268Q8—48no—
269Q8—41no—
270Q8—41no—
271Q8—41no—
272Q8—41no—
273Q8—41no—
274Q8—41no—
275D4—44no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—22no—
282C2×C4—22no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—22no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—48no—
291C2×C4—41no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—41no—
295C2×C4—41no—
296C2×C4—41no—
297C2×C4—41no—
298C2×C4—41no—
299C2×C4—41no—
300C2×C4—41no—
301C2×C4—41no—
302C2×C4—41no—
303C2×C4—41no—
304C2×C4—41no—
305C2×C4—41no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4#13712yes—
321C2×C4C22×D411yescharacteristic
322C2×C4C2×#4111yescharacteristic
323C2×C4H(5,𝔽2)11yescharacteristic
324C2×C4H(5,𝔽2)11yescharacteristic
325C2×C4H(5,𝔽2)11yescharacteristic
326C2×C4H(5,𝔽2)11yescharacteristic
327C8—22no—
328C8—41no—
329C8—41no—
330C8—41no—
331C22—22no—
332C22—22no—
333C22—22no—
334C22—41no—
335C22—41no—
336C22—41no—
337C22—41no—
338C22—41no—
339C22—41no—
340C22—41no—
341C22—41no—
342C22—41no—
343C22—41no—
344C22—41no—
345C22#53411yescharacteristic
346C22D4211yescharacteristic
347C22#54611yescharacteristic
348C22C2×#13711yescharacteristic
349C22C2×#13711yescharacteristic
350C22#57711yescharacteristic
351C22#57811yescharacteristic
352C4—22no—
353C4—41no—
354C4—41no—
355C4—42no—
356C4—42no—
357C4—42no—
358C4—41no—
359C4—81no—
360C4—81no—
361C4—81no—
362C4—81no—
363C4—81no—
364C4#53411yescharacteristic
365C4#53811yescharacteristic
366C4#54611yescharacteristic
367C4#56011yescharacteristic
368C2—42no—
369C2—81no—
370C2#266811yescharacteristic
371C2#319511yescharacteristic
372C2#320111yescharacteristic
373C2#327011yescharacteristic
374C2#327811yescharacteristic
375C2#331611yescharacteristic
376C2#332811yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC22×D4D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#470C2semidirect
2. C2×#451C2semidirect
5. #2096C2semidirect
6. #2077C2semidirect
7. #2072C2semidirect
8. #2035C2semidirect
9. #2035C2semidirect
10. #2001C2semidirect
11. #1894C2semidirect
12. #1878C2semidirect
13. #1878C2semidirect
15. #1787C2semidirect
80. C2×Q16D4semidirect

Maximal subgroups

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

subgroupindex
C2×#4702
C2×#4512
#25382
#24092
#20962
#20772
#20722
#20352
#20012
#18942
#18782
#18702
#17872