Finite Group Databasebeta

← #24734

Subgroups — #24734

Order 256.

494 conjugacy classes of subgroups.

The subgroup lattice has 365 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#464C211yesmaximal, characteristic, complemented
2C2×#464C211yesmaximal, characteristic, complemented
3C2×#460C211yesmaximal, characteristic
4C2×C4×SD16C211yesmaximal, characteristic, complemented
5#2544C211yesmaximal, characteristic
6#2351C211yesmaximal, characteristic, complemented
7#2066C211yesmaximal, characteristic, complemented
8#2065C211yesmaximal, characteristic, complemented
9#2042C211yesmaximal, characteristic, complemented
10#2041C211yesmaximal, characteristic
11#1958C211yesmaximal, characteristic, complemented
12#1939C211yesmaximal, characteristic, complemented
13#1880C211yesmaximal, characteristic, complemented
14#1878C211yesmaximal, characteristic, complemented
15#1804C211yesmaximal, characteristic, complemented
16C22×Q16C2211yescharacteristic
17C22×SD16C2211yescharacteristic
18C22×SD16C2211yescharacteristic
19C2×#127C2211yescharacteristic
20C2×#124C2211yescharacteristic
21C2×#124C2211yescharacteristic
22C2×C4×Q8C2211yescharacteristic
23C2×C4×Q8C2211yescharacteristic
24C2×C4×Q8C2211yescharacteristic
25C2×C4×D4C2211yescharacteristic
26#464—24no—
27#464—24no—
28#460—24no—
29C4×SD16—24no—
30C2×#106C2211yescharacteristic
31C2×C4⋊C8C2211yescharacteristic
32C2×C4⋊C8C2211yescharacteristic
33C2×C4⋊C8C2211yescharacteristic
34C2×#103C2211yescharacteristic
35C2×#103C2211yescharacteristic
36C2×#103C2211yescharacteristic
37C2×#103C2211yescharacteristic
38C2×#102C2211yescharacteristic
39C2×#102C2211yescharacteristic
40C2×#102C2211yescharacteristic
41C2×#102C2211yescharacteristic
42C2×#102C2211yescharacteristic
43C2×#102C2211yescharacteristic
44C2×C4×C8C2211yescharacteristic
45#394—21no—
46#394—21no—
47#391C2211yescharacteristic
48#390C2211yescharacteristic
49#389—21no—
50#389—21no—
51#389C2211yescharacteristic
52#389C2211yescharacteristic
53#386—21no—
54#386—21no—
55#382—21no—
56#382C2211yescharacteristic
57C4×C4⋊C4—21no—
58C4×C4⋊C4C2211yescharacteristic
59#340C2211yescharacteristic
60#340C2211yescharacteristic
61#336C2211yescharacteristic
62#336C2211yescharacteristic
63C22×Q8C2311yescharacteristic
64C22×Q8C2311yescharacteristic
65C22×D4—21no—
66C22×D4—21no—
67C22×D4C2311yescharacteristic
68C22×D4C2311yescharacteristic
69C23×C4—21no—
70C2×Q16—22no—
71C2×Q16—22no—
72C2×Q16—22no—
73C2×SD16—22no—
74C2×SD16—22no—
75C2×SD16—22no—
76C2×SD16—22no—
77C2×SD16—22no—
78C2×SD16D414yes—
79C22×C8C2311yescharacteristic
80C22×C8C2311yescharacteristic
81C22×C8C2311yescharacteristic
82C22×C8C2311yescharacteristic
83#127—22no—
84#127—41no—
85#124—22no—
86#124—22no—
87#124—22no—
88#124—41no—
89C4×Q8—22no—
90C4×Q8—22no—
91C4×Q8—22no—
92C4×Q8—41no—
93C4×Q8—41no—
94C4×Q8D414yescomplemented
95C4×D4—22no—
96C4×D4—22no—
97C2×C4⋊C4—21no—
98C2×C4⋊C4—21no—
99C2×C4⋊C4—21no—
100C2×C4⋊C4—21no—
101C2×C4⋊C4—21no—
102C2×C4⋊C4D411yescharacteristic
103C2×C4⋊C4D411yescharacteristic
104C2×C4⋊C4C2311yescharacteristic
105C2×C4⋊C4C2311yescharacteristic
106C2×C4⋊C4C2311yescharacteristic
107C2×C4⋊C4C2311yescharacteristic
108C2×C22⋊C4—21no—
109C2×C22⋊C4—21no—
110C2×C22⋊C4—21no—
111C2×C22⋊C4—21no—
112C2×C22⋊C4—21no—
113C2×C22⋊C4—21no—
114C2×C22⋊C4—21no—
115C2×C42—21no—
116C2×C42—21no—
117C2×C42—21no—
118C2×C42C2311yescharacteristic
119C2×C42C2311yescharacteristic
120C2×C42C2311yescharacteristic
121#106—22no—
122C4⋊C8—22no—
123C4⋊C8—22no—
124C4⋊C8—22no—
125#103—22no—
126#103—22no—
127#103—22no—
128#103—41no—
129#102—22no—
130#102—22no—
131#102—22no—
132#102—22no—
133#102—22no—
134#102—41no—
135C4×C8—22no—
136#95—21no—
137#95—21no—
138#95—21no—
139#95—21no—
140#95—21no—
141#95—21no—
142#95D411yescharacteristic
143#95D411yescharacteristic
144C24—21no—
145C24—21no—
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×Q8C2×D412yes—
156C2×Q8C2×D412yes—
157C2×Q8#4112yes—
158C2×D4—21no—
159C2×D4—21no—
160C2×D4—21no—
161C2×D4—21no—
162C2×D4—21no—
163C2×D4—21no—
164C2×D4—21no—
165C2×D4—21no—
166C2×D4—21no—
167C2×D4—21no—
168C2×D4—21no—
169C2×D4—42no—
170C2×D4—42no—
171C2×D4—42no—
172C2×D4—42no—
173C2×D4—81no—
174C2×D4—81no—
175C2×D4C2×D412yes—
176C22×C4—21no—
177C22×C4—21no—
178C22×C4—21no—
179C22×C4—21no—
180C22×C4—21no—
181C22×C4—21no—
182C22×C4—42no—
183C22×C4—81no—
184C22×C4—81no—
185C22×C4C2×D411yescharacteristic
186C22×C4C2×D411yescharacteristic
187C22×C4C2×D411yescharacteristic
188C22×C4C2×D411yescharacteristic
189C22×C4C2×D411yescharacteristic
190C22×C4C2×D411yescharacteristic
191C22×C4C2411yescharacteristic
192Q16—44no—
193SD16—28no—
194SD16—44no—
195C2×C8—21no—
196C2×C8—21no—
197C2×C8—21no—
198C2×C8—21no—
199C2×C8—21no—
200C2×C8—21no—
201C2×C8—21no—
202C2×C8—21no—
203C2×C8—21no—
204C2×C8—21no—
205C2×C8—21no—
206C2×C8C2×D412yes—
207C4⋊C4—24no—
208C4⋊C4—22no—
209C4⋊C4—21no—
210C4⋊C4—21no—
211C4⋊C4—21no—
212C4⋊C4—21no—
213C4⋊C4—21no—
214C4⋊C4—21no—
215C4⋊C4—21no—
216C4⋊C4—41no—
217C4⋊C4—42no—
218C4⋊C4—42no—
219C4⋊C4—42no—
220C4⋊C4—42no—
221C4⋊C4C2×D412yes—
222C22⋊C4—42no—
223C22⋊C4—42no—
224C22⋊C4—42no—
225C22⋊C4—42no—
226C22⋊C4—42no—
227C22⋊C4—81no—
228C22⋊C4—81no—
229C42—24no—
230C42—21no—
231C42—21no—
232C42—21no—
233C42—21no—
234C42—42no—
235C42—42no—
236C42C2×D412yes—
237C42#4112yes—
238C23—22no—
239C23—41no—
240C23—41no—
241C23—41no—
242C23—41no—
243C23—41no—
244C23—41no—
245C23—41no—
246C23—41no—
247C23—41no—
248C23—41no—
249C23—41no—
250C23—41no—
251C23—41no—
252C23C22×D411yescharacteristic
253Q8—22no—
254Q8—22no—
255Q8—22no—
256Q8—41no—
257Q8—41no—
258Q8—41no—
259Q8—41no—
260Q8#12114yes—
261D4—22no—
262D4—22no—
263D4—41no—
264D4—41no—
265D4—41no—
266D4—41no—
267D4—41no—
268D4—41no—
269D4—84no—
270D4—84no—
271C2×C4—22no—
272C2×C4—21no—
273C2×C4—22no—
274C2×C4—22no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—22no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—41no—
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—84no—
305C2×C4#12112yes—
306C2×C4#12112yes—
307C2×C4#13512yes—
308C2×C4C22×D411yescharacteristic
309C2×C4C22×D411yescharacteristic
310C2×C4C2×#4111yescharacteristic
311C2×C4C2×#4111yescharacteristic
312C2×C4H(5,𝔽2)11yescharacteristic
313C2×C42−1+411yescharacteristic
314C8—22no—
315C8—41no—
316C8—41no—
317C8—41no—
318C22—42no—
319C22—42no—
320C22—42no—
321C22—81no—
322C22—81no—
323C22—81no—
324C22—81no—
325C22—81no—
326C22—81no—
327C22—81no—
328C22—81no—
329C22—81no—
330C22—81no—
331C22—81no—
332C22C2×#12111yescharacteristic
333C22D4211yescharacteristic
334C22#54811yescharacteristic
335C22C2×#13511yescharacteristic
336C22#57511yescharacteristic
337C22#57711yescharacteristic
338C22#57711yescharacteristic
339C4—24no—
340C4—22no—
341C4—22no—
342C4—22no—
343C4—41no—
344C4—41no—
345C4—41no—
346C4—42no—
347C4—81no—
348C4—81no—
349C4—81no—
350C4—81no—
351C4C2×#12111yescharacteristic
352C4#53711yescharacteristic
353C4#54711yescharacteristic
354C4#55311yescharacteristic
355C2—82no—
356C2—161no—
357C2#241611yescharacteristic
358C2#303511yescharacteristic
359C2#304311yescharacteristic
360C2#327511yescharacteristic
361C2#328211yescharacteristic
362C2#336511yescharacteristic
363C2#337111yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC2×#127C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#464C2semidirect
2. C2×#464C2semidirect
4. C2×C4×SD16C2semidirect
6. #2351C2semidirect
7. #2066C2semidirect
8. #2065C2semidirect
9. #2042C2semidirect
11. #1958C2semidirect
12. #1939C2semidirect
13. #1880C2semidirect
14. #1878C2semidirect
15. #1804C2semidirect
94. C4×Q8D4semidirect

Maximal subgroups

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

subgroupindex
C2×#4642
C2×#4602
C2×C4×SD162
#25442
#23512
#20662
#20652
#20422
#20412
#19582
#19392
#18802
#18782
#18042