Finite Group Databasebeta

← #24736

Subgroups — #24736

Order 256.

462 conjugacy classes of subgroups.

The subgroup lattice has 345 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#464C211yesmaximal, characteristic
2C2×#464C211yesmaximal, characteristic
3C2×#462C211yesmaximal, characteristic, complemented
4C2×C4×Q16C211yesmaximal, characteristic, complemented
5#2545C211yesmaximal, characteristic
6#2351C211yesmaximal, characteristic, complemented
7#2068C211yesmaximal, characteristic, complemented
8#2067C211yesmaximal, characteristic
9#2043C211yesmaximal, characteristic, complemented
10#2042C211yesmaximal, characteristic
11#1958C211yesmaximal, characteristic
12#1939C211yesmaximal, characteristic, complemented
13#1880C211yesmaximal, characteristic
14#1878C211yesmaximal, characteristic, complemented
15#1805C211yesmaximal, characteristic, complemented
16C22×Q16C2211yescharacteristic
17C22×Q16C2211yescharacteristic
18C22×SD16C2211yescharacteristic
19C2×C4⋊Q8C2211yescharacteristic
20C2×#124C2211yescharacteristic
21C2×#124C2211yescharacteristic
22C2×C4×Q8C2211yescharacteristic
23C2×C4×Q8C2211yescharacteristic
24C2×C4×Q8C2211yescharacteristic
25C2×C4×Q8C2211yescharacteristic
26#464—24no—
27#464—24no—
28#462—24no—
29C4×Q16—24no—
30C2×#107C2211yescharacteristic
31C2×C4⋊C8C2211yescharacteristic
32C2×C4⋊C8C2211yescharacteristic
33C2×C4⋊C8C2211yescharacteristic
34C2×#103C2211yescharacteristic
35C2×#103C2211yescharacteristic
36C2×#103C2211yescharacteristic
37C2×#103C2211yescharacteristic
38C2×#103C2211yescharacteristic
39C2×#103C2211yescharacteristic
40C2×#103C2211yescharacteristic
41C2×#102C2211yescharacteristic
42C2×#102C2211yescharacteristic
43C2×#102C2211yescharacteristic
44C2×C4×C8C2211yescharacteristic
45#396—21no—
46#396—21no—
47#391C2211yescharacteristic
48#390C2211yescharacteristic
49#389—21no—
50#389—21no—
51#389C2211yescharacteristic
52#389C2211yescharacteristic
53#388—21no—
54#388—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×Q8C2311yescharacteristic
66C22×D4C2311yescharacteristic
67C2×Q16—22no—
68C2×Q16—22no—
69C2×Q16—22no—
70C2×Q16—22no—
71C2×Q16—22no—
72C2×Q16D414yes—
73C2×SD16—22no—
74C2×SD16—22no—
75C2×SD16—22no—
76C22×C8C2311yescharacteristic
77C22×C8C2311yescharacteristic
78C22×C8C2311yescharacteristic
79C22×C8C2311yescharacteristic
80C4⋊Q8—22no—
81C4⋊Q8—41no—
82#124—22no—
83#124—22no—
84#124—22no—
85#124—41no—
86C4×Q8—22no—
87C4×Q8—22no—
88C4×Q8—22no—
89C4×Q8—22no—
90C4×Q8—22no—
91C4×Q8—41no—
92C4×Q8—41no—
93C4×Q8D414yes—
94C2×C4⋊C4—21no—
95C2×C4⋊C4—21no—
96C2×C4⋊C4—21no—
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×C42—21no—
115C2×C42—21no—
116C2×C42—21no—
117C2×C42—21no—
118C2×C42C2311yescharacteristic
119C2×C42C2311yescharacteristic
120C2×C42C2311yescharacteristic
121#107—22no—
122C4⋊C8—22no—
123C4⋊C8—22no—
124C4⋊C8—22no—
125#103—22no—
126#103—22no—
127#103—22no—
128#103—22no—
129#103—22no—
130#103—22no—
131#103—41no—
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—
145C2×Q8—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×Q8—21no—
156C2×Q8—21no—
157C2×Q8—21no—
158C2×Q8—21no—
159C2×Q8C2×D412yes—
160C2×Q8C2×D412yes—
161C2×Q8C2×D412yes—
162C2×Q8#4112yes—
163C2×D4—21no—
164C2×D4—21no—
165C2×D4—21no—
166C2×D4—21no—
167C2×D4—21no—
168C2×D4—21no—
169C22×C4—21no—
170C22×C4—21no—
171C22×C4—21no—
172C22×C4—21no—
173C22×C4—21no—
174C22×C4—21no—
175C22×C4—21no—
176C22×C4C2×D411yescharacteristic
177C22×C4C2×D411yescharacteristic
178C22×C4C2×D411yescharacteristic
179C22×C4C2×D411yescharacteristic
180C22×C4C2×D411yescharacteristic
181C22×C4C2×D411yescharacteristic
182C22×C4C2411yescharacteristic
183Q16—28no—
184Q16—44no—
185SD16—44no—
186C2×C8—21no—
187C2×C8—21no—
188C2×C8—21no—
189C2×C8—21no—
190C2×C8—21no—
191C2×C8—21no—
192C2×C8—21no—
193C2×C8—21no—
194C2×C8—21no—
195C2×C8—21no—
196C2×C8—21no—
197C2×C8C2×D412yes—
198C4⋊C4—24no—
199C4⋊C4—22no—
200C4⋊C4—21no—
201C4⋊C4—21no—
202C4⋊C4—21no—
203C4⋊C4—21no—
204C4⋊C4—21no—
205C4⋊C4—21no—
206C4⋊C4—21no—
207C4⋊C4—41no—
208C4⋊C4—42no—
209C4⋊C4—42no—
210C4⋊C4—42no—
211C4⋊C4—42no—
212C4⋊C4—42no—
213C4⋊C4—42no—
214C4⋊C4—42no—
215C4⋊C4C2×D412yes—
216C22⋊C4—42no—
217C22⋊C4—42no—
218C22⋊C4—42no—
219C22⋊C4—42no—
220C22⋊C4—42no—
221C22⋊C4—42no—
222C42—24no—
223C42—21no—
224C42—21no—
225C42—21no—
226C42—21no—
227C42—42no—
228C42—42no—
229C42—42no—
230C42C2×D412yes—
231C42#4112yes—
232C23—41no—
233C23—41no—
234C23—41no—
235C23—41no—
236C23—41no—
237C23—41no—
238C23—41no—
239C23C22×D411yescharacteristic
240Q8—22no—
241Q8—22no—
242Q8—22no—
243Q8—22no—
244Q8—22no—
245Q8—41no—
246Q8—41no—
247Q8—41no—
248Q8—41no—
249Q8—41no—
250Q8—41no—
251Q8#12114yes—
252D4—41no—
253D4—41no—
254D4—41no—
255D4—41no—
256C2×C4—22no—
257C2×C4—21no—
258C2×C4—22no—
259C2×C4—22no—
260C2×C4—21no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—22no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—22no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—41no—
277C2×C4—41no—
278C2×C4—41no—
279C2×C4—41no—
280C2×C4—41no—
281C2×C4—41no—
282C2×C4—41no—
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#12112yes—
293C2×C4#12112yes—
294C2×C4#13512yes—
295C2×C4C22×D411yescharacteristic
296C2×C4C22×D411yescharacteristic
297C2×C4C2×#4111yescharacteristic
298C2×C4C2×#4111yescharacteristic
299C2×C4H(5,𝔽2)11yescharacteristic
300C2×C42−1+411yescharacteristic
301C8—22no—
302C8—41no—
303C8—41no—
304C8—41no—
305C22—81no—
306C22—81no—
307C22—81no—
308C22—81no—
309C22—81no—
310C22—81no—
311C22—81no—
312C22C2×#12111yescharacteristic
313C22D4211yescharacteristic
314C22#54811yescharacteristic
315C22C2×#13511yescharacteristic
316C22#57511yescharacteristic
317C22#57811yescharacteristic
318C22#57811yescharacteristic
319C4—24no—
320C4—22no—
321C4—22no—
322C4—22no—
323C4—42no—
324C4—41no—
325C4—41no—
326C4—41no—
327C4—42no—
328C4—81no—
329C4—81no—
330C4—81no—
331C4—81no—
332C4C2×#12111yescharacteristic
333C4#53711yescharacteristic
334C4#54711yescharacteristic
335C4#55311yescharacteristic
336C2—161no—
337C2#241611yescharacteristic
338C2#303511yescharacteristic
339C2#304411yescharacteristic
340C2#327711yescharacteristic
341C2#328511yescharacteristic
342C2#336411yescharacteristic
343C2#337711yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC22×D4C23
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
3. C2×#462C2semidirect
4. C2×C4×Q16C2semidirect
6. #2351C2semidirect
7. #2068C2semidirect
9. #2043C2semidirect
12. #1939C2semidirect
14. #1878C2semidirect
15. #1805C2semidirect

Maximal subgroups

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

subgroupindex
C2×#4642
C2×#4622
C2×C4×Q162
#25452
#23512
#20682
#20672
#20432
#20422
#19582
#19392
#18802
#18782
#18052