Finite Group Databasebeta

← #22549

Subgroups — #22549

Order 256.

559 conjugacy classes of subgroups.

The subgroup lattice has 358 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#559C211yesmaximal, characteristic
2C2×#446C211yesmaximal, characteristic, complemented
3#2355C211yesmaximal, characteristic, complemented
4#1983C211yesmaximal, characteristic, complemented
5#1983C211yesmaximal, characteristic, complemented
6#1982C211yesmaximal, characteristic
7#1961C211yesmaximal, characteristic
8#1959C211yesmaximal, characteristic, complemented
9#1958C211yesmaximal, characteristic, complemented
10#1805C211yesmaximal, characteristic, complemented
11#1804C211yesmaximal, characteristic, complemented
12#1802C211yesmaximal, characteristic, complemented
13#1763C211yesmaximal, characteristic, complemented
14#1762C211yesmaximal, characteristic, complemented
15C4×#102C211yesmaximal, characteristic, complemented
16#559—24no—
17#559C418yescomplemented
18C2×#127C2211yescharacteristic
19C2×#124—21no—
20C2×#124C2211yescharacteristic
21C2×#124C2211yescharacteristic
22C2×#123—21no—
23C2×#123—21no—
24C2×C4×Q8C2211yescharacteristic
25C2×C4×Q8C2211yescharacteristic
26C2×C4×D4—21no—
27C2×C4×D4C2211yescharacteristic
28C2×C4×D4C2211yescharacteristic
29#446—24no—
30C2×C4⋊C8C2211yescharacteristic
31C2×C4⋊C8C2211yescharacteristic
32C2×C4⋊C8C2211yescharacteristic
33C2×#103C2211yescharacteristic
34C2×#103C2211yescharacteristic
35C2×#103C2211yescharacteristic
36C2×#103C2211yescharacteristic
37C2×#102C2211yescharacteristic
38C2×#102C2211yescharacteristic
39C2×#102C2211yescharacteristic
40C2×#102C2211yescharacteristic
41C2×#102C2211yescharacteristic
42C2×#102C2211yescharacteristic
43C2×#102C2211yescharacteristic
44C2×#102C2211yescharacteristic
45C2×#97C2211yescharacteristic
46C2×#97C2211yescharacteristic
47C2×C4×C8C2211yescharacteristic
48#391—21no—
49#387—21no—
50#387—21no—
51#384C2211yescharacteristic
52#382C2211yescharacteristic
53#382C2211yescharacteristic
54C4×C4⋊C4—21no—
55C4×C4⋊C4C2211yescharacteristic
56C4×C4⋊C4C2211yescharacteristic
57C4×C4⋊C4C2211yescharacteristic
58#340C2211yescharacteristic
59#340C2211yescharacteristic
60#340C2211yescharacteristic
61#340C2211yescharacteristic
62C22×Q8D411yescharacteristic
63C22×Q8C2311yescharacteristic
64C22×D4—21no—
65C22×D4—21no—
66C22×D4C2311yescharacteristic
67C22×D4C2311yescharacteristic
68C23×C4—21no—
69C23×C4—21no—
70C22×C8C2311yescharacteristic
71C22×C8C2311yescharacteristic
72C22×C8C2311yescharacteristic
73C22×C8C2311yescharacteristic
74#127—22no—
75#127C2×C414yes—
76#124—28no—
77#124—22no—
78#124—22no—
79#124C2×C414yes—
80#124C2×C414yes—
81#123—28no—
82#123—28no—
83C4×Q8—22no—
84C4×Q8C2×C414yes—
85C4×Q8C2×C414yes—
86C4×Q8D414yes—
87C4×D4—28no—
88C4×D4—22no—
89C4×D4—22no—
90C4×D4C2×C414yes—
91C4×D4C2×C414yes—
92C2×C4⋊C4—21no—
93C2×C4⋊C4—21no—
94C2×C4⋊C4—21no—
95C2×C4⋊C4D411yescharacteristic
96C2×C4⋊C4D411yescharacteristic
97C2×C4⋊C4D411yescharacteristic
98C2×C4⋊C4C2311yescharacteristic
99C2×C4⋊C4C2311yescharacteristic
100C2×C4⋊C4C2311yescharacteristic
101C2×C4⋊C4C2311yescharacteristic
102C2×C4⋊C4C2311yescharacteristic
103C2×C22⋊C4—21no—
104C2×C22⋊C4—21no—
105C2×C22⋊C4—21no—
106C2×C22⋊C4—21no—
107C2×C22⋊C4—21no—
108C2×C22⋊C4—21no—
109C2×C42—21no—
110C2×C42—21no—
111C2×C42—21no—
112C2×C42—21no—
113C2×C42C2311yescharacteristic
114C2×C42C2311yescharacteristic
115C2×C42C2311yescharacteristic
116C4⋊C8—22no—
117C4⋊C8—22no—
118C4⋊C8—22no—
119#103—22no—
120#103—22no—
121#103—41no—
122#103—41no—
123#102—22no—
124#102—22no—
125#102—22no—
126#102—22no—
127#102—41no—
128#102—41no—
129#102—41no—
130#102—41no—
131#97—22no—
132#97—22no—
133C4×C8—22no—
134#95—21no—
135#95—21no—
136#95—21no—
137#95—21no—
138#95—21no—
139#95—21no—
140#95—21no—
141C24—21no—
142C24—21no—
143C2×Q8—21no—
144C2×Q8—21no—
145C2×Q8—21no—
146C2×Q8—21no—
147C2×Q8—21no—
148C2×Q8—41no—
149C2×Q8—41no—
150C2×Q8C22⋊C414yes—
151C2×Q8C22×C412yes—
152C2×D4—24no—
153C2×D4—24no—
154C2×D4—21no—
155C2×D4—21no—
156C2×D4—21no—
157C2×D4—21no—
158C2×D4—21no—
159C2×D4—21no—
160C2×D4—21no—
161C2×D4—21no—
162C2×D4—21no—
163C2×D4—21no—
164C2×D4—42no—
165C2×D4—42no—
166C2×D4—81no—
167C2×D4—81no—
168C2×D4C22×C412yes—
169C2×D4C22×C412yes—
170C22×C4—21no—
171C22×C4—21no—
172C22×C4—21no—
173C22×C4—24no—
174C22×C4—24no—
175C22×C4—21no—
176C22×C4—21no—
177C22×C4—21no—
178C22×C4—81no—
179C22×C4—81no—
180C22×C4—81no—
181C22×C4—81no—
182C22×C4C2×D411yescharacteristic
183C22×C4C2×D411yescharacteristic
184C22×C4C2×D411yescharacteristic
185C22×C4C2×D411yescharacteristic
186C22×C4C2×D411yescharacteristic
187C22×C4C2×D411yescharacteristic
188C22×C4C2411yescharacteristic
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×C8—21no—
198C2×C8—21no—
199C2×C8—21no—
200C2×C8#4112yes—
201C4⋊C4—24no—
202C4⋊C4—21no—
203C4⋊C4—21no—
204C4⋊C4—21no—
205C4⋊C4—21no—
206C4⋊C4—21no—
207C4⋊C4—21no—
208C4⋊C4—21no—
209C4⋊C4—21no—
210C4⋊C4—42no—
211C4⋊C4—81no—
212C4⋊C4C22⋊C414yes—
213C4⋊C4C22⋊C414yes—
214C4⋊C4C22⋊C414yes—
215C4⋊C4C22×C412yes—
216C4⋊C4#4112yes—
217C22⋊C4—24no—
218C22⋊C4—24no—
219C22⋊C4—24no—
220C22⋊C4—24no—
221C22⋊C4—24no—
222C22⋊C4—24no—
223C42—24no—
224C42—42no—
225C42—81no—
226C42—81no—
227C42C22×C412yes—
228C42C22×C412yes—
229C42C22×C412yes—
230C42C2×D412yes—
231C42C2×D412yes—
232C42C2×D412yes—
233C23—22no—
234C23—22no—
235C23—41no—
236C23—41no—
237C23—41no—
238C23—41no—
239C23—41no—
240C23—41no—
241C23—41no—
242C23—41no—
243C23—41no—
244C23—41no—
245C23—41no—
246C23—41no—
247C23C22×D411yescharacteristic
248Q8—22no—
249Q8—22no—
250Q8—44no—
251Q8—41no—
252Q8—41no—
253D4—22no—
254D4—22no—
255D4—22no—
256D4—22no—
257D4—48no—
258D4—41no—
259D4—41no—
260D4—41no—
261D4—41no—
262D4—84no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—22no—
276C2×C4—22no—
277C2×C4—22no—
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—41no—
293C2×C4—84no—
294C2×C4—84no—
295C2×C4C2×C22⋊C412yes—
296C2×C4C2×C22⋊C412yes—
297C2×C4C2×C22⋊C412yes—
298C2×C4C2×C22⋊C412yes—
299C2×C4C2×C22⋊C412yes—
300C2×C4C2×C22⋊C412yes—
301C2×C4C23×C411yescharacteristic
302C2×C4C22×D411yescharacteristic
303C2×C4C2×#4111yescharacteristic
304C2×C4C2×#4111yescharacteristic
305C2×C4H(5,𝔽2)11yescharacteristic
306C2×C42−1+411yescharacteristic
307C8—22no—
308C8—41no—
309C8—41no—
310C8—41no—
311C22—42no—
312C22—42no—
313C22—42no—
314C22—42no—
315C22—42no—
316C22—42no—
317C22—81no—
318C22—81no—
319C22—81no—
320C22—81no—
321C22—81no—
322C22—81no—
323C22—81no—
324C22—81no—
325C22C22×C22⋊C411yescharacteristic
326C22#54811yescharacteristic
327C22#55011yescharacteristic
328C22#57511yescharacteristic
329C22#57511yescharacteristic
330C22#57611yescharacteristic
331C22#57711yescharacteristic
332C4—22no—
333C4—22no—
334C4—22no—
335C4—22no—
336C4—22no—
337C4—22no—
338C4—42no—
339C4—42no—
340C4—42no—
341C4—81no—
342C4—81no—
343C4—81no—
344C4C4×#4111yescharacteristic
345C4#52011yescharacteristic
346C4#54811yescharacteristic
347C4#55011yescharacteristic
348C2—82no—
349C2—82no—
350C2#234111yescharacteristic
351C2#289611yescharacteristic
352C2#289711yescharacteristic
353C2#337011yescharacteristic
354C2#337111yescharacteristic
355C2#340111yescharacteristic
356C2#340311yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutatorC2×C4C23×C4
Frattini, cosocleC22×C4C24
prime radicalC2×#127C22
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×Q8 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#446C2semidirect
3. #2355C2semidirect
4. #1983C2semidirect
5. #1983C2semidirect
8. #1959C2semidirect
9. #1958C2semidirect
10. #1805C2semidirect
11. #1804C2semidirect
12. #1802C2semidirect
13. #1763C2semidirect
14. #1762C2semidirect
15. C4×#102C2semidirect
17. #559C4semidirect

Maximal subgroups

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

subgroupindex
C2×#5592
C2×#4462
#23552
#19832
#19822
#19612
#19592
#19582
#18052
#18042
#18022
#17632
#17622
C4×#1022