Finite Group Databasebeta

← #22988

Subgroups — #22988

Order 256.

525 conjugacy classes of subgroups.

The subgroup lattice has 386 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#436C211yesmaximal, characteristic, complemented
2C2×#435C211yesmaximal, characteristic, complemented
3C2×#435C211yesmaximal, characteristic, complemented
4C2×#423C211yesmaximal, characteristic, complemented
5C4×#121C211yesmaximal, characteristic
6#1955C211yesmaximal, characteristic, complemented
7#1940C211yesmaximal, characteristic, complemented
8#1924C211yesmaximal, characteristic, complemented
9#1923C211yesmaximal, characteristic, complemented
10#1870C211yesmaximal, characteristic, complemented
11#1870C211yesmaximal, characteristic, complemented
12#1851C211yesmaximal, characteristic, complemented
13#1850C211yesmaximal, characteristic, complemented
14#1771C211yesmaximal, characteristic, complemented
15#1749C211yesmaximal, characteristic, complemented
16C22×M16C2211yescharacteristic
17C22×M16C2211yescharacteristic
18C22×M16C2211yescharacteristic
19C23×C8C2211yescharacteristic
20C2×#121C411yescharacteristic
21C2×C4×D4C2211yescharacteristic
22C2×C4×D4C2211yescharacteristic
23C2×C4×D4C2211yescharacteristic
24C22×C42C2211yescharacteristic
25#436—24no—
26#435—24no—
27#435—24no—
28#423—24no—
29C2×C4⋊C8C2211yescharacteristic
30C2×C4⋊C8C2211yescharacteristic
31C2×C4⋊C8C2211yescharacteristic
32C2×C4⋊C8C2211yescharacteristic
33C2×C4⋊C8C2211yescharacteristic
34C2×C4⋊C8C2211yescharacteristic
35C2×C22⋊C8C2211yescharacteristic
36C2×C22⋊C8C2211yescharacteristic
37C2×C22⋊C8C2211yescharacteristic
38C2×C22⋊C8C2211yescharacteristic
39C2×C22⋊C8C2211yescharacteristic
40C2×C22⋊C8C2211yescharacteristic
41C2×C22⋊C8C2211yescharacteristic
42C2×C22⋊C8C2211yescharacteristic
43C2×#97C2211yescharacteristic
44C2×#97C2211yescharacteristic
45C2×#97C2211yescharacteristic
46C2×C4×C8C2211yescharacteristic
47#390C411yescharacteristic
48#389C411yescharacteristic
49#388C411yescharacteristic
50#388C411yescharacteristic
51#386C411yescharacteristic
52#386C411yescharacteristic
53#380C411yescharacteristic
54C4×C4⋊C4C2211yescharacteristic
55C4×C22⋊C4C2211yescharacteristic
56C4×C22⋊C4C2211yescharacteristic
57#336C2211yescharacteristic
58#336C2211yescharacteristic
59#336C2211yescharacteristic
60#336C2211yescharacteristic
61#336C2211yescharacteristic
62#336C2211yescharacteristic
63C22×D4C2×C411yescharacteristic
64C22×D4C2×C411yescharacteristic
65C22×D4C2×C411yescharacteristic
66C23×C4C2×C411yescharacteristic
67C23×C4C2×C411yescharacteristic
68C23×C4C2311yescharacteristic
69C23×C4C2311yescharacteristic
70C23×C4C2311yescharacteristic
71C2×M16—22no—
72C2×M16—22no—
73C2×M16—22no—
74C2×M16—22no—
75C2×M16—22no—
76C2×M16—22no—
77C2×M16—41no—
78C2×M16—41no—
79C2×M16—41no—
80C22×C8—22no—
81C22×C8—22no—
82C22×C8—41no—
83C22×C8C2311yescharacteristic
84C22×C8C2311yescharacteristic
85C22×C8C2311yescharacteristic
86C22×C8C2311yescharacteristic
87C22×C8C2311yescharacteristic
88C22×C8C2311yescharacteristic
89C22×C8C2311yescharacteristic
90C22×C8C2311yescharacteristic
91#121—24no—
92C4×D4—22no—
93C4×D4—22no—
94C4×D4—22no—
95C4×D4—22no—
96C4×D4—22no—
97C4×D4—22no—
98C2×C4⋊C4C2×C411yescharacteristic
99C2×C4⋊C4C2×C411yescharacteristic
100C2×C4⋊C4C2×C411yescharacteristic
101C2×C4⋊C4C2×C411yescharacteristic
102C2×C4⋊C4C2×C411yescharacteristic
103C2×C22⋊C4C2×C411yescharacteristic
104C2×C22⋊C4C2×C411yescharacteristic
105C2×C22⋊C4C2×C411yescharacteristic
106C2×C22⋊C4C2×C411yescharacteristic
107C2×C22⋊C4C2×C411yescharacteristic
108C2×C22⋊C4C2×C411yescharacteristic
109C2×C22⋊C4C2×C411yescharacteristic
110C2×C22⋊C4C2×C411yescharacteristic
111C2×C22⋊C4C2×C411yescharacteristic
112C2×C22⋊C4C2×C411yescharacteristic
113C2×C42—22no—
114C2×C42—22no—
115C2×C42C2×C411yescharacteristic
116C2×C42C2×C411yescharacteristic
117C2×C42C2311yescharacteristic
118C2×C42C2311yescharacteristic
119C2×C42C2311yescharacteristic
120C2×C42C2311yescharacteristic
121C4⋊C8—22no—
122C4⋊C8—22no—
123C4⋊C8—22no—
124C4⋊C8—22no—
125C4⋊C8—22no—
126C4⋊C8D414yescomplemented
127C22⋊C8—22no—
128C22⋊C8—22no—
129C22⋊C8—22no—
130C22⋊C8—22no—
131C22⋊C8—22no—
132C22⋊C8—41no—
133C22⋊C8—41no—
134C22⋊C8D414yescomplemented
135#97—22no—
136#97—22no—
137#97—22no—
138C4×C8—22no—
139#95C2×C411yescharacteristic
140#95C2×C411yescharacteristic
141#95C2×C411yescharacteristic
142#95C2×C411yescharacteristic
143#95C2×C411yescharacteristic
144#95C2×C411yescharacteristic
145C24C22×C411yescharacteristic
146C24C22×C411yescharacteristic
147C24C22×C411yescharacteristic
148C2×D4—22no—
149C2×D4—22no—
150C2×D4—22no—
151C2×D4—22no—
152C2×D4—22no—
153C2×D4—22no—
154C2×D4—41no—
155C2×D4—41no—
156C2×D4—41no—
157C22×C4—21no—
158C22×C4—21no—
159C22×C4—21no—
160C22×C4—21no—
161C22×C4—22no—
162C22×C4—22no—
163C22×C4—22no—
164C22×C4—21no—
165C22×C4—21no—
166C22×C4—21no—
167C22×C4—21no—
168C22×C4—21no—
169C22×C4—21no—
170C22×C4—21no—
171C22×C4—21no—
172C22×C4—21no—
173C22×C4—21no—
174C22×C4—21no—
175C22×C4—21no—
176C22×C4—41no—
177C22×C4—41no—
178C22×C4—41no—
179C22×C4C22×C411yescharacteristic
180C22×C4C22×C411yescharacteristic
181C22×C4C22×C411yescharacteristic
182C22×C4C22×C411yescharacteristic
183C22×C4C22×C411yescharacteristic
184C22×C4C22×C411yescharacteristic
185C22×C4C22×C411yescharacteristic
186C22×C4C22×C411yescharacteristic
187C22×C4C22×C411yescharacteristic
188C22×C4C22×C411yescharacteristic
189C22×C4C22×C411yescharacteristic
190C22×C4C2×D412yes—
191C22×C4#4112yes—
192C22×C4C2411yescharacteristic
193M16—44no—
194M16—44no—
195M16—44no—
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×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×C8—44no—
217C2×C8C2×D412yes—
218C2×C8C2×D412yes—
219C2×C8C2×D412yes—
220C2×C8C2×D412yes—
221C4⋊C4—22no—
222C4⋊C4—22no—
223C4⋊C4—22no—
224C4⋊C4—22no—
225C4⋊C4—41no—
226C22⋊C4—22no—
227C22⋊C4—22no—
228C22⋊C4—22no—
229C22⋊C4—22no—
230C22⋊C4—22no—
231C22⋊C4—22no—
232C22⋊C4—22no—
233C22⋊C4—41no—
234C22⋊C4—41no—
235C22⋊C4M1614yescomplemented
236C42—22no—
237C42—22no—
238C42—21no—
239C42—21no—
240C42—21no—
241C42—21no—
242C42—21no—
243C42—21no—
244C42C2×D412yes—
245C42#4112yes—
246C23—21no—
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—21no—
261C23—21no—
262C23—21no—
263C23—21no—
264C23—21no—
265C23C4×D412yes—
266C23C2×M1612yes—
267C23C23×C411yescharacteristic
268D4—44no—
269D4—44no—
270D4—44no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—22no—
282C2×C4—22no—
283C2×C4—22no—
284C2×C4—22no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—44no—
304C2×C4—44no—
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×C4C4×D412yes—
314C2×C4C4×D412yes—
315C2×C4C4×D412yes—
316C2×C4C2×M1612yes—
317C2×C4C2×M1612yes—
318C2×C4C22×D411yescharacteristic
319C2×C4C22×D411yescharacteristic
320C2×C4C2×#4111yescharacteristic
321C2×C4C2×#4111yescharacteristic
322C2×C4H(5,𝔽2)11yescharacteristic
323C2×C42−1+411yescharacteristic
324C8—22no—
325C8—22no—
326C8—22no—
327C8—22no—
328C8—41no—
329C8—41no—
330C8—41no—
331C8—41no—
332C22—22no—
333C22—22no—
334C22—22no—
335C22—22no—
336C22—22no—
337C22—22no—
338C22—41no—
339C22—41no—
340C22—41no—
341C22—41no—
342C22—41no—
343C22—41no—
344C22—41no—
345C22—41no—
346C22—41no—
347C22—41no—
348C22—41no—
349C22—41no—
350C22—41no—
351C22—41no—
352C22—41no—
353C22C2×C4×D411yescharacteristic
354C22C2×C4×D411yescharacteristic
355C22#52011yescharacteristic
356C22C22×M1611yescharacteristic
357C22#56811yescharacteristic
358C22#56811yescharacteristic
359C22#56811yescharacteristic
360C4—22no—
361C4—22no—
362C4—22no—
363C4—22no—
364C4—22no—
365C4—41no—
366C4—41no—
367C4—41no—
368C4—41no—
369C4—41no—
370C4—41no—
371C4D4211yescharacteristic
372C4#54711yescharacteristic
373C4#54811yescharacteristic
374C4#55311yescharacteristic
375C2—42no—
376C2—42no—
377C2—81no—
378C2#236111yescharacteristic
379C2#293011yescharacteristic
380C2#293211yescharacteristic
381C2D4×M1611yescharacteristic
382C2D4×M1611yescharacteristic
383C2#298911yescharacteristic
384C2#299511yescharacteristic
subgroupG/H
center, Frattini, cosocleC22×C4C24
commutator, socleC23C23×C4
prime radicalC2×#121C4
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C22×C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#436C2semidirect
2. C2×#435C2semidirect
3. C2×#435C2semidirect
4. C2×#423C2semidirect
6. #1955C2semidirect
7. #1940C2semidirect
8. #1924C2semidirect
9. #1923C2semidirect
10. #1870C2semidirect
11. #1870C2semidirect
12. #1851C2semidirect
13. #1850C2semidirect
14. #1771C2semidirect
15. #1749C2semidirect
126. C4⋊C8D4semidirect
134. C22⋊C8D4semidirect
235. C22⋊C4M16semidirect

Maximal subgroups

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

subgroupindex
C2×#4362
C2×#4352
C2×#4232
C4×#1212
#19552
#19402
#19242
#19232
#18702
#18512
#18502
#17712
#17492