Finite Group Databasebeta

← #22493

Subgroups — #22493

Order 256.

629 conjugacy classes of subgroups.

The subgroup lattice has 382 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#553C211yesmaximal, characteristic
2C2×#435C211yesmaximal, characteristic, complemented
3D4×C4⋊C4C211yesmaximal, characteristic, complemented
4#1992C211yesmaximal, characteristic, complemented
5#1989C211yesmaximal, characteristic, complemented
6#1895C211yesmaximal, characteristic, complemented
7#1893C211yesmaximal, characteristic, complemented
8#1874C211yesmaximal, characteristic, complemented
9#1874C211yesmaximal, characteristic, complemented
10#1825C211yesmaximal, characteristic, complemented
11#1824C211yesmaximal, characteristic, complemented
12#1802C211yesmaximal, characteristic, complemented
13#1762C211yesmaximal, characteristic, complemented
14#1736C211yesmaximal, characteristic, complemented
15C2×#340C211yesmaximal, characteristic, complemented
16C22×M16C2211yescharacteristic
17C23×C8C2211yescharacteristic
18#553—24no—
19#553C418yescomplemented
20C2×#126—21no—
21C2×#125C2211yescharacteristic
22C2×#123—21no—
23C2×#121—21no—
24C2×#121C2211yescharacteristic
25C2×#121C2211yescharacteristic
26C2×C4×D4—21no—
27C2×C4×D4—21no—
28C2×C4×D4C2211yescharacteristic
29C2×C4×D4C2211yescharacteristic
30C2×#117C2211yescharacteristic
31C22×C4⋊C4C2211yescharacteristic
32C22×C4⋊C4C2211yescharacteristic
33C22×C4⋊C4C2211yescharacteristic
34#435—24no—
35C2×C4⋊C8C2211yescharacteristic
36C2×#102C2211yescharacteristic
37C2×#102C2211yescharacteristic
38C2×#102C2211yescharacteristic
39C2×#102C2211yescharacteristic
40C2×C22⋊C8C2211yescharacteristic
41C2×C22⋊C8C2211yescharacteristic
42C2×#97C2211yescharacteristic
43#389—21no—
44#385—21no—
45#385—21no—
46#384C2211yescharacteristic
47#380C2211yescharacteristic
48#380C2211yescharacteristic
49C4×C4⋊C4C2211yescharacteristic
50#340C2211yescharacteristic
51#340C2211yescharacteristic
52#340C2211yescharacteristic
53#340C2211yescharacteristic
54#340C2211yescharacteristic
55#340C2211yescharacteristic
56#340C2211yescharacteristic
57#340C2211yescharacteristic
58#340C2211yescharacteristic
59#340C2211yescharacteristic
60#340C2211yescharacteristic
61#340C2211yescharacteristic
62C22×D4—21no—
63C22×D4—21no—
64C22×D4D411yescharacteristic
65C22×D4C2311yescharacteristic
66C23×C4—21no—
67C23×C4—21no—
68C23×C4—21no—
69C23×C4—21no—
70C23×C4C2311yescharacteristic
71C23×C4C2311yescharacteristic
72C2×M16—22no—
73C2×M16—22no—
74C2×M16—41no—
75C22×C8—22no—
76C22×C8—22no—
77C22×C8—41no—
78C22×C8C2311yescharacteristic
79C22×C8C2311yescharacteristic
80C22×C8C2311yescharacteristic
81C22×C8C2311yescharacteristic
82#126—28no—
83#125—22no—
84#125C2×C414yescomplemented
85#123—28no—
86#121—28no—
87#121—22no—
88#121—22no—
89#121C2×C414yescomplemented
90#121C2×C414yescomplemented
91C4×D4—28no—
92C4×D4—22no—
93C4×D4—44no—
94C4×D4C2×C414yes—
95C4×D4C2×C414yescomplemented
96C4×D4D414yes—
97#117—22no—
98#117C2×C414yescomplemented
99C2×C4⋊C4—21no—
100C2×C4⋊C4—22no—
101C2×C4⋊C4—21no—
102C2×C4⋊C4—21no—
103C2×C4⋊C4—22no—
104C2×C4⋊C4—22no—
105C2×C4⋊C4—21no—
106C2×C4⋊C4—21no—
107C2×C4⋊C4—21no—
108C2×C4⋊C4—21no—
109C2×C4⋊C4—41no—
110C2×C4⋊C4—41no—
111C2×C4⋊C4C2×C414yescomplemented
112C2×C4⋊C4D411yescharacteristic
113C2×C4⋊C4C2311yescharacteristic
114C2×C4⋊C4C2311yescharacteristic
115C2×C4⋊C4C2311yescharacteristic
116C2×C4⋊C4C2311yescharacteristic
117C2×C4⋊C4C2311yescharacteristic
118C2×C4⋊C4C2311yescharacteristic
119C2×C4⋊C4C2311yescharacteristic
120C2×C22⋊C4—21no—
121C2×C22⋊C4—21no—
122C2×C22⋊C4—21no—
123C2×C22⋊C4—21no—
124C2×C22⋊C4—21no—
125C2×C22⋊C4—21no—
126C2×C22⋊C4D411yescharacteristic
127C2×C22⋊C4D411yescharacteristic
128C2×C42—21no—
129C2×C42—21no—
130C2×C42C2311yescharacteristic
131C4⋊C8—22no—
132#102—22no—
133#102—22no—
134#102—41no—
135#102—41no—
136C22⋊C8—22no—
137C22⋊C8—22no—
138#97—22no—
139#95—21no—
140#95—21no—
141#95—21no—
142C24—21no—
143C24C2×D411yescharacteristic
144C24C2×D411yescharacteristic
145C2×D4—22no—
146C2×D4—22no—
147C2×D4—24no—
148C2×D4—24no—
149C2×D4—21no—
150C2×D4—21no—
151C2×D4—21no—
152C2×D4—21no—
153C2×D4—21no—
154C2×D4—42no—
155C2×D4—42no—
156C2×D4—81no—
157C2×D4—81no—
158C2×D4C22⋊C414yes—
159C2×D4C22×C412yes—
160C22×C4—21no—
161C22×C4—21no—
162C22×C4—21no—
163C22×C4—21no—
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—24no—
173C22×C4—24no—
174C22×C4—21no—
175C22×C4—21no—
176C22×C4—21no—
177C22×C4—42no—
178C22×C4—42no—
179C22×C4—42no—
180C22×C4—42no—
181C22×C4—81no—
182C22×C4—81no—
183C22×C4—81no—
184C22×C4—81no—
185C22×C4—81no—
186C22×C4—81no—
187C22×C4C22×C412yes—
188C22×C4C22×C412yes—
189C22×C4C2×D411yescharacteristic
190C22×C4C2×D411yescharacteristic
191C22×C4C2×D411yescharacteristic
192C22×C4C2×D411yescharacteristic
193C22×C4C2×D412yes—
194C22×C4C2×D412yes—
195C22×C4C2411yescharacteristic
196M16—44no—
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—44no—
209C2×C8#4112yes—
210C4⋊C4—24no—
211C4⋊C4—24no—
212C4⋊C4—24no—
213C4⋊C4—21no—
214C4⋊C4—21no—
215C4⋊C4—21no—
216C4⋊C4—21no—
217C4⋊C4—21no—
218C4⋊C4—21no—
219C4⋊C4—21no—
220C4⋊C4—21no—
221C4⋊C4—21no—
222C4⋊C4—21no—
223C4⋊C4—42no—
224C4⋊C4—42no—
225C4⋊C4—42no—
226C4⋊C4—81no—
227C4⋊C4C22⋊C414yes—
228C4⋊C4C22×C412yes—
229C4⋊C4C22×C412yes—
230C4⋊C4C22×C412yes—
231C4⋊C4#4112yes—
232C22⋊C4—24no—
233C22⋊C4—24no—
234C22⋊C4—24no—
235C22⋊C4—24no—
236C22⋊C4—42no—
237C22⋊C4—42no—
238C22⋊C4C22⋊C414yes—
239C22⋊C4C22⋊C414yes—
240C42—24no—
241C42—42no—
242C42C22×C412yes—
243C42C2×D412yes—
244C23—21no—
245C23—21no—
246C23—21no—
247C23—21no—
248C23—21no—
249C23—21no—
250C23—21no—
251C23—21no—
252C23—21no—
253C23—21no—
254C23—22no—
255C23—41no—
256C23—41no—
257C23—41no—
258C23—41no—
259C23—41no—
260C23—41no—
261C23C2×C22⋊C412yes—
262C23C2×C22⋊C412yes—
263C23AGL(1,ℤ/8)12yes—
264C23#13712yes—
265C23C22×D411yescharacteristic
266D4—28no—
267D4—22no—
268D4—22no—
269D4—48no—
270D4—41no—
271D4—41no—
272D4—84no—
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—22no—
286C2×C4—22no—
287C2×C4—22no—
288C2×C4—22no—
289C2×C4—22no—
290C2×C4—22no—
291C2×C4—22no—
292C2×C4—22no—
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—84no—
311C2×C4—84no—
312C2×C4—84no—
313C2×C4—84no—
314C2×C4C2×C22⋊C412yes—
315C2×C4C2×C22⋊C412yes—
316C2×C4C2×C22⋊C412yes—
317C2×C4C2×C22⋊C412yes—
318C2×C4C23×C411yescharacteristic
319C2×C4C22×D411yescharacteristic
320C2×C4C2×#4111yescharacteristic
321C2×C4C2×#4111yescharacteristic
322C2×C4H(5,𝔽2)11yescharacteristic
323C2×C42−1+411yescharacteristic
324C8—22no—
325C8—41no—
326C8—41no—
327C8—41no—
328C22—22no—
329C22—22no—
330C22—22no—
331C22—22no—
332C22—22no—
333C22—22no—
334C22—22no—
335C22—22no—
336C22—22no—
337C22—41no—
338C22—41no—
339C22—41no—
340C22—41no—
341C22—42no—
342C22—42no—
343C22—42no—
344C22—81no—
345C22—81no—
346C22—81no—
347C22—81no—
348C22#41714yes—
349C22C22×C22⋊C411yescharacteristic
350C22#54611yescharacteristic
351C22#54711yescharacteristic
352C22C2×AGL(1,ℤ/8)11yescharacteristic
353C22C2×#13711yescharacteristic
354C22#57611yescharacteristic
355C22#57711yescharacteristic
356C4—22no—
357C4—22no—
358C4—22no—
359C4—22no—
360C4—42no—
361C4—42no—
362C4—42no—
363C4—42no—
364C4—81no—
365C4—81no—
366C4—81no—
367C4C4×#4111yescharacteristic
368C4#52011yescharacteristic
369C4#54611yescharacteristic
370C4#54711yescharacteristic
371C2—24no—
372C2—42no—
373C2—82no—
374C2#233911yescharacteristic
375C2C2×#41711yescharacteristic
376C2#289711yescharacteristic
377C2#331811yescharacteristic
378C2#332011yescharacteristic
379C2#334511yescharacteristic
380C2#334611yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutatorC2×C4C23×C4
Frattini, cosocleC22×C4C24
prime radicalC2×#553C2
LCSG ▷ C2×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
2. C2×#435C2semidirect
3. D4×C4⋊C4C2semidirect
4. #1992C2semidirect
5. #1989C2semidirect
6. #1895C2semidirect
7. #1893C2semidirect
8. #1874C2semidirect
9. #1874C2semidirect
10. #1825C2semidirect
11. #1824C2semidirect
12. #1802C2semidirect
13. #1762C2semidirect
14. #1736C2semidirect
15. C2×#340C2semidirect
19. #553C4semidirect
84. #125C2×C4semidirect
89. #121C2×C4semidirect
90. #121C2×C4semidirect
95. C4×D4C2×C4semidirect
98. #117C2×C4semidirect
111. C2×C4⋊C4C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#5532
C2×#4352
D4×C4⋊C42
#19922
#19892
#18952
#18932
#18742
#18252
#18242
#18022
#17622
#17362
C2×#3402