Finite Group Databasebeta

← C22×#1294 #181493

Subgroups — C22×#1294 #181493

Order 512.

5130 conjugacy classes of subgroups.

The subgroup lattice has 387 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×C22⋊C8C211yesmaximal, characteristic
2C22×#380C211yesmaximal, characteristic
3C23×#95C211yesmaximal, characteristic
4C2×#1294C2112yesmaximal, direct product factor
5C25×C4—21no—
6C25×C4C2211yescharacteristic
7C24×C8—21no—
8C23×C4⋊C4C412yescomplemented
9C23×C22⋊C4—21no—
10C22×C22⋊C8—24no—
11C22×C22⋊C8—23no—
12C22×C22⋊C8C418yescomplemented
13C22×C22⋊C8C2216yes—
14C2×#380—23no—
15C2×#380—23no—
16C2×#380C416yescomplemented
17C2×#380C2216yes—
18C22×#95—24no—
19C22×#95—23no—
20C22×#95—21no—
21C22×#95C418yescomplemented
22C22×#95C2216yes—
23#1294C22116yesdirect product factor
24C26D411yescharacteristic
25C24×C4—216no—
26C24×C4—28no—
27C24×C4—26no—
28C24×C4—22no—
29C24×C4—48no—
30C24×C4—48no—
31C24×C4—44no—
32C24×C4—44no—
33C24×C4C2×C411yescharacteristic
34C24×C4C2×C414yes—
35C24×C4C2×C414yes—
36C24×C4C2×C413yes—
37C24×C4D411yescharacteristic
38C24×C4D413yes—
39C24×C4D414yes—
40C24×C4D413yes—
41C24×C4Q814yes—
42C24×C4C2313yes—
43C23×C8—28no—
44C23×C8—26no—
45C23×C8—48no—
46C22×C4⋊C4—26no—
47C22×C4⋊C4—48no—
48C22×C4⋊C4C2×C4112yescomplemented
49C22×C22⋊C4—26no—
50C22×C22⋊C4—26no—
51C22×C22⋊C4—44no—
52C22×C22⋊C4—44no—
53C2×C22⋊C8—224no—
54C2×C22⋊C8—26no—
55C2×C22⋊C8—212no—
56C2×C22⋊C8C2×C4124yescomplemented
57C2×C22⋊C8C2314yes—
58#380—212no—
59#380—26no—
60#380—26no—
61#380C2×C4112yescomplemented
62#380C2314yes—
63C2×#95—224no—
64C2×#95—26no—
65C2×#95—26no—
66C2×#95—212no—
67C2×#95—26no—
68C2×#95C2×C4124yescomplemented
69C2×#95C2314yes—
70C25—28no—
71C25—28no—
72C25C22⋊C413yes—
73C25C22⋊C414yes—
74C25C22⋊C411yescharacteristic
75C25C4⋊C414yes—
76C25D814yes—
77C25SD1614yes—
78C25SD1614yes—
79C25Q1614yes—
80C25C2×D413yes—
81C23×C4—28no—
82C23×C4—28no—
83C23×C4—264no—
84C23×C4—248no—
85C23×C4—224no—
86C23×C4—24no—
87C23×C4—212no—
88C23×C4—416no—
89C23×C4—416no—
90C23×C4—412no—
91C23×C4—416no—
92C23×C4—412no—
93C23×C4—412no—
94C23×C4—416no—
95C23×C4—412no—
96C23×C4—464no—
97C23×C4—464no—
98C23×C4—432no—
99C23×C4—424no—
100C23×C4—432no—
101C23×C4—412no—
102C23×C4—412no—
103C23×C4—48no—
104C23×C4—48no—
105C23×C4—48no—
106C23×C4C4214yes—
107C23×C4C42112yes—
108C23×C4C22⋊C413yes—
109C23×C4C22⋊C413yes—
110C23×C4C22⋊C414yes—
111C23×C4C22⋊C4112yes—
112C23×C4C22⋊C416yes—
113C23×C4C22⋊C4112yes—
114C23×C4C4⋊C414yes—
115C23×C4C4⋊C414yes—
116C23×C4C4⋊C4112yes—
117C23×C4C4⋊C4112yes—
118C23×C4C4⋊C4112yes—
119C23×C4C22×C413yes—
120C23×C4C22×C416yes—
121C23×C4C22×C416yes—
122C23×C4C22×C413yes—
123C23×C4C2×D413yes—
124C23×C4C2×D413yes—
125C23×C4C2×D416yes—
126C23×C4C2×D413yes—
127C23×C4C2×Q816yes—
128C23×C4C2411yescharacteristic
129C22×C8—224no—
130C22×C8—24no—
131C22×C8—432no—
132C22×C8—424no—
133C2×C4⋊C4—212no—
134C2×C4⋊C4—424no—
135C2×C4⋊C4—424no—
136C2×C4⋊C4C22×C418yescomplemented
137C2×C22⋊C4—212no—
138C2×C22⋊C4—24no—
139C2×C22⋊C4—424no—
140C2×C22⋊C4—412no—
141C2×C22⋊C4—412no—
142C22⋊C8—224no—
143C22⋊C8—24no—
144C22⋊C8C22×C418yescomplemented
145#95—224no—
146#95—212no—
147#95—24no—
148#95—24no—
149#95C22×C418yescomplemented
150C24—232no—
151C24—232no—
152C24—232no—
153C24—232no—
154C24—216no—
155C24—216no—
156C24—216no—
157C24—216no—
158C24—212no—
159C24—212no—
160C24—212no—
161C24—212no—
162C24#95112yes—
163C24#9514yes—
164C24#9918yes—
165C24#102112yes—
166C24#102116yes—
167C24#10214yes—
168C24#103116yes—
169C24#103112yes—
170C24#10314yes—
171C24#10418yes—
172C24#106116yes—
173C24#107116yes—
174C24C2×C22⋊C413yes—
175C24C2×C22⋊C416yes—
176C24C2×C22⋊C413yes—
177C24C2×C4⋊C416yes—
178C24C2×D816yes—
179C24C2×SD1616yes—
180C24C2×SD1616yes—
181C24C2×Q1616yes—
182C24C22×D411yescharacteristic
183C22×C4—216no—
184C22×C4—216no—
185C22×C4—216no—
186C22×C4—216no—
187C22×C4—212no—
188C22×C4—212no—
189C22×C4—212no—
190C22×C4—212no—
191C22×C4—216no—
192C22×C4—296no—
193C22×C4—28no—
194C22×C4—28no—
195C22×C4—424no—
196C22×C4—412no—
197C22×C4—424no—
198C22×C4—424no—
199C22×C4—424no—
200C22×C4—424no—
201C22×C4—424no—
202C22×C4—46no—
203C22×C4—412no—
204C22×C4—412no—
205C22×C4—46no—
206C22×C4—46no—
207C22×C4—412no—
208C22×C4—412no—
209C22×C4—42no—
210C22×C4—4256no—
211C22×C4—48no—
212C22×C4—448no—
213C22×C4—496no—
214C22×C4—496no—
215C22×C4—44no—
216C22×C4—44no—
217C22×C4—448no—
218C22×C4—424no—
219C22×C4—424no—
220C22×C4—424no—
221C22×C4—832no—
222C22×C4#95112yes—
223C22×C4#95112yes—
224C22×C4#95124yes—
225C22×C4C2×C4216yes—
226C22×C4C2×C4216yes—
227C22×C4C2×C22⋊C416yes—
228C22×C4C2×C22⋊C413yes—
229C22×C4C2×C22⋊C413yes—
230C22×C4C2×C22⋊C416yes—
231C22×C4C2×C22⋊C416yes—
232C22×C4C2×C4⋊C416yes—
233C22×C4C2×C4⋊C416yes—
234C22×C4C2×C4⋊C416yes—
235C22×C4C2×C4⋊C416yes—
236C22×C4C2×C4⋊C416yes—
237C22×C4C23×C411yescharacteristic
238C22×C4C23×C411yescharacteristic
239C22×C4C23×C411yescharacteristic
240C22×C4C22×D411yescharacteristic
241C22×C4C22×D411yescharacteristic
242C22×C4C22×Q811yescharacteristic
243C2×C8—28no—
244C2×C8—48no—
245C2×C8—448no—
246C4⋊C4—48no—
247C4⋊C4—424no—
248C22⋊C4—44no—
249C22⋊C4—44no—
250C22⋊C4—424no—
251C23—264no—
252C23—264no—
253C23—264no—
254C23—264no—
255C23—22no—
256C23—26no—
257C23—22no—
258C23—26no—
259C23—212no—
260C23—224no—
261C23—212no—
262C23—212no—
263C23—224no—
264C23—212no—
265C23—224no—
266C23—248no—
267C23—224no—
268C23—224no—
269C23—248no—
270C23—224no—
271C23#327132yes—
272C23#328132yes—
273C23#339116yes—
274C23#340148yes—
275C23#340116yes—
276C23#342116yes—
277C23C2×#9516yes—
278C23C2×#9516yes—
279C23C2×#99112yes—
280C23C2×#10216yes—
281C23C2×#102112yes—
282C23C2×#10216yes—
283C23C2×#103112yes—
284C23C2×#10316yes—
285C23C2×#10316yes—
286C23C2×#104112yes—
287C23C2×#106112yes—
288C23C2×#107112yes—
289C23C22×C22⋊C411yescharacteristic
290C23C22×C22⋊C411yescharacteristic
291C23C22×C4⋊C411yescharacteristic
292C23C22×D811yescharacteristic
293C23C22×SD1611yescharacteristic
294C23C22×SD1611yescharacteristic
295C23C22×Q1611yescharacteristic
296C2×C4—26no—
297C2×C4—22no—
298C2×C4—22no—
299C2×C4—26no—
300C2×C4—212no—
301C2×C4—224no—
302C2×C4—224no—
303C2×C4—212no—
304C2×C4—212no—
305C2×C4—212no—
306C2×C4—216no—
307C2×C4—424no—
308C2×C4—46no—
309C2×C4—46no—
310C2×C4—46no—
311C2×C4—46no—
312C2×C4—46no—
313C2×C4—46no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—4192no—
319C2×C4—416no—
320C2×C4—416no—
321C2×C4—48no—
322C2×C4—48no—
323C2×C4—48no—
324C2×C4—48no—
325C2×C4—48no—
326C2×C4—848no—
327C2×C4C2×#9516yes—
328C2×C4C2×#9516yes—
329C2×C4C22×C4211yescharacteristic
330C2×C4C22×C22⋊C411yescharacteristic
331C2×C4C22×C4⋊C411yescharacteristic
332C2×C4C22×C4⋊C411yescharacteristic
333C8—48no—
334C22—224no—
335C22—296no—
336C22—224no—
337C22—224no—
338C22—224no—
339C22—21no—
340C22—21no—
341C22—26no—
342C22—22no—
343C22—22no—
344C22—212no—
345C22—22no—
346C22—22no—
347C22—212no—
348C22—21no—
349C22—21no—
350C22—26no—
351C22#1294164yesdirect product factor
352C22C2×#327124yes—
353C22C2×#328124yes—
354C22C2×#339112yes—
355C22C2×#340112yes—
356C22C2×#340112yes—
357C22C2×#342112yes—
358C22C22×#9511yescharacteristic
359C22C22×#9912yes—
360C22C22×#10211yescharacteristic
361C22C22×#10211yescharacteristic
362C22C22×#10311yescharacteristic
363C22C22×#10311yescharacteristic
364C22C22×#10412yes—
365C22C22×#10611yescharacteristic
366C22C22×#10711yescharacteristic
367C4—21no—
368C4—21no—
369C4—21no—
370C4—26no—
371C4—26no—
372C4—21no—
373C4—416no—
374C4—88no—
375C2—21no—
376C2—21no—
377C2—21no—
378C2—21no—
379C2—212no—
380C2C2×#1294124yesdirect product factor
381C2C22×#32712yes—
382C2C22×#32812yes—
383C2C22×#33911yescharacteristic
384C2C22×#34011yescharacteristic
385C2C22×#34211yescharacteristic
subgroupG/H
center, socleC25C22⋊C4
commutatorC2×C4C22×C42
Frattini, cosocleC23×C4C24
prime radicalC26D4
LCSG ▷ C2×C4 ▷ C22 ▷ C1
UCSC1 ◁ C25 ◁ C25×C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. C2×#1294C2direct (N × H)
8. C23×C4⋊C4C4semidirect
12. C22×C22⋊C8C4semidirect
16. C2×#380C4semidirect
21. C22×#95C4semidirect
23. #1294C22direct (N × H)
48. C22×C4⋊C4C2×C4semidirect
56. C2×C22⋊C8C2×C4semidirect
61. #380C2×C4semidirect
68. C2×#95C2×C4semidirect
136. C2×C4⋊C4C22×C4semidirect
144. C22⋊C8C22×C4semidirect
149. #95C22×C4semidirect
351. C22#1294direct (N × H)
380. C2C2×#1294direct (N × H)

Maximal subgroups

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

subgroupindex
C23×C22⋊C82
C22×#3802
C23×#952
C2×#12942