Finite Group Databasebeta

← C22×#1518 #6343341

Subgroups — C22×#1518 #6343341

Order 512.

3562 conjugacy classes of subgroups.

The subgroup lattice has 338 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#533C211yesmaximal, characteristic
2C22×#431C211yesmaximal, characteristic
3C22×#382C211yesmaximal, characteristic
4C2×#1518C2124yesmaximal, direct product factor
5C22×#328C214yesmaximal
6C2×#533—26no—
7C2×#533—26no—
8C2×#533C4112yescomplemented
9C2×#533C22112yes—
10C22×C4⋊Q8C411yescharacteristic, complemented
11C22×C4⋊Q8C411yescharacteristic, complemented
12C22×#125—21no—
13C22×C22⋊Q8—21no—
14C22×C22⋊Q8C2212yes—
15C22×C4×Q8—21no—
16C22×C4×Q8C412yescomplemented
17C22×#117—21no—
18C23×C4⋊C4—21no—
19C23×C42C2211yescharacteristic
20C2×#431—26no—
21C2×#431C22112yes—
22C22×C4⋊C8—21no—
23C22×C22⋊C8C2212yes—
24C22×C4×C8—21no—
25C2×#382—26no—
26C2×#382C22112yes—
27C22×#95—21no—
28C22×#95C2212yes—
29#1518C22164yesdirect product factor
30C2×#328C22148yes—
31C23×Q8—21no—
32C23×Q8C2×C412yes—
33C24×C4—22no—
34C24×C4D411yescharacteristic
35C24×C4D411yescharacteristic
36C24×C4C2311yescharacteristic
37C23×C8—22no—
38#533—248no—
39#533—224no—
40#533—224no—
41#533C2×C4148yescomplemented
42#533C23116yes—
43C2×C4⋊Q8—26no—
44C2×C4⋊Q8—26no—
45C2×C4⋊Q8C2×C4112yescomplemented
46C2×C4⋊Q8C2×C4112yescomplemented
47C2×#125—212no—
48C2×#125—212no—
49C2×C22⋊Q8—212no—
50C2×C22⋊Q8—212no—
51C2×C22⋊Q8—26no—
52C2×C22⋊Q8—26no—
53C2×C22⋊Q8C2×C4112yes—
54C2×C22⋊Q8C23112yes—
55C2×C4×Q8—212no—
56C2×C4×Q8—212no—
57C2×C4×Q8—212no—
58C2×C4×Q8C2×C4124yescomplemented
59C2×#117—212no—
60C2×#117—212no—
61C22×C4⋊C4—21no—
62C22×C4⋊C4—21no—
63C22×C4⋊C4—21no—
64C22×C4⋊C4—22no—
65C22×C4⋊C4—22no—
66C22×C4⋊C4—212no—
67C22×C4⋊C4—48no—
68C22×C4⋊C4C2×C412yes—
69C22×C22⋊C4—22no—
70C22×C42—22no—
71C22×C42C2×C411yescharacteristic
72C22×C42C2×C411yescharacteristic
73C22×C42C2×C416yes—
74C22×C42D416yes—
75C22×C42D411yescharacteristic
76C22×C42D411yescharacteristic
77C22×C42D416yes—
78C22×C42C2316yes—
79#431—224no—
80#431C23116yes—
81C2×C4⋊C8—212no—
82C2×C22⋊C8—26no—
83C2×C22⋊C8C23112yes—
84C2×C4×C8—212no—
85#382—224no—
86#382C23116yes—
87C2×#95—212no—
88C2×#95—26no—
89C2×#95C23112yes—
90#328C23164yes—
91C25C2×D411yescharacteristic
92C22×Q8—26no—
93C22×Q8—212no—
94C22×Q8—48no—
95C22×Q8—48no—
96C22×Q8C22×C4112yes—
97C23×C4—22no—
98C23×C4—22no—
99C23×C4—212no—
100C23×C4—44no—
101C23×C4—44no—
102C23×C4—48no—
103C23×C4—48no—
104C23×C4—48no—
105C23×C4C22⋊C411yescharacteristic
106C23×C4C22⋊C411yescharacteristic
107C23×C4C22⋊C416yes—
108C23×C4C22⋊C416yes—
109C23×C4C22⋊C411yescharacteristic
110C23×C4C22⋊C411yescharacteristic
111C23×C4SD1618yes—
112C23×C4Q1618yes—
113C23×C4C22×C411yescharacteristic
114C23×C4C22×C413yes—
115C23×C4C2×D411yescharacteristic
116C23×C4C2×D416yes—
117C23×C4C2×D413yes—
118C23×C4C2×D413yes—
119C23×C4C2×D416yes—
120C23×C4C2413yes—
121C22×C8—212no—
122C22×C8—48no—
123C4⋊Q8—224no—
124C4⋊Q8—224no—
125C4⋊Q8C22×C4116yescomplemented
126C4⋊Q8C22×C4116yescomplemented
127#125—248no—
128#125—216no—
129C22⋊Q8—248no—
130C22⋊Q8—224no—
131C22⋊Q8—216no—
132C22⋊Q8—212no—
133C22⋊Q8—212no—
134C22⋊Q8C22×C4124yes—
135C22⋊Q8C2418yes—
136C4×Q8—248no—
137C4×Q8—248no—
138C4×Q8—216no—
139C4×Q8C22×C4132yescomplemented
140#117—248no—
141#117—216no—
142C2×C4⋊C4—212no—
143C2×C4⋊C4—212no—
144C2×C4⋊C4—212no—
145C2×C4⋊C4—212no—
146C2×C4⋊C4—224no—
147C2×C4⋊C4—26no—
148C2×C4⋊C4—212no—
149C2×C4⋊C4—216no—
150C2×C4⋊C4—448no—
151C2×C4⋊C4C22×C4112yes—
152C2×C22⋊C4—212no—
153C2×C22⋊C4—212no—
154C2×C42—224no—
155C2×C42C22⋊C416yes—
156C2×C42C22⋊C416yes—
157C2×C42C22⋊C416yes—
158C2×C42C22⋊C416yes—
159C2×C42C22⋊C4124yes—
160C2×C42C22×C416yes—
161C2×C42C22×C416yes—
162C2×C42C22×C4112yes—
163C2×C42C2×D4112yes—
164C2×C42C2×D416yes—
165C2×C42C2×D416yes—
166C2×C42C2×D4112yes—
167C2×C42C2414yes—
168C4⋊C8—216no—
169C22⋊C8—212no—
170C22⋊C8C2418yes—
171C4×C8—216no—
172#95—216no—
173#95—212no—
174#95C2418yes—
175C24—24no—
176C24—24no—
177C24C2×C22⋊C413yes—
178C24C2×C22⋊C411yescharacteristic
179C24C2×SD1614yes—
180C24C2×Q1614yes—
181C24C22×D413yes—
182C2×Q8—212no—
183C2×Q8—216no—
184C2×Q8—424no—
185C2×Q8—448no—
186C2×Q8—424no—
187C2×Q8C23×C418yes—
188C22×C4—24no—
189C22×C4—24no—
190C22×C4—24no—
191C22×C4—24no—
192C22×C4—212no—
193C22×C4—212no—
194C22×C4—28no—
195C22×C4—412no—
196C22×C4—46no—
197C22×C4—46no—
198C22×C4—46no—
199C22×C4—412no—
200C22×C4—46no—
201C22×C4—48no—
202C22×C4—48no—
203C22×C4—424no—
204C22×C4—424no—
205C22×C4—424no—
206C22×C4—832no—
207C22×C4#103148yes—
208C22×C4#10318yes—
209C22×C4#10318yes—
210C22×C4C2×C22⋊C413yes—
211C22×C4C2×C22⋊C413yes—
212C22×C4C2×C22⋊C416yes—
213C22×C4C2×C22⋊C416yes—
214C22×C4C2×C22⋊C416yes—
215C22×C4C2×C22⋊C4112yes—
216C22×C4C2×C22⋊C416yes—
217C22×C4C2×C22⋊C416yes—
218C22×C4C2×C22⋊C4112yes—
219C22×C4C2×SD16124yes—
220C22×C4C2×Q16124yes—
221C22×C4C23×C413yes—
222C22×C4C23×C413yes—
223C22×C4C22×D413yes—
224C22×C4C22×D413yes—
225C22×C4C22×D414yes—
226C22×C4C22×D413yes—
227C22×C4C22×D414yes—
228C22×C4C2511yescharacteristic
229C2×C8—28no—
230C2×C8—424no—
231C4⋊C4—216no—
232C4⋊C4—216no—
233C4⋊C4—216no—
234C4⋊C4—224no—
235C4⋊C4—232no—
236C4⋊C4—212no—
237C4⋊C4—28no—
238C4⋊C4—432no—
239C4⋊C4C23×C418yes—
240C22⋊C4—224no—
241C22⋊C4—28no—
242C42—232no—
243C42C2×C22⋊C4112yes—
244C42C2×C22⋊C4112yes—
245C42C2×C22⋊C4112yes—
246C42C2×C22⋊C4112yes—
247C42C23×C414yes—
248C42C23×C414yes—
249C42C22×D414yes—
250C42C22×D414yes—
251C23—26no—
252C23—26no—
253C23—26no—
254C23—26no—
255C23—216no—
256C23—216no—
257C23#41014yes—
258C23C2×#103112yes—
259C23C2×#10314yes—
260C23#42114yes—
261C23C22×C22⋊C413yes—
262C23C22×C22⋊C413yes—
263C23C22×SD1616yes—
264C23C22×Q1616yes—
265C23C23×D411yescharacteristic
266Q8—48no—
267Q8—424no—
268Q8—432no—
269C2×C4—26no—
270C2×C4—26no—
271C2×C4—26no—
272C2×C4—26no—
273C2×C4—212no—
274C2×C4—212no—
275C2×C4—28no—
276C2×C4—28no—
277C2×C4—46no—
278C2×C4—412no—
279C2×C4—43no—
280C2×C4—43no—
281C2×C4—43no—
282C2×C4—44no—
283C2×C4—41no—
284C2×C4—424no—
285C2×C4—424no—
286C2×C4—48no—
287C2×C4—48no—
288C2×C4—48no—
289C2×C4—848no—
290C2×C4C2×#103148yes—
291C2×C4C2×#103124yes—
292C2×C4C2×#103124yes—
293C2×C4C22×C22⋊C413yes—
294C2×C4C22×C22⋊C413yes—
295C2×C4C22×C22⋊C414yes—
296C2×C4C22×C22⋊C414yes—
297C2×C4C22×C22⋊C414yes—
298C2×C4C22×C22⋊C414yes—
299C2×C4C22×SD1618yes—
300C2×C4C22×Q1618yes—
301C2×C4C24×C411yescharacteristic
302C2×C4C23×D411yescharacteristic
303C8—48no—
304C22—212no—
305C22—212no—
306C22—224no—
307C22—21no—
308C22—23no—
309C22—21no—
310C22—23no—
311C22#1518116yesdirect product factor
312C22C2×#41016yes—
313C22C22×#10316yes—
314C22C22×#10316yes—
315C22C2×#42116yes—
316C22C23×C22⋊C411yescharacteristic
317C22C23×SD1611yescharacteristic
318C22C23×Q1611yescharacteristic
319C4—21no—
320C4—23no—
321C4—23no—
322C4—21no—
323C4—24no—
324C4—24no—
325C4—48no—
326C4—48no—
327C4—88no—
328C4C22×#10318yes—
329C4C22×#10318yes—
330C2—21no—
331C2—26no—
332C2—21no—
333C2C2×#1518112yesdirect product factor
334C2C22×#41011yescharacteristic
335C2C23×#10311yescharacteristic
336C2C22×#42111yescharacteristic
subgroupG/H
center, socleC24C2×C22⋊C4
commutatorC2×C4C24×C4
Frattini, cosocleC22×C4C25
prime radicalC25C2×D4
LCSG ▷ C2×C4 ▷ C22 ▷ C1
UCSC1 ◁ C24 ◁ C23×C42 ◁ G
LECSG ▷ C22×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. C2×#1518C2direct (N × H)
8. C2×#533C4semidirect
10. C22×C4⋊Q8C4semidirect
11. C22×C4⋊Q8C4semidirect
16. C22×C4×Q8C4semidirect
29. #1518C22direct (N × H)
41. #533C2×C4semidirect
45. C2×C4⋊Q8C2×C4semidirect
46. C2×C4⋊Q8C2×C4semidirect
58. C2×C4×Q8C2×C4semidirect
125. C4⋊Q8C22×C4semidirect
126. C4⋊Q8C22×C4semidirect
139. C4×Q8C22×C4semidirect
311. C22#1518direct (N × H)
333. C2C2×#1518direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#5332
C22×#4312
C22×#3822
C2×#15182
C22×#3282