Finite Group Databasebeta

← C4×#1958 #354549

Subgroups — C4×#1958 #354549

Order 512.

1966 conjugacy classes of subgroups.

The subgroup lattice has 416 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×#124C211yesmaximal, characteristic
2C2×#1958C212yesmaximal
3C2×C4×#103C212yesmaximal, complemented
4C2×C4×#102C212yesmaximal
5C2×C42×C8C211yesmaximal, characteristic, complemented
6C4×#382C211yesmaximal, characteristic, complemented
7#10926C212yesmaximal, complemented
8#10925C212yesmaximal
9#10842C212yesmaximal, complemented
10C22×#124C2211yescharacteristic
11C22×C4×Q8C2211yescharacteristic
12C22×C4×D4C2211yescharacteristic
13C22×#103C2214yes—
14C22×#102C2214yes—
15C22×C4×C8C2212yes—
16C22×C4×C8C2212yes—
17C22×C4×C8C2212yes—
18C4×#124—24no—
19C4×#124C418yescomplemented
20C2×#391C2211yescharacteristic
21C2×#390C2211yescharacteristic
22C2×#388—22no—
23C2×#382C2211yescharacteristic
24C2×#382C2212yes—
25C2×C4×C4⋊C4C2212yes—
26C2×C4×C22⋊C4—22no—
27C2×C43C2211yescharacteristic
28#1958C4116yesdirect product factor
29C4×#103—28no—
30C4×#102—28no—
31C2×#340C2214yes—
32C2×#340C2214yes—
33C42×C8—24no—
34C4⋊#95C2212yes—
35#1434—22no—
36C4×#95—22no—
37C23×Q8C2311yescharacteristic
38C23×D4C2311yescharacteristic
39C24×C4—21no—
40C23×C8C2314yes—
41C2×#124—22no—
42C2×#124—24no—
43C2×#124C2×C418yes—
44C2×#124C2×C414yes—
45C2×C4×Q8—22no—
46C2×C4×Q8—24no—
47C2×C4×Q8—24no—
48C2×C4×Q8C2×C414yes—
49C2×C4×D4—22no—
50C2×C4×D4—24no—
51C2×C4×D4—24no—
52C2×C4×D4C2×C414yes—
53C22×C4⋊C4—21no—
54C22×C4⋊C4C2314yes—
55C22×C4⋊C4C2311yescharacteristic
56C22×C4⋊C4C2311yescharacteristic
57C22×C22⋊C4—22no—
58C22×C22⋊C4—21no—
59C22×C22⋊C4—22no—
60C22×C42—21no—
61C22×C42—22no—
62C22×C42D412yes—
63C22×C42D412yes—
64C22×C42C2311yescharacteristic
65C22×C42C2311yescharacteristic
66C22×C42C2311yescharacteristic
67C2×#103—28no—
68C2×#103—28no—
69C2×#103C2×C4116yescomplemented
70C2×#102—28no—
71C2×#102—28no—
72C2×#102C2×C4116yes—
73C2×C4×C8—28no—
74C2×C4×C8—24no—
75C2×C4×C8—24no—
76C2×C4×C8—24no—
77C2×C4×C8—24no—
78C2×C4×C8—24no—
79C2×C4×C8C2×C418yescomplemented
80C2×C4×C8D418yes—
81#391—22no—
82#391C2×C414yes—
83#390—22no—
84#390C2×C414yes—
85#388—216no—
86#382—22no—
87#382—24no—
88#382C2×C414yes—
89#382C2×C418yescomplemented
90C4×C4⋊C4—24no—
91C4×C4⋊C4—24no—
92C4×C22⋊C4—216no—
93C2×#95—24no—
94C2×#95—22no—
95C2×#95—22no—
96C2×#95—24no—
97C2×#95—22no—
98C43C2×C414yes—
99C43D414yes—
100#340—28no—
101#340—28no—
102C25—21no—
103C22×Q8—21no—
104C22×Q8—22no—
105C22×Q8—22no—
106C22×Q8—22no—
107C22×Q8—24no—
108C22×Q8C22×C412yes—
109C22×Q8C22×C414yes—
110C22×D4—22no—
111C22×D4—21no—
112C22×D4—22no—
113C22×D4—24no—
114C22×D4—22no—
115C22×D4C22×C412yes—
116C22×D4C22×C414yes—
117C23×C4—21no—
118C23×C4—21no—
119C23×C4—21no—
120C23×C4—24no—
121C23×C4—24no—
122C23×C4—42no—
123C23×C4—42no—
124C23×C4—48no—
125C23×C4C2×D412yes—
126C23×C4C2×D411yescharacteristic
127C23×C4C2×D412yes—
128C23×C4C2×D411yescharacteristic
129C23×C4C2411yescharacteristic
130C22×C8—24no—
131C22×C8—24no—
132C22×C8—24no—
133C22×C8—24no—
134C22×C8C22×C418yes—
135C22×C8C2×D418yes—
136C22×C8#4118yes—
137#124—28no—
138#124—216no—
139#124C42116yescomplemented
140C4×Q8—216no—
141C4×Q8—48no—
142C4×D4—216no—
143C4×D4—48no—
144C2×C4⋊C4—22no—
145C2×C4⋊C4—22no—
146C2×C4⋊C4—24no—
147C2×C4⋊C4—24no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—24no—
151C2×C4⋊C4—24no—
152C2×C4⋊C4—24no—
153C2×C4⋊C4—21no—
154C2×C4⋊C4—21no—
155C2×C4⋊C4—24no—
156C2×C4⋊C4—42no—
157C2×C4⋊C4—42no—
158C2×C4⋊C4C22×C412yes—
159C2×C4⋊C4C22×C412yes—
160C2×C4⋊C4C22×C418yes—
161C2×C22⋊C4—28no—
162C2×C22⋊C4—216no—
163C2×C22⋊C4—24no—
164C2×C22⋊C4—28no—
165C2×C22⋊C4—42no—
166C2×C22⋊C4—42no—
167C2×C22⋊C4—48no—
168C2×C42—28no—
169C2×C42—28no—
170C2×C42—24no—
171C2×C42—24no—
172C2×C42—42no—
173C2×C42—42no—
174C2×C42—44no—
175C2×C42—48no—
176C2×C42C22⋊C418yes—
177C2×C42C22⋊C418yes—
178C2×C42C22×C412yes—
179C2×C42C22×C412yes—
180C2×C42C22×C414yes—
181C2×C42C22×C412yes—
182C2×C42C2×D412yes—
183C2×C42C2×D412yes—
184C2×C42C2×D414yes—
185C2×C42C2×D412yes—
186C2×C42#4114yes—
187C2×C42#4114yes—
188#103—232no—
189#102—232no—
190C4×C8—216no—
191C4×C8—216no—
192C4×C8—216no—
193#95—28no—
194#95—28no—
195#95—216no—
196#95—44no—
197#95—48no—
198C24—22no—
199C24—24no—
200C24—41no—
201C24—41no—
202C24—42no—
203C24—44no—
204C24—44no—
205C24C22×D411yescharacteristic
206C2×Q8—22no—
207C2×Q8—24no—
208C2×Q8—28no—
209C2×Q8—24no—
210C2×Q8—24no—
211C2×Q8—28no—
212C2×Q8—42no—
213C2×Q8—42no—
214C2×Q8—44no—
215C2×Q8—44no—
216C2×Q8C2×C4214yes—
217C2×D4—22no—
218C2×D4—28no—
219C2×D4—24no—
220C2×D4—24no—
221C2×D4—24no—
222C2×D4—28no—
223C2×D4—42no—
224C2×D4—42no—
225C2×D4—44no—
226C2×D4—44no—
227C2×D4C2×C4214yes—
228C22×C4—24no—
229C22×C4—24no—
230C22×C4—24no—
231C22×C4—24no—
232C22×C4—24no—
233C22×C4—22no—
234C22×C4—24no—
235C22×C4—22no—
236C22×C4—22no—
237C22×C4—28no—
238C22×C4—41no—
239C22×C4—41no—
240C22×C4—42no—
241C22×C4—41no—
242C22×C4—41no—
243C22×C4—44no—
244C22×C4—41no—
245C22×C4—41no—
246C22×C4—44no—
247C22×C4—48no—
248C22×C4—48no—
249C22×C4—44no—
250C22×C4—44no—
251C22×C4—48no—
252C22×C4—416no—
253C22×C4—88no—
254C22×C4—88no—
255C22×C4C2×C22⋊C414yes—
256C22×C4C2×C22⋊C412yes—
257C22×C4C2×C22⋊C414yes—
258C22×C4C2×C22⋊C412yes—
259C22×C4C4×D418yes—
260C22×C4#12412yes—
261C22×C4#12412yes—
262C22×C4#12712yes—
263C22×C4#12712yes—
264C22×C4#13518yes—
265C22×C4#13518yes—
266C22×C4C23×C411yescharacteristic
267C22×C4C23×C412yes—
268C22×C4C22×D412yes—
269C22×C4C22×D411yescharacteristic
270C22×C4C2×#4112yes—
271C22×C4C2×#4112yes—
272C22×C4C2×#4114yes—
273C2×C8—28no—
274C2×C8—28no—
275C2×C8—28no—
276C2×C8—28no—
277C2×C8—28no—
278C2×C8—28no—
279C2×C8C4×D4116yes—
280C4⋊C4—28no—
281C4⋊C4—28no—
282C4⋊C4—24no—
283C4⋊C4—24no—
284C4⋊C4—42no—
285C4⋊C4—42no—
286C4⋊C4—44no—
287C4⋊C4—44no—
288C4⋊C4—48no—
289C22⋊C4—232no—
290C22⋊C4—48no—
291C22⋊C4—416no—
292C42—24no—
293C42—24no—
294C42—216no—
295C42—216no—
296C42—48no—
297C42—88no—
298C42C2×C4214yes—
299C42#11714yes—
300C42C4×D418yes—
301C42C4×D418yes—
302C42#12118yes—
303C23—24no—
304C23—44no—
305C23—44no—
306C23—44no—
307C23—416no—
308C23—42no—
309C23—42no—
310C23—44no—
311C23—82no—
312C23—82no—
313C23—82no—
314C23—82no—
315C23—88no—
316C23C22×C22⋊C411yescharacteristic
317C23C2×C4×D412yes—
318C23C2×#12411yescharacteristic
319C23C2×#12711yescharacteristic
320C23#53212yes—
321C23C2×#13514yes—
322C23C2×#13514yes—
323Q8—28no—
324Q8—44no—
325Q8—48no—
326D4—28no—
327D4—44no—
328D4—48no—
329C2×C4—24no—
330C2×C4—24no—
331C2×C4—24no—
332C2×C4—24no—
333C2×C4—24no—
334C2×C4—24no—
335C2×C4—216no—
336C2×C4—216no—
337C2×C4—24no—
338C2×C4—24no—
339C2×C4—44no—
340C2×C4—42no—
341C2×C4—42no—
342C2×C4—44no—
343C2×C4—48no—
344C2×C4—44no—
345C2×C4—44no—
346C2×C4—416no—
347C2×C4—42no—
348C2×C4—42no—
349C2×C4—82no—
350C2×C4—82no—
351C2×C4—84no—
352C2×C4—82no—
353C2×C4—84no—
354C2×C4—82no—
355C2×C4—88no—
356C2×C4—816no—
357C2×C4C4×C22⋊C418yes—
358C2×C4#39014yes—
359C2×C4#39014yes—
360C2×C4#41618yes—
361C2×C4#41618yes—
362C2×C4#48818yes—
363C2×C4#49518yes—
364C2×C4C22×C4211yescharacteristic
365C2×C4C2×#11711yescharacteristic
366C2×C4C2×C4×D412yes—
367C2×C4C2×C4×D414yes—
368C2×C4C2×C4×D412yes—
369C2×C4C4×#4112yes—
370C2×C4C4×#4114yes—
371C2×C4C2×#12112yes—
372C2×C4#52512yes—
373C2×C4#52914yes—
374C2×C4#53214yes—
375C8—216no—
376C22—48no—
377C22—44no—
378C22—88no—
379C22—84no—
380C22—88no—
381C22—82no—
382C22—82no—
383C22C2×C4×C22⋊C411yescharacteristic
384C22C2×#39011yescharacteristic
385C22#229711yescharacteristic
386C22C4×#12412yes—
387C22C4×#12712yes—
388C22C2×#41612yes—
389C22C2×#41612yes—
390C22C4×#13518yes—
391C22C2×#48812yes—
392C22#313112yes—
393C22C2×#49512yes—
394C22#314712yes—
395C22#316818yes—
396C4—28no—
397C4—28no—
398C4—24no—
399C4—44no—
400C4—88no—
401C4—82no—
402C4—82no—
403C4#195818yesdirect product factor
404C4C42×D412yes—
405C4#229312yes—
406C4C4×#12114yes—
407C2—84no—
408C2C4×#39011yescharacteristic
409C2C4×#41612yes—
410C2C2×#195811yescharacteristic
411C2#2119511yescharacteristic
412C2#2140212yes—
413C2C4×#48814yes—
414C2C4×#49514yes—
subgroupG/H
centerC23×C4C2×D4
commutatorC2×C4C22×C42
Frattini, cosocleC23×C4C24
socleC24C22×D4
prime radicalC23×D4C23
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C4 ◁ C2×C43 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. C2×C4×#103C2semidirect
5. C2×C42×C8C2semidirect
6. C4×#382C2semidirect
7. #10926C2semidirect
9. #10842C2semidirect
19. C4×#124C4semidirect
28. #1958C4direct (N × H)
69. C2×#103C2×C4semidirect
79. C2×C4×C8C2×C4semidirect
89. #382C2×C4semidirect
139. #124C42semidirect
403. C4#1958direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C4×#1242
C2×#19582
C2×C4×#1032
C2×C4×#1022
C2×C42×C82
C4×#3822
#109262
#109252
#108422