Finite Group Databasebeta

← C2×#9465 #201283

Subgroups — C2×#9465 #201283

Order 512.

1520 conjugacy classes of subgroups.

The subgroup lattice has 488 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2504C211yesmaximal, characteristic
2C2×#2313C211yesmaximal, characteristic, complemented
3C2×#1821C211yesmaximal, characteristic, complemented
4C2×#1510C212yesmaximal
5#9465C218yesmaximal, direct product factor
6C2×#1284C212yesmaximal, complemented
7C22×#121C2211yescharacteristic
8C23×C4⋊C4C2211yescharacteristic
9C2×#407C2212yes—
10#2504—22no—
11#2504—22no—
12#2504C414yescomplemented
13#2504C2214yes—
14C2×#397—22no—
15#2313—24no—
16#2313—22no—
17#2313C2214yescomplemented
18C2×#390C412yescomplemented
19C2×#387—24no—
20C2×#386—21no—
21C2×#386—21no—
22C2×#382C412yescomplemented
23C2×#381C2212yes—
24C2×#380—21no—
25C2×C4×C22⋊C4—21no—
26C2×C4×C22⋊C4C2211yescharacteristic
27#1821—22no—
28#1821C2214yescomplemented
29C2×#343—22no—
30#1510—24no—
31#1510C2218yes—
32C2×#327—24no—
33#1284C2218yescomplemented
34C23×D4—21no—
35C23×D4C2×C411yescharacteristic
36C24×C4D411yescharacteristic
37C24×C4D411yescharacteristic
38C24×C4C2311yescharacteristic
39C22×M16—22no—
40C2×#121—21no—
41C2×#121—21no—
42C2×#121—22no—
43C2×#121—22no—
44C2×#121—22no—
45C2×#121—22no—
46C2×#121C2×C412yes—
47C2×#121C2312yes—
48C22×C4⋊C4—22no—
49C22×C4⋊C4—22no—
50C22×C4⋊C4—22no—
51C22×C4⋊C4—22no—
52C22×C4⋊C4—22no—
53C22×C4⋊C4C2×C411yescharacteristic
54C22×C4⋊C4C2×C412yes—
55C22×C4⋊C4C2×C412yes—
56C22×C4⋊C4D412yes—
57C22×C4⋊C4D412yes—
58C22×C4⋊C4D412yes—
59C22×C4⋊C4C2312yes—
60C22×C22⋊C4—22no—
61C22×C22⋊C4—21no—
62C22×C22⋊C4—21no—
63C22×C22⋊C4—22no—
64C22×C22⋊C4—22no—
65C22×C42—22no—
66C22×C42C2×C412yes—
67#407—22no—
68#407—22no—
69#407—22no—
70#407C2314yes—
71C2×C22⋊C8—22no—
72C2×C22⋊C8—22no—
73#397—28no—
74#397—28no—
75#390—24no—
76#390C2×C418yescomplemented
77#387—216no—
78#387—216no—
79#386—24no—
80#386—24no—
81#386—24no—
82#386—24no—
83#382—24no—
84#382C2×C418yescomplemented
85#381—22no—
86#381—22no—
87#381D414yes—
88#381C2314yes—
89#380—24no—
90#380—24no—
91C4×C22⋊C4—24no—
92C4×C22⋊C4—24no—
93C4×C22⋊C4—21no—
94C4×C22⋊C4—21no—
95C4×C22⋊C4C2×C412yes—
96C4×C22⋊C4C2312yes—
97C2×#95—22no—
98C2×#95—22no—
99C2×#95—22no—
100C2×#95—22no—
101C2×#95—21no—
102C2×#95—22no—
103C2×#95D414yes—
104#343—28no—
105#327—216no—
106C25—21no—
107C25C2×D411yescharacteristic
108C22×D4—22no—
109C22×D4—22no—
110C22×D4—21no—
111C22×D4—22no—
112C22×D4—22no—
113C22×D4—44no—
114C22×D4—48no—
115C22×D4—41no—
116C22×D4—41no—
117C22×D4—41no—
118C22×D4—41no—
119C22×D4C22×C412yes—
120C23×C4—21no—
121C23×C4—21no—
122C23×C4—21no—
123C23×C4—21no—
124C23×C4—21no—
125C23×C4—21no—
126C23×C4—21no—
127C23×C4—21no—
128C23×C4—22no—
129C23×C4—21no—
130C23×C4—24no—
131C23×C4—44no—
132C23×C4—82no—
133C23×C4—82no—
134C23×C4C22⋊C412yes—
135C23×C4C22⋊C412yes—
136C23×C4C22⋊C412yes—
137C23×C4C22⋊C412yes—
138C23×C4C22×C411yescharacteristic
139C23×C4C22×C411yescharacteristic
140C23×C4C2×D411yescharacteristic
141C23×C4C2×D412yes—
142C23×C4C2×D412yes—
143C23×C4C2×D411yescharacteristic
144C23×C4C2×D411yescharacteristic
145C23×C4C2×D412yes—
146C23×C4C2×D412yes—
147C23×C4#4112yes—
148C23×C4#4112yes—
149C23×C4C2411yescharacteristic
150C2×M16—24no—
151C2×M16—42no—
152C2×M16—48no—
153C22×C8—42no—
154#121—24no—
155#121—24no—
156#121—24no—
157#121—24no—
158#121—42no—
159#121—42no—
160#121—42no—
161#121—42no—
162C2×C4⋊C4—24no—
163C2×C4⋊C4—24no—
164C2×C4⋊C4—24no—
165C2×C4⋊C4—24no—
166C2×C4⋊C4—24no—
167C2×C4⋊C4—24no—
168C2×C4⋊C4—24no—
169C2×C4⋊C4—24no—
170C2×C4⋊C4—24no—
171C2×C4⋊C4—24no—
172C2×C4⋊C4—24no—
173C2×C4⋊C4—24no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—28no—
178C2×C4⋊C4—84no—
179C2×C4⋊C4C22⋊C414yes—
180C2×C4⋊C4C22⋊C414yes—
181C2×C4⋊C4C22×C412yes—
182C2×C4⋊C4C22×C414yes—
183C2×C4⋊C4C2×D414yes—
184C2×C22⋊C4—22no—
185C2×C22⋊C4—24no—
186C2×C22⋊C4—24no—
187C2×C22⋊C4—22no—
188C2×C22⋊C4—22no—
189C2×C22⋊C4—22no—
190C2×C22⋊C4—28no—
191C2×C22⋊C4—28no—
192C2×C22⋊C4—48no—
193C2×C22⋊C4—44no—
194C2×C22⋊C4—42no—
195C2×C22⋊C4—42no—
196C2×C22⋊C4—84no—
197C2×C22⋊C4—84no—
198C2×C42—22no—
199C2×C42—28no—
200C2×C42—44no—
201C2×C42—84no—
202C2×C42C22×C414yes—
203C22⋊C8—24no—
204C22⋊C8—24no—
205C22⋊C8—42no—
206C22⋊C8—42no—
207#95—28no—
208#95—28no—
209#95—22no—
210#95—28no—
211#95—28no—
212#95—24no—
213#95—24no—
214#95—22no—
215#95—24no—
216#95C2×D418yes—
217C24—24no—
218C24—24no—
219C24—22no—
220C24—42no—
221C24—44no—
222C24—44no—
223C24—41no—
224C24—41no—
225C24—44no—
226C24C2×C22⋊C411yescharacteristic
227C24C2×C22⋊C411yescharacteristic
228C24#12011yescharacteristic
229C24#12011yescharacteristic
230C24#12311yescharacteristic
231C24#12311yescharacteristic
232C24C22×D411yescharacteristic
233C2×D4—24no—
234C2×D4—44no—
235C2×D4—44no—
236C2×D4—48no—
237C2×D4—48no—
238C2×D4—42no—
239C2×D4—42no—
240C2×D4—42no—
241C2×D4—42no—
242C2×D4—42no—
243C2×D4—84no—
244C2×D4—84no—
245C2×D4—84no—
246C2×D4—84no—
247C2×D4—84no—
248C2×D4—84no—
249C22×C4—24no—
250C22×C4—24no—
251C22×C4—22no—
252C22×C4—22no—
253C22×C4—22no—
254C22×C4—22no—
255C22×C4—22no—
256C22×C4—22no—
257C22×C4—22no—
258C22×C4—22no—
259C22×C4—22no—
260C22×C4—22no—
261C22×C4—22no—
262C22×C4—22no—
263C22×C4—22no—
264C22×C4—24no—
265C22×C4—28no—
266C22×C4—42no—
267C22×C4—42no—
268C22×C4—42no—
269C22×C4—44no—
270C22×C4—44no—
271C22×C4—44no—
272C22×C4—44no—
273C22×C4—42no—
274C22×C4—44no—
275C22×C4—44no—
276C22×C4—44no—
277C22×C4—44no—
278C22×C4—48no—
279C22×C4—44no—
280C22×C4—48no—
281C22×C4—44no—
282C22×C4—44no—
283C22×C4—41no—
284C22×C4—41no—
285C22×C4—44no—
286C22×C4—48no—
287C22×C4—82no—
288C22×C4—82no—
289C22×C4—82no—
290C22×C4—82no—
291C22×C4—82no—
292C22×C4—82no—
293C22×C4—82no—
294C22×C4—82no—
295C22×C4—82no—
296C22×C4—82no—
297C22×C4C2×C22⋊C411yescharacteristic
298C22×C4C2×C22⋊C411yescharacteristic
299C22×C4C2×C22⋊C414yes—
300C22×C4C2×C22⋊C414yes—
301C22×C4C4×D412yes—
302C22×C4C4×D412yes—
303C22×C4C4×D412yes—
304C22×C4C4×D412yes—
305C22×C4#12012yes—
306C22×C4#12112yes—
307C22×C4#12112yes—
308C22×C4#12112yes—
309C22×C4#12112yes—
310C22×C4#12312yes—
311C22×C4C23×C411yescharacteristic
312C22×C4C22×D411yescharacteristic
313C22×C4C22×D412yes—
314C22×C4C2×#4112yes—
315M16—48no—
316M16—84no—
317C2×C8—44no—
318C2×C8—84no—
319C4⋊C4—28no—
320C4⋊C4—28no—
321C4⋊C4—28no—
322C4⋊C4—28no—
323C4⋊C4—24no—
324C4⋊C4—42no—
325C4⋊C4—88no—
326C22⋊C4—48no—
327C22⋊C4—44no—
328C22⋊C4—48no—
329C22⋊C4—44no—
330C22⋊C4—44no—
331C22⋊C4—44no—
332C22⋊C4—88no—
333C22⋊C4—88no—
334C42—44no—
335C42—44no—
336C42—88no—
337C23—22no—
338C23—22no—
339C23—22no—
340C23—22no—
341C23—22no—
342C23—22no—
343C23—22no—
344C23—22no—
345C23—22no—
346C23—22no—
347C23—22no—
348C23—22no—
349C23—22no—
350C23—22no—
351C23—22no—
352C23—41no—
353C23—41no—
354C23—41no—
355C23—41no—
356C23—416no—
357C23—44no—
358C23—44no—
359C23—44no—
360C23—42no—
361C23—82no—
362C23—82no—
363C23—816no—
364C23—82no—
365C23—82no—
366C23—82no—
367C23—82no—
368C23#38611yescharacteristic
369C23#38611yescharacteristic
370C23#38611yescharacteristic
371C23#38611yescharacteristic
372C23#41014yes—
373C23#45414yes—
374C23C22×C22⋊C411yescharacteristic
375C23C2×#12011yescharacteristic
376C23C2×#12311yescharacteristic
377D4—84no—
378D4—84no—
379D4—84no—
380D4—84no—
381D4—84no—
382D4—84no—
383C2×C4—22no—
384C2×C4—22no—
385C2×C4—22no—
386C2×C4—22no—
387C2×C4—24no—
388C2×C4—24no—
389C2×C4—24no—
390C2×C4—24no—
391C2×C4—24no—
392C2×C4—24no—
393C2×C4—42no—
394C2×C4—42no—
395C2×C4—42no—
396C2×C4—42no—
397C2×C4—42no—
398C2×C4—42no—
399C2×C4—42no—
400C2×C4—42no—
401C2×C4—44no—
402C2×C4—44no—
403C2×C4—44no—
404C2×C4—44no—
405C2×C4—44no—
406C2×C4—44no—
407C2×C4—44no—
408C2×C4—44no—
409C2×C4—48no—
410C2×C4—44no—
411C2×C4—48no—
412C2×C4—44no—
413C2×C4—44no—
414C2×C4—44no—
415C2×C4—48no—
416C2×C4—42no—
417C2×C4—82no—
418C2×C4—82no—
419C2×C4—82no—
420C2×C4—82no—
421C2×C4—82no—
422C2×C4—82no—
423C2×C4—816no—
424C2×C4—88no—
425C2×C4—82no—
426C2×C4—82no—
427C2×C4#38612yes—
428C2×C4#38612yes—
429C2×C4C2×C4×D412yes—
430C2×C4C2×#12112yes—
431C8—84no—
432C22—21no—
433C22—21no—
434C22—21no—
435C22—21no—
436C22—21no—
437C22—21no—
438C22—21no—
439C22—21no—
440C22—28no—
441C22—28no—
442C22—21no—
443C22—21no—
444C22—21no—
445C22—21no—
446C22—21no—
447C22—21no—
448C22—44no—
449C22—44no—
450C22—44no—
451C22—44no—
452C22—44no—
453C22—88no—
454C22—82no—
455C22—82no—
456C22—82no—
457C22—164no—
458C22—164no—
459C22#188812yes—
460C22#188812yes—
461C22#189912yes—
462C22#189912yes—
463C22#191112yes—
464C22C2×#38611yescharacteristic
465C22C2×#41012yes—
466C22C2×#45412yes—
467C4—22no—
468C4—22no—
469C4—22no—
470C4—22no—
471C4—44no—
472C4—44no—
473C4—88no—
474C4—84no—
475C4—82no—
476C2—22no—
477C2—22no—
478C2—22no—
479C2—22no—
480C2—41no—
481C2—41no—
482C2—162no—
483C2#946514yesdirect product factor
484C2C2×#188811yescharacteristic
485C2C2×#189911yescharacteristic
486C2C2×#191111yescharacteristic
subgroupG/H
center, socleC23#386
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC22×#121C22
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C24×C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2313C2semidirect
3. C2×#1821C2semidirect
5. #9465C2direct (N × H)
6. C2×#1284C2semidirect
12. #2504C4semidirect
17. #2313C22semidirect
18. C2×#390C4semidirect
22. C2×#382C4semidirect
28. #1821C22semidirect
33. #1284C22semidirect
76. #390C2×C4semidirect
84. #382C2×C4semidirect
483. C2#9465direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#25042
C2×#23132
C2×#18212
C2×#15102
#94652
C2×#12842