Finite Group Databasebeta

← C2×#9206 #197504

Subgroups — C2×#9206 #197504

Order 512.

1608 conjugacy classes of subgroups.

The subgroup lattice has 518 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2618C211yesmaximal, characteristic
2C2×#2313C211yesmaximal, characteristic, complemented
3C2×#1519C212yesmaximal
4C22×#324C211yesmaximal, characteristic, complemented
5#9206C218yesmaximal, direct product factor
6C2×#1289C212yesmaximal, complemented
7C22×#123—21no—
8C22×#121C2211yescharacteristic
9C23×C4⋊C4C2211yescharacteristic
10C22×C22⋊C8C2212yes—
11#2618—22no—
12#2618—22no—
13#2618C414yescomplemented
14#2618C2214yes—
15C2×#398C412yescomplemented
16C2×#396—22no—
17C2×#394C412yescomplemented
18#2313—24no—
19#2313—22no—
20#2313C2214yescomplemented
21C2×#387—24no—
22C2×#386—21no—
23C2×#381—21no—
24C2×#380C2211yescharacteristic
25C2×#380C2212yes—
26C2×C4×C22⋊C4—21no—
27#1519—24no—
28#1519C2218yes—
29C2×#327—24no—
30C2×#324—22no—
31C2×#324—22no—
32C2×#324C2214yescomplemented
33#1289C2218yescomplemented
34C23×D4—21no—
35C23×D4C2×C411yescharacteristic
36C24×C4D411yescharacteristic
37C24×C4D411yescharacteristic
38C24×C4C2311yescharacteristic
39C23×C8—22no—
40C2×#123—24no—
41C2×#123—24no—
42C2×#123—48no—
43C2×#121—21no—
44C2×#121—21no—
45C2×#121—22no—
46C2×#121—22no—
47C2×#121—22no—
48C2×#121—22no—
49C2×#121C2×C412yes—
50C2×#121C2312yes—
51C22×C4⋊C4—22no—
52C22×C4⋊C4—22no—
53C22×C4⋊C4—22no—
54C22×C4⋊C4—22no—
55C22×C4⋊C4—22no—
56C22×C4⋊C4—21no—
57C22×C4⋊C4—22no—
58C22×C4⋊C4C2×C411yescharacteristic
59C22×C4⋊C4C2×C412yes—
60C22×C4⋊C4C2×C412yes—
61C22×C4⋊C4D412yes—
62C22×C4⋊C4D412yes—
63C22×C4⋊C4D412yes—
64C22×C4⋊C4C2312yes—
65C22×C22⋊C4—21no—
66C22×C22⋊C4—22no—
67C22×C22⋊C4—22no—
68C22×C22⋊C4—21no—
69C22×C22⋊C4—22no—
70C22×C42—22no—
71C2×C22⋊C8—22no—
72C2×C22⋊C8—22no—
73C2×C22⋊C8—22no—
74C2×C22⋊C8—22no—
75C2×C22⋊C8—22no—
76C2×C22⋊C8C2314yes—
77#398—24no—
78#398C2×C418yescomplemented
79#396—28no—
80#396—28no—
81#394—24no—
82#394C2×C418yescomplemented
83#387—216no—
84#387—216no—
85#386—24no—
86#386—24no—
87#381—24no—
88#381—24no—
89#380—22no—
90#380—22no—
91#380—21no—
92#380—21no—
93#380C2×C412yes—
94#380D414yes—
95#380C2312yes—
96#380C2314yes—
97C4×C22⋊C4—24no—
98C4×C22⋊C4—24no—
99C2×#95—22no—
100C2×#95—24no—
101C2×#95—22no—
102C2×#95C2×C412yes—
103C2×#95D414yes—
104#327—216no—
105#324—28no—
106C25—21no—
107C25C2×D411yescharacteristic
108C22×D4—22no—
109C22×D4—22no—
110C22×D4—21no—
111C22×D4—22no—
112C22×D4—22no—
113C22×D4—41no—
114C22×D4—41no—
115C22×D4—41no—
116C22×D4—41no—
117C22×D4—44no—
118C22×D4—44no—
119C22×D4—82no—
120C22×D4C22×C412yes—
121C23×C4—21no—
122C23×C4—21no—
123C23×C4—21no—
124C23×C4—21no—
125C23×C4—21no—
126C23×C4—24no—
127C23×C4—24no—
128C23×C4—21no—
129C23×C4—21no—
130C23×C4—21no—
131C23×C4—22no—
132C23×C4—21no—
133C23×C4—24no—
134C23×C4—82no—
135C23×C4—82no—
136C23×C4C22⋊C412yes—
137C23×C4C22⋊C412yes—
138C23×C4C22⋊C412yes—
139C23×C4C22⋊C412yes—
140C23×C4C22×C411yescharacteristic
141C23×C4C22×C411yescharacteristic
142C23×C4C2×D412yes—
143C23×C4C2×D412yes—
144C23×C4C2×D411yescharacteristic
145C23×C4C2×D412yes—
146C23×C4C2×D411yescharacteristic
147C23×C4C2×D411yescharacteristic
148C23×C4C2×D412yes—
149C23×C4#4112yes—
150C23×C4#4112yes—
151C23×C4C2411yescharacteristic
152C22×C8—24no—
153C22×C8—42no—
154C22×C8—42no—
155C22×C8—84no—
156#123—416no—
157#123—416no—
158#121—24no—
159#121—24no—
160#121—24no—
161#121—24no—
162#121—42no—
163#121—42no—
164#121—42no—
165#121—42no—
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—24no—
175C2×C4⋊C4—24no—
176C2×C4⋊C4—24no—
177C2×C4⋊C4—24no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—28no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—24no—
184C2×C4⋊C4—24no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—48no—
187C2×C4⋊C4—84no—
188C2×C4⋊C4—82no—
189C2×C4⋊C4C22⋊C414yes—
190C2×C4⋊C4C22⋊C414yes—
191C2×C4⋊C4C22×C412yes—
192C2×C4⋊C4C22×C414yes—
193C2×C4⋊C4C2×D414yes—
194C2×C22⋊C4—22no—
195C2×C22⋊C4—24no—
196C2×C22⋊C4—22no—
197C2×C22⋊C4—24no—
198C2×C22⋊C4—22no—
199C2×C22⋊C4—28no—
200C2×C22⋊C4—22no—
201C2×C22⋊C4—28no—
202C2×C22⋊C4—48no—
203C2×C22⋊C4—44no—
204C2×C22⋊C4—44no—
205C2×C22⋊C4—44no—
206C2×C22⋊C4—42no—
207C2×C22⋊C4—42no—
208C2×C22⋊C4—84no—
209C2×C42—28no—
210C2×C42—84no—
211C22⋊C8—24no—
212C22⋊C8—24no—
213C22⋊C8—42no—
214C22⋊C8—42no—
215#95—28no—
216#95—216no—
217#95—28no—
218#95—22no—
219#95—24no—
220#95C22×C414yes—
221#95C2×D418yes—
222C24—22no—
223C24—42no—
224C24—42no—
225C24—44no—
226C24—44no—
227C24—41no—
228C24—41no—
229C24—42no—
230C24—42no—
231C24C2×C22⋊C411yescharacteristic
232C24C2×C22⋊C411yescharacteristic
233C24#12011yescharacteristic
234C24#12011yescharacteristic
235C24#12311yescharacteristic
236C24#12311yescharacteristic
237C24#13514yes—
238C24AGL(1,ℤ/8)14yes—
239C24C22×D411yescharacteristic
240C2×D4—24no—
241C2×D4—42no—
242C2×D4—42no—
243C2×D4—44no—
244C2×D4—44no—
245C2×D4—42no—
246C2×D4—42no—
247C2×D4—44no—
248C2×D4—44no—
249C2×D4—42no—
250C2×D4—84no—
251C2×D4—82no—
252C2×D4—84no—
253C2×D4—82no—
254C2×D4—84no—
255C2×D4—84no—
256C2×D4—84no—
257C2×D4—84no—
258C22×C4—22no—
259C22×C4—22no—
260C22×C4—22no—
261C22×C4—24no—
262C22×C4—24no—
263C22×C4—22no—
264C22×C4—22no—
265C22×C4—22no—
266C22×C4—22no—
267C22×C4—22no—
268C22×C4—22no—
269C22×C4—24no—
270C22×C4—24no—
271C22×C4—24no—
272C22×C4—24no—
273C22×C4—22no—
274C22×C4—22no—
275C22×C4—22no—
276C22×C4—22no—
277C22×C4—24no—
278C22×C4—28no—
279C22×C4—42no—
280C22×C4—44no—
281C22×C4—44no—
282C22×C4—42no—
283C22×C4—44no—
284C22×C4—44no—
285C22×C4—44no—
286C22×C4—44no—
287C22×C4—44no—
288C22×C4—44no—
289C22×C4—48no—
290C22×C4—44no—
291C22×C4—48no—
292C22×C4—44no—
293C22×C4—41no—
294C22×C4—41no—
295C22×C4—42no—
296C22×C4—44no—
297C22×C4—42no—
298C22×C4—48no—
299C22×C4—81no—
300C22×C4—81no—
301C22×C4—82no—
302C22×C4—81no—
303C22×C4—82no—
304C22×C4—81no—
305C22×C4—82no—
306C22×C4—82no—
307C22×C4—82no—
308C22×C4—82no—
309C22×C4—82no—
310C22×C4C2×C22⋊C414yes—
311C22×C4C2×C22⋊C414yes—
312C22×C4C2×C22⋊C411yescharacteristic
313C22×C4C2×C22⋊C411yescharacteristic
314C22×C4C4×D412yes—
315C22×C4C4×D412yes—
316C22×C4C4×D412yes—
317C22×C4C4×D412yes—
318C22×C4#12012yes—
319C22×C4#12112yes—
320C22×C4#12112yes—
321C22×C4#12112yes—
322C22×C4#12112yes—
323C22×C4#12312yes—
324C22×C4C23×C411yescharacteristic
325C22×C4C22×D412yes—
326C22×C4C22×D411yescharacteristic
327C22×C4C2×#4112yes—
328C2×C8—44no—
329C2×C8—84no—
330C2×C8—84no—
331C2×C8—84no—
332C4⋊C4—28no—
333C4⋊C4—28no—
334C4⋊C4—28no—
335C4⋊C4—28no—
336C4⋊C4—24no—
337C4⋊C4—416no—
338C4⋊C4—42no—
339C4⋊C4—82no—
340C4⋊C4—84no—
341C4⋊C4—84no—
342C4⋊C4—82no—
343C22⋊C4—416no—
344C22⋊C4—44no—
345C22⋊C4—44no—
346C22⋊C4—44no—
347C22⋊C4—44no—
348C22⋊C4—48no—
349C22⋊C4—44no—
350C22⋊C4—44no—
351C22⋊C4—88no—
352C42—88no—
353C23—22no—
354C23—22no—
355C23—22no—
356C23—22no—
357C23—22no—
358C23—22no—
359C23—41no—
360C23—41no—
361C23—41no—
362C23—41no—
363C23—41no—
364C23—41no—
365C23—41no—
366C23—48no—
367C23—48no—
368C23—41no—
369C23—44no—
370C23—44no—
371C23—42no—
372C23—42no—
373C23—42no—
374C23—82no—
375C23—82no—
376C23—88no—
377C23—88no—
378C23—82no—
379C23—82no—
380C23—82no—
381C23—82no—
382C23#38611yescharacteristic
383C23#38611yescharacteristic
384C23#38611yescharacteristic
385C23#38611yescharacteristic
386C23#41612yes—
387C23#41612yes—
388C23#41812yes—
389C23#41812yes—
390C23#44914yes—
391C23#44914yes—
392C23#48214yes—
393C23#48214yes—
394C23C22×C22⋊C411yescharacteristic
395C23C2×#12011yescharacteristic
396C23C2×#12311yescharacteristic
397C23C2×#13512yes—
398C23C2×AGL(1,ℤ/8)12yes—
399D4—84no—
400D4—84no—
401D4—84no—
402D4—84no—
403D4—84no—
404D4—84no—
405C2×C4—24no—
406C2×C4—24no—
407C2×C4—24no—
408C2×C4—24no—
409C2×C4—24no—
410C2×C4—24no—
411C2×C4—41no—
412C2×C4—42no—
413C2×C4—41no—
414C2×C4—42no—
415C2×C4—42no—
416C2×C4—42no—
417C2×C4—41no—
418C2×C4—42no—
419C2×C4—42no—
420C2×C4—42no—
421C2×C4—42no—
422C2×C4—42no—
423C2×C4—42no—
424C2×C4—41no—
425C2×C4—44no—
426C2×C4—44no—
427C2×C4—44no—
428C2×C4—44no—
429C2×C4—44no—
430C2×C4—44no—
431C2×C4—44no—
432C2×C4—44no—
433C2×C4—44no—
434C2×C4—44no—
435C2×C4—48no—
436C2×C4—44no—
437C2×C4—48no—
438C2×C4—44no—
439C2×C4—42no—
440C2×C4—44no—
441C2×C4—42no—
442C2×C4—48no—
443C2×C4—42no—
444C2×C4—82no—
445C2×C4—82no—
446C2×C4—82no—
447C2×C4—88no—
448C2×C4—816no—
449C2×C4—82no—
450C2×C4—82no—
451C2×C4#38612yes—
452C2×C4#38612yes—
453C2×C4C2×C4×D412yes—
454C2×C4C2×#12112yes—
455C8—84no—
456C22—21no—
457C22—21no—
458C22—21no—
459C22—21no—
460C22—21no—
461C22—21no—
462C22—21no—
463C22—21no—
464C22—21no—
465C22—21no—
466C22—21no—
467C22—21no—
468C22—24no—
469C22—28no—
470C22—24no—
471C22—28no—
472C22—42no—
473C22—42no—
474C22—42no—
475C22—42no—
476C22—44no—
477C22—44no—
478C22—84no—
479C22—84no—
480C22—82no—
481C22—82no—
482C22—82no—
483C22—164no—
484C22—164no—
485C22#187512yes—
486C22#187512yes—
487C22#187512yes—
488C22#187512yes—
489C22#191212yes—
490C22#191312yes—
491C22C2×#38611yescharacteristic
492C22C2×#41611yescharacteristic
493C22C2×#41811yescharacteristic
494C22C2×#44912yes—
495C22C2×#48212yes—
496C4—22no—
497C4—22no—
498C4—22no—
499C4—22no—
500C4—44no—
501C4—44no—
502C4—84no—
503C4—88no—
504C4—82no—
505C2—21no—
506C2—22no—
507C2—21no—
508C2—22no—
509C2—22no—
510C2—22no—
511C2—22no—
512C2—162no—
513C2#920614yesdirect product factor
514C2C2×#187511yescharacteristic
515C2C2×#191211yescharacteristic
516C2C2×#191311yescharacteristic
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
4. C22×#324C2semidirect
5. #9206C2direct (N × H)
6. C2×#1289C2semidirect
13. #2618C4semidirect
15. C2×#398C4semidirect
17. C2×#394C4semidirect
20. #2313C22semidirect
32. C2×#324C22semidirect
33. #1289C22semidirect
78. #398C2×C4semidirect
82. #394C2×C4semidirect
513. C2#9206direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#26182
C2×#23132
C2×#15192
C22×#3242
#92062
C2×#12892