Finite Group Databasebeta

← C2×#9204 #197498

Subgroups — C2×#9204 #197498

Order 512.

1760 conjugacy classes of subgroups.

The subgroup lattice has 573 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2618C211yesmaximal, characteristic
2C2×#2316C211yesmaximal, characteristic, complemented
3C2×#1519C212yesmaximal
4C22×#324C211yesmaximal, characteristic, complemented
5#9204C218yesmaximal, direct product factor
6C2×#1289C212yesmaximal, complemented
7C22×#123—21no—
8C22×#121C2211yescharacteristic
9C22×#117—21no—
10C23×C4⋊C4C2211yescharacteristic
11C22×C22⋊C8C2212yes—
12#2618—22no—
13#2618—22no—
14#2618C414yescomplemented
15#2618C2214yes—
16C2×#398C412yescomplemented
17C2×#396—22no—
18C2×#394C412yescomplemented
19#2316—24no—
20#2316—22no—
21#2316C2214yescomplemented
22C2×#389—24no—
23C2×#386—21no—
24C2×#380—21no—
25C2×#380C2211yescharacteristic
26C2×#380C2212yes—
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—44no—
43C2×#123—44no—
44C2×#121—21no—
45C2×#121—21no—
46C2×#121—22no—
47C2×#121—22no—
48C2×#121—22no—
49C2×#121—22no—
50C2×#121C2×C412yes—
51C2×#121C2312yes—
52C2×#117—24no—
53C2×#117—24no—
54C2×#117—44no—
55C2×#117—44no—
56C22×C4⋊C4—22no—
57C22×C4⋊C4—22no—
58C22×C4⋊C4—22no—
59C22×C4⋊C4—22no—
60C22×C4⋊C4—22no—
61C22×C4⋊C4—22no—
62C22×C4⋊C4—22no—
63C22×C4⋊C4—21no—
64C22×C4⋊C4—22no—
65C22×C4⋊C4C2×C411yescharacteristic
66C22×C4⋊C4C2×C412yes—
67C22×C4⋊C4C2×C412yes—
68C22×C4⋊C4D412yes—
69C22×C4⋊C4D412yes—
70C22×C4⋊C4D412yes—
71C22×C4⋊C4C2312yes—
72C22×C22⋊C4—22no—
73C22×C22⋊C4—21no—
74C22×C22⋊C4—22no—
75C22×C22⋊C4—21no—
76C22×C22⋊C4—22no—
77C22×C42—22no—
78C2×C22⋊C8—22no—
79C2×C22⋊C8—22no—
80C2×C22⋊C8—22no—
81C2×C22⋊C8—22no—
82C2×C22⋊C8—22no—
83C2×C22⋊C8C2314yes—
84#398—24no—
85#398C2×C418yescomplemented
86#396—28no—
87#396—28no—
88#394—24no—
89#394C2×C418yescomplemented
90#389—216no—
91#389—216no—
92#386—24no—
93#386—24no—
94#380—24no—
95#380—22no—
96#380—22no—
97#380—24no—
98#380—21no—
99#380—21no—
100#380C2×C412yes—
101#380D414yes—
102#380C2312yes—
103#380C2314yes—
104C2×#95—22no—
105C2×#95—22no—
106C2×#95C2×C412yes—
107C2×#95D414yes—
108#327—216no—
109#324—28no—
110C25—21no—
111C25C2×D411yescharacteristic
112C22×D4—22no—
113C22×D4—22no—
114C22×D4—21no—
115C22×D4—22no—
116C22×D4—22no—
117C22×D4—41no—
118C22×D4—41no—
119C22×D4—41no—
120C22×D4—41no—
121C22×D4—42no—
122C22×D4—42no—
123C22×D4—42no—
124C22×D4—42no—
125C22×D4—82no—
126C22×D4C22×C412yes—
127C23×C4—21no—
128C23×C4—21no—
129C23×C4—21no—
130C23×C4—21no—
131C23×C4—22no—
132C23×C4—22no—
133C23×C4—21no—
134C23×C4—21no—
135C23×C4—21no—
136C23×C4—21no—
137C23×C4—22no—
138C23×C4—21no—
139C23×C4—24no—
140C23×C4—82no—
141C23×C4—82no—
142C23×C4C22⋊C412yes—
143C23×C4C22⋊C412yes—
144C23×C4C22⋊C412yes—
145C23×C4C22⋊C412yes—
146C23×C4D814yes—
147C23×C4SD1614yes—
148C23×C4C22×C411yescharacteristic
149C23×C4C22×C411yescharacteristic
150C23×C4C2×D412yes—
151C23×C4C2×D412yes—
152C23×C4C2×D411yescharacteristic
153C23×C4C2×D412yes—
154C23×C4C2×D411yescharacteristic
155C23×C4C2×D411yescharacteristic
156C23×C4C2×D412yes—
157C23×C4#4112yes—
158C23×C4#4112yes—
159C23×C4C2411yescharacteristic
160C22×C8—24no—
161C22×C8—42no—
162C22×C8—42no—
163C22×C8—84no—
164#123—416no—
165#123—48no—
166#123—48no—
167#121—24no—
168#121—24no—
169#121—24no—
170#121—24no—
171#121—42no—
172#121—42no—
173#121—42no—
174#121—42no—
175#117—48no—
176#117—416no—
177#117—48no—
178C2×C4⋊C4—24no—
179C2×C4⋊C4—24no—
180C2×C4⋊C4—24no—
181C2×C4⋊C4—24no—
182C2×C4⋊C4—24no—
183C2×C4⋊C4—24no—
184C2×C4⋊C4—24no—
185C2×C4⋊C4—24no—
186C2×C4⋊C4—24no—
187C2×C4⋊C4—24no—
188C2×C4⋊C4—24no—
189C2×C4⋊C4—24no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—28no—
193C2×C4⋊C4—28no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—28no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—24no—
198C2×C4⋊C4—24no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—48no—
201C2×C4⋊C4—48no—
202C2×C4⋊C4—82no—
203C2×C4⋊C4—84no—
204C2×C4⋊C4—84no—
205C2×C4⋊C4C22⋊C414yes—
206C2×C4⋊C4C22⋊C414yes—
207C2×C4⋊C4C22×C412yes—
208C2×C4⋊C4C22×C414yes—
209C2×C4⋊C4C2×D414yes—
210C2×C22⋊C4—22no—
211C2×C22⋊C4—24no—
212C2×C22⋊C4—24no—
213C2×C22⋊C4—22no—
214C2×C22⋊C4—22no—
215C2×C22⋊C4—28no—
216C2×C22⋊C4—22no—
217C2×C22⋊C4—28no—
218C2×C22⋊C4—44no—
219C2×C22⋊C4—42no—
220C2×C22⋊C4—42no—
221C2×C22⋊C4—44no—
222C2×C22⋊C4—48no—
223C2×C22⋊C4—42no—
224C2×C22⋊C4—42no—
225C2×C22⋊C4—84no—
226C2×C42—28no—
227C2×C42—48no—
228C22⋊C8—24no—
229C22⋊C8—24no—
230C22⋊C8—42no—
231C22⋊C8—42no—
232#95—28no—
233#95—28no—
234#95—22no—
235#95—24no—
236#95C22×C414yes—
237#95C2×D418yes—
238C24—22no—
239C24—42no—
240C24—42no—
241C24—44no—
242C24—44no—
243C24—41no—
244C24—41no—
245C24—42no—
246C24—42no—
247C24C2×C22⋊C411yescharacteristic
248C24C2×C22⋊C411yescharacteristic
249C24#12011yescharacteristic
250C24#12011yescharacteristic
251C24#12311yescharacteristic
252C24#12311yescharacteristic
253C24C2×D812yes—
254C24C2×SD1612yes—
255C24AGL(1,ℤ/8)12yes—
256C24#13712yes—
257C24C22×D411yescharacteristic
258C2×D4—24no—
259C2×D4—42no—
260C2×D4—42no—
261C2×D4—42no—
262C2×D4—42no—
263C2×D4—42no—
264C2×D4—42no—
265C2×D4—42no—
266C2×D4—42no—
267C2×D4—44no—
268C2×D4—44no—
269C2×D4—42no—
270C2×D4—84no—
271C2×D4—82no—
272C2×D4—84no—
273C2×D4—82no—
274C2×D4—82no—
275C2×D4—82no—
276C2×D4—82no—
277C2×D4—82no—
278C2×D4—82no—
279C2×D4—82no—
280C2×D4—82no—
281C2×D4—82no—
282C22×C4—22no—
283C22×C4—22no—
284C22×C4—22no—
285C22×C4—22no—
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—22no—
289C22×C4—22no—
290C22×C4—24no—
291C22×C4—24no—
292C22×C4—24no—
293C22×C4—24no—
294C22×C4—24no—
295C22×C4—24no—
296C22×C4—24no—
297C22×C4—24no—
298C22×C4—24no—
299C22×C4—24no—
300C22×C4—22no—
301C22×C4—22no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—22no—
306C22×C4—22no—
307C22×C4—24no—
308C22×C4—28no—
309C22×C4—42no—
310C22×C4—44no—
311C22×C4—44no—
312C22×C4—42no—
313C22×C4—44no—
314C22×C4—44no—
315C22×C4—44no—
316C22×C4—48no—
317C22×C4—48no—
318C22×C4—44no—
319C22×C4—41no—
320C22×C4—41no—
321C22×C4—44no—
322C22×C4—42no—
323C22×C4—42no—
324C22×C4—48no—
325C22×C4—81no—
326C22×C4—81no—
327C22×C4—82no—
328C22×C4—81no—
329C22×C4—82no—
330C22×C4—81no—
331C22×C4—82no—
332C22×C4—82no—
333C22×C4—82no—
334C22×C4—82no—
335C22×C4—82no—
336C22×C4#10218yes—
337C22×C4#10218yes—
338C22×C4C2×C22⋊C414yes—
339C22×C4C2×C22⋊C414yes—
340C22×C4C2×C22⋊C411yescharacteristic
341C22×C4C2×C22⋊C411yescharacteristic
342C22×C4C4×D412yes—
343C22×C4C4×D412yes—
344C22×C4C4×D412yes—
345C22×C4C4×D412yes—
346C22×C4#12012yes—
347C22×C4#12112yes—
348C22×C4#12112yes—
349C22×C4#12112yes—
350C22×C4#12112yes—
351C22×C4#12312yes—
352C22×C4C2×D814yes—
353C22×C4C2×SD1614yes—
354C22×C4C23×C411yescharacteristic
355C22×C4C22×D412yes—
356C22×C4C22×D411yescharacteristic
357C22×C4C2×#4112yes—
358C2×C8—44no—
359C2×C8—84no—
360C2×C8—84no—
361C2×C8—84no—
362C4⋊C4—28no—
363C4⋊C4—28no—
364C4⋊C4—28no—
365C4⋊C4—28no—
366C4⋊C4—24no—
367C4⋊C4—416no—
368C4⋊C4—42no—
369C4⋊C4—416no—
370C4⋊C4—82no—
371C4⋊C4—84no—
372C4⋊C4—82no—
373C4⋊C4—84no—
374C4⋊C4—88no—
375C22⋊C4—44no—
376C22⋊C4—42no—
377C22⋊C4—42no—
378C22⋊C4—44no—
379C22⋊C4—44no—
380C22⋊C4—48no—
381C22⋊C4—44no—
382C22⋊C4—44no—
383C22⋊C4—416no—
384C22⋊C4—88no—
385C42—416no—
386C23—22no—
387C23—22no—
388C23—22no—
389C23—22no—
390C23—22no—
391C23—22no—
392C23—41no—
393C23—41no—
394C23—41no—
395C23—41no—
396C23—41no—
397C23—41no—
398C23—41no—
399C23—44no—
400C23—44no—
401C23—44no—
402C23—44no—
403C23—41no—
404C23—44no—
405C23—44no—
406C23—42no—
407C23—42no—
408C23—42no—
409C23—82no—
410C23—82no—
411C23—88no—
412C23—88no—
413C23—82no—
414C23—82no—
415C23—82no—
416C23—82no—
417C23#38611yescharacteristic
418C23#38611yescharacteristic
419C23#38611yescharacteristic
420C23#38611yescharacteristic
421C23C2×#10212yes—
422C23C2×#10212yes—
423C23#41712yes—
424C23#41712yes—
425C23#44712yes—
426C23#44712yes—
427C23#44812yes—
428C23#44812yes—
429C23#48012yes—
430C23#48012yes—
431C23#48112yes—
432C23#48112yes—
433C23C22×C22⋊C411yescharacteristic
434C23C2×#12011yescharacteristic
435C23C2×#12311yescharacteristic
436C23C22×D811yescharacteristic
437C23C22×SD1611yescharacteristic
438C23C2×AGL(1,ℤ/8)11yescharacteristic
439C23C2×#13711yescharacteristic
440D4—84no—
441D4—84no—
442D4—82no—
443D4—82no—
444D4—84no—
445D4—84no—
446D4—82no—
447D4—82no—
448C2×C4—24no—
449C2×C4—24no—
450C2×C4—24no—
451C2×C4—28no—
452C2×C4—28no—
453C2×C4—28no—
454C2×C4—28no—
455C2×C4—24no—
456C2×C4—24no—
457C2×C4—24no—
458C2×C4—24no—
459C2×C4—24no—
460C2×C4—24no—
461C2×C4—41no—
462C2×C4—42no—
463C2×C4—41no—
464C2×C4—42no—
465C2×C4—42no—
466C2×C4—42no—
467C2×C4—41no—
468C2×C4—42no—
469C2×C4—42no—
470C2×C4—42no—
471C2×C4—42no—
472C2×C4—42no—
473C2×C4—42no—
474C2×C4—41no—
475C2×C4—44no—
476C2×C4—44no—
477C2×C4—44no—
478C2×C4—44no—
479C2×C4—44no—
480C2×C4—48no—
481C2×C4—48no—
482C2×C4—44no—
483C2×C4—44no—
484C2×C4—42no—
485C2×C4—42no—
486C2×C4—48no—
487C2×C4—42no—
488C2×C4—82no—
489C2×C4—82no—
490C2×C4—82no—
491C2×C4—816no—
492C2×C4—88no—
493C2×C4—82no—
494C2×C4—82no—
495C2×C4#38612yes—
496C2×C4#38612yes—
497C2×C4C2×#10218yes—
498C2×C4C2×C4×D412yes—
499C2×C4C2×#12112yes—
500C8—84no—
501C22—21no—
502C22—21no—
503C22—21no—
504C22—21no—
505C22—21no—
506C22—21no—
507C22—21no—
508C22—21no—
509C22—21no—
510C22—21no—
511C22—21no—
512C22—21no—
513C22—24no—
514C22—24no—
515C22—24no—
516C22—24no—
517C22—24no—
518C22—24no—
519C22—41no—
520C22—41no—
521C22—41no—
522C22—41no—
523C22—42no—
524C22—42no—
525C22—42no—
526C22—42no—
527C22—42no—
528C22—42no—
529C22—84no—
530C22—84no—
531C22—82no—
532C22—82no—
533C22—82no—
534C22—164no—
535C22—164no—
536C22#187412yes—
537C22#187412yes—
538C22#187412yes—
539C22#187412yes—
540C22#191212yes—
541C22#191312yes—
542C22C2×#38611yescharacteristic
543C22C22×#10211yescharacteristic
544C22C2×#41711yescharacteristic
545C22C2×#44711yescharacteristic
546C22C2×#44811yescharacteristic
547C22C2×#48011yescharacteristic
548C22C2×#48111yescharacteristic
549C4—22no—
550C4—22no—
551C4—22no—
552C4—22no—
553C4—28no—
554C4—44no—
555C4—88no—
556C4—84no—
557C4—82no—
558C2—21no—
559C2—21no—
560C2—21no—
561C2—22no—
562C2—21no—
563C2—21no—
564C2—22no—
565C2—22no—
566C2—21no—
567C2—162no—
568C2#920414yesdirect product factor
569C2C2×#187411yescharacteristic
570C2C2×#191211yescharacteristic
571C2C2×#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×#2316C2semidirect
4. C22×#324C2semidirect
5. #9204C2direct (N × H)
6. C2×#1289C2semidirect
14. #2618C4semidirect
16. C2×#398C4semidirect
18. C2×#394C4semidirect
21. #2316C22semidirect
32. C2×#324C22semidirect
33. #1289C22semidirect
85. #398C2×C4semidirect
89. #394C2×C4semidirect
568. C2#9204direct (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×#23162
C2×#15192
C22×#3242
#92042
C2×#12892