Finite Group Databasebeta

← C2×#7215 #127191

Subgroups — C2×#7215 #127191

Order 512.

1150 conjugacy classes of subgroups.

The subgroup lattice has 586 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1821C211yesmaximal, characteristic
2C2×#1463C211yesmaximal, characteristic
3C2×#1436C211yesmaximal, characteristic
4#7215C214yesmaximal, direct product factor
5C23×C4⋊C4C2211yescharacteristic
6C23×C42C411yescharacteristic
7C2×#407—21no—
8C2×#407—21no—
9C22×#95—21no—
10C22×#95—21no—
11C22×#95C411yescharacteristic
12#1821—21no—
13#1821—21no—
14#1821C412yescomplemented
15#1821C2212yes—
16C2×#343—21no—
17C2×#343C412yescomplemented
18#1463—21no—
19#1463—21no—
20#1463C412yescomplemented
21#1463C2212yes—
22C2×#324—21no—
23C2×#324C412yescomplemented
24#1436—21no—
25#1436—21no—
26#1436C412yes—
27#1436C2212yes—
28C24×C4—21no—
29C24×C4—21no—
30C24×C4C2×C411yescharacteristic
31C24×C4D411yescharacteristic
32C24×C4D411yescharacteristic
33C22×M16—22no—
34C22×M16—41no—
35C22×C4⋊C4—41no—
36C22×C4⋊C4—41no—
37C22×C4⋊C4—41no—
38C22×C4⋊C4—41no—
39C22×C4⋊C4C2×C411yescharacteristic
40C22×C4⋊C4C2×C411yescharacteristic
41C22×C4⋊C4C2×C411yescharacteristic
42C22×C4⋊C4C2×C411yescharacteristic
43C22×C4⋊C4C2×C411yescharacteristic
44C22×C4⋊C4D411yescharacteristic
45C22×C4⋊C4D411yescharacteristic
46C22×C4⋊C4D411yescharacteristic
47C22×C4⋊C4D411yescharacteristic
48C22×C4⋊C4Q811yescharacteristic
49C22×C4⋊C4Q811yescharacteristic
50C22×C4⋊C4C2311yescharacteristic
51C22×C42—21no—
52C22×C42—41no—
53C22×C42—41no—
54C22×C42—44no—
55C22×C42C2×C412yes—
56#407—22no—
57#407—22no—
58#407—22no—
59#407—22no—
60#407—41no—
61#407—41no—
62#407—41no—
63#407—41no—
64C2×C22⋊C8—22no—
65C2×C22⋊C8—41no—
66C2×C22⋊C8—41no—
67C2×C22⋊C8—41no—
68C2×#95—28no—
69C2×#95—22no—
70C2×#95—22no—
71C2×#95—21no—
72C2×#95—22no—
73C2×#95—22no—
74C2×#95—41no—
75C2×#95—41no—
76C2×#95—44no—
77C2×#95C2×C412yes—
78#343—22no—
79#343—22no—
80#343—22no—
81#343C2×C414yescomplemented
82#324—22no—
83#324—22no—
84#324—22no—
85#324C2×C414yescomplemented
86C25C22⋊C411yescharacteristic
87C23×C4—24no—
88C23×C4—24no—
89C23×C4—24no—
90C23×C4—21no—
91C23×C4—21no—
92C23×C4—21no—
93C23×C4—21no—
94C23×C4—21no—
95C23×C4—21no—
96C23×C4—21no—
97C23×C4—21no—
98C23×C4—21no—
99C23×C4—22no—
100C23×C4—21no—
101C23×C4—21no—
102C23×C4—21no—
103C23×C4—24no—
104C23×C4—24no—
105C23×C4—22no—
106C23×C4—42no—
107C23×C4—42no—
108C23×C4—42no—
109C23×C4—42no—
110C23×C4—41no—
111C23×C4—41no—
112C23×C4—42no—
113C23×C4—42no—
114C23×C4—42no—
115C23×C4—42no—
116C23×C4—42no—
117C23×C4—42no—
118C23×C4—42no—
119C23×C4—42no—
120C23×C4—42no—
121C23×C4—41no—
122C23×C4—41no—
123C23×C4C4211yescharacteristic
124C23×C4C4211yescharacteristic
125C23×C4C22⋊C411yescharacteristic
126C23×C4C22⋊C411yescharacteristic
127C23×C4C22⋊C411yescharacteristic
128C23×C4C22⋊C411yescharacteristic
129C23×C4C22⋊C411yescharacteristic
130C23×C4C22⋊C411yescharacteristic
131C23×C4C22⋊C411yescharacteristic
132C23×C4C4⋊C411yescharacteristic
133C23×C4C4⋊C411yescharacteristic
134C23×C4C4⋊C411yescharacteristic
135C23×C4C4⋊C411yescharacteristic
136C23×C4C4⋊C411yescharacteristic
137C23×C4C4⋊C411yescharacteristic
138C23×C4C22×C411yescharacteristic
139C23×C4C2×D411yescharacteristic
140C23×C4C2×D411yescharacteristic
141C2×M16—24no—
142C2×M16—42no—
143C2×M16—42no—
144C2×M16—42no—
145C2×M16—44no—
146C2×M16—44no—
147C2×M16—82no—
148C2×M16—82no—
149C22×C8—42no—
150C22×C8—42no—
151C2×C4⋊C4—41no—
152C2×C4⋊C4—41no—
153C2×C4⋊C4—41no—
154C2×C4⋊C4—41no—
155C2×C4⋊C4—41no—
156C2×C4⋊C4—41no—
157C2×C4⋊C4—41no—
158C2×C4⋊C4—41no—
159C2×C4⋊C4—41no—
160C2×C4⋊C4—41no—
161C2×C4⋊C4—41no—
162C2×C4⋊C4—41no—
163C2×C4⋊C4—41no—
164C2×C4⋊C4—41no—
165C2×C4⋊C4—41no—
166C2×C4⋊C4—41no—
167C2×C4⋊C4—41no—
168C2×C4⋊C4—41no—
169C2×C4⋊C4—41no—
170C2×C4⋊C4—41no—
171C2×C4⋊C4—41no—
172C2×C4⋊C4—41no—
173C2×C4⋊C4—41no—
174C2×C4⋊C4—41no—
175C2×C4⋊C4C4211yescharacteristic
176C2×C4⋊C4C4211yescharacteristic
177C2×C4⋊C4C22⋊C411yescharacteristic
178C2×C4⋊C4C22⋊C411yescharacteristic
179C2×C4⋊C4C22⋊C411yescharacteristic
180C2×C4⋊C4C22⋊C411yescharacteristic
181C2×C4⋊C4C4⋊C411yescharacteristic
182C2×C4⋊C4C4⋊C411yescharacteristic
183C2×C4⋊C4C4⋊C411yescharacteristic
184C2×C4⋊C4C4⋊C411yescharacteristic
185C2×C4⋊C4C4⋊C411yescharacteristic
186C2×C4⋊C4C4⋊C411yescharacteristic
187C2×C4⋊C4C22×C411yescharacteristic
188C2×C4⋊C4C22×C411yescharacteristic
189C2×C4⋊C4C2×D411yescharacteristic
190C2×C4⋊C4C2×Q811yescharacteristic
191C2×C42—42no—
192C2×C42—44no—
193C2×C42—42no—
194C2×C42—44no—
195C2×C42—48no—
196C2×C42—44no—
197C2×C42—44no—
198C22⋊C8—24no—
199C22⋊C8—42no—
200C22⋊C8—42no—
201C22⋊C8—42no—
202C22⋊C8—42no—
203C22⋊C8—42no—
204C22⋊C8—42no—
205C22⋊C8—42no—
206#95—28no—
207#95—28no—
208#95—42no—
209#95—44no—
210#95—44no—
211#95—42no—
212C24—22no—
213C24—22no—
214C24—22no—
215C24—22no—
216C24—22no—
217C24—22no—
218C24#9511yescharacteristic
219C24#9511yescharacteristic
220C24#9911yescharacteristic
221C24#9911yescharacteristic
222C24#9911yescharacteristic
223C24#9911yescharacteristic
224C24C2×C22⋊C411yescharacteristic
225C22×C4—22no—
226C22×C4—21no—
227C22×C4—21no—
228C22×C4—24no—
229C22×C4—24no—
230C22×C4—216no—
231C22×C4—24no—
232C22×C4—22no—
233C22×C4—22no—
234C22×C4—21no—
235C22×C4—21no—
236C22×C4—21no—
237C22×C4—21no—
238C22×C4—21no—
239C22×C4—21no—
240C22×C4—21no—
241C22×C4—21no—
242C22×C4—21no—
243C22×C4—21no—
244C22×C4—21no—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—21no—
248C22×C4—21no—
249C22×C4—21no—
250C22×C4—21no—
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—21no—
256C22×C4—21no—
257C22×C4—21no—
258C22×C4—21no—
259C22×C4—21no—
260C22×C4—21no—
261C22×C4—21no—
262C22×C4—21no—
263C22×C4—21no—
264C22×C4—21no—
265C22×C4—21no—
266C22×C4—21no—
267C22×C4—21no—
268C22×C4—22no—
269C22×C4—22no—
270C22×C4—22no—
271C22×C4—22no—
272C22×C4—24no—
273C22×C4—41no—
274C22×C4—41no—
275C22×C4—41no—
276C22×C4—41no—
277C22×C4—44no—
278C22×C4—44no—
279C22×C4—44no—
280C22×C4—44no—
281C22×C4—42no—
282C22×C4—42no—
283C22×C4—44no—
284C22×C4—48no—
285C22×C4—48no—
286C22×C4—48no—
287C22×C4—44no—
288C22×C4—44no—
289C22×C4—42no—
290C22×C4—44no—
291C22×C4—44no—
292C22×C4—42no—
293C22×C4—42no—
294C22×C4—44no—
295C22×C4—44no—
296C22×C4—42no—
297C22×C4—44no—
298C22×C4—42no—
299C22×C4—42no—
300C22×C4—42no—
301C22×C4—42no—
302C22×C4—41no—
303C22×C4—41no—
304C22×C4—42no—
305C22×C4—42no—
306C22×C4—41no—
307C22×C4—42no—
308C22×C4—42no—
309C22×C4—41no—
310C22×C4—41no—
311C22×C4—41no—
312C22×C4—41no—
313C22×C4—42no—
314C22×C4—42no—
315C22×C4—42no—
316C22×C4—42no—
317C22×C4—41no—
318C22×C4—42no—
319C22×C4—48no—
320C22×C4—48no—
321C22×C4—44no—
322C22×C4—44no—
323C22×C4—42no—
324C22×C4—42no—
325C22×C4—42no—
326C22×C4—42no—
327C22×C4—42no—
328C22×C4—81no—
329C22×C4—81no—
330C22×C4—82no—
331C22×C4—82no—
332C22×C4—84no—
333C22×C4—84no—
334C22×C4—82no—
335C22×C4—82no—
336C22×C4—82no—
337C22×C4—82no—
338C22×C4#9511yescharacteristic
339C22×C4#9511yescharacteristic
340C22×C4#9511yescharacteristic
341C22×C4#9511yescharacteristic
342C22×C4#9511yescharacteristic
343C22×C4#9511yescharacteristic
344C22×C4#10414yes—
345C22×C4C2×C4211yescharacteristic
346C22×C4C2×C22⋊C411yescharacteristic
347C22×C4C2×C22⋊C411yescharacteristic
348C22×C4C2×C4⋊C411yescharacteristic
349C22×C4C2×C4⋊C411yescharacteristic
350C22×C4C2×C4⋊C411yescharacteristic
351M16—44no—
352M16—44no—
353M16—84no—
354M16—82no—
355M16—82no—
356M16—84no—
357C2×C8—44no—
358C2×C8—44no—
359C2×C8—84no—
360C2×C8—84no—
361C4⋊C4—41no—
362C4⋊C4—41no—
363C4⋊C4—41no—
364C4⋊C4—41no—
365C4⋊C4—41no—
366C4⋊C4—41no—
367C4⋊C4—41no—
368C4⋊C4—41no—
369C4⋊C4—41no—
370C4⋊C4—41no—
371C4⋊C4—41no—
372C4⋊C4—41no—
373C4⋊C4—41no—
374C4⋊C4—41no—
375C4⋊C4—41no—
376C4⋊C4—41no—
377C42—44no—
378C42—44no—
379C42—48no—
380C23—24no—
381C23—24no—
382C23—24no—
383C23—24no—
384C23—24no—
385C23—24no—
386C23—24no—
387C23—24no—
388C23—22no—
389C23—22no—
390C23—22no—
391C23—22no—
392C23—21no—
393C23—21no—
394C23—21no—
395C23—21no—
396C23—21no—
397C23—21no—
398C23—21no—
399C23—21no—
400C23—22no—
401C23—22no—
402C23—22no—
403C23—22no—
404C23—22no—
405C23—22no—
406C23—22no—
407C23—22no—
408C23—21no—
409C23—21no—
410C23—21no—
411C23—21no—
412C23H(3,ℤ/4)12yes—
413C23#33912yes—
414C23#34211yescharacteristic
415C23#34211yescharacteristic
416C23#34211yescharacteristic
417C23#34211yescharacteristic
418C23AΣL(1,𝔽16)12yes—
419C23#35212yes—
420C23#35512yes—
421C23#35612yes—
422C23C2×#9511yescharacteristic
423C23C2×#9911yescharacteristic
424C23C2×#9911yescharacteristic
425C2×C4—24no—
426C2×C4—24no—
427C2×C4—21no—
428C2×C4—21no—
429C2×C4—21no—
430C2×C4—21no—
431C2×C4—28no—
432C2×C4—21no—
433C2×C4—21no—
434C2×C4—21no—
435C2×C4—21no—
436C2×C4—21no—
437C2×C4—21no—
438C2×C4—21no—
439C2×C4—21no—
440C2×C4—21no—
441C2×C4—21no—
442C2×C4—21no—
443C2×C4—21no—
444C2×C4—21no—
445C2×C4—21no—
446C2×C4—42no—
447C2×C4—42no—
448C2×C4—42no—
449C2×C4—42no—
450C2×C4—42no—
451C2×C4—42no—
452C2×C4—42no—
453C2×C4—42no—
454C2×C4—44no—
455C2×C4—44no—
456C2×C4—416no—
457C2×C4—44no—
458C2×C4—44no—
459C2×C4—44no—
460C2×C4—416no—
461C2×C4—42no—
462C2×C4—42no—
463C2×C4—42no—
464C2×C4—42no—
465C2×C4—42no—
466C2×C4—42no—
467C2×C4—42no—
468C2×C4—42no—
469C2×C4—41no—
470C2×C4—41no—
471C2×C4—44no—
472C2×C4—41no—
473C2×C4—41no—
474C2×C4—41no—
475C2×C4—41no—
476C2×C4—41no—
477C2×C4—41no—
478C2×C4—42no—
479C2×C4—42no—
480C2×C4—42no—
481C2×C4—42no—
482C2×C4—42no—
483C2×C4—42no—
484C2×C4—42no—
485C2×C4—42no—
486C2×C4—41no—
487C2×C4—41no—
488C2×C4—41no—
489C2×C4—41no—
490C2×C4—41no—
491C2×C4—41no—
492C2×C4—41no—
493C2×C4—41no—
494C2×C4—44no—
495C2×C4—44no—
496C2×C4—44no—
497C2×C4—81no—
498C2×C4—81no—
499C2×C4—81no—
500C2×C4—81no—
501C2×C4—81no—
502C2×C4—81no—
503C2×C4—82no—
504C2×C4—88no—
505C2×C4—82no—
506C2×C4—82no—
507C2×C4—88no—
508C2×C4—82no—
509C2×C4#33912yes—
510C2×C4C2×#10412yes—
511C8—84no—
512C8—84no—
513C22—21no—
514C22—21no—
515C22—21no—
516C22—24no—
517C22—21no—
518C22—21no—
519C22—24no—
520C22—21no—
521C22—24no—
522C22—24no—
523C22—21no—
524C22—21no—
525C22—24no—
526C22—24no—
527C22—21no—
528C22—21no—
529C22—21no—
530C22—21no—
531C22—21no—
532C22—21no—
533C22—21no—
534C22—21no—
535C22—21no—
536C22—21no—
537C22—21no—
538C22—21no—
539C22—24no—
540C22—24no—
541C22—24no—
542C22—24no—
543C22—24no—
544C22—24no—
545C22#128412yes—
546C22#128412yes—
547C22#139012yes—
548C22#139012yes—
549C22#139412yes—
550C22#139412yes—
551C22C2×H(3,ℤ/4)11yescharacteristic
552C22C2×#33911yescharacteristic
553C22C2×#34211yescharacteristic
554C22C2×AΣL(1,𝔽16)11yescharacteristic
555C22C2×#35211yescharacteristic
556C22C2×#35511yescharacteristic
557C22C2×#35611yescharacteristic
558C4—22no—
559C4—22no—
560C4—48no—
561C4—41no—
562C4—41no—
563C4—41no—
564C4—41no—
565C4—41no—
566C4—41no—
567C4—41no—
568C4—41no—
569C4—41no—
570C4—41no—
571C4—84no—
572C2—21no—
573C2—21no—
574C2—22no—
575C2—21no—
576C2—21no—
577C2—22no—
578C2—22no—
579C2—21no—
580C2—21no—
581C2#721514yesdirect product factor
582C2C2×#128411yescharacteristic
583C2C2×#139011yescharacteristic
584C2C2×#139411yescharacteristic
subgroupG/H
center, socleC23#342
commutatorC22×C4C2×C42
Frattini, cosocleC22×C4⋊C4C23
prime radicalC25C22⋊C4
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C25 ◁ C23×C4⋊C4 ◁ G
LECSG ▷ C22×C4⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #7215C2direct (N × H)
14. #1821C4semidirect
17. C2×#343C4semidirect
20. #1463C4semidirect
23. C2×#324C4semidirect
81. #343C2×C4semidirect
85. #324C2×C4semidirect
581. C2#7215direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#18212
C2×#14632
C2×#14362
#72152