Finite Group Databasebeta

← C2×#13297 #495405

Subgroups — C2×#13297 #495405

Order 512.

1314 conjugacy classes of subgroups.

The subgroup lattice has 554 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2579C211yesmaximal, characteristic
2C2×#1800C211yesmaximal, characteristic
3C2×C4×#99C211yesmaximal, characteristic
4#13297C218yesmaximal, direct product factor
5C2×#1390C212yesmaximal
6C2×#1390C212yesmaximal
7C22×C4×D4C2211yescharacteristic
8C23×C4⋊C4C411yescharacteristic, complemented
9C2×#423—21no—
10C22×#100C2212yes—
11C22×#99C2212yes—
12#2579—22no—
13#2579—22no—
14#2579C414yescomplemented
15#2579C2214yes—
16C2×#398—21no—
17C2×#396—21no—
18C2×#387—21no—
19C2×#386C2211yescharacteristic
20C2×#386C2211yescharacteristic
21C2×#385—21no—
22C2×#385C411yescharacteristic, complemented
23C2×#385C411yescharacteristic, complemented
24C2×#380C411yescharacteristic, complemented
25C2×C4×C22⋊C4—21no—
26#1800—22no—
27#1800C2214yes—
28C4×#99—22no—
29C4×#99C2214yes—
30C2×#343—21no—
31C2×#342—21no—
32#1390C2218yes—
33#1390C2218yes—
34C23×D4C2311yescharacteristic
35C24×C4C2×C411yescharacteristic
36C24×C4C2×C411yescharacteristic
37C24×C4C2×C411yescharacteristic
38C24×C4D411yescharacteristic
39C22×M16—22no—
40C2×C4×D4—41no—
41C2×C4×D4—41no—
42C2×C4×D4—42no—
43C2×C4×D4C2×C412yes—
44C2×C4×D4D412yes—
45C2×C4×D4D412yes—
46C2×C4×D4C2312yes—
47C22×C4⋊C4—22no—
48C22×C4⋊C4—21no—
49C22×C4⋊C4—21no—
50C22×C4⋊C4—21no—
51C22×C4⋊C4—21no—
52C22×C4⋊C4—21no—
53C22×C4⋊C4—42no—
54C22×C4⋊C4—41no—
55C22×C4⋊C4—41no—
56C22×C4⋊C4C2×C414yescomplemented
57C22×C4⋊C4D411yescharacteristic
58C22×C22⋊C4—22no—
59C22×C22⋊C4—21no—
60C22×C22⋊C4—21no—
61C22×C22⋊C4C2×C411yescharacteristic
62C22×C22⋊C4C2×C411yescharacteristic
63C22×C22⋊C4C2×C411yescharacteristic
64C22×C22⋊C4D411yescharacteristic
65C22×C42—41no—
66C22×C42D411yescharacteristic
67#423—24no—
68#423—42no—
69C2×C4⋊C8—41no—
70C2×C4⋊C8—41no—
71C2×#100—28no—
72C2×#100—22no—
73C2×#100C2314yes—
74C2×#99—22no—
75C2×#99—24no—
76C2×#99Q818yes—
77C2×#99C2314yes—
78#398—24no—
79#398—24no—
80#396—24no—
81#396—24no—
82#387—24no—
83#387—24no—
84#386—21no—
85#386—21no—
86#386—21no—
87#386—21no—
88#386C2×C412yes—
89#386C2×C412yes—
90#386C2312yes—
91#386C2312yes—
92#385—24no—
93#385—24no—
94#385—22no—
95#385—22no—
96#385C2×C414yescomplemented
97#385C2×C414yescomplemented
98#380—22no—
99#380C2×C414yescomplemented
100C4×C22⋊C4—24no—
101C4×C22⋊C4—42no—
102C2×#95—21no—
103C2×#95—21no—
104C2×#95—21no—
105C2×#95—21no—
106C2×#95—21no—
107C2×#95—41no—
108#343—24no—
109#342—24no—
110C25C22×C411yescharacteristic
111C25C2×D411yescharacteristic
112C22×D4—41no—
113C22×D4—41no—
114C22×D4—42no—
115C22×D4C22×C411yescharacteristic
116C22×D4C2×D411yescharacteristic
117C22×D4C2×D411yescharacteristic
118C22×D4C2×Q814yes—
119C22×D4#4114yes—
120C22×D4C2411yescharacteristic
121C23×C4—21no—
122C23×C4—21no—
123C23×C4—22no—
124C23×C4—21no—
125C23×C4—21no—
126C23×C4—21no—
127C23×C4—22no—
128C23×C4—21no—
129C23×C4—21no—
130C23×C4—21no—
131C23×C4—21no—
132C23×C4—21no—
133C23×C4—44no—
134C23×C4—41no—
135C23×C4—41no—
136C23×C4—42no—
137C23×C4—41no—
138C23×C4—41no—
139C23×C4—82no—
140C23×C4—82no—
141C23×C4—82no—
142C23×C4C22⋊C411yescharacteristic
143C23×C4C22⋊C411yescharacteristic
144C23×C4C22⋊C411yescharacteristic
145C23×C4C22⋊C412yes—
146C23×C4C22⋊C412yes—
147C23×C4C22⋊C411yescharacteristic
148C23×C4C22×C412yes—
149C23×C4C22×C412yes—
150C23×C4C22×C412yes—
151C23×C4C2×D411yescharacteristic
152C23×C4C2×D412yes—
153C2×M16—24no—
154C2×M16—42no—
155C2×M16—44no—
156C2×M16—44no—
157C22×C8—42no—
158C4×D4—42no—
159C4×D4—42no—
160C4×D4—42no—
161C4×D4—42no—
162C4×D4—42no—
163C4×D4—42no—
164C4×D4—42no—
165C4×D4—42no—
166C2×C4⋊C4—24no—
167C2×C4⋊C4—24no—
168C2×C4⋊C4—24no—
169C2×C4⋊C4—24no—
170C2×C4⋊C4—24no—
171C2×C4⋊C4—41no—
172C2×C4⋊C4—41no—
173C2×C4⋊C4—44no—
174C2×C4⋊C4—42no—
175C2×C4⋊C4—42no—
176C2×C4⋊C4—44no—
177C2×C4⋊C4—44no—
178C2×C4⋊C4—42no—
179C2×C4⋊C4—41no—
180C2×C4⋊C4—41no—
181C2×C4⋊C4—44no—
182C2×C4⋊C4—44no—
183C2×C4⋊C4—82no—
184C2×C4⋊C4—82no—
185C2×C4⋊C4C22⋊C412yes—
186C2×C4⋊C4C2×D412yes—
187C2×C22⋊C4—28no—
188C2×C22⋊C4—24no—
189C2×C22⋊C4—22no—
190C2×C22⋊C4—21no—
191C2×C22⋊C4—21no—
192C2×C22⋊C4—22no—
193C2×C22⋊C4—22no—
194C2×C22⋊C4—22no—
195C2×C22⋊C4—41no—
196C2×C22⋊C4—41no—
197C2×C22⋊C4—41no—
198C2×C22⋊C4—41no—
199C2×C22⋊C4—44no—
200C2×C22⋊C4—44no—
201C2×C22⋊C4—42no—
202C2×C22⋊C4—42no—
203C2×C22⋊C4—42no—
204C2×C22⋊C4—82no—
205C2×C22⋊C4C22⋊C412yes—
206C2×C22⋊C4C22⋊C412yes—
207C2×C22⋊C4C22×C412yes—
208C2×C22⋊C4C22×C412yes—
209C2×C22⋊C4C22×C412yes—
210C2×C22⋊C4C2×D412yes—
211C2×C42—21no—
212C2×C42—21no—
213C2×C42—22no—
214C2×C42—44no—
215C2×C42—84no—
216C2×C42C22⋊C412yes—
217C2×C42C2×D412yes—
218C4⋊C8—44no—
219C4⋊C8—44no—
220#100—28no—
221#100—44no—
222#99—28no—
223#99—24no—
224#99C2×Q818yes—
225#95—24no—
226#95—24no—
227#95—24no—
228#95—22no—
229#95—22no—
230#95—22no—
231#95—22no—
232#95—44no—
233C24—21no—
234C24—21no—
235C24—21no—
236C24—22no—
237C24—21no—
238C24—22no—
239C24—42no—
240C24—42no—
241C24—42no—
242C24—42no—
243C24C2×C22⋊C411yescharacteristic
244C24C2×C22⋊C411yescharacteristic
245C24C2×C22⋊C411yescharacteristic
246C24C4×Q814yes—
247C24C22⋊Q814yes—
248C24C23×C411yescharacteristic
249C24C22×D411yescharacteristic
250C2×D4—41no—
251C2×D4—41no—
252C2×D4—41no—
253C2×D4—41no—
254C2×D4—41no—
255C2×D4—41no—
256C2×D4—41no—
257C2×D4—44no—
258C2×D4—44no—
259C2×D4—44no—
260C2×D4—44no—
261C2×D4—41no—
262C2×D4C4×Q814yes—
263C2×D4C22⋊Q814yes—
264C2×D4#12412yes—
265C2×D4C4⋊Q812yes—
266C2×D4C22×Q812yes—
267C2×D4C2×#4112yes—
268C22×C4—21no—
269C22×C4—21no—
270C22×C4—22no—
271C22×C4—24no—
272C22×C4—22no—
273C22×C4—21no—
274C22×C4—21no—
275C22×C4—21no—
276C22×C4—21no—
277C22×C4—21no—
278C22×C4—21no—
279C22×C4—22no—
280C22×C4—22no—
281C22×C4—22no—
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—42no—
291C22×C4—44no—
292C22×C4—44no—
293C22×C4—44no—
294C22×C4—42no—
295C22×C4—42no—
296C22×C4—48no—
297C22×C4—44no—
298C22×C4—41no—
299C22×C4—41no—
300C22×C4—42no—
301C22×C4—42no—
302C22×C4—44no—
303C22×C4—42no—
304C22×C4—42no—
305C22×C4—44no—
306C22×C4—48no—
307C22×C4—42no—
308C22×C4—42no—
309C22×C4—42no—
310C22×C4—42no—
311C22×C4—42no—
312C22×C4—42no—
313C22×C4—42no—
314C22×C4—41no—
315C22×C4—41no—
316C22×C4—42no—
317C22×C4—42no—
318C22×C4—42no—
319C22×C4—82no—
320C22×C4—82no—
321C22×C4—82no—
322C22×C4—82no—
323C22×C4—82no—
324C22×C4—84no—
325C22×C4—84no—
326C22×C4—82no—
327C22×C4—82no—
328C22×C4—82no—
329C22×C4—82no—
330C22×C4—84no—
331C22×C4—84no—
332C22×C4—84no—
333C22×C4—84no—
334C22×C4#9911yescharacteristic
335C22×C4#9911yescharacteristic
336C22×C4#9912yes—
337C22×C4C2×C22⋊C411yescharacteristic
338C22×C4C2×C22⋊C412yes—
339C22×C4C2×C22⋊C412yes—
340C22×C4C2×C22⋊C412yes—
341C22×C4C2×C22⋊C412yes—
342C22×C4#12412yes—
343C22×C4C4⋊Q812yes—
344C22×C4C22×D411yescharacteristic
345M16—44no—
346M16—44no—
347M16—84no—
348C2×C8—44no—
349C2×C8—84no—
350C4⋊C4—42no—
351C4⋊C4—42no—
352C4⋊C4—48no—
353C4⋊C4—48no—
354C4⋊C4—44no—
355C4⋊C4—44no—
356C4⋊C4—84no—
357C4⋊C4—84no—
358C22⋊C4—28no—
359C22⋊C4—42no—
360C22⋊C4—42no—
361C22⋊C4—42no—
362C22⋊C4—42no—
363C22⋊C4—44no—
364C22⋊C4—44no—
365C22⋊C4—44no—
366C22⋊C4—48no—
367C22⋊C4—42no—
368C22⋊C4—42no—
369C22⋊C4—42no—
370C22⋊C4—42no—
371C22⋊C4—82no—
372C22⋊C4—82no—
373C42—22no—
374C42—22no—
375C42—22no—
376C42—22no—
377C42—88no—
378C23—22no—
379C23—22no—
380C23—21no—
381C23—21no—
382C23—21no—
383C23—21no—
384C23—21no—
385C23—21no—
386C23—24no—
387C23—24no—
388C23—24no—
389C23—21no—
390C23—21no—
391C23—24no—
392C23—41no—
393C23—41no—
394C23—41no—
395C23—41no—
396C23—48no—
397C23—42no—
398C23—42no—
399C23—48no—
400C23—41no—
401C23—41no—
402C23—41no—
403C23—41no—
404C23—48no—
405C23—42no—
406C23—42no—
407C23—42no—
408C23—42no—
409C23—81no—
410C23—81no—
411C23—84no—
412C23#39112yes—
413C23#39112yes—
414C23#39112yes—
415C23C2×#9911yescharacteristic
416C23C2×#9911yescharacteristic
417C23#41012yes—
418C23C22×C22⋊C411yescharacteristic
419C23C2×C4×Q812yes—
420C23C2×C22⋊Q812yes—
421D4—44no—
422D4—44no—
423D4—42no—
424D4—42no—
425D4—42no—
426D4—42no—
427C2×C4—21no—
428C2×C4—21no—
429C2×C4—21no—
430C2×C4—21no—
431C2×C4—22no—
432C2×C4—22no—
433C2×C4—22no—
434C2×C4—22no—
435C2×C4—22no—
436C2×C4—22no—
437C2×C4—24no—
438C2×C4—22no—
439C2×C4—22no—
440C2×C4—22no—
441C2×C4—42no—
442C2×C4—42no—
443C2×C4—44no—
444C2×C4—44no—
445C2×C4—42no—
446C2×C4—44no—
447C2×C4—42no—
448C2×C4—48no—
449C2×C4—44no—
450C2×C4—42no—
451C2×C4—42no—
452C2×C4—44no—
453C2×C4—42no—
454C2×C4—42no—
455C2×C4—42no—
456C2×C4—42no—
457C2×C4—42no—
458C2×C4—44no—
459C2×C4—48no—
460C2×C4—42no—
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—82no—
470C2×C4—82no—
471C2×C4—82no—
472C2×C4—82no—
473C2×C4—82no—
474C2×C4—82no—
475C2×C4—81no—
476C2×C4—81no—
477C2×C4—84no—
478C2×C4—84no—
479C2×C4—82no—
480C2×C4—84no—
481C2×C4—84no—
482C2×C4—816no—
483C2×C4—84no—
484C2×C4—82no—
485C2×C4—84no—
486C2×C4#39112yes—
487C2×C4C2×#9912yes—
488C2×C4C2×#9912yes—
489C2×C4C2×#12411yescharacteristic
490C2×C4C2×C4⋊Q811yescharacteristic
491C8—84no—
492C22—21no—
493C22—21no—
494C22—21no—
495C22—21no—
496C22—24no—
497C22—24no—
498C22—22no—
499C22—22no—
500C22—22no—
501C22—22no—
502C22—42no—
503C22—42no—
504C22—42no—
505C22—42no—
506C22—42no—
507C22—42no—
508C22—48no—
509C22—41no—
510C22—41no—
511C22—41no—
512C22—41no—
513C22—41no—
514C22—41no—
515C22—41no—
516C22—41no—
517C22—48no—
518C22—48no—
519C22—81no—
520C22—81no—
521C22—84no—
522C22#199612yes—
523C22#199612yes—
524C22#212012yes—
525C22#212112yes—
526C22C2×#39111yescharacteristic
527C22C22×#9911yescharacteristic
528C22C2×#41011yescharacteristic
529C4—22no—
530C4—22no—
531C4—21no—
532C4—21no—
533C4—22no—
534C4—44no—
535C4—42no—
536C4—42no—
537C4—42no—
538C4—82no—
539C4—82no—
540C4—88no—
541C4—82no—
542C2—21no—
543C2—21no—
544C2—22no—
545C2—42no—
546C2—42no—
547C2—42no—
548C2—42no—
549C2#1329714yesdirect product factor
550C2C2×#199611yescharacteristic
551C2C2×#212011yescharacteristic
552C2C2×#212111yescharacteristic
subgroupG/H
center, socleC23C2×#99
commutatorC24C23×C4
Frattini, cosocleC22×D4C24
prime radicalC23×D4C23
LCSG ▷ C24 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C4 ◁ C22×C4×D4 ◁ G
LECSG ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #13297C2direct (N × H)
8. C23×C4⋊C4C4semidirect
14. #2579C4semidirect
22. C2×#385C4semidirect
23. C2×#385C4semidirect
24. C2×#380C4semidirect
56. C22×C4⋊C4C2×C4semidirect
96. #385C2×C4semidirect
97. #385C2×C4semidirect
99. #380C2×C4semidirect
549. C2#13297direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#25792
C2×#18002
C2×C4×#992
#132972
C2×#13902