Finite Group Databasebeta

← C2×#8760 #190198

Subgroups — C2×#8760 #190198

Order 512.

1472 conjugacy classes of subgroups.

The subgroup lattice has 545 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2349C211yesmaximal, characteristic
2C2×#1799C211yesmaximal, characteristic
3C2×#1459C211yesmaximal, characteristic
4#8760C218yesmaximal, direct product factor
5C2×#1294C212yesmaximal, complemented
6C2×#1289C212yesmaximal, complemented
7C22×C4×D4—21no—
8C22×C4×D4C2211yescharacteristic
9C22×#99—21no—
10C22×C22⋊C8C2212yes—
11#2349—22no—
12#2349—22no—
13#2349C414yescomplemented
14#2349C2214yes—
15C2×#389—21no—
16C2×#385—21no—
17C2×#385—21no—
18C2×#382C411yescharacteristic, complemented
19C2×#380C411yescharacteristic, complemented
20C2×#380C2212yes—
21C2×#380C2211yescharacteristic
22C2×C4×C4⋊C4C411yescharacteristic, complemented
23C22×#95C411yescharacteristic, complemented
24C22×#95C2211yescharacteristic
25#1799—22no—
26#1799C2214yes—
27C2×#342—21no—
28#1459—22no—
29#1459C2214yes—
30C2×#324—21no—
31C2×#323—21no—
32#1294C2218yescomplemented
33#1289C2218yescomplemented
34C23×D4D411yescharacteristic
35C24×C4—21no—
36C24×C4—21no—
37C24×C4C2×C411yescharacteristic
38C24×C4C2311yescharacteristic
39C23×C8—22no—
40C2×C4×D4—22no—
41C2×C4×D4—22no—
42C2×C4×D4—22no—
43C2×C4×D4—22no—
44C2×C4×D4—24no—
45C2×C4×D4—24no—
46C2×C4×D4—44no—
47C2×C4×D4—44no—
48C2×C4×D4C2×C412yes—
49C2×C4×D4D412yes—
50C2×C4×D4D412yes—
51C2×C4×D4C2312yes—
52C22×C4⋊C4—21no—
53C22×C4⋊C4—21no—
54C22×C4⋊C4—21no—
55C22×C4⋊C4—21no—
56C22×C4⋊C4—22no—
57C22×C4⋊C4D411yescharacteristic
58C22×C22⋊C4—21no—
59C22×C22⋊C4—21no—
60C22×C22⋊C4—22no—
61C22×C22⋊C4D411yescharacteristic
62C22×C22⋊C4D411yescharacteristic
63C22×C42—21no—
64C22×C42C2×C411yescharacteristic
65C2×#99—24no—
66C2×#99—44no—
67C2×C22⋊C8—22no—
68C2×C22⋊C8—22no—
69C2×C22⋊C8—22no—
70C2×C22⋊C8—22no—
71C2×C22⋊C8—22no—
72C2×C22⋊C8C2314yes—
73#389—24no—
74#389—24no—
75#385—24no—
76#385—24no—
77#385—24no—
78#385—24no—
79#382—22no—
80#382C2×C414yescomplemented
81#380—22no—
82#380—21no—
83#380—21no—
84#380—22no—
85#380—22no—
86#380—22no—
87#380C2×C412yes—
88#380C2×C414yescomplemented
89#380C2314yes—
90#380C2312yes—
91C4×C4⋊C4—22no—
92C4×C4⋊C4C2×C414yescomplemented
93C2×#95—22no—
94C2×#95—21no—
95C2×#95—21no—
96C2×#95—21no—
97C2×#95—21no—
98C2×#95—21no—
99C2×#95—22no—
100C2×#95C2×C412yes—
101C2×#95C2×C411yescharacteristic
102C2×#95C2×C411yescharacteristic
103C2×#95C2×C411yescharacteristic
104C2×#95C2×C411yescharacteristic
105C2×#95C2×C414yescomplemented
106C2×#95C2312yes—
107#342—24no—
108#324—24no—
109#323—24no—
110C25C22⋊C411yescharacteristic
111C25C2×D411yescharacteristic
112C22×D4—22no—
113C22×D4—22no—
114C22×D4—24no—
115C22×D4C22⋊C412yes—
116C22×D4D814yes—
117C22×D4SD1614yes—
118C22×D4C2×D412yes—
119C23×C4—21no—
120C23×C4—21no—
121C23×C4—21no—
122C23×C4—21no—
123C23×C4—21no—
124C23×C4—21no—
125C23×C4—21no—
126C23×C4—21no—
127C23×C4—22no—
128C23×C4—24no—
129C23×C4—22no—
130C23×C4—44no—
131C23×C4—44no—
132C23×C4—44no—
133C23×C4—42no—
134C23×C4—42no—
135C23×C4—42no—
136C23×C4—82no—
137C23×C4—82no—
138C23×C4—82no—
139C23×C4—82no—
140C23×C4C22⋊C411yescharacteristic
141C23×C4C22⋊C411yescharacteristic
142C23×C4C22⋊C411yescharacteristic
143C23×C4C22⋊C412yes—
144C23×C4C22×C412yes—
145C23×C4C22×C411yescharacteristic
146C23×C4C22×C411yescharacteristic
147C23×C4C2×D411yescharacteristic
148C23×C4C2×D411yescharacteristic
149C23×C4C2×D411yescharacteristic
150C23×C4#4112yes—
151C23×C4#4112yes—
152C23×C4#4112yes—
153C23×C4#4112yes—
154C23×C4C2411yescharacteristic
155C22×C8—24no—
156C22×C8—42no—
157C22×C8—42no—
158C22×C8—84no—
159C4×D4—24no—
160C4×D4—24no—
161C4×D4—24no—
162C4×D4—24no—
163C4×D4—24no—
164C4×D4—24no—
165C4×D4—24no—
166C4×D4—24no—
167C4×D4—416no—
168C4×D4—48no—
169C4×D4—48no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—24no—
173C2×C4⋊C4—24no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—24no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—24no—
178C2×C4⋊C4—44no—
179C2×C4⋊C4—44no—
180C2×C4⋊C4—42no—
181C2×C4⋊C4—82no—
182C2×C4⋊C4—82no—
183C2×C4⋊C4—84no—
184C2×C4⋊C4C22⋊C412yes—
185C2×C4⋊C4C2×D412yes—
186C2×C22⋊C4—22no—
187C2×C22⋊C4—22no—
188C2×C22⋊C4—22no—
189C2×C22⋊C4—22no—
190C2×C22⋊C4—22no—
191C2×C22⋊C4—24no—
192C2×C22⋊C4—22no—
193C2×C22⋊C4—24no—
194C2×C22⋊C4—44no—
195C2×C22⋊C4—42no—
196C2×C22⋊C4—42no—
197C2×C22⋊C4—44no—
198C2×C22⋊C4—44no—
199C2×C22⋊C4—42no—
200C2×C22⋊C4C22⋊C412yes—
201C2×C22⋊C4C22⋊C412yes—
202C2×C22⋊C4C2×D412yes—
203C2×C22⋊C4C2×D412yes—
204C2×C42—21no—
205C2×C42—21no—
206C2×C42—21no—
207C2×C42—21no—
208C2×C42—24no—
209C2×C42—44no—
210C2×C42C22⋊C412yes—
211C2×C42C22×C412yes—
212#99—48no—
213C22⋊C8—24no—
214C22⋊C8—24no—
215C22⋊C8—42no—
216C22⋊C8—42no—
217#95—21no—
218#95—21no—
219#95—21no—
220#95—22no—
221#95—24no—
222#95—24no—
223#95—21no—
224#95—22no—
225#95—24no—
226#95—42no—
227#95C22×C412yes—
228#95C22×C412yes—
229#95C22×C412yes—
230#95C22×C412yes—
231C24—21no—
232C24—21no—
233C24—22no—
234C24—42no—
235C24—42no—
236C24—42no—
237C24—42no—
238C24#10214yes—
239C24C2×C22⋊C411yescharacteristic
240C24C2×C22⋊C412yes—
241C24C2×C22⋊C411yescharacteristic
242C24#12312yes—
243C24#12312yes—
244C24C2×D812yes—
245C24C2×SD1612yes—
246C24#13714yes—
247C24C22×D411yescharacteristic
248C2×D4—24no—
249C2×D4—24no—
250C2×D4—24no—
251C2×D4—24no—
252C2×D4—28no—
253C2×D4—28no—
254C2×D4—28no—
255C2×D4—28no—
256C2×D4#10218yes—
257C2×D4C2×D814yes—
258C2×D4C2×SD1614yes—
259C22×C4—21no—
260C22×C4—21no—
261C22×C4—22no—
262C22×C4—24no—
263C22×C4—22no—
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—22no—
273C22×C4—22no—
274C22×C4—22no—
275C22×C4—42no—
276C22×C4—44no—
277C22×C4—44no—
278C22×C4—42no—
279C22×C4—44no—
280C22×C4—44no—
281C22×C4—48no—
282C22×C4—42no—
283C22×C4—42no—
284C22×C4—42no—
285C22×C4—42no—
286C22×C4—42no—
287C22×C4—42no—
288C22×C4—42no—
289C22×C4—42no—
290C22×C4—42no—
291C22×C4—42no—
292C22×C4—42no—
293C22×C4—42no—
294C22×C4—42no—
295C22×C4—42no—
296C22×C4—42no—
297C22×C4—48no—
298C22×C4—44no—
299C22×C4—44no—
300C22×C4—82no—
301C22×C4—82no—
302C22×C4—81no—
303C22×C4—81no—
304C22×C4—81no—
305C22×C4—82no—
306C22×C4—82no—
307C22×C4—81no—
308C22×C4—82no—
309C22×C4—82no—
310C22×C4—82no—
311C22×C4—82no—
312C22×C4—84no—
313C22×C4—82no—
314C22×C4—82no—
315C22×C4—88no—
316C22×C4—88no—
317C22×C4—816no—
318C22×C4—84no—
319C22×C4—84no—
320C22×C4—84no—
321C22×C4#10214yes—
322C22×C4C2×C22⋊C412yes—
323C22×C4C2×C22⋊C412yes—
324C22×C4C2×C22⋊C412yes—
325C22×C4C2×C22⋊C411yescharacteristic
326C22×C4C2×C22⋊C411yescharacteristic
327C22×C4#11711yescharacteristic
328C22×C4#11711yescharacteristic
329C22×C4#11711yescharacteristic
330C22×C4#11711yescharacteristic
331C22×C4#11712yes—
332C22×C4#11712yes—
333C22×C4#12312yes—
334C22×C4#12312yes—
335C22×C4#12312yes—
336C22×C4#12312yes—
337C22×C4#12312yes—
338C22×C4#12312yes—
339C22×C4C23×C411yescharacteristic
340C22×C4C22×D411yescharacteristic
341C22×C4C2×#4112yes—
342C22×C4C2×#4112yes—
343C2×C8—44no—
344C2×C8—84no—
345C2×C8—84no—
346C2×C8—84no—
347C4⋊C4—24no—
348C4⋊C4—24no—
349C4⋊C4—48no—
350C4⋊C4—48no—
351C4⋊C4—82no—
352C4⋊C4—82no—
353C4⋊C4—84no—
354C4⋊C4—84no—
355C4⋊C4—84no—
356C22⋊C4—24no—
357C22⋊C4—24no—
358C22⋊C4—24no—
359C22⋊C4—24no—
360C22⋊C4—42no—
361C22⋊C4—42no—
362C22⋊C4—48no—
363C22⋊C4—44no—
364C22⋊C4—44no—
365C22⋊C4—44no—
366C22⋊C4—84no—
367C42—22no—
368C42—22no—
369C42—22no—
370C42—22no—
371C42—48no—
372C23—22no—
373C23—24no—
374C23—22no—
375C23—24no—
376C23—24no—
377C23—24no—
378C23—22no—
379C23—22no—
380C23—41no—
381C23—41no—
382C23—41no—
383C23—41no—
384C23—42no—
385C23—44no—
386C23—44no—
387C23—42no—
388C23—48no—
389C23—42no—
390C23—42no—
391C23—48no—
392C23—42no—
393C23—48no—
394C23—42no—
395C23—42no—
396C23—42no—
397C23#38112yes—
398C23#38111yescharacteristic
399C23#38111yescharacteristic
400C23C2×#10212yes—
401C23C2×#10214yes—
402C23C2×#10212yes—
403C23#41912yes—
404C23#41912yes—
405C23#48014yes—
406C23#48014yes—
407C23#48314yes—
408C23#48314yes—
409C23C22×C22⋊C411yescharacteristic
410C23C2×#12312yes—
411C23C22×D811yescharacteristic
412C23C22×SD1611yescharacteristic
413C23C2×#13712yes—
414D4—28no—
415D4—28no—
416D4—216no—
417C2×C4—24no—
418C2×C4—24no—
419C2×C4—24no—
420C2×C4—22no—
421C2×C4—22no—
422C2×C4—22no—
423C2×C4—22no—
424C2×C4—24no—
425C2×C4—22no—
426C2×C4—22no—
427C2×C4—41no—
428C2×C4—42no—
429C2×C4—41no—
430C2×C4—42no—
431C2×C4—42no—
432C2×C4—42no—
433C2×C4—41no—
434C2×C4—41no—
435C2×C4—41no—
436C2×C4—41no—
437C2×C4—42no—
438C2×C4—42no—
439C2×C4—42no—
440C2×C4—42no—
441C2×C4—42no—
442C2×C4—42no—
443C2×C4—42no—
444C2×C4—48no—
445C2×C4—42no—
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—42no—
455C2×C4—42no—
456C2×C4—42no—
457C2×C4—42no—
458C2×C4—42no—
459C2×C4—42no—
460C2×C4—82no—
461C2×C4—82no—
462C2×C4—82no—
463C2×C4—82no—
464C2×C4—84no—
465C2×C4—84no—
466C2×C4—88no—
467C2×C4—816no—
468C2×C4—84no—
469C2×C4—84no—
470C2×C4—84no—
471C2×C4—84no—
472C2×C4—84no—
473C2×C4—84no—
474C2×C4—168no—
475C2×C4#38111yescharacteristic
476C2×C4#38111yescharacteristic
477C2×C4#38112yes—
478C2×C4C2×#10214yes—
479C2×C4C2×#11711yescharacteristic
480C2×C4C2×#11711yescharacteristic
481C2×C4C2×#12312yes—
482C8—84no—
483C22—24no—
484C22—28no—
485C22—24no—
486C22—28no—
487C22—21no—
488C22—21no—
489C22—22no—
490C22—22no—
491C22—21no—
492C22—21no—
493C22—22no—
494C22—22no—
495C22—24no—
496C22—41no—
497C22—41no—
498C22—42no—
499C22—42no—
500C22—42no—
501C22—44no—
502C22—44no—
503C22—44no—
504C22—48no—
505C22—44no—
506C22—48no—
507C22—42no—
508C22—42no—
509C22#182312yes—
510C22#182314yes—
511C22#182312yes—
512C22#182712yes—
513C22#182812yes—
514C22C2×#38111yescharacteristic
515C22C22×#10211yescharacteristic
516C22C2×#41911yescharacteristic
517C22C2×#48012yes—
518C22C2×#48312yes—
519C4—24no—
520C4—22no—
521C4—21no—
522C4—21no—
523C4—21no—
524C4—21no—
525C4—22no—
526C4—44no—
527C4—42no—
528C4—42no—
529C4—82no—
530C4—82no—
531C4—82no—
532C4—164no—
533C2—21no—
534C2—22no—
535C2—22no—
536C2—21no—
537C2—22no—
538C2—24no—
539C2—44no—
540C2#876014yesdirect product factor
541C2C2×#182311yescharacteristic
542C2C2×#182711yescharacteristic
543C2C2×#182811yescharacteristic
subgroupG/H
center, socleC23#381
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC23×D4D4
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
4. #8760C2direct (N × H)
5. C2×#1294C2semidirect
6. C2×#1289C2semidirect
13. #2349C4semidirect
18. C2×#382C4semidirect
19. C2×#380C4semidirect
22. C2×C4×C4⋊C4C4semidirect
23. C22×#95C4semidirect
32. #1294C22semidirect
33. #1289C22semidirect
80. #382C2×C4semidirect
88. #380C2×C4semidirect
92. C4×C4⋊C4C2×C4semidirect
105. C2×#95C2×C4semidirect
540. C2#8760direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#23492
C2×#17992
C2×#14592
#87602
C2×#12942
C2×#12892