Finite Group Databasebeta

← C2×#30390 #6434385

Subgroups — C2×#30390 #6434385

Order 512.

1562 conjugacy classes of subgroups.

The subgroup lattice has 611 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2644C211yesmaximal, characteristic
2C2×#2647C211yesmaximal, characteristic, complemented
3#30390C2116yesmaximal, direct product factor
4C2×#2101C211yesmaximal, characteristic, complemented
5C2×#2092C211yesmaximal, characteristic
6C2×#2087C211yesmaximal, characteristic, complemented
7C2×#2088C211yesmaximal, characteristic, complemented
8C2×#2083C211yesmaximal, characteristic, complemented
9C2×#2071C211yesmaximal, characteristic
10C2×#2067C211yesmaximal, characteristic
11C2×#2067C211yesmaximal, characteristic
12C2×#1944C211yesmaximal, characteristic, complemented
13C2×#1942C211yesmaximal, characteristic, complemented
14C2×#1936C211yesmaximal, characteristic
15C2×#1934C211yesmaximal, characteristic
16C2×#1768C211yesmaximal, characteristic, complemented
17C22×#107C2211yescharacteristic
18C22×#107C2211yescharacteristic
19C22×#107C2211yescharacteristic
20C22×#106C2211yescharacteristic
21C22×C4⋊C8C2211yescharacteristic
22C22×C4⋊C8C2211yescharacteristic
23C22×#102C2211yescharacteristic
24C22×#102C2211yescharacteristic
25C22×#102C2211yescharacteristic
26C22×#102C2211yescharacteristic
27C22×#97C2211yescharacteristic
28C22×C4×C8C2211yescharacteristic
29#2647—24no—
30#2647C2218yescomplemented
31#2644—24no—
32#2644C2218yes—
33C2×#400—21no—
34C2×#400—21no—
35C2×#398—21no—
36C2×#398—21no—
37C2×#397—21no—
38C2×#397—21no—
39C2×#397—21no—
40C2×#396—21no—
41C2×#390C2211yescharacteristic
42C2×#390C2211yescharacteristic
43C2×#390C2211yescharacteristic
44C2×#389C2211yescharacteristic
45C2×#389C2211yescharacteristic
46C2×#389C2211yescharacteristic
47C2×#389C2211yescharacteristic
48C2×#389C2211yescharacteristic
49C2×#389C2211yescharacteristic
50C2×#389C2211yescharacteristic
51C2×#384C2211yescharacteristic
52C2×#382C2211yescharacteristic
53C2×#376C2211yescharacteristic
54#2101C2218yescomplemented
55#2092C2218yes—
56#2088C2218yescomplemented
57#2087C2218yescomplemented
58#2083C2218yescomplemented
59#2071C2218yes—
60#2067C2218yes—
61#2067C2218yes—
62#1944C2218yescomplemented
63#1942C2218yescomplemented
64#1936C2218yes—
65#1934C2218yes—
66#1768C2218yescomplemented
67C2×#340C2211yescharacteristic
68C2×#340C2211yescharacteristic
69C2×#340C2211yescharacteristic
70C2×#340C2211yescharacteristic
71C2×#340C2211yescharacteristic
72C2×#340C2211yescharacteristic
73C2×#340C2211yescharacteristic
74C2×#340C2211yescharacteristic
75C2×#336C2211yescharacteristic
76C2×#336C2211yescharacteristic
77C23×D4C2311yescharacteristic
78C23×C8C2311yescharacteristic
79C23×C8C2311yescharacteristic
80C23×C8C2311yescharacteristic
81C23×C8C2311yescharacteristic
82C22×C4⋊C4—21no—
83C22×C4⋊C4—21no—
84C22×C4⋊C4—21no—
85C22×C4⋊C4—21no—
86C22×C4⋊C4—21no—
87C22×C4⋊C4—21no—
88C22×C4⋊C4—21no—
89C22×C4⋊C4—21no—
90C22×C4⋊C4—21no—
91C22×C4⋊C4C2311yescharacteristic
92C22×C4⋊C4C2311yescharacteristic
93C22×C4⋊C4C2311yescharacteristic
94C22×C4⋊C4C2311yescharacteristic
95C22×C4⋊C4C2311yescharacteristic
96C22×C4⋊C4C2311yescharacteristic
97C22×C4⋊C4C2311yescharacteristic
98C22×C22⋊C4—21no—
99C22×C22⋊C4—21no—
100C22×C22⋊C4—21no—
101C22×C22⋊C4—21no—
102C22×C22⋊C4—21no—
103C22×C22⋊C4—21no—
104C22×C42C2311yescharacteristic
105C22×C42C2311yescharacteristic
106C22×C42C2311yescharacteristic
107C2×#107—24no—
108C2×#107—42no—
109C2×#107—42no—
110C2×#107C2314yes—
111C2×#107C2314yes—
112C2×#107C2314yes—
113C2×#106—24no—
114C2×#106C2314yes—
115C2×C4⋊C8—24no—
116C2×C4⋊C8—24no—
117C2×C4⋊C8C2314yes—
118C2×C4⋊C8C2314yes—
119C2×#102—24no—
120C2×#102—24no—
121C2×#102—42no—
122C2×#102—42no—
123C2×#102C2314yes—
124C2×#102C2314yes—
125C2×#102C2314yes—
126C2×#102C2314yes—
127C2×#97—42no—
128C2×#97C2314yes—
129C2×C4×C8—42no—
130C2×C4×C8C2314yes—
131#400—28no—
132#400—28no—
133#398—28no—
134#398—28no—
135#397—28no—
136#397—28no—
137#397—28no—
138#396—28no—
139#390—22no—
140#390—22no—
141#390—22no—
142#390C2314yes—
143#390C2314yes—
144#390C2314yes—
145#389—22no—
146#389—22no—
147#389—22no—
148#389—22no—
149#389—22no—
150#389—22no—
151#389—22no—
152#389C2314yes—
153#389C2314yes—
154#389C2314yes—
155#389C2314yes—
156#389C2314yes—
157#389C2314yes—
158#389C2314yes—
159#384—22no—
160#384C2314yes—
161#382—22no—
162#382C2314yes—
163#376D414yes—
164#376C2314yes—
165C2×#95—21no—
166C2×#95—21no—
167C2×#95—21no—
168C2×#95—21no—
169C2×#95—21no—
170C2×#95—21no—
171C2×#95—21no—
172C2×#95—21no—
173C2×#95—21no—
174C2×#95D411yescharacteristic
175C2×#95D411yescharacteristic
176C2×#95D411yescharacteristic
177C2×#95D411yescharacteristic
178#340C2314yes—
179#340C2314yes—
180#340C2314yes—
181#340C2314yes—
182#340C2314yes—
183#340C2314yes—
184#340C2314yes—
185#340C2314yes—
186#336C2314yes—
187#336C2314yes—
188C25—21no—
189C22×D4—21no—
190C22×D4—22no—
191C22×D4—22no—
192C22×D4—22no—
193C22×D4—22no—
194C22×D4—22no—
195C22×D4—22no—
196C22×D4C2412yes—
197C23×C4—21no—
198C23×C4—21no—
199C23×C4—21no—
200C23×C4—21no—
201C23×C4—21no—
202C23×C4—21no—
203C23×C4—21no—
204C23×C4C2×D411yescharacteristic
205C23×C4C2×D411yescharacteristic
206C23×C4C2×D411yescharacteristic
207C23×C4C2×D411yescharacteristic
208C23×C4C2×D411yescharacteristic
209C23×C4C2×D411yescharacteristic
210C23×C4C2411yescharacteristic
211C22×C8—22no—
212C22×C8—22no—
213C22×C8—22no—
214C22×C8—22no—
215C22×C8—22no—
216C22×C8—22no—
217C22×C8—22no—
218C22×C8—22no—
219C22×C8—22no—
220C22×C8—22no—
221C22×C8#4114yes—
222C22×C8#4114yes—
223C22×C8C2412yes—
224C22×C8C2412yes—
225C22×C8C2412yes—
226C22×C8C2412yes—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—22no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—21no—
242C2×C4⋊C4—21no—
243C2×C4⋊C4—21no—
244C2×C4⋊C4—21no—
245C2×C4⋊C4—21no—
246C2×C4⋊C4—21no—
247C2×C4⋊C4—21no—
248C2×C4⋊C4—24no—
249C2×C4⋊C4—24no—
250C2×C4⋊C4—24no—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—24no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—24no—
256C2×C4⋊C4—24no—
257C2×C4⋊C4—82no—
258C2×C4⋊C4—82no—
259C2×C4⋊C4—82no—
260C2×C4⋊C4—82no—
261C2×C4⋊C4—82no—
262C2×C4⋊C4—82no—
263C2×C4⋊C4—82no—
264C2×C4⋊C4—82no—
265C2×C4⋊C4—82no—
266C2×C4⋊C4C2412yes—
267C2×C4⋊C4C2412yes—
268C2×C4⋊C4C2412yes—
269C2×C4⋊C4C2412yes—
270C2×C4⋊C4C2412yes—
271C2×C4⋊C4C2412yes—
272C2×C4⋊C4C2412yes—
273C2×C22⋊C4—24no—
274C2×C22⋊C4—24no—
275C2×C22⋊C4—24no—
276C2×C22⋊C4—24no—
277C2×C22⋊C4—24no—
278C2×C22⋊C4—24no—
279C2×C22⋊C4—82no—
280C2×C22⋊C4—82no—
281C2×C22⋊C4—82no—
282C2×C22⋊C4—82no—
283C2×C22⋊C4—82no—
284C2×C22⋊C4—82no—
285C2×C42—22no—
286C2×C42—22no—
287C2×C42—22no—
288C2×C42—22no—
289C2×C42—22no—
290C2×C42—22no—
291C2×C42C2×D412yes—
292C2×C42C2×D412yes—
293C2×C42C2×D412yes—
294C2×C42C2412yes—
295C2×C42C2412yes—
296C2×C42C2412yes—
297#107—28no—
298#107—44no—
299#107—44no—
300#106—28no—
301C4⋊C8—28no—
302C4⋊C8—28no—
303#102—28no—
304#102—28no—
305#102—44no—
306#102—44no—
307#97—44no—
308C4×C8—44no—
309#95—24no—
310#95—24no—
311#95—24no—
312#95—24no—
313#95—24no—
314#95—24no—
315#95—24no—
316#95—24no—
317#95—24no—
318#95C2×D414yes—
319#95C2×D414yes—
320#95C2×D414yes—
321#95C2×D414yes—
322C24—22no—
323C24—42no—
324C24—42no—
325C24—42no—
326C24—42no—
327C24—42no—
328C24—42no—
329C24—42no—
330C24C22×D411yescharacteristic
331C2×D4—22no—
332C2×D4—22no—
333C2×D4—22no—
334C2×D4—22no—
335C2×D4—22no—
336C2×D4—22no—
337C2×D4—42no—
338C2×D4—42no—
339C2×D4—42no—
340C2×D4—44no—
341C2×D4—44no—
342C2×D4—44no—
343C2×D4—44no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—42no—
370C22×C4—42no—
371C22×C4—42no—
372C22×C4—42no—
373C22×C4—42no—
374C22×C4—42no—
375C22×C4—42no—
376C22×C4—42no—
377C22×C4—42no—
378C22×C4—42no—
379C22×C4—42no—
380C22×C4—42no—
381C22×C4—42no—
382C22×C4—42no—
383C22×C4—42no—
384C22×C4—42no—
385C22×C4—42no—
386C22×C4—42no—
387C22×C4—42no—
388C22×C4—42no—
389C22×C4—42no—
390C22×C4C22×D412yes—
391C22×C4C22×D412yes—
392C22×C4C22×D412yes—
393C22×C4C22×D412yes—
394C22×C4C22×D412yes—
395C22×C4C22×D412yes—
396C22×C4C22×D411yescharacteristic
397C22×C4C2×#4112yes—
398C22×C4C2×#4112yes—
399C22×C4H(5,𝔽2)12yes—
400C22×C42−1+412yes—
401C22×C42−1+412yes—
402C22×C42−1+412yes—
403C22×C4C2511yescharacteristic
404C2×C8—28no—
405C2×C8—28no—
406C2×C8—22no—
407C2×C8—22no—
408C2×C8—22no—
409C2×C8—22no—
410C2×C8—22no—
411C2×C8—22no—
412C2×C8—22no—
413C2×C8—22no—
414C2×C8—22no—
415C2×C8—22no—
416C2×C8—44no—
417C2×C8—44no—
418C2×C8C2×#4114yes—
419C2×C8C2×#4114yes—
420C4⋊C4—22no—
421C4⋊C4—22no—
422C4⋊C4—22no—
423C4⋊C4—22no—
424C4⋊C4—22no—
425C4⋊C4—22no—
426C4⋊C4—22no—
427C4⋊C4—22no—
428C4⋊C4—22no—
429C4⋊C4—22no—
430C4⋊C4—22no—
431C4⋊C4—22no—
432C4⋊C4—22no—
433C4⋊C4—22no—
434C4⋊C4—42no—
435C4⋊C4—42no—
436C4⋊C4—42no—
437C4⋊C4—42no—
438C4⋊C4—42no—
439C4⋊C4—42no—
440C4⋊C4—42no—
441C4⋊C4—84no—
442C4⋊C4—84no—
443C4⋊C4—84no—
444C4⋊C4—84no—
445C4⋊C4—84no—
446C4⋊C4—84no—
447C4⋊C4—84no—
448C4⋊C4—84no—
449C4⋊C4—84no—
450C22⋊C4—84no—
451C22⋊C4—84no—
452C22⋊C4—84no—
453C22⋊C4—84no—
454C22⋊C4—84no—
455C22⋊C4—84no—
456C42—24no—
457C42—24no—
458C42—24no—
459C42—22no—
460C42—22no—
461C42—22no—
462C42—22no—
463C42—22no—
464C42—22no—
465C23—42no—
466C23—42no—
467C23—42no—
468C23—42no—
469C23—42no—
470C23—42no—
471C23—42no—
472C23—84no—
473C23—84no—
474C23—84no—
475C23—84no—
476C23—84no—
477C23—84no—
478C23—84no—
479C23#53212yes—
480C23#53612yes—
481C23#53712yes—
482C23#57612yes—
483C23#57612yes—
484C23#57712yes—
485C23#57812yes—
486C23C23×D411yescharacteristic
487D4—42no—
488D4—42no—
489D4—42no—
490D4—42no—
491D4—84no—
492C2×C4—22no—
493C2×C4—22no—
494C2×C4—22no—
495C2×C4—22no—
496C2×C4—22no—
497C2×C4—22no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—22no—
503C2×C4—22no—
504C2×C4—22no—
505C2×C4—22no—
506C2×C4—22no—
507C2×C4—22no—
508C2×C4—22no—
509C2×C4—22no—
510C2×C4—44no—
511C2×C4—44no—
512C2×C4—44no—
513C2×C4—44no—
514C2×C4—44no—
515C2×C4—44no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—84no—
538C2×C4—84no—
539C2×C4—84no—
540C2×C4—84no—
541C2×C4—84no—
542C2×C4—84no—
543C2×C4—84no—
544C2×C4#53212yes—
545C2×C4#53612yes—
546C2×C4#53712yes—
547C2×C4#53914yes—
548C2×C4#53914yes—
549C2×C4#54114yes—
550C2×C4#54114yes—
551C2×C4C22×#4111yescharacteristic
552C2×C4C22×#4111yescharacteristic
553C2×C4C2×H(5,𝔽2)11yescharacteristic
554C2×C4C2×2−1+411yescharacteristic
555C2×C4C2×2−1+411yescharacteristic
556C2×C4C2×2−1+411yescharacteristic
557C8—24no—
558C8—24no—
559C8—42no—
560C8—42no—
561C22—82no—
562C22—82no—
563C22—82no—
564C22—82no—
565C22—82no—
566C22—82no—
567C22—82no—
568C22—168no—
569C22#266414yes—
570C22#317514yes—
571C22#317914yes—
572C22#324114yes—
573C22#324314yes—
574C22#325814yes—
575C22#325914yes—
576C22C2×#53211yescharacteristic
577C22C2×#53611yescharacteristic
578C22C2×#53711yescharacteristic
579C22C2×#57611yescharacteristic
580C22C2×#57611yescharacteristic
581C22C2×#57711yescharacteristic
582C22C2×#57811yescharacteristic
583C4—42no—
584C4—42no—
585C4—42no—
586C4—42no—
587C4—42no—
588C4—42no—
589C4—82no—
590C4—82no—
591C4—82no—
592C4—82no—
593C4—82no—
594C4—82no—
595C4—82no—
596C4#266414yes—
597C4C2×#53911yescharacteristic
598C4C2×#53911yescharacteristic
599C4C2×#54111yescharacteristic
600C4C2×#54111yescharacteristic
601C2—162no—
602C2#3039018yesdirect product factor
603C2C2×#266411yescharacteristic
604C2C2×#317511yescharacteristic
605C2C2×#317911yescharacteristic
606C2C2×#324111yescharacteristic
607C2C2×#324311yescharacteristic
608C2C2×#325811yescharacteristic
609C2C2×#325911yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC23×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×#376 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2647C2semidirect
3. #30390C2direct (N × H)
4. C2×#2101C2semidirect
6. C2×#2087C2semidirect
7. C2×#2088C2semidirect
8. C2×#2083C2semidirect
12. C2×#1944C2semidirect
13. C2×#1942C2semidirect
16. C2×#1768C2semidirect
30. #2647C22semidirect
54. #2101C22semidirect
56. #2088C22semidirect
57. #2087C22semidirect
58. #2083C22semidirect
62. #1944C22semidirect
63. #1942C22semidirect
66. #1768C22semidirect
602. C2#30390direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#26442
C2×#26472
#303902
C2×#21012
C2×#20922
C2×#20872
C2×#20882
C2×#20832
C2×#20712
C2×#20672
C2×#19442
C2×#19422
C2×#19362
C2×#19342
C2×#17682