Finite Group Databasebeta

← C4×#2683 #521215

Subgroups — C4×#2683 #521215

Order 512.

3618 conjugacy classes of subgroups.

The subgroup lattice has 618 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2683C211yesmaximal, characteristic
2C2×C4×C22⋊Q8C214yesmaximal, complemented
3#14549C214yesmaximal, complemented
4#14538C212yesmaximal, complemented
5#14441C212yesmaximal, complemented
6#14440C211yesmaximal, characteristic
7#14401C212yesmaximal
8#14362C214yesmaximal, complemented
9C4×#396C214yesmaximal, complemented
10C4×#395C212yesmaximal, complemented
11C4×#389C212yesmaximal, complemented
12C4×#390C211yesmaximal, characteristic
13C4×#386C212yesmaximal
14C22×C22⋊Q8C2214yes—
15C22×C4×Q8C2212yes—
16C22×C4×D4C2211yescharacteristic
17C23×C42C2212yes—
18#2683C4116yesdirect product factor
19C2×#396C2214yes—
20C2×#395C2212yes—
21C4×C22⋊Q8—232no—
22C2×#391C2214yes—
23C2×#390C2211yescharacteristic
24C2×#390C2211yescharacteristic
25C2×#389C2212yes—
26C2×#389C2214yes—
27C2×#389C2212yes—
28C2×#387C2214yes—
29C2×#387C2214yes—
30C2×#386C2212yes—
31C2×#386C2212yes—
32C2×#385C2214yes—
33C2×#385C2214yes—
34C2×#380C2214yes—
35C2×C4×C4⋊C4C2214yes—
36C2×C4×C4⋊C4C2214yes—
37C2×C4×C4⋊C4C2211yescharacteristic
38C2×C4×C4⋊C4C2212yes—
39C2×C4×C4⋊C4C2212yes—
40C2×C4×C22⋊C4C2214yes—
41C2×C4×C22⋊C4C2214yes—
42C2×C4×C22⋊C4C2212yes—
43C2×C43C2211yescharacteristic
44#1446C2212yes—
45#1444C2214yes—
46#1444C2212yes—
47#1443C2214yes—
48#1441C2214yes—
49#1441C2214yes—
50#1441C2214yes—
51#1441C2212yes—
52#1441C2214yes—
53#1439C2214yes—
54#1439C2214yes—
55#1439C2214yes—
56#1437C2214yes—
57#1437C2214yes—
58#1436C2214yes—
59#1436C2214yes—
60#1436C2214yes—
61C4⋊#95C2212yes—
62C4⋊#95C2211yescharacteristic
63C4⋊#95C2211yescharacteristic
64C4⋊#95C2212yes—
65C22⋊#95C2212yes—
66C22⋊#95C2212yes—
67C4×#95C2212yes—
68C4×#95C2214yes—
69C23×Q8C2312yes—
70C23×D4C2311yescharacteristic
71C24×C4C2312yes—
72C24×C4C2312yes—
73C24×C4C2312yes—
74C2×C22⋊Q8—216no—
75C2×C22⋊Q8—216no—
76C2×C22⋊Q8C2×C4132yescomplemented
77C2×C4×Q8—216no—
78C2×C4×Q8—44no—
79C2×C4×D4—42no—
80C2×C4×D4—42no—
81C2×C4×D4Q818yescomplemented
82C22×C4⋊C4C2314yes—
83C22×C4⋊C4C2312yes—
84C22×C4⋊C4C2314yes—
85C22×C4⋊C4C2314yes—
86C22×C4⋊C4C2314yes—
87C22×C4⋊C4C2312yes—
88C22×C4⋊C4C2311yescharacteristic
89C22×C4⋊C4C2314yes—
90C22×C4⋊C4C2311yescharacteristic
91C22×C4⋊C4C2312yes—
92C22×C4⋊C4C2312yes—
93C22×C4⋊C4C2311yescharacteristic
94C22×C4⋊C4C2312yes—
95C22×C22⋊C4C2314yes—
96C22×C22⋊C4C2314yes—
97C22×C22⋊C4C2314yes—
98C22×C22⋊C4C2314yes—
99C22×C22⋊C4C2312yes—
100C22×C22⋊C4C2312yes—
101C22×C22⋊C4C2312yes—
102C22×C42—216no—
103C22×C42—44no—
104C22×C42C2312yes—
105C22×C42C2314yes—
106C22×C42C2314yes—
107C22×C42C2314yes—
108C22×C42C2312yes—
109C22×C42C2311yescharacteristic
110C22×C42C2311yescharacteristic
111C22×C42C2312yes—
112C22×C42C2311yescharacteristic
113#396C2×C4132yescomplemented
114#395C2×C4116yescomplemented
115#391—216no—
116#390—24no—
117#390C2×C418yes—
118#389—216no—
119#389—28no—
120#389C2×C4116yescomplemented
121#387—216no—
122#387—216no—
123#386—28no—
124#386C2×C4116yes—
125#385—216no—
126#385—216no—
127#380—216no—
128C4×C4⋊C4—216no—
129C4×C4⋊C4—216no—
130C4×C4⋊C4—42no—
131C4×C4⋊C4—44no—
132C4×C4⋊C4D4116yescomplemented
133C4×C22⋊C4—216no—
134C4×C22⋊C4—216no—
135C4×C22⋊C4—44no—
136C2×#95C2314yes—
137C2×#95C2314yes—
138C2×#95C2314yes—
139C2×#95C2314yes—
140C2×#95C2314yes—
141C2×#95C2314yes—
142C2×#95C2314yes—
143C2×#95C2314yes—
144C2×#95C2314yes—
145C2×#95C2314yes—
146C2×#95C2314yes—
147C2×#95C2314yes—
148C2×#95C2312yes—
149C2×#95C2314yes—
150C2×#95C2314yes—
151C2×#95C2312yes—
152C2×#95C2312yes—
153C2×#95C2314yes—
154C2×#95C2314yes—
155C43—42no—
156C25C2412yes—
157C22×Q8—28no—
158C22×Q8—28no—
159C22×Q8—42no—
160C22×Q8—42no—
161C22×Q8C22×C418yes—
162C22×D4—41no—
163C22×D4—41no—
164C22×D4—41no—
165C22×D4—41no—
166C22×D4C22×C414yes—
167C22×D4C2×Q814yes—
168C22×D4#4114yes—
169C23×C4—28no—
170C23×C4—28no—
171C23×C4—28no—
172C23×C4—24no—
173C23×C4—24no—
174C23×C4—28no—
175C23×C4—24no—
176C23×C4—28no—
177C23×C4—42no—
178C23×C4—44no—
179C23×C4—42no—
180C23×C4—42no—
181C23×C4—42no—
182C23×C4C22×C418yes—
183C23×C4C2×Q818yes—
184C23×C4C2412yes—
185C23×C4C2414yes—
186C23×C4C2412yes—
187C23×C4C2414yes—
188C23×C4C2412yes—
189C23×C4C2414yes—
190C23×C4C2411yescharacteristic
191C23×C4C2412yes—
192C23×C4C2412yes—
193C23×C4C2414yes—
194C23×C4C2411yescharacteristic
195C23×C4C2411yescharacteristic
196C22⋊Q8—2128no—
197C4×Q8—432no—
198C4×D4—416no—
199C2×C4⋊C4—28no—
200C2×C4⋊C4—28no—
201C2×C4⋊C4—28no—
202C2×C4⋊C4—28no—
203C2×C4⋊C4—28no—
204C2×C4⋊C4—28no—
205C2×C4⋊C4—28no—
206C2×C4⋊C4—28no—
207C2×C4⋊C4—28no—
208C2×C4⋊C4—28no—
209C2×C4⋊C4—24no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—42no—
212C2×C4⋊C4—41no—
213C2×C4⋊C4—42no—
214C2×C4⋊C4—42no—
215C2×C4⋊C4—44no—
216C2×C4⋊C4—42no—
217C2×C4⋊C4—41no—
218C2×C4⋊C4—44no—
219C2×C4⋊C4—44no—
220C2×C4⋊C4—41no—
221C2×C4⋊C4—41no—
222C2×C4⋊C4—44no—
223C2×C4⋊C4—42no—
224C2×C4⋊C4—42no—
225C2×C4⋊C4—41no—
226C2×C4⋊C4—42no—
227C2×C4⋊C4—41no—
228C2×C4⋊C4C22×C4116yes—
229C2×C4⋊C4C22×C4116yes—
230C2×C4⋊C4C22×C418yes—
231C2×C4⋊C4C22×C414yes—
232C2×C4⋊C4C22×C418yes—
233C2×C4⋊C4C2×D418yes—
234C2×C4⋊C4C2×D418yes—
235C2×C4⋊C4C2×Q814yes—
236C2×C4⋊C4#4118yes—
237C2×C22⋊C4—28no—
238C2×C22⋊C4—28no—
239C2×C22⋊C4—28no—
240C2×C22⋊C4—28no—
241C2×C22⋊C4—28no—
242C2×C22⋊C4—24no—
243C2×C22⋊C4—28no—
244C2×C22⋊C4—28no—
245C2×C22⋊C4—28no—
246C2×C22⋊C4—42no—
247C2×C22⋊C4—44no—
248C2×C22⋊C4—44no—
249C2×C22⋊C4—42no—
250C2×C22⋊C4—42no—
251C2×C22⋊C4—42no—
252C2×C22⋊C4—42no—
253C2×C22⋊C4—42no—
254C2×C22⋊C4C22×C4116yes—
255C2×C22⋊C4C22×C4116yes—
256C2×C22⋊C4C22×C418yes—
257C2×C22⋊C4C2×Q818yes—
258C2×C42—28no—
259C2×C42—22no—
260C2×C42—22no—
261C2×C42—28no—
262C2×C42—24no—
263C2×C42—24no—
264C2×C42—28no—
265C2×C42—28no—
266C2×C42—28no—
267C2×C42—28no—
268C2×C42—24no—
269C2×C42—22no—
270C2×C42—24no—
271C2×C42—28no—
272C2×C42—28no—
273C2×C42—432no—
274C2×C42—41no—
275C2×C42—42no—
276C2×C42—41no—
277C2×C42—44no—
278C2×C42—41no—
279C2×C42—42no—
280C2×C42—41no—
281C2×C42C22×C414yes—
282C2×C42C2×D4116yes—
283C2×C42C2×D418yes—
284C2×C42C2×Q814yes—
285#95—28no—
286#95—28no—
287#95—28no—
288#95—28no—
289#95—28no—
290#95—28no—
291#95—28no—
292#95—28no—
293#95—28no—
294#95—28no—
295#95—44no—
296#95—44no—
297#95—42no—
298#95—44no—
299#95—44no—
300#95—42no—
301#95C22×C4116yes—
302#95C22×C418yes—
303#95C2×D4116yes—
304C24—24no—
305C24—24no—
306C24—22no—
307C24—22no—
308C24—24no—
309C24—24no—
310C24—22no—
311C24—22no—
312C24C23×C414yes—
313C24C22×Q814yes—
314C24C2×#4114yes—
315C24C2511yescharacteristic
316C2×Q8—232no—
317C2×Q8—48no—
318C2×Q8—48no—
319C2×Q8—416no—
320C2×Q8—48no—
321C2×D4—44no—
322C2×D4—44no—
323C2×D4—44no—
324C2×D4—44no—
325C2×D4—44no—
326C2×D4—44no—
327C2×D4C4×Q8116yescomplemented
328C22×C4—24no—
329C22×C4—24no—
330C22×C4—22no—
331C22×C4—21no—
332C22×C4—24no—
333C22×C4—21no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—21no—
338C22×C4—21no—
339C22×C4—24no—
340C22×C4—24no—
341C22×C4—24no—
342C22×C4—24no—
343C22×C4—24no—
344C22×C4—22no—
345C22×C4—24no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—24no—
349C22×C4—24no—
350C22×C4—22no—
351C22×C4—21no—
352C22×C4—24no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—21no—
356C22×C4—24no—
357C22×C4—22no—
358C22×C4—24no—
359C22×C4—22no—
360C22×C4—24no—
361C22×C4—22no—
362C22×C4—24no—
363C22×C4—24no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—24no—
368C22×C4—24no—
369C22×C4—24no—
370C22×C4—232no—
371C22×C4—216no—
372C22×C4—232no—
373C22×C4—24no—
374C22×C4—24no—
375C22×C4—24no—
376C22×C4—24no—
377C22×C4—22no—
378C22×C4—24no—
379C22×C4—22no—
380C22×C4—24no—
381C22×C4—24no—
382C22×C4—24no—
383C22×C4—24no—
384C22×C4—22no—
385C22×C4—48no—
386C22×C4—48no—
387C22×C4—48no—
388C22×C4—416no—
389C22×C4—48no—
390C22×C4—416no—
391C22×C4—416no—
392C22×C4—48no—
393C22×C4—48no—
394C22×C4—416no—
395C22×C4—48no—
396C22×C4—48no—
397C22×C4C23×C418yes—
398C22×C4C23×C414yes—
399C22×C4C23×C418yes—
400C22×C4C23×C414yes—
401C22×C4C23×C412yes—
402C22×C4C22×D414yes—
403C22×C4C22×D418yes—
404C22×C4C22×D414yes—
405C22×C4C22×D418yes—
406C22×C4C22×D414yes—
407C22×C4C22×Q812yes—
408C22×C4C22×Q814yes—
409C22×C4C22×Q812yes—
410C22×C4C22×Q812yes—
411C22×C4C2×#4118yes—
412C22×C4C2×#4114yes—
413C22×C4C2×#4112yes—
414C22×C4H(5,𝔽2)12yes—
415C22×C4H(5,𝔽2)12yes—
416C22×C42−1+414yes—
417C4⋊C4—232no—
418C4⋊C4—232no—
419C4⋊C4—48no—
420C4⋊C4—44no—
421C4⋊C4—48no—
422C4⋊C4—44no—
423C4⋊C4—416no—
424C4⋊C4—416no—
425C4⋊C4—48no—
426C4⋊C4—44no—
427C4⋊C4—48no—
428C4⋊C4—416no—
429C4⋊C4C4×D4132yescomplemented
430C22⋊C4—232no—
431C22⋊C4—232no—
432C22⋊C4—48no—
433C22⋊C4—48no—
434C22⋊C4—416no—
435C22⋊C4—416no—
436C22⋊C4—48no—
437C42—216no—
438C42—28no—
439C42—232no—
440C42—44no—
441C42—416no—
442C42—44no—
443C42—48no—
444C42—416no—
445C42—48no—
446C23—28no—
447C23—24no—
448C23—24no—
449C23—28no—
450C23—24no—
451C23—24no—
452C23—24no—
453C23—24no—
454C23—28no—
455C23—28no—
456C23—28no—
457C23—28no—
458C23—44no—
459C23—44no—
460C23—44no—
461C23—44no—
462C23—42no—
463C23—44no—
464C23—42no—
465C23—42no—
466C23—42no—
467C23—44no—
468C23C2×C4×Q818yes—
469C23C24×C411yescharacteristic
470C23C23×D412yes—
471C23C23×Q811yescharacteristic
472C23C22×#4112yes—
473C23C22×#4111yescharacteristic
474C23C2×H(5,𝔽2)11yescharacteristic
475C23C2×H(5,𝔽2)11yescharacteristic
476C23C2×2−1+412yes—
477C23#58511yescharacteristic
478C23#58511yescharacteristic
479C23#58512yes—
480Q8—432no—
481D4—416no—
482C2×C4—28no—
483C2×C4—24no—
484C2×C4—22no—
485C2×C4—22no—
486C2×C4—28no—
487C2×C4—24no—
488C2×C4—24no—
489C2×C4—24no—
490C2×C4—22no—
491C2×C4—22no—
492C2×C4—22no—
493C2×C4—22no—
494C2×C4—24no—
495C2×C4—24no—
496C2×C4—28no—
497C2×C4—28no—
498C2×C4—28no—
499C2×C4—28no—
500C2×C4—28no—
501C2×C4—28no—
502C2×C4—28no—
503C2×C4—24no—
504C2×C4—28no—
505C2×C4—24no—
506C2×C4—28no—
507C2×C4—22no—
508C2×C4—24no—
509C2×C4—28no—
510C2×C4—28no—
511C2×C4—24no—
512C2×C4—28no—
513C2×C4—24no—
514C2×C4—24no—
515C2×C4—432no—
516C2×C4—432no—
517C2×C4—44no—
518C2×C4—44no—
519C2×C4—44no—
520C2×C4—44no—
521C2×C4—42no—
522C2×C4—44no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—44no—
526C2×C4—44no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—44no—
530C2×C4—44no—
531C2×C4—41no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—41no—
536C2×C4—44no—
537C2×C4—42no—
538C2×C4—432no—
539C2×C4—44no—
540C2×C4—44no—
541C2×C4—44no—
542C2×C4—42no—
543C2×C4—44no—
544C2×C4—42no—
545C2×C4—44no—
546C2×C4—44no—
547C2×C4—44no—
548C2×C4—44no—
549C2×C4—42no—
550C2×C4—44no—
551C2×C4C2×C4×D418yes—
552C2×C4C2×C4×D4116yes—
553C2×C4C2×C4×Q814yes—
554C2×C4#53718yes—
555C2×C4#54314yes—
556C2×C4D4214yes—
557C2×C4D4×Q818yes—
558C2×C4#56314yes—
559C22—28no—
560C22—216no—
561C22—48no—
562C22—44no—
563C22—44no—
564C22—44no—
565C22—44no—
566C22—44no—
567C22—44no—
568C22—48no—
569C22—48no—
570C22C22×C4×D412yes—
571C22C22×C4×Q811yescharacteristic
572C22C4×H(5,𝔽2)11yescharacteristic
573C22C4×H(5,𝔽2)11yescharacteristic
574C22C4×2−1+412yes—
575C22C2×#53712yes—
576C22#344912yes—
577C22C2×#54311yescharacteristic
578C22#345811yescharacteristic
579C22#345911yescharacteristic
580C22#345911yescharacteristic
581C22#346012yes—
582C22#346112yes—
583C22C2×D4211yescharacteristic
584C22C2×D4×Q812yes—
585C22D4×#4112yes—
586C22D4×#4112yes—
587C22Q8×#4112yes—
588C22#348111yescharacteristic
589C22#348212yes—
590C22C2×#56311yescharacteristic
591C22#352911yescharacteristic
592C22#353112yes—
593C4—28no—
594C4—24no—
595C4—216no—
596C4—48no—
597C4—42no—
598C4—44no—
599C4—44no—
600C4—48no—
601C4—44no—
602C4—82no—
603C4—84no—
604C4#268318yesdirect product factor
605C2—48no—
606C2C4×#53712yes—
607C2C4×#54311yescharacteristic
608C2C4×D4211yescharacteristic
609C2C4×D4×Q812yes—
610C2C4×#56311yescharacteristic
611C2C2×#268311yescharacteristic
612C2#5210311yescharacteristic
613C2#5216212yes—
614C2#5216912yes—
615C2#5228811yescharacteristic
616C2#5230411yescharacteristic
subgroupG/H
centerC23×C4C24
commutatorC23C24×C4
Frattini, socle, cosocleC24C25
prime radicalC23×D4C23
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C23×C4 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×C4×C22⋊Q8C2semidirect
3. #14549C2semidirect
4. #14538C2semidirect
5. #14441C2semidirect
8. #14362C2semidirect
9. C4×#396C2semidirect
10. C4×#395C2semidirect
11. C4×#389C2semidirect
18. #2683C4direct (N × H)
76. C2×C22⋊Q8C2×C4semidirect
81. C2×C4×D4Q8semidirect
113. #396C2×C4semidirect
114. #395C2×C4semidirect
120. #389C2×C4semidirect
132. C4×C4⋊C4D4semidirect
327. C2×D4C4×Q8semidirect
429. C4⋊C4C4×D4semidirect
604. C4#2683direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#26832
C2×C4×C22⋊Q82
#145492
#145382
#144412
#144402
#144012
#143622
C4×#3962
C4×#3952
C4×#3892
C4×#3902
C4×#3862