Finite Group Databasebeta

← #535067

Subgroups — #535067

Order 512.

3530 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×#2360C214yesmaximal, complemented
3#14441C212yesmaximal, complemented
4#14440C211yesmaximal, characteristic
5#14236C212yesmaximal, complemented
6#14222C214yesmaximal, complemented
7#14204C214yesmaximal, complemented
8#14174C212yesmaximal, complemented
9#14099C214yesmaximal, complemented
10#14015C212yesmaximal
11#13824C212yesmaximal
12C4×#389C212yesmaximal, complemented
13C4×#390C211yesmaximal, characteristic
14C22×C22⋊Q8C2214yes—
15C22×C4×Q8C2212yes—
16C22×C4×D4C2211yescharacteristic
17C23×C4⋊C4C2212yes—
18#2683C4116yescomplemented
19C2×#396C2214yes—
20C2×#395C2212yes—
21#2360—232no—
22C2×#391C2214yes—
23C2×#390C2211yescharacteristic
24C2×#390C2211yescharacteristic
25C2×#389C2214yes—
26C2×#389C2214yes—
27C2×#389C2212yes—
28C2×#389C2212yes—
29C2×#387C2214yes—
30C2×#387C2214yes—
31C2×#386C2212yes—
32C2×#386C2212yes—
33C2×#385C2214yes—
34C2×#385C2214yes—
35C2×#385C2212yes—
36C2×C4×C4⋊C4C2212yes—
37C2×C4×C4⋊C4C2214yes—
38C2×C4×C4⋊C4C2211yescharacteristic
39C2×C4×C4⋊C4C2212yes—
40C2×C4×C22⋊C4C2214yes—
41C2×C4×C22⋊C4C2214yes—
42C2×C4×C22⋊C4C2214yes—
43C2×C43C2211yescharacteristic
44#1446C2212yes—
45#1444C2214yes—
46#1442C2212yes—
47#1441C2214yes—
48#1441C2212yes—
49#1441C2214yes—
50#1439C2214yes—
51#1439C2214yes—
52#1437C2214yes—
53#1437C2214yes—
54#1437C2214yes—
55#1436C2214yes—
56#1436C2214yes—
57#1436C2214yes—
58#1436C2214yes—
59#1436C2214yes—
60#1436C2214yes—
61C4⋊#95C2212yes—
62C4⋊#95C2211yescharacteristic
63C4⋊#95C2212yes—
64C4⋊#95C2211yescharacteristic
65#1434C2212yes—
66C22⋊#95C2212yes—
67C22⋊#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—24no—
80C2×C4×D4—42no—
81C2×C4×D4—42no—
82C22×C4⋊C4—216no—
83C22×C4⋊C4—44no—
84C22×C4⋊C4C2314yes—
85C22×C4⋊C4C2312yes—
86C22×C4⋊C4C2314yes—
87C22×C4⋊C4C2314yes—
88C22×C4⋊C4C2314yes—
89C22×C4⋊C4C2311yescharacteristic
90C22×C4⋊C4C2312yes—
91C22×C4⋊C4C2314yes—
92C22×C4⋊C4C2311yescharacteristic
93C22×C4⋊C4C2312yes—
94C22×C4⋊C4C2312yes—
95C22×C4⋊C4C2311yescharacteristic
96C22×C4⋊C4C2312yes—
97C22×C22⋊C4C2312yes—
98C22×C22⋊C4C2314yes—
99C22×C22⋊C4C2314yes—
100C22×C22⋊C4C2314yes—
101C22×C22⋊C4C2314yes—
102C22×C22⋊C4C2312yes—
103C22×C22⋊C4C2312yes—
104C22×C42C2311yescharacteristic
105C22×C42C2312yes—
106C22×C42C2312yes—
107C22×C42C2314yes—
108C22×C42C2314yes—
109C22×C42C2314yes—
110C22×C42C2311yescharacteristic
111C22×C42C2312yes—
112C22×C42C2311yescharacteristic
113#396C2×C4132yescomplemented
114#395C2×C4116yescomplemented
115#391—216no—
116#390C2×C418yes—
117#390Q818yescomplemented
118#389—216no—
119#389—216no—
120#389C2×C4116yescomplemented
121#389D4116yescomplemented
122#387—216no—
123#387—216no—
124#386—28no—
125#386C2×C4116yes—
126#385—216no—
127#385—216no—
128#385—44no—
129C4×C4⋊C4—28no—
130C4×C4⋊C4—216no—
131C4×C4⋊C4—44no—
132C4×C4⋊C4—42no—
133C4×C22⋊C4—216no—
134C4×C22⋊C4—216no—
135C4×C22⋊C4—216no—
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×#95C2314yes—
149C2×#95C2314yes—
150C2×#95C2312yes—
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—24no—
172C23×C4—24no—
173C23×C4—28no—
174C23×C4—28no—
175C23×C4—24no—
176C23×C4—42no—
177C23×C4—42no—
178C23×C4—42no—
179C23×C4—42no—
180C23×C4—44no—
181C23×C4—44no—
182C23×C4—42no—
183C23×C4C22×C418yes—
184C23×C4C2414yes—
185C23×C4C2412yes—
186C23×C4C2412yes—
187C23×C4C2414yes—
188C23×C4C2414yes—
189C23×C4C2412yes—
190C23×C4C2411yescharacteristic
191C23×C4C2412yes—
192C23×C4C2414yes—
193C23×C4C2411yescharacteristic
194C23×C4C2411yescharacteristic
195C23×C4C2412yes—
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—22no—
209C2×C4⋊C4—24no—
210C2×C4⋊C4—28no—
211C2×C4⋊C4—432no—
212C2×C4⋊C4—44no—
213C2×C4⋊C4—42no—
214C2×C4⋊C4—41no—
215C2×C4⋊C4—42no—
216C2×C4⋊C4—41no—
217C2×C4⋊C4—42no—
218C2×C4⋊C4—42no—
219C2×C4⋊C4—41no—
220C2×C4⋊C4—41no—
221C2×C4⋊C4—42no—
222C2×C4⋊C4—42no—
223C2×C4⋊C4—41no—
224C2×C4⋊C4—42no—
225C2×C4⋊C4—41no—
226C2×C4⋊C4—44no—
227C2×C4⋊C4—44no—
228C2×C4⋊C4C22×C4116yes—
229C2×C4⋊C4C22×C4116yes—
230C2×C4⋊C4C22×C418yes—
231C2×C4⋊C4C22×C414yes—
232C2×C4⋊C4C22×C418yes—
233C2×C4⋊C4C2×D4116yes—
234C2×C4⋊C4C2×D418yes—
235C2×C4⋊C4C2×D418yes—
236C2×C4⋊C4C2×Q814yes—
237C2×C4⋊C4#4118yes—
238C2×C22⋊C4—28no—
239C2×C22⋊C4—28no—
240C2×C22⋊C4—28no—
241C2×C22⋊C4—28no—
242C2×C22⋊C4—28no—
243C2×C22⋊C4—28no—
244C2×C22⋊C4—24no—
245C2×C22⋊C4—28no—
246C2×C22⋊C4—28no—
247C2×C22⋊C4—24no—
248C2×C22⋊C4—24no—
249C2×C22⋊C4—42no—
250C2×C22⋊C4—44no—
251C2×C22⋊C4—42no—
252C2×C22⋊C4—42no—
253C2×C22⋊C4—42no—
254C2×C22⋊C4—42no—
255C2×C22⋊C4C22×C4116yes—
256C2×C22⋊C4C22×C4116yes—
257C2×C22⋊C4C22×C418yes—
258C2×C22⋊C4C2×Q8116yes—
259C2×C42—22no—
260C2×C42—24no—
261C2×C42—22no—
262C2×C42—22no—
263C2×C42—24no—
264C2×C42—24no—
265C2×C42—28no—
266C2×C42—28no—
267C2×C42—28no—
268C2×C42—28no—
269C2×C42—28no—
270C2×C42—24no—
271C2×C42—28no—
272C2×C42—44no—
273C2×C42—41no—
274C2×C42—42no—
275C2×C42—41no—
276C2×C42—41no—
277C2×C42—42no—
278C2×C42—41no—
279C2×C42—44no—
280C2×C42—44no—
281C2×C42—44no—
282C2×C42C22×C414yes—
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—28no—
296#95—28no—
297#95—44no—
298#95—44no—
299#95—42no—
300#95—42no—
301#95C22×C4116yes—
302#95C22×C418yes—
303#95C2×D4116yes—
304C24—24no—
305C24—24no—
306C24—22no—
307C24—22no—
308C24—24no—
309C24—22no—
310C24—24no—
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—22no—
330C22×C4—21no—
331C22×C4—24no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—21no—
335C22×C4—21no—
336C22×C4—21no—
337C22×C4—24no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—21no—
344C22×C4—21no—
345C22×C4—22no—
346C22×C4—24no—
347C22×C4—24no—
348C22×C4—24no—
349C22×C4—24no—
350C22×C4—24no—
351C22×C4—22no—
352C22×C4—24no—
353C22×C4—21no—
354C22×C4—24no—
355C22×C4—24no—
356C22×C4—24no—
357C22×C4—24no—
358C22×C4—21no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—24no—
362C22×C4—24no—
363C22×C4—22no—
364C22×C4—24no—
365C22×C4—22no—
366C22×C4—24no—
367C22×C4—24no—
368C22×C4—22no—
369C22×C4—24no—
370C22×C4—24no—
371C22×C4—232no—
372C22×C4—24no—
373C22×C4—24no—
374C22×C4—24no—
375C22×C4—24no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—24no—
380C22×C4—24no—
381C22×C4—24no—
382C22×C4—24no—
383C22×C4—24no—
384C22×C4—48no—
385C22×C4—48no—
386C22×C4—416no—
387C22×C4—48no—
388C22×C4—48no—
389C22×C4—48no—
390C22×C4—416no—
391C22×C4—416no—
392C22×C4—48no—
393C22×C4—88no—
394C22×C4—84no—
395C22×C4—88no—
396C22×C4—84no—
397C22×C4—84no—
398C22×C4C23×C418yes—
399C22×C4C23×C414yes—
400C22×C4C23×C418yes—
401C22×C4C23×C414yes—
402C22×C4C23×C412yes—
403C22×C4C22×D418yes—
404C22×C4C22×D414yes—
405C22×C4C22×D418yes—
406C22×C4C22×D414yes—
407C22×C4C22×D414yes—
408C22×C4C22×Q814yes—
409C22×C4C22×Q812yes—
410C22×C4C22×Q814yes—
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—232no—
420C4⋊C4—48no—
421C4⋊C4—44no—
422C4⋊C4—48no—
423C4⋊C4—44no—
424C4⋊C4—416no—
425C4⋊C4—48no—
426C4⋊C4—48no—
427C4⋊C4—44no—
428C4⋊C4—88no—
429C4⋊C4C4×D4132yescomplemented
430C22⋊C4—232no—
431C22⋊C4—232no—
432C22⋊C4—232no—
433C22⋊C4—48no—
434C22⋊C4—416no—
435C22⋊C4—48no—
436C22⋊C4—84no—
437C42—216no—
438C42—28no—
439C42—44no—
440C42—48no—
441C42—44no—
442C42—416no—
443C42—416no—
444C42—48no—
445C42—88no—
446C23—28no—
447C23—24no—
448C23—24no—
449C23—28no—
450C23—28no—
451C23—28no—
452C23—28no—
453C23—28no—
454C23—24no—
455C23—24no—
456C23—24no—
457C23—24no—
458C23—42no—
459C23—44no—
460C23—42no—
461C23—44no—
462C23—44no—
463C23—44no—
464C23—44no—
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—28no—
489C2×C4—24no—
490C2×C4—28no—
491C2×C4—24no—
492C2×C4—28no—
493C2×C4—24no—
494C2×C4—28no—
495C2×C4—22no—
496C2×C4—22no—
497C2×C4—22no—
498C2×C4—24no—
499C2×C4—24no—
500C2×C4—24no—
501C2×C4—24no—
502C2×C4—24no—
503C2×C4—24no—
504C2×C4—24no—
505C2×C4—24no—
506C2×C4—22no—
507C2×C4—22no—
508C2×C4—22no—
509C2×C4—24no—
510C2×C4—28no—
511C2×C4—28no—
512C2×C4—28no—
513C2×C4—24no—
514C2×C4—28no—
515C2×C4—28no—
516C2×C4—28no—
517C2×C4—28no—
518C2×C4—22no—
519C2×C4—24no—
520C2×C4—28no—
521C2×C4—24no—
522C2×C4—432no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—44no—
526C2×C4—44no—
527C2×C4—41no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—44no—
531C2×C4—44no—
532C2×C4—44no—
533C2×C4—44no—
534C2×C4—44no—
535C2×C4—44no—
536C2×C4—44no—
537C2×C4—44no—
538C2×C4—41no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—44no—
542C2×C4—44no—
543C2×C4—44no—
544C2×C4—44no—
545C2×C4—42no—
546C2×C4—44no—
547C2×C4—44no—
548C2×C4—44no—
549C2×C4—44no—
550C2×C4—42no—
551C2×C4—44no—
552C2×C4—816no—
553C2×C4—816no—
554C2×C4C2×C4×D418yes—
555C2×C4C2×C4×D4116yes—
556C2×C4C2×C4×Q814yes—
557C2×C4D4214yes—
558C2×C4D4×Q818yes—
559C22—216no—
560C22—28no—
561C22—44no—
562C22—44no—
563C22—44no—
564C22—44no—
565C22—44no—
566C22—48no—
567C22—48no—
568C22—44no—
569C22—48no—
570C22C22×C4×D412yes—
571C22C22×C4×Q811yescharacteristic
572C22C4×H(5,𝔽2)11yescharacteristic
573C22C4×H(5,𝔽2)11yescharacteristic
574C22C4×2−1+412yes—
575C22C2×D4211yescharacteristic
576C22C2×D4×Q812yes—
577C22D4×#4112yes—
578C22D4×#4112yes—
579C22Q8×#4112yes—
580C22#348111yescharacteristic
581C22#348212yes—
582C22#349011yescharacteristic
583C22#349111yescharacteristic
584C22#349212yes—
585C22#349312yes—
586C22#349612yes—
587C22#349712yes—
588C22#350111yescharacteristic
589C22#350111yescharacteristic
590C22#354511yescharacteristic
591C22#355011yescharacteristic
592C22#355312yes—
593C4—216no—
594C4—24no—
595C4—28no—
596C4—24no—
597C4—28no—
598C4—44no—
599C4—42no—
600C4—48no—
601C4—44no—
602C4—48no—
603C4—82no—
604C4—84no—
605C2—48no—
606C2C4×D4211yescharacteristic
607C2C4×D4×Q812yes—
608C2#3427011yescharacteristic
609C2#3427412yes—
610C2#3452311yescharacteristic
611C2#5981411yescharacteristic
612C2#5981711yescharacteristic
613C2#5983012yes—
614C2#5983512yes—
615C2#5992611yescharacteristic
616C2#6004611yescharacteristic
subgroupG/H
center, Frattini, socle, cosocleC24C25
commutatorC23C24×C4
prime radicalC23×D4C23
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2360C2semidirect
3. #14441C2semidirect
5. #14236C2semidirect
6. #14222C2semidirect
7. #14204C2semidirect
8. #14174C2semidirect
9. #14099C2semidirect
12. C4×#389C2semidirect
18. #2683C4semidirect
76. C2×C22⋊Q8C2×C4semidirect
113. #396C2×C4semidirect
114. #395C2×C4semidirect
117. #390Q8semidirect
120. #389C2×C4semidirect
121. #389D4semidirect
327. C2×D4C4×Q8semidirect
429. C4⋊C4C4×D4semidirect

Maximal subgroups

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

subgroupindex
C2×#26832
C2×#23602
#144412
#144402
#142362
#142222
#142042
#141742
#140992
#140152
#138242
C4×#3892
C4×#3902