Finite Group Databasebeta

← #612286

Subgroups — #612286

Order 512.

2882 conjugacy classes of subgroups.

The subgroup lattice has 593 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2396C214yesmaximal, complemented
2#17072C211yesmaximal, characteristic, complemented
3#17071C212yesmaximal
4#17070C211yesmaximal, characteristic, complemented
5#16882C214yesmaximal, complemented
6#16877C212yesmaximal, complemented
7#16556C212yesmaximal
8#16550C214yesmaximal, complemented
9C4×#396C214yesmaximal, complemented
10C4×#395C212yesmaximal, complemented
11#13839C212yesmaximal
12C4×#389C212yesmaximal, complemented
13C4×#390C211yesmaximal, characteristic
14C22×C4×Q8C2212yes—
15C22×C4×D4C2211yescharacteristic
16C22×#117C2212yes—
17#2396—232no—
18C2×#400C2214yes—
19C2×#399C2214yes—
20C2×#398C2214yes—
21C2×#398C2214yes—
22C2×#396C2214yes—
23C2×#396C2214yes—
24C2×#396C2214yes—
25C2×#395C2212yes—
26C2×#395C2214yes—
27C2×#393C2214yes—
28C2×#390C2211yescharacteristic
29C2×#390C2211yescharacteristic
30C2×#389C2212yes—
31C2×#389C2212yes—
32C2×C4×C4⋊C4C2214yes—
33C2×C4×C4⋊C4C2214yes—
34C2×C4×C4⋊C4C2211yescharacteristic
35C2×C4×C4⋊C4C2212yes—
36C2×C4×C4⋊C4C2212yes—
37C2×C4×C22⋊C4C2214yes—
38C2×C4×C22⋊C4C2214yes—
39C2×C4×C22⋊C4C2212yes—
40C2×C43C2211yescharacteristic
41#1446C2212yes—
42#1444C2212yes—
43#1444C2214yes—
44#1443C2214yes—
45#1441C2212yes—
46#1441C2212yes—
47#1441C2214yes—
48#1441C2212yes—
49#1441C2214yes—
50#1441C2214yes—
51#1441C2214yes—
52#1440C2214yes—
53#1439C2212yes—
54#1439C2212yes—
55#1439C2214yes—
56#1439C2214yes—
57#1439C2214yes—
58#1438C2214yes—
59#1437C2214yes—
60#1436C2214yes—
61#1436C2214yes—
62C4⋊#95C2211yescharacteristic
63C4⋊#95C2212yes—
64C4⋊#95C2211yescharacteristic
65C4⋊#95C2212yes—
66C4×#95C2214yes—
67C4×#95C2212yes—
68C23×Q8C2312yes—
69C23×D4C2311yescharacteristic
70C24×C4C2312yes—
71C2×C4×Q8—216no—
72C2×C4×Q8—44no—
73C2×C4×D4—42no—
74C2×C4×D4—42no—
75C2×C4×D4Q818yescomplemented
76C2×#117—216no—
77C2×#117—44no—
78C22×C4⋊C4C2314yes—
79C22×C4⋊C4C2314yes—
80C22×C4⋊C4C2312yes—
81C22×C4⋊C4C2312yes—
82C22×C4⋊C4C2314yes—
83C22×C4⋊C4C2314yes—
84C22×C4⋊C4C2314yes—
85C22×C4⋊C4C2312yes—
86C22×C4⋊C4C2312yes—
87C22×C4⋊C4C2312yes—
88C22×C4⋊C4C2311yescharacteristic
89C22×C4⋊C4C2312yes—
90C22×C4⋊C4C2311yescharacteristic
91C22×C4⋊C4C2311yescharacteristic
92C22×C4⋊C4C2312yes—
93C22×C22⋊C4C2314yes—
94C22×C22⋊C4C2314yes—
95C22×C22⋊C4C2314yes—
96C22×C22⋊C4C2314yes—
97C22×C22⋊C4C2314yes—
98C22×C22⋊C4C2314yes—
99C22×C22⋊C4C2312yes—
100C22×C42C2312yes—
101C22×C42C2314yes—
102C22×C42C2314yes—
103C22×C42C2312yes—
104C22×C42C2311yescharacteristic
105C22×C42C2311yescharacteristic
106C22×C42C2312yes—
107C22×C42C2311yescharacteristic
108#400—216no—
109#399—216no—
110#398—216no—
111#398—216no—
112#396—216no—
113#396—216no—
114#396—216no—
115#395—216no—
116#395—28no—
117#393—216no—
118#390—24no—
119#390—24no—
120#389—28no—
121#389—28no—
122C4×C4⋊C4—216no—
123C4×C4⋊C4—216no—
124C4×C4⋊C4—42no—
125C4×C4⋊C4—44no—
126C4×C4⋊C4D4116yescomplemented
127C4×C22⋊C4—216no—
128C4×C22⋊C4—216no—
129C4×C22⋊C4—44no—
130C2×#95C2314yes—
131C2×#95C2314yes—
132C2×#95C2314yes—
133C2×#95C2314yes—
134C2×#95C2314yes—
135C2×#95C2314yes—
136C2×#95C2312yes—
137C2×#95C2314yes—
138C2×#95C2314yes—
139C2×#95C2314yes—
140C2×#95C2314yes—
141C2×#95C2314yes—
142C2×#95C2314yes—
143C2×#95C2314yes—
144C2×#95C2312yes—
145C2×#95C2314yes—
146C2×#95C2312yes—
147C2×#95C2312yes—
148C2×#95C2312yes—
149C2×#95C2314yes—
150C43—42no—
151C25C2412yes—
152C22×Q8—28no—
153C22×Q8—24no—
154C22×Q8—42no—
155C22×Q8—44no—
156C22×Q8—42no—
157C22×D4—22no—
158C22×D4—22no—
159C22×D4—41no—
160C22×D4—41no—
161C22×D4—41no—
162C22×D4—41no—
163C22×D4C2×Q814yes—
164C23×C4—28no—
165C23×C4—24no—
166C23×C4—24no—
167C23×C4—28no—
168C23×C4C2×Q818yes—
169C23×C4C2412yes—
170C23×C4C2414yes—
171C23×C4C2414yes—
172C23×C4C2414yes—
173C23×C4C2412yes—
174C23×C4C2411yescharacteristic
175C23×C4C2412yes—
176C23×C4C2412yes—
177C23×C4C2414yes—
178C23×C4C2412yes—
179C23×C4C2411yescharacteristic
180C23×C4C2411yescharacteristic
181C4×Q8—432no—
182C4×D4—416no—
183#117—432no—
184C2×C4⋊C4—24no—
185C2×C4⋊C4—24no—
186C2×C4⋊C4—28no—
187C2×C4⋊C4—28no—
188C2×C4⋊C4—28no—
189C2×C4⋊C4—28no—
190C2×C4⋊C4—28no—
191C2×C4⋊C4—28no—
192C2×C4⋊C4—28no—
193C2×C4⋊C4—28no—
194C2×C4⋊C4—28no—
195C2×C4⋊C4—28no—
196C2×C4⋊C4—24no—
197C2×C4⋊C4—28no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—28no—
200C2×C4⋊C4—28no—
201C2×C4⋊C4—28no—
202C2×C4⋊C4—42no—
203C2×C4⋊C4—42no—
204C2×C4⋊C4—42no—
205C2×C4⋊C4—41no—
206C2×C4⋊C4—44no—
207C2×C4⋊C4—44no—
208C2×C4⋊C4—44no—
209C2×C4⋊C4—42no—
210C2×C4⋊C4—41no—
211C2×C4⋊C4—42no—
212C2×C4⋊C4—42no—
213C2×C4⋊C4—41no—
214C2×C4⋊C4—41no—
215C2×C4⋊C4—42no—
216C2×C4⋊C4—44no—
217C2×C4⋊C4—42no—
218C2×C4⋊C4—41no—
219C2×C4⋊C4—42no—
220C2×C4⋊C4—41no—
221C2×C4⋊C4C2×D418yes—
222C2×C4⋊C4C2×D418yes—
223C2×C4⋊C4C2×Q814yes—
224C2×C4⋊C4#4118yes—
225C2×C4⋊C4#4114yes—
226C2×C22⋊C4—28no—
227C2×C22⋊C4—28no—
228C2×C22⋊C4—28no—
229C2×C22⋊C4—28no—
230C2×C22⋊C4—28no—
231C2×C22⋊C4—28no—
232C2×C22⋊C4—28no—
233C2×C22⋊C4—28no—
234C2×C22⋊C4—28no—
235C2×C22⋊C4—28no—
236C2×C22⋊C4—28no—
237C2×C22⋊C4—44no—
238C2×C22⋊C4—44no—
239C2×C22⋊C4—44no—
240C2×C22⋊C4—44no—
241C2×C22⋊C4—44no—
242C2×C22⋊C4—44no—
243C2×C22⋊C4—42no—
244C2×C22⋊C4—42no—
245C2×C22⋊C4—44no—
246C2×C22⋊C4C2×Q818yes—
247C2×C42—28no—
248C2×C42—28no—
249C2×C42—22no—
250C2×C42—22no—
251C2×C42—24no—
252C2×C42—24no—
253C2×C42—28no—
254C2×C42—28no—
255C2×C42—24no—
256C2×C42—22no—
257C2×C42—24no—
258C2×C42—28no—
259C2×C42—28no—
260C2×C42—41no—
261C2×C42—42no—
262C2×C42—41no—
263C2×C42—41no—
264C2×C42—42no—
265C2×C42—41no—
266C2×C42C2×D4116yes—
267C2×C42C2×D418yes—
268C2×C42C2×Q814yes—
269C2×C42#4114yes—
270#95—28no—
271#95—28no—
272#95—28no—
273#95—28no—
274#95—28no—
275#95—28no—
276#95—28no—
277#95—28no—
278#95—28no—
279#95—28no—
280#95—24no—
281#95—28no—
282#95—28no—
283#95—44no—
284#95—42no—
285#95—42no—
286#95—42no—
287#95—44no—
288#95—42no—
289#95C2×D4116yes—
290C24—24no—
291C24—24no—
292C24—22no—
293C24—22no—
294C24—22no—
295C24—22no—
296C24—24no—
297C24—24no—
298C24—22no—
299C24—22no—
300C24C22×Q814yes—
301C24C2511yescharacteristic
302C2×Q8—48no—
303C2×Q8—416no—
304C2×Q8—416no—
305C2×Q8—48no—
306C2×Q8—48no—
307C2×D4—28no—
308C2×D4—44no—
309C2×D4—44no—
310C2×D4—44no—
311C2×D4—44no—
312C2×D4—44no—
313C2×D4—44no—
314C22×C4—22no—
315C22×C4—21no—
316C22×C4—21no—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—21no—
321C22×C4—21no—
322C22×C4—24no—
323C22×C4—24no—
324C22×C4—24no—
325C22×C4—24no—
326C22×C4—24no—
327C22×C4—22no—
328C22×C4—24no—
329C22×C4—24no—
330C22×C4—24no—
331C22×C4—24no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—24no—
335C22×C4—22no—
336C22×C4—21no—
337C22×C4—24no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—21no—
341C22×C4—24no—
342C22×C4—24no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—24no—
346C22×C4—24no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—24no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—24no—
353C22×C4—24no—
354C22×C4—24no—
355C22×C4—24no—
356C22×C4—24no—
357C22×C4—24no—
358C22×C4—216no—
359C22×C4—232no—
360C22×C4—24no—
361C22×C4—24no—
362C22×C4—24no—
363C22×C4—24no—
364C22×C4—24no—
365C22×C4—22no—
366C22×C4—24no—
367C22×C4—22no—
368C22×C4—24no—
369C22×C4—24no—
370C22×C4—24no—
371C22×C4—22no—
372C22×C4—416no—
373C22×C4—48no—
374C22×C4—48no—
375C22×C4C22×D414yes—
376C22×C4C22×D418yes—
377C22×C4C22×D418yes—
378C22×C4C22×D414yes—
379C22×C4C22×D414yes—
380C22×C4C22×Q812yes—
381C22×C4C22×Q814yes—
382C22×C4C22×Q812yes—
383C22×C4C22×Q812yes—
384C22×C4C2×#4118yes—
385C22×C4C2×#4114yes—
386C22×C4C2×#4114yes—
387C22×C4C2×#4114yes—
388C22×C4C2×#4112yes—
389C22×C4C2×#4112yes—
390C22×C4H(5,𝔽2)12yes—
391C22×C4H(5,𝔽2)12yes—
392C22×C42−1+414yes—
393C4⋊C4—216no—
394C4⋊C4—48no—
395C4⋊C4—44no—
396C4⋊C4—48no—
397C4⋊C4—44no—
398C4⋊C4—416no—
399C4⋊C4—416no—
400C4⋊C4—416no—
401C4⋊C4—48no—
402C4⋊C4—48no—
403C4⋊C4—48no—
404C4⋊C4—416no—
405C4⋊C4—48no—
406C4⋊C4—44no—
407C4⋊C4—416no—
408C22⋊C4—48no—
409C22⋊C4—416no—
410C22⋊C4—416no—
411C22⋊C4—416no—
412C22⋊C4—416no—
413C22⋊C4—416no—
414C22⋊C4—416no—
415C42—216no—
416C42—28no—
417C42—232no—
418C42—44no—
419C42—48no—
420C42—44no—
421C42—416no—
422C42—48no—
423C23—24no—
424C23—24no—
425C23—24no—
426C23—28no—
427C23—28no—
428C23—44no—
429C23—44no—
430C23—44no—
431C23—44no—
432C23—42no—
433C23—42no—
434C23—42no—
435C23—42no—
436C23—44no—
437C23—44no—
438C23—42no—
439C23—44no—
440C23—44no—
441C23—44no—
442C23—42no—
443C23—42no—
444C23—42no—
445C23—44no—
446C23C23×D412yes—
447C23C23×Q811yescharacteristic
448C23C22×#4111yescharacteristic
449C23C22×#4112yes—
450C23C22×#4111yescharacteristic
451C23C2×H(5,𝔽2)11yescharacteristic
452C23C2×H(5,𝔽2)11yescharacteristic
453C23C2×2−1+412yes—
454C23#58511yescharacteristic
455C23#58512yes—
456C23#58511yescharacteristic
457Q8—432no—
458D4—416no—
459C2×C4—24no—
460C2×C4—24no—
461C2×C4—22no—
462C2×C4—22no—
463C2×C4—24no—
464C2×C4—24no—
465C2×C4—22no—
466C2×C4—22no—
467C2×C4—22no—
468C2×C4—22no—
469C2×C4—24no—
470C2×C4—24no—
471C2×C4—28no—
472C2×C4—24no—
473C2×C4—28no—
474C2×C4—22no—
475C2×C4—24no—
476C2×C4—28no—
477C2×C4—24no—
478C2×C4—28no—
479C2×C4—28no—
480C2×C4—28no—
481C2×C4—24no—
482C2×C4—28no—
483C2×C4—28no—
484C2×C4—28no—
485C2×C4—24no—
486C2×C4—28no—
487C2×C4—24no—
488C2×C4—28no—
489C2×C4—44no—
490C2×C4—44no—
491C2×C4—44no—
492C2×C4—44no—
493C2×C4—44no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—44no—
497C2×C4—42no—
498C2×C4—44no—
499C2×C4—44no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4—44no—
503C2×C4—44no—
504C2×C4—44no—
505C2×C4—42no—
506C2×C4—41no—
507C2×C4—44no—
508C2×C4—42no—
509C2×C4—41no—
510C2×C4—42no—
511C2×C4—44no—
512C2×C4—42no—
513C2×C4—44no—
514C2×C4—432no—
515C2×C4—44no—
516C2×C4—44no—
517C2×C4—44no—
518C2×C4—44no—
519C2×C4—44no—
520C2×C4—44no—
521C2×C4—42no—
522C2×C4—44no—
523C2×C4—44no—
524C2×C4—42no—
525C2×C4—44no—
526C2×C4—42no—
527C2×C4#53218yes—
528C2×C4#53314yes—
529C2×C4#53718yes—
530C2×C4#54314yes—
531C2×C4D4214yes—
532C2×C4D4×Q818yes—
533C2×C4#56314yes—
534C22—44no—
535C22—44no—
536C22—48no—
537C22—44no—
538C22—48no—
539C22—84no—
540C22—82no—
541C22—82no—
542C22—82no—
543C22—82no—
544C22—84no—
545C22C2×#53212yes—
546C22C2×#53311yescharacteristic
547C22C2×#53712yes—
548C22#344912yes—
549C22C2×#54311yescharacteristic
550C22#345811yescharacteristic
551C22#345911yescharacteristic
552C22#345911yescharacteristic
553C22#345912yes—
554C22#346011yescharacteristic
555C22#346011yescharacteristic
556C22#346112yes—
557C22#346112yes—
558C22C2×D4211yescharacteristic
559C22C2×D4×Q812yes—
560C22D4×#4112yes—
561C22D4×#4112yes—
562C22Q8×#4112yes—
563C22#348212yes—
564C22#348311yescharacteristic
565C22C2×#56311yescharacteristic
566C22#352911yescharacteristic
567C22#353112yes—
568C4—24no—
569C4—28no—
570C4—42no—
571C4—48no—
572C4—44no—
573C4—48no—
574C4—44no—
575C4—44no—
576C4—48no—
577C4—84no—
578C4—82no—
579C4#268318yes—
580C2—84no—
581C2#3457811yescharacteristic
582C2#3459412yes—
583C2#3492511yescharacteristic
584C2#3492912yes—
585C2#3669811yescharacteristic
586C2C2×#268311yescharacteristic
587C2#5210311yescharacteristic
588C2#5216212yes—
589C2#5216912yes—
590C2#5229111yescharacteristic
591C2#5230611yescharacteristic
subgroupG/H
centerC23×C4C24
commutator, Frattini, socle, cosocleC24C25
prime radicalC23×D4C23
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C23×C4 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#2396C2semidirect
2. #17072C2semidirect
4. #17070C2semidirect
5. #16882C2semidirect
6. #16877C2semidirect
8. #16550C2semidirect
9. C4×#396C2semidirect
10. C4×#395C2semidirect
12. C4×#389C2semidirect
75. C2×C4×D4Q8semidirect
126. C4×C4⋊C4D4semidirect

Maximal subgroups

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

subgroupindex
C2×#23962
#170722
#170712
#170702
#168822
#168772
#165562
#165502
C4×#3962
C4×#3952
#138392
C4×#3892
C4×#3902