Finite Group Databasebeta

← #387061

Subgroups — #387061

Order 512.

1374 conjugacy classes of subgroups.

The subgroup lattice has 576 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2538C211yesmaximal, characteristic
2#21303C212yesmaximal, complemented
3#11354C212yesmaximal, complemented
4#11204C212yesmaximal, complemented
5#11044C212yesmaximal
6#10962C212yesmaximal
7#10913C212yesmaximal, complemented
8#10762C211yesmaximal, characteristic
9#10762C211yesmaximal, characteristic
10C22×C22⋊Q8C412yes—
11C22×C4×Q8C2211yescharacteristic
12C22×C4×D4C2211yescharacteristic
13C2×#423C2212yes—
14C22×#100C2212yes—
15C22×#100C2211yescharacteristic
16C22×#100C2211yescharacteristic
17C2×C4×M16C2212yes—
18#2538—22no—
19#2538—22no—
20#2538—22no—
21#2538—22no—
22C2×#396C414yes—
23C2×#390C2212yes—
24C2×#389C2212yes—
25C2×#384C2211yescharacteristic
26C2×#380C412yes—
27#1985—22no—
28#1985—22no—
29#1984—22no—
30#1984—22no—
31C2×#343C2212yes—
32C2×#343C2212yes—
33C2×#343C2211yescharacteristic
34C2×#343C2212yes—
35C2×#343C2211yescharacteristic
36C2×#343C2212yes—
37C2×#343C2212yes—
38#1728C2212yes—
39#1728C2212yes—
40#1728C2212yes—
41#1728C2212yes—
42C23×Q8C2×C411yescharacteristic
43C23×D4C2311yescharacteristic
44C24×C4C2×C412yes—
45C22×M16C2312yes—
46C22×M16C2312yes—
47C22×M16C2312yes—
48C22×M16C2312yes—
49C2×C22⋊Q8—24no—
50C2×C22⋊Q8—22no—
51C2×C22⋊Q8—22no—
52C2×C22⋊Q8—48no—
53C2×C4×Q8—41no—
54C2×C4×Q8—42no—
55C2×C4×Q8—41no—
56C2×C4×Q8D412yes—
57C2×C4×Q8D412yes—
58C2×C4×Q8D412yes—
59C2×C4×Q8D412yes—
60C2×C4×D4—22no—
61C2×C4×D4—22no—
62C2×C4×D4—41no—
63C2×C4×D4—41no—
64C2×C4×D4—42no—
65C22×C4⋊C4C2×C414yes—
66C22×C4⋊C4C2×C412yes—
67C22×C4⋊C4C2×C412yes—
68C22×C4⋊C4C2×C412yes—
69C22×C4⋊C4C2×C411yescharacteristic
70C22×C4⋊C4C2311yescharacteristic
71C22×C4⋊C4C2312yes—
72C22×C22⋊C4C2×C412yes—
73C22×C22⋊C4C2×C414yes—
74C22×C22⋊C4C2×C414yes—
75C22×C42C2312yes—
76C22×C42C2311yescharacteristic
77#423—42no—
78#423—42no—
79C2×C4⋊C8—22no—
80C2×C4⋊C8—22no—
81C2×#100—24no—
82C2×#100—22no—
83C2×#100—24no—
84C2×#100—41no—
85C2×#100—41no—
86C2×#100—42no—
87C2×#100—41no—
88C2×#100—41no—
89C2×#100Q814yescomplemented
90C4×M16—42no—
91C4×M16—42no—
92C2×#97—22no—
93C2×C4×C8—22no—
94#396—24no—
95#396—24no—
96#396—24no—
97#396—24no—
98#390—22no—
99#390—22no—
100#390—22no—
101#390—22no—
102#389—22no—
103#389—22no—
104#389—22no—
105#389—22no—
106#384—22no—
107#384—22no—
108#380—22no—
109#380—22no—
110#380—24no—
111C2×#95C2×C414yes—
112#343—24no—
113#343—24no—
114#343—41no—
115#343—42no—
116#343—41no—
117#343—42no—
118#343—42no—
119#336—22no—
120#336—22no—
121#336—22no—
122#336—22no—
123#336—22no—
124#336—22no—
125#336—22no—
126#336—22no—
127C25C22×C412yes—
128C22×Q8—24no—
129C22×Q8—41no—
130C22×Q8—41no—
131C22×Q8—42no—
132C22×Q8C22⋊C412yes—
133C22×Q8C22⋊C411yescharacteristic
134C22×Q8C22⋊C411yescharacteristic
135C22×D4—22no—
136C22×D4—22no—
137C22×D4—21no—
138C22×D4—21no—
139C22×D4—22no—
140C22×D4—21no—
141C22×D4—22no—
142C22×D4—21no—
143C22×D4C2×Q812yes—
144C22×D4#4112yes—
145C23×C4—22no—
146C23×C4—22no—
147C23×C4—42no—
148C23×C4—42no—
149C23×C4—44no—
150C23×C4—44no—
151C23×C4C22×C411yescharacteristic
152C23×C4C22×C412yes—
153C23×C4C22×C414yes—
154C23×C4C22×C412yes—
155C23×C4C22×C411yescharacteristic
156C23×C4C22×C412yes—
157C23×C4C2411yescharacteristic
158C2×M16—22no—
159C2×M16—22no—
160C2×M16—22no—
161C2×M16—22no—
162C2×M16—22no—
163C2×M16—22no—
164C2×M16—22no—
165C2×M16—22no—
166C2×M16—22no—
167C2×M16—22no—
168C2×M16—22no—
169C2×M16—22no—
170C2×M16—22no—
171C2×M16—22no—
172C2×M16—22no—
173C2×M16—22no—
174C2×M16—22no—
175C2×M16—22no—
176C2×M16—22no—
177C2×M16—22no—
178C2×M16—22no—
179C2×M16—22no—
180C2×M16C2×Q814yes—
181C2×M16#4114yes—
182C22×C8—22no—
183C22×C8—22no—
184C22×C8—22no—
185C22×C8—22no—
186C22⋊Q8—48no—
187C22⋊Q8—416no—
188C22⋊Q8—48no—
189C4×Q8—42no—
190C4×Q8—42no—
191C4×Q8—42no—
192C4×Q8—42no—
193C4×Q8—42no—
194C4×Q8—42no—
195C4×Q8—42no—
196C4×Q8—42no—
197C4×D4—44no—
198C4×D4—44no—
199C4×D4—44no—
200C4×D4—44no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—24no—
204C2×C4⋊C4—24no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—24no—
216C2×C4⋊C4—41no—
217C2×C4⋊C4—41no—
218C2×C4⋊C4—42no—
219C2×C4⋊C4—42no—
220C2×C4⋊C4—44no—
221C2×C4⋊C4—48no—
222C2×C4⋊C4—82no—
223C2×C4⋊C4C22⋊C412yes—
224C2×C4⋊C4C22⋊C411yescharacteristic
225C2×C4⋊C4C22⋊C411yescharacteristic
226C2×C4⋊C4C22⋊C412yes—
227C2×C4⋊C4C22⋊C412yes—
228C2×C4⋊C4C22⋊C412yes—
229C2×C4⋊C4C22⋊C412yes—
230C2×C22⋊C4—24no—
231C2×C22⋊C4—24no—
232C2×C22⋊C4—24no—
233C2×C22⋊C4—24no—
234C2×C22⋊C4—24no—
235C2×C22⋊C4—42no—
236C2×C22⋊C4—42no—
237C2×C22⋊C4—48no—
238C2×C22⋊C4—84no—
239C2×C42—21no—
240C2×C42—21no—
241C2×C42—22no—
242C2×C42—22no—
243C2×C42—22no—
244C2×C42—22no—
245C2×C42—22no—
246C2×C42C2×D412yes—
247C2×C42C2×D412yes—
248C2×C42C2×D412yes—
249C2×C42C2×D412yes—
250C2×C42C2×D412yes—
251C2×C42C2×D412yes—
252C4⋊C8—44no—
253C4⋊C8—44no—
254#100—44no—
255#100—48no—
256#100—44no—
257#97—44no—
258C4×C8—44no—
259#95—24no—
260#95—24no—
261C24—22no—
262C24—22no—
263C24—22no—
264C24—22no—
265C24—22no—
266C24—22no—
267C24—42no—
268C24—42no—
269C24—42no—
270C24—42no—
271C24C4×Q814yes—
272C24C23×C411yescharacteristic
273C2×Q8—24no—
274C2×Q8—22no—
275C2×Q8—24no—
276C2×Q8—41no—
277C2×Q8—42no—
278C2×Q8—41no—
279C2×Q8—41no—
280C2×Q8—41no—
281C2×Q8—48no—
282C2×Q8—48no—
283C2×Q8#10014yescomplemented
284C2×D4—22no—
285C2×D4—22no—
286C2×D4—22no—
287C2×D4—22no—
288C2×D4—24no—
289C2×D4—24no—
290C2×D4—24no—
291C2×D4—24no—
292C2×D4—42no—
293C2×D4—42no—
294C2×D4—42no—
295C2×D4—42no—
296C2×D4—42no—
297C2×D4—42no—
298C2×D4—42no—
299C2×D4—42no—
300C22×C4—21no—
301C22×C4—21no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—21no—
305C22×C4—21no—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—24no—
311C22×C4—22no—
312C22×C4—24no—
313C22×C4—24no—
314C22×C4—22no—
315C22×C4—22no—
316C22×C4—24no—
317C22×C4—41no—
318C22×C4—41no—
319C22×C4—42no—
320C22×C4—42no—
321C22×C4—42no—
322C22×C4—44no—
323C22×C4—44no—
324C22×C4—44no—
325C22×C4—44no—
326C22×C4—44no—
327C22×C4—44no—
328C22×C4—44no—
329C22×C4—44no—
330C22×C4—42no—
331C22×C4—42no—
332C22×C4—88no—
333C22×C4—88no—
334C22×C4C2×C22⋊C412yes—
335C22×C4C2×C22⋊C412yes—
336C22×C4C2×C22⋊C412yes—
337C22×C4C2×C22⋊C412yes—
338C22×C4C2×C22⋊C412yes—
339C22×C4C2×C22⋊C411yescharacteristic
340C22×C4C2×C22⋊C411yescharacteristic
341C22×C4C22×D412yes—
342C22×C4C22×Q811yescharacteristic
343C22×C4C2×#4111yescharacteristic
344C22×C4C2×#4112yes—
345C22×C4H(5,𝔽2)11yescharacteristic
346C22×C4H(5,𝔽2)11yescharacteristic
347C22×C4H(5,𝔽2)12yes—
348C22×C4H(5,𝔽2)11yescharacteristic
349C22×C42−1+412yes—
350C22×C42−1+411yescharacteristic
351M16—24no—
352M16—24no—
353M16—24no—
354M16—24no—
355M16—42no—
356M16—42no—
357M16—42no—
358M16—42no—
359M16—42no—
360M16—42no—
361M16—42no—
362M16—42no—
363M16—42no—
364M16—42no—
365M16—42no—
366M16—42no—
367C2×C8—24no—
368C2×C8—24no—
369C2×C8—42no—
370C2×C8—42no—
371C2×C8—42no—
372C2×C8—42no—
373C2×C8—42no—
374C2×C8—42no—
375C2×C8—42no—
376C2×C8—42no—
377C2×C8—42no—
378C2×C8—42no—
379C4⋊C4—42no—
380C4⋊C4—41no—
381C4⋊C4—41no—
382C4⋊C4—42no—
383C4⋊C4—42no—
384C4⋊C4—42no—
385C4⋊C4—42no—
386C4⋊C4—44no—
387C4⋊C4—48no—
388C4⋊C4—42no—
389C4⋊C4—42no—
390C4⋊C4—42no—
391C4⋊C4—42no—
392C4⋊C4—42no—
393C4⋊C4—42no—
394C4⋊C4—44no—
395C4⋊C4—48no—
396C4⋊C4—84no—
397C22⋊C4—48no—
398C22⋊C4—48no—
399C22⋊C4—48no—
400C22⋊C4—84no—
401C22⋊C4—84no—
402C42—22no—
403C42—22no—
404C42—22no—
405C42—22no—
406C42—22no—
407C42—22no—
408C42—22no—
409C42—22no—
410C42—22no—
411C42—22no—
412C42—22no—
413C42—22no—
414C23—21no—
415C23—21no—
416C23—21no—
417C23—21no—
418C23—24no—
419C23—24no—
420C23—24no—
421C23—44no—
422C23—44no—
423C23—44no—
424C23—44no—
425C23—42no—
426C23—42no—
427C23—42no—
428C23—42no—
429C23—42no—
430C23—42no—
431C23—42no—
432C23—42no—
433C23—82no—
434C23—82no—
435C23—82no—
436C23—82no—
437C23—82no—
438C23—82no—
439C23—82no—
440C23—82no—
441C23C22×C22⋊C411yescharacteristic
442C23C2×C4×Q811yescharacteristic
443C23C4×#4111yescharacteristic
444C23#51811yescharacteristic
445C23#51811yescharacteristic
446C23#51911yescharacteristic
447C23#52011yescharacteristic
448Q8—44no—
449Q8—48no—
450Q8—44no—
451D4—44no—
452D4—44no—
453D4—44no—
454D4—44no—
455C2×C4—24no—
456C2×C4—24no—
457C2×C4—21no—
458C2×C4—21no—
459C2×C4—22no—
460C2×C4—21no—
461C2×C4—21no—
462C2×C4—22no—
463C2×C4—22no—
464C2×C4—22no—
465C2×C4—22no—
466C2×C4—22no—
467C2×C4—22no—
468C2×C4—22no—
469C2×C4—22no—
470C2×C4—22no—
471C2×C4—22no—
472C2×C4—22no—
473C2×C4—22no—
474C2×C4—22no—
475C2×C4—44no—
476C2×C4—44no—
477C2×C4—42no—
478C2×C4—42no—
479C2×C4—42no—
480C2×C4—44no—
481C2×C4—44no—
482C2×C4—42no—
483C2×C4—42no—
484C2×C4—44no—
485C2×C4—88no—
486C2×C4—88no—
487C2×C4—82no—
488C2×C4—82no—
489C2×C4—84no—
490C2×C4—84no—
491C2×C4—82no—
492C2×C4—82no—
493C2×C4C2×#10014yes—
494C2×C4C2×#10012yes—
495C2×C4#41312yes—
496C2×C4#53412yes—
497C2×C4#53412yes—
498C2×C4#53811yescharacteristic
499C2×C4#53911yescharacteristic
500C2×C4#53912yes—
501C2×C4#54212yes—
502C2×C4#54311yescharacteristic
503C2×C4#54411yescharacteristic
504C2×C4#54812yes—
505C2×C4#54812yes—
506C2×C4D4×Q812yes—
507C2×C4#55012yes—
508C2×C4#55112yes—
509C2×C4#55412yes—
510C2×C4#56212yes—
511C2×C4#56312yes—
512C8—44no—
513C8—44no—
514C8—82no—
515C8—82no—
516C22—21no—
517C22—21no—
518C22—21no—
519C22—21no—
520C22—21no—
521C22—21no—
522C22—21no—
523C22—21no—
524C22—21no—
525C22—21no—
526C22—21no—
527C22—21no—
528C22—44no—
529C22—44no—
530C22—44no—
531C22—84no—
532C22—84no—
533C22—84no—
534C22—84no—
535C22—82no—
536C22—82no—
537C22—82no—
538C22—82no—
539C22#230811yescharacteristic
540C22#232911yescharacteristic
541C22#233211yescharacteristic
542C22Q8×C22⋊C411yescharacteristic
543C22#234111yescharacteristic
544C22#236811yescharacteristic
545C22#237811yescharacteristic
546C22C22×#10011yescharacteristic
547C22C2×#41311yescharacteristic
548C22#288811yescharacteristic
549C22#288911yescharacteristic
550C4—22no—
551C4—22no—
552C4—24no—
553C4—42no—
554C4—42no—
555C4—42no—
556C4—42no—
557C4—84no—
558C4—82no—
559C4#272812yes—
560C4#272912yes—
561C4#273012yes—
562C4#273212yes—
563C2—21no—
564C2—21no—
565C2—21no—
566C2—21no—
567C2—84no—
568C2#1444911yescharacteristic
569C2#2175211yescharacteristic
570C2#2175711yescharacteristic
571C2Q8×#10011yescharacteristic
572C2#2251511yescharacteristic
573C2#2253611yescharacteristic
574C2#2253711yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC24C23×C4
Frattini, cosocleC23×C4C24
prime radicalC23×D4C23
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C4×Q8 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #21303C2semidirect
3. #11354C2semidirect
4. #11204C2semidirect
7. #10913C2semidirect
89. C2×#100Q8semidirect
283. C2×Q8#100semidirect

Maximal subgroups

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

subgroupindex
C2×#25382
#213032
#113542
#112042
#110442
#109622
#109132
#107622