Finite Group Databasebeta

← #195203

Subgroups — #195203

Order 512.

2078 conjugacy classes of subgroups.

The subgroup lattice has 568 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#451C211yesmaximal, characteristic
2#34212C211yesmaximal, characteristic
3C2×#1787C211yesmaximal, characteristic
4C2×#1520C211yesmaximal, characteristic
5#13922C211yesmaximal, characteristic
6#9040C218yesmaximal, complemented
7#8023C211yesmaximal, characteristic
8#7907C211yesmaximal, characteristic
9C24×Q8C2211yescharacteristic
10C23×Q16—21no—
11C22×C22⋊Q8C2211yescharacteristic
12C2×#451—24no—
13C2×#451—22no—
14C2×#451C418yescomplemented
15C2×#451C2214yes—
16C22×#103—21no—
17C22×#103—22no—
18C22×C22⋊C8C2211yescharacteristic
19C22×C22⋊C8C2211yescharacteristic
20#2368—22no—
21#2360—22no—
22#2345—22no—
23#2345—21no—
24Q8×C22⋊C4—21no—
25#2329C2214yes—
26#2315—21no—
27C4×C22⋊Q8—22no—
28C2×#391—22no—
29C2×#380C2211yescharacteristic
30C2×#380C2211yescharacteristic
31#2273—21no—
32C2×C4×C22⋊C4C2211yescharacteristic
33#1787—22no—
34#1787C2214yes—
35#1520—24no—
36C2×#328C2214yes—
37#1446C2214yes—
38C22⋊#95—21no—
39C22⋊#95—21no—
40#1294C2214yes—
41#1280C2214yes—
42C23×Q8—21no—
43C23×Q8—22no—
44C23×Q8—21no—
45C23×Q8—21no—
46C23×Q8—21no—
47C23×Q8—22no—
48C23×Q8—21no—
49C23×Q8—22no—
50C23×Q8—48no—
51C23×Q8C2×C414yes—
52C23×Q8D411yescharacteristic
53C23×Q8C2312yes—
54C24×C4—21no—
55C24×C4C2311yescharacteristic
56C22×Q16—28no—
57C22×Q16—24no—
58C22×Q16—48no—
59C23×C8—21no—
60C23×C8—21no—
61C2×C22⋊Q8—21no—
62C2×C22⋊Q8—21no—
63C2×C22⋊Q8—22no—
64C2×C22⋊Q8—22no—
65C2×C22⋊Q8C2×C414yes—
66C2×C22⋊Q8D414yes—
67C2×C22⋊Q8D412yes—
68C2×C22⋊Q8C2312yes—
69C2×C4×Q8—22no—
70C22×C4⋊C4—21no—
71C22×C4⋊C4—21no—
72C22×C4⋊C4D411yescharacteristic
73C22×C4⋊C4D412yes—
74C22×C22⋊C4—21no—
75C22×C22⋊C4—21no—
76C22×C22⋊C4—21no—
77C22×C22⋊C4D411yescharacteristic
78C22×C22⋊C4D411yescharacteristic
79C22×C42D412yes—
80#451—216no—
81#451—28no—
82#451C2×C4116yescomplemented
83C2×#103—28no—
84C2×#103—24no—
85C2×#103—28no—
86C2×#103—48no—
87C2×C22⋊C8—21no—
88C2×C22⋊C8—22no—
89C2×C22⋊C8—22no—
90C2×C22⋊C8—22no—
91C2×C22⋊C8—21no—
92C2×C22⋊C8C2×C414yes—
93C2×C22⋊C8C2312yes—
94C2×C22⋊C8C2312yes—
95#391—22no—
96#391—24no—
97#391—48no—
98#389—22no—
99#389—24no—
100#387—24no—
101#387—24no—
102#387—24no—
103#385—24no—
104#385—24no—
105#380—22no—
106#380—22no—
107#380—21no—
108#380—21no—
109#380—22no—
110#380—22no—
111#380D412yes—
112#380C2312yes—
113C4×C4⋊C4—22no—
114C4×C4⋊C4—24no—
115C4×C22⋊C4—22no—
116C4×C22⋊C4—22no—
117C4×C22⋊C4—24no—
118C4×C22⋊C4—24no—
119C4×C22⋊C4—21no—
120C4×C22⋊C4—21no—
121C4×C22⋊C4—21no—
122C4×C22⋊C4—21no—
123C2×#95—21no—
124C2×#95—24no—
125C2×#95—24no—
126C2×#95C2312yes—
127C2×#95C2312yes—
128#328—28no—
129C25C2×D411yescharacteristic
130C22×Q8—28no—
131C22×Q8—24no—
132C22×Q8—21no—
133C22×Q8—22no—
134C22×Q8—22no—
135C22×Q8—28no—
136C22×Q8—28no—
137C22×Q8—24no—
138C22×Q8—24no—
139C22×Q8—24no—
140C22×Q8—24no—
141C22×Q8—22no—
142C22×Q8—22no—
143C22×Q8—22no—
144C22×Q8—22no—
145C22×Q8—48no—
146C22×Q8—44no—
147C22×Q8—44no—
148C22×Q8—44no—
149C22×Q8—44no—
150C22×Q8—416no—
151C22×Q8—416no—
152C22×Q8—416no—
153C22×Q8—416no—
154C22×Q8—48no—
155C22×Q8—48no—
156C22×Q8—41no—
157C22×Q8—41no—
158C22×Q8—42no—
159C22×Q8—42no—
160C22×Q8—42no—
161C22×Q8—42no—
162C22×Q8—42no—
163C22×Q8C22⋊C414yes—
164C22×Q8C22×C414yes—
165C22×Q8C2×D412yes—
166C23×C4—22no—
167C23×C4—21no—
168C23×C4—21no—
169C23×C4—24no—
170C23×C4—22no—
171C23×C4—22no—
172C23×C4—44no—
173C23×C4—48no—
174C23×C4—41no—
175C23×C4—41no—
176C23×C4—42no—
177C23×C4C22×C412yes—
178C23×C4C2×D411yescharacteristic
179C23×C4C2×D411yescharacteristic
180C23×C4C2×D411yescharacteristic
181C23×C4C2×D412yes—
182C23×C4C2×D412yes—
183C23×C4C2×D411yescharacteristic
184C23×C4C2×D412yes—
185C23×C4C2×D411yescharacteristic
186C23×C4#4112yes—
187C23×C4#4112yes—
188C23×C4C2411yescharacteristic
189C2×Q16—216no—
190C2×Q16—416no—
191C2×Q16—432no—
192C22×C8—24no—
193C22×C8—22no—
194C22×C8—22no—
195C22×C8—44no—
196C22×C8—44no—
197C22×C8—42no—
198C22⋊Q8—24no—
199C22⋊Q8—24no—
200C22⋊Q8—24no—
201C22⋊Q8—24no—
202C22⋊Q8—24no—
203C22⋊Q8—22no—
204C22⋊Q8—22no—
205C22⋊Q8C22⋊C418yes—
206C22⋊Q8C22×C414yes—
207C22⋊Q8C2×D414yes—
208C4×Q8—84no—
209C2×C4⋊C4—24no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—24no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—24no—
217C2×C4⋊C4—24no—
218C2×C4⋊C4—44no—
219C2×C4⋊C4—44no—
220C2×C4⋊C4—42no—
221C2×C4⋊C4C22⋊C414yes—
222C2×C4⋊C4C2×D412yes—
223C2×C4⋊C4C2×D414yes—
224C2×C22⋊C4—24no—
225C2×C22⋊C4—22no—
226C2×C22⋊C4—21no—
227C2×C22⋊C4—21no—
228C2×C22⋊C4—24no—
229C2×C22⋊C4—22no—
230C2×C22⋊C4—22no—
231C2×C22⋊C4—22no—
232C2×C22⋊C4—21no—
233C2×C22⋊C4—21no—
234C2×C22⋊C4—22no—
235C2×C22⋊C4—22no—
236C2×C22⋊C4—22no—
237C2×C22⋊C4—42no—
238C2×C22⋊C4—42no—
239C2×C22⋊C4—42no—
240C2×C22⋊C4—42no—
241C2×C22⋊C4—42no—
242C2×C22⋊C4Q1614yescomplemented
243C2×C42—22no—
244C2×C42—24no—
245C2×C42—24no—
246C2×C42—22no—
247C2×C42—44no—
248#103—216no—
249#103—416no—
250C22⋊C8—24no—
251C22⋊C8—22no—
252C22⋊C8—22no—
253C22⋊C8—22no—
254C22⋊C8—42no—
255C22⋊C8C22×C414yes—
256#95—24no—
257#95—24no—
258#95—24no—
259#95—24no—
260#95—22no—
261#95—22no—
262#95—22no—
263#95—22no—
264#95—22no—
265#95—22no—
266#95—48no—
267#95—48no—
268C24—42no—
269C24—42no—
270C24C4×D412yes—
271C24#12011yescharacteristic
272C24#12011yescharacteristic
273C24#12112yes—
274C24C2×Q1612yes—
275C24#13512yes—
276C24#13712yes—
277C24#13712yes—
278C24C22×D411yescharacteristic
279C2×Q8—216no—
280C2×Q8—22no—
281C2×Q8—24no—
282C2×Q8—28no—
283C2×Q8—28no—
284C2×Q8—216no—
285C2×Q8—216no—
286C2×Q8—24no—
287C2×Q8—24no—
288C2×Q8—48no—
289C2×Q8—48no—
290C2×Q8—44no—
291C2×Q8—44no—
292C2×Q8—416no—
293C2×Q8—48no—
294C2×Q8—416no—
295C2×Q8—48no—
296C2×Q8—48no—
297C2×Q8—416no—
298C2×Q8—48no—
299C2×Q8—44no—
300C2×Q8—48no—
301C2×Q8—416no—
302C2×Q8—416no—
303C2×Q8—48no—
304C2×Q8—48no—
305C2×Q8—416no—
306C2×Q8—416no—
307C2×Q8—42no—
308C2×Q8—42no—
309C2×Q8—44no—
310C2×Q8—44no—
311C2×Q8—44no—
312C2×Q8—48no—
313C2×Q8—84no—
314C2×Q8C2×C22⋊C414yes—
315C22×C4—24no—
316C22×C4—24no—
317C22×C4—24no—
318C22×C4—28no—
319C22×C4—24no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—24no—
325C22×C4—24no—
326C22×C4—22no—
327C22×C4—24no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—44no—
331C22×C4—44no—
332C22×C4—44no—
333C22×C4—44no—
334C22×C4—42no—
335C22×C4—42no—
336C22×C4—42no—
337C22×C4—416no—
338C22×C4—44no—
339C22×C4—48no—
340C22×C4—48no—
341C22×C4—416no—
342C22×C4—42no—
343C22×C4—42no—
344C22×C4—44no—
345C22×C4—44no—
346C22×C4—42no—
347C22×C4—42no—
348C22×C4—42no—
349C22×C4—42no—
350C22×C4—44no—
351C22×C4—44no—
352C22×C4—44no—
353C22×C4—42no—
354C22×C4—42no—
355C22×C4—42no—
356C22×C4—82no—
357C22×C4—82no—
358C22×C4—82no—
359C22×C4—82no—
360C22×C4C2×C22⋊C412yes—
361C22×C4C2×C22⋊C412yes—
362C22×C4C4×D412yes—
363C22×C4C4×D412yes—
364C22×C4C4×D412yes—
365C22×C4#12011yescharacteristic
366C22×C4#12011yescharacteristic
367C22×C4#12112yes—
368C22×C4#12112yes—
369C22×C4#12112yes—
370C22×C4#12312yes—
371C22×C4#12312yes—
372C22×C4C2×Q1614yescomplemented
373C22×C4C23×C411yescharacteristic
374C22×C4C22×D411yescharacteristic
375C22×C4C22×D411yescharacteristic
376C22×C4C22×D411yescharacteristic
377C22×C4C2×#4111yescharacteristic
378C22×C4C2×#4111yescharacteristic
379Q16—432no—
380C2×C8—24no—
381C2×C8—44no—
382C2×C8—48no—
383C2×C8—44no—
384C2×C8—42no—
385C2×C8—84no—
386C4⋊C4—24no—
387C4⋊C4—24no—
388C4⋊C4—22no—
389C4⋊C4—28no—
390C4⋊C4—24no—
391C4⋊C4—44no—
392C4⋊C4—44no—
393C4⋊C4—84no—
394C4⋊C4—84no—
395C4⋊C4—84no—
396C4⋊C4C2×C22⋊C414yes—
397C22⋊C4—24no—
398C22⋊C4—24no—
399C22⋊C4—28no—
400C22⋊C4—24no—
401C22⋊C4—24no—
402C22⋊C4—42no—
403C22⋊C4—42no—
404C22⋊C4—42no—
405C22⋊C4—42no—
406C22⋊C4—44no—
407C22⋊C4—84no—
408C42—44no—
409C42—82no—
410C42—84no—
411C42—84no—
412C23—22no—
413C23—22no—
414C23—22no—
415C23—22no—
416C23—22no—
417C23—22no—
418C23—42no—
419C23—42no—
420C23—42no—
421C23—42no—
422C23—44no—
423C23—44no—
424C23—44no—
425C23—44no—
426C23#38612yes—
427C23#38612yes—
428C23C4×Q1614yes—
429C23#44114yes—
430C23#45112yes—
431C23#45112yes—
432C23#45212yes—
433C23#45212yes—
434C23#46214yes—
435C23#46414yes—
436C23C2×C4×D411yescharacteristic
437C23C2×#12011yescharacteristic
438C23C2×#12111yescharacteristic
439C23C22×Q1611yescharacteristic
440C23C2×#13511yescharacteristic
441C23C2×#13711yescharacteristic
442C23C2×#13711yescharacteristic
443Q8—28no—
444Q8—28no—
445Q8—44no—
446Q8—44no—
447Q8—48no—
448Q8—44no—
449Q8—48no—
450Q8—416no—
451Q8—416no—
452Q8—48no—
453Q8—82no—
454C2×C4—22no—
455C2×C4—22no—
456C2×C4—22no—
457C2×C4—24no—
458C2×C4—24no—
459C2×C4—24no—
460C2×C4—24no—
461C2×C4—24no—
462C2×C4—24no—
463C2×C4—28no—
464C2×C4—24no—
465C2×C4—24no—
466C2×C4—24no—
467C2×C4—42no—
468C2×C4—42no—
469C2×C4—42no—
470C2×C4—42no—
471C2×C4—41no—
472C2×C4—41no—
473C2×C4—41no—
474C2×C4—41no—
475C2×C4—44no—
476C2×C4—48no—
477C2×C4—416no—
478C2×C4—416no—
479C2×C4—42no—
480C2×C4—44no—
481C2×C4—42no—
482C2×C4—44no—
483C2×C4—48no—
484C2×C4—44no—
485C2×C4—44no—
486C2×C4—44no—
487C2×C4—42no—
488C2×C4—44no—
489C2×C4—44no—
490C2×C4—42no—
491C2×C4—42no—
492C2×C4—44no—
493C2×C4—82no—
494C2×C4—84no—
495C2×C4—84no—
496C2×C4—82no—
497C2×C4—82no—
498C2×C4—84no—
499C2×C4—84no—
500C2×C4—82no—
501C2×C4—82no—
502C2×C4—164no—
503C2×C4#38612yes—
504C2×C4#38612yes—
505C2×C4B(4,𝔽2)14yes—
506C2×C4C22×C22⋊C411yescharacteristic
507C2×C4C2×C4×D411yescharacteristic
508C2×C4C2×#12111yescharacteristic
509C2×C4C2×#12311yescharacteristic
510C8—44no—
511C8—82no—
512C22—22no—
513C22—22no—
514C22—24no—
515C22—24no—
516C22—22no—
517C22—22no—
518C22—24no—
519C22—24no—
520C22—21no—
521C22—22no—
522C22—21no—
523C22—22no—
524C22—21no—
525C22—21no—
526C22—22no—
527C22—22no—
528C22—44no—
529C22—44no—
530C22—44no—
531C22—44no—
532C22#187814yes—
533C22#187814yes—
534C22C2×#38611yescharacteristic
535C22C2×C4×Q1611yescharacteristic
536C22C2×#44111yescharacteristic
537C22C2×#45111yescharacteristic
538C22C2×#45211yescharacteristic
539C22#302212yes—
540C22C2×B(4,𝔽2)12yes—
541C22C2×#46211yescharacteristic
542C22C2×#46411yescharacteristic
543C4—21no—
544C4—21no—
545C4—21no—
546C4—21no—
547C4—22no—
548C4—44no—
549C4—44no—
550C4—48no—
551C4—44no—
552C4—82no—
553C4—84no—
554C4—162no—
555C4#188914yes—
556C2—21no—
557C2—24no—
558C2—24no—
559C2—21no—
560C2—21no—
561C2—21no—
562C2#1819712yes—
563C2#1819912yes—
564C2C2×#187811yescharacteristic
565C2#2094411yescharacteristic
566C2C2×#188911yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C24×C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
6. #9040C2semidirect
14. C2×#451C4semidirect
82. #451C2×C4semidirect
242. C2×C22⋊C4Q16semidirect
372. C22×C4C2×Q16semidirect

Maximal subgroups

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

subgroupindex
C22×#4512
#342122
C2×#17872
C2×#15202
#139222
#90402
#80232
#79072