Finite Group Databasebeta

← #96925

Subgroups — #96925

Order 512.

1122 conjugacy classes of subgroups.

The subgroup lattice has 578 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2855C211yesmaximal, characteristic
2#10712C211yesmaximal, characteristic
3#10712C211yesmaximal, characteristic
4C2×#1272C212yesmaximal, complemented
5C2×#1271C212yesmaximal, complemented
6C22×C22⋊Q8C411yescharacteristic
7C23×C42C2211yescharacteristic
8C22×C22⋊C8—21no—
9C22×C22⋊C8—21no—
10#2855—24no—
11#2855—24no—
12C2×#391C411yescharacteristic
13C2×#391C2211yescharacteristic
14C2×#389C411yescharacteristic
15C2×#389C2211yescharacteristic
16C2×#387—21no—
17C2×#387—21no—
18C2×#385—21no—
19C2×#381—21no—
20C2×#380C411yescharacteristic
21C2×#336—21no—
22C2×#336—21no—
23C2×#336C2212yes—
24C2×#336C2212yes—
25#1272—24no—
26#1271—24no—
27C23×Q8C2×C411yescharacteristic
28C24×C4C2×C411yescharacteristic
29C24×C4D411yescharacteristic
30C24×C4D411yescharacteristic
31C23×C8—22no—
32C23×C8—22no—
33C2×C22⋊Q8—22no—
34C2×C22⋊Q8—22no—
35C2×C22⋊Q8—42no—
36C2×C22⋊Q8—42no—
37C22×C4⋊C4—21no—
38C22×C4⋊C4—21no—
39C22×C4⋊C4—21no—
40C22×C4⋊C4C2×C411yescharacteristic
41C22×C4⋊C4C2×C411yescharacteristic
42C22×C4⋊C4C2×C411yescharacteristic
43C22×C22⋊C4—21no—
44C22×C22⋊C4—21no—
45C22×C42—22no—
46C22×C42—22no—
47C22×C42—42no—
48C22×C42—42no—
49C22×C42C2×C411yescharacteristic
50C22×C42D411yescharacteristic
51C22×C42D411yescharacteristic
52C22×C42C2311yescharacteristic
53C2×C22⋊C8—24no—
54C2×C22⋊C8—24no—
55C2×C22⋊C8—42no—
56C2×C22⋊C8—42no—
57C2×C22⋊C8—42no—
58C2×C22⋊C8—42no—
59#391—22no—
60#391—22no—
61#391—22no—
62#391D414yes—
63#389—22no—
64#389—22no—
65#389—22no—
66#389D414yes—
67#387—28no—
68#387—28no—
69#385—28no—
70#381—28no—
71#380—22no—
72#380—22no—
73C2×#95—21no—
74C2×#95—21no—
75C2×#95—21no—
76C2×#95—21no—
77C2×#95—21no—
78C2×#95—21no—
79C2×#95—21no—
80#336—22no—
81#336—22no—
82#336—22no—
83#336—22no—
84#336—42no—
85#336—42no—
86C25C22⋊C411yescharacteristic
87C22×Q8—21no—
88C22×Q8—22no—
89C22×Q8—22no—
90C22×Q8—41no—
91C22×Q8—41no—
92C22×Q8—41no—
93C22×Q8—41no—
94C22×Q8C22⋊C412yes—
95C23×C4—22no—
96C23×C4—22no—
97C23×C4—22no—
98C23×C4—22no—
99C23×C4—22no—
100C23×C4—22no—
101C23×C4—21no—
102C23×C4—21no—
103C23×C4—22no—
104C23×C4—21no—
105C23×C4—21no—
106C23×C4—22no—
107C23×C4—21no—
108C23×C4—21no—
109C23×C4—21no—
110C23×C4—21no—
111C23×C4—22no—
112C23×C4—22no—
113C23×C4—22no—
114C23×C4—22no—
115C23×C4—41no—
116C23×C4—41no—
117C23×C4—41no—
118C23×C4—41no—
119C23×C4—41no—
120C23×C4—41no—
121C23×C4C22⋊C411yescharacteristic
122C23×C4C22⋊C411yescharacteristic
123C23×C4C22⋊C411yescharacteristic
124C23×C4SD1612yes—
125C23×C4Q1612yes—
126C23×C4C22×C411yescharacteristic
127C23×C4C2×D411yescharacteristic
128C23×C4C2×D411yescharacteristic
129C22×C8—24no—
130C22×C8—24no—
131C22×C8—42no—
132C22×C8—42no—
133C22×C8—42no—
134C22×C8—42no—
135C22×C8—42no—
136C22×C8—42no—
137C22×C8—42no—
138C22×C8—42no—
139C22×C8—42no—
140C22×C8—42no—
141C22×C8—42no—
142C22×C8—42no—
143C22⋊Q8—48no—
144C22⋊Q8—48no—
145C2×C4⋊C4—22no—
146C2×C4⋊C4—22no—
147C2×C4⋊C4—24no—
148C2×C4⋊C4—24no—
149C2×C4⋊C4—21no—
150C2×C4⋊C4—21no—
151C2×C4⋊C4—21no—
152C2×C4⋊C4—24no—
153C2×C4⋊C4—41no—
154C2×C4⋊C4—41no—
155C2×C4⋊C4—41no—
156C2×C4⋊C4—41no—
157C2×C4⋊C4—42no—
158C2×C4⋊C4—42no—
159C2×C4⋊C4—42no—
160C2×C4⋊C4—42no—
161C2×C4⋊C4—42no—
162C2×C4⋊C4—41no—
163C2×C4⋊C4—41no—
164C2×C4⋊C4C22⋊C412yes—
165C2×C4⋊C4C22⋊C412yes—
166C2×C4⋊C4C22⋊C412yes—
167C2×C22⋊C4—24no—
168C2×C22⋊C4—24no—
169C2×C22⋊C4—42no—
170C2×C22⋊C4—42no—
171C2×C22⋊C4—42no—
172C2×C22⋊C4—42no—
173C2×C42—22no—
174C2×C42—22no—
175C2×C42—21no—
176C2×C42—21no—
177C2×C42—22no—
178C2×C42—22no—
179C2×C42—41no—
180C2×C42—41no—
181C2×C42—41no—
182C2×C42—41no—
183C2×C42—41no—
184C2×C42—41no—
185C2×C42—48no—
186C2×C42—48no—
187C2×C42C2×D412yes—
188C2×C42C2×D412yes—
189C22⋊C8—48no—
190C22⋊C8—48no—
191#95—24no—
192#95—24no—
193#95—24no—
194#95—24no—
195#95—24no—
196#95—24no—
197#95—24no—
198C24—21no—
199C24—21no—
200C24—21no—
201C24—21no—
202C24—21no—
203C24—21no—
204C24—21no—
205C24—21no—
206C24—21no—
207C24—21no—
208C24—21no—
209C24—21no—
210C24#9912yes—
211C24#10312yes—
212C24#10412yes—
213C24C2×C22⋊C411yescharacteristic
214C2×Q8—24no—
215C2×Q8—42no—
216C2×Q8—42no—
217C2×Q8—42no—
218C2×Q8—42no—
219C2×Q8—42no—
220C2×Q8—82no—
221C2×Q8—82no—
222C2×Q8—82no—
223C2×Q8—82no—
224C22×C4—21no—
225C22×C4—21no—
226C22×C4—21no—
227C22×C4—21no—
228C22×C4—21no—
229C22×C4—21no—
230C22×C4—21no—
231C22×C4—21no—
232C22×C4—21no—
233C22×C4—21no—
234C22×C4—21no—
235C22×C4—21no—
236C22×C4—24no—
237C22×C4—24no—
238C22×C4—24no—
239C22×C4—24no—
240C22×C4—24no—
241C22×C4—24no—
242C22×C4—21no—
243C22×C4—21no—
244C22×C4—21no—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—21no—
248C22×C4—21no—
249C22×C4—21no—
250C22×C4—21no—
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—21no—
256C22×C4—21no—
257C22×C4—21no—
258C22×C4—21no—
259C22×C4—21no—
260C22×C4—24no—
261C22×C4—24no—
262C22×C4—22no—
263C22×C4—22no—
264C22×C4—22no—
265C22×C4—22no—
266C22×C4—42no—
267C22×C4—42no—
268C22×C4—44no—
269C22×C4—44no—
270C22×C4—44no—
271C22×C4—44no—
272C22×C4—44no—
273C22×C4—44no—
274C22×C4—44no—
275C22×C4—44no—
276C22×C4—44no—
277C22×C4—44no—
278C22×C4—44no—
279C22×C4—44no—
280C22×C4—42no—
281C22×C4—42no—
282C22×C4—42no—
283C22×C4—42no—
284C22×C4—42no—
285C22×C4—42no—
286C22×C4—42no—
287C22×C4—42no—
288C22×C4—41no—
289C22×C4—41no—
290C22×C4—41no—
291C22×C4—41no—
292C22×C4—41no—
293C22×C4—41no—
294C22×C4—41no—
295C22×C4—41no—
296C22×C4—41no—
297C22×C4—41no—
298C22×C4—41no—
299C22×C4—41no—
300C22×C4—41no—
301C22×C4—41no—
302C22×C4—41no—
303C22×C4—41no—
304C22×C4—41no—
305C22×C4—41no—
306C22×C4—41no—
307C22×C4—41no—
308C22×C4—41no—
309C22×C4—41no—
310C22×C4—41no—
311C22×C4—41no—
312C22×C4#9912yes—
313C22×C4#10312yes—
314C22×C4#10412yes—
315C22×C4C2×C22⋊C411yescharacteristic
316C22×C4C2×C22⋊C411yescharacteristic
317C22×C4#12012yes—
318C22×C4#12012yes—
319C22×C4#13112yes—
320C22×C4#13112yes—
321C22×C4C2×SD1611yescharacteristic
322C22×C4C2×Q1611yescharacteristic
323C22×C4#13512yes—
324C22×C4AGL(1,ℤ/8)11yescharacteristic
325C22×C4#13712yes—
326C22×C4#13711yescharacteristic
327C2×C8—44no—
328C2×C8—44no—
329C2×C8—44no—
330C2×C8—44no—
331C2×C8—44no—
332C2×C8—44no—
333C2×C8—82no—
334C2×C8—82no—
335C2×C8—82no—
336C2×C8—82no—
337C2×C8—82no—
338C2×C8—82no—
339C2×C8—82no—
340C2×C8—82no—
341C4⋊C4—24no—
342C4⋊C4—42no—
343C4⋊C4—42no—
344C4⋊C4—42no—
345C4⋊C4—48no—
346C4⋊C4—42no—
347C4⋊C4—48no—
348C4⋊C4—42no—
349C4⋊C4—44no—
350C4⋊C4—44no—
351C22⋊C4—48no—
352C22⋊C4—48no—
353C42—42no—
354C42—42no—
355C42—44no—
356C42—44no—
357C42—42no—
358C42—42no—
359C23—22no—
360C23—22no—
361C23—22no—
362C23—22no—
363C23—22no—
364C23—22no—
365C23—22no—
366C23—22no—
367C23—22no—
368C23—22no—
369C23—22no—
370C23—22no—
371C23—22no—
372C23—22no—
373C23—22no—
374C23—22no—
375C23—22no—
376C23—22no—
377C23—41no—
378C23—41no—
379C23—41no—
380C23—41no—
381C23—41no—
382C23—41no—
383C23—41no—
384C23—41no—
385C23—41no—
386C23—41no—
387C23—41no—
388C23—41no—
389C23—41no—
390C23—41no—
391C23—41no—
392C23—41no—
393C23#32814yes—
394C23#37911yescharacteristic
395C23#40811yescharacteristic
396C23#40811yescharacteristic
397C23C2×#9911yescharacteristic
398C23C2×#9911yescharacteristic
399C23#41311yescharacteristic
400C23#41311yescharacteristic
401C23C2×#10311yescharacteristic
402C23#41611yescharacteristic
403C23#41711yescharacteristic
404C23#41911yescharacteristic
405C23C2×#10411yescharacteristic
406C23C2×#10411yescharacteristic
407C23#42111yescharacteristic
408C23#42111yescharacteristic
409Q8—84no—
410Q8—84no—
411C2×C4—22no—
412C2×C4—22no—
413C2×C4—22no—
414C2×C4—22no—
415C2×C4—21no—
416C2×C4—21no—
417C2×C4—22no—
418C2×C4—21no—
419C2×C4—21no—
420C2×C4—22no—
421C2×C4—22no—
422C2×C4—21no—
423C2×C4—21no—
424C2×C4—22no—
425C2×C4—21no—
426C2×C4—21no—
427C2×C4—21no—
428C2×C4—21no—
429C2×C4—22no—
430C2×C4—22no—
431C2×C4—21no—
432C2×C4—21no—
433C2×C4—22no—
434C2×C4—22no—
435C2×C4—22no—
436C2×C4—22no—
437C2×C4—22no—
438C2×C4—22no—
439C2×C4—22no—
440C2×C4—22no—
441C2×C4—22no—
442C2×C4—22no—
443C2×C4—22no—
444C2×C4—48no—
445C2×C4—48no—
446C2×C4—48no—
447C2×C4—48no—
448C2×C4—48no—
449C2×C4—48no—
450C2×C4—41no—
451C2×C4—41no—
452C2×C4—41no—
453C2×C4—41no—
454C2×C4—41no—
455C2×C4—41no—
456C2×C4—41no—
457C2×C4—41no—
458C2×C4—41no—
459C2×C4—41no—
460C2×C4—41no—
461C2×C4—41no—
462C2×C4—42no—
463C2×C4—42no—
464C2×C4—42no—
465C2×C4—42no—
466C2×C4—42no—
467C2×C4—42no—
468C2×C4—42no—
469C2×C4—42no—
470C2×C4—42no—
471C2×C4—42no—
472C2×C4—42no—
473C2×C4—42no—
474C2×C4—81no—
475C2×C4—81no—
476C2×C4—81no—
477C2×C4—81no—
478C2×C4—81no—
479C2×C4—81no—
480C2×C4—81no—
481C2×C4—81no—
482C2×C4—81no—
483C2×C4—81no—
484C2×C4—81no—
485C2×C4—81no—
486C2×C4—81no—
487C2×C4—81no—
488C2×C4—81no—
489C2×C4—81no—
490C2×C4#40812yes—
491C2×C4#40812yes—
492C2×C4#44812yes—
493C2×C4#44912yes—
494C2×C4#45112yes—
495C2×C4#45212yes—
496C8—84no—
497C8—84no—
498C22—24no—
499C22—24no—
500C22—24no—
501C22—42no—
502C22—42no—
503C22—42no—
504C22—42no—
505C22—42no—
506C22—42no—
507C22—42no—
508C22—42no—
509C22—42no—
510C22—42no—
511C22—42no—
512C22—42no—
513C22#146811yescharacteristic
514C22#146811yescharacteristic
515C22#146811yescharacteristic
516C22#146811yescharacteristic
517C22#148111yescharacteristic
518C22#148211yescharacteristic
519C22#148311yescharacteristic
520C22#148511yescharacteristic
521C22#149311yescharacteristic
522C22#149411yescharacteristic
523C22#149511yescharacteristic
524C22#149711yescharacteristic
525C22C2×#32811yescharacteristic
526C22#150111yescharacteristic
527C22#150411yescharacteristic
528C22#150811yescharacteristic
529C22#151111yescharacteristic
530C22#151311yescharacteristic
531C22#151611yescharacteristic
532C22#152011yescharacteristic
533C22#152411yescharacteristic
534C22#152911yescharacteristic
535C22#153211yescharacteristic
536C22#153511yescharacteristic
537C22#154211yescharacteristic
538C22#155011yescharacteristic
539C22#155411yescharacteristic
540C22#155911yescharacteristic
541C22#178011yescharacteristic
542C22#178111yescharacteristic
543C22#178511yescharacteristic
544C22#178711yescharacteristic
545C22#178811yescharacteristic
546C22#178911yescharacteristic
547C22#178911yescharacteristic
548C4—22no—
549C4—22no—
550C4—22no—
551C4—22no—
552C4—22no—
553C4—22no—
554C4—42no—
555C4—42no—
556C4—42no—
557C4—82no—
558C4—82no—
559C4—82no—
560C4—82no—
561C2—44no—
562C2#765511yescharacteristic
563C2#766111yescharacteristic
564C2#767511yescharacteristic
565C2#768111yescharacteristic
566C2#769511yescharacteristic
567C2#770111yescharacteristic
568C2#771511yescharacteristic
569C2#772111yescharacteristic
570C2#830411yescharacteristic
571C2#831411yescharacteristic
572C2#831511yescharacteristic
573C2#832911yescharacteristic
574C2#833111yescharacteristic
575C2#833611yescharacteristic
576C2#834311yescharacteristic
subgroupG/H
center, socleC24C2×C22⋊C4
commutatorC23×C4C22×C4
Frattini, cosocleC22×C42C23
prime radicalC25C22⋊C4
LCSG ▷ C23×C4 ▷ C22 ▷ C1
UCSC1 ◁ C24 ◁ C23×C42 ◁ G
LECSG ▷ C22×C42 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
4. C2×#1272C2semidirect
5. C2×#1271C2semidirect

Maximal subgroups

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

subgroupindex
C2×#28552
#107122
C2×#12722
C2×#12712