Finite Group Databasebeta

← #192093

Subgroups — #192093

Order 512.

1642 conjugacy classes of subgroups.

The subgroup lattice has 559 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34181C211yesmaximal, characteristic
2C2×#1852C212yesmaximal, complemented
3C2×#1459C211yesmaximal, characteristic
4#13872C211yesmaximal, characteristic
5#8856C214yesmaximal, complemented
6#8853C214yesmaximal, complemented
7#8023C212yesmaximal, complemented
8C2×#518—21no—
9C22×C4×D4C2211yescharacteristic
10C23×C4⋊C4C2211yescharacteristic
11C22×#107—22no—
12C22×C22⋊C8C2212yes—
13#2349—24no—
14D4×C4⋊C4—22no—
15D4×C4⋊C4—22no—
16#2316C2212yes—
17#2315—21no—
18#2315C2212yes—
19C2×#386—21no—
20#2285—21no—
21C2×#380—21no—
22C2×#380—21no—
23C2×#380C2211yescharacteristic
24C2×#380C2212yes—
25#1852—24no—
26#1852C2218yes—
27C2×#340—22no—
28#1459C418yescomplemented
29C2×#324C2212yes—
30C2×#323C2212yes—
31#1441C2212yes—
32#1439C2212yes—
33C4⋊#95—21no—
34C22⋊#95—21no—
35#1294C2218yes—
36C23×D4D411yescharacteristic
37C24×C4—21no—
38C24×C4D411yescharacteristic
39C24×C4C2311yescharacteristic
40C23×C8—22no—
41#518—44no—
42#518—44no—
43C2×C4×D4—22no—
44C2×C4×D4—22no—
45C2×C4×D4—24no—
46C2×C4×D4—24no—
47C2×C4×D4—24no—
48C2×C4×D4C2×C414yes—
49C2×C4×D4Q814yes—
50C2×#117—22no—
51C22×C4⋊C4—22no—
52C22×C4⋊C4—21no—
53C22×C4⋊C4—21no—
54C22×C4⋊C4—21no—
55C22×C4⋊C4—22no—
56C22×C4⋊C4—21no—
57C22×C4⋊C4—44no—
58C22×C4⋊C4D412yes—
59C22×C4⋊C4D412yes—
60C22×C4⋊C4C2311yescharacteristic
61C22×C4⋊C4C2312yes—
62C22×C22⋊C4—22no—
63C22×C22⋊C4—22no—
64C22×C22⋊C4—21no—
65C22×C22⋊C4—22no—
66C22×C22⋊C4D411yescharacteristic
67C22×C22⋊C4C2311yescharacteristic
68C22×C42D411yescharacteristic
69C2×#107—28no—
70C2×#107—48no—
71C2×C22⋊C8—22no—
72C2×C22⋊C8—24no—
73C2×C22⋊C8C2×C418yescomplemented
74C2×C22⋊C8C2314yes—
75#389—24no—
76#389—24no—
77#386—44no—
78#385—24no—
79#385—24no—
80#385—24no—
81#385—24no—
82#384—21no—
83#384—21no—
84#382—22no—
85#380—22no—
86#380—22no—
87#380—22no—
88#380—22no—
89#380—22no—
90#380—22no—
91#380—24no—
92#380—44no—
93#380D414yes—
94#380C2314yes—
95C4×C4⋊C4—22no—
96C4×C4⋊C4—22no—
97C2×#95—22no—
98C2×#95—22no—
99C2×#95—22no—
100C2×#95—22no—
101C2×#95—22no—
102C2×#95C2312yes—
103#340—28no—
104#324C2×C418yescomplemented
105#323C2×C418yescomplemented
106C25C2×D411yescharacteristic
107C25C2×D411yescharacteristic
108C22×D4—22no—
109C22×D4—22no—
110C22×D4—22no—
111C22×D4—22no—
112C22×D4C4⋊C414yes—
113C22×D4D814yes—
114C22×D4Q1614yes—
115C23×C4—21no—
116C23×C4—21no—
117C23×C4—22no—
118C23×C4—22no—
119C23×C4—22no—
120C23×C4—22no—
121C23×C4—22no—
122C23×C4—22no—
123C23×C4—22no—
124C23×C4—42no—
125C23×C4—42no—
126C23×C4—42no—
127C23×C4—42no—
128C23×C4—44no—
129C23×C4—42no—
130C23×C4—42no—
131C23×C4—42no—
132C23×C4—82no—
133C23×C4C4⋊C414yes—
134C23×C4C22×C412yes—
135C23×C4C2×D411yescharacteristic
136C23×C4C2×D411yescharacteristic
137C23×C4C2×D412yes—
138C23×C4C2×D411yescharacteristic
139C23×C4C2×D411yescharacteristic
140C23×C4C2×D412yes—
141C23×C4C2×Q812yes—
142C23×C4#4114yes—
143C23×C4C2411yescharacteristic
144C22×C8—28no—
145C22×C8—24no—
146C22×C8—44no—
147C22×C8—44no—
148C4×D4—28no—
149C4×D4—28no—
150C4×D4—28no—
151C4×D4—28no—
152C4×D4—416no—
153C4×D4—416no—
154#117—44no—
155#117—44no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—28no—
159C2×C4⋊C4—24no—
160C2×C4⋊C4—24no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—44no—
170C2×C4⋊C4—48no—
171C2×C4⋊C4—44no—
172C2×C4⋊C4—42no—
173C2×C4⋊C4—41no—
174C2×C4⋊C4—41no—
175C2×C4⋊C4—44no—
176C2×C4⋊C4—44no—
177C2×C4⋊C4—42no—
178C2×C4⋊C4—42no—
179C2×C4⋊C4—42no—
180C2×C4⋊C4—82no—
181C2×C4⋊C4—82no—
182C2×C4⋊C4C22×C412yes—
183C2×C4⋊C4C2×D414yes—
184C2×C4⋊C4C2×D414yes—
185C2×C4⋊C4C2×Q812yes—
186C2×C22⋊C4—22no—
187C2×C22⋊C4—22no—
188C2×C22⋊C4—22no—
189C2×C22⋊C4—22no—
190C2×C22⋊C4—24no—
191C2×C22⋊C4—24no—
192C2×C22⋊C4—24no—
193C2×C22⋊C4—44no—
194C2×C22⋊C4—44no—
195C2×C22⋊C4—44no—
196C2×C22⋊C4—44no—
197C2×C22⋊C4—44no—
198C2×C22⋊C4—42no—
199C2×C22⋊C4—44no—
200C2×C22⋊C4—42no—
201C2×C22⋊C4—42no—
202C2×C22⋊C4—82no—
203C2×C22⋊C4C4⋊C414yes—
204C2×C22⋊C4C22×C412yes—
205C2×C22⋊C4C2×Q812yes—
206C2×C42—21no—
207C2×C42—21no—
208C2×C42—22no—
209C2×C42—24no—
210C2×C42C4⋊C414yes—
211#107—416no—
212C22⋊C8—28no—
213C22⋊C8—24no—
214C22⋊C8C22×C418yes—
215#95—24no—
216#95—24no—
217#95—24no—
218#95—24no—
219#95—24no—
220#95—44no—
221#95—44no—
222#95—82no—
223C24—21no—
224C24—21no—
225C24—21no—
226C24—21no—
227C24—41no—
228C24—41no—
229C24—41no—
230C24—41no—
231C24—41no—
232C24—41no—
233C24—41no—
234C24—41no—
235C24C2×C4⋊C412yes—
236C24C2×C4⋊C412yes—
237C24#12012yes—
238C24#12312yes—
239C24C2×D812yes—
240C24C2×D812yes—
241C24C2×Q1612yes—
242C24C2×Q1612yes—
243C24AGL(1,ℤ/8)12yes—
244C24#13712yes—
245C24C22×D411yescharacteristic
246C2×D4—28no—
247C2×D4—28no—
248C2×D4—28no—
249C2×D4—28no—
250C2×D4—28no—
251C2×D4—28no—
252C2×D4#107116yescomplemented
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—21no—
256C22×C4—21no—
257C22×C4—24no—
258C22×C4—22no—
259C22×C4—24no—
260C22×C4—22no—
261C22×C4—21no—
262C22×C4—21no—
263C22×C4—21no—
264C22×C4—21no—
265C22×C4—24no—
266C22×C4—24no—
267C22×C4—24no—
268C22×C4—24no—
269C22×C4—24no—
270C22×C4—44no—
271C22×C4—44no—
272C22×C4—48no—
273C22×C4—42no—
274C22×C4—41no—
275C22×C4—42no—
276C22×C4—41no—
277C22×C4—48no—
278C22×C4—42no—
279C22×C4—42no—
280C22×C4—41no—
281C22×C4—41no—
282C22×C4—41no—
283C22×C4—41no—
284C22×C4—42no—
285C22×C4—42no—
286C22×C4—48no—
287C22×C4—42no—
288C22×C4—42no—
289C22×C4—42no—
290C22×C4—42no—
291C22×C4—44no—
292C22×C4—44no—
293C22×C4—44no—
294C22×C4—44no—
295C22×C4—42no—
296C22×C4—42no—
297C22×C4—42no—
298C22×C4—42no—
299C22×C4—42no—
300C22×C4—42no—
301C22×C4—42no—
302C22×C4—84no—
303C22×C4—81no—
304C22×C4—81no—
305C22×C4—82no—
306C22×C4—82no—
307C22×C4—82no—
308C22×C4—82no—
309C22×C4—82no—
310C22×C4—82no—
311C22×C4—82no—
312C22×C4—82no—
313C22×C4—88no—
314C22×C4—88no—
315C22×C4—84no—
316C22×C4—84no—
317C22×C4C2×C4⋊C412yes—
318C22×C4C2×C4⋊C412yes—
319C22×C4C2×C4⋊C412yes—
320C22×C4C2×C4⋊C412yes—
321C22×C4C4×D414yes—
322C22×C4C4×D414yes—
323C22×C4#12011yescharacteristic
324C22×C4#12011yescharacteristic
325C22×C4C22⋊Q814yes—
326C22×C4C22⋊Q814yes—
327C22×C4#12312yes—
328C22×C4C2×D812yes—
329C22×C4C2×Q1612yes—
330C22×C4C23×C411yescharacteristic
331C22×C4C22×D412yes—
332C22×C4C22×Q811yescharacteristic
333C22×C4C2×#4112yes—
334C2×C8—28no—
335C2×C8—44no—
336C2×C8—44no—
337C2×C8—416no—
338C4⋊C4—24no—
339C4⋊C4—24no—
340C4⋊C4—28no—
341C4⋊C4—44no—
342C4⋊C4—44no—
343C4⋊C4—44no—
344C4⋊C4—44no—
345C4⋊C4—44no—
346C4⋊C4—44no—
347C4⋊C4—44no—
348C4⋊C4—44no—
349C4⋊C4—44no—
350C4⋊C4—84no—
351C4⋊C4—82no—
352C4⋊C4—82no—
353C4⋊C4—82no—
354C4⋊C4—82no—
355C22⋊C4—28no—
356C22⋊C4—24no—
357C22⋊C4—24no—
358C22⋊C4—48no—
359C22⋊C4—44no—
360C22⋊C4—44no—
361C22⋊C4—48no—
362C22⋊C4—88no—
363C22⋊C4—84no—
364C42—24no—
365C42—24no—
366C42—48no—
367C23—21no—
368C23—22no—
369C23—21no—
370C23—22no—
371C23—21no—
372C23—22no—
373C23—21no—
374C23—22no—
375C23—22no—
376C23—22no—
377C23—22no—
378C23—22no—
379C23—22no—
380C23—22no—
381C23—42no—
382C23—42no—
383C23—42no—
384C23—42no—
385C23—42no—
386C23—42no—
387C23—42no—
388C23—42no—
389C23—42no—
390C23—42no—
391C23—42no—
392C23—42no—
393C23—42no—
394C23—42no—
395C23—42no—
396C23—42no—
397C23—42no—
398C23—42no—
399C23—41no—
400C23—41no—
401C23—41no—
402C23—41no—
403C23—81no—
404C23—81no—
405C23—81no—
406C23—81no—
407C23—81no—
408C23—81no—
409C23#38514yes—
410C23C2×#10714yes—
411C23C2×#10714yescomplemented
412C23#42814yes—
413C23#44714yes—
414C23#45114yes—
415C23#48014yes—
416C23#48414yes—
417C23C22×C4⋊C411yescharacteristic
418C23C2×#12011yescharacteristic
419C23C2×#12311yescharacteristic
420C23C22×D811yescharacteristic
421C23C22×Q1611yescharacteristic
422C23C2×AGL(1,ℤ/8)11yescharacteristic
423C23C2×#13711yescharacteristic
424D4—232no—
425C2×C4—22no—
426C2×C4—22no—
427C2×C4—22no—
428C2×C4—22no—
429C2×C4—22no—
430C2×C4—22no—
431C2×C4—21no—
432C2×C4—22no—
433C2×C4—21no—
434C2×C4—22no—
435C2×C4—24no—
436C2×C4—28no—
437C2×C4—24no—
438C2×C4—28no—
439C2×C4—22no—
440C2×C4—22no—
441C2×C4—44no—
442C2×C4—44no—
443C2×C4—41no—
444C2×C4—41no—
445C2×C4—42no—
446C2×C4—41no—
447C2×C4—41no—
448C2×C4—41no—
449C2×C4—41no—
450C2×C4—48no—
451C2×C4—42no—
452C2×C4—42no—
453C2×C4—44no—
454C2×C4—44no—
455C2×C4—44no—
456C2×C4—44no—
457C2×C4—44no—
458C2×C4—44no—
459C2×C4—44no—
460C2×C4—44no—
461C2×C4—44no—
462C2×C4—84no—
463C2×C4—81no—
464C2×C4—81no—
465C2×C4—82no—
466C2×C4—82no—
467C2×C4—816no—
468C2×C4—82no—
469C2×C4—82no—
470C2×C4—82no—
471C2×C4—82no—
472C2×C4—82no—
473C2×C4—82no—
474C2×C4—82no—
475C2×C4—82no—
476C2×C4—82no—
477C2×C4—164no—
478C2×C4—164no—
479C2×C4—164no—
480C2×C4—164no—
481C2×C4#38512yes—
482C2×C4#38512yes—
483C2×C4C2×#10714yescomplemented
484C2×C4B(4,𝔽2)12yes—
485C2×C4#45812yes—
486C2×C4C2×C4×D412yes—
487C2×C4C2×C22⋊Q812yes—
488C8—48no—
489C22—21no—
490C22—21no—
491C22—24no—
492C22—21no—
493C22—21no—
494C22—24no—
495C22—21no—
496C22—24no—
497C22—21no—
498C22—21no—
499C22—21no—
500C22—24no—
501C22—21no—
502C22—21no—
503C22—24no—
504C22—21no—
505C22—21no—
506C22—24no—
507C22—24no—
508C22—44no—
509C22—44no—
510C22—44no—
511C22—44no—
512C22—44no—
513C22—42no—
514C22—42no—
515C22—42no—
516C22—42no—
517C22—42no—
518C22—42no—
519C22—82no—
520C22—82no—
521C22—82no—
522C22#185218yes—
523C22C2×#38511yescharacteristic
524C22C22×#10711yescharacteristic
525C22C2×#42811yescharacteristic
526C22C2×#44711yescharacteristic
527C22C2×#45111yescharacteristic
528C22#302111yescharacteristic
529C22#302211yescharacteristic
530C22C2×B(4,𝔽2)11yescharacteristic
531C22C2×#45811yescharacteristic
532C22C2×#48011yescharacteristic
533C22C2×#48411yescharacteristic
534C4—24no—
535C4—22no—
536C4—22no—
537C4—22no—
538C4—42no—
539C4—42no—
540C4—84no—
541C4—84no—
542C4—84no—
543C4—162no—
544C4#186714yes—
545C2—21no—
546C2—21no—
547C2—21no—
548C2—21no—
549C2—28no—
550C2—44no—
551C2#1819411yescharacteristic
552C2#1819711yescharacteristic
553C2#1820211yescharacteristic
554C2#1820511yescharacteristic
555C2C2×#185211yescharacteristic
556C2#2085811yescharacteristic
557C2C2×#186711yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC23×D4D4
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
2. C2×#1852C2semidirect
5. #8856C2semidirect
6. #8853C2semidirect
7. #8023C2semidirect
28. #1459C4semidirect
73. C2×C22⋊C8C2×C4semidirect
104. #324C2×C4semidirect
105. #323C2×C4semidirect
252. C2×D4#107semidirect
411. C23C2×#107semidirect
483. C2×C4C2×#107semidirect

Maximal subgroups

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

subgroupindex
#341812
C2×#18522
C2×#14592
#138722
#88562
#88532
#80232