Finite Group Databasebeta

← #368286

Subgroups — #368286

Order 512.

1250 conjugacy classes of subgroups.

The subgroup lattice has 610 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34216C211yesmaximal, characteristic
2C2×#1997C211yesmaximal, characteristic
3#21170C211yesmaximal, characteristic
4C4×#428C211yesmaximal, characteristic
5#14155C211yesmaximal, characteristic
6#11342C212yesmaximal
7#11296C212yesmaximal
8#10899C212yesmaximal
9#10899C212yesmaximal
10#10852C211yesmaximal, characteristic
11#10851C211yesmaximal, characteristic
12C22×C4⋊Q8C2211yescharacteristic
13C2×#428C2211yescharacteristic
14C2×#428C2212yes—
15C2×#428C2211yescharacteristic
16C2×#423C2211yescharacteristic
17C2×C4×M16C2211yescharacteristic
18#2369—21no—
19#2369—21no—
20#2365—21no—
21#2365—21no—
22#2345—22no—
23Q8×C22⋊C4—22no—
24#2331—22no—
25#2316C2212yes—
26C4×C4⋊Q8—21no—
27C4×C4⋊Q8—21no—
28C2×#391—22no—
29#2285C2211yescharacteristic
30#2274C2212yes—
31C4×#117C2211yescharacteristic
32C2×C4×C4⋊C4C2211yescharacteristic
33C2×C4×C4⋊C4C2211yescharacteristic
34#1997C418yescomplemented
35#1949—22no—
36#1948—21no—
37#1947—21no—
38#1776—21no—
39#1776—21no—
40C4×#107—21no—
41C4×#106—21no—
42C2×#343C2212yes—
43C2×#343C2212yes—
44#1736C2211yescharacteristic
45#1736C2212yes—
46#1736C2211yescharacteristic
47C2×#339C2212yes—
48C2×#339C2212yes—
49C2×#339C2212yes—
50C2×#339C2212yes—
51#1445—22no—
52C4⋊#95C2212yes—
53C4⋊#95C2212yes—
54C23×Q8D412yes—
55C22×M16C2311yescharacteristic
56C22×M16C2312yes—
57C22×M16C2311yescharacteristic
58C2×C4⋊Q8—21no—
59C2×C4⋊Q8—21no—
60C2×C4⋊Q8—21no—
61C2×C4⋊Q8—21no—
62C2×C4⋊Q8—22no—
63C2×C4⋊Q8—22no—
64C2×C4⋊Q8C2×C414yes—
65C2×C4⋊Q8Q814yes—
66C2×C4×Q8—22no—
67C2×C4×Q8—22no—
68C2×#117C2311yescharacteristic
69C2×#117C2312yes—
70C2×#117C2311yescharacteristic
71C22×C4⋊C4—21no—
72C22×C4⋊C4D412yes—
73C22×C4⋊C4C2311yescharacteristic
74C22×C4⋊C4C2311yescharacteristic
75C22×C4⋊C4C2312yes—
76C22×C42—21no—
77C22×C42C2312yes—
78C22×C42C2311yescharacteristic
79#428—22no—
80#428—22no—
81#428—22no—
82#428—24no—
83#428—42no—
84#428Q814yescomplemented
85C2×#107—21no—
86C2×#107—21no—
87C2×#107—22no—
88C2×#106—21no—
89C2×#106—21no—
90C2×#106—22no—
91#423—22no—
92#423C2×C414yes—
93C2×C4⋊C8—21no—
94C2×C4⋊C8—21no—
95C4×M16—22no—
96C4×M16C2×C414yes—
97C2×#97—21no—
98C2×C4×C8—21no—
99#391—22no—
100#391—22no—
101#391—24no—
102#391—24no—
103#391—48no—
104#389—22no—
105#389—22no—
106#389—22no—
107#389—22no—
108#389—22no—
109#389—22no—
110#387—24no—
111#387—24no—
112#384D411yescharacteristic
113#384D411yescharacteristic
114#382D412yes—
115#380—24no—
116#380—21no—
117#380—21no—
118C4×C4⋊C4—22no—
119C4×C4⋊C4—22no—
120C4×C4⋊C4—22no—
121C4×C4⋊C4—22no—
122C4×C4⋊C4—22no—
123C4×C4⋊C4—22no—
124C4×C4⋊C4—21no—
125C4×C4⋊C4—21no—
126C4×C4⋊C4—22no—
127C4×C4⋊C4—22no—
128C4×C4⋊C4—21no—
129C4×C4⋊C4—21no—
130C4×C4⋊C4—21no—
131C4×C4⋊C4—21no—
132C4×C4⋊C4D411yescharacteristic
133C4×C4⋊C4D411yescharacteristic
134C4×C4⋊C4D411yescharacteristic
135C4×C4⋊C4D411yescharacteristic
136C4×C22⋊C4—21no—
137C4×C22⋊C4—24no—
138C4×C22⋊C4—21no—
139#376—21no—
140C2×#95—24no—
141C2×#95—24no—
142C2×#95—21no—
143C2×#95—21no—
144C43—21no—
145#343C2×C418yes—
146#343C2×C418yes—
147#340—22no—
148#340—22no—
149#340—21no—
150#340—21no—
151#340—21no—
152#340—21no—
153#339—24no—
154#339—24no—
155#339—42no—
156#339—42no—
157C22×Q8—22no—
158C22×Q8—22no—
159C22×Q8—24no—
160C22×Q8—44no—
161C22×Q8—44no—
162C22×Q8C4⋊C418yes—
163C23×C4—22no—
164C23×C4—21no—
165C23×C4—21no—
166C23×C4C2×D411yescharacteristic
167C23×C4C2×D414yes—
168C23×C4C2×D411yescharacteristic
169C23×C4C2411yescharacteristic
170C2×M16—21no—
171C2×M16—21no—
172C2×M16—21no—
173C2×M16—21no—
174C2×M16—22no—
175C2×M16—22no—
176C2×M16—22no—
177C2×M16—21no—
178C2×M16—22no—
179C2×M16—21no—
180C2×M16—21no—
181C2×M16—21no—
182C2×M16—22no—
183C2×M16C22×C412yes—
184C2×M16C22×C414yes—
185C2×M16C22×C412yes—
186C2×M16C2×Q812yes—
187C2×M16#4112yes—
188C22×C8—21no—
189C22×C8—21no—
190C22×C8—22no—
191C4⋊Q8—24no—
192C4⋊Q8—24no—
193C4⋊Q8—24no—
194C4⋊Q8—24no—
195C4⋊Q8—24no—
196C4⋊Q8—24no—
197C4⋊Q8—24no—
198C4⋊Q8—24no—
199C4×Q8—84no—
200C4×Q8—84no—
201#117—21no—
202#117—21no—
203#117—22no—
204#117—22no—
205#117—21no—
206#117—21no—
207#117—21no—
208#117—22no—
209#117—22no—
210#117—21no—
211#117C2×Q812yes—
212#117#4112yes—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—24no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—21no—
221C2×C4⋊C4—21no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4—21no—
224C2×C4⋊C4—21no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4—21no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4—24no—
229C2×C4⋊C4—21no—
230C2×C4⋊C4—21no—
231C2×C4⋊C4—21no—
232C2×C4⋊C4—21no—
233C2×C4⋊C4—21no—
234C2×C4⋊C4—21no—
235C2×C4⋊C4—24no—
236C2×C4⋊C4—24no—
237C2×C4⋊C4—42no—
238C2×C4⋊C4—42no—
239C2×C4⋊C4C4⋊C418yes—
240C2×C4⋊C4C22×C414yes—
241C2×C4⋊C4C2×D412yes—
242C2×C4⋊C4C2×D412yes—
243C2×C4⋊C4C2×D411yescharacteristic
244C2×C4⋊C4C2×D411yescharacteristic
245C2×C4⋊C4C2×D411yescharacteristic
246C2×C4⋊C4C2×D411yescharacteristic
247C2×C4⋊C4C2×Q812yes—
248C2×C4⋊C4C2×Q814yes—
249C2×C4⋊C4#4112yes—
250C2×C22⋊C4—21no—
251C2×C22⋊C4—21no—
252C2×C22⋊C4—22no—
253C2×C42—21no—
254C2×C42—21no—
255C2×C42—21no—
256C2×C42—21no—
257C2×C42—21no—
258C2×C42—21no—
259C2×C42—22no—
260C2×C42—22no—
261C2×C42—22no—
262C2×C42—22no—
263C2×C42—22no—
264C2×C42—22no—
265C2×C42—22no—
266C2×C42—24no—
267C2×C42—24no—
268C2×C42—44no—
269C2×C42—41no—
270C2×C42—41no—
271C2×C42C22×C412yes—
272C2×C42C2×D411yescharacteristic
273C2×C42C2×D411yescharacteristic
274C2×C42C2×D411yescharacteristic
275C2×C42C2×D411yescharacteristic
276C2×C42C2×Q812yes—
277C2×C42#4112yes—
278C2×C42#4112yes—
279#107—24no—
280#107—24no—
281#107—44no—
282#106—24no—
283#106—24no—
284#106—44no—
285C4⋊C8—24no—
286C4⋊C8—24no—
287#97—24no—
288C4×C8—24no—
289#95—24no—
290#95—24no—
291#95—24no—
292#95—24no—
293#95—21no—
294#95—21no—
295#95—21no—
296#95—21no—
297#95—22no—
298#95—22no—
299#95—48no—
300#95—48no—
301#95—41no—
302#95—41no—
303#95—42no—
304C24C22×D411yescharacteristic
305C2×Q8—28no—
306C2×Q8—28no—
307C2×Q8—416no—
308C2×Q8—44no—
309C2×Q8—44no—
310C2×Q8—82no—
311C2×Q8—82no—
312C2×Q8—88no—
313C22×C4—22no—
314C22×C4—22no—
315C22×C4—21no—
316C22×C4—21no—
317C22×C4—24no—
318C22×C4—24no—
319C22×C4—24no—
320C22×C4—21no—
321C22×C4—22no—
322C22×C4—21no—
323C22×C4—22no—
324C22×C4—22no—
325C22×C4—22no—
326C22×C4—41no—
327C22×C4—41no—
328C22×C4—44no—
329C22×C4—44no—
330C22×C4—42no—
331C22×C4—42no—
332C22×C4—42no—
333C22×C4—42no—
334C22×C4—42no—
335C22×C4—42no—
336C22×C4—42no—
337C22×C4—41no—
338C22×C4—41no—
339C22×C4—42no—
340C22×C4—42no—
341C22×C4—41no—
342C22×C4—42no—
343C22×C4—41no—
344C22×C4C2×C4⋊C412yes—
345C22×C4C2×C4⋊C418yes—
346C22×C4C2×C4⋊C412yes—
347C22×C4#12011yescharacteristic
348C22×C4#12011yescharacteristic
349C22×C4#12011yescharacteristic
350C22×C4#12011yescharacteristic
351C22×C4#12312yes—
352C22×C4#12312yes—
353C22×C4C23×C411yescharacteristic
354C22×C4C22×D411yescharacteristic
355C22×C4C22×D411yescharacteristic
356C22×C4C22×Q811yescharacteristic
357C22×C4C22×Q811yescharacteristic
358C22×C4C2×#4111yescharacteristic
359C22×C4C2×#4111yescharacteristic
360C22×C4C2×#4111yescharacteristic
361C22×C4C2×#4111yescharacteristic
362C22×C4C2×#4111yescharacteristic
363C22×C4H(5,𝔽2)11yescharacteristic
364C22×C42−1+411yescharacteristic
365C22×C42−1+411yescharacteristic
366C22×C42−1+411yescharacteristic
367M16—22no—
368M16—22no—
369M16—22no—
370M16—24no—
371M16—22no—
372M16—22no—
373M16—24no—
374M16—41no—
375M16—41no—
376M16—42no—
377M16—42no—
378M16C4×Q814yescomplemented
379C2×C8—22no—
380C2×C8—22no—
381C2×C8—22no—
382C2×C8—22no—
383C2×C8—24no—
384C2×C8—41no—
385C2×C8—41no—
386C2×C8—42no—
387C2×C8—42no—
388C4⋊C4—24no—
389C4⋊C4—24no—
390C4⋊C4—24no—
391C4⋊C4—24no—
392C4⋊C4—21no—
393C4⋊C4—21no—
394C4⋊C4—28no—
395C4⋊C4—28no—
396C4⋊C4—22no—
397C4⋊C4—21no—
398C4⋊C4—22no—
399C4⋊C4—21no—
400C4⋊C4—21no—
401C4⋊C4—21no—
402C4⋊C4—22no—
403C4⋊C4—22no—
404C4⋊C4—22no—
405C4⋊C4—22no—
406C4⋊C4—22no—
407C4⋊C4—22no—
408C4⋊C4—42no—
409C4⋊C4—42no—
410C4⋊C4—48no—
411C4⋊C4—41no—
412C4⋊C4—41no—
413C4⋊C4—41no—
414C4⋊C4—41no—
415C4⋊C4—42no—
416C4⋊C4—42no—
417C4⋊C4—84no—
418C4⋊C4—84no—
419C4⋊C4C22⋊Q812yes—
420C4⋊C4C22⋊Q812yes—
421C22⋊C4—22no—
422C22⋊C4—22no—
423C22⋊C4—42no—
424C22⋊C4—41no—
425C22⋊C4—42no—
426C22⋊C4—41no—
427C42—22no—
428C42—22no—
429C42—42no—
430C42—42no—
431C42—42no—
432C42—42no—
433C42—41no—
434C42—41no—
435C42—42no—
436C42—42no—
437C42—42no—
438C42—42no—
439C42—42no—
440C42—42no—
441C42—82no—
442C42—84no—
443C42—84no—
444C42—82no—
445C42C4×D412yes—
446C42C4×D412yes—
447C42C4×D412yes—
448C42C4×D412yes—
449C42C22⋊Q812yes—
450C42C22⋊Q812yes—
451C42C22⋊Q812yes—
452C42C22⋊Q812yes—
453C23—21no—
454C23—21no—
455C23—21no—
456C23—21no—
457C23C22×C4⋊C411yescharacteristic
458C23C2×#12011yescharacteristic
459C23C2×#12311yescharacteristic
460C23#54811yescharacteristic
461C23#54811yescharacteristic
462C23D4×Q811yescharacteristic
463C23#55011yescharacteristic
464Q8—416no—
465Q8—88no—
466C2×C4—21no—
467C2×C4—21no—
468C2×C4—21no—
469C2×C4—21no—
470C2×C4—21no—
471C2×C4—21no—
472C2×C4—24no—
473C2×C4—24no—
474C2×C4—22no—
475C2×C4—22no—
476C2×C4—28no—
477C2×C4—42no—
478C2×C4—42no—
479C2×C4—41no—
480C2×C4—44no—
481C2×C4—41no—
482C2×C4—44no—
483C2×C4—41no—
484C2×C4—41no—
485C2×C4—48no—
486C2×C4—44no—
487C2×C4—44no—
488C2×C4—44no—
489C2×C4—44no—
490C2×C4—44no—
491C2×C4—42no—
492C2×C4—41no—
493C2×C4—42no—
494C2×C4—41no—
495C2×C4—42no—
496C2×C4—42no—
497C2×C4—41no—
498C2×C4—42no—
499C2×C4—41no—
500C2×C4—42no—
501C2×C4—41no—
502C2×C4—41no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—84no—
506C2×C4—82no—
507C2×C4—82no—
508C2×C4—82no—
509C2×C4—82no—
510C2×C4—82no—
511C2×C4—82no—
512C2×C4—81no—
513C2×C4—81no—
514C2×C4—81no—
515C2×C4—81no—
516C2×C4#38512yes—
517C2×C4#38512yes—
518C2×C4#38512yes—
519C2×C4#38512yes—
520C2×C4#45312yes—
521C2×C4#45512yes—
522C2×C4C2×C4×D411yescharacteristic
523C2×C4C2×C4×D411yescharacteristic
524C2×C4C2×C4×Q811yescharacteristic
525C2×C4C4×#4111yescharacteristic
526C2×C4#51911yescharacteristic
527C2×C4#52011yescharacteristic
528C2×C4C2×C22⋊Q811yescharacteristic
529C2×C4C2×C22⋊Q811yescharacteristic
530C2×C4C2×C22⋊Q811yescharacteristic
531C2×C4#52511yescharacteristic
532C2×C4#52911yescharacteristic
533C2×C4#53311yescharacteristic
534C2×C4#53611yescharacteristic
535C2×C4#53711yescharacteristic
536C2×C4#54111yescharacteristic
537C2×C4#54211yescharacteristic
538C2×C4#54611yescharacteristic
539C2×C4#54711yescharacteristic
540C2×C4#54811yescharacteristic
541C2×C4D4×Q811yescharacteristic
542C2×C4#55611yescharacteristic
543C2×C4#55611yescharacteristic
544C2×C4#55711yescharacteristic
545C2×C4#55711yescharacteristic
546C2×C4#55711yescharacteristic
547C2×C4#55711yescharacteristic
548C2×C4Q8211yescharacteristic
549C2×C4#55911yescharacteristic
550C8—24no—
551C8—42no—
552C8—44no—
553C22—21no—
554C22—21no—
555C22—21no—
556C22—21no—
557C22—21no—
558C22—21no—
559C22—21no—
560C22—21no—
561C22—21no—
562C22—21no—
563C22—21no—
564C22—21no—
565C22C2×#38511yescharacteristic
566C22Q8×C4⋊C411yescharacteristic
567C22#235111yescharacteristic
568C22#240711yescharacteristic
569C22#240911yescharacteristic
570C22#244611yescharacteristic
571C22#244811yescharacteristic
572C22C2×#45311yescharacteristic
573C22C2×#45511yescharacteristic
574C22#302011yescharacteristic
575C22#302011yescharacteristic
576C4—22no—
577C4—22no—
578C4—22no—
579C4—42no—
580C4—42no—
581C4—48no—
582C4—84no—
583C4—82no—
584C4—81no—
585C4—82no—
586C4—82no—
587C4—81no—
588C4—82no—
589C4#185614yes—
590C4C4×C22⋊Q811yescharacteristic
591C4#232311yescharacteristic
592C4#236511yescharacteristic
593C4#236711yescharacteristic
594C4#245111yescharacteristic
595C4#245311yescharacteristic
596C4#258111yescharacteristic
597C4#258211yescharacteristic
598C2—21no—
599C2—21no—
600C2—21no—
601C2—21no—
602C2#1401711yescharacteristic
603C2C2×#185611yescharacteristic
604C2#2085711yescharacteristic
605C2#2417011yescharacteristic
606C2#2417211yescharacteristic
607C2#2432011yescharacteristic
608C2#2432211yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC24C22×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
34. #1997C4semidirect
84. #428Q8semidirect
378. M16C4×Q8semidirect

Maximal subgroups

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

subgroupindex
#342162
C2×#19972
#211702
C4×#4282
#141552
#113422
#112962
#108992
#108522
#108512