Finite Group Databasebeta

← #219812

Subgroups — #219812

Order 512.

1238 conjugacy classes of subgroups.

The subgroup lattice has 608 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#547C211yesmaximal, characteristic
2#22799C211yesmaximal, characteristic, complemented
3#21264C211yesmaximal, characteristic, complemented
4C2×#1488C212yesmaximal, complemented
5#9673C211yesmaximal, characteristic, complemented
6#9637C211yesmaximal, characteristic, complemented
7#9075C212yesmaximal, complemented
8#8941C212yesmaximal, complemented
9#8563C211yesmaximal, characteristic, complemented
10#8312C212yesmaximal, complemented
11C4×D4⋊C8C211yesmaximal, characteristic, complemented
12C2×#547C411yescharacteristic
13C2×C4×#41C2212yes—
14C2×#432C2212yes—
15C2×#431C2212yes—
16#2365C411yescharacteristic
17#2362C412yes—
18#2361—21no—
19#2359—22no—
20#2354C411yescharacteristic
21D4×C4⋊C4C411yescharacteristic
22#2344—21no—
23#2339C412yes—
24C4×C4⋊Q8C2211yescharacteristic
25C4×C22⋊Q8C2212yes—
26C4×#123—22no—
27C4×#121—21no—
28C42×D4C2211yescharacteristic
29C42×D4C2211yescharacteristic
30C2×C4×C4⋊C4—21no—
31#1954—21no—
32#1954—21no—
33#1939—21no—
34#1924—22no—
35#1923—22no—
36#1842C2211yescharacteristic
37#1797C2212yes—
38#1770—21no—
39C4×C4⋊C8C2211yescharacteristic
40C4×C4⋊C8C2211yescharacteristic
41C4×C22⋊C8C2212yes—
42C42×C8C2211yescharacteristic
43#1488—28no—
44C2×Q8⋊C8C2212yes—
45C2×Q8⋊C8C2212yes—
46C2×Q8⋊C8C2212yes—
47C2×Q8⋊C8C2212yes—
48C2×D4⋊C8C2212yes—
49C2×D4⋊C8C2211yescharacteristic
50C2×D4⋊C8C2211yescharacteristic
51#1279C2212yes—
52#1279C2211yescharacteristic
53#1279C2211yescharacteristic
54C22×#41C2×C412yes—
55#547—24no—
56#547—24no—
57C2×C4⋊Q8C2×C411yescharacteristic
58C2×#123—22no—
59C2×C22⋊Q8C2×C412yes—
60C2×#121—21no—
61C4×#41—24no—
62C4×#41—24no—
63C4×#41—42no—
64C4×#41—42no—
65C2×C4×Q8C2×C412yes—
66C2×C4×Q8C2312yes—
67C2×C4×D4—22no—
68C2×C4×D4—22no—
69C2×C4×D4C2×C411yescharacteristic
70C2×C4×D4C2×C412yes—
71C2×C4×D4C2×C411yescharacteristic
72C2×C4×D4C2×C411yescharacteristic
73C2×C4×D4D411yescharacteristic
74C2×C4×D4D411yescharacteristic
75C2×C4×D4C2311yescharacteristic
76C2×#117—22no—
77C2×#117C2×C412yes—
78C22×C4⋊C4—21no—
79C22×C4⋊C4—21no—
80C22×C42—22no—
81C22×C42—21no—
82C22×C42C2312yes—
83#432—24no—
84#432—42no—
85#431—24no—
86#431—42no—
87C2×C4⋊C8—21no—
88C2×C4⋊C8—22no—
89C2×C4⋊C8—21no—
90C2×C4⋊C8—21no—
91C2×C4⋊C8C2311yescharacteristic
92C2×C4⋊C8C2312yes—
93C2×C4⋊C8C2311yescharacteristic
94C2×C22⋊C8—22no—
95C2×C22⋊C8—22no—
96C2×C22⋊C8—22no—
97C2×C22⋊C8—22no—
98C2×#97—22no—
99C2×C4×C8—21no—
100C2×C4×C8—21no—
101C2×C4×C8—21no—
102C2×C4×C8C2312yes—
103C2×C4×C8C2311yescharacteristic
104C2×C4×C8C2311yescharacteristic
105#391C2×C412yes—
106#391C2×C412yes—
107#390—21no—
108#389—21no—
109#389—21no—
110#389—21no—
111#389C2×C412yes—
112#388—22no—
113#388—22no—
114#388—22no—
115#387—22no—
116#387—22no—
117#387—22no—
118#386—22no—
119#386—22no—
120#385—22no—
121#385—22no—
122#384C2×C411yescharacteristic
123#381—22no—
124#380—21no—
125#380C2×C412yes—
126C4×C4⋊C4—21no—
127C4×C4⋊C4—21no—
128C4×C4⋊C4—21no—
129C4×C4⋊C4—21no—
130C4×C4⋊C4—22no—
131C4×C4⋊C4—22no—
132C4×C4⋊C4C2×C411yescharacteristic
133C4×C4⋊C4C2×C411yescharacteristic
134C4×C4⋊C4C2×C411yescharacteristic
135C4×C4⋊C4D411yescharacteristic
136C4×C4⋊C4D411yescharacteristic
137C4×C4⋊C4C2311yescharacteristic
138C4×C22⋊C4—22no—
139C4×C22⋊C4—21no—
140C4×C22⋊C4—21no—
141C4×C22⋊C4—21no—
142C4×C22⋊C4C2×C412yes—
143C4×C22⋊C4D412yes—
144C4×C22⋊C4D412yes—
145C2×#95—22no—
146C43C2311yescharacteristic
147#336—22no—
148#336—21no—
149#336—22no—
150#336—21no—
151Q8⋊C8—24no—
152Q8⋊C8—24no—
153Q8⋊C8—42no—
154Q8⋊C8—42no—
155D4⋊C8—24no—
156D4⋊C8—22no—
157D4⋊C8D414yescomplemented
158C2×#41—24no—
159C2×#41—24no—
160C2×#41—24no—
161C2×#41—24no—
162C2×#41—42no—
163C2×#41—42no—
164C22×Q8C22×C412yes—
165C22×D4—22no—
166C22×D4C22⋊C411yescharacteristic
167C22×D4C22×C411yescharacteristic
168C23×C4—21no—
169C23×C4—22no—
170C23×C4—21no—
171C23×C4—22no—
172C23×C4—21no—
173C23×C4C22×C412yes—
174C23×C4C2×D412yes—
175C23×C4C2×D412yes—
176C22×C8—21no—
177C22×C8—21no—
178C22×C8—21no—
179C22×C8—22no—
180C22×C8—21no—
181C22×C8—22no—
182C4⋊Q8—22no—
183C4⋊Q8M1614yes—
184#123—216no—
185C22⋊Q8—24no—
186C22⋊Q8—24no—
187#121—28no—
188C4×Q8—24no—
189C4×Q8—22no—
190C4×Q8—22no—
191C4×Q8—22no—
192C4×Q8—22no—
193C4×Q8—42no—
194C4×D4—24no—
195C4×D4—22no—
196C4×D4—22no—
197C4×D4—22no—
198C4×D4—22no—
199C4×D4—22no—
200C4×D4—22no—
201C4×D4—21no—
202C4×D4—21no—
203C4×D4—41no—
204C4×D4—42no—
205C4×D4—42no—
206C4×D4—44no—
207C4×D4—44no—
208C4×D4—44no—
209C4×D4—44no—
210C4×D4M1614yes—
211C4×D4C2×D412yes—
212C4×D4#4112yes—
213#117—24no—
214#117—44no—
215#117—44no—
216#117—42no—
217C2×C4⋊C4—28no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—21no—
221C2×C4⋊C4—21no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4—44no—
227C2×C4⋊C4C22⋊C411yescharacteristic
228C2×C4⋊C4C22⋊C411yescharacteristic
229C2×C4⋊C4C22⋊C411yescharacteristic
230C2×C4⋊C4C22×C412yes—
231C2×C4⋊C4C22×C411yescharacteristic
232C2×C4⋊C4C22×C411yescharacteristic
233C2×C4⋊C4C22×C411yescharacteristic
234C2×C22⋊C4—22no—
235C2×C22⋊C4—21no—
236C2×C22⋊C4—21no—
237C2×C22⋊C4—22no—
238C2×C22⋊C4—22no—
239C2×C22⋊C4—21no—
240C2×C22⋊C4—22no—
241C2×C22⋊C4C22⋊C412yes—
242C2×C22⋊C4C22⋊C412yes—
243C2×C22⋊C4C22⋊C412yes—
244C2×C42—22no—
245C2×C42—22no—
246C2×C42—22no—
247C2×C42—22no—
248C2×C42—22no—
249C2×C42—22no—
250C2×C42—21no—
251C2×C42—21no—
252C2×C42—22no—
253C2×C42—21no—
254C2×C42—44no—
255C2×C42—44no—
256C2×C42—42no—
257C2×C42—42no—
258C2×C42C22×C412yes—
259C2×C42C22×C411yescharacteristic
260C2×C42C22×C411yescharacteristic
261C2×C42C2×D412yes—
262C2×C42C2×D411yescharacteristic
263C2×C42C2×D411yescharacteristic
264C2×C42C2×D412yes—
265C2×C42C2×D411yescharacteristic
266C2×C42C2×D411yescharacteristic
267C2×C42C2411yescharacteristic
268C4⋊C8—22no—
269C4⋊C8—21no—
270C4⋊C8—21no—
271C4⋊C8—21no—
272C4⋊C8—22no—
273C4⋊C8—44no—
274C4⋊C8—42no—
275C4⋊C8—42no—
276C4⋊C8—81no—
277C4⋊C8C2×D412yes—
278C22⋊C8—44no—
279C22⋊C8—44no—
280C22⋊C8—44no—
281C22⋊C8—82no—
282#97—44no—
283C4×C8—22no—
284C4×C8—21no—
285C4×C8—21no—
286C4×C8—21no—
287C4×C8—22no—
288C4×C8—42no—
289C4×C8—42no—
290C4×C8—42no—
291C4×C8C2×D412yes—
292#95—22no—
293#95—22no—
294#95—22no—
295#95—21no—
296#95—21no—
297#95—22no—
298#95—22no—
299#95—22no—
300#95—21no—
301#95C22⋊C412yes—
302#95C22⋊C412yes—
303#95C22⋊C412yes—
304C24—21no—
305C24C2×C22⋊C412yes—
306#41—48no—
307#41—48no—
308#41—48no—
309#41—48no—
310C2×Q8—22no—
311C2×Q8—22no—
312C2×Q8—22no—
313C2×Q8—22no—
314C2×Q8—22no—
315C2×Q8C2×M1614yes—
316C2×D4—28no—
317C2×D4—21no—
318C2×D4—21no—
319C2×D4—21no—
320C2×D4—22no—
321C2×D4—22no—
322C2×D4—22no—
323C2×D4—44no—
324C2×D4—44no—
325C2×D4C4×D412yes—
326C2×D4C2×M1612yes—
327C2×D4#13112yes—
328C22×C4—28no—
329C22×C4—21no—
330C22×C4—22no—
331C22×C4—21no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—24no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—21no—
350C22×C4—22no—
351C22×C4—42no—
352C22×C4—42no—
353C22×C4—44no—
354C22×C4—44no—
355C22×C4—44no—
356C22×C4—42no—
357C22×C4—44no—
358C22×C4—41no—
359C22×C4—41no—
360C22×C4—41no—
361C22×C4—41no—
362C22×C4—42no—
363C22×C4—42no—
364C22×C4—42no—
365C22×C4—81no—
366C22×C4—82no—
367C22×C4C2×C22⋊C411yescharacteristic
368C22×C4C2×C22⋊C412yes—
369C22×C4C2×C22⋊C412yes—
370C22×C4C2×C22⋊C412yes—
371C22×C4C2×C22⋊C411yescharacteristic
372C22×C4C2×C22⋊C411yescharacteristic
373C22×C4C2×C22⋊C411yescharacteristic
374C22×C4C23×C411yescharacteristic
375C22×C4C22×D411yescharacteristic
376C22×C4C22×D411yescharacteristic
377C2×C8—22no—
378C2×C8—22no—
379C2×C8—22no—
380C2×C8—41no—
381C2×C8—41no—
382C2×C8—41no—
383C2×C8—41no—
384C2×C8—41no—
385C2×C8—41no—
386C2×C8—42no—
387C2×C8—42no—
388C2×C8—42no—
389C2×C8—42no—
390C2×C8—41no—
391C2×C8—41no—
392C2×C8—42no—
393C2×C8—41no—
394C2×C8—42no—
395C4⋊C4—28no—
396C4⋊C4—21no—
397C4⋊C4—21no—
398C4⋊C4—22no—
399C4⋊C4—24no—
400C4⋊C4—24no—
401C4⋊C4—22no—
402C4⋊C4—22no—
403C4⋊C4—22no—
404C4⋊C4—22no—
405C4⋊C4—21no—
406C4⋊C4—21no—
407C4⋊C4—44no—
408C4⋊C4—44no—
409C4⋊C4—42no—
410C4⋊C4—42no—
411C4⋊C4—42no—
412C4⋊C4—81no—
413C4⋊C4C4×D412yes—
414C4⋊C4C2×M1612yes—
415C22⋊C4—28no—
416C22⋊C4—22no—
417C22⋊C4—22no—
418C22⋊C4—24no—
419C22⋊C4—24no—
420C22⋊C4—24no—
421C22⋊C4—42no—
422C22⋊C4—44no—
423C22⋊C4—44no—
424C22⋊C4—42no—
425C22⋊C4—82no—
426C42—22no—
427C42—22no—
428C42—22no—
429C42—22no—
430C42—22no—
431C42—22no—
432C42—21no—
433C42—21no—
434C42—21no—
435C42—21no—
436C42—22no—
437C42—41no—
438C42—44no—
439C42—44no—
440C42—42no—
441C42—42no—
442C42—41no—
443C42—41no—
444C42—42no—
445C42—42no—
446C42C2×M1612yes—
447C42C2×M1612yes—
448C42#13512yes—
449C42#13512yes—
450C42C22×D411yescharacteristic
451C42C2×#4111yescharacteristic
452C42H(5,𝔽2)11yescharacteristic
453C42H(5,𝔽2)11yescharacteristic
454C23—22no—
455C23—22no—
456C23—22no—
457C23—22no—
458C23—22no—
459C23—22no—
460C23—22no—
461C23—22no—
462C23—22no—
463C23—22no—
464C23—41no—
465C23—41no—
466C23—41no—
467C23—41no—
468C23C22×C22⋊C411yescharacteristic
469Q8—24no—
470Q8—24no—
471Q8—42no—
472Q8—42no—
473D4—22no—
474D4—22no—
475D4—22no—
476D4—416no—
477D4—44no—
478D4#43614yes—
479C2×C4—21no—
480C2×C4—21no—
481C2×C4—21no—
482C2×C4—21no—
483C2×C4—21no—
484C2×C4—21no—
485C2×C4—22no—
486C2×C4—22no—
487C2×C4—22no—
488C2×C4—22no—
489C2×C4—22no—
490C2×C4—22no—
491C2×C4—24no—
492C2×C4—21no—
493C2×C4—22no—
494C2×C4—22no—
495C2×C4—21no—
496C2×C4—22no—
497C2×C4—21no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—416no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—41no—
504C2×C4—41no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—48no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—42no—
524C2×C4—41no—
525C2×C4—42no—
526C2×C4—82no—
527C2×C4—82no—
528C2×C4—84no—
529C2×C4—88no—
530C2×C4#40712yes—
531C2×C4#40712yes—
532C2×C4#41612yes—
533C2×C4#41612yes—
534C2×C4#42112yes—
535C2×C4C2×C4×D411yescharacteristic
536C2×C4#51811yescharacteristic
537C2×C4D4211yescharacteristic
538C2×C4D4211yescharacteristic
539C2×C4#54611yescharacteristic
540C2×C4#54611yescharacteristic
541C2×C4C22×M1611yescharacteristic
542C2×C4C22×M1611yescharacteristic
543C2×C4C2×#13111yescharacteristic
544C2×C4#56811yescharacteristic
545C2×C4C2×#13511yescharacteristic
546C2×C4C2×#13511yescharacteristic
547C2×C4#57711yescharacteristic
548C2×C4#57811yescharacteristic
549C8—42no—
550C8—42no—
551C8—42no—
552C8—82no—
553C8—82no—
554C8—81no—
555C22—24no—
556C22—42no—
557C22—42no—
558C22—42no—
559C22—42no—
560C22—42no—
561C22—42no—
562C22—42no—
563C22—42no—
564C22—42no—
565C22—42no—
566C22—42no—
567C22—42no—
568C22—42no—
569C22D4×C22⋊C411yescharacteristic
570C22C2×#40711yescharacteristic
571C22#288011yescharacteristic
572C22C2×#41611yescharacteristic
573C22#289811yescharacteristic
574C22C2×#42111yescharacteristic
575C22#290111yescharacteristic
576C4—22no—
577C4—22no—
578C4—21no—
579C4—21no—
580C4—42no—
581C4—41no—
582C4—42no—
583C4—44no—
584C4—42no—
585C4—42no—
586C4—42no—
587C4—81no—
588C4—82no—
589C4#148814yescomplemented
590C4C2×#43611yescharacteristic
591C4D4×M1611yescharacteristic
592C4#297411yescharacteristic
593C4#297511yescharacteristic
594C4#328311yescharacteristic
595C4#328411yescharacteristic
596C4#330211yescharacteristic
597C4#330311yescharacteristic
598C2—44no—
599C2—82no—
600C2C2×#148811yescharacteristic
601C2#1745411yescharacteristic
602C2#1750811yescharacteristic
603C2#1752311yescharacteristic
604C2#2240111yescharacteristic
605C2#2241711yescharacteristic
606C2#2243511yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC23C22×C22⋊C4
prime radicalC2×#547C4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C42×D4 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #22799C2semidirect
3. #21264C2semidirect
4. C2×#1488C2semidirect
5. #9673C2semidirect
6. #9637C2semidirect
7. #9075C2semidirect
8. #8941C2semidirect
9. #8563C2semidirect
10. #8312C2semidirect
11. C4×D4⋊C8C2semidirect
157. D4⋊C8D4semidirect
589. C4#1488semidirect

Maximal subgroups

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

subgroupindex
C4×#5472
#227992
#212642
C2×#14882
#96732
#96372
#90752
#89412
#85632
#83122
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