Finite Group Databasebeta

← #224050

Subgroups — #224050

Order 512.

1048 conjugacy classes of subgroups.

The subgroup lattice has 599 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2423C211yesmaximal, characteristic
2#22510C211yesmaximal, characteristic, complemented
3Q8×C22⋊C8C211yesmaximal, characteristic, complemented
4#9881C212yesmaximal, complemented
5#9880C212yesmaximal, complemented
6#9879C211yesmaximal, characteristic, complemented
7#9879C211yesmaximal, characteristic, complemented
8#8561C212yesmaximal
9#8557C211yesmaximal, characteristic
10#8155C212yesmaximal
11C4×#323C211yesmaximal, characteristic
12C22×C22⋊Q8C411yescharacteristic
13C22×C4×Q8C2211yescharacteristic
14C2×#446—21no—
15C2×C8×Q8—21no—
16#2423—21no—
17#2423—21no—
18#2423—21no—
19#2423—21no—
20#2423—22no—
21#2423—22no—
22C2×#399C412yes—
23C2×#398C412yes—
24C2×#391C411yescharacteristic
25C2×#389C2211yescharacteristic
26C2×#388C2212yes—
27C2×#387C2212yes—
28C2×#384C411yescharacteristic
29C2×#380C411yescharacteristic
30C2×C4×C22⋊C4C2211yescharacteristic
31#1985—22no—
32#1984—21no—
33#1984—22no—
34#1984—21no—
35#1797C2211yescharacteristic
36#1797C2211yescharacteristic
37#1797C2212yes—
38#1797C2212yes—
39#1749C2212yes—
40C4×C22⋊C8C2211yescharacteristic
41C4×C22⋊C8C2211yescharacteristic
42C4×C22⋊C8C2212yes—
43C2×#323C2212yes—
44C2×#323C2211yescharacteristic
45C2×#323C2211yescharacteristic
46#1280C2212yes—
47#1280C2212yes—
48#1280C2212yes—
49#1280C2212yes—
50#1280C2212yes—
51#1280C2211yescharacteristic
52#1280C2211yescharacteristic
53C23×Q8C2×C411yescharacteristic
54C24×C4C2×C411yescharacteristic
55C2×C22⋊Q8—21no—
56C2×C22⋊Q8—22no—
57C2×C22⋊Q8—21no—
58C2×C22⋊Q8—44no—
59C2×C4×Q8—44no—
60C2×C4×Q8D411yescharacteristic
61C2×C4×Q8D411yescharacteristic
62C2×C4×Q8D411yescharacteristic
63C2×C4×Q8D411yescharacteristic
64C2×C4×Q8D412yes—
65C2×C4×Q8D412yes—
66C22×C4⋊C4C2×C411yescharacteristic
67C22×C4⋊C4C2×C412yes—
68C22×C4⋊C4C2×C412yes—
69C22×C4⋊C4C2×C411yescharacteristic
70C22×C4⋊C4C2×C412yes—
71C22×C4⋊C4C2×C412yes—
72C22×C4⋊C4C2×C412yes—
73C22×C4⋊C4C2×C411yescharacteristic
74C22×C22⋊C4C2×C411yescharacteristic
75C22×C22⋊C4C2×C412yes—
76C22×C22⋊C4C2×C412yes—
77C22×C22⋊C4C2311yescharacteristic
78C22×C42C2×C411yescharacteristic
79C22×C42C2311yescharacteristic
80C22×C42C2312yes—
81#446—44no—
82C8×Q8—44no—
83C2×C4⋊C8—21no—
84C2×C4⋊C8—22no—
85C2×C4⋊C8—21no—
86C2×C4⋊C8—22no—
87C2×C22⋊C8C2312yes—
88C2×C22⋊C8C2311yescharacteristic
89C2×C22⋊C8C2311yescharacteristic
90C2×C22⋊C8C2312yes—
91C2×C22⋊C8C2311yescharacteristic
92C2×C22⋊C8C2311yescharacteristic
93C2×#97—22no—
94C2×C4×C8—21no—
95C2×C4×C8—22no—
96C2×C4×C8—21no—
97#399—22no—
98#399—22no—
99#399—22no—
100#399—22no—
101#398—22no—
102#398—22no—
103#398—22no—
104#398—22no—
105#391—21no—
106#391—22no—
107#391—21no—
108#389—21no—
109#389—21no—
110#389—21no—
111#389—21no—
112#388—22no—
113#388—22no—
114#388—22no—
115#388—22no—
116#387—22no—
117#387—22no—
118#387—22no—
119#387—22no—
120#384—21no—
121#384—21no—
122#384—22no—
123#380—21no—
124#380—21no—
125#380—22no—
126C4×C22⋊C4—21no—
127C4×C22⋊C4—21no—
128C4×C22⋊C4—21no—
129C4×C22⋊C4—21no—
130C2×#95C2×C411yescharacteristic
131C2×#95C2×C412yes—
132C2×#95C2×C412yes—
133C2×#95C2×C412yes—
134C2×#95C2311yescharacteristic
135C2×#95C2312yes—
136#336—21no—
137#336—22no—
138#336—22no—
139#336—21no—
140#336—21no—
141#336—22no—
142#336—22no—
143#336—21no—
144#323—24no—
145#323—22no—
146#323Q814yescomplemented
147C25C22×C411yescharacteristic
148C22×Q8—24no—
149C22×Q8—22no—
150C22×Q8—22no—
151C22×Q8—42no—
152C22×Q8C22⋊C411yescharacteristic
153C22×Q8C22⋊C412yes—
154C22×Q8C22⋊C411yescharacteristic
155C23×C4—21no—
156C23×C4—21no—
157C23×C4—22no—
158C23×C4—22no—
159C23×C4—42no—
160C23×C4—42no—
161C23×C4C22×C412yes—
162C23×C4C22×C411yescharacteristic
163C23×C4C22×C412yes—
164C23×C4C22×C412yes—
165C23×C4C22×C411yescharacteristic
166C23×C4C22×C411yescharacteristic
167C23×C4C22×C412yes—
168C23×C4C22×C411yescharacteristic
169C23×C4C22×C411yescharacteristic
170C23×C4C2411yescharacteristic
171C22×C8—21no—
172C22×C8—21no—
173C22×C8—22no—
174C22×C8—21no—
175C22×C8—21no—
176C22×C8—22no—
177C22⋊Q8—44no—
178C22⋊Q8—48no—
179C22⋊Q8—44no—
180C4×Q8—24no—
181C4×Q8—24no—
182C4×Q8—44no—
183C4×Q8—42no—
184C4×Q8—42no—
185C4×Q8—44no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—24no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—42no—
202C2×C4⋊C4—44no—
203C2×C4⋊C4—42no—
204C2×C4⋊C4—42no—
205C2×C4⋊C4—44no—
206C2×C4⋊C4—82no—
207C2×C4⋊C4—82no—
208C2×C4⋊C4C22⋊C411yescharacteristic
209C2×C4⋊C4C22⋊C411yescharacteristic
210C2×C4⋊C4C22⋊C412yes—
211C2×C4⋊C4C22⋊C412yes—
212C2×C4⋊C4C22⋊C412yes—
213C2×C4⋊C4C22⋊C412yes—
214C2×C4⋊C4C22⋊C412yes—
215C2×C22⋊C4—21no—
216C2×C22⋊C4—21no—
217C2×C22⋊C4—21no—
218C2×C22⋊C4—21no—
219C2×C22⋊C4—22no—
220C2×C22⋊C4—22no—
221C2×C22⋊C4—21no—
222C2×C22⋊C4—21no—
223C2×C22⋊C4—22no—
224C2×C22⋊C4—22no—
225C2×C22⋊C4—22no—
226C2×C22⋊C4—41no—
227C2×C22⋊C4—44no—
228C2×C22⋊C4—82no—
229C2×C22⋊C4C2×Q812yes—
230C2×C22⋊C4#4112yes—
231C2×C42—21no—
232C2×C42—21no—
233C2×C42—42no—
234C2×C42—44no—
235C2×C42—42no—
236C2×C42C2×D411yescharacteristic
237C2×C42C2×D411yescharacteristic
238C2×C42C2×D412yes—
239C2×C42C2×D412yes—
240C2×C42C2×D411yescharacteristic
241C2×C42C2×D411yescharacteristic
242C2×C42C2×D412yes—
243C2×C42C2×D412yes—
244C4⋊C8—44no—
245C4⋊C8—42no—
246C4⋊C8—42no—
247C4⋊C8—44no—
248C22⋊C8—21no—
249C22⋊C8—21no—
250C22⋊C8—22no—
251C22⋊C8—21no—
252C22⋊C8—21no—
253C22⋊C8—22no—
254C22⋊C8—22no—
255C22⋊C8—21no—
256C22⋊C8—21no—
257C22⋊C8—22no—
258C22⋊C8C2×Q812yes—
259C22⋊C8C2×Q812yes—
260#97—44no—
261C4×C8—44no—
262C4×C8—42no—
263C4×C8—42no—
264#95—21no—
265#95—21no—
266#95—22no—
267#95—22no—
268#95—22no—
269#95—22no—
270#95—22no—
271#95—22no—
272#95—21no—
273#95—21no—
274#95—22no—
275#95—22no—
276C24—21no—
277C24—21no—
278C24—21no—
279C24—21no—
280C24—21no—
281C24—21no—
282C24—41no—
283C24—41no—
284C24—41no—
285C24—41no—
286C24C4×Q812yes—
287C24C23×C411yescharacteristic
288C2×Q8—22no—
289C2×Q8—24no—
290C2×Q8—22no—
291C2×Q8—24no—
292C2×Q8—22no—
293C2×Q8—22no—
294C2×Q8—22no—
295C2×Q8—42no—
296C2×Q8—48no—
297C2×Q8—48no—
298C2×Q8#9914yes—
299C22×C4—22no—
300C22×C4—22no—
301C22×C4—21no—
302C22×C4—21no—
303C22×C4—21no—
304C22×C4—22no—
305C22×C4—22no—
306C22×C4—21no—
307C22×C4—21no—
308C22×C4—22no—
309C22×C4—21no—
310C22×C4—22no—
311C22×C4—24no—
312C22×C4—22no—
313C22×C4—21no—
314C22×C4—22no—
315C22×C4—22no—
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—21no—
319C22×C4—21no—
320C22×C4—22no—
321C22×C4—41no—
322C22×C4—42no—
323C22×C4—41no—
324C22×C4—41no—
325C22×C4—42no—
326C22×C4—42no—
327C22×C4—41no—
328C22×C4—42no—
329C22×C4—42no—
330C22×C4—41no—
331C22×C4—41no—
332C22×C4—41no—
333C22×C4—41no—
334C22×C4—42no—
335C22×C4—42no—
336C22×C4—42no—
337C22×C4—42no—
338C22×C4—42no—
339C22×C4—42no—
340C22×C4—41no—
341C22×C4—42no—
342C22×C4—41no—
343C22×C4—42no—
344C22×C4—84no—
345C22×C4—84no—
346C22×C4C2×C22⋊C412yes—
347C22×C4C2×C22⋊C411yescharacteristic
348C22×C4C2×C22⋊C411yescharacteristic
349C22×C4C2×C22⋊C412yes—
350C22×C4C2×C22⋊C412yes—
351C22×C4C2×C22⋊C411yescharacteristic
352C22×C4C2×C22⋊C412yes—
353C22×C4C2×C22⋊C411yescharacteristic
354C22×C4C4×Q812yes—
355C22×C4#13112yes—
356C22×C4#13112yes—
357C22×C4#13112yes—
358C22×C4C22×D411yescharacteristic
359C22×C4C22×D411yescharacteristic
360C22×C4C22×Q811yescharacteristic
361C22×C4C2×#4111yescharacteristic
362C22×C42−1+411yescharacteristic
363C22×C42−1+411yescharacteristic
364C2×C8—22no—
365C2×C8—22no—
366C2×C8—41no—
367C2×C8—41no—
368C2×C8—41no—
369C2×C8—41no—
370C2×C8—42no—
371C2×C8—41no—
372C2×C8—41no—
373C2×C8—42no—
374C2×C8—42no—
375C2×C8—41no—
376C2×C8—41no—
377C2×C8—42no—
378C2×C8—42no—
379C2×C8—41no—
380C2×C8—41no—
381C2×C8—42no—
382C4⋊C4—24no—
383C4⋊C4—24no—
384C4⋊C4—22no—
385C4⋊C4—22no—
386C4⋊C4—42no—
387C4⋊C4—42no—
388C4⋊C4—42no—
389C4⋊C4—44no—
390C4⋊C4—44no—
391C4⋊C4—42no—
392C4⋊C4—44no—
393C4⋊C4—42no—
394C4⋊C4—42no—
395C4⋊C4—42no—
396C4⋊C4—44no—
397C4⋊C4—82no—
398C4⋊C4—82no—
399C4⋊C4—82no—
400C4⋊C4—82no—
401C22⋊C4—22no—
402C22⋊C4—22no—
403C22⋊C4—22no—
404C22⋊C4—22no—
405C22⋊C4—42no—
406C22⋊C4—42no—
407C22⋊C4—44no—
408C22⋊C4—44no—
409C22⋊C4—82no—
410C22⋊C4—82no—
411C42—24no—
412C42—22no—
413C42—22no—
414C42—24no—
415C42—41no—
416C42—41no—
417C42—42no—
418C42—42no—
419C42—42no—
420C42—42no—
421C23—21no—
422C23—21no—
423C23—21no—
424C23—21no—
425C23—22no—
426C23—22no—
427C23—22no—
428C23—42no—
429C23—42no—
430C23—42no—
431C23—42no—
432C23—41no—
433C23—41no—
434C23—41no—
435C23—41no—
436C23—41no—
437C23—41no—
438C23—41no—
439C23—41no—
440C23—81no—
441C23—81no—
442C23—81no—
443C23—81no—
444C23—81no—
445C23—81no—
446C23—81no—
447C23—81no—
448C23C22×C22⋊C411yescharacteristic
449C23C2×C4×Q811yescharacteristic
450C23#51911yescharacteristic
451C23C2×#13111yescharacteristic
452C23C2×#13111yescharacteristic
453C23C2×#13111yescharacteristic
454C23#56811yescharacteristic
455Q8—28no—
456Q8—48no—
457Q8—44no—
458C2×C4—22no—
459C2×C4—21no—
460C2×C4—21no—
461C2×C4—21no—
462C2×C4—21no—
463C2×C4—21no—
464C2×C4—21no—
465C2×C4—22no—
466C2×C4—24no—
467C2×C4—24no—
468C2×C4—22no—
469C2×C4—22no—
470C2×C4—22no—
471C2×C4—21no—
472C2×C4—21no—
473C2×C4—22no—
474C2×C4—22no—
475C2×C4—21no—
476C2×C4—21no—
477C2×C4—22no—
478C2×C4—22no—
479C2×C4—22no—
480C2×C4—22no—
481C2×C4—22no—
482C2×C4—22no—
483C2×C4—21no—
484C2×C4—21no—
485C2×C4—41no—
486C2×C4—41no—
487C2×C4—41no—
488C2×C4—41no—
489C2×C4—44no—
490C2×C4—44no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—41no—
494C2×C4—41no—
495C2×C4—41no—
496C2×C4—41no—
497C2×C4—42no—
498C2×C4—42no—
499C2×C4—42no—
500C2×C4—41no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—41no—
504C2×C4—42no—
505C2×C4—41no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—84no—
509C2×C4—81no—
510C2×C4—82no—
511C2×C4—81no—
512C2×C4—82no—
513C2×C4—84no—
514C2×C4—82no—
515C2×C4—82no—
516C2×C4—82no—
517C2×C4—82no—
518C2×C4—81no—
519C2×C4—81no—
520C2×C4#40812yes—
521C2×C4#40812yes—
522C2×C4C2×#9914yes—
523C2×C4C2×#9912yes—
524C2×C4#41312yes—
525C2×C4#54811yescharacteristic
526C2×C4#54811yescharacteristic
527C2×C4D4×Q811yescharacteristic
528C2×C4D4×Q811yescharacteristic
529C8—42no—
530C8—42no—
531C8—82no—
532C8—81no—
533C8—81no—
534C8—82no—
535C22—21no—
536C22—21no—
537C22—21no—
538C22—21no—
539C22—21no—
540C22—21no—
541C22—21no—
542C22—21no—
543C22—21no—
544C22—21no—
545C22—21no—
546C22—21no—
547C22—42no—
548C22—42no—
549C22—42no—
550C22—82no—
551C22—82no—
552C22—82no—
553C22—82no—
554C22—81no—
555C22—81no—
556C22—81no—
557C22—81no—
558C22Q8×C22⋊C411yescharacteristic
559C22C2×#40811yescharacteristic
560C22#288011yescharacteristic
561C22C22×#9911yescharacteristic
562C22#288411yescharacteristic
563C22C2×#41311yescharacteristic
564C22#288811yescharacteristic
565C22#296011yescharacteristic
566C22#296211yescharacteristic
567C22#298511yescharacteristic
568C22#298611yescharacteristic
569C4—22no—
570C4—24no—
571C4—21no—
572C4—21no—
573C4—21no—
574C4—21no—
575C4—41no—
576C4—41no—
577C4—42no—
578C4—42no—
579C4—42no—
580C4—42no—
581C4—42no—
582C4—82no—
583C4—81no—
584C4—82no—
585C4#146814yes—
586C2—21no—
587C2—21no—
588C2—21no—
589C2—21no—
590C2—82no—
591C2C2×#146811yescharacteristic
592C2#1739011yescharacteristic
593C2#1741911yescharacteristic
594C2#1742311yescharacteristic
595C2#2250911yescharacteristic
596C2Q8×#9911yescharacteristic
597C2#2251511yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC24C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25C22×C4
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C4×Q8 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #22510C2semidirect
3. Q8×C22⋊C8C2semidirect
4. #9881C2semidirect
5. #9880C2semidirect
6. #9879C2semidirect
7. #9879C2semidirect
146. #323Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#24232
#225102
Q8×C22⋊C82
#98812
#98802
#98792
#85612
#85572
#81552
C4×#3232