Finite Group Databasebeta

← #224379

Subgroups — #224379

Order 512.

1161 conjugacy classes of subgroups.

The subgroup lattice has 616 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1Q8×#121C211yesmaximal, characteristic
2#22510C211yesmaximal, characteristic, complemented
3#13931C211yesmaximal, characteristic, complemented
4#9911C212yesmaximal, complemented
5#9909C211yesmaximal, characteristic, complemented
6#9889C212yesmaximal, complemented
7#9885C211yesmaximal, characteristic, complemented
8#8575C212yesmaximal
9#8571C211yesmaximal, characteristic
10#8216C212yesmaximal
11C4×#327C211yesmaximal, characteristic
12C2×D4×Q8—21no—
13C2×D4×Q8C411yescharacteristic, complemented
14C22×C4×Q8C2211yescharacteristic
15C2×#446—21no—
16#2857C2211yescharacteristic
17#2857C2212yes—
18#2546C411yescharacteristic, complemented
19#2546C412yescomplemented
20#2538C411yescharacteristic, complemented
21#2538C412yescomplemented
22#2429—21no—
23#2429—22no—
24#2417—21no—
25#2417—22no—
26Q8×C4⋊C4C411yescharacteristic, complemented
27Q8×C22⋊C4—21no—
28C4×#121C2211yescharacteristic
29C4×#121C2212yes—
30C2×#391—21no—
31#1985—22no—
32#1984—21no—
33#1797C2211yescharacteristic
34#1797C2212yes—
35#1749C2212yes—
36C4×C22⋊C8C2211yescharacteristic
37C2×#327C2211yescharacteristic
38C2×#327C2212yes—
39C2×#327C2211yescharacteristic
40#1445—22no—
41C4⋊#95—21no—
42C4⋊#95C2211yescharacteristic
43#1434C2212yes—
44#1434C2212yes—
45C4×#95C2211yescharacteristic
46#1294C2211yescharacteristic
47#1294C2211yescharacteristic
48#1294C2212yes—
49#1294C2212yes—
50#1294C2212yes—
51#1294C2212yes—
52#1294C2212yes—
53C23×Q8—21no—
54C23×Q8D411yescharacteristic
55D4×Q8—44no—
56D4×Q8—44no—
57D4×Q8—44no—
58C2×C4⋊Q8—21no—
59C2×C4⋊Q8—22no—
60C2×C4⋊Q8C2×C411yescharacteristic
61C2×C4⋊Q8C2×C412yes—
62C2×C22⋊Q8—21no—
63C2×C22⋊Q8—22no—
64C2×C22⋊Q8—22no—
65C2×C22⋊Q8—21no—
66C2×C22⋊Q8—22no—
67C2×C22⋊Q8—21no—
68C2×#121C2311yescharacteristic
69C2×C4×Q8—21no—
70C2×C4×Q8—44no—
71C2×C4×Q8C2×C411yescharacteristic
72C2×C4×Q8C2×C411yescharacteristic
73C2×C4×Q8C2×C412yes—
74C2×C4×Q8D411yescharacteristic
75C2×C4×Q8D411yescharacteristic
76C2×C4×Q8D412yes—
77C2×C4×D4—22no—
78C2×C4×D4—21no—
79C2×C4×D4C2×C411yescharacteristic
80C2×C4×D4C2×C412yes—
81C22×C4⋊C4—21no—
82C22×C4⋊C4D411yescharacteristic
83C22×C4⋊C4D412yes—
84C22×C42—21no—
85C22×C42C2311yescharacteristic
86C22×C42C2312yes—
87#446—44no—
88C2×C4⋊C8—21no—
89C2×C4⋊C8—22no—
90C2×C22⋊C8C2311yescharacteristic
91C2×C22⋊C8C2312yes—
92C2×C22⋊C8C2311yescharacteristic
93C2×#97—22no—
94C2×C4×C8—21no—
95#398—22no—
96#398—22no—
97#398—22no—
98#396—22no—
99#396—22no—
100#396—22no—
101#391—22no—
102#391—21no—
103#391—21no—
104#391—22no—
105#391—44no—
106#390C2×C411yescharacteristic
107#390C2×C412yes—
108#389—22no—
109#389—21no—
110#389—21no—
111#389—22no—
112#389C2×C412yes—
113#389C2×C412yes—
114#389C2×C412yes—
115#389C2×C411yescharacteristic
116#389C2×C412yes—
117#388—21no—
118#388—22no—
119#386—21no—
120#386—22no—
121#384C2×C411yescharacteristic
122#384C2×C412yes—
123#380—21no—
124#380—22no—
125#380—22no—
126#380—21no—
127#380C2311yescharacteristic
128#380C2312yes—
129C4×C4⋊C4C2×C411yescharacteristic
130C4×C4⋊C4C2×C412yes—
131C4×C22⋊C4—21no—
132C4×C22⋊C4—22no—
133C2×#95—21no—
134C2×#95—22no—
135C2×#95—22no—
136C2×#95—22no—
137C2×#95—22no—
138C2×#95—21no—
139C2×#95C2311yescharacteristic
140C2×#95C2312yes—
141C2×#95C2311yescharacteristic
142#336—22no—
143#336—21no—
144#336—22no—
145#336—21no—
146#327—22no—
147#327—24no—
148#327Q814yescomplemented
149C22×Q8—24no—
150C22×Q8—22no—
151C22×Q8—21no—
152C22×Q8—22no—
153C22×Q8—22no—
154C22×Q8—42no—
155C22×Q8—44no—
156C22×Q8—44no—
157C22×Q8—44no—
158C22×Q8C22⋊C411yescharacteristic
159C22×Q8C22⋊C412yes—
160C22×Q8C22⋊C411yescharacteristic
161C22×D4—21no—
162C22×D4C22×C411yescharacteristic
163C23×C4—21no—
164C23×C4—21no—
165C23×C4—22no—
166C23×C4—21no—
167C23×C4—22no—
168C23×C4C2×D412yes—
169C23×C4C2×D411yescharacteristic
170C23×C4C2×D412yes—
171C23×C4C2×D411yescharacteristic
172C23×C4C2411yescharacteristic
173C22×C8—21no—
174C22×C8—21no—
175C22×C8—22no—
176C4⋊Q8—44no—
177C4⋊Q8—42no—
178C4⋊Q8—48no—
179C4⋊Q8—44no—
180C22⋊Q8—48no—
181C22⋊Q8—44no—
182C22⋊Q8—44no—
183C22⋊Q8—44no—
184C22⋊Q8—44no—
185C22⋊Q8—48no—
186C22⋊Q8—42no—
187C22⋊Q8—42no—
188#121—21no—
189#121—21no—
190#121C2×Q812yes—
191#121#4112yes—
192C4×Q8—24no—
193C4×Q8—24no—
194C4×Q8—44no—
195C4×Q8—42no—
196C4×Q8—42no—
197C4×Q8—44no—
198C4×Q8—44no—
199C4×D4—44no—
200C4×D4—42no—
201C4×D4—44no—
202C4×D4—44no—
203C4×D4—42no—
204C4×D4—42no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—42no—
217C2×C4⋊C4—44no—
218C2×C4⋊C4—42no—
219C2×C4⋊C4—42no—
220C2×C4⋊C4—42no—
221C2×C4⋊C4C22⋊C412yes—
222C2×C4⋊C4C22⋊C411yescharacteristic
223C2×C4⋊C4C22⋊C412yes—
224C2×C4⋊C4C22⋊C411yescharacteristic
225C2×C4⋊C4C22⋊C412yes—
226C2×C4⋊C4C22⋊C412yes—
227C2×C4⋊C4C22⋊C412yes—
228C2×C4⋊C4C22×C411yescharacteristic
229C2×C4⋊C4C22×C411yescharacteristic
230C2×C4⋊C4C22×C412yes—
231C2×C4⋊C4C22×C411yescharacteristic
232C2×C4⋊C4C22×C412yes—
233C2×C22⋊C4—21no—
234C2×C22⋊C4—22no—
235C2×C22⋊C4—21no—
236C2×C22⋊C4—22no—
237C2×C22⋊C4—21no—
238C2×C22⋊C4—21no—
239C2×C22⋊C4—22no—
240C2×C22⋊C4—21no—
241C2×C22⋊C4—22no—
242C2×C42—22no—
243C2×C42—21no—
244C2×C42—42no—
245C2×C42—44no—
246C2×C42—42no—
247C2×C42—42no—
248C2×C42—42no—
249C2×C42C22×C412yes—
250C2×C42C22×C411yescharacteristic
251C2×C42C22×C412yes—
252C2×C42C22×C411yescharacteristic
253C2×C42C2×D411yescharacteristic
254C2×C42C2×D412yes—
255C2×C42C2×D412yes—
256C2×C42C2×D411yescharacteristic
257C4⋊C8—42no—
258C4⋊C8—44no—
259C22⋊C8—21no—
260C22⋊C8—21no—
261C22⋊C8—22no—
262C22⋊C8—21no—
263C22⋊C8—22no—
264C22⋊C8C2×Q812yescomplemented
265#97—44no—
266C4×C8—42no—
267#95—21no—
268#95—22no—
269#95—21no—
270#95—22no—
271#95—22no—
272#95—22no—
273#95—22no—
274#95—21no—
275#95—22no—
276#95—21no—
277#95—21no—
278#95—22no—
279#95—21no—
280#95—22no—
281#95—42no—
282#95—44no—
283#95—44no—
284#95—44no—
285#95—82no—
286#95—81no—
287#95C2×Q812yescomplemented
288C24—21no—
289C24C22×D411yescharacteristic
290C2×Q8—24no—
291C2×Q8—22no—
292C2×Q8—22no—
293C2×Q8—24no—
294C2×Q8—22no—
295C2×Q8—22no—
296C2×Q8—22no—
297C2×Q8—42no—
298C2×Q8—48no—
299C2×Q8—48no—
300C2×Q8—44no—
301C2×Q8—44no—
302C2×Q8—44no—
303C2×Q8—42no—
304C2×Q8—42no—
305C2×Q8—42no—
306C2×Q8—816no—
307C2×Q8#9914yes—
308C2×D4—22no—
309C2×D4—22no—
310C2×D4—21no—
311C2×D4—41no—
312C2×D4—41no—
313C2×D4—41no—
314C2×D4—41no—
315C2×D4—41no—
316C2×D4—41no—
317C2×D4C4×Q812yes—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—21no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—21no—
325C22×C4—22no—
326C22×C4—21no—
327C22×C4—24no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—21no—
331C22×C4—22no—
332C22×C4—21no—
333C22×C4—22no—
334C22×C4—41no—
335C22×C4—42no—
336C22×C4—41no—
337C22×C4—41no—
338C22×C4—42no—
339C22×C4—41no—
340C22×C4—42no—
341C22×C4—44no—
342C22×C4—44no—
343C22×C4—41no—
344C22×C4—41no—
345C22×C4—41no—
346C22×C4—41no—
347C22×C4—41no—
348C22×C4—41no—
349C22×C4—41no—
350C22×C4—41no—
351C22×C4—41no—
352C22×C4—41no—
353C22×C4—41no—
354C22×C4—41no—
355C22×C4—44no—
356C22×C4—42no—
357C22×C4—42no—
358C22×C4—42no—
359C22×C4—42no—
360C22×C4—42no—
361C22×C4—42no—
362C22×C4—42no—
363C22×C4—42no—
364C22×C4—42no—
365C22×C4—42no—
366C22×C4C2×C22⋊C411yescharacteristic
367C22×C4C2×C22⋊C412yes—
368C22×C4C2×C22⋊C412yes—
369C22×C4C2×C22⋊C411yescharacteristic
370C22×C4C2×C22⋊C412yes—
371C22×C4C2×C22⋊C412yes—
372C22×C4C2×C22⋊C411yescharacteristic
373C22×C4C2×C22⋊C411yescharacteristic
374C22×C4AGL(1,ℤ/8)12yes—
375C22×C4AGL(1,ℤ/8)12yes—
376C22×C4C23×C411yescharacteristic
377C22×C4C22×D411yescharacteristic
378C22×C4C22×Q811yescharacteristic
379C22×C4C2×#4111yescharacteristic
380C22×C42−1+411yescharacteristic
381C22×C42−1+411yescharacteristic
382C2×C8—22no—
383C2×C8—41no—
384C2×C8—41no—
385C2×C8—41no—
386C2×C8—42no—
387C2×C8—41no—
388C2×C8—42no—
389C2×C8—41no—
390C2×C8—42no—
391C4⋊C4—21no—
392C4⋊C4—24no—
393C4⋊C4—22no—
394C4⋊C4—24no—
395C4⋊C4—22no—
396C4⋊C4—42no—
397C4⋊C4—42no—
398C4⋊C4—42no—
399C4⋊C4—41no—
400C4⋊C4—42no—
401C4⋊C4—42no—
402C4⋊C4—41no—
403C4⋊C4—44no—
404C4⋊C4—44no—
405C4⋊C4—44no—
406C4⋊C4—42no—
407C4⋊C4—44no—
408C4⋊C4—44no—
409C4⋊C4—42no—
410C4⋊C4—42no—
411C4⋊C4—44no—
412C4⋊C4—81no—
413C4⋊C4—82no—
414C4⋊C4—84no—
415C4⋊C4C4×Q812yes—
416C22⋊C4—22no—
417C22⋊C4—22no—
418C22⋊C4—44no—
419C22⋊C4—42no—
420C22⋊C4—44no—
421C22⋊C4—42no—
422C22⋊C4—41no—
423C22⋊C4—42no—
424C22⋊C4—41no—
425C22⋊C4—42no—
426C22⋊C4—81no—
427C22⋊C4—82no—
428C42—22no—
429C42—24no—
430C42—24no—
431C42—22no—
432C42—41no—
433C42—42no—
434C42—41no—
435C42—42no—
436C42—44no—
437C42—42no—
438C42—84no—
439C23—21no—
440C23—21no—
441C23—21no—
442C23—21no—
443C23—22no—
444C23—41no—
445C23—41no—
446C23—41no—
447C23—41no—
448C23—41no—
449C23—41no—
450C23C22×C22⋊C411yescharacteristic
451C23#54811yescharacteristic
452C23D4×Q811yescharacteristic
453C23C2×AGL(1,ℤ/8)11yescharacteristic
454C23C2×AGL(1,ℤ/8)11yescharacteristic
455C23#57711yescharacteristic
456C23#57811yescharacteristic
457Q8—28no—
458Q8—48no—
459Q8—44no—
460Q8—88no—
461Q8—84no—
462D4—42no—
463D4—42no—
464D4—42no—
465D4—42no—
466D4—42no—
467D4—42no—
468C2×C4—21no—
469C2×C4—21no—
470C2×C4—21no—
471C2×C4—21no—
472C2×C4—21no—
473C2×C4—22no—
474C2×C4—21no—
475C2×C4—22no—
476C2×C4—24no—
477C2×C4—24no—
478C2×C4—22no—
479C2×C4—22no—
480C2×C4—21no—
481C2×C4—21no—
482C2×C4—22no—
483C2×C4—22no—
484C2×C4—22no—
485C2×C4—21no—
486C2×C4—22no—
487C2×C4—22no—
488C2×C4—21no—
489C2×C4—22no—
490C2×C4—21no—
491C2×C4—22no—
492C2×C4—21no—
493C2×C4—22no—
494C2×C4—41no—
495C2×C4—41no—
496C2×C4—41no—
497C2×C4—41no—
498C2×C4—44no—
499C2×C4—44no—
500C2×C4—41no—
501C2×C4—41no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—41no—
510C2×C4—42no—
511C2×C4—41no—
512C2×C4—42no—
513C2×C4—41no—
514C2×C4—42no—
515C2×C4—41no—
516C2×C4—42no—
517C2×C4—41no—
518C2×C4—42no—
519C2×C4—41no—
520C2×C4—88no—
521C2×C4—81no—
522C2×C4—82no—
523C2×C4—81no—
524C2×C4—82no—
525C2×C4—81no—
526C2×C4—81no—
527C2×C4—81no—
528C2×C4—81no—
529C2×C4—81no—
530C2×C4—81no—
531C2×C4—81no—
532C2×C4—81no—
533C2×C4—84no—
534C2×C4—82no—
535C2×C4—82no—
536C2×C4—82no—
537C2×C4—82no—
538C2×C4—82no—
539C2×C4—82no—
540C2×C4—82no—
541C2×C4C2×#9914yes—
542C2×C4C2×#9912yes—
543C2×C4#41812yes—
544C2×C4#41812yes—
545C2×C4#42112yes—
546C2×C4C2×C4×Q811yescharacteristic
547C2×C4#51911yescharacteristic
548C2×C4#54811yescharacteristic
549C2×C4D4×Q811yescharacteristic
550C8—42no—
551C8—81no—
552C8—82no—
553C22—21no—
554C22—21no—
555C22—21no—
556C22—21no—
557C22—21no—
558C22—21no—
559C22—21no—
560C22—21no—
561C22—21no—
562C22—21no—
563C22—21no—
564C22—21no—
565C22—42no—
566C22—42no—
567C22—42no—
568C22—81no—
569C22—81no—
570C22—81no—
571C22—81no—
572C22Q8×C22⋊C411yescharacteristic
573C22C22×#9911yescharacteristic
574C22#288411yescharacteristic
575C22C2×#41811yescharacteristic
576C22#289811yescharacteristic
577C22C2×#42111yescharacteristic
578C22#290111yescharacteristic
579C22#336711yescharacteristic
580C22#337311yescharacteristic
581C22#338411yescharacteristic
582C22#338611yescharacteristic
583C4—22no—
584C4—24no—
585C4—21no—
586C4—21no—
587C4—21no—
588C4—21no—
589C4—42no—
590C4—42no—
591C4—41no—
592C4—41no—
593C4—42no—
594C4—42no—
595C4—82no—
596C4—82no—
597C4—82no—
598C4—81no—
599C4—82no—
600C4—81no—
601C4—162no—
602C4#151014yes—
603C2—21no—
604C2—21no—
605C2—21no—
606C2—21no—
607C2—82no—
608C2C2×#151011yescharacteristic
609C2#1760111yescharacteristic
610C2#1760611yescharacteristic
611C2#1763111yescharacteristic
612C2Q8×#9911yescharacteristic
613C2#2252311yescharacteristic
614C2#2252911yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC2×#121C23
LCSG ▷ C22×C4 ▷ C22 ▷ 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. #13931C2semidirect
4. #9911C2semidirect
5. #9909C2semidirect
6. #9889C2semidirect
7. #9885C2semidirect
13. C2×D4×Q8C4semidirect
18. #2546C4semidirect
19. #2546C4semidirect
20. #2538C4semidirect
21. #2538C4semidirect
26. Q8×C4⋊C4C4semidirect
148. #327Q8semidirect
264. C22⋊C8C2×Q8semidirect
287. #95C2×Q8semidirect

Maximal subgroups

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

subgroupindex
Q8×#1212
#225102
#139312
#99112
#99092
#98892
#98852
#85752
#85712
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C4×#3272