Finite Group Databasebeta

← #231128

Subgroups — #231128

Order 512.

743 conjugacy classes of subgroups.

The subgroup lattice has 613 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23696C211yesmaximal, characteristic
2#22990C211yesmaximal, characteristic, complemented
3#22985C211yesmaximal, characteristic, complemented
4#9558C211yesmaximal, characteristic, complemented
5#9554C211yesmaximal, characteristic, complemented
6#9533C211yesmaximal, characteristic, complemented
7#9533C211yesmaximal, characteristic, complemented
8#8951C211yesmaximal, characteristic, complemented
9#8947C211yesmaximal, characteristic, complemented
10#8944C211yesmaximal, characteristic, complemented
11#8939C211yesmaximal, characteristic, complemented
12#8210C211yesmaximal, characteristic, complemented
13C8×#102C211yesmaximal, characteristic, complemented
14#8148C211yesmaximal, characteristic, complemented
15#7877C211yesmaximal, characteristic, complemented
16C2×C4×D8C2211yescharacteristic
17C2×#436—21no—
18C2×#436C2211yescharacteristic
19C2×C8×D4—21no—
20C2×C8×D4C2211yescharacteristic
21C2×#431C2211yescharacteristic
22C2×#423C2211yescharacteristic
23C22×C4⋊C8C2211yescharacteristic
24C4×#121C2211yescharacteristic
25C4×#121C2211yescharacteristic
26#2086C411yescharacteristic
27#2075C411yescharacteristic
28#2067C411yescharacteristic
29#2067C411yescharacteristic
30#2054C411yescharacteristic
31#2033C411yescharacteristic
32#2033C411yescharacteristic
33#1999C411yescharacteristic
34#1954C2211yescharacteristic
35#1954C2211yescharacteristic
36#1939C2211yescharacteristic
37#1939C2211yescharacteristic
38#1923—21no—
39#1923—21no—
40#1870—21no—
41#1869—21no—
42#1851—21no—
43#1850—21no—
44C4×#107C2211yescharacteristic
45C8×C4⋊C4C2211yescharacteristic
46C8×C4⋊C4C2211yescharacteristic
47C4×#102C2211yescharacteristic
48C4×#102C2211yescharacteristic
49C4×C22⋊C8C2211yescharacteristic
50C4×C22⋊C8C2211yescharacteristic
51C2×D4⋊C8C2211yescharacteristic
52C2×D4⋊C8C2211yescharacteristic
53C2×D4⋊C8C2211yescharacteristic
54C2×D4⋊C8C2211yescharacteristic
55C2×#322C2211yescharacteristic
56C2×C82C2211yescharacteristic
57#1279C2211yescharacteristic
58#1279C2211yescharacteristic
59#1279C2211yescharacteristic
60#1279C2211yescharacteristic
61#1277C2211yescharacteristic
62#1277C2211yescharacteristic
63#1277C2211yescharacteristic
64#1277C2211yescharacteristic
65#1273C2211yescharacteristic
66#1273C2211yescharacteristic
67C22×D8C2×C411yescharacteristic
68C22×M16—21no—
69C22×M16—21no—
70C23×C8—21no—
71C23×C8—21no—
72C2×#121C2×C411yescharacteristic
73C2×#121C2×C411yescharacteristic
74C2×C4×D4—21no—
75C2×C4×D4—21no—
76C2×C4×D4C2311yescharacteristic
77C2×C4×D4C2311yescharacteristic
78C22×C42C2311yescharacteristic
79C4×D8—41no—
80C4×D8—41no—
81#436—22no—
82#436—41no—
83#436—44no—
84C8×D4—22no—
85C8×D4—41no—
86C8×D4—44no—
87#431—41no—
88#431—41no—
89C2×#107C2×C411yescharacteristic
90C2×#107C2×C411yescharacteristic
91C2×#107C2×C411yescharacteristic
92#423—22no—
93#423—41no—
94C2×C4⋊C8—22no—
95C2×C4⋊C8—21no—
96C2×C4⋊C8—21no—
97C2×C4⋊C8—21no—
98C2×C4⋊C8—21no—
99C2×C4⋊C8—41no—
100C2×C4⋊C8C2×C411yescharacteristic
101C2×C4⋊C8C2×C411yescharacteristic
102C2×C4⋊C8C2311yescharacteristic
103C2×C4⋊C8C2311yescharacteristic
104C2×C4⋊C8C2311yescharacteristic
105C2×C4⋊C8C2311yescharacteristic
106C2×#102C2×C411yescharacteristic
107C2×#102C2×C411yescharacteristic
108C2×#102C2×C411yescharacteristic
109C2×#102C2×C411yescharacteristic
110C2×#102C2×C411yescharacteristic
111C2×#102C2×C411yescharacteristic
112C2×C22⋊C8—21no—
113C2×C22⋊C8—21no—
114C2×C22⋊C8—21no—
115C2×C22⋊C8—21no—
116C2×C22⋊C8—21no—
117C2×C22⋊C8—21no—
118C2×C22⋊C8—21no—
119C2×C22⋊C8C2×C411yescharacteristic
120C2×C22⋊C8C2×C411yescharacteristic
121C2×C22⋊C8D411yescharacteristic
122C2×C22⋊C8D411yescharacteristic
123C2×C4×C8—21no—
124C2×C4×C8—21no—
125C2×C4×C8C2311yescharacteristic
126C2×C4×C8C2311yescharacteristic
127C2×C4×C8C2311yescharacteristic
128C2×C4×C8C2311yescharacteristic
129C2×C4×C8C2311yescharacteristic
130C2×C4×C8C2311yescharacteristic
131#390C2×C411yescharacteristic
132#390C2×C411yescharacteristic
133#389C2×C411yescharacteristic
134#389C2×C411yescharacteristic
135#388—21no—
136#388—21no—
137#386—21no—
138#386—21no—
139#380C2×C411yescharacteristic
140#380C2×C411yescharacteristic
141C4×C4⋊C4C2311yescharacteristic
142C4×C4⋊C4C2311yescharacteristic
143C4×C22⋊C4—21no—
144C4×C22⋊C4—21no—
145#340C2×C411yescharacteristic
146#340C2×C411yescharacteristic
147#340C2×C411yescharacteristic
148#340C2×C411yescharacteristic
149#340C2×C411yescharacteristic
150#340C2×C411yescharacteristic
151#336—21no—
152#336—21no—
153#336—21no—
154#336—21no—
155#336—21no—
156#336D411yescharacteristic
157#336D411yescharacteristic
158D4⋊C8—22no—
159D4⋊C8—22no—
160D4⋊C8—41no—
161D4⋊C8—41no—
162#322—41no—
163C82—41no—
164C22×D4—21no—
165C22×D4—21no—
166C22×D4C22×C411yescharacteristic
167C22×D4C22×C411yescharacteristic
168C23×C4—21no—
169C23×C4—21no—
170C23×C4C22×C411yescharacteristic
171C23×C4C2×D411yescharacteristic
172C23×C4#4111yescharacteristic
173C2×D8—22no—
174C2×D8—41no—
175C2×D8—41no—
176C2×M16—42no—
177C2×M16—42no—
178C2×M16—42no—
179C2×M16—42no—
180C2×M16—81no—
181C2×M16—81no—
182C22×C8—21no—
183C22×C8—21no—
184C22×C8—21no—
185C22×C8—21no—
186C22×C8—21no—
187C22×C8—21no—
188C22×C8—42no—
189C22×C8—42no—
190C22×C8—42no—
191C22×C8—42no—
192C22×C8—81no—
193C22×C8—81no—
194C22×C8C22×C411yescharacteristic
195C22×C8C22×C411yescharacteristic
196C22×C8C22×C411yescharacteristic
197C22×C8C22×C411yescharacteristic
198C22×C8C2×D411yescharacteristic
199C22×C8C2×D411yescharacteristic
200C22×C8C2×D411yescharacteristic
201C22×C8C2×D411yescharacteristic
202#121—22no—
203#121—22no—
204#121—22no—
205#121—41no—
206C4×D4—21no—
207C4×D4—21no—
208C4×D4—21no—
209C4×D4—21no—
210C4×D4—21no—
211C4×D4—21no—
212C4×D4—21no—
213C4×D4—21no—
214C4×D4—42no—
215C4×D4—42no—
216C4×D4—42no—
217C4×D4—42no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4C22×C411yescharacteristic
221C2×C4⋊C4C22×C411yescharacteristic
222C2×C4⋊C4C22×C411yescharacteristic
223C2×C4⋊C4C22×C411yescharacteristic
224C2×C4⋊C4C22×C411yescharacteristic
225C2×C4⋊C4C22×C411yescharacteristic
226C2×C22⋊C4—21no—
227C2×C22⋊C4—21no—
228C2×C22⋊C4—21no—
229C2×C22⋊C4—21no—
230C2×C22⋊C4—21no—
231C2×C22⋊C4—21no—
232C2×C22⋊C4—21no—
233C2×C22⋊C4—21no—
234C2×C22⋊C4—21no—
235C2×C22⋊C4—21no—
236C2×C42—21no—
237C2×C42—21no—
238C2×C42—21no—
239C2×C42—21no—
240C2×C42—21no—
241C2×C42—21no—
242C2×C42C22×C411yescharacteristic
243C2×C42C2×D411yescharacteristic
244C2×C42#4111yescharacteristic
245C2×C42C2411yescharacteristic
246#107—22no—
247#107—22no—
248#107—41no—
249C4⋊C8—21no—
250C4⋊C8—21no—
251C4⋊C8—21no—
252C4⋊C8—21no—
253C4⋊C8—21no—
254C4⋊C8—21no—
255C4⋊C8—41no—
256C4⋊C8—41no—
257C4⋊C8—42no—
258C4⋊C8—42no—
259C4⋊C8—42no—
260C4⋊C8—42no—
261C4⋊C8#4112yes—
262C4⋊C8#4112yes—
263#102—22no—
264#102—22no—
265#102—22no—
266#102—41no—
267#102—41no—
268#102—41no—
269C22⋊C8—22no—
270C22⋊C8—41no—
271C22⋊C8—41no—
272C22⋊C8—42no—
273C22⋊C8—42no—
274C22⋊C8—42no—
275C22⋊C8—42no—
276C22⋊C8—42no—
277C22⋊C8—81no—
278C22⋊C8—81no—
279C22⋊C8D814yescomplemented
280C4×C8—21no—
281C4×C8—21no—
282C4×C8—21no—
283C4×C8—21no—
284C4×C8—21no—
285C4×C8—21no—
286C4×C8—21no—
287C4×C8—21no—
288C4×C8—21no—
289C4×C8—21no—
290C4×C8—21no—
291C4×C8—21no—
292C4×C8—42no—
293C4×C8—42no—
294#95—21no—
295#95—21no—
296#95—21no—
297#95—21no—
298#95—21no—
299#95—21no—
300C24—21no—
301C24—21no—
302C24C4×D411yescharacteristic
303C2×D4—24no—
304C2×D4—21no—
305C2×D4—21no—
306C2×D4—21no—
307C2×D4—21no—
308C2×D4—21no—
309C2×D4—21no—
310C2×D4—21no—
311C2×D4—21no—
312C2×D4—21no—
313C2×D4—21no—
314C2×D4—21no—
315C2×D4—42no—
316C2×D4—42no—
317C2×D4—42no—
318C2×D4—42no—
319C2×D4—81no—
320C2×D4#13112yes—
321C22×C4—21no—
322C22×C4—21no—
323C22×C4—21no—
324C22×C4—21no—
325C22×C4—21no—
326C22×C4—21no—
327C22×C4—21no—
328C22×C4—21no—
329C22×C4—21no—
330C22×C4—21no—
331C22×C4—21no—
332C22×C4—21no—
333C22×C4—21no—
334C22×C4—21no—
335C22×C4—21no—
336C22×C4—21no—
337C22×C4—21no—
338C22×C4—21no—
339C22×C4—41no—
340C22×C4—41no—
341C22×C4—41no—
342C22×C4—41no—
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×C4C4×D411yescharacteristic
354C22×C4C4×D411yescharacteristic
355C22×C4C4×D411yescharacteristic
356C22×C4C2×D812yes—
357C22×C4#13512yes—
358C22×C4C23×C411yescharacteristic
359C22×C4C22×D411yescharacteristic
360C22×C4C2×#4111yescharacteristic
361D8—44no—
362M16—84no—
363M16—84no—
364C2×C8—21no—
365C2×C8—21no—
366C2×C8—21no—
367C2×C8—21no—
368C2×C8—22no—
369C2×C8—21no—
370C2×C8—21no—
371C2×C8—21no—
372C2×C8—21no—
373C2×C8—21no—
374C2×C8—21no—
375C2×C8—22no—
376C2×C8—22no—
377C2×C8—21no—
378C2×C8—21no—
379C2×C8—21no—
380C2×C8—21no—
381C2×C8—21no—
382C2×C8—21no—
383C2×C8—21no—
384C2×C8—21no—
385C2×C8—21no—
386C2×C8—21no—
387C2×C8—21no—
388C2×C8—21no—
389C2×C8—41no—
390C2×C8—41no—
391C2×C8—41no—
392C2×C8—41no—
393C2×C8—41no—
394C2×C8—41no—
395C2×C8—41no—
396C2×C8—41no—
397C2×C8—41no—
398C2×C8—41no—
399C2×C8—41no—
400C2×C8—41no—
401C2×C8—41no—
402C2×C8—41no—
403C2×C8—41no—
404C2×C8—84no—
405C2×C8—84no—
406C2×C8C2×D812yes—
407C2×C8C2×D812yes—
408C4⋊C4—21no—
409C4⋊C4—21no—
410C4⋊C4—21no—
411C4⋊C4—21no—
412C4⋊C4—21no—
413C4⋊C4—21no—
414C4⋊C4—21no—
415C4⋊C4—21no—
416C4⋊C4—21no—
417C4⋊C4—21no—
418C4⋊C4—42no—
419C4⋊C4—42no—
420C4⋊C4#13112yes—
421C4⋊C4#13112yes—
422C22⋊C4—24no—
423C22⋊C4—42no—
424C22⋊C4—42no—
425C22⋊C4—42no—
426C22⋊C4—42no—
427C22⋊C4—41no—
428C22⋊C4—41no—
429C22⋊C4—41no—
430C22⋊C4—41no—
431C22⋊C4—42no—
432C22⋊C4—42no—
433C22⋊C4—81no—
434C42—21no—
435C42—21no—
436C42—21no—
437C42—21no—
438C42—21no—
439C42—21no—
440C42—41no—
441C42—41no—
442C42—41no—
443C42—41no—
444C42C2×#4111yescharacteristic
445C42C2×#4111yescharacteristic
446C42H(5,𝔽2)11yescharacteristic
447C42H(5,𝔽2)11yescharacteristic
448C23—21no—
449C23—21no—
450C23—21no—
451C23—21no—
452C23—21no—
453C23—21no—
454C23—22no—
455C23—41no—
456C23—41no—
457C23—41no—
458C23—41no—
459C23—41no—
460C23—41no—
461C23—41no—
462C23—41no—
463C23—41no—
464C23—41no—
465C23—41no—
466C23—41no—
467C23—41no—
468C23C4×D812yes—
469C23C2×C4×D411yescharacteristic
470D4—22no—
471D4—22no—
472D4—48no—
473D4—41no—
474D4—41no—
475D4—41no—
476D4—41no—
477D4—41no—
478D4—41no—
479D4—84no—
480C2×C4—21no—
481C2×C4—21no—
482C2×C4—21no—
483C2×C4—21no—
484C2×C4—22no—
485C2×C4—22no—
486C2×C4—22no—
487C2×C4—22no—
488C2×C4—21no—
489C2×C4—21no—
490C2×C4—21no—
491C2×C4—22no—
492C2×C4—21no—
493C2×C4—22no—
494C2×C4—21no—
495C2×C4—21no—
496C2×C4—41no—
497C2×C4—41no—
498C2×C4—41no—
499C2×C4—41no—
500C2×C4—41no—
501C2×C4—41no—
502C2×C4—41no—
503C2×C4—41no—
504C2×C4—41no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—81no—
517C2×C4—81no—
518C2×C4—81no—
519C2×C4—81no—
520C2×C4—81no—
521C2×C4—81no—
522C2×C4—81no—
523C2×C4—81no—
524C2×C4—81no—
525C2×C4—81no—
526C2×C4C4×D812yes—
527C2×C4C4×#4111yescharacteristic
528C2×C4#51811yescharacteristic
529C2×C4#54611yescharacteristic
530C2×C4#54611yescharacteristic
531C2×C4#55311yescharacteristic
532C2×C4#55311yescharacteristic
533C2×C4C2×#13111yescharacteristic
534C2×C4C2×#13111yescharacteristic
535C2×C4C2×#13111yescharacteristic
536C2×C4#56811yescharacteristic
537C2×C4C22×D811yescharacteristic
538C2×C4C2×#13511yescharacteristic
539C2×C4#57711yescharacteristic
540C2×C4#57711yescharacteristic
541C8—22no—
542C8—22no—
543C8—42no—
544C8—41no—
545C8—41no—
546C8—41no—
547C8—41no—
548C8—42no—
549C8—42no—
550C8—41no—
551C8—41no—
552C8—81no—
553C8—81no—
554C8—81no—
555C22—22no—
556C22—22no—
557C22—22no—
558C22—41no—
559C22—41no—
560C22—41no—
561C22—41no—
562C22—42no—
563C22—42no—
564C22—42no—
565C22—81no—
566C22—81no—
567C22—81no—
568C22—81no—
569C22—81no—
570C22—81no—
571C22—81no—
572C22—81no—
573C22—81no—
574C22—81no—
575C22—81no—
576C22#236411yescharacteristic
577C22#292911yescharacteristic
578C22#293311yescharacteristic
579C22C2×C4×D811yescharacteristic
580C22#295211yescharacteristic
581C22#295511yescharacteristic
582C22#295611yescharacteristic
583C4—22no—
584C4—21no—
585C4—21no—
586C4—42no—
587C4—41no—
588C4—41no—
589C4—42no—
590C4—81no—
591C4—81no—
592C4—81no—
593C4—81no—
594C4#296611yescharacteristic
595C4#296911yescharacteristic
596C4#297611yescharacteristic
597C4#297911yescharacteristic
598C4#329411yescharacteristic
599C4#329411yescharacteristic
600C4#330111yescharacteristic
601C4#330111yescharacteristic
602C2—42no—
603C2—82no—
604C2—161no—
605C2#1790011yescharacteristic
606C2#1792711yescharacteristic
607C2#1799411yescharacteristic
608C2#1803111yescharacteristic
609C2#2312211yescharacteristic
610C2#2312711yescharacteristic
611C2#2314811yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC23C2×C4×D4
prime radical#1999C4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C4×C22⋊C8 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #22990C2semidirect
3. #22985C2semidirect
4. #9558C2semidirect
5. #9554C2semidirect
6. #9533C2semidirect
7. #9533C2semidirect
8. #8951C2semidirect
9. #8947C2semidirect
10. #8944C2semidirect
11. #8939C2semidirect
12. #8210C2semidirect
13. C8×#102C2semidirect
14. #8148C2semidirect
15. #7877C2semidirect
279. C22⋊C8D8semidirect

Maximal subgroups

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

subgroupindex
#236962
#229902
#229852
#95582
#95542
#95332
#89512
#89472
#89442
#89392
#82102
C8×#1022
#81482
#78772