Finite Group Databasebeta

← #382248

Subgroups — #382248

Order 512.

825 conjugacy classes of subgroups.

The subgroup lattice has 604 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23627C211yesmaximal, characteristic
2#21144C211yesmaximal, characteristic, complemented
3#21047C211yesmaximal, characteristic
4#11359C211yesmaximal, characteristic, complemented
5#11291C211yesmaximal, characteristic, complemented
6#11291C211yesmaximal, characteristic, complemented
7#11269C211yesmaximal, characteristic, complemented
8#11068C211yesmaximal, characteristic
9#11068C211yesmaximal, characteristic
10#11053C211yesmaximal, characteristic
11#10930C211yesmaximal, characteristic
12#10929C211yesmaximal, characteristic
13#10883C211yesmaximal, characteristic, complemented
14#10882C211yesmaximal, characteristic, complemented
15#10696C211yesmaximal, characteristic, complemented
16C2×#424C2211yescharacteristic
17C2×#424C2211yescharacteristic
18C2×#424C2211yescharacteristic
19C2×#423C2211yescharacteristic
20C22×#104C2211yescharacteristic
21C22×#104C2211yescharacteristic
22C2×#416C2211yescharacteristic
23C2×#416C2211yescharacteristic
24C2×#408C2211yescharacteristic
25C2×#408C2211yescharacteristic
26C22×#97C2211yescharacteristic
27C22×#97C2211yescharacteristic
28C22×C4×C8C2211yescharacteristic
29#2297C2211yescharacteristic
30#2285C2211yescharacteristic
31#2074C412yescomplemented
32#2073C412yescomplemented
33#2072C412yescomplemented
34#2071C412yescomplemented
35#1940—24no—
36#1924—24no—
37C2×#341C2211yescharacteristic
38C2×#341C2211yescharacteristic
39C2×#341C2211yescharacteristic
40C2×#341C2211yescharacteristic
41#1735C2211yescharacteristic
42#1735C2211yescharacteristic
43C2×#339C2211yescharacteristic
44C2×#339C2211yescharacteristic
45C2×#339C2211yescharacteristic
46C2×#339C2211yescharacteristic
47C2×#339C2211yescharacteristic
48C2×#339C2211yescharacteristic
49#1729C2211yescharacteristic
50#1729C2211yescharacteristic
51#1729C2211yescharacteristic
52#1728C2211yescharacteristic
53C2×#336C2211yescharacteristic
54C2×#336C2211yescharacteristic
55C2×#336C2211yescharacteristic
56C4×#95C2211yescharacteristic
57C22×#41C2311yescharacteristic
58C22×M16C2311yescharacteristic
59C22×M16C2311yescharacteristic
60C22×M16C2311yescharacteristic
61C22×M16C2311yescharacteristic
62C23×C8C2311yescharacteristic
63C23×C8C2311yescharacteristic
64C23×C8C2311yescharacteristic
65C23×C8C2311yescharacteristic
66C2×#117C2311yescharacteristic
67C2×#117C2311yescharacteristic
68C2×#117C2311yescharacteristic
69C22×C42C2311yescharacteristic
70C22×C42C2311yescharacteristic
71C22×C42C2311yescharacteristic
72#424—22no—
73#424—22no—
74#424—41no—
75#424—41no—
76#424—41no—
77#424—41no—
78#423—41no—
79#423—41no—
80C2×C4⋊C8—22no—
81C2×C4⋊C8—22no—
82C2×C4⋊C8—22no—
83C2×C4⋊C8—22no—
84C2×#104—22no—
85C2×#104—22no—
86C2×#104—41no—
87C2×#104—41no—
88C2×#104—41no—
89C2×#104—41no—
90#416—41no—
91#416—41no—
92#416—41no—
93#416—41no—
94C2×#103C2×C412yes—
95C2×#103C2×C412yes—
96C2×#102C2×C412yes—
97C2×#102C2×C412yes—
98#408—22no—
99#408—22no—
100#408—41no—
101#408—41no—
102C2×C22⋊C8—22no—
103C2×C22⋊C8—22no—
104C2×#97—22no—
105C2×#97—41no—
106C2×#97—41no—
107C2×#97—41no—
108C2×#97—41no—
109C2×#97D414yes—
110C2×C4×C8—22no—
111C2×C4×C8—41no—
112C2×C4×C8—41no—
113#391C2×C411yescharacteristic
114#391C2×C411yescharacteristic
115#390C2×C411yescharacteristic
116#390C2×C411yescharacteristic
117#386—22no—
118#384C2×C412yes—
119#382C2×C412yes—
120#380—22no—
121C4×C4⋊C4—22no—
122C4×C22⋊C4—22no—
123C2×#95—22no—
124#341—41no—
125#341—41no—
126#341—41no—
127#341—41no—
128#340C2×C412yes—
129#340C2×C412yes—
130#340C2×C412yes—
131#340C2×C412yes—
132#339—22no—
133#339—22no—
134#339—41no—
135#339—41no—
136#339—41no—
137#339—41no—
138#336—22no—
139#336—22no—
140#336—22no—
141#336—22no—
142#336—22no—
143#336—22no—
144#336C2×C412yes—
145#336C2×C412yes—
146C2×#41—21no—
147C2×#41—21no—
148C2×#41—21no—
149C2×#41—21no—
150C2×#41—21no—
151C2×#41—21no—
152C2×#41—21no—
153C2×#41—21no—
154C2×#41—21no—
155C2×#41—21no—
156C2×#41—21no—
157C2×#41—21no—
158C22×Q8C22×C411yescharacteristic
159C22×D4—21no—
160C22×D4C22×C411yescharacteristic
161C23×C4—21no—
162C23×C4—21no—
163C23×C4—21no—
164C23×C4#4112yes—
165C23×C4C2411yescharacteristic
166C2×M16—21no—
167C2×M16—21no—
168C2×M16—21no—
169C2×M16—21no—
170C2×M16—21no—
171C2×M16—21no—
172C2×M16—21no—
173C2×M16—21no—
174C2×M16—21no—
175C2×M16—21no—
176C2×M16—21no—
177C2×M16—21no—
178C2×M16—21no—
179C2×M16—21no—
180C2×M16—21no—
181C2×M16—21no—
182C2×M16—21no—
183C2×M16—21no—
184C2×M16—21no—
185C2×M16—21no—
186C2×M16—21no—
187C2×M16—21no—
188C2×M16—21no—
189C2×M16—21no—
190C22×C8—21no—
191C22×C8—21no—
192C22×C8—21no—
193C22×C8—21no—
194C22×C8—21no—
195C22×C8—21no—
196C22×C8—21no—
197C22×C8—21no—
198C22×C8—21no—
199C22×C8—21no—
200C22×C8—21no—
201C22×C8—21no—
202C22×C8—21no—
203C22×C8—21no—
204C22×C8—21no—
205C22×C8—21no—
206C22×C8—21no—
207C22×C8—21no—
208C22×C8—21no—
209C22×C8—21no—
210C22×C8—21no—
211C22×C8—21no—
212C22×C8—21no—
213C22×C8—21no—
214C22×C8—21no—
215C22×C8—21no—
216C22×C8C22×C412yes—
217C22×C8C22×C412yes—
218C22×C8C2×D412yes—
219C22×C8C2×D412yes—
220C22×C8#4112yes—
221C22×C8#4112yes—
222#117—21no—
223#117—21no—
224#117—21no—
225#117—21no—
226#117—21no—
227#117—21no—
228#117—21no—
229#117—21no—
230#117—21no—
231#117—21no—
232#117—21no—
233#117—21no—
234C2×C4⋊C4—22no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4C22×C411yescharacteristic
237C2×C4⋊C4C22×C411yescharacteristic
238C2×C4⋊C4C22×C412yes—
239C2×C4⋊C4C22×C412yes—
240C2×C22⋊C4—21no—
241C2×C22⋊C4—21no—
242C2×C22⋊C4—21no—
243C2×C22⋊C4—22no—
244C2×C22⋊C4—22no—
245C2×C42—21no—
246C2×C42—21no—
247C2×C42—21no—
248C2×C42—21no—
249C2×C42—21no—
250C2×C42—21no—
251C2×C42—21no—
252C2×C42—21no—
253C2×C42—21no—
254C2×C42—21no—
255C2×C42—21no—
256C2×C42—21no—
257C2×C42—21no—
258C2×C42—21no—
259C2×C42—21no—
260C2×C42—21no—
261C2×C42—21no—
262C2×C42C22×C411yescharacteristic
263C2×C42C22×C411yescharacteristic
264C2×C42C2×D412yes—
265C2×C42#4112yes—
266C2×C42#4112yes—
267C4⋊C8—44no—
268C4⋊C8—44no—
269C4⋊C8—44no—
270C4⋊C8—44no—
271#104—44no—
272#104—44no—
273#103—42no—
274#103—42no—
275#102—42no—
276#102—42no—
277C22⋊C8—44no—
278C22⋊C8—44no—
279#97—44no—
280#97—44no—
281C4×C8—44no—
282#95—22no—
283#95—22no—
284#95—24no—
285#95—44no—
286C24—21no—
287C24C2×#4111yescharacteristic
288#41—41no—
289#41—41no—
290#41—41no—
291#41—41no—
292#41—41no—
293#41—41no—
294#41—41no—
295#41—41no—
296#41—41no—
297#41—41no—
298#41—41no—
299#41—41no—
300#41—41no—
301#41—41no—
302#41—41no—
303#41—41no—
304C2×Q8—21no—
305C2×Q8—21no—
306C2×Q8—21no—
307C2×Q8—21no—
308C2×Q8—21no—
309C2×Q8—21no—
310C2×D4—21no—
311C2×D4—21no—
312C2×D4—21no—
313C2×D4—21no—
314C2×D4—21no—
315C2×D4—21no—
316C2×D4—41no—
317C2×D4—41no—
318C2×D4—41no—
319C2×D4—41no—
320C2×D4—41no—
321C2×D4—41no—
322C22×C4—24no—
323C22×C4—22no—
324C22×C4—22no—
325C22×C4—22no—
326C22×C4—21no—
327C22×C4—21no—
328C22×C4—21no—
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—42no—
333C22×C4—42no—
334C22×C4—42no—
335C22×C4—41no—
336C22×C4—41no—
337C22×C4—42no—
338C22×C4—42no—
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×C4C4×D412yes—
348C22×C4C4×D412yes—
349C22×C4C23×C411yescharacteristic
350C22×C4C22×D411yescharacteristic
351C22×C4C2×#4111yescharacteristic
352C22×C4C2×#4111yescharacteristic
353C22×C4C2×#4111yescharacteristic
354C22×C4C2×#4111yescharacteristic
355C22×C4H(5,𝔽2)11yescharacteristic
356C22×C4H(5,𝔽2)11yescharacteristic
357C22×C4H(5,𝔽2)11yescharacteristic
358C22×C4H(5,𝔽2)11yescharacteristic
359C22×C4H(5,𝔽2)11yescharacteristic
360C22×C42−1+411yescharacteristic
361C22×C42−1+411yescharacteristic
362C22×C42−1+411yescharacteristic
363M16—41no—
364M16—41no—
365M16—41no—
366M16—41no—
367M16—41no—
368M16—41no—
369M16—41no—
370M16—41no—
371M16—41no—
372M16—41no—
373M16—41no—
374M16—41no—
375M16—41no—
376M16—41no—
377M16—41no—
378M16—41no—
379C2×C8—22no—
380C2×C8—22no—
381C2×C8—22no—
382C2×C8—22no—
383C2×C8—22no—
384C2×C8—22no—
385C2×C8—22no—
386C2×C8—22no—
387C2×C8—22no—
388C2×C8—22no—
389C2×C8—22no—
390C2×C8—22no—
391C2×C8—22no—
392C2×C8—22no—
393C2×C8—22no—
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—41no—
405C2×C8—41no—
406C2×C8—41no—
407C2×C8—41no—
408C2×C8—41no—
409C2×C8—41no—
410C2×C8—41no—
411C2×C8—41no—
412C2×C8—41no—
413C2×C8—41no—
414C2×C8—41no—
415C2×C8—41no—
416C2×C8—41no—
417C2×C8—41no—
418C2×C8#12414yes—
419C4⋊C4—21no—
420C4⋊C4—21no—
421C4⋊C4—21no—
422C4⋊C4—21no—
423C4⋊C4—22no—
424C4⋊C4—22no—
425C4⋊C4—22no—
426C4⋊C4—22no—
427C4⋊C4—82no—
428C4⋊C4—82no—
429C22⋊C4—41no—
430C22⋊C4—41no—
431C22⋊C4—41no—
432C22⋊C4—41no—
433C22⋊C4—41no—
434C22⋊C4—41no—
435C22⋊C4—82no—
436C22⋊C4—82no—
437C42—21no—
438C42—21no—
439C42—21no—
440C42—21no—
441C42—22no—
442C42—22no—
443C42—22no—
444C42—22no—
445C42—22no—
446C42—22no—
447C42—41no—
448C42—41no—
449C42—41no—
450C42—41no—
451C42—41no—
452C42—41no—
453C42—41no—
454C42—41no—
455C42—41no—
456C42—41no—
457C23—22no—
458C23—22no—
459C23—41no—
460C23—41no—
461C23—42no—
462C23—42no—
463C23—41no—
464C23C2×C4×D411yescharacteristic
465C23#52911yescharacteristic
466C23#52911yescharacteristic
467C23#53911yescharacteristic
468C23#53911yescharacteristic
469C23#54011yescharacteristic
470C23#54211yescharacteristic
471Q8—41no—
472Q8—41no—
473Q8—41no—
474Q8—41no—
475D4—41no—
476D4—41no—
477D4—41no—
478D4—41no—
479D4—81no—
480D4—81no—
481D4—81no—
482D4—81no—
483C2×C4—24no—
484C2×C4—22no—
485C2×C4—22no—
486C2×C4—22no—
487C2×C4—22no—
488C2×C4—22no—
489C2×C4—22no—
490C2×C4—44no—
491C2×C4—44no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—41no—
497C2×C4—42no—
498C2×C4—42no—
499C2×C4—41no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—82no—
511C2×C4—82no—
512C2×C4—82no—
513C2×C4—82no—
514C2×C4—81no—
515C2×C4—81no—
516C2×C4—81no—
517C2×C4—81no—
518C2×C4C4×#4111yescharacteristic
519C2×C4C4×#4111yescharacteristic
520C2×C4#51811yescharacteristic
521C2×C4#52011yescharacteristic
522C2×C4#52011yescharacteristic
523C2×C4#52011yescharacteristic
524C2×C4C2×#12411yescharacteristic
525C2×C4#53211yescharacteristic
526C2×C4#53411yescharacteristic
527C2×C4#53511yescharacteristic
528C2×C4#53611yescharacteristic
529C2×C4#53711yescharacteristic
530C2×C4#53811yescharacteristic
531C2×C4#53811yescharacteristic
532C2×C4#53811yescharacteristic
533C2×C4#53811yescharacteristic
534C2×C4#53911yescharacteristic
535C2×C4#53911yescharacteristic
536C2×C4#53911yescharacteristic
537C2×C4#53911yescharacteristic
538C2×C4#54011yescharacteristic
539C2×C4#54011yescharacteristic
540C2×C4#54111yescharacteristic
541C2×C4#54111yescharacteristic
542C2×C4#54211yescharacteristic
543C2×C4#54211yescharacteristic
544C2×C4#54211yescharacteristic
545C2×C4#54211yescharacteristic
546C8—24no—
547C8—42no—
548C8—42no—
549C8—42no—
550C8—81no—
551C8—81no—
552C8—81no—
553C8—81no—
554C22—22no—
555C22—22no—
556C22—22no—
557C22—22no—
558C22—21no—
559C22—21no—
560C22—21no—
561C22—21no—
562C22—82no—
563C22—82no—
564C22—81no—
565C22—81no—
566C22—81no—
567C22C4×#12411yescharacteristic
568C22#232211yescharacteristic
569C22#232411yescharacteristic
570C22#264911yescharacteristic
571C22#264911yescharacteristic
572C22#265411yescharacteristic
573C22#265411yescharacteristic
574C22#295612yes—
575C22#295712yes—
576C4—44no—
577C4—42no—
578C4—42no—
579C4—82no—
580C4—82no—
581C4—82no—
582C4—81no—
583C4—81no—
584C4—81no—
585C4#233411yescharacteristic
586C4#233411yescharacteristic
587C4#233511yescharacteristic
588C4#233511yescharacteristic
589C4#263911yescharacteristic
590C4#264011yescharacteristic
591C4#266111yescharacteristic
592C4#266411yescharacteristic
593C2—21no—
594C2—21no—
595C2—21no—
596C2—21no—
597C2—161no—
598C2#1432411yescharacteristic
599C2#2165911yescharacteristic
600C2#2166211yescharacteristic
601C2#2209712yes—
602C2#2216712yes—
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
socleC23C2×C4×D4
prime radicalC22×#41C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C22×#97 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #21144C2semidirect
4. #11359C2semidirect
5. #11291C2semidirect
6. #11291C2semidirect
7. #11269C2semidirect
13. #10883C2semidirect
14. #10882C2semidirect
15. #10696C2semidirect
31. #2074C4semidirect
32. #2073C4semidirect
33. #2072C4semidirect
34. #2071C4semidirect

Maximal subgroups

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

subgroupindex
#236272
#211442
#210472
#113592
#112912
#112692
#110682
#110532
#109302
#109292
#108832
#108822
#106962