Finite Group Databasebeta

← #371089

Subgroups — #371089

Order 512.

892 conjugacy classes of subgroups.

The subgroup lattice has 583 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1937C211yesmaximal, characteristic
2#20405C211yesmaximal, characteristic
3#20366C211yesmaximal, characteristic, complemented
4#11369C211yesmaximal, characteristic, complemented
5#11305C211yesmaximal, characteristic, complemented
6#11177C211yesmaximal, characteristic
7#11112C211yesmaximal, characteristic
8#10946C211yesmaximal, characteristic
9#10920C211yesmaximal, characteristic, complemented
10#10902C211yesmaximal, characteristic, complemented
11#10897C211yesmaximal, characteristic, complemented
12#10824C211yesmaximal, characteristic, complemented
13#10818C211yesmaximal, characteristic, complemented
14#10779C211yesmaximal, characteristic
15#10747C211yesmaximal, characteristic
16#3428C2211yescharacteristic
17C2×#430C2211yescharacteristic
18C2×#424C2211yescharacteristic
19C2×#421C2211yescharacteristic
20C2×#421C2211yescharacteristic
21C22×#104C2211yescharacteristic
22C2×#416C2211yescharacteristic
23C2×#413C2211yescharacteristic
24C2×#408C2211yescharacteristic
25#2873C2211yescharacteristic
26C2×C4×M16C2211yescharacteristic
27C22×C4×C8C2211yescharacteristic
28C4×#117C2211yescharacteristic
29#1937C418yescomplemented
30#1759—22no—
31C4×#104—22no—
32#1746—22no—
33#1744—22no—
34C2×#344C2211yescharacteristic
35C2×#344C2211yescharacteristic
36#1742C2211yescharacteristic
37#1742C2211yescharacteristic
38#1740C2211yescharacteristic
39#1738C2211yescharacteristic
40C2×#341C2211yescharacteristic
41C2×#341C2211yescharacteristic
42#1735C2211yescharacteristic
43#1733C2211yescharacteristic
44#1733C2211yescharacteristic
45C2×#339C2211yescharacteristic
46C2×#339C2211yescharacteristic
47C2×#339C2211yescharacteristic
48C2×H(3,ℤ/4)C2211yescharacteristic
49C2×H(3,ℤ/4)C2211yescharacteristic
50#1729C2211yescharacteristic
51#1729C2211yescharacteristic
52#1729C2211yescharacteristic
53#1729C2211yescharacteristic
54#1729C2211yescharacteristic
55#1729C2211yescharacteristic
56C22×#41C2311yescharacteristic
57C22×M16C2311yescharacteristic
58C22×M16C2311yescharacteristic
59C22×M16C2311yescharacteristic
60C22×M16C2311yescharacteristic
61C22×M16C2311yescharacteristic
62C22×M16C2311yescharacteristic
63C23×C8C2311yescharacteristic
64C23×C8C2311yescharacteristic
65#520—22no—
66#520—22no—
67#519—21no—
68#518—22no—
69#518—21no—
70C4×#41—22no—
71C4×#41—41no—
72C2×#117—41no—
73C2×#117C2311yescharacteristic
74C2×#117C2311yescharacteristic
75C2×#117C2311yescharacteristic
76C2×#117C2311yescharacteristic
77C2×#117C2311yescharacteristic
78C22×C42C2311yescharacteristic
79#430—22no—
80#430—22no—
81C2×#108—41no—
82#424—41no—
83#424C2×C414yescomplemented
84C2×C4⋊C8—41no—
85#421—22no—
86#421—22no—
87#421—22no—
88#421—22no—
89C2×#104—21no—
90C2×#104—21no—
91C2×#104—22no—
92C2×#104—22no—
93C2×#104—41no—
94C2×#104C2×C414yes—
95#416—22no—
96#416C2×C414yes—
97C2×#103—21no—
98C2×#102—21no—
99#413—22no—
100#413—41no—
101C2×#101—21no—
102C2×#100—21no—
103#408—41no—
104#408C2×C414yes—
105C2×C22⋊C8—41no—
106#405—41no—
107#405—41no—
108C4×M16—22no—
109C4×M16—22no—
110C4×M16—22no—
111C2×#97—21no—
112C2×C4×C8—21no—
113C2×C4×C8—21no—
114C2×C4×C8—21no—
115C2×C4×C8—22no—
116C2×C4×C8C2×C414yescomplemented
117C4×C4⋊C4—22no—
118C4×C22⋊C4—22no—
119#376D412yes—
120C43D412yes—
121#344—22no—
122#344—41no—
123#343—22no—
124#343—22no—
125#342—22no—
126#341—22no—
127#341—41no—
128#341C2×C414yescomplemented
129#340—22no—
130#339—21no—
131#339—21no—
132#339—22no—
133#339—22no—
134#339—41no—
135#339C2×C414yescomplemented
136H(3,ℤ/4)—41no—
137H(3,ℤ/4)—41no—
138#336—22no—
139#336—22no—
140#336—22no—
141#336—22no—
142#336—41no—
143#336—41no—
144C2×#41—21no—
145C2×#41—21no—
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×#41C22×C412yes—
155C2×#41#4112yes—
156C22×Q8#4111yescharacteristic
157C22×D4—21no—
158C22×D4#4111yescharacteristic
159C23×C4—21no—
160C23×C4—21no—
161C23×C4C2411yescharacteristic
162C2×M16—21no—
163C2×M16—21no—
164C2×M16—21no—
165C2×M16—21no—
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—
190C2×M16—21no—
191C2×M16—21no—
192C2×M16—21no—
193C2×M16C22×C412yes—
194C2×M16C22×C412yes—
195C2×M16#4112yes—
196C2×M16#4112yes—
197C2×M16#4112yes—
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—
218C4×Q8—24no—
219C4×Q8—42no—
220C4×D4—24no—
221C4×D4—44no—
222C4×D4—42no—
223C4×D4—44no—
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—
234#117—21no—
235#117—21no—
236#117—21no—
237#117—21no—
238#117—21no—
239#117—21no—
240#117—21no—
241#117—21no—
242#117—21no—
243#117—42no—
244#117—42no—
245#117—42no—
246#117C22×C412yes—
247C2×C4⋊C4—21no—
248C2×C4⋊C4—21no—
249C2×C4⋊C4—21no—
250C2×C4⋊C4—42no—
251C2×C4⋊C4#4112yes—
252C2×C4⋊C4#4112yes—
253C2×C22⋊C4—21no—
254C2×C22⋊C4—21no—
255C2×C22⋊C4—21no—
256C2×C22⋊C4—21no—
257C2×C22⋊C4—42no—
258C2×C22⋊C4#4112yes—
259C2×C42—21no—
260C2×C42—21no—
261C2×C42—21no—
262C2×C42—21no—
263C2×C42—24no—
264C2×C42—21no—
265C2×C42—21no—
266C2×C42—42no—
267C2×C42C22×C412yes—
268C2×C42C2×D412yes—
269C2×C42C2×D412yes—
270C2×C42C2×D411yescharacteristic
271C2×C42C2×D411yescharacteristic
272C2×C42#4112yes—
273#108—44no—
274C4⋊C8—44no—
275#104—28no—
276#104—24no—
277#104—24no—
278#104—44no—
279#103—24no—
280#102—24no—
281#101—42no—
282#100—42no—
283C22⋊C8—44no—
284#97—24no—
285#97—42no—
286C4×C8—24no—
287C4×C8—24no—
288C4×C8—24no—
289C4×C8—42no—
290#95—24no—
291C24—21no—
292C24C2×#4111yescharacteristic
293#41—22no—
294#41—22no—
295#41—22no—
296#41—22no—
297#41—22no—
298#41—22no—
299#41—22no—
300#41—22no—
301#41—41no—
302#41—41no—
303#41—41no—
304#41—41no—
305#41—41no—
306#41—41no—
307#41—41no—
308#41—41no—
309C2×Q8—21no—
310C2×Q8—21no—
311C2×Q8—21no—
312C2×Q8—21no—
313C2×Q8—21no—
314C2×Q8#11712yes—
315C2×D4—22no—
316C2×D4—22no—
317C2×D4—21no—
318C2×D4—21no—
319C2×D4—21no—
320C2×D4—21no—
321C2×D4—21no—
322C2×D4—41no—
323C2×D4—41no—
324C2×D4—41no—
325C2×D4—41no—
326C2×D4#11712yes—
327C22×C4—21no—
328C22×C4—22no—
329C22×C4—21no—
330C22×C4—22no—
331C22×C4—21no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—44no—
337C22×C4—44no—
338C22×C4—41no—
339C22×C4—41no—
340C22×C4—41no—
341C22×C4—41no—
342C22×C4—41no—
343C22×C4—41no—
344C22×C4#12314yes—
345C22×C4C23×C411yescharacteristic
346C22×C4C22×D411yescharacteristic
347C22×C4C2×#4111yescharacteristic
348C22×C4C2×#4111yescharacteristic
349C22×C4C2×#4111yescharacteristic
350C22×C4C2×#4111yescharacteristic
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×C42−1+411yescharacteristic
359M16—22no—
360M16—22no—
361M16—22no—
362M16—22no—
363M16—22no—
364M16—22no—
365M16—22no—
366M16—22no—
367M16—22no—
368M16—22no—
369M16—41no—
370M16—41no—
371M16—41no—
372M16—41no—
373M16—41no—
374M16—41no—
375M16—41no—
376M16—41no—
377M16—41no—
378M16—41no—
379M16—41no—
380M16—41no—
381M16—41no—
382M16—41no—
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—
415C4⋊C4—22no—
416C4⋊C4—22no—
417C4⋊C4—22no—
418C4⋊C4—22no—
419C4⋊C4—22no—
420C4⋊C4—44no—
421C4⋊C4—41no—
422C4⋊C4—41no—
423C4⋊C4—41no—
424C4⋊C4—41no—
425C4⋊C4—41no—
426C22⋊C4—22no—
427C22⋊C4—22no—
428C22⋊C4—22no—
429C22⋊C4—44no—
430C22⋊C4—41no—
431C22⋊C4—41no—
432C22⋊C4—41no—
433C22⋊C4—41no—
434C22⋊C4—41no—
435C22⋊C4—41no—
436C22⋊C4—41no—
437C42—22no—
438C42—21no—
439C42—21no—
440C42—22no—
441C42—22no—
442C42—22no—
443C42—22no—
444C42—22no—
445C42—22no—
446C42—48no—
447C42—44no—
448C42—41no—
449C42—41no—
450C42—41no—
451C42—41no—
452C42—41no—
453C42C4×D412yes—
454C42C4×D412yes—
455C23—22no—
456C23—42no—
457C23—41no—
458C23—41no—
459C23—82no—
460C23C4×#4111yescharacteristic
461C23C2×#12311yescharacteristic
462C23#52911yescharacteristic
463C23#55111yescharacteristic
464C23#55111yescharacteristic
465C23#55311yescharacteristic
466C23#55611yescharacteristic
467Q8—22no—
468Q8—22no—
469Q8—41no—
470Q8—41no—
471D4—22no—
472D4—22no—
473D4—42no—
474D4—42no—
475D4—42no—
476D4—42no—
477D4—41no—
478D4—41no—
479C2×C4—24no—
480C2×C4—22no—
481C2×C4—22no—
482C2×C4—24no—
483C2×C4—22no—
484C2×C4—44no—
485C2×C4—44no—
486C2×C4—42no—
487C2×C4—41no—
488C2×C4—41no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—44no—
494C2×C4—44no—
495C2×C4—42no—
496C2×C4—42no—
497C2×C4—42no—
498C2×C4—82no—
499C2×C4—82no—
500C2×C4—82no—
501C2×C4—81no—
502C2×C4—164no—
503C2×C4C2×#11711yescharacteristic
504C2×C4C2×C4×D411yescharacteristic
505C2×C4C4×#4111yescharacteristic
506C2×C4C4×#4111yescharacteristic
507C2×C4#51811yescharacteristic
508C2×C4#52011yescharacteristic
509C2×C4#52511yescharacteristic
510C2×C4#52511yescharacteristic
511C2×C4#52511yescharacteristic
512C2×C4#52911yescharacteristic
513C2×C4#52911yescharacteristic
514C2×C4#52911yescharacteristic
515C2×C4#53511yescharacteristic
516C2×C4#53711yescharacteristic
517C2×C4#54011yescharacteristic
518C2×C4#54211yescharacteristic
519C2×C4#55111yescharacteristic
520C2×C4#55111yescharacteristic
521C2×C4#55111yescharacteristic
522C2×C4#55111yescharacteristic
523C2×C4#55111yescharacteristic
524C2×C4#55211yescharacteristic
525C2×C4#55211yescharacteristic
526C2×C4#55211yescharacteristic
527C2×C4#55311yescharacteristic
528C2×C4#55311yescharacteristic
529C2×C4#55511yescharacteristic
530C2×C4#55511yescharacteristic
531C8—42no—
532C8—42no—
533C8—42no—
534C8—42no—
535C8—42no—
536C8—42no—
537C8—42no—
538C8—81no—
539C22—22no—
540C22—22no—
541C22—22no—
542C22—41no—
543C22—41no—
544C22—41no—
545C22—41no—
546C22—42no—
547C22—82no—
548C22—82no—
549C22—161no—
550C22C4×#12311yescharacteristic
551C22#235311yescharacteristic
552C22#237311yescharacteristic
553C22#248211yescharacteristic
554C22#248511yescharacteristic
555C22#252711yescharacteristic
556C22#253111yescharacteristic
557C4—24no—
558C4—42no—
559C4—44no—
560C4—42no—
561C4—82no—
562C4—82no—
563C4—82no—
564C4—161no—
565C4#229011yescharacteristic
566C4#232411yescharacteristic
567C4#234311yescharacteristic
568C4#237211yescharacteristic
569C4#252611yescharacteristic
570C4#253011yescharacteristic
571C4#259811yescharacteristic
572C4#260911yescharacteristic
573C2—21no—
574C2—21no—
575C2—21no—
576C2—21no—
577C2—21no—
578C2—21no—
579C2—82no—
580C2#1406611yescharacteristic
581C2#2759812yes—
subgroupG/H
centerC2×C4#535
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
socleC22#2482
prime radicalC22×#41C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #20366C2semidirect
4. #11369C2semidirect
5. #11305C2semidirect
9. #10920C2semidirect
10. #10902C2semidirect
11. #10897C2semidirect
12. #10824C2semidirect
13. #10818C2semidirect
29. #1937C4semidirect
83. #424C2×C4semidirect
116. C2×C4×C8C2×C4semidirect
128. #341C2×C4semidirect
135. #339C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#19372
#204052
#203662
#113692
#113052
#111772
#111122
#109462
#109202
#109022
#108972
#108242
#108182
#107792
#107472