Finite Group Databasebeta

← #391580

Subgroups — #391580

Order 512.

676 conjugacy classes of subgroups.

The subgroup lattice has 556 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23652C211yesmaximal, characteristic
2#11335C211yesmaximal, characteristic, complemented
3#11307C211yesmaximal, characteristic, complemented
4#11273C211yesmaximal, characteristic, complemented
5#11263C211yesmaximal, characteristic, complemented
6#11263C211yesmaximal, characteristic, complemented
7#11251C211yesmaximal, characteristic, complemented
8#11186C211yesmaximal, characteristic
9#11146C211yesmaximal, characteristic
10#11138C211yesmaximal, characteristic
11#11060C211yesmaximal, characteristic
12#11060C211yesmaximal, characteristic
13#10951C211yesmaximal, characteristic
14#10905C211yesmaximal, characteristic, complemented
15#10901C211yesmaximal, characteristic, complemented
16C2×#430C2211yescharacteristic
17C2×#430C2211yescharacteristic
18#2911C2211yescharacteristic
19C2×#421C2211yescharacteristic
20C22×#104C2211yescharacteristic
21#2896C2211yescharacteristic
22#2896C2211yescharacteristic
23C2×#413C2211yescharacteristic
24C2×#410C2211yescharacteristic
25C2×#410C2211yescharacteristic
26#2873C2211yescharacteristic
27#2296C2211yescharacteristic
28#2296C2211yescharacteristic
29#2082C414yescomplemented
30#2081C414yescomplemented
31C2×#344C2211yescharacteristic
32C2×#344C2211yescharacteristic
33C2×#344C2211yescharacteristic
34C2×#344C2211yescharacteristic
35C2×#344C2211yescharacteristic
36C2×#344C2211yescharacteristic
37#1740C2211yescharacteristic
38#1740C2211yescharacteristic
39#1733C2211yescharacteristic
40#1733C2211yescharacteristic
41#1733C2211yescharacteristic
42#1733C2211yescharacteristic
43#1733C2211yescharacteristic
44#1733C2211yescharacteristic
45#1733C2211yescharacteristic
46C2×#339C2211yescharacteristic
47C2×#339C2211yescharacteristic
48C2×#339C2211yescharacteristic
49C2×H(3,ℤ/4)C2211yescharacteristic
50C2×H(3,ℤ/4)C2211yescharacteristic
51C2×H(3,ℤ/4)C2211yescharacteristic
52#1728C2211yescharacteristic
53C22×#41C2311yescharacteristic
54C22×M16C2311yescharacteristic
55C22×M16C2311yescharacteristic
56C22×M16C2311yescharacteristic
57C22×M16C2311yescharacteristic
58C22×M16C2311yescharacteristic
59C22×M16C2311yescharacteristic
60C2×#117C2311yescharacteristic
61C2×#117C2311yescharacteristic
62C2×#117C2311yescharacteristic
63C2×#117C2311yescharacteristic
64C2×#117C2311yescharacteristic
65C2×#117C2311yescharacteristic
66C2×#117C2311yescharacteristic
67C22×C42C2311yescharacteristic
68#430—41no—
69#430—41no—
70#430—41no—
71#430C2×C414yes—
72C2×#108—22no—
73C2×#108—41no—
74#428—41no—
75#428—41no—
76#427—41no—
77#421—41no—
78#421—41no—
79C2×#104—22no—
80C2×#104—41no—
81C2×#104—41no—
82C2×#104—41no—
83#419C2×C412yes—
84#419C2×C412yes—
85#418C2×C412yes—
86#418C2×C412yes—
87#417—41no—
88#417—41no—
89#416—41no—
90#416—41no—
91#413—41no—
92#413—41no—
93C2×#101C2×C412yes—
94C2×#100C2×C412yes—
95#410—22no—
96#410—41no—
97#410—41no—
98#410—41no—
99C2×#99—41no—
100C2×#99—41no—
101#405—41no—
102#405—41no—
103C4×M16—41no—
104#389—41no—
105#389—41no—
106#385—41no—
107#385—41no—
108C4×C4⋊C4—41no—
109C4×C4⋊C4—41no—
110#344—22no—
111#344—41no—
112#344—41no—
113#344—41no—
114#344C2×C414yes—
115#344C2×C414yes—
116#342—41no—
117#342—41no—
118#339—22no—
119#339—41no—
120#339—41no—
121#339—41no—
122#339—41no—
123#339—41no—
124#339—41no—
125#339—41no—
126#339—41no—
127#339—41no—
128H(3,ℤ/4)—22no—
129H(3,ℤ/4)—41no—
130H(3,ℤ/4)C2×C414yes—
131#336—41no—
132C2×#41—21no—
133C2×#41—21no—
134C2×#41—21no—
135C2×#41—21no—
136C2×#41—21no—
137C2×#41—21no—
138C2×#41—21no—
139C2×#41—21no—
140C2×#41—21no—
141C2×#41—21no—
142C2×#41—21no—
143C2×#41—21no—
144C22×Q8C22×C411yescharacteristic
145C22×D4—21no—
146C22×D4C22×C411yescharacteristic
147C23×C4—21no—
148C23×C4—21no—
149C23×C4C2411yescharacteristic
150C2×M16—21no—
151C2×M16—21no—
152C2×M16—21no—
153C2×M16—21no—
154C2×M16—21no—
155C2×M16—21no—
156C2×M16—21no—
157C2×M16—21no—
158C2×M16—21no—
159C2×M16—21no—
160C2×M16—21no—
161C2×M16—21no—
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×M16C22×C412yes—
182C2×M16C22×C412yes—
183C2×M16C22×C412yes—
184C2×M16C22×C412yes—
185C2×M16#4112yes—
186C22×C8—21no—
187C22×C8—21no—
188C22×C8—21no—
189C22×C8—21no—
190C22×C8—21no—
191C22×C8—21no—
192#117—21no—
193#117—21no—
194#117—21no—
195#117—21no—
196#117—21no—
197#117—21no—
198#117—21no—
199#117—21no—
200#117—21no—
201#117—21no—
202#117—21no—
203#117—21no—
204#117—21no—
205#117—21no—
206#117—21no—
207#117—21no—
208#117—21no—
209#117—21no—
210#117—21no—
211#117—21no—
212#117—21no—
213#117—21no—
214#117—21no—
215#117—21no—
216#117—21no—
217#117C22×C412yes—
218#117C22×C412yes—
219#117#4112yes—
220C2×C4⋊C4—21no—
221C2×C4⋊C4—21no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4—21no—
224C2×C4⋊C4—21no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4—21no—
227C2×C4⋊C4—42no—
228C2×C4⋊C4—42no—
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×C42—21no—
243C2×C42—21no—
244C2×C42—21no—
245C2×C42—21no—
246C2×C42—21no—
247C2×C42—21no—
248C2×C42—21no—
249#108—44no—
250#108—44no—
251#107—42no—
252#106—42no—
253#104—44no—
254#104—44no—
255#103—42no—
256#103—42no—
257#102—42no—
258#102—42no—
259#101—42no—
260#100—42no—
261#99—44no—
262#99—44no—
263#97—42no—
264C4×C8—42no—
265#95—42no—
266#95—42no—
267C24—21no—
268C24H(5,𝔽2)11yescharacteristic
269#41—41no—
270#41—41no—
271#41—41no—
272#41—41no—
273#41—41no—
274#41—41no—
275#41—41no—
276#41—41no—
277#41—41no—
278#41—41no—
279#41—41no—
280#41—41no—
281#41—41no—
282#41—41no—
283#41—41no—
284#41—41no—
285C2×Q8—21no—
286C2×Q8—21no—
287C2×Q8—21no—
288C2×Q8—21no—
289C2×Q8—21no—
290C2×Q8—21no—
291C2×D4—21no—
292C2×D4—21no—
293C2×D4—21no—
294C2×D4—21no—
295C2×D4—21no—
296C2×D4—21no—
297C2×D4—41no—
298C2×D4—41no—
299C2×D4—41no—
300C2×D4—41no—
301C2×D4—41no—
302C2×D4—41no—
303C22×C4—21no—
304C22×C4—41no—
305C22×C4—41no—
306C22×C4—41no—
307C22×C4—41no—
308C22×C4—41no—
309C22×C4—41no—
310C22×C4—41no—
311C22×C4—41no—
312C22×C4—41no—
313C22×C4—41no—
314C22×C4—41no—
315C22×C4—41no—
316C22×C4—41no—
317C22×C4—41no—
318C22×C4—41no—
319C22×C4—41no—
320C22×C4—82no—
321C22×C4C23×C411yescharacteristic
322C22×C4C2×#4111yescharacteristic
323C22×C4C2×#4111yescharacteristic
324C22×C4H(5,𝔽2)11yescharacteristic
325C22×C4H(5,𝔽2)11yescharacteristic
326C22×C4H(5,𝔽2)11yescharacteristic
327C22×C4H(5,𝔽2)11yescharacteristic
328C22×C4H(5,𝔽2)11yescharacteristic
329C22×C4H(5,𝔽2)11yescharacteristic
330C22×C4H(5,𝔽2)11yescharacteristic
331C22×C42−1+411yescharacteristic
332C22×C42−1+411yescharacteristic
333C22×C42−1+411yescharacteristic
334C22×C42−1+411yescharacteristic
335M16—22no—
336M16—22no—
337M16—22no—
338M16—22no—
339M16—22no—
340M16—22no—
341M16—22no—
342M16—22no—
343M16—22no—
344M16—22no—
345M16—41no—
346M16—41no—
347M16—41no—
348M16—41no—
349M16—41no—
350M16—41no—
351M16—41no—
352M16—41no—
353M16—41no—
354M16—41no—
355M16—41no—
356M16—41no—
357M16—41no—
358M16—41no—
359C2×C8—22no—
360C2×C8—22no—
361C2×C8—22no—
362C2×C8—22no—
363C2×C8—22no—
364C2×C8—41no—
365C2×C8—41no—
366C2×C8—41no—
367C2×C8—41no—
368C2×C8—41no—
369C2×C8—41no—
370C2×C8—41no—
371C2×C8—41no—
372C2×C8—41no—
373C2×C8—41no—
374C2×C8—41no—
375C2×C8—41no—
376C2×C8—41no—
377C4⋊C4—22no—
378C4⋊C4—22no—
379C4⋊C4—22no—
380C4⋊C4—41no—
381C4⋊C4—41no—
382C4⋊C4—41no—
383C4⋊C4—41no—
384C4⋊C4—41no—
385C4⋊C4—41no—
386C4⋊C4—41no—
387C4⋊C4—41no—
388C4⋊C4—41no—
389C4⋊C4—41no—
390C4⋊C4—41no—
391C4⋊C4—84no—
392C4⋊C4—84no—
393C22⋊C4—22no—
394C22⋊C4—22no—
395C22⋊C4—22no—
396C22⋊C4—41no—
397C22⋊C4—41no—
398C22⋊C4—41no—
399C22⋊C4—41no—
400C22⋊C4—41no—
401C22⋊C4—41no—
402C22⋊C4—41no—
403C22⋊C4—41no—
404C22⋊C4—41no—
405C22⋊C4—41no—
406C22⋊C4—41no—
407C42—22no—
408C42—22no—
409C42—22no—
410C42—41no—
411C42—41no—
412C42—41no—
413C42—41no—
414C42—41no—
415C42—41no—
416C42—41no—
417C42—41no—
418C42—41no—
419C42—41no—
420C42—41no—
421C42—41no—
422C42—41no—
423C42—41no—
424C42—41no—
425C23—41no—
426C23—41no—
427C23—41no—
428C23—81no—
429C23—161no—
430C23#52011yescharacteristic
431C23#53811yescharacteristic
432C23#54211yescharacteristic
433C23#56011yescharacteristic
434C23H(3,𝔽4)11yescharacteristic
435C23#56211yescharacteristic
436C23#56211yescharacteristic
437Q8—41no—
438Q8—41no—
439Q8—41no—
440Q8—41no—
441D4—41no—
442D4—41no—
443D4—41no—
444D4—41no—
445D4—81no—
446D4—81no—
447D4—81no—
448D4—81no—
449C2×C4—42no—
450C2×C4—42no—
451C2×C4—42no—
452C2×C4—42no—
453C2×C4—41no—
454C2×C4—42no—
455C2×C4—41no—
456C2×C4—42no—
457C2×C4—41no—
458C2×C4—82no—
459C2×C4—82no—
460C2×C4—81no—
461C2×C4—81no—
462C2×C4—81no—
463C2×C4—81no—
464C2×C4—81no—
465C2×C4—81no—
466C2×C4—81no—
467C2×C4—81no—
468C2×C4—82no—
469C2×C4—82no—
470C2×C4—81no—
471C2×C4—81no—
472C2×C4—81no—
473C2×C4—81no—
474C2×C4—161no—
475C2×C4—161no—
476C2×C4—161no—
477C2×C4C4×#4111yescharacteristic
478C2×C4#51811yescharacteristic
479C2×C4#51811yescharacteristic
480C2×C4#52011yescharacteristic
481C2×C4#52011yescharacteristic
482C2×C4#52011yescharacteristic
483C2×C4#53811yescharacteristic
484C2×C4#53811yescharacteristic
485C2×C4#53911yescharacteristic
486C2×C4#53911yescharacteristic
487C2×C4#53911yescharacteristic
488C2×C4#54011yescharacteristic
489C2×C4#54011yescharacteristic
490C2×C4#54111yescharacteristic
491C2×C4#54211yescharacteristic
492C2×C4#54211yescharacteristic
493C2×C4#56011yescharacteristic
494C2×C4#56011yescharacteristic
495C2×C4#56011yescharacteristic
496C2×C4#56211yescharacteristic
497C2×C4#56211yescharacteristic
498C2×C4#56211yescharacteristic
499C2×C4#56211yescharacteristic
500C2×C4#56311yescharacteristic
501C2×C4#56311yescharacteristic
502C2×C4#56311yescharacteristic
503C2×C4#56311yescharacteristic
504C2×C4#56311yescharacteristic
505C8—42no—
506C8—42no—
507C8—42no—
508C8—42no—
509C8—42no—
510C8—81no—
511C22—41no—
512C22—41no—
513C22—41no—
514C22—41no—
515C22—41no—
516C22—41no—
517C22—41no—
518C22—81no—
519C22—161no—
520C22—161no—
521C22—161no—
522C22#233411yescharacteristic
523C22#237611yescharacteristic
524C22#237911yescharacteristic
525C22#282711yescharacteristic
526C22#282811yescharacteristic
527C22#283011yescharacteristic
528C22#283211yescharacteristic
529C4—82no—
530C4—82no—
531C4—82no—
532C4—82no—
533C4—81no—
534C4—161no—
535C4—161no—
536C4—161no—
537C4—161no—
538C4#233511yescharacteristic
539C4#233511yescharacteristic
540C4#237711yescharacteristic
541C4#238311yescharacteristic
542C4#282611yescharacteristic
543C4#283611yescharacteristic
544C4#283611yescharacteristic
545C4#283811yescharacteristic
546C2—21no—
547C2—21no—
548C2—21no—
549C2—21no—
550C2—21no—
551C2—21no—
552C2—21no—
553C2—161no—
554C2#1464311yescharacteristic
subgroupG/H
centerC4#2826
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
socleC2#14643
prime radicalC22×#41C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C4 ◁ C23×C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #11335C2semidirect
3. #11307C2semidirect
4. #11273C2semidirect
5. #11263C2semidirect
6. #11263C2semidirect
7. #11251C2semidirect
14. #10905C2semidirect
15. #10901C2semidirect
29. #2082C4semidirect
30. #2081C4semidirect

Maximal subgroups

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

subgroupindex
#236522
#113352
#113072
#112732
#112632
#112512
#111862
#111462
#111382
#110602
#109512
#109052
#109012