Finite Group Databasebeta

← #381698

Subgroups — #381698

Order 512.

742 conjugacy classes of subgroups.

The subgroup lattice has 561 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23652C211yesmaximal, characteristic
2#21100C211yesmaximal, characteristic
3#20459C211yesmaximal, characteristic, complemented
4#11363C211yesmaximal, characteristic, complemented
5#11334C211yesmaximal, characteristic, complemented
6#11281C211yesmaximal, characteristic, complemented
7#11264C211yesmaximal, characteristic, complemented
8#11137C211yesmaximal, characteristic
9#11131C211yesmaximal, characteristic
10#11065C211yesmaximal, characteristic
11#10955C211yesmaximal, characteristic
12#10950C211yesmaximal, characteristic
13#10905C211yesmaximal, characteristic, complemented
14#10890C211yesmaximal, characteristic, complemented
15#10804C211yesmaximal, characteristic, complemented
16C2×#430C2211yescharacteristic
17C2×#430C2211yescharacteristic
18C2×#430C2211yescharacteristic
19#2905C2211yescharacteristic
20C2×#424C2211yescharacteristic
21C2×#421C2211yescharacteristic
22C2×#421C2211yescharacteristic
23#2896C2211yescharacteristic
24#2896C2211yescharacteristic
25C2×#413C2211yescharacteristic
26#2879C2211yescharacteristic
27C2×#408C2211yescharacteristic
28#2873C2211yescharacteristic
29#2873C2211yescharacteristic
30C2×#405C2211yescharacteristic
31#2082C414yescomplemented
32#2081C414yescomplemented
33#1938—22no—
34#1937—22no—
35#1778—22no—
36C4×#108—22no—
37C2×#344C2211yescharacteristic
38C2×#344C2211yescharacteristic
39C2×#344C2211yescharacteristic
40C2×#344C2211yescharacteristic
41C2×#344C2211yescharacteristic
42C2×#344C2211yescharacteristic
43#1742C2211yescharacteristic
44#1738C2211yescharacteristic
45#1738C2211yescharacteristic
46#1738C2211yescharacteristic
47C2×#341C2211yescharacteristic
48C2×#341C2211yescharacteristic
49C2×#341C2211yescharacteristic
50#1735C2211yescharacteristic
51#1733C2211yescharacteristic
52#1733C2211yescharacteristic
53C2×#338C2211yescharacteristic
54C2×H(3,ℤ/4)C2211yescharacteristic
55C2×H(3,ℤ/4)C2211yescharacteristic
56C2×H(3,ℤ/4)C2211yescharacteristic
57C22×#41C2311yescharacteristic
58C22×M16C2311yescharacteristic
59C22×M16C2311yescharacteristic
60C22×M16C2311yescharacteristic
61C22×M16C2311yescharacteristic
62C22×M16C2311yescharacteristic
63C22×M16C2311yescharacteristic
64C22×M16C2311yescharacteristic
65C22×M16C2311yescharacteristic
66C22×M16C2311yescharacteristic
67C23×C8C2311yescharacteristic
68C2×#117C2311yescharacteristic
69C2×#117C2311yescharacteristic
70C2×#117C2311yescharacteristic
71C2×#117C2311yescharacteristic
72#430—22no—
73#430—22no—
74#430—22no—
75#430—41no—
76#430—41no—
77#430C2×C414yes—
78C2×#108—22no—
79C2×#108—22no—
80C2×#108—22no—
81#424—21no—
82#424—21no—
83#424—41no—
84#424—41no—
85#424—41no—
86#423—22no—
87C2×C4⋊C8—41no—
88#421—22no—
89#421—22no—
90#421—41no—
91#421—41no—
92C2×#104—22no—
93C2×#104—22no—
94#419C2×C412yes—
95#419C2×C412yes—
96#418C2×C412yes—
97#418C2×C412yes—
98#417—22no—
99#417—41no—
100#416—22no—
101#416—41no—
102#413—41no—
103#413—41no—
104C2×#101C2×C412yes—
105C2×#100C2×C412yes—
106#408—22no—
107#408—21no—
108#408—21no—
109#408—41no—
110#408—41no—
111#407—41no—
112C2×C22⋊C8—41no—
113#405—22no—
114#405—22no—
115#405—41no—
116#405—41no—
117#405—41no—
118#405—41no—
119C4×M16—22no—
120C4×M16—22no—
121C2×#97D412yes—
122C2×C4×C8D412yes—
123#344—22no—
124#344—41no—
125#344—41no—
126#344—41no—
127#344C2×C414yes—
128#344C2×C414yes—
129#343—41no—
130#341—22no—
131#341—21no—
132#341—21no—
133#341—21no—
134#341—21no—
135#341—41no—
136#341—41no—
137#341—41no—
138#340—41no—
139#339—22no—
140#339—41no—
141#338—41no—
142H(3,ℤ/4)—22no—
143H(3,ℤ/4)—22no—
144H(3,ℤ/4)C2×C414yes—
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×#41—21no—
155C2×#41—21no—
156C2×#41—21no—
157C22×Q8C22×C411yescharacteristic
158C22×D4—21no—
159C22×D4C22×C411yescharacteristic
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×M16—21no—
194C2×M16—21no—
195C2×M16—21no—
196C2×M16—21no—
197C2×M16—21no—
198C2×M16—21no—
199C2×M16—21no—
200C2×M16—21no—
201C2×M16—21no—
202C2×M16—21no—
203C2×M16—21no—
204C2×M16—21no—
205C2×M16—21no—
206C2×M16—21no—
207C2×M16—21no—
208C2×M16—21no—
209C2×M16C22×C412yes—
210C2×M16C22×C412yes—
211C2×M16C22×C412yes—
212C2×M16C22×C412yes—
213C2×M16#4112yes—
214C2×M16#4112yes—
215C2×M16#4112yes—
216C22×C8—21no—
217C22×C8—21no—
218C22×C8—21no—
219C22×C8—21no—
220C22×C8—21no—
221C22×C8—21no—
222C22×C8—21no—
223C22×C8—21no—
224C22×C8—21no—
225C22×C8—21no—
226C22×C8—21no—
227C22×C8—21no—
228C22×C8—21no—
229C22×C8—21no—
230C22×C8C2×D411yescharacteristic
231C22×C8C2×D411yescharacteristic
232C22×C8C2×D412yes—
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—21no—
244#117—21no—
245#117C22×C412yes—
246#117C22×C412yes—
247#117#4112yes—
248#117#4112yes—
249C2×C4⋊C4—21no—
250C2×C4⋊C4—21no—
251C2×C4⋊C4—21no—
252C2×C4⋊C4—21no—
253C2×C22⋊C4—21no—
254C2×C22⋊C4—21no—
255C2×C22⋊C4—21no—
256C2×C22⋊C4—21no—
257C2×C42—21no—
258C2×C42—21no—
259C2×C42—21no—
260C2×C42C2×D412yes—
261#108—28no—
262#108—28no—
263#108—44no—
264C4⋊C8—42no—
265C4⋊C8—42no—
266C4⋊C8—44no—
267#104—44no—
268#104—44no—
269#103—24no—
270#103—42no—
271#102—24no—
272#102—42no—
273#101—42no—
274#100—42no—
275C22⋊C8—42no—
276C22⋊C8—42no—
277C22⋊C8—44no—
278#97—24no—
279#97—42no—
280#97—42no—
281C4×C8—24no—
282C4×C8—42no—
283C4×C8—42no—
284C24—21no—
285C242−1+411yescharacteristic
286#41—41no—
287#41—41no—
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—
302C2×Q8—21no—
303C2×Q8—21no—
304C2×Q8—21no—
305C2×Q8—21no—
306C2×Q8—21no—
307C2×Q8—21no—
308C2×D4—21no—
309C2×D4—21no—
310C2×D4—21no—
311C2×D4—21no—
312C2×D4—21no—
313C2×D4—21no—
314C2×D4—41no—
315C2×D4—41no—
316C2×D4—41no—
317C2×D4—41no—
318C2×D4—41no—
319C2×D4—41no—
320C22×C4—22no—
321C22×C4—21no—
322C22×C4—41no—
323C22×C4—41no—
324C22×C4—41no—
325C22×C4—41no—
326C22×C4—41no—
327C22×C4—41no—
328C22×C4—41no—
329C22×C4—41no—
330C22×C4—41no—
331C22×C4C23×C411yescharacteristic
332C22×C4C22×D411yescharacteristic
333C22×C4C2×#4111yescharacteristic
334C22×C4C2×#4111yescharacteristic
335C22×C4C2×#4111yescharacteristic
336C22×C4C2×#4111yescharacteristic
337C22×C4C2×#4111yescharacteristic
338C22×C4H(5,𝔽2)11yescharacteristic
339C22×C4H(5,𝔽2)11yescharacteristic
340C22×C4H(5,𝔽2)11yescharacteristic
341C22×C4H(5,𝔽2)11yescharacteristic
342C22×C4H(5,𝔽2)11yescharacteristic
343C22×C42−1+411yescharacteristic
344C22×C42−1+411yescharacteristic
345M16—22no—
346M16—22no—
347M16—22no—
348M16—22no—
349M16—22no—
350M16—22no—
351M16—22no—
352M16—22no—
353M16—22no—
354M16—22no—
355M16—22no—
356M16—22no—
357M16—22no—
358M16—22no—
359M16—41no—
360M16—41no—
361M16—41no—
362M16—41no—
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—
379M16—41no—
380M16—41no—
381C2×C8—22no—
382C2×C8—22no—
383C2×C8—22no—
384C2×C8—22no—
385C2×C8—22no—
386C2×C8—21no—
387C2×C8—21no—
388C2×C8—21no—
389C2×C8—21no—
390C2×C8—21no—
391C2×C8—21no—
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×C8C4×D414yes—
418C4⋊C4—22no—
419C4⋊C4—22no—
420C4⋊C4—22no—
421C4⋊C4—22no—
422C4⋊C4—41no—
423C4⋊C4—41no—
424C4⋊C4—41no—
425C4⋊C4—41no—
426C22⋊C4—22no—
427C22⋊C4—22no—
428C22⋊C4—22no—
429C22⋊C4—22no—
430C22⋊C4—41no—
431C22⋊C4—41no—
432C22⋊C4—41no—
433C22⋊C4—41no—
434C42—22no—
435C42—22no—
436C42—22no—
437C42—41no—
438C42—41no—
439C42—41no—
440C42—41no—
441C42#12314yes—
442C23—41no—
443C23—41no—
444C23—41no—
445C23—81no—
446C23—82no—
447C23#52011yescharacteristic
448C23#53611yescharacteristic
449C23#53911yescharacteristic
450C23#53911yescharacteristic
451C23#54111yescharacteristic
452C23#54211yescharacteristic
453C23#54211yescharacteristic
454Q8—41no—
455Q8—41no—
456Q8—41no—
457Q8—41no—
458D4—41no—
459D4—41no—
460D4—41no—
461D4—41no—
462D4—81no—
463D4—81no—
464D4—81no—
465D4—81no—
466C2×C4—24no—
467C2×C4—42no—
468C2×C4—41no—
469C2×C4—41no—
470C2×C4—42no—
471C2×C4—41no—
472C2×C4—82no—
473C2×C4—82no—
474C2×C4—81no—
475C2×C4—81no—
476C2×C4—81no—
477C2×C4—81no—
478C2×C4—81no—
479C2×C4—161no—
480C2×C4—161no—
481C2×C4C2×C4×D411yescharacteristic
482C2×C4C4×#4111yescharacteristic
483C2×C4C4×#4111yescharacteristic
484C2×C4#51811yescharacteristic
485C2×C4#52011yescharacteristic
486C2×C4#52011yescharacteristic
487C2×C4C2×#12311yescharacteristic
488C2×C4#52511yescharacteristic
489C2×C4#52911yescharacteristic
490C2×C4#52911yescharacteristic
491C2×C4#53511yescharacteristic
492C2×C4#53511yescharacteristic
493C2×C4#53711yescharacteristic
494C2×C4#53811yescharacteristic
495C2×C4#53811yescharacteristic
496C2×C4#53811yescharacteristic
497C2×C4#53811yescharacteristic
498C2×C4#53911yescharacteristic
499C2×C4#53911yescharacteristic
500C2×C4#54011yescharacteristic
501C2×C4#54011yescharacteristic
502C2×C4#54011yescharacteristic
503C2×C4#54111yescharacteristic
504C2×C4#54211yescharacteristic
505C2×C4#54211yescharacteristic
506C2×C4#54211yescharacteristic
507C2×C4#54211yescharacteristic
508C2×C4#54211yescharacteristic
509C8—24no—
510C8—42no—
511C8—42no—
512C8—42no—
513C8—41no—
514C8—41no—
515C8—42no—
516C8—42no—
517C8—81no—
518C8—81no—
519C8—81no—
520C22—41no—
521C22—41no—
522C22—41no—
523C22—41no—
524C22—41no—
525C22—41no—
526C22—41no—
527C22—82no—
528C22—82no—
529C22—81no—
530C22—161no—
531C22#232611yescharacteristic
532C22#233411yescharacteristic
533C22#233511yescharacteristic
534C22#262311yescharacteristic
535C22#263711yescharacteristic
536C22#265011yescharacteristic
537C22#265111yescharacteristic
538C4—44no—
539C4—82no—
540C4—82no—
541C4—81no—
542C4—161no—
543C4C4×#12311yescharacteristic
544C4#232411yescharacteristic
545C4#233411yescharacteristic
546C4#233511yescharacteristic
547C4#262411yescharacteristic
548C4#263111yescharacteristic
549C4#265011yescharacteristic
550C4#265111yescharacteristic
551C2—21no—
552C2—21no—
553C2—21no—
554C2—21no—
555C2—21no—
556C2—21no—
557C2—21no—
558C2—161no—
559C2#1430611yescharacteristic
subgroupG/H
centerC4#2631
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
socleC2#14306
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
3. #20459C2semidirect
4. #11363C2semidirect
5. #11334C2semidirect
6. #11281C2semidirect
7. #11264C2semidirect
13. #10905C2semidirect
14. #10890C2semidirect
15. #10804C2semidirect
31. #2082C4semidirect
32. #2081C4semidirect

Maximal subgroups

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

subgroupindex
#236522
#211002
#204592
#113632
#113342
#112812
#112642
#111372
#111312
#110652
#109552
#109502
#109052
#108902
#108042