Finite Group Databasebeta

← #387573

Subgroups — #387573

Order 512.

755 conjugacy classes of subgroups.

The subgroup lattice has 565 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21100C211yesmaximal, characteristic
2#11369C211yesmaximal, characteristic, complemented
3#11309C211yesmaximal, characteristic, complemented
4#11278C211yesmaximal, characteristic, complemented
5#11266C211yesmaximal, characteristic, complemented
6#11179C211yesmaximal, characteristic
7#11112C211yesmaximal, characteristic
8#10955C211yesmaximal, characteristic
9#10920C211yesmaximal, characteristic, complemented
10#10810C211yesmaximal, characteristic, complemented
11#10803C211yesmaximal, characteristic, complemented
12#10800C211yesmaximal, characteristic, complemented
13#10792C211yesmaximal, characteristic
14#10783C211yesmaximal, characteristic
15#10747C211yesmaximal, characteristic
16#3428C2211yescharacteristic
17C2×#430C2211yescharacteristic
18C2×#424C2211yescharacteristic
19C2×#421C2211yescharacteristic
20C2×#421C2211yescharacteristic
21C2×#421C2211yescharacteristic
22#2896C2211yescharacteristic
23C2×#413C2211yescharacteristic
24C2×#408C2211yescharacteristic
25#2873C2211yescharacteristic
26C2×#405C2211yescharacteristic
27#1938C414yescomplemented
28#1937C414yescomplemented
29C2×#344C2211yescharacteristic
30C2×#344C2211yescharacteristic
31C2×#344C2211yescharacteristic
32C2×#344C2211yescharacteristic
33#1742C2211yescharacteristic
34#1742C2211yescharacteristic
35#1742C2211yescharacteristic
36#1740C2211yescharacteristic
37#1740C2211yescharacteristic
38#1738C2211yescharacteristic
39#1738C2211yescharacteristic
40#1733C2211yescharacteristic
41#1733C2211yescharacteristic
42#1733C2211yescharacteristic
43C2×#338C2211yescharacteristic
44C2×H(3,ℤ/4)C2211yescharacteristic
45C2×H(3,ℤ/4)C2211yescharacteristic
46C2×H(3,ℤ/4)C2211yescharacteristic
47C2×H(3,ℤ/4)C2211yescharacteristic
48C2×H(3,ℤ/4)C2211yescharacteristic
49#1729C2211yescharacteristic
50#1729C2211yescharacteristic
51#1729C2211yescharacteristic
52#1729C2211yescharacteristic
53C22×#41C2311yescharacteristic
54C22×M16C2311yescharacteristic
55C22×M16C2311yescharacteristic
56C22×M16C2311yescharacteristic
57C22×M16C2311yescharacteristic
58C22×M16C2311yescharacteristic
59C22×M16C2311yescharacteristic
60C22×M16C2311yescharacteristic
61C23×C8C2311yescharacteristic
62#520—41no—
63#520—41no—
64#519—21no—
65#518—22no—
66#518—21no—
67C4×#41—22no—
68C4×#41—41no—
69C2×#117—41no—
70C2×#117C2311yescharacteristic
71C2×#117C2311yescharacteristic
72C2×#117C2311yescharacteristic
73C2×#117C2311yescharacteristic
74C2×#117C2311yescharacteristic
75C2×#117C2311yescharacteristic
76#430—41no—
77#430—41no—
78C2×#108—41no—
79#424—41no—
80#424C2×C412yescomplemented
81#424C2×C412yescomplemented
82C2×C4⋊C8—41no—
83#421—22no—
84#421—22no—
85#421—41no—
86#421—41no—
87#421—41no—
88#421—41no—
89C2×#104—22no—
90C2×#104—41no—
91C2×#104C2×C414yes—
92#419—21no—
93#418—21no—
94#417—22no—
95#416C2×C414yes—
96#413—22no—
97#413—41no—
98C2×#101—21no—
99C2×#100—21no—
100#408—41no—
101#408C2×C412yes—
102#408C2×C412yes—
103C2×C22⋊C8—41no—
104#405—41no—
105#405—41no—
106#405—41no—
107#405—41no—
108C4×M16—22no—
109C2×#97C2×C412yescomplemented
110C2×C4×C8C2×C412yescomplemented
111#344—22no—
112#344—41no—
113#344—41no—
114#344—41no—
115#343—22no—
116#343—41no—
117#343—41no—
118#342—22no—
119#342—22no—
120#341—41no—
121#341C2×C414yescomplemented
122#339—22no—
123#339—41no—
124#339C2×C414yescomplemented
125#338—41no—
126H(3,ℤ/4)—41no—
127H(3,ℤ/4)—41no—
128H(3,ℤ/4)—41no—
129H(3,ℤ/4)—41no—
130H(3,ℤ/4)—41no—
131#336—22no—
132#336—22no—
133#336—41no—
134#336—41no—
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—
144C2×#41—21no—
145C2×#41C22×C412yes—
146C2×#41#4112yes—
147C22×Q8#4111yescharacteristic
148C22×D4—21no—
149C22×D4#4111yescharacteristic
150C23×C4—21no—
151C23×C4C2411yescharacteristic
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×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×M16C22×C412yes—
192C2×M16C22×C412yes—
193C2×M16#4112yes—
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×C8C22×C411yescharacteristic
207C22×C8C22×C411yescharacteristic
208C22×C8C22×C412yes—
209C4×Q8—42no—
210C4×Q8—42no—
211C4×D4—42no—
212C4×D4—44no—
213C4×D4—44no—
214C4×D4—42no—
215#117—21no—
216#117—21no—
217#117—21no—
218#117—21no—
219#117—21no—
220#117—21no—
221#117—21no—
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—
234#117—21no—
235#117—21no—
236#117—21no—
237#117—21no—
238#117—42no—
239#117—42no—
240#117—42no—
241#117C22×C412yes—
242C2×C4⋊C4—21no—
243C2×C4⋊C4—21no—
244C2×C4⋊C4—21no—
245C2×C4⋊C4—21no—
246C2×C4⋊C4—21no—
247C2×C4⋊C4—21no—
248C2×C4⋊C4—42no—
249C2×C22⋊C4—21no—
250C2×C22⋊C4—21no—
251C2×C22⋊C4—21no—
252C2×C22⋊C4—21no—
253C2×C22⋊C4—21no—
254C2×C22⋊C4—42no—
255C2×C22⋊C4#4112yes—
256C2×C42—21no—
257C2×C42—21no—
258C2×C42—21no—
259C2×C42—42no—
260C2×C42C22×C412yes—
261C2×C42#4112yes—
262C2×C42#4112yes—
263#108—44no—
264C4⋊C8—42no—
265C4⋊C8—42no—
266#104—44no—
267#104—44no—
268#104—44no—
269#103—24no—
270#102—24no—
271#101—42no—
272#100—42no—
273C22⋊C8—42no—
274C22⋊C8—42no—
275#97—42no—
276#97—42no—
277C4×C8—42no—
278C4×C8—42no—
279C24—21no—
280C24C2×#4111yescharacteristic
281#41—22no—
282#41—22no—
283#41—22no—
284#41—22no—
285#41—22no—
286#41—22no—
287#41—22no—
288#41—22no—
289#41—41no—
290#41—41no—
291#41—41no—
292#41—41no—
293#41—41no—
294#41—41no—
295#41—41no—
296#41—41no—
297C2×Q8—21no—
298C2×Q8—21no—
299C2×Q8—21no—
300C2×Q8—21no—
301C2×Q8—21no—
302C2×Q8#11712yes—
303C2×D4—22no—
304C2×D4—22no—
305C2×D4—21no—
306C2×D4—21no—
307C2×D4—21no—
308C2×D4—21no—
309C2×D4—21no—
310C2×D4—41no—
311C2×D4—41no—
312C2×D4—41no—
313C2×D4—41no—
314C2×D4#11712yes—
315C22×C4—21no—
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—41no—
323C22×C4—41no—
324C22×C4—41no—
325C22×C4—41no—
326C22×C4—41no—
327C22×C4—44no—
328C22×C4—41no—
329C22×C4C23×C411yescharacteristic
330C22×C4C2×#4111yescharacteristic
331C22×C4C2×#4111yescharacteristic
332C22×C4C2×#4111yescharacteristic
333C22×C4C2×#4111yescharacteristic
334C22×C4C2×#4111yescharacteristic
335C22×C4H(5,𝔽2)11yescharacteristic
336C22×C4H(5,𝔽2)11yescharacteristic
337C22×C4H(5,𝔽2)11yescharacteristic
338C22×C4H(5,𝔽2)11yescharacteristic
339C22×C4H(5,𝔽2)11yescharacteristic
340C22×C4H(5,𝔽2)11yescharacteristic
341C22×C42−1+411yescharacteristic
342C22×C42−1+411yescharacteristic
343M16—22no—
344M16—22no—
345M16—22no—
346M16—22no—
347M16—22no—
348M16—22no—
349M16—41no—
350M16—41no—
351M16—41no—
352M16—41no—
353M16—41no—
354M16—41no—
355M16—41no—
356M16—41no—
357M16—41no—
358M16—41no—
359M16—41no—
360M16—41no—
361M16—41no—
362M16—41no—
363M16—41no—
364M16—41no—
365M16—41no—
366M16—41no—
367M16—41no—
368M16—41no—
369M16—41no—
370M16—41no—
371C2×C8—22no—
372C2×C8—21no—
373C2×C8—21no—
374C2×C8—21no—
375C2×C8—22no—
376C2×C8—21no—
377C2×C8—21no—
378C2×C8—21no—
379C2×C8—22no—
380C2×C8—22no—
381C2×C8—22no—
382C2×C8—22no—
383C2×C8—41no—
384C2×C8—41no—
385C2×C8—41no—
386C2×C8—41no—
387C2×C8—41no—
388C2×C8—41no—
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—
402C4⋊C4—22no—
403C4⋊C4—44no—
404C4⋊C4—41no—
405C4⋊C4—41no—
406C4⋊C4—41no—
407C4⋊C4—41no—
408C4⋊C4—41no—
409C4⋊C4—41no—
410C4⋊C4—41no—
411C4⋊C4—41no—
412C4⋊C4—41no—
413C4⋊C4—41no—
414C4⋊C4—41no—
415C22⋊C4—22no—
416C22⋊C4—22no—
417C22⋊C4—22no—
418C22⋊C4—44no—
419C22⋊C4—41no—
420C22⋊C4—41no—
421C22⋊C4—41no—
422C22⋊C4—41no—
423C22⋊C4—41no—
424C22⋊C4—41no—
425C22⋊C4—41no—
426C22⋊C4—41no—
427C22⋊C4—41no—
428C42—22no—
429C42—22no—
430C42—22no—
431C42—22no—
432C42—22no—
433C42—22no—
434C42—22no—
435C42—44no—
436C42—41no—
437C42—41no—
438C42—41no—
439C42—41no—
440C42—41no—
441C23—22no—
442C23—41no—
443C23—41no—
444C23—81no—
445C23—82no—
446C23C4×#4111yescharacteristic
447C23#53811yescharacteristic
448C23#54211yescharacteristic
449C23#55111yescharacteristic
450C23#55111yescharacteristic
451C23#55211yescharacteristic
452C23#55311yescharacteristic
453Q8—22no—
454Q8—22no—
455Q8—41no—
456Q8—41no—
457D4—22no—
458D4—22no—
459D4—42no—
460D4—42no—
461D4—42no—
462D4—42no—
463D4—41no—
464D4—41no—
465C2×C4—22no—
466C2×C4—41no—
467C2×C4—41no—
468C2×C4—42no—
469C2×C4—42no—
470C2×C4—42no—
471C2×C4—42no—
472C2×C4—42no—
473C2×C4—42no—
474C2×C4—42no—
475C2×C4—42no—
476C2×C4—42no—
477C2×C4—82no—
478C2×C4—82no—
479C2×C4—82no—
480C2×C4—82no—
481C2×C4—82no—
482C2×C4—82no—
483C2×C4—81no—
484C2×C4—164no—
485C2×C4C2×#11711yescharacteristic
486C2×C4C4×#4111yescharacteristic
487C2×C4#51811yescharacteristic
488C2×C4#51811yescharacteristic
489C2×C4#52011yescharacteristic
490C2×C4#52011yescharacteristic
491C2×C4#53811yescharacteristic
492C2×C4#53811yescharacteristic
493C2×C4#53811yescharacteristic
494C2×C4#53811yescharacteristic
495C2×C4#53911yescharacteristic
496C2×C4#53911yescharacteristic
497C2×C4#53911yescharacteristic
498C2×C4#54011yescharacteristic
499C2×C4#54011yescharacteristic
500C2×C4#54011yescharacteristic
501C2×C4#55111yescharacteristic
502C2×C4#55111yescharacteristic
503C2×C4#55111yescharacteristic
504C2×C4#55211yescharacteristic
505C2×C4#55211yescharacteristic
506C2×C4#55311yescharacteristic
507C2×C4#55311yescharacteristic
508C2×C4#55611yescharacteristic
509C2×C4H(3,𝔽4)11yescharacteristic
510C2×C4#56211yescharacteristic
511C2×C4#56211yescharacteristic
512C2×C4#56311yescharacteristic
513C8—42no—
514C8—41no—
515C8—41no—
516C8—42no—
517C8—42no—
518C8—42no—
519C8—81no—
520C8—81no—
521C8—81no—
522C22—41no—
523C22—41no—
524C22—41no—
525C22—41no—
526C22—41no—
527C22—41no—
528C22—41no—
529C22—42no—
530C22—82no—
531C22—82no—
532C22—161no—
533C22#233511yescharacteristic
534C22#235311yescharacteristic
535C22#237311yescharacteristic
536C22#273611yescharacteristic
537C22#273811yescharacteristic
538C22#274111yescharacteristic
539C22#275011yescharacteristic
540C4—42no—
541C4—82no—
542C4—82no—
543C4—82no—
544C4—82no—
545C4—82no—
546C4—161no—
547C4#231811yescharacteristic
548C4#233311yescharacteristic
549C4#234311yescharacteristic
550C4#237911yescharacteristic
551C4#274011yescharacteristic
552C4#275011yescharacteristic
553C4#275311yescharacteristic
554C4#278711yescharacteristic
555C2—21no—
556C2—21no—
557C2—21no—
558C2—21no—
559C2—21no—
560C2—21no—
561C2—21no—
562C2—82no—
563C2#1448111yescharacteristic
subgroupG/H
centerC4#2333
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
socleC2#14481
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. #11369C2semidirect
3. #11309C2semidirect
4. #11278C2semidirect
5. #11266C2semidirect
9. #10920C2semidirect
10. #10810C2semidirect
11. #10803C2semidirect
12. #10800C2semidirect
27. #1938C4semidirect
28. #1937C4semidirect
80. #424C2×C4semidirect
81. #424C2×C4semidirect
109. C2×#97C2×C4semidirect
110. C2×C4×C8C2×C4semidirect
121. #341C2×C4semidirect
124. #339C2×C4semidirect

Maximal subgroups

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

subgroupindex
#211002
#113692
#113092
#112782
#112662
#111792
#111122
#109552
#109202
#108102
#108032
#108002
#107922
#107832
#107472