Finite Group Databasebeta

← #388118

Subgroups — #388118

Order 512.

765 conjugacy classes of subgroups.

The subgroup lattice has 547 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1858C211yesmaximal, characteristic
2#11368C211yesmaximal, characteristic, complemented
3#11367C211yesmaximal, characteristic, complemented
4#11306C211yesmaximal, characteristic, complemented
5#11305C211yesmaximal, characteristic, complemented
6#11151C211yesmaximal, characteristic
7#11150C211yesmaximal, characteristic
8#10956C211yesmaximal, characteristic
9#10929C211yesmaximal, characteristic
10#10902C211yesmaximal, characteristic, complemented
11#10901C211yesmaximal, characteristic, complemented
12#10822C211yesmaximal, characteristic, complemented
13#10818C211yesmaximal, characteristic, complemented
14#10782C211yesmaximal, characteristic
15#10780C211yesmaximal, characteristic
16C2×#568C2211yescharacteristic
17C2×#430C2211yescharacteristic
18C2×#430C2211yescharacteristic
19C2×#421C2211yescharacteristic
20C22×#104C2211yescharacteristic
21#2896C2211yescharacteristic
22C2×#416C2211yescharacteristic
23C2×#413C2211yescharacteristic
24C2×#408C2211yescharacteristic
25#2873C2211yescharacteristic
26C22×C4×C8C2211yescharacteristic
27#1858C418yescomplemented
28C2×#344C2211yescharacteristic
29C2×#344C2211yescharacteristic
30C2×#344C2211yescharacteristic
31C2×#344C2211yescharacteristic
32#1742C2211yescharacteristic
33#1738C2211yescharacteristic
34C2×#341C2211yescharacteristic
35C2×#341C2211yescharacteristic
36C2×#341C2211yescharacteristic
37C2×#341C2211yescharacteristic
38C2×#341C2211yescharacteristic
39C2×#341C2211yescharacteristic
40#1735C2211yescharacteristic
41#1733C2211yescharacteristic
42C2×#339C2211yescharacteristic
43C2×#339C2211yescharacteristic
44C2×#339C2211yescharacteristic
45C2×#338C2211yescharacteristic
46C2×H(3,ℤ/4)C2211yescharacteristic
47#1729C2211yescharacteristic
48#1729C2211yescharacteristic
49#1729C2211yescharacteristic
50#1728C2211yescharacteristic
51#1728C2211yescharacteristic
52C22×#41C2311yescharacteristic
53#568—41no—
54#568—41no—
55#568—41no—
56#568C2×C414yes—
57C2×#131—41no—
58C2×#131—41no—
59C22×M16—41no—
60C22×M16C2311yescharacteristic
61C22×M16C2311yescharacteristic
62C22×M16C2311yescharacteristic
63C22×M16C2311yescharacteristic
64C22×M16C2311yescharacteristic
65C22×M16C2311yescharacteristic
66C22×M16C2311yescharacteristic
67C22×M16C2311yescharacteristic
68C23×C8C2311yescharacteristic
69C23×C8C2311yescharacteristic
70C2×#117C2311yescharacteristic
71C2×#117C2311yescharacteristic
72C2×#117C2311yescharacteristic
73C22×C42C2311yescharacteristic
74#430—41no—
75#430—41no—
76#430C2×C414yes—
77#430C2×C414yes—
78C2×#108—41no—
79C2×#108—41no—
80#421—22no—
81#421—22no—
82C2×#104—22no—
83C2×#104—22no—
84C2×#104—41no—
85C2×#104—41no—
86#419—21no—
87#418—21no—
88#417—22no—
89#416—22no—
90#416—41no—
91#416—41no—
92C2×#103—21no—
93C2×#102—21no—
94#413—22no—
95#413C2×C414yes—
96C2×#101—21no—
97C2×#100—21no—
98#408—41no—
99#408C2×C414yes—
100C2×C22⋊C8—41no—
101#405—41no—
102#405—41no—
103C4×M16—41no—
104C2×C4×C8—22no—
105C2×C4×C8—22no—
106C2×C4×C8—41no—
107#344—22no—
108#344—41no—
109#344—41no—
110#344—41no—
111#343—22no—
112#341—22no—
113#341—22no—
114#341—41no—
115#341—41no—
116#341—41no—
117#341C2×C414yes—
118#341C2×C414yes—
119#340—22no—
120#339—22no—
121#339—41no—
122#339—41no—
123#339—41no—
124#338—41no—
125H(3,ℤ/4)—41no—
126#336—41no—
127#336—41no—
128#336—41no—
129#336—41no—
130#336—41no—
131C2×#41—21no—
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×#41C22×C412yes—
142C2×#41#4112yes—
143C22×Q8#4111yescharacteristic
144C22×D4—21no—
145C22×D4#4111yescharacteristic
146C23×C4—21no—
147C23×C4—21no—
148C23×C4C2411yescharacteristic
149#131—44no—
150#131—44no—
151#131—44no—
152#131—44no—
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×M16—21no—
192C2×M16—21no—
193C2×M16—21no—
194C2×M16—21no—
195C2×M16—44no—
196C2×M16—42no—
197C2×M16—44no—
198C2×M16C22×C412yes—
199C2×M16C22×C412yes—
200C2×M16C22×C412yes—
201C2×M16C22×C412yes—
202C2×M16C22×C412yes—
203C2×M16#4112yes—
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×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—42no—
225C22×C8C22×C412yes—
226C22×C8#4112yes—
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—
239C2×C4⋊C4—21no—
240C2×C4⋊C4—21no—
241C2×C4⋊C4#4112yes—
242C2×C22⋊C4—21no—
243C2×C22⋊C4—21no—
244C2×C22⋊C4—21no—
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#4112yes—
254#108—44no—
255#108—44no—
256#104—44no—
257#104—44no—
258#103—42no—
259#103—42no—
260#102—42no—
261#102—42no—
262#101—24no—
263#100—24no—
264C22⋊C8—44no—
265#97—42no—
266C4×C8—44no—
267C4×C8—42no—
268C24—21no—
269C242−1+411yescharacteristic
270#41—22no—
271#41—22no—
272#41—22no—
273#41—22no—
274#41—22no—
275#41—22no—
276#41—22no—
277#41—22no—
278#41—41no—
279#41—41no—
280#41—41no—
281#41—41no—
282#41—41no—
283#41—41no—
284#41—41no—
285#41—41no—
286C2×Q8—21no—
287C2×Q8—21no—
288C2×Q8—21no—
289C2×Q8—21no—
290C2×Q8—21no—
291C2×Q8#11712yes—
292C2×D4—22no—
293C2×D4—22no—
294C2×D4—21no—
295C2×D4—21no—
296C2×D4—21no—
297C2×D4—21no—
298C2×D4—21no—
299C2×D4—41no—
300C2×D4—41no—
301C2×D4—41no—
302C2×D4—41no—
303C2×D4#11712yes—
304C22×C4—22no—
305C22×C4—21no—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—44no—
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×C4C23×C411yescharacteristic
319C22×C4C2×#4111yescharacteristic
320C22×C4C2×#4111yescharacteristic
321C22×C4C2×#4111yescharacteristic
322C22×C4C2×#4111yescharacteristic
323C22×C4C2×#4111yescharacteristic
324C22×C4C2×#4111yescharacteristic
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
332M16—22no—
333M16—22no—
334M16—22no—
335M16—22no—
336M16—22no—
337M16—22no—
338M16—22no—
339M16—22no—
340M16—22no—
341M16—22no—
342M16—22no—
343M16—22no—
344M16—48no—
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—
359M16—41no—
360M16—41no—
361M16—41no—
362M16—41no—
363M16—41no—
364M16—41no—
365C2×C8—22no—
366C2×C8—22no—
367C2×C8—22no—
368C2×C8—22no—
369C2×C8—22no—
370C2×C8—22no—
371C2×C8—22no—
372C2×C8—22no—
373C2×C8—22no—
374C2×C8—22no—
375C2×C8—22no—
376C2×C8—22no—
377C2×C8—48no—
378C2×C8—41no—
379C2×C8—41no—
380C2×C8—41no—
381C2×C8—41no—
382C2×C8—41no—
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—
402C2×C8—41no—
403C2×C8—41no—
404C4⋊C4—22no—
405C4⋊C4—22no—
406C4⋊C4—41no—
407C4⋊C4—41no—
408C4⋊C4—41no—
409C4⋊C4—41no—
410C22⋊C4—41no—
411C22⋊C4—41no—
412C22⋊C4—41no—
413C22⋊C4—41no—
414C22⋊C4—41no—
415C22⋊C4—41no—
416C42—22no—
417C42—22no—
418C42—41no—
419C42—41no—
420C42—41no—
421C42—41no—
422C42—41no—
423C42—41no—
424C42—41no—
425C42—41no—
426C23—22no—
427C23—42no—
428C23—41no—
429C23—41no—
430C23—82no—
431C23#51911yescharacteristic
432C23#53911yescharacteristic
433C23#53911yescharacteristic
434C23#53911yescharacteristic
435C23#53911yescharacteristic
436C23#54211yescharacteristic
437C23#54211yescharacteristic
438Q8—22no—
439Q8—22no—
440Q8—41no—
441Q8—41no—
442D4—22no—
443D4—22no—
444D4—42no—
445D4—42no—
446D4—42no—
447D4—42no—
448D4—41no—
449D4—41no—
450C2×C4—24no—
451C2×C4—22no—
452C2×C4—42no—
453C2×C4—44no—
454C2×C4—44no—
455C2×C4—42no—
456C2×C4—41no—
457C2×C4—41no—
458C2×C4—42no—
459C2×C4—42no—
460C2×C4—42no—
461C2×C4—42no—
462C2×C4—42no—
463C2×C4—82no—
464C2×C4—82no—
465C2×C4—82no—
466C2×C4—81no—
467C2×C4—81no—
468C2×C4C2×#11711yescharacteristic
469C2×C4C4×#4111yescharacteristic
470C2×C4C4×#4111yescharacteristic
471C2×C4#51811yescharacteristic
472C2×C4#51811yescharacteristic
473C2×C4#51811yescharacteristic
474C2×C4#53911yescharacteristic
475C2×C4#53911yescharacteristic
476C2×C4#53911yescharacteristic
477C2×C4#53911yescharacteristic
478C2×C4#54211yescharacteristic
479C2×C4#54211yescharacteristic
480C2×C4#55111yescharacteristic
481C2×C4#55111yescharacteristic
482C2×C4#55111yescharacteristic
483C2×C4#55111yescharacteristic
484C2×C4#55111yescharacteristic
485C2×C4#55311yescharacteristic
486C2×C4#55311yescharacteristic
487C2×C4#55311yescharacteristic
488C2×C4#55311yescharacteristic
489C2×C4#55311yescharacteristic
490C2×C4#55311yescharacteristic
491C2×C4#55911yescharacteristic
492C2×C4#56011yescharacteristic
493C2×C4#56011yescharacteristic
494C2×C4#56011yescharacteristic
495C2×C4H(3,𝔽4)11yescharacteristic
496C8—42no—
497C8—42no—
498C8—42no—
499C8—42no—
500C8—42no—
501C8—42no—
502C8—42no—
503C8—42no—
504C8—81no—
505C8—81no—
506C22—22no—
507C22—22no—
508C22—22no—
509C22—41no—
510C22—41no—
511C22—41no—
512C22—41no—
513C22—42no—
514C22—82no—
515C22—82no—
516C22—161no—
517C22#231311yescharacteristic
518C22#233211yescharacteristic
519C22#233211yescharacteristic
520C22#274611yescharacteristic
521C22#275711yescharacteristic
522C22#278211yescharacteristic
523C22#279611yescharacteristic
524C4—44no—
525C4—42no—
526C4—82no—
527C4—161no—
528C4—161no—
529C4#235311yescharacteristic
530C4#235311yescharacteristic
531C4#237411yescharacteristic
532C4#237511yescharacteristic
533C4#274611yescharacteristic
534C4#275711yescharacteristic
535C4#278211yescharacteristic
536C4#279611yescharacteristic
537C2—21no—
538C2—21no—
539C2—21no—
540C2—21no—
541C2—21no—
542C2—21no—
543C2—82no—
544C2#1451511yescharacteristic
545C2#3270212yes—
subgroupG/H
centerC2×C4H(3,𝔽4)
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
socleC22#2782
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
2. #11368C2semidirect
3. #11367C2semidirect
4. #11306C2semidirect
5. #11305C2semidirect
10. #10902C2semidirect
11. #10901C2semidirect
12. #10822C2semidirect
13. #10818C2semidirect
27. #1858C4semidirect

Maximal subgroups

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

subgroupindex
C2×#18582
#113682
#113672
#113062
#113052
#111512
#111502
#109562
#109292
#109022
#109012
#108222
#108182
#107822
#107802