Finite Group Databasebeta

← #391175

Subgroups — #391175

Order 512.

686 conjugacy classes of subgroups.

The subgroup lattice has 553 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23652C211yesmaximal, characteristic
2#11285C211yesmaximal, characteristic, complemented
3#11251C211yesmaximal, characteristic, complemented
4#11180C211yesmaximal, characteristic
5#11141C211yesmaximal, characteristic
6#11065C211yesmaximal, characteristic
7#10961C211yesmaximal, characteristic
8#10956C211yesmaximal, characteristic
9#10951C211yesmaximal, characteristic
10#10917C211yesmaximal, characteristic, complemented
11#10902C211yesmaximal, characteristic, complemented
12#10890C211yesmaximal, characteristic, complemented
13#10831C211yesmaximal, characteristic, complemented
14#10828C211yesmaximal, characteristic, complemented
15#10804C211yesmaximal, characteristic, complemented
16C2×#430C2211yescharacteristic
17#2911C2211yescharacteristic
18#2905C2211yescharacteristic
19C2×#421C2211yescharacteristic
20C2×#421C2211yescharacteristic
21#2896C2211yescharacteristic
22#2896C2211yescharacteristic
23C2×#413C2211yescharacteristic
24C2×#410C2211yescharacteristic
25C2×#408C2211yescharacteristic
26#2873C2211yescharacteristic
27#2873C2211yescharacteristic
28C2×#405C2211yescharacteristic
29#2082C414yescomplemented
30#2081C414yescomplemented
31C2×#344C2211yescharacteristic
32C2×#344C2211yescharacteristic
33C2×#344C2211yescharacteristic
34C2×#344C2211yescharacteristic
35C2×#344C2211yescharacteristic
36C2×#344C2211yescharacteristic
37C2×#344C2211yescharacteristic
38C2×#344C2211yescharacteristic
39#1742C2211yescharacteristic
40#1740C2211yescharacteristic
41#1740C2211yescharacteristic
42#1738C2211yescharacteristic
43C2×#341C2211yescharacteristic
44C2×#341C2211yescharacteristic
45#1735C2211yescharacteristic
46#1735C2211yescharacteristic
47#1733C2211yescharacteristic
48#1733C2211yescharacteristic
49C2×H(3,ℤ/4)C2211yescharacteristic
50C2×H(3,ℤ/4)C2211yescharacteristic
51C2×H(3,ℤ/4)C2211yescharacteristic
52#1729C2211yescharacteristic
53C22×#41C2311yescharacteristic
54C22×M16C2311yescharacteristic
55C22×M16C2311yescharacteristic
56C22×M16C2311yescharacteristic
57C22×M16C2311yescharacteristic
58C22×M16C2311yescharacteristic
59C22×M16C2311yescharacteristic
60C22×M16C2311yescharacteristic
61C23×C8C2311yescharacteristic
62C2×#117C2311yescharacteristic
63C2×#117C2311yescharacteristic
64C2×#117C2311yescharacteristic
65C2×#117C2311yescharacteristic
66C2×#117C2311yescharacteristic
67C2×#117C2311yescharacteristic
68#430—41no—
69#430C2×C414yes—
70C2×#108—41no—
71#428—41no—
72#428—41no—
73#427—41no—
74#424—22no—
75#424—41no—
76#423—22no—
77#421—22no—
78#421—41no—
79#421—41no—
80#421—41no—
81C2×#104—22no—
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—
97C2×#99—22no—
98#408—21no—
99#408—21no—
100#408—41no—
101C2×C22⋊C8—22no—
102#405—22no—
103#405—22no—
104#405—41no—
105#405—41no—
106#405—41no—
107#405—41no—
108C4×M16—41no—
109C4×M16—41no—
110C2×#97—21no—
111C2×C4×C8—21no—
112#344—22no—
113#344—22no—
114#344—41no—
115#344—41no—
116#344—41no—
117#344—41no—
118#344C2×C414yes—
119#344C2×C414yes—
120#343—22no—
121#342—22no—
122#342—22no—
123#341—21no—
124#341—21no—
125#341—41no—
126#341—41no—
127#340—41no—
128#340—41no—
129#339—41no—
130#339—41no—
131H(3,ℤ/4)—22no—
132H(3,ℤ/4)—41no—
133H(3,ℤ/4)C2×C414yes—
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×#41—21no—
146C2×#41—21no—
147C22×Q8C22×C411yescharacteristic
148C22×D4—21no—
149C22×D4C22×C411yescharacteristic
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×M16C22×C412yes—
189C2×M16C22×C412yes—
190C2×M16C22×C412yes—
191C2×M16C22×C412yes—
192C2×M16#4112yes—
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×C8#4111yescharacteristic
207C22×C8#4111yescharacteristic
208C22×C8#4112yes—
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#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#117C22×C412yes—
231#117C22×C412yes—
232#117#4112yes—
233C2×C4⋊C4—21no—
234C2×C4⋊C4—21no—
235C2×C4⋊C4—21no—
236C2×C4⋊C4—21no—
237C2×C4⋊C4—21no—
238C2×C4⋊C4—21no—
239C2×C22⋊C4—21no—
240C2×C22⋊C4—21no—
241C2×C22⋊C4—21no—
242C2×C22⋊C4—21no—
243C2×C22⋊C4—21no—
244C2×C22⋊C4#4112yes—
245C2×C42—21no—
246C2×C42—21no—
247C2×C42—21no—
248C2×C42—21no—
249C2×C42—21no—
250C2×C42—21no—
251#108—44no—
252#107—42no—
253#106—42no—
254C4⋊C8—44no—
255#104—44no—
256#104—44no—
257#103—42no—
258#103—42no—
259#102—42no—
260#102—42no—
261#101—42no—
262#100—42no—
263#99—44no—
264C22⋊C8—42no—
265C22⋊C8—42no—
266#97—42no—
267#97—42no—
268#97—42no—
269C4×C8—42no—
270C4×C8—42no—
271C4×C8—42no—
272C24—21no—
273C24C2×#4111yescharacteristic
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—
285#41—41no—
286#41—41no—
287#41—41no—
288#41—41no—
289#41—41no—
290C2×Q8—21no—
291C2×Q8—21no—
292C2×Q8—21no—
293C2×Q8—21no—
294C2×Q8—21no—
295C2×Q8—21no—
296C2×D4—21no—
297C2×D4—21no—
298C2×D4—21no—
299C2×D4—21no—
300C2×D4—21no—
301C2×D4—21no—
302C2×D4—41no—
303C2×D4—41no—
304C2×D4—41no—
305C2×D4—41no—
306C2×D4—41no—
307C2×D4—41no—
308C22×C4—21no—
309C22×C4—22no—
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—41no—
321C22×C4C23×C411yescharacteristic
322C22×C4C2×#4111yescharacteristic
323C22×C4C2×#4111yescharacteristic
324C22×C4C2×#4111yescharacteristic
325C22×C4C2×#4111yescharacteristic
326C22×C4C2×#4111yescharacteristic
327C22×C4H(5,𝔽2)11yescharacteristic
328C22×C4H(5,𝔽2)11yescharacteristic
329C22×C4H(5,𝔽2)11yescharacteristic
330C22×C4H(5,𝔽2)11yescharacteristic
331C22×C4H(5,𝔽2)11yescharacteristic
332C22×C4H(5,𝔽2)11yescharacteristic
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—22no—
346M16—22no—
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—
363C2×C8—22no—
364C2×C8—22no—
365C2×C8—22no—
366C2×C8—22no—
367C2×C8—22no—
368C2×C8—21no—
369C2×C8—21no—
370C2×C8—21no—
371C2×C8—21no—
372C2×C8—21no—
373C2×C8—21no—
374C2×C8—22no—
375C2×C8—22no—
376C2×C8—22no—
377C2×C8—22no—
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—
394C4⋊C4—22no—
395C4⋊C4—22no—
396C4⋊C4—22no—
397C4⋊C4—41no—
398C4⋊C4—41no—
399C4⋊C4—41no—
400C4⋊C4—41no—
401C4⋊C4—41no—
402C4⋊C4—41no—
403C4⋊C4—41no—
404C4⋊C4—41no—
405C4⋊C4—41no—
406C22⋊C4—22no—
407C22⋊C4—22no—
408C22⋊C4—22no—
409C22⋊C4—22no—
410C22⋊C4—22no—
411C22⋊C4—41no—
412C22⋊C4—41no—
413C22⋊C4—41no—
414C22⋊C4—41no—
415C22⋊C4—41no—
416C22⋊C4—41no—
417C22⋊C4—41no—
418C42—22no—
419C42—22no—
420C42—22no—
421C42—41no—
422C42—41no—
423C42—41no—
424C42—41no—
425C42—41no—
426C42—41no—
427C42—41no—
428C42—41no—
429C42—41no—
430C23—41no—
431C23—41no—
432C23—41no—
433C23—81no—
434C23—82no—
435C23C4×#4111yescharacteristic
436C23#53911yescharacteristic
437C23#54011yescharacteristic
438C23#55111yescharacteristic
439C23#55311yescharacteristic
440C23#55511yescharacteristic
441C23#55611yescharacteristic
442Q8—41no—
443Q8—41no—
444Q8—41no—
445Q8—41no—
446D4—41no—
447D4—41no—
448D4—41no—
449D4—41no—
450D4—81no—
451D4—81no—
452D4—81no—
453D4—81no—
454C2×C4—42no—
455C2×C4—41no—
456C2×C4—42no—
457C2×C4—41no—
458C2×C4—41no—
459C2×C4—42no—
460C2×C4—42no—
461C2×C4—82no—
462C2×C4—81no—
463C2×C4—81no—
464C2×C4—81no—
465C2×C4—81no—
466C2×C4—82no—
467C2×C4—81no—
468C2×C4—81no—
469C2×C4—161no—
470C2×C4—161no—
471C2×C4—161no—
472C2×C4—161no—
473C2×C4C4×#4111yescharacteristic
474C2×C4C4×#4111yescharacteristic
475C2×C4#51811yescharacteristic
476C2×C4#51811yescharacteristic
477C2×C4#52011yescharacteristic
478C2×C4#52011yescharacteristic
479C2×C4#53811yescharacteristic
480C2×C4#53811yescharacteristic
481C2×C4#53811yescharacteristic
482C2×C4#53911yescharacteristic
483C2×C4#53911yescharacteristic
484C2×C4#54011yescharacteristic
485C2×C4#54111yescharacteristic
486C2×C4#54211yescharacteristic
487C2×C4#54211yescharacteristic
488C2×C4#54211yescharacteristic
489C2×C4#55111yescharacteristic
490C2×C4#55111yescharacteristic
491C2×C4#55111yescharacteristic
492C2×C4#55211yescharacteristic
493C2×C4#55311yescharacteristic
494C2×C4#55311yescharacteristic
495C2×C4#55511yescharacteristic
496C2×C4#55911yescharacteristic
497C2×C4#56011yescharacteristic
498C2×C4#56011yescharacteristic
499C2×C4#56211yescharacteristic
500C2×C4#56211yescharacteristic
501C8—42no—
502C8—42no—
503C8—42no—
504C8—42no—
505C8—41no—
506C8—41no—
507C8—42no—
508C8—42no—
509C8—42no—
510C22—41no—
511C22—41no—
512C22—41no—
513C22—41no—
514C22—41no—
515C22—41no—
516C22—41no—
517C22—82no—
518C22—81no—
519C22—161no—
520C22—161no—
521C22#233511yescharacteristic
522C22#237211yescharacteristic
523C22#237311yescharacteristic
524C22#274011yescharacteristic
525C22#275711yescharacteristic
526C22#278311yescharacteristic
527C22#280311yescharacteristic
528C4—82no—
529C4—82no—
530C4—82no—
531C4—82no—
532C4—81no—
533C4—161no—
534C4—161no—
535C4#233411yescharacteristic
536C4#233511yescharacteristic
537C4#237411yescharacteristic
538C4#237711yescharacteristic
539C4#273711yescharacteristic
540C4#274911yescharacteristic
541C4#275411yescharacteristic
542C4#278111yescharacteristic
543C2—21no—
544C2—21no—
545C2—21no—
546C2—21no—
547C2—21no—
548C2—21no—
549C2—21no—
550C2—161no—
551C2#1461911yescharacteristic
subgroupG/H
centerC4#2749
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
socleC2#14619
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. #11285C2semidirect
3. #11251C2semidirect
10. #10917C2semidirect
11. #10902C2semidirect
12. #10890C2semidirect
13. #10831C2semidirect
14. #10828C2semidirect
15. #10804C2semidirect
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
#112852
#112512
#111802
#111412
#110652
#109612
#109562
#109512
#109172
#109022
#108902
#108312
#108282
#108042