Finite Group Databasebeta

← #474311

Subgroups — #474311

Order 512.

637 conjugacy classes of subgroups.

The subgroup lattice has 532 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11306C211yesmaximal, characteristic, complemented
2#11300C211yesmaximal, characteristic, complemented
3#11299C211yesmaximal, characteristic, complemented
4#11272C211yesmaximal, characteristic, complemented
5#11264C211yesmaximal, characteristic, complemented
6#11259C211yesmaximal, characteristic, complemented
7#11177C211yesmaximal, characteristic
8#11165C211yesmaximal, characteristic
9#11165C211yesmaximal, characteristic
10#11108C211yesmaximal, characteristic
11#11099C211yesmaximal, characteristic
12#11086C211yesmaximal, characteristic
13#10929C211yesmaximal, characteristic
14#10882C211yesmaximal, characteristic, complemented
15#10864C211yesmaximal, characteristic, complemented
16C22×#108C2211yescharacteristic
17C22×#108C2211yescharacteristic
18C2×#424C2211yescharacteristic
19C2×#421C2211yescharacteristic
20C2×#421C2211yescharacteristic
21C22×#104C2211yescharacteristic
22C2×#416C2211yescharacteristic
23C2×#413C2211yescharacteristic
24C2×#413C2211yescharacteristic
25C2×#408C2211yescharacteristic
26C22×C4×C8C2211yescharacteristic
27C2×#344C2211yescharacteristic
28C2×#344C2211yescharacteristic
29C2×#344C2211yescharacteristic
30C2×#344C2211yescharacteristic
31C2×#344C2211yescharacteristic
32C2×#344C2211yescharacteristic
33#1738C2211yescharacteristic
34#1738C2211yescharacteristic
35C2×#341C2211yescharacteristic
36C2×#341C2211yescharacteristic
37C2×#341C2211yescharacteristic
38C2×#341C2211yescharacteristic
39#1735C2211yescharacteristic
40C2×#339C2211yescharacteristic
41C2×#339C2211yescharacteristic
42C2×#339C2211yescharacteristic
43C2×#338C2211yescharacteristic
44C2×#338C2211yescharacteristic
45C2×H(3,ℤ/4)C2211yescharacteristic
46C2×H(3,ℤ/4)C2211yescharacteristic
47#1729C2211yescharacteristic
48#1729C2211yescharacteristic
49#1728C2211yescharacteristic
50#1728C2211yescharacteristic
51C22×#41C2311yescharacteristic
52C22×M16C2311yescharacteristic
53C22×M16C2311yescharacteristic
54C22×M16C2311yescharacteristic
55C22×M16C2311yescharacteristic
56C22×M16C2311yescharacteristic
57C22×M16C2311yescharacteristic
58C22×M16C2311yescharacteristic
59C22×M16C2311yescharacteristic
60C23×C8C2311yescharacteristic
61C23×C8C2311yescharacteristic
62C2×#117C2311yescharacteristic
63C2×#117C2311yescharacteristic
64C2×#117C2311yescharacteristic
65C22×C42C2311yescharacteristic
66C2×#108—22no—
67C2×#108—22no—
68C2×#108—41no—
69C2×#108—41no—
70C2×#108—41no—
71C2×#108—41no—
72#424—22no—
73#424—41no—
74C2×C4⋊C8—41no—
75#421—22no—
76#421—41no—
77#421—41no—
78#421—41no—
79C2×#104—22no—
80C2×#104—41no—
81C2×#104—41no—
82C2×#104—41no—
83C2×#104—41no—
84#416—22no—
85#416—41no—
86C2×#103—21no—
87C2×#102—21no—
88#413—41no—
89#413—41no—
90#413—41no—
91#413—41no—
92C2×#101—21no—
93C2×#101—21no—
94C2×#100—21no—
95C2×#100—21no—
96#408—22no—
97#408—22no—
98C2×C22⋊C8—22no—
99C2×C4×C8—22no—
100C2×C4×C8—41no—
101C2×C4×C8—41no—
102#344—22no—
103#344—41no—
104#344—41no—
105#344—41no—
106#344—41no—
107#344—41no—
108#341—22no—
109#341—22no—
110#341—41no—
111#341—41no—
112#341—41no—
113#341—41no—
114#340—22no—
115#339—22no—
116#339—41no—
117#339—41no—
118#338—41no—
119#338—41no—
120H(3,ℤ/4)—41no—
121H(3,ℤ/4)—41no—
122#336—22no—
123#336—22no—
124#336—22no—
125#336—41no—
126C2×#41—21no—
127C2×#41—21no—
128C2×#41—21no—
129C2×#41—21no—
130C2×#41—21no—
131C2×#41—21no—
132C2×#41—21no—
133C2×#41—21no—
134C2×#41—21no—
135C2×#41—21no—
136C2×#41—21no—
137C2×#41—21no—
138C22×Q8#4111yescharacteristic
139C22×D4—21no—
140C22×D4#4111yescharacteristic
141C23×C4—21no—
142C23×C4—21no—
143C23×C4C2411yescharacteristic
144C2×M16—21no—
145C2×M16—21no—
146C2×M16—21no—
147C2×M16—21no—
148C2×M16—21no—
149C2×M16—21no—
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×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—
192C22×C8—21no—
193C22×C8—21no—
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—21no—
207C22×C8—21no—
208C22×C8—21no—
209C22×C8—21no—
210C22×C8#4112yes—
211C22×C8#4112yes—
212C22×C8#4112yes—
213C22×C8#4112yes—
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#4112yes—
226C2×C4⋊C4—21no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4—21no—
229C2×C22⋊C4—21no—
230C2×C22⋊C4—21no—
231C2×C22⋊C4—21no—
232C2×C42—21no—
233C2×C42—21no—
234C2×C42—21no—
235C2×C42—21no—
236C2×C42—21no—
237C2×C42—21no—
238C2×C42—21no—
239C2×C42—21no—
240C2×C42#4112yes—
241#108—44no—
242#108—44no—
243C4⋊C8—44no—
244#104—44no—
245#104—44no—
246#104—44no—
247#103—42no—
248#102—42no—
249#101—42no—
250#101—42no—
251#100—42no—
252#100—42no—
253C22⋊C8—44no—
254C4×C8—44no—
255C24—21no—
256C242−1+411yescharacteristic
257#41—41no—
258#41—41no—
259#41—41no—
260#41—41no—
261#41—41no—
262#41—41no—
263#41—41no—
264#41—41no—
265#41—41no—
266#41—41no—
267#41—41no—
268#41—41no—
269#41—41no—
270#41—41no—
271#41—41no—
272#41—41no—
273C2×Q8—21no—
274C2×Q8—21no—
275C2×Q8—21no—
276C2×Q8—21no—
277C2×Q8—21no—
278C2×Q8—21no—
279C2×D4—21no—
280C2×D4—21no—
281C2×D4—21no—
282C2×D4—21no—
283C2×D4—21no—
284C2×D4—21no—
285C2×D4—41no—
286C2×D4—41no—
287C2×D4—41no—
288C2×D4—41no—
289C2×D4—41no—
290C2×D4—41no—
291C22×C4—22no—
292C22×C4—21no—
293C22×C4—44no—
294C22×C4—41no—
295C22×C4—41no—
296C22×C4—41no—
297C22×C4—41no—
298C22×C4—41no—
299C22×C4—41no—
300C22×C4—41no—
301C22×C4—41no—
302C22×C4—41no—
303C22×C4—41no—
304C22×C4—41no—
305C22×C4C2×#4111yescharacteristic
306C22×C4C2×#4111yescharacteristic
307C22×C4C2×#4111yescharacteristic
308C22×C4C2×#4111yescharacteristic
309C22×C4C2×#4111yescharacteristic
310C22×C4C2×#4111yescharacteristic
311C22×C4C2×#4111yescharacteristic
312C22×C4H(5,𝔽2)11yescharacteristic
313C22×C4H(5,𝔽2)11yescharacteristic
314C22×C4H(5,𝔽2)11yescharacteristic
315C22×C4H(5,𝔽2)11yescharacteristic
316C22×C4H(5,𝔽2)11yescharacteristic
317C22×C4H(5,𝔽2)11yescharacteristic
318C22×C42−1+411yescharacteristic
319M16—41no—
320M16—41no—
321M16—41no—
322M16—41no—
323M16—41no—
324M16—41no—
325M16—41no—
326M16—41no—
327M16—41no—
328M16—41no—
329M16—41no—
330M16—41no—
331M16—41no—
332M16—41no—
333M16—41no—
334M16—41no—
335M16—41no—
336M16—41no—
337M16—41no—
338M16—41no—
339M16—41no—
340M16—41no—
341M16—41no—
342M16—41no—
343M16—41no—
344M16—41no—
345M16—41no—
346M16—41no—
347M16—41no—
348M16—41no—
349M16—41no—
350M16—41no—
351C2×C8—22no—
352C2×C8—22no—
353C2×C8—22no—
354C2×C8—22no—
355C2×C8—22no—
356C2×C8—22no—
357C2×C8—22no—
358C2×C8—22no—
359C2×C8—22no—
360C2×C8—41no—
361C2×C8—41no—
362C2×C8—41no—
363C2×C8—41no—
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—
377C2×C8—41no—
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#12614yes—
389C4⋊C4—22no—
390C4⋊C4—41no—
391C4⋊C4—41no—
392C4⋊C4—41no—
393C4⋊C4—41no—
394C4⋊C4—41no—
395C22⋊C4—22no—
396C22⋊C4—41no—
397C22⋊C4—41no—
398C22⋊C4—41no—
399C22⋊C4—41no—
400C22⋊C4—41no—
401C42—22no—
402C42—22no—
403C42—22no—
404C42—41no—
405C42—41no—
406C42—41no—
407C42—41no—
408C42—41no—
409C42—41no—
410C42—41no—
411C23—42no—
412C23—41no—
413C23—41no—
414C23—41no—
415C23—82no—
416C23#53911yescharacteristic
417C23#54211yescharacteristic
418C23#54211yescharacteristic
419C23#55211yescharacteristic
420C23#55211yescharacteristic
421C23#56211yescharacteristic
422C23#56311yescharacteristic
423Q8—41no—
424Q8—41no—
425Q8—41no—
426Q8—41no—
427D4—41no—
428D4—41no—
429D4—41no—
430D4—41no—
431D4—81no—
432D4—81no—
433D4—81no—
434D4—81no—
435C2×C4—24no—
436C2×C4—42no—
437C2×C4—44no—
438C2×C4—44no—
439C2×C4—42no—
440C2×C4—42no—
441C2×C4—41no—
442C2×C4—41no—
443C2×C4—41no—
444C2×C4—82no—
445C2×C4—81no—
446C2×C4—81no—
447C2×C4—81no—
448C2×C4—81no—
449C2×C4—82no—
450C2×C4—82no—
451C2×C4—81no—
452C2×C4—81no—
453C2×C4C2×#12611yescharacteristic
454C2×C4#52911yescharacteristic
455C2×C4#52911yescharacteristic
456C2×C4#53811yescharacteristic
457C2×C4#53811yescharacteristic
458C2×C4#53811yescharacteristic
459C2×C4#53811yescharacteristic
460C2×C4#53811yescharacteristic
461C2×C4#53811yescharacteristic
462C2×C4#53811yescharacteristic
463C2×C4#53811yescharacteristic
464C2×C4#53911yescharacteristic
465C2×C4#53911yescharacteristic
466C2×C4#53911yescharacteristic
467C2×C4#54011yescharacteristic
468C2×C4#54011yescharacteristic
469C2×C4#55111yescharacteristic
470C2×C4#55211yescharacteristic
471C2×C4#55311yescharacteristic
472C2×C4#55311yescharacteristic
473C2×C4#55311yescharacteristic
474C2×C4#55511yescharacteristic
475C2×C4#55511yescharacteristic
476C2×C4#55611yescharacteristic
477C2×C4#55911yescharacteristic
478C2×C4#55911yescharacteristic
479C2×C4#56011yescharacteristic
480C2×C4#56211yescharacteristic
481C8—24no—
482C8—42no—
483C8—81no—
484C8—81no—
485C8—81no—
486C8—81no—
487C8—81no—
488C8—81no—
489C8—81no—
490C8—81no—
491C22—22no—
492C22—22no—
493C22—22no—
494C22—41no—
495C22—41no—
496C22—41no—
497C22—41no—
498C22—81no—
499C22—82no—
500C22—161no—
501C22—161no—
502C22#249511yescharacteristic
503C22#265311yescharacteristic
504C22#275111yescharacteristic
505C22#277111yescharacteristic
506C22#278111yescharacteristic
507C22#279311yescharacteristic
508C22#283711yescharacteristic
509C4—44no—
510C4—82no—
511C4—81no—
512C4—161no—
513C4—161no—
514C4#251311yescharacteristic
515C4#257311yescharacteristic
516C4#264311yescharacteristic
517C4#264811yescharacteristic
518C4#277811yescharacteristic
519C4#278111yescharacteristic
520C4#280311yescharacteristic
521C4#280811yescharacteristic
522C2—21no—
523C2—21no—
524C2—21no—
525C2—21no—
526C2—21no—
527C2—21no—
528C2—161no—
529C2#1694011yescharacteristic
530C2#2783312yes—
subgroupG/H
centerC2×C4#555
commutator, Frattini, cosocleC23×C4C24
socleC22#2495
prime radicalC22×#41C23
LCSG ▷ C23×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
1. #11306C2semidirect
2. #11300C2semidirect
3. #11299C2semidirect
4. #11272C2semidirect
5. #11264C2semidirect
6. #11259C2semidirect
14. #10882C2semidirect
15. #10864C2semidirect

Maximal subgroups

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

subgroupindex
#113062
#113002
#112992
#112722
#112642
#112592
#111772
#111652
#111082
#110992
#110862
#109292
#108822
#108642