Finite Group Databasebeta

← #229305

Subgroups — #229305

Order 512.

672 conjugacy classes of subgroups.

The subgroup lattice has 517 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23163C211yesmaximal, characteristic, complemented
2#23094C211yesmaximal, characteristic
3#23061C211yesmaximal, characteristic
4#22658C211yesmaximal, characteristic
5C2×#1671C211yesmaximal, characteristic
6#9841C211yesmaximal, characteristic, complemented
7#9621C211yesmaximal, characteristic, complemented
8#9567C211yesmaximal, characteristic, complemented
9#9562C211yesmaximal, characteristic, complemented
10#8988C211yesmaximal, characteristic
11#8976C211yesmaximal, characteristic
12#8928C211yesmaximal, characteristic, complemented
13#8925C211yesmaximal, characteristic, complemented
14#8284C211yesmaximal, characteristic, complemented
15#8202C211yesmaximal, characteristic
16C2×#498C2211yescharacteristic
17C2×#476C2211yescharacteristic
18C2×#461C2211yescharacteristic
19C2×C8×D4C2211yescharacteristic
20C2×#427—21no—
21#2369C2211yescharacteristic
22#2365C2211yescharacteristic
23#2354C2211yescharacteristic
24#2354C2211yescharacteristic
25#1990C2211yescharacteristic
26#1990C2211yescharacteristic
27#1988C2211yescharacteristic
28#1987C2211yescharacteristic
29#1968—21no—
30#1956C2211yescharacteristic
31#1956C2211yescharacteristic
32#1948C2211yescharacteristic
33#1948C2211yescharacteristic
34#1948C2211yescharacteristic
35#1948C2211yescharacteristic
36#1944—21no—
37#1934—21no—
38#1927—21no—
39#1926—21no—
40#1920—21no—
41#1920—21no—
42#1919—21no—
43#1880—21no—
44#1854—21no—
45#1854—21no—
46#1845C2211yescharacteristic
47#1808—21no—
48#1776—21no—
49C4×#107C2211yescharacteristic
50C4×#106—21no—
51#1770C2211yescharacteristic
52C8×C4⋊C4C2211yescharacteristic
53C8×C4⋊C4C2211yescharacteristic
54C4×#103—21no—
55C4×#102C2211yescharacteristic
56#1671C418yescomplemented
57C2×#334C2211yescharacteristic
58C2×#333C2211yescharacteristic
59C2×#332C2211yescharacteristic
60C2×D4⋊C8C2211yescharacteristic
61C2×#322C2211yescharacteristic
62#1296C2211yescharacteristic
63#1296C2211yescharacteristic
64#1279C2211yescharacteristic
65#1279C2211yescharacteristic
66#1279C2211yescharacteristic
67#1277C2211yescharacteristic
68C22×SD16—21no—
69C23×C8—21no—
70C2×C4⋊Q8C2311yescharacteristic
71C2×C4⋊Q8C2311yescharacteristic
72C2×C4×Q8—21no—
73C2×C4×D4C2311yescharacteristic
74C2×#117—21no—
75C2×#117—21no—
76#498—22no—
77#498C2×C414yes—
78#476—22no—
79#476C2×C414yes—
80#461—22no—
81#461C2×C414yes—
82C8×D4—22no—
83C8×D4C2×C414yes—
84#427—44no—
85C2×#107—21no—
86C2×#107—21no—
87C2×#107D411yescharacteristic
88C2×#107D411yescharacteristic
89C2×#106—21no—
90C2×#106—21no—
91C2×#106—21no—
92C2×#106—21no—
93C2×#106—21no—
94C2×#106—21no—
95C2×#106—21no—
96C2×C4⋊C8C2311yescharacteristic
97C2×C4⋊C8C2311yescharacteristic
98C2×C4⋊C8C2311yescharacteristic
99C2×C4⋊C8C2311yescharacteristic
100C2×#103—21no—
101C2×#103—21no—
102C2×#103—21no—
103C2×#103—41no—
104C2×#102—21no—
105C2×#102D411yescharacteristic
106C2×#102D411yescharacteristic
107C2×C22⋊C8—21no—
108C2×#97—21no—
109C2×C4×C8—21no—
110C2×C4×C8—21no—
111C2×C4×C8C2311yescharacteristic
112C2×C4×C8C2311yescharacteristic
113C2×C4×C8C2311yescharacteristic
114#391—21no—
115#391—41no—
116#390—21no—
117#390—21no—
118#389—21no—
119#389—21no—
120#389—21no—
121#388—41no—
122#388—41no—
123#387—41no—
124#384C2311yescharacteristic
125#384C2311yescharacteristic
126#380—21no—
127#380—21no—
128C4×C4⋊C4—21no—
129C4×C4⋊C4—21no—
130C4×C4⋊C4—21no—
131C4×C4⋊C4—21no—
132C4×C4⋊C4—21no—
133C4×C4⋊C4C2311yescharacteristic
134C4×C4⋊C4C2311yescharacteristic
135C4×C4⋊C4C2311yescharacteristic
136#340—21no—
137#340—21no—
138#340—21no—
139#340—21no—
140#340—21no—
141#340—21no—
142#340—21no—
143#340—21no—
144#340—21no—
145#340—21no—
146#340—41no—
147#340D411yescharacteristic
148#340D411yescharacteristic
149#340D411yescharacteristic
150#340D411yescharacteristic
151#336—21no—
152#336—21no—
153#336—21no—
154#334C2×C414yes—
155#333C2×C414yes—
156#332C2×C414yes—
157D4⋊C8—22no—
158#322—22no—
159C22×Q8—21no—
160C22×Q8—21no—
161C22×D4C2×D411yescharacteristic
162C23×C4—21no—
163C2×SD16—24no—
164C2×SD16—81no—
165C2×SD16—81no—
166C22×C8—21no—
167C22×C8—21no—
168C22×C8—21no—
169C22×C8—24no—
170C22×C8—21no—
171C22×C8—21no—
172C22×C8—42no—
173C22×C8—42no—
174C22×C8C2×D411yescharacteristic
175C22×C8C2×D411yescharacteristic
176C22×C8C2×D411yescharacteristic
177C22×C8C2×D411yescharacteristic
178C4⋊Q8—21no—
179C4⋊Q8—21no—
180C4⋊Q8—21no—
181C4⋊Q8—21no—
182C4⋊Q8—21no—
183C4⋊Q8—21no—
184C4⋊Q8C22×C412yes—
185C4⋊Q8C22×C412yes—
186C4×Q8—42no—
187C4×Q8—81no—
188C4×D4—21no—
189C4×D4—21no—
190C4×D4C22×C412yes—
191C4×D4#4112yes—
192#117—42no—
193#117—42no—
194#117—81no—
195#117—81no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—41no—
207C2×C4⋊C4—41no—
208C2×C4⋊C4C2×D411yescharacteristic
209C2×C4⋊C4C2×D411yescharacteristic
210C2×C4⋊C4C2×D411yescharacteristic
211C2×C4⋊C4C2×D411yescharacteristic
212C2×C4⋊C4C2×D411yescharacteristic
213C2×C4⋊C4C2×D411yescharacteristic
214C2×C4⋊C4C2×D411yescharacteristic
215C2×C22⋊C4—21no—
216C2×C22⋊C4—41no—
217C2×C22⋊C4—41no—
218C2×C22⋊C4—41no—
219C2×C22⋊C4—41no—
220C2×C42—21no—
221C2×C42—21no—
222C2×C42—21no—
223C2×C42—41no—
224C2×C42—41no—
225C2×C42—41no—
226C2×C42—41no—
227C2×C42C2411yescharacteristic
228#107—22no—
229#107—42no—
230#107—81no—
231#107SD1614yescomplemented
232#106—24no—
233#106—24no—
234#106—24no—
235#106—42no—
236#106—42no—
237#106—42no—
238#106—81no—
239C4⋊C8—21no—
240C4⋊C8—21no—
241C4⋊C8—21no—
242C4⋊C8—21no—
243C4⋊C8—21no—
244C4⋊C8C22×C412yes—
245C4⋊C8C22×C412yes—
246C4⋊C8C22×C412yes—
247#103—24no—
248#103—42no—
249#103—81no—
250#103—82no—
251#102—24no—
252#102—41no—
253#102—41no—
254C22⋊C8—24no—
255#97—42no—
256C4×C8—21no—
257C4×C8—21no—
258C4×C8—21no—
259C4×C8—21no—
260C4×C8—42no—
261C4×C8—81no—
262C4×C8C22×C412yes—
263C4×C8#4112yes—
264#95—21no—
265#95—21no—
266#95—21no—
267#95—41no—
268#95—41no—
269#95—41no—
270#95—41no—
271#95—41no—
272#95—41no—
273C24—21no—
274C2×Q8—22no—
275C2×Q8—22no—
276C2×Q8—41no—
277C2×Q8—41no—
278C2×Q8—41no—
279C2×Q8—41no—
280C2×Q8—41no—
281C2×Q8—41no—
282C2×Q8—81no—
283C2×Q8—81no—
284C2×D4—21no—
285C2×D4—21no—
286C2×D4—21no—
287C2×D4—21no—
288C2×D4—21no—
289C2×D4C4×D412yes—
290C22×C4—21no—
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—22no—
294C22×C4—22no—
295C22×C4—21no—
296C22×C4—21no—
297C22×C4—21no—
298C22×C4—21no—
299C22×C4—41no—
300C22×C4—41no—
301C22×C4—41no—
302C22×C4—41no—
303C22×C4—41no—
304C22×C4—41no—
305C22×C4—41no—
306C22×C4—41no—
307C22×C4C22×D411yescharacteristic
308C22×C4C22×D411yescharacteristic
309C22×C4H(5,𝔽2)11yescharacteristic
310SD16—84no—
311C2×C8—21no—
312C2×C8—21no—
313C2×C8—22no—
314C2×C8—22no—
315C2×C8—22no—
316C2×C8—21no—
317C2×C8—21no—
318C2×C8—21no—
319C2×C8—21no—
320C2×C8—21no—
321C2×C8—21no—
322C2×C8—41no—
323C2×C8—41no—
324C2×C8—41no—
325C2×C8—41no—
326C2×C8—41no—
327C2×C8—41no—
328C2×C8—48no—
329C2×C8—41no—
330C2×C8—41no—
331C2×C8—41no—
332C2×C8—41no—
333C2×C8—41no—
334C2×C8—41no—
335C2×C8C4×D412yes—
336C2×C8C4×D412yes—
337C2×C8C2×SD1612yes—
338C2×C8#13512yes—
339C4⋊C4—21no—
340C4⋊C4—21no—
341C4⋊C4—21no—
342C4⋊C4—22no—
343C4⋊C4—22no—
344C4⋊C4—22no—
345C4⋊C4—22no—
346C4⋊C4—21no—
347C4⋊C4—21no—
348C4⋊C4—21no—
349C4⋊C4—21no—
350C4⋊C4—21no—
351C4⋊C4—21no—
352C4⋊C4—21no—
353C4⋊C4—21no—
354C4⋊C4—41no—
355C4⋊C4—41no—
356C4⋊C4—41no—
357C4⋊C4—41no—
358C4⋊C4—41no—
359C4⋊C4—41no—
360C4⋊C4—41no—
361C4⋊C4—41no—
362C4⋊C4—41no—
363C4⋊C4—41no—
364C4⋊C4—41no—
365C4⋊C4—41no—
366C4⋊C4—81no—
367C4⋊C4—81no—
368C4⋊C4—82no—
369C4⋊C4—161no—
370C4⋊C4C4×D412yes—
371C4⋊C4C2×SD1612yes—
372C4⋊C4C2×SD1612yes—
373C22⋊C4—22no—
374C22⋊C4—22no—
375C22⋊C4—82no—
376C22⋊C4—82no—
377C22⋊C4—161no—
378C22⋊C4—161no—
379C42—41no—
380C42—41no—
381C42—41no—
382C42—41no—
383C42—41no—
384C42—41no—
385C42—82no—
386C42—82no—
387C42—82no—
388C42—161no—
389C42C23×C411yescharacteristic
390C42C2×#4111yescharacteristic
391C42C2×#4111yescharacteristic
392C422−1+411yescharacteristic
393C23—22no—
394C23—41no—
395C23—41no—
396C23—41no—
397C23—41no—
398C23—41no—
399C23—41no—
400C23D4211yescharacteristic
401Q8—42no—
402Q8—42no—
403Q8—82no—
404Q8—81no—
405Q8—81no—
406Q8—161no—
407D4—22no—
408D4—22no—
409D4—41no—
410D4—41no—
411C2×C4—21no—
412C2×C4—21no—
413C2×C4—22no—
414C2×C4—22no—
415C2×C4—22no—
416C2×C4—41no—
417C2×C4—41no—
418C2×C4—41no—
419C2×C4—41no—
420C2×C4—41no—
421C2×C4—41no—
422C2×C4—42no—
423C2×C4—42no—
424C2×C4—42no—
425C2×C4—42no—
426C2×C4—42no—
427C2×C4—42no—
428C2×C4—41no—
429C2×C4—41no—
430C2×C4—41no—
431C2×C4—41no—
432C2×C4—41no—
433C2×C4—41no—
434C2×C4—41no—
435C2×C4—41no—
436C2×C4—41no—
437C2×C4—41no—
438C2×C4—41no—
439C2×C4—41no—
440C2×C4—81no—
441C2×C4—81no—
442C2×C4—81no—
443C2×C4—81no—
444C2×C4—81no—
445C2×C4—81no—
446C2×C4—81no—
447C2×C4—81no—
448C2×C4—81no—
449C2×C4—81no—
450C2×C4C2×C4×D411yescharacteristic
451C2×C4C2×C4×D411yescharacteristic
452C2×C4#52011yescharacteristic
453C2×C4#54711yescharacteristic
454C2×C4#54811yescharacteristic
455C2×C4#55311yescharacteristic
456C2×C4C22×SD1611yescharacteristic
457C2×C4C2×#13511yescharacteristic
458C2×C4#57511yescharacteristic
459C2×C4#57511yescharacteristic
460C2×C4#57711yescharacteristic
461C2×C4#57711yescharacteristic
462C2×C4#57711yescharacteristic
463C2×C4#57811yescharacteristic
464C8—22no—
465C8—41no—
466C8—41no—
467C8—42no—
468C8—42no—
469C8—42no—
470C8—81no—
471C8—81no—
472C8C4×SD1614yes—
473C22—42no—
474C22—42no—
475C22—42no—
476C22—81no—
477C22—81no—
478C22—81no—
479C22—81no—
480C22#236111yescharacteristic
481C22#326711yescharacteristic
482C22#327111yescharacteristic
483C22D4×SD1611yescharacteristic
484C22#328311yescharacteristic
485C22#328811yescharacteristic
486C22#328911yescharacteristic
487C4—21no—
488C4—21no—
489C4—42no—
490C4—42no—
491C4—42no—
492C4—82no—
493C4—81no—
494C4—81no—
495C4—81no—
496C4—81no—
497C4—82no—
498C4—161no—
499C4—161no—
500C4C2×C4×SD1611yescharacteristic
501C4#294311yescharacteristic
502C4#294711yescharacteristic
503C4#294911yescharacteristic
504C4#334811yescharacteristic
505C4#335011yescharacteristic
506C4#335911yescharacteristic
507C4#336011yescharacteristic
508C2—82no—
509C2#1976611yescharacteristic
510C2#1977011yescharacteristic
511C2#1986211yescharacteristic
512C2#1986611yescharacteristic
513C2#2299611yescharacteristic
514C2#2303311yescharacteristic
515C2#2304911yescharacteristic
subgroupG/H
center, socleC23D42
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
prime radicalC22×D4C2×D4
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C2×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. #23163C2semidirect
6. #9841C2semidirect
7. #9621C2semidirect
8. #9567C2semidirect
9. #9562C2semidirect
12. #8928C2semidirect
13. #8925C2semidirect
14. #8284C2semidirect
56. #1671C4semidirect
231. #107SD16semidirect

Maximal subgroups

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

subgroupindex
#231632
#230942
#230612
#226582
C2×#16712
#98412
#96212
#95672
#95622
#89882
#89762
#89282
#89252
#82842
#82022