Finite Group Databasebeta

← #230235

Subgroups — #230235

Order 512.

674 conjugacy classes of subgroups.

The subgroup lattice has 524 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23222C211yesmaximal, characteristic, complemented
2#23100C211yesmaximal, characteristic
3#23060C211yesmaximal, characteristic
4#22658C211yesmaximal, characteristic
5C2×#1679C211yesmaximal, characteristic
6#9833C211yesmaximal, characteristic, complemented
7#9620C211yesmaximal, characteristic, complemented
8#9573C211yesmaximal, characteristic, complemented
9#9562C211yesmaximal, characteristic, complemented
10#8988C211yesmaximal, characteristic
11#8984C211yesmaximal, characteristic
12#8926C211yesmaximal, characteristic, complemented
13#8923C211yesmaximal, characteristic, complemented
14#8288C211yesmaximal, characteristic, complemented
15#8204C211yesmaximal, characteristic
16C2×#487C2211yescharacteristic
17C2×#474C2211yescharacteristic
18C2×#461C2211yescharacteristic
19C2×C8×D4C2211yescharacteristic
20C2×#427—21no—
21#2365C2211yescharacteristic
22#2365C2211yescharacteristic
23#2354C2211yescharacteristic
24#2354C2211yescharacteristic
25#1990C2211yescharacteristic
26#1990C2211yescharacteristic
27#1990C2211yescharacteristic
28#1988C2211yescharacteristic
29#1981C2211yescharacteristic
30#1973—21no—
31#1956C2211yescharacteristic
32#1948C2211yescharacteristic
33#1948C2211yescharacteristic
34#1947C2211yescharacteristic
35#1947C2211yescharacteristic
36#1935—21no—
37#1934—21no—
38#1928—21no—
39#1928—21no—
40#1925—21no—
41#1918—21no—
42#1880—21no—
43#1863—21no—
44#1854—21no—
45#1854—21no—
46#1845C2211yescharacteristic
47#1808—21no—
48C4×#107—21no—
49C4×#107C2211yescharacteristic
50#1770C2211yescharacteristic
51#1770C2211yescharacteristic
52C8×C4⋊C4C2211yescharacteristic
53#1763—21no—
54#1763—21no—
55C4×#102C2211yescharacteristic
56#1679C418yescomplemented
57C2×#334C2211yescharacteristic
58C2×#332C2211yescharacteristic
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×Q8—21no—
74C2×C4×D4C2311yescharacteristic
75C2×#117—21no—
76C2×#117—21no—
77#487—22no—
78#487C2×C414yes—
79#474—22no—
80#474C2×C414yes—
81#461—22no—
82#461C2×C414yes—
83C8×D4—22no—
84C8×D4C2×C414yes—
85#427—44no—
86C2×#107—21no—
87C2×#107—21no—
88C2×#107—21no—
89C2×#107—21no—
90C2×#107—21no—
91C2×#107D411yescharacteristic
92C2×#107D411yescharacteristic
93C2×#106—21no—
94C2×#106—21no—
95C2×C4⋊C8C2311yescharacteristic
96C2×C4⋊C8C2311yescharacteristic
97C2×C4⋊C8C2311yescharacteristic
98C2×C4⋊C8C2311yescharacteristic
99C2×#103—21no—
100C2×#103—21no—
101C2×#103—21no—
102C2×#103—21no—
103C2×#103—21no—
104C2×#103—21no—
105C2×#103—21no—
106C2×#102—21no—
107C2×#102D411yescharacteristic
108C2×#102D411yescharacteristic
109C2×C22⋊C8—21no—
110C2×#97—21no—
111C2×#97—21no—
112C2×C4×C8—21no—
113C2×C4×C8C2311yescharacteristic
114C2×C4×C8C2311yescharacteristic
115C2×C4×C8C2311yescharacteristic
116#391—21no—
117#391—21no—
118#390—21no—
119#390—21no—
120#389—21no—
121#389—21no—
122#388—41no—
123#388—41no—
124#387—41no—
125#387—41no—
126#384C2311yescharacteristic
127#384C2311yescharacteristic
128#380—21no—
129#380—21no—
130C4×C4⋊C4—21no—
131C4×C4⋊C4—21no—
132C4×C4⋊C4—21no—
133C4×C4⋊C4—21no—
134C4×C4⋊C4C2311yescharacteristic
135C4×C4⋊C4C2311yescharacteristic
136C4×C4⋊C4C2311yescharacteristic
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—41no—
146#340—41no—
147#340D411yescharacteristic
148#340D411yescharacteristic
149#340D411yescharacteristic
150#340D411yescharacteristic
151#336—21no—
152#336—21no—
153#336—21no—
154#334C2×C414yes—
155#332C2×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—21no—
170C22×C8—24no—
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—42no—
188C4×Q8—81no—
189C4×Q8—81no—
190C4×D4—21no—
191C4×D4—21no—
192C4×D4C22×C412yes—
193C4×D4#4112yes—
194#117—42no—
195#117—42no—
196#117—81no—
197#117—81no—
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—21no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—41no—
209C2×C4⋊C4—41no—
210C2×C4⋊C4C2×D411yescharacteristic
211C2×C4⋊C4C2×D411yescharacteristic
212C2×C4⋊C4C2×D411yescharacteristic
213C2×C4⋊C4C2×D411yescharacteristic
214C2×C4⋊C4C2×D411yescharacteristic
215C2×C4⋊C4C2×D411yescharacteristic
216C2×C4⋊C4C2×D411yescharacteristic
217C2×C22⋊C4—21no—
218C2×C22⋊C4—41no—
219C2×C22⋊C4—41no—
220C2×C22⋊C4—41no—
221C2×C22⋊C4—41no—
222C2×C42—21no—
223C2×C42—21no—
224C2×C42—21no—
225C2×C42—41no—
226C2×C42—41no—
227C2×C42—41no—
228C2×C42—41no—
229C2×C42C2411yescharacteristic
230#107—24no—
231#107—22no—
232#107—41no—
233#107—42no—
234#107—42no—
235#107—81no—
236#107—81no—
237#106—42no—
238#106—42no—
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—24no—
249#103—24no—
250#103—42no—
251#103—42no—
252#103—81no—
253#103—81no—
254#102—24no—
255#102—41no—
256#102—41no—
257C22⋊C8—24no—
258#97—81no—
259#97—81no—
260C4×C8—21no—
261C4×C8—21no—
262C4×C8—21no—
263C4×C8—21no—
264C4×C8—42no—
265C4×C8C22×C412yes—
266C4×C8#4112yes—
267#95—21no—
268#95—21no—
269#95—21no—
270#95—41no—
271#95—41no—
272#95—41no—
273#95—41no—
274#95—41no—
275#95—41no—
276C24—21no—
277C2×Q8—22no—
278C2×Q8—22no—
279C2×Q8—41no—
280C2×Q8—41no—
281C2×Q8—41no—
282C2×Q8—41no—
283C2×Q8—41no—
284C2×Q8—41no—
285C2×Q8—41no—
286C2×Q8—41no—
287C2×Q8—41no—
288C2×Q8—41no—
289C2×D4—21no—
290C2×D4—21no—
291C2×D4—21no—
292C2×D4—21no—
293C2×D4—21no—
294C2×D4C4×D412yes—
295C22×C4—21no—
296C22×C4—21no—
297C22×C4—22no—
298C22×C4—22no—
299C22×C4—21no—
300C22×C4—21no—
301C22×C4—21no—
302C22×C4—21no—
303C22×C4—21no—
304C22×C4—41no—
305C22×C4—41no—
306C22×C4—41no—
307C22×C4—41no—
308C22×C4—41no—
309C22×C4—41no—
310C22×C4—41no—
311C22×C4—41no—
312C22×C4C22×D411yescharacteristic
313C22×C4C22×D411yescharacteristic
314C22×C4H(5,𝔽2)11yescharacteristic
315SD16—84no—
316C2×C8—21no—
317C2×C8—21no—
318C2×C8—21no—
319C2×C8—21no—
320C2×C8—22no—
321C2×C8—22no—
322C2×C8—22no—
323C2×C8—22no—
324C2×C8—21no—
325C2×C8—21no—
326C2×C8—21no—
327C2×C8—21no—
328C2×C8—21no—
329C2×C8—21no—
330C2×C8—48no—
331C2×C8—41no—
332C2×C8—41no—
333C2×C8—41no—
334C2×C8—41no—
335C2×C8—41no—
336C2×C8—41no—
337C2×C8—41no—
338C2×C8—41no—
339C2×C8—41no—
340C2×C8—41no—
341C2×C8—41no—
342C2×C8C4×D412yes—
343C2×C8C4×D412yes—
344C4⋊C4—21no—
345C4⋊C4—21no—
346C4⋊C4—22no—
347C4⋊C4—22no—
348C4⋊C4—22no—
349C4⋊C4—22no—
350C4⋊C4—21no—
351C4⋊C4—21no—
352C4⋊C4—21no—
353C4⋊C4—21no—
354C4⋊C4—21no—
355C4⋊C4—21no—
356C4⋊C4—21no—
357C4⋊C4—21no—
358C4⋊C4—21no—
359C4⋊C4—21no—
360C4⋊C4—41no—
361C4⋊C4—41no—
362C4⋊C4—41no—
363C4⋊C4—41no—
364C4⋊C4—41no—
365C4⋊C4—41no—
366C4⋊C4—41no—
367C4⋊C4—41no—
368C4⋊C4—41no—
369C4⋊C4—41no—
370C4⋊C4—41no—
371C4⋊C4—41no—
372C4⋊C4—82no—
373C4⋊C4—81no—
374C4⋊C4—81no—
375C4⋊C4—82no—
376C4⋊C4C4×D412yes—
377C4⋊C4AGL(1,ℤ/8)12yes—
378C22⋊C4—22no—
379C22⋊C4—22no—
380C22⋊C4—82no—
381C22⋊C4—82no—
382C22⋊C4—161no—
383C22⋊C4—161no—
384C42—41no—
385C42—41no—
386C42—41no—
387C42—41no—
388C42—41no—
389C42—41no—
390C42—82no—
391C42—82no—
392C42—82no—
393C42—82no—
394C42C23×C411yescharacteristic
395C42C2×#4111yescharacteristic
396C42C2×#4111yescharacteristic
397C422−1+411yescharacteristic
398C23—22no—
399C23—41no—
400C23—41no—
401C23—41no—
402C23—41no—
403C23—41no—
404C23—41no—
405C23D4211yescharacteristic
406Q8—42no—
407Q8—42no—
408Q8—42no—
409Q8—42no—
410Q8—81no—
411Q8—81no—
412Q8—81no—
413Q8—81no—
414D4—22no—
415D4—22no—
416D4—41no—
417D4—41no—
418C2×C4—21no—
419C2×C4—21no—
420C2×C4—22no—
421C2×C4—22no—
422C2×C4—41no—
423C2×C4—41no—
424C2×C4—41no—
425C2×C4—41no—
426C2×C4—42no—
427C2×C4—42no—
428C2×C4—42no—
429C2×C4—42no—
430C2×C4—42no—
431C2×C4—42no—
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—41no—
441C2×C4—41no—
442C2×C4—41no—
443C2×C4—41no—
444C2×C4—41no—
445C2×C4—41no—
446C2×C4—41no—
447C2×C4—81no—
448C2×C4—81no—
449C2×C4—81no—
450C2×C4—81no—
451C2×C4—81no—
452C2×C4—81no—
453C2×C4—81no—
454C2×C4—81no—
455C2×C4—81no—
456C2×C4—81no—
457C2×C4C2×C4×D411yescharacteristic
458C2×C4C2×C4×D411yescharacteristic
459C2×C4#52011yescharacteristic
460C2×C4#54711yescharacteristic
461C2×C4#54811yescharacteristic
462C2×C4#55311yescharacteristic
463C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
464C2×C4#57511yescharacteristic
465C2×C4#57511yescharacteristic
466C2×C4#57511yescharacteristic
467C2×C4#57611yescharacteristic
468C2×C4#57711yescharacteristic
469C2×C4#57711yescharacteristic
470C2×C4#57811yescharacteristic
471C8—22no—
472C8—22no—
473C8—42no—
474C8—41no—
475C8—41no—
476C8—42no—
477C8—42no—
478C8—42no—
479C8—81no—
480C22—42no—
481C22—42no—
482C22—42no—
483C22—81no—
484C22—81no—
485C22—81no—
486C22—81no—
487C22#236111yescharacteristic
488C22#326411yescharacteristic
489C22#326711yescharacteristic
490C22#326911yescharacteristic
491C22#327111yescharacteristic
492C22#329011yescharacteristic
493C22#329311yescharacteristic
494C4—21no—
495C4—21no—
496C4—42no—
497C4—42no—
498C4—81no—
499C4—81no—
500C4—81no—
501C4—81no—
502C4—81no—
503C4—82no—
504C4—82no—
505C4—161no—
506C4—161no—
507C4#294311yescharacteristic
508C4C4×AGL(1,ℤ/8)11yescharacteristic
509C4#294911yescharacteristic
510C4#295111yescharacteristic
511C4#334811yescharacteristic
512C4#335011yescharacteristic
513C4#336711yescharacteristic
514C4#337011yescharacteristic
515C2—82no—
516C2#2005611yescharacteristic
517C2#2006611yescharacteristic
518C2#2012311yescharacteristic
519C2#2012911yescharacteristic
520C2#2301111yescharacteristic
521C2#2303311yescharacteristic
522C2#2304511yescharacteristic
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. #23222C2semidirect
6. #9833C2semidirect
7. #9620C2semidirect
8. #9573C2semidirect
9. #9562C2semidirect
12. #8926C2semidirect
13. #8923C2semidirect
14. #8288C2semidirect
56. #1679C4semidirect

Maximal subgroups

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

subgroupindex
#232222
#231002
#230602
#226582
C2×#16792
#98332
#96202
#95732
#95622
#89882
#89842
#89262
#89232
#82882
#82042