Finite Group Databasebeta

← #330145

Subgroups — #330145

Order 512.

574 conjugacy classes of subgroups.

The subgroup lattice has 544 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#29942C211yesmaximal, characteristic, complemented
2#29925C211yesmaximal, characteristic
3#29842C211yesmaximal, characteristic
4#21973C211yesmaximal, characteristic, complemented
5#10652C211yesmaximal, characteristic, complemented
6#10649C211yesmaximal, characteristic, complemented
7#10599C211yesmaximal, characteristic, complemented
8#10597C211yesmaximal, characteristic, complemented
9#10593C211yesmaximal, characteristic, complemented
10#10587C211yesmaximal, characteristic, complemented
11#10539C211yesmaximal, characteristic, complemented
12#10539C211yesmaximal, characteristic, complemented
13#9133C211yesmaximal, characteristic
14#9127C211yesmaximal, characteristic, complemented
15#8600C211yesmaximal, characteristic, complemented
16C2×#486C2211yescharacteristic
17C2×#431C2211yescharacteristic
18C22×C4⋊C8C2211yescharacteristic
19#2857C2211yescharacteristic
20#2856C2211yescharacteristic
21#2283C2211yescharacteristic
22#2101—21no—
23#2092—21no—
24#2085—21no—
25#2077—21no—
26#2075—21no—
27#2073—21no—
28#2067—21no—
29#2067—21no—
30#2034—21no—
31#2033—21no—
32#2033—21no—
33#2033—21no—
34#1990C2211yescharacteristic
35#1987C2211yescharacteristic
36#1987C2211yescharacteristic
37#1982C2211yescharacteristic
38#1982C2211yescharacteristic
39#1957C2211yescharacteristic
40#1948C2211yescharacteristic
41#1948C2211yescharacteristic
42#1947C2211yescharacteristic
43#1947C2211yescharacteristic
44#1939C2211yescharacteristic
45#1939C2211yescharacteristic
46#1939C2211yescharacteristic
47#1939C2211yescharacteristic
48#1850—21no—
49#1850—21no—
50#1845C2211yescharacteristic
51#1843—21no—
52#1832—21no—
53#1797C2211yescharacteristic
54#1797C2211yescharacteristic
55C2×#335C2211yescharacteristic
56C2×#334C2211yescharacteristic
57C2×#333C2211yescharacteristic
58C2×#333C2211yescharacteristic
59C2×#331C2211yescharacteristic
60C2×#331C2211yescharacteristic
61#1296C2211yescharacteristic
62#1296C2211yescharacteristic
63#1296C2211yescharacteristic
64#1296C2211yescharacteristic
65#1277C2211yescharacteristic
66#1277C2211yescharacteristic
67C23×C8—21no—
68C2×C4⋊Q8C2311yescharacteristic
69C2×#127C2311yescharacteristic
70C2×C22⋊Q8—21no—
71C2×#121—21no—
72C2×#121—21no—
73C2×#121—21no—
74C22×C42C2311yescharacteristic
75#486—41no—
76#486—41no—
77#431—41no—
78#431—41no—
79C2×#107—21no—
80C2×#107—21no—
81C2×#107—21no—
82C2×#107—21no—
83C2×#106—21no—
84C2×#106—21no—
85C2×C4⋊C8—21no—
86C2×C4⋊C8—21no—
87C2×C4⋊C8—21no—
88C2×C4⋊C8—21no—
89C2×C4⋊C8—21no—
90C2×C4⋊C8—41no—
91C2×C4⋊C8—41no—
92C2×C4⋊C8D411yescharacteristic
93C2×C4⋊C8D411yescharacteristic
94C2×C4⋊C8C2311yescharacteristic
95C2×C4⋊C8C2311yescharacteristic
96C2×C4⋊C8C2311yescharacteristic
97C2×C4⋊C8C2311yescharacteristic
98C2×C4⋊C8C2311yescharacteristic
99C2×C4⋊C8C2311yescharacteristic
100C2×#102—21no—
101C2×#102—21no—
102C2×#102—21no—
103C2×#102—21no—
104C2×#102—21no—
105C2×#102—21no—
106C2×#102—21no—
107C2×#102—21no—
108C2×C22⋊C8—21no—
109C2×C22⋊C8—21no—
110C2×C22⋊C8D411yescharacteristic
111C2×C22⋊C8D411yescharacteristic
112C2×C4×C8C2311yescharacteristic
113C2×C4×C8C2311yescharacteristic
114#391—21no—
115#390—21no—
116#390—21no—
117#390—21no—
118#389—21no—
119#389—21no—
120#384C2311yescharacteristic
121#384C2311yescharacteristic
122#383—21no—
123#382D411yescharacteristic
124#382D411yescharacteristic
125#381—21no—
126#380—21no—
127#380—21no—
128#380D411yescharacteristic
129#380D411yescharacteristic
130C4×C4⋊C4C2311yescharacteristic
131C4×C4⋊C4C2311yescharacteristic
132C4×C22⋊C4—21no—
133#376—21no—
134#340—21no—
135#340—21no—
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#336—21no—
145#336—21no—
146#336—21no—
147#336—21no—
148#336—21no—
149#336—21no—
150#335—41no—
151#334—41no—
152#333—41no—
153#333—41no—
154#331—41no—
155#331—41no—
156C22×Q8—21no—
157C22×D4—21no—
158C22×D4—21no—
159C22×D4—21no—
160C22×D4—41no—
161C23×C4C2×D411yescharacteristic
162C23×C4C2×D411yescharacteristic
163C23×C4#4111yescharacteristic
164C22×C8—21no—
165C22×C8—21no—
166C22×C8—21no—
167C22×C8—21no—
168C22×C8—21no—
169C22×C8—21no—
170C22×C8—42no—
171C22×C8—81no—
172C22×C8—81no—
173C22×C8C2×D411yescharacteristic
174C22×C8C2×D411yescharacteristic
175C22×C8C2×D411yescharacteristic
176C22×C8C2×D411yescharacteristic
177C4⋊Q8—21no—
178C4⋊Q8—21no—
179C4⋊Q8—21no—
180C4⋊Q8—21no—
181#127—21no—
182#127—21no—
183#127—21no—
184#127—21no—
185C22⋊Q8—82no—
186#121—81no—
187#121—81no—
188#121—81no—
189#121—81no—
190#121—82no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—41no—
200C2×C4⋊C4C2×D411yescharacteristic
201C2×C4⋊C4C2×D411yescharacteristic
202C2×C4⋊C4C2×D411yescharacteristic
203C2×C4⋊C4C2×D411yescharacteristic
204C2×C22⋊C4—21no—
205C2×C22⋊C4—21no—
206C2×C22⋊C4—41no—
207C2×C22⋊C4—41no—
208C2×C22⋊C4—41no—
209C2×C22⋊C4—41no—
210C2×C42—21no—
211C2×C42—21no—
212C2×C42—21no—
213C2×C42—21no—
214C2×C42—21no—
215C2×C42—21no—
216C2×C42C2×D411yescharacteristic
217C2×C42C2×D411yescharacteristic
218C2×C42#4111yescharacteristic
219C2×C42C2411yescharacteristic
220#107—81no—
221#107—81no—
222#107—81no—
223#107—81no—
224#106—81no—
225#106—81no—
226C4⋊C8—21no—
227C4⋊C8—21no—
228C4⋊C8—21no—
229C4⋊C8—21no—
230C4⋊C8—21no—
231C4⋊C8—21no—
232C4⋊C8—21no—
233C4⋊C8—21no—
234C4⋊C8—21no—
235C4⋊C8—21no—
236C4⋊C8—21no—
237C4⋊C8—21no—
238C4⋊C8—41no—
239C4⋊C8—41no—
240C4⋊C8—42no—
241C4⋊C8—81no—
242C4⋊C8—81no—
243C4⋊C8—81no—
244C4⋊C8—81no—
245#102—42no—
246#102—42no—
247#102—81no—
248#102—81no—
249#102—81no—
250#102—81no—
251#102—81no—
252#102—81no—
253C22⋊C8—41no—
254C22⋊C8—41no—
255C22⋊C8—81no—
256C22⋊C8—81no—
257C4×C8—21no—
258C4×C8—21no—
259C4×C8—21no—
260C4×C8—21no—
261#95—21no—
262#95—21no—
263#95—21no—
264#95—21no—
265#95—21no—
266#95—21no—
267#95—21no—
268#95—21no—
269#95—41no—
270#95—41no—
271C24—41no—
272C24#12111yescharacteristic
273C2×Q8—41no—
274C2×Q8—41no—
275C2×Q8—81no—
276C2×Q8—81no—
277C2×D4—41no—
278C2×D4—41no—
279C2×D4—41no—
280C2×D4—41no—
281C2×D4—41no—
282C2×D4—41no—
283C2×D4—41no—
284C2×D4—41no—
285C2×D4—41no—
286C2×D4—41no—
287C2×D4—41no—
288C2×D4—41no—
289C2×D4—41no—
290C2×D4—41no—
291C2×D4—81no—
292C2×D4—81no—
293C2×D4—82no—
294C2×D4—82no—
295C2×D4—82no—
296C22×C4—22no—
297C22×C4—21no—
298C22×C4—21no—
299C22×C4—21no—
300C22×C4—21no—
301C22×C4—21no—
302C22×C4—21no—
303C22×C4—21no—
304C22×C4—21no—
305C22×C4—21no—
306C22×C4—21no—
307C22×C4—21no—
308C22×C4—21no—
309C22×C4—21no—
310C22×C4—21no—
311C22×C4—21no—
312C22×C4—21no—
313C22×C4—21no—
314C22×C4—21no—
315C22×C4—41no—
316C22×C4—41no—
317C22×C4—41no—
318C22×C4—41no—
319C22×C4#12111yescharacteristic
320C22×C4#12111yescharacteristic
321C22×C4#12111yescharacteristic
322C22×C4C22×D411yescharacteristic
323C22×C4C22×D411yescharacteristic
324C22×C4C2×#4111yescharacteristic
325C2×C8—21no—
326C2×C8—21no—
327C2×C8—21no—
328C2×C8—21no—
329C2×C8—21no—
330C2×C8—21no—
331C2×C8—21no—
332C2×C8—21no—
333C2×C8—21no—
334C2×C8—21no—
335C2×C8—21no—
336C2×C8—21no—
337C2×C8—41no—
338C2×C8—41no—
339C2×C8—41no—
340C2×C8—41no—
341C2×C8—41no—
342C2×C8—41no—
343C2×C8—41no—
344C2×C8—41no—
345C2×C8—41no—
346C2×C8—41no—
347C2×C8—41no—
348C2×C8—41no—
349C2×C8—41no—
350C2×C8—41no—
351C2×C8—41no—
352C2×C8—41no—
353C2×C8—41no—
354C2×C8—41no—
355C2×C8—84no—
356C4⋊C4—21no—
357C4⋊C4—21no—
358C4⋊C4—21no—
359C4⋊C4—21no—
360C4⋊C4—21no—
361C4⋊C4—21no—
362C4⋊C4—21no—
363C4⋊C4—21no—
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—41no—
373C4⋊C4—41no—
374C4⋊C4—41no—
375C4⋊C4—41no—
376C4⋊C4—81no—
377C4⋊C4—81no—
378C4⋊C4—82no—
379C22⋊C4—42no—
380C22⋊C4—42no—
381C22⋊C4—82no—
382C22⋊C4—82no—
383C22⋊C4—82no—
384C22⋊C4—82no—
385C42—21no—
386C42—21no—
387C42—21no—
388C42—21no—
389C42—21no—
390C42—21no—
391C42—41no—
392C42—41no—
393C42—41no—
394C42—41no—
395C42H(5,𝔽2)11yescharacteristic
396C42H(5,𝔽2)11yescharacteristic
397C422−1+411yescharacteristic
398C422−1+411yescharacteristic
399C23—21no—
400C23—21no—
401C23—21no—
402C23—21no—
403C23—21no—
404C23—21no—
405C23—81no—
406C23—81no—
407C23—81no—
408C23—81no—
409C23—81no—
410C23—81no—
411C23—81no—
412C23#47212yes—
413C23C2×#12111yescharacteristic
414Q8—161no—
415Q8—161no—
416D4—81no—
417D4—81no—
418D4—81no—
419D4—81no—
420D4—81no—
421D4—81no—
422D4—81no—
423D4—81no—
424D4—161no—
425D4—161no—
426D4—164no—
427C2×C4—21no—
428C2×C4—21no—
429C2×C4—21no—
430C2×C4—21no—
431C2×C4—21no—
432C2×C4—21no—
433C2×C4—21no—
434C2×C4—21no—
435C2×C4—21no—
436C2×C4—21no—
437C2×C4—21no—
438C2×C4—42no—
439C2×C4—42no—
440C2×C4—41no—
441C2×C4—41no—
442C2×C4—41no—
443C2×C4—41no—
444C2×C4—41no—
445C2×C4—41no—
446C2×C4—41no—
447C2×C4—41no—
448C2×C4—41no—
449C2×C4—41no—
450C2×C4—41no—
451C2×C4—41no—
452C2×C4—41no—
453C2×C4—41no—
454C2×C4—41no—
455C2×C4—41no—
456C2×C4—41no—
457C2×C4—41no—
458C2×C4—41no—
459C2×C4—41no—
460C2×C4—81no—
461C2×C4—81no—
462C2×C4—81no—
463C2×C4—81no—
464C2×C4—81no—
465C2×C4—81no—
466C2×C4—81no—
467C2×C4—81no—
468C2×C4—81no—
469C2×C4#53411yescharacteristic
470C2×C4#53611yescharacteristic
471C2×C4#53711yescharacteristic
472C2×C4#53711yescharacteristic
473C2×C4#53911yescharacteristic
474C2×C4#53911yescharacteristic
475C2×C4#57611yescharacteristic
476C2×C4#57611yescharacteristic
477C2×C4#57711yescharacteristic
478C2×C4#57711yescharacteristic
479C2×C4#57711yescharacteristic
480C2×C4#57711yescharacteristic
481C2×C4#57811yescharacteristic
482C2×C4#57811yescharacteristic
483C8—41no—
484C8—41no—
485C8—41no—
486C8—41no—
487C8—81no—
488C8—81no—
489C8—81no—
490C8—81no—
491C8—81no—
492C8—81no—
493C22—22no—
494C22—22no—
495C22—22no—
496C22—41no—
497C22—41no—
498C22—41no—
499C22—41no—
500C22—161no—
501C22—161no—
502C22—161no—
503C22—161no—
504C22—161no—
505C22—161no—
506C22—161no—
507C22#261811yescharacteristic
508C22#304511yescharacteristic
509C22#304611yescharacteristic
510C22#306211yescharacteristic
511C22#306311yescharacteristic
512C22C2×#47211yescharacteristic
513C22#306711yescharacteristic
514C4—21no—
515C4—21no—
516C4—41no—
517C4—41no—
518C4—41no—
519C4—81no—
520C4—81no—
521C4—81no—
522C4—81no—
523C4—161no—
524C4—161no—
525C4—161no—
526C4#319811yescharacteristic
527C4#320111yescharacteristic
528C4#324311yescharacteristic
529C4#324411yescharacteristic
530C4#325711yescharacteristic
531C4#325811yescharacteristic
532C4#326011yescharacteristic
533C4#326211yescharacteristic
534C2—42no—
535C2—321no—
536C2#1957711yescharacteristic
537C2#1958211yescharacteristic
538C2#1958411yescharacteristic
539C2#1959111yescharacteristic
540C2#2976911yescharacteristic
541C2#2978711yescharacteristic
542C2#2978811yescharacteristic
subgroupG/H
center, socleC23C2×#121
commutator, Frattini, cosocleC2×C42C24
prime radical#2856C22
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #29942C2semidirect
4. #21973C2semidirect
5. #10652C2semidirect
6. #10649C2semidirect
7. #10599C2semidirect
8. #10597C2semidirect
9. #10593C2semidirect
10. #10587C2semidirect
11. #10539C2semidirect
12. #10539C2semidirect
14. #9127C2semidirect
15. #8600C2semidirect

Maximal subgroups

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

subgroupindex
#299422
#299252
#298422
#219732
#106522
#106492
#105992
#105972
#105932
#105872
#105392
#91332
#91272
#86002