Finite Group Databasebeta

← #225097

Subgroups — #225097

Order 512.

632 conjugacy classes of subgroups.

The subgroup lattice has 561 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22636C211yesmaximal, characteristic
2#22509C211yesmaximal, characteristic
3#21389C211yesmaximal, characteristic
4#9557C211yesmaximal, characteristic, complemented
5#9555C211yesmaximal, characteristic, complemented
6#9548C211yesmaximal, characteristic, complemented
7#9547C211yesmaximal, characteristic, complemented
8#8850C211yesmaximal, characteristic, complemented
9#8844C211yesmaximal, characteristic, complemented
10#8708C211yesmaximal, characteristic
11#8600C211yesmaximal, characteristic
12#8314C211yesmaximal, characteristic
13#8209C211yesmaximal, characteristic, complemented
14#8203C211yesmaximal, characteristic, complemented
15#8149C211yesmaximal, characteristic
16C2×#533C2211yescharacteristic
17C2×#446C2211yescharacteristic
18C2×C8×Q8C2211yescharacteristic
19C2×#431C2211yescharacteristic
20C4×#117C2211yescharacteristic
21#1988C411yescharacteristic
22#1988C411yescharacteristic
23#1985—21no—
24#1984—21no—
25#1984C2211yescharacteristic
26#1984C2211yescharacteristic
27#1981C411yescharacteristic
28#1981C411yescharacteristic
29#1939C2211yescharacteristic
30#1939C2211yescharacteristic
31#1924—21no—
32#1924—21no—
33#1873C411yescharacteristic
34#1873C411yescharacteristic
35#1873C411yescharacteristic
36#1873C411yescharacteristic
37#1831—21no—
38#1798—21no—
39#1797C2211yescharacteristic
40#1797C2211yescharacteristic
41#1770C2211yescharacteristic
42#1770C2211yescharacteristic
43C4×#103C2211yescharacteristic
44C4×#103C2211yescharacteristic
45C4×#103C2211yescharacteristic
46C4×#103C2211yescharacteristic
47#1750—21no—
48C4×C22⋊C8C2211yescharacteristic
49C4×C22⋊C8C2211yescharacteristic
50#1745—21no—
51C2×#335C2211yescharacteristic
52C2×#334C2211yescharacteristic
53C2×Q8⋊C8C2211yescharacteristic
54C2×Q8⋊C8C2211yescharacteristic
55C2×Q8⋊C8C2211yescharacteristic
56C2×Q8⋊C8C2211yescharacteristic
57C2×#322C2211yescharacteristic
58C2×#322C2211yescharacteristic
59#1279C2211yescharacteristic
60#1279C2211yescharacteristic
61#1279C2211yescharacteristic
62#1279C2211yescharacteristic
63#1277C2211yescharacteristic
64#1277C2211yescharacteristic
65#1273C2211yescharacteristic
66#1273C2211yescharacteristic
67#533—22no—
68#533—22no—
69#533—41no—
70#533—41no—
71C2×C4⋊Q8C2×C411yescharacteristic
72C2×C4⋊Q8C2×C411yescharacteristic
73C2×#125—21no—
74C2×C22⋊Q8—21no—
75C2×C22⋊Q8C2×C411yescharacteristic
76C2×C22⋊Q8C2×C411yescharacteristic
77C2×C4×Q8—21no—
78C2×C4×Q8C2311yescharacteristic
79C2×C4×Q8C2311yescharacteristic
80C2×#117—21no—
81C2×#117C2×C411yescharacteristic
82C2×#117C2×C411yescharacteristic
83C22×C42C2311yescharacteristic
84#446—22no—
85#446—41no—
86C8×Q8—22no—
87C8×Q8—41no—
88#431—41no—
89#431—41no—
90C2×C4⋊C8—21no—
91C2×C4⋊C8—21no—
92C2×C4⋊C8—21no—
93C2×C4⋊C8—21no—
94C2×C4⋊C8C2×C411yescharacteristic
95C2×C4⋊C8C2×C411yescharacteristic
96C2×C4⋊C8C2311yescharacteristic
97C2×C4⋊C8C2311yescharacteristic
98C2×C4⋊C8C2311yescharacteristic
99C2×C4⋊C8C2311yescharacteristic
100C2×#103C2×C411yescharacteristic
101C2×#103C2×C411yescharacteristic
102C2×#103C2×C411yescharacteristic
103C2×#103C2×C411yescharacteristic
104C2×#103C2×C411yescharacteristic
105C2×#103C2×C411yescharacteristic
106C2×#103C2×C411yescharacteristic
107C2×#103C2×C411yescharacteristic
108C2×C22⋊C8—21no—
109C2×C22⋊C8—21no—
110C2×C22⋊C8—21no—
111C2×C22⋊C8C2×C411yescharacteristic
112C2×C22⋊C8C2×C411yescharacteristic
113C2×C22⋊C8D411yescharacteristic
114C2×C22⋊C8D411yescharacteristic
115C2×#97—21no—
116C2×#97—21no—
117C2×#97—21no—
118C2×C4×C8—21no—
119C2×C4×C8C2311yescharacteristic
120C2×C4×C8C2311yescharacteristic
121C2×C4×C8C2311yescharacteristic
122C2×C4×C8C2311yescharacteristic
123C2×C4×C8C2311yescharacteristic
124C2×C4×C8C2311yescharacteristic
125C4×C4⋊C4C2×C411yescharacteristic
126C4×C4⋊C4C2×C411yescharacteristic
127C4×C4⋊C4C2311yescharacteristic
128C4×C4⋊C4C2311yescharacteristic
129C4×C22⋊C4—21no—
130C4×C22⋊C4—21no—
131#376—21no—
132C43—21no—
133#340C2×C411yescharacteristic
134#340C2×C411yescharacteristic
135#340C2×C411yescharacteristic
136#340C2×C411yescharacteristic
137#340C2×C411yescharacteristic
138#340C2×C411yescharacteristic
139#340C2×C411yescharacteristic
140#340C2×C411yescharacteristic
141#336—21no—
142#336—21no—
143#336—21no—
144#336—21no—
145#336—21no—
146#336—21no—
147#336—21no—
148#336—21no—
149#336—21no—
150#336—21no—
151#336—21no—
152#336D411yescharacteristic
153#336D411yescharacteristic
154#335—41no—
155#334—41no—
156Q8⋊C8—22no—
157Q8⋊C8—22no—
158Q8⋊C8—41no—
159Q8⋊C8—41no—
160#322—41no—
161#322—41no—
162C22×Q8—21no—
163C22×Q8C22×C411yescharacteristic
164C22×Q8C22×C411yescharacteristic
165C23×C4C22×C411yescharacteristic
166C23×C4C2×D411yescharacteristic
167C23×C4#4111yescharacteristic
168C22×C8—21no—
169C22×C8—21no—
170C22×C8—21no—
171C22×C8—21no—
172C22×C8—21no—
173C22×C8—21no—
174C22×C8C22×C411yescharacteristic
175C22×C8C22×C411yescharacteristic
176C22×C8C22×C411yescharacteristic
177C22×C8C22×C411yescharacteristic
178C22×C8C2×D411yescharacteristic
179C22×C8C2×D411yescharacteristic
180C22×C8C2×D411yescharacteristic
181C22×C8C2×D411yescharacteristic
182C4⋊Q8—22no—
183C4⋊Q8—22no—
184C4⋊Q8—41no—
185C4⋊Q8—41no—
186#125—42no—
187#125—42no—
188C22⋊Q8—22no—
189C22⋊Q8—22no—
190C22⋊Q8—41no—
191C22⋊Q8—41no—
192C22⋊Q8—42no—
193C22⋊Q8—42no—
194C4×Q8—21no—
195C4×Q8—21no—
196C4×Q8—21no—
197C4×Q8—21no—
198C4×Q8—21no—
199C4×Q8—21no—
200C4×Q8—42no—
201C4×Q8—42no—
202C4×Q8#4112yes—
203C4×Q8#4112yes—
204#117—22no—
205#117—42no—
206#117—42no—
207#117—41no—
208#117—41no—
209#117—41no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4C22×C411yescharacteristic
218C2×C4⋊C4C22×C411yescharacteristic
219C2×C4⋊C4C22×C411yescharacteristic
220C2×C4⋊C4C22×C411yescharacteristic
221C2×C4⋊C4C22×C411yescharacteristic
222C2×C4⋊C4C22×C411yescharacteristic
223C2×C22⋊C4—21no—
224C2×C22⋊C4—21no—
225C2×C22⋊C4—21no—
226C2×C22⋊C4—21no—
227C2×C42—21no—
228C2×C42—21no—
229C2×C42—21no—
230C2×C42—21no—
231C2×C42—21no—
232C2×C42—21no—
233C2×C42—21no—
234C2×C42—21no—
235C2×C42—21no—
236C2×C42—21no—
237C2×C42—21no—
238C2×C42—21no—
239C2×C42C22×C411yescharacteristic
240C2×C42C2×D411yescharacteristic
241C2×C42#4111yescharacteristic
242C2×C42C2411yescharacteristic
243C4⋊C8—21no—
244C4⋊C8—21no—
245C4⋊C8—21no—
246C4⋊C8—21no—
247C4⋊C8—21no—
248C4⋊C8—21no—
249C4⋊C8—21no—
250C4⋊C8—21no—
251C4⋊C8—41no—
252C4⋊C8—41no—
253C4⋊C8—42no—
254C4⋊C8—42no—
255C4⋊C8—81no—
256C4⋊C8—81no—
257#103—22no—
258#103—22no—
259#103—41no—
260#103—41no—
261#103—41no—
262#103—41no—
263#103—41no—
264#103—41no—
265C22⋊C8—22no—
266C22⋊C8—41no—
267C22⋊C8—41no—
268C22⋊C8—41no—
269C22⋊C8—42no—
270C22⋊C8—42no—
271C22⋊C8—81no—
272#97—42no—
273#97—42no—
274#97—42no—
275C4×C8—21no—
276C4×C8—21no—
277C4×C8—21no—
278C4×C8—21no—
279C4×C8—21no—
280C4×C8—21no—
281C4×C8—21no—
282C4×C8—21no—
283C4×C8—21no—
284C4×C8—21no—
285C4×C8—21no—
286C4×C8—21no—
287C4×C8—42no—
288#95—21no—
289#95—21no—
290#95—21no—
291#95—21no—
292C24C4×D411yescharacteristic
293C2×Q8—21no—
294C2×Q8—21no—
295C2×Q8—21no—
296C2×Q8—21no—
297C2×Q8—21no—
298C2×Q8—21no—
299C2×Q8—21no—
300C2×Q8—21no—
301C2×Q8—21no—
302C2×Q8—21no—
303C2×Q8—21no—
304C2×Q8—42no—
305C2×Q8—42no—
306C2×Q8—81no—
307C2×Q8#11712yes—
308C22×C4—21no—
309C22×C4—21no—
310C22×C4—21no—
311C22×C4—21no—
312C22×C4—21no—
313C22×C4—21no—
314C22×C4—21no—
315C22×C4—21no—
316C22×C4—21no—
317C22×C4—21no—
318C22×C4—21no—
319C22×C4—21no—
320C22×C4—21no—
321C22×C4—21no—
322C22×C4—21no—
323C22×C4—21no—
324C22×C4—21no—
325C22×C4—22no—
326C22×C4—21no—
327C22×C4—21no—
328C22×C4—21no—
329C22×C4—21no—
330C22×C4—21no—
331C22×C4—41no—
332C22×C4C4×D411yescharacteristic
333C22×C4C4×D411yescharacteristic
334C22×C4C4×D411yescharacteristic
335C22×C4C23×C411yescharacteristic
336C22×C4C22×D411yescharacteristic
337C22×C4C2×#4111yescharacteristic
338C2×C8—21no—
339C2×C8—21no—
340C2×C8—21no—
341C2×C8—21no—
342C2×C8—21no—
343C2×C8—21no—
344C2×C8—21no—
345C2×C8—21no—
346C2×C8—21no—
347C2×C8—21no—
348C2×C8—21no—
349C2×C8—21no—
350C2×C8—21no—
351C2×C8—21no—
352C2×C8—21no—
353C2×C8—21no—
354C2×C8—21no—
355C2×C8—21no—
356C2×C8—21no—
357C2×C8—21no—
358C2×C8—21no—
359C2×C8—21no—
360C2×C8—21no—
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#13712yes—
380C4⋊C4—21no—
381C4⋊C4—22no—
382C4⋊C4—22no—
383C4⋊C4—21no—
384C4⋊C4—21no—
385C4⋊C4—21no—
386C4⋊C4—21no—
387C4⋊C4—21no—
388C4⋊C4—21no—
389C4⋊C4—21no—
390C4⋊C4—21no—
391C4⋊C4—21no—
392C4⋊C4—42no—
393C4⋊C4—42no—
394C4⋊C4—42no—
395C4⋊C4—42no—
396C4⋊C4—42no—
397C4⋊C4—41no—
398C4⋊C4—41no—
399C4⋊C4#11712yes—
400C4⋊C4#13112yes—
401C22⋊C4—42no—
402C22⋊C4—42no—
403C22⋊C4—42no—
404C22⋊C4—42no—
405C42—21no—
406C42—21no—
407C42—22no—
408C42—22no—
409C42—21no—
410C42—21no—
411C42—21no—
412C42—21no—
413C42—42no—
414C42—42no—
415C42—41no—
416C42—41no—
417C42—41no—
418C42—41no—
419C42—41no—
420C42—41no—
421C42—42no—
422C42—81no—
423C42C2×#4111yescharacteristic
424C42C2×#4111yescharacteristic
425C42H(5,𝔽2)11yescharacteristic
426C42H(5,𝔽2)11yescharacteristic
427C23—21no—
428C23—21no—
429C23—21no—
430C23—21no—
431C23—21no—
432C23—21no—
433C23—21no—
434C23C2×C4×D411yescharacteristic
435Q8—22no—
436Q8—22no—
437Q8—41no—
438Q8—41no—
439Q8—41no—
440Q8—41no—
441Q8—41no—
442Q8—41no—
443Q8—84no—
444C2×C4—21no—
445C2×C4—21no—
446C2×C4—21no—
447C2×C4—21no—
448C2×C4—21no—
449C2×C4—21no—
450C2×C4—22no—
451C2×C4—22no—
452C2×C4—21no—
453C2×C4—21no—
454C2×C4—22no—
455C2×C4—21no—
456C2×C4—21no—
457C2×C4—42no—
458C2×C4—42no—
459C2×C4—41no—
460C2×C4—41no—
461C2×C4—41no—
462C2×C4—41no—
463C2×C4—41no—
464C2×C4—41no—
465C2×C4—41no—
466C2×C4—41no—
467C2×C4—41no—
468C2×C4—41no—
469C2×C4—41no—
470C2×C4—41no—
471C2×C4—41no—
472C2×C4—41no—
473C2×C4—41no—
474C2×C4—41no—
475C2×C4—41no—
476C2×C4—41no—
477C2×C4—41no—
478C2×C4—41no—
479C2×C4—41no—
480C2×C4—41no—
481C2×C4—41no—
482C2×C4—41no—
483C2×C4—41no—
484C2×C4—41no—
485C2×C4—41no—
486C2×C4—41no—
487C2×C4—41no—
488C2×C4#44412yes—
489C2×C4C2×#11711yescharacteristic
490C2×C4#51811yescharacteristic
491C2×C4#54611yescharacteristic
492C2×C4#54611yescharacteristic
493C2×C4#55111yescharacteristic
494C2×C4#55111yescharacteristic
495C2×C4C2×#13111yescharacteristic
496C2×C4#56811yescharacteristic
497C2×C4#56811yescharacteristic
498C2×C4#56811yescharacteristic
499C2×C4C2×#13711yescharacteristic
500C2×C4#57511yescharacteristic
501C2×C4#57711yescharacteristic
502C2×C4#57811yescharacteristic
503C8—22no—
504C8—41no—
505C8—41no—
506C8—41no—
507C8—41no—
508C8—41no—
509C8—41no—
510C8—41no—
511C8—81no—
512C8—81no—
513C8—81no—
514C8—81no—
515C8—81no—
516C8—81no—
517C22—41no—
518C22—41no—
519C22—41no—
520C22—41no—
521C22—41no—
522C22—41no—
523C22—41no—
524C22#234211yescharacteristic
525C22#293011yescharacteristic
526C22#293311yescharacteristic
527C22C4×#13711yescharacteristic
528C22#294711yescharacteristic
529C22C2×#44411yescharacteristic
530C22#295711yescharacteristic
531C4—21no—
532C4—21no—
533C4—22no—
534C4—41no—
535C4—42no—
536C4—41no—
537C4—42no—
538C4—42no—
539C4—81no—
540C4—81no—
541C4—81no—
542C4—81no—
543C4—81no—
544C4#292311yescharacteristic
545C4#292511yescharacteristic
546C4#298011yescharacteristic
547C4#298111yescharacteristic
548C4#331211yescharacteristic
549C4#332111yescharacteristic
550C4#332711yescharacteristic
551C4#332911yescharacteristic
552C2—81no—
553C2#1794111yescharacteristic
554C2#1794911yescharacteristic
555C2#1801311yescharacteristic
556C2#1802111yescharacteristic
557C2#2255711yescharacteristic
558C2#2258511yescharacteristic
559C2#2258611yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC23C2×C4×D4
prime radicalC24C4×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C4×C22⋊C8 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #9557C2semidirect
5. #9555C2semidirect
6. #9548C2semidirect
7. #9547C2semidirect
8. #8850C2semidirect
9. #8844C2semidirect
13. #8209C2semidirect
14. #8203C2semidirect

Maximal subgroups

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

subgroupindex
#226362
#225092
#213892
#95572
#95552
#95482
#95472
#88502
#88442
#87082
#86002
#83142
#82092
#82032
#81492