Finite Group Databasebeta

← #226575

Subgroups — #226575

Order 512.

649 conjugacy classes of subgroups.

The subgroup lattice has 539 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22839C211yesmaximal, characteristic
2#22722C211yesmaximal, characteristic, complemented
3C4×#440C211yesmaximal, characteristic
4#9556C211yesmaximal, characteristic
5#9554C211yesmaximal, characteristic
6#9540C211yesmaximal, characteristic, complemented
7#9537C211yesmaximal, characteristic, complemented
8#8952C211yesmaximal, characteristic
9#8952C211yesmaximal, characteristic
10#8786C211yesmaximal, characteristic, complemented
11#8566C211yesmaximal, characteristic, complemented
12#8563C211yesmaximal, characteristic
13#8262C211yesmaximal, characteristic, complemented
14#8209C211yesmaximal, characteristic, complemented
15#8207C211yesmaximal, characteristic, complemented
16C2×#446C2211yescharacteristic
17C2×#446C2211yescharacteristic
18C2×#440C411yescharacteristic
19C2×#440C411yescharacteristic
20C2×C4×SD16C2211yescharacteristic
21C42×Q8C2211yescharacteristic
22C42×D4C2211yescharacteristic
23#1954C2211yescharacteristic
24#1954C2211yescharacteristic
25#1940—21no—
26#1940—21no—
27#1939C2211yescharacteristic
28#1939C2211yescharacteristic
29#1924—21no—
30#1924—21no—
31#1842—21no—
32#1842C2211yescharacteristic
33#1831C2211yescharacteristic
34#1817—21no—
35#1776C411yescharacteristic
36#1776C411yescharacteristic
37C4×#107C2211yescharacteristic
38#1770C2211yescharacteristic
39#1770C2211yescharacteristic
40#1767—21no—
41C4×C4⋊C8C2211yescharacteristic
42C4×C4⋊C8C2211yescharacteristic
43#1763C411yescharacteristic
44#1763C411yescharacteristic
45#1762C411yescharacteristic
46#1762C411yescharacteristic
47C4×#103C2211yescharacteristic
48C4×#102C2211yescharacteristic
49#1749—21no—
50C4×#97C2211yescharacteristic
51C2×Q8⋊C8C2211yescharacteristic
52C2×Q8⋊C8C2211yescharacteristic
53C2×D4⋊C8C2211yescharacteristic
54C2×D4⋊C8C2211yescharacteristic
55C2×#322C2211yescharacteristic
56C2×C82C2211yescharacteristic
57#1279C2211yescharacteristic
58#1279C2211yescharacteristic
59#1279C2211yescharacteristic
60#1279C2211yescharacteristic
61#1277C2211yescharacteristic
62#1277C2211yescharacteristic
63#1277C2211yescharacteristic
64#1277C2211yescharacteristic
65#1273C2211yescharacteristic
66#1273C2211yescharacteristic
67C22×SD16C2×C411yescharacteristic
68C2×C4×Q8C2×C411yescharacteristic
69C2×C4×Q8C2×C411yescharacteristic
70C2×C4×Q8C2311yescharacteristic
71C2×C4×D4C2×C411yescharacteristic
72C2×C4×D4C2×C411yescharacteristic
73C2×C4×D4C2311yescharacteristic
74C22×C42—21no—
75#446—22no—
76#446—22no—
77#446—41no—
78#446—41no—
79#440—24no—
80#440—24no—
81C4×SD16—22no—
82C4×SD16—22no—
83C2×#107C2×C411yescharacteristic
84C2×#107C2×C411yescharacteristic
85C2×#106C2×C411yescharacteristic
86C2×C4⋊C8—21no—
87C2×C4⋊C8—21no—
88C2×C4⋊C8—21no—
89C2×C4⋊C8—21no—
90C2×C4⋊C8—21no—
91C2×C4⋊C8—21no—
92C2×C4⋊C8—21no—
93C2×C4⋊C8D411yescharacteristic
94C2×C4⋊C8D411yescharacteristic
95C2×C4⋊C8C2311yescharacteristic
96C2×C4⋊C8C2311yescharacteristic
97C2×C4⋊C8C2311yescharacteristic
98C2×C4⋊C8C2311yescharacteristic
99C2×#103C2×C411yescharacteristic
100C2×#103C2×C411yescharacteristic
101C2×#103C2×C411yescharacteristic
102C2×#102C2×C411yescharacteristic
103C2×#102C2×C411yescharacteristic
104C2×#102C2×C411yescharacteristic
105C2×C22⋊C8—21no—
106C2×C22⋊C8—21no—
107C2×C22⋊C8—21no—
108C2×C22⋊C8—21no—
109C2×#97—21no—
110C2×#97—21no—
111C2×#97—21no—
112C2×#97—21no—
113C2×#97C2×C411yescharacteristic
114C2×#97C2×C411yescharacteristic
115C2×#97C2×C411yescharacteristic
116C2×#97C2×C411yescharacteristic
117C2×C4×C8C2311yescharacteristic
118C2×C4×C8C2311yescharacteristic
119C2×C4×C8C2311yescharacteristic
120C2×C4×C8C2311yescharacteristic
121C2×C4×C8C2311yescharacteristic
122C2×C4×C8C2311yescharacteristic
123C4×C4⋊C4—21no—
124C4×C4⋊C4—21no—
125C4×C4⋊C4—21no—
126C4×C4⋊C4C2×C411yescharacteristic
127C4×C4⋊C4C2×C411yescharacteristic
128C4×C4⋊C4C2×C411yescharacteristic
129C4×C4⋊C4C2×C411yescharacteristic
130C4×C4⋊C4C2311yescharacteristic
131C4×C4⋊C4C2311yescharacteristic
132C4×C22⋊C4—21no—
133C4×C22⋊C4—21no—
134C4×C22⋊C4—21no—
135C43—21no—
136C43C2311yescharacteristic
137#340C2×C411yescharacteristic
138#340C2×C411yescharacteristic
139#340C2×C411yescharacteristic
140#340C2×C411yescharacteristic
141#340C2×C411yescharacteristic
142#340C2×C411yescharacteristic
143#336—21no—
144#336—21no—
145#336—21no—
146#336—21no—
147#336—21no—
148#336—21no—
149#336—21no—
150#336D411yescharacteristic
151#336D411yescharacteristic
152Q8⋊C8—22no—
153Q8⋊C8—22no—
154D4⋊C8—41no—
155D4⋊C8—41no—
156#322—41no—
157C82—41no—
158C22×Q8C22×C411yescharacteristic
159C22×D4C22×C411yescharacteristic
160C23×C4—21no—
161C23×C4—21no—
162C23×C4—21no—
163C2×SD16—22no—
164C2×SD16—22no—
165C2×SD16—22no—
166C22×C8—21no—
167C22×C8—21no—
168C22×C8—21no—
169C22×C8—21no—
170C22×C8—21no—
171C22×C8—21no—
172C22×C8C22×C411yescharacteristic
173C22×C8C22×C411yescharacteristic
174C22×C8C22×C411yescharacteristic
175C22×C8C22×C411yescharacteristic
176C22×C8C2×D411yescharacteristic
177C22×C8C2×D411yescharacteristic
178C22×C8C2×D411yescharacteristic
179C22×C8C2×D411yescharacteristic
180C4×Q8—21no—
181C4×Q8—21no—
182C4×Q8—22no—
183C4×Q8—22no—
184C4×Q8—22no—
185C4×Q8—41no—
186C4×Q8#4112yes—
187C4×Q8#4112yes—
188C4×D4—22no—
189C4×D4—22no—
190C4×D4—22no—
191C4×D4—21no—
192C4×D4—21no—
193C4×D4—21no—
194C4×D4—21no—
195C4×D4—41no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4C22×C411yescharacteristic
200C2×C4⋊C4C22×C411yescharacteristic
201C2×C4⋊C4C22×C411yescharacteristic
202C2×C4⋊C4C22×C411yescharacteristic
203C2×C4⋊C4C22×C411yescharacteristic
204C2×C4⋊C4C22×C411yescharacteristic
205C2×C22⋊C4—21no—
206C2×C22⋊C4—21no—
207C2×C22⋊C4—21no—
208C2×C42—21no—
209C2×C42—21no—
210C2×C42—21no—
211C2×C42—21no—
212C2×C42—21no—
213C2×C42—21no—
214C2×C42—21no—
215C2×C42—21no—
216C2×C42—21no—
217C2×C42—81no—
218C2×C42—81no—
219C2×C42C22×C411yescharacteristic
220C2×C42C22×C411yescharacteristic
221C2×C42C2×D411yescharacteristic
222C2×C42C2×D411yescharacteristic
223C2×C42#4111yescharacteristic
224C2×C42#4111yescharacteristic
225C2×C42C2411yescharacteristic
226#107—22no—
227#107—22no—
228#106—22no—
229C4⋊C8—22no—
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—42no—
240C4⋊C8—42no—
241C4⋊C8—42no—
242C4⋊C8—42no—
243C4⋊C8—81no—
244C4⋊C8—81no—
245C4⋊C8—81no—
246#103—22no—
247#103—22no—
248#103—22no—
249#102—22no—
250#102—22no—
251#102M1614yes—
252C22⋊C8—42no—
253C22⋊C8—42no—
254C22⋊C8—81no—
255C22⋊C8—81no—
256#97—22no—
257#97—22no—
258#97—41no—
259#97—42no—
260#97—42no—
261#97—81no—
262#97—81no—
263#97M1614yes—
264C4×C8—21no—
265C4×C8—21no—
266C4×C8—21no—
267C4×C8—21no—
268C4×C8—21no—
269C4×C8—21no—
270C4×C8—21no—
271C4×C8—21no—
272C4×C8—21no—
273C4×C8—21no—
274C4×C8—21no—
275C4×C8—21no—
276#95—21no—
277#95—21no—
278#95—21no—
279#95—21no—
280#95—21no—
281#95—21no—
282C24—21no—
283C2×Q8—21no—
284C2×Q8—21no—
285C2×Q8—21no—
286C2×Q8—21no—
287C2×Q8—21no—
288C2×Q8#11712yes—
289C2×D4—21no—
290C2×D4—21no—
291C2×D4—21no—
292C2×D4—21no—
293C2×D4—21no—
294C2×D4C2×M1612yes—
295C22×C4—21no—
296C22×C4—21no—
297C22×C4—21no—
298C22×C4—21no—
299C22×C4—21no—
300C22×C4—21no—
301C22×C4—21no—
302C22×C4—42no—
303C22×C4—42no—
304C22×C4—41no—
305C22×C4—41no—
306C22×C4—41no—
307C22×C4—41no—
308C22×C4—41no—
309C22×C4—41no—
310C22×C4—42no—
311C22×C4—81no—
312C22×C4—81no—
313C22×C4—81no—
314C22×C4C4×D411yescharacteristic
315C22×C4C4×D411yescharacteristic
316C22×C4C4×D411yescharacteristic
317C22×C4C4×D411yescharacteristic
318C22×C4C23×C411yescharacteristic
319C22×C4C22×D411yescharacteristic
320C22×C4C2×#4111yescharacteristic
321SD16—28no—
322C2×C8—21no—
323C2×C8—21no—
324C2×C8—21no—
325C2×C8—21no—
326C2×C8—21no—
327C2×C8—21no—
328C2×C8—21no—
329C2×C8—21no—
330C2×C8—22no—
331C2×C8—21no—
332C2×C8—21no—
333C2×C8—21no—
334C2×C8—21no—
335C2×C8—21no—
336C2×C8—21no—
337C2×C8—21no—
338C2×C8—21no—
339C2×C8—21no—
340C2×C8—21no—
341C2×C8—21no—
342C2×C8—21no—
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—41no—
356C2×C8—41no—
357C2×C8—41no—
358C2×C8—41no—
359C2×C8—41no—
360C2×C8C2×M1612yes—
361C2×C8C2×M1612yes—
362C2×C8C2×M1612yes—
363C2×C8#13712yes—
364C4⋊C4—21no—
365C4⋊C4—22no—
366C4⋊C4—22no—
367C4⋊C4—21no—
368C4⋊C4—21no—
369C4⋊C4—21no—
370C4⋊C4—21no—
371C4⋊C4—21no—
372C4⋊C4—21no—
373C4⋊C4—21no—
374C4⋊C4—21no—
375C4⋊C4—21no—
376C4⋊C4—42no—
377C4⋊C4—42no—
378C4⋊C4#11712yes—
379C4⋊C4C2×M1612yes—
380C22⋊C4—42no—
381C22⋊C4—41no—
382C22⋊C4—41no—
383C22⋊C4—42no—
384C42—22no—
385C42—22no—
386C42—21no—
387C42—21no—
388C42—21no—
389C42—21no—
390C42—21no—
391C42—21no—
392C42—41no—
393C42—41no—
394C42—42no—
395C42—42no—
396C42—41no—
397C42—41no—
398C42—41no—
399C42—41no—
400C42—42no—
401C42—81no—
402C42—81no—
403C42—81no—
404C42C2×M1612yes—
405C42#13112yes—
406C42C2×#4111yescharacteristic
407C42C2×#4111yescharacteristic
408C42H(5,𝔽2)11yescharacteristic
409C422−1+411yescharacteristic
410C23—22no—
411C23—41no—
412C23—41no—
413C23—41no—
414C23—41no—
415C23—41no—
416C23—41no—
417C23C2×C4×D411yescharacteristic
418Q8—22no—
419Q8—22no—
420Q8—41no—
421Q8—41no—
422D4—22no—
423D4—22no—
424D4—41no—
425D4—41no—
426C2×C4—21no—
427C2×C4—21no—
428C2×C4—21no—
429C2×C4—21no—
430C2×C4—21no—
431C2×C4—21no—
432C2×C4—22no—
433C2×C4—22no—
434C2×C4—21no—
435C2×C4—21no—
436C2×C4—22no—
437C2×C4—21no—
438C2×C4—21no—
439C2×C4—21no—
440C2×C4—21no—
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—84no—
460C2×C4—84no—
461C2×C4—81no—
462C2×C4—81no—
463C2×C4—81no—
464C2×C4—81no—
465C2×C4#43612yes—
466C2×C4#43612yes—
467C2×C4C2×#11711yescharacteristic
468C2×C4#52011yescharacteristic
469C2×C4#54611yescharacteristic
470C2×C4#54811yescharacteristic
471C2×C4#55211yescharacteristic
472C2×C4#55311yescharacteristic
473C2×C4C22×M1611yescharacteristic
474C2×C4C22×M1611yescharacteristic
475C2×C4C2×#13111yescharacteristic
476C2×C4#56811yescharacteristic
477C2×C4C2×#13711yescharacteristic
478C2×C4#57511yescharacteristic
479C2×C4#57611yescharacteristic
480C2×C4#57711yescharacteristic
481C8—22no—
482C8—22no—
483C8—22no—
484C8—22no—
485C8—42no—
486C8—41no—
487C8—41no—
488C8—41no—
489C8—41no—
490C8—81no—
491C8—81no—
492C8—81no—
493C8—81no—
494C8—81no—
495C22—42no—
496C22—42no—
497C22—42no—
498C22—81no—
499C22—81no—
500C22—81no—
501C22—81no—
502C22#235211yescharacteristic
503C22C2×#43611yescharacteristic
504C22D4×M1611yescharacteristic
505C22C4×#13711yescharacteristic
506C22#294611yescharacteristic
507C22#295511yescharacteristic
508C22#295611yescharacteristic
509C4—21no—
510C4—21no—
511C4—22no—
512C4—22no—
513C4—42no—
514C4—41no—
515C4—41no—
516C4—42no—
517C4—42no—
518C4—81no—
519C4—81no—
520C4—81no—
521C4—81no—
522C4#292011yescharacteristic
523C4#292511yescharacteristic
524C4#298911yescharacteristic
525C4#299011yescharacteristic
526C4#330911yescharacteristic
527C4#332011yescharacteristic
528C4#337111yescharacteristic
529C4#337511yescharacteristic
530C2—82no—
531C2#1789211yescharacteristic
532C2#1789311yescharacteristic
533C2#1804711yescharacteristic
534C2#1805011yescharacteristic
535C2#2275111yescharacteristic
536C2#2277511yescharacteristic
537C2#2278411yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC23C2×C4×D4
prime radicalC22×D4C22×C4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C4×C4⋊C8 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #22722C2semidirect
6. #9540C2semidirect
7. #9537C2semidirect
10. #8786C2semidirect
11. #8566C2semidirect
13. #8262C2semidirect
14. #8209C2semidirect
15. #8207C2semidirect

Maximal subgroups

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

subgroupindex
#228392
#227222
C4×#4402
#95562
#95542
#95402
#95372
#89522
#87862
#85662
#85632
#82622
#82092
#82072