Finite Group Databasebeta

← #228301

Subgroups — #228301

Order 512.

634 conjugacy classes of subgroups.

The subgroup lattice has 552 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23754C211yesmaximal, characteristic
2#23083C211yesmaximal, characteristic
3#22626C211yesmaximal, characteristic, complemented
4#9594C211yesmaximal, characteristic, complemented
5#9560C211yesmaximal, characteristic, complemented
6#9545C211yesmaximal, characteristic, complemented
7#9536C211yesmaximal, characteristic, complemented
8#9533C211yesmaximal, characteristic, complemented
9#9078C211yesmaximal, characteristic
10#8845C211yesmaximal, characteristic, complemented
11#8831C211yesmaximal, characteristic, complemented
12#8288C211yesmaximal, characteristic, complemented
13C8×#102C211yesmaximal, characteristic, complemented
14#8146C211yesmaximal, characteristic, complemented
15#7896C211yesmaximal, characteristic, complemented
16C2×C8×D4C2211yescharacteristic
17C2×#431C2211yescharacteristic
18C2×C4×M16C2211yescharacteristic
19C4×#121C2211yescharacteristic
20#2285C2211yescharacteristic
21#2099C411yescharacteristic
22#2093C411yescharacteristic
23#2083C411yescharacteristic
24#2075C411yescharacteristic
25#2075C411yescharacteristic
26#2071C411yescharacteristic
27#2033C411yescharacteristic
28#2033C411yescharacteristic
29#1954—21no—
30#1954C2211yescharacteristic
31#1939C2211yescharacteristic
32#1939C2211yescharacteristic
33#1939C2211yescharacteristic
34#1923—21no—
35#1917—21no—
36#1916—21no—
37#1869—21no—
38#1798—21no—
39#1797C2211yescharacteristic
40C4×#107C2211yescharacteristic
41C4×#107C2211yescharacteristic
42#1770C2211yescharacteristic
43C8×C4⋊C4C2211yescharacteristic
44C8×C4⋊C4C2211yescharacteristic
45C4×#102C2211yescharacteristic
46C4×#102C2211yescharacteristic
47C8×C22⋊C4—21no—
48#1750—21no—
49C4×C22⋊C8C2211yescharacteristic
50C4×C22⋊C8C2211yescharacteristic
51C2×#335C2211yescharacteristic
52C2×#334C2211yescharacteristic
53C2×D4⋊C8C2211yescharacteristic
54C2×D4⋊C8C2211yescharacteristic
55C2×#322C2211yescharacteristic
56C2×C82C2211yescharacteristic
57#1279C2211yescharacteristic
58#1279C2211yescharacteristic
59#1279C2211yescharacteristic
60#1279C2211yescharacteristic
61#1279C2211yescharacteristic
62#1279C2211yescharacteristic
63#1277C2211yescharacteristic
64#1277C2211yescharacteristic
65#1273C2211yescharacteristic
66#1273C2211yescharacteristic
67C22×M16—21no—
68C23×C8—21no—
69C2×#121C2×C411yescharacteristic
70C2×C4×D4—21no—
71C2×C4×D4C2311yescharacteristic
72C2×#117—21no—
73C22×C42C2311yescharacteristic
74C8×D4—22no—
75C8×D4—41no—
76#431—22no—
77#431—41no—
78C2×#107C2×C411yescharacteristic
79C2×#107C2×C411yescharacteristic
80C2×#107C2×C411yescharacteristic
81C2×#107C2×C411yescharacteristic
82C2×C4⋊C8—21no—
83C2×C4⋊C8—21no—
84C2×C4⋊C8—21no—
85C2×C4⋊C8C2311yescharacteristic
86C2×C4⋊C8C2311yescharacteristic
87C2×C4⋊C8C2311yescharacteristic
88C2×C4⋊C8C2311yescharacteristic
89C2×#102C2×C411yescharacteristic
90C2×#102C2×C411yescharacteristic
91C2×#102C2×C411yescharacteristic
92C2×#102C2×C411yescharacteristic
93C2×C22⋊C8—21no—
94C2×C22⋊C8—21no—
95C2×C22⋊C8—21no—
96C2×C22⋊C8—21no—
97C2×C22⋊C8—21no—
98C2×C22⋊C8C2×C411yescharacteristic
99C2×C22⋊C8C2×C411yescharacteristic
100C2×C22⋊C8D411yescharacteristic
101C2×C22⋊C8D411yescharacteristic
102C4×M16—22no—
103C4×M16—41no—
104C2×#97—21no—
105C2×#97—21no—
106C2×C4×C8—21no—
107C2×C4×C8—21no—
108C2×C4×C8—21no—
109C2×C4×C8C2311yescharacteristic
110C2×C4×C8C2311yescharacteristic
111C2×C4×C8C2311yescharacteristic
112C2×C4×C8C2311yescharacteristic
113C2×C4×C8C2311yescharacteristic
114C2×C4×C8C2311yescharacteristic
115#390C2×C411yescharacteristic
116#389C2×C411yescharacteristic
117#388—21no—
118#386—21no—
119#384C2×C411yescharacteristic
120#384C2×C411yescharacteristic
121#382—21no—
122#380—21no—
123#380C2×C411yescharacteristic
124#380C2×C411yescharacteristic
125#380C2×C411yescharacteristic
126C4×C4⋊C4—21no—
127C4×C4⋊C4C2311yescharacteristic
128C4×C4⋊C4C2311yescharacteristic
129C4×C4⋊C4C2311yescharacteristic
130C4×C22⋊C4—21no—
131#340C2×C411yescharacteristic
132#340C2×C411yescharacteristic
133#340C2×C411yescharacteristic
134#340C2×C411yescharacteristic
135#340C2×C411yescharacteristic
136#340C2×C411yescharacteristic
137#340C2×C411yescharacteristic
138#340C2×C411yescharacteristic
139#336—21no—
140#336—21no—
141#336—21no—
142#336—21no—
143#336—21no—
144#336—21no—
145#336—21no—
146#336C2×C411yescharacteristic
147#336C2×C411yescharacteristic
148#336D411yescharacteristic
149#336D411yescharacteristic
150#335—41no—
151#334—22no—
152D4⋊C8—22no—
153D4⋊C8—41no—
154#322—22no—
155C82—41no—
156C22×D4—21no—
157C22×D4C22×C411yescharacteristic
158C23×C4—21no—
159C23×C4C22×C411yescharacteristic
160C23×C4C2×D411yescharacteristic
161C23×C4#4111yescharacteristic
162C2×M16—24no—
163C2×M16—42no—
164C2×M16—42no—
165C22×C8—21no—
166C22×C8—21no—
167C22×C8—21no—
168C22×C8—21no—
169C22×C8—21no—
170C22×C8—21no—
171C22×C8—42no—
172C22×C8—42no—
173C22×C8—81no—
174C22×C8C22×C411yescharacteristic
175C22×C8C22×C411yescharacteristic
176C22×C8C22×C411yescharacteristic
177C22×C8C22×C411yescharacteristic
178C22×C8C2×D411yescharacteristic
179C22×C8C2×D411yescharacteristic
180C22×C8C2×D411yescharacteristic
181C22×C8C2×D411yescharacteristic
182#121—22no—
183#121—22no—
184C4×D4—21no—
185C4×D4—21no—
186C4×D4—21no—
187C4×D4—21no—
188C4×D4—81no—
189C4×D4—81no—
190#117—81no—
191#117—81no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4C22×C411yescharacteristic
198C2×C4⋊C4C22×C411yescharacteristic
199C2×C4⋊C4C22×C411yescharacteristic
200C2×C4⋊C4C22×C411yescharacteristic
201C2×C4⋊C4C22×C411yescharacteristic
202C2×C4⋊C4C22×C411yescharacteristic
203C2×C4⋊C4C22×C411yescharacteristic
204C2×C22⋊C4—21no—
205C2×C22⋊C4—21no—
206C2×C22⋊C4—21no—
207C2×C22⋊C4—21no—
208C2×C22⋊C4—21no—
209C2×C22⋊C4—21no—
210C2×C22⋊C4—21no—
211C2×C42—21no—
212C2×C42—21no—
213C2×C42—21no—
214C2×C42—21no—
215C2×C42—21no—
216C2×C42—21no—
217C2×C42—21no—
218C2×C42C22×C411yescharacteristic
219C2×C42C2×D411yescharacteristic
220C2×C42#4111yescharacteristic
221C2×C42C2411yescharacteristic
222#107—22no—
223#107—22no—
224#107—22no—
225#107—41no—
226C4⋊C8—21no—
227C4⋊C8—21no—
228C4⋊C8—21no—
229C4⋊C8—21no—
230C4⋊C8—21no—
231C4⋊C8—21no—
232C4⋊C8—21no—
233C4⋊C8—81no—
234C4⋊C8—81no—
235C4⋊C8—81no—
236C4⋊C8#4112yes—
237#102—22no—
238#102—22no—
239#102—22no—
240#102—41no—
241C22⋊C8—22no—
242C22⋊C8—22no—
243C22⋊C8—41no—
244C22⋊C8—41no—
245C22⋊C8—42no—
246C22⋊C8—42no—
247C22⋊C8—42no—
248C22⋊C8—42no—
249C22⋊C8—81no—
250#97—42no—
251#97—81no—
252C4×C8—21no—
253C4×C8—21no—
254C4×C8—21no—
255C4×C8—21no—
256C4×C8—21no—
257C4×C8—21no—
258C4×C8—21no—
259C4×C8—21no—
260C4×C8—21no—
261C4×C8—21no—
262C4×C8—21no—
263C4×C8—42no—
264C4×C8—81no—
265C4×C8—81no—
266C4×C8#4112yes—
267#95—21no—
268#95—21no—
269#95—21no—
270#95—21no—
271#95—21no—
272#95—21no—
273#95—21no—
274C24—21no—
275C24C4×D411yescharacteristic
276C2×D4—24no—
277C2×D4—21no—
278C2×D4—21no—
279C2×D4—21no—
280C2×D4—21no—
281C2×D4—21no—
282C2×D4—81no—
283C2×D4—81no—
284C2×D4#13112yes—
285C22×C4—21no—
286C22×C4—21no—
287C22×C4—21no—
288C22×C4—21no—
289C22×C4—21no—
290C22×C4—21no—
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—21no—
294C22×C4—21no—
295C22×C4—21no—
296C22×C4—21no—
297C22×C4—21no—
298C22×C4—21no—
299C22×C4—21no—
300C22×C4—22no—
301C22×C4—21no—
302C22×C4—21no—
303C22×C4—21no—
304C22×C4—21no—
305C22×C4—21no—
306C22×C4—21no—
307C22×C4—41no—
308C22×C4—41no—
309C22×C4—41no—
310C22×C4—41no—
311C22×C4—41no—
312C22×C4—41no—
313C22×C4—41no—
314C22×C4C4×D411yescharacteristic
315C22×C4C4×D411yescharacteristic
316C22×C4C4×D411yescharacteristic
317C22×C4C23×C411yescharacteristic
318C22×C4C22×D411yescharacteristic
319C22×C4C2×#4111yescharacteristic
320M16—48no—
321C2×C8—22no—
322C2×C8—22no—
323C2×C8—21no—
324C2×C8—21no—
325C2×C8—22no—
326C2×C8—21no—
327C2×C8—21no—
328C2×C8—22no—
329C2×C8—21no—
330C2×C8—21no—
331C2×C8—21no—
332C2×C8—21no—
333C2×C8—22no—
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—21no—
344C2×C8—21no—
345C2×C8—21no—
346C2×C8—21no—
347C2×C8—21no—
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×C8—41no—
361C2×C8—84no—
362C2×C8#13112yes—
363C2×C8#13712yes—
364C4⋊C4—21no—
365C4⋊C4—21no—
366C4⋊C4—21no—
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—21no—
377C4⋊C4—42no—
378C4⋊C4—81no—
379C4⋊C4—81no—
380C4⋊C4—81no—
381C4⋊C4—81no—
382C4⋊C4#13112yes—
383C22⋊C4—24no—
384C22⋊C4—41no—
385C22⋊C4—41no—
386C22⋊C4—42no—
387C22⋊C4—81no—
388C22⋊C4—81no—
389C22⋊C4—81no—
390C22⋊C4—81no—
391C42—21no—
392C42—21no—
393C42—21no—
394C42—21no—
395C42—21no—
396C42—21no—
397C42—41no—
398C42—41no—
399C42—41no—
400C42—41no—
401C42—41no—
402C42—41no—
403C42C2×#4111yescharacteristic
404C42C2×#4111yescharacteristic
405C42H(5,𝔽2)11yescharacteristic
406C422−1+411yescharacteristic
407C23—21no—
408C23—22no—
409C23—21no—
410C23—21no—
411C23—21no—
412C23—21no—
413C23—21no—
414C23—21no—
415C23—41no—
416C23—41no—
417C23—41no—
418C23—41no—
419C23—41no—
420C23—41no—
421C23C2×C4×D411yescharacteristic
422D4—22no—
423D4—22no—
424D4—41no—
425D4—41no—
426D4—84no—
427C2×C4—21no—
428C2×C4—21no—
429C2×C4—21no—
430C2×C4—21no—
431C2×C4—21no—
432C2×C4—21no—
433C2×C4—21no—
434C2×C4—22no—
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—41no—
461C2×C4—41no—
462C2×C4—41no—
463C2×C4—41no—
464C2×C4—41no—
465C2×C4—41no—
466C2×C4—41no—
467C2×C4—41no—
468C2×C4—81no—
469C2×C4—81no—
470C2×C4—81no—
471C2×C4—81no—
472C2×C4#44312yes—
473C2×C4C4×#4111yescharacteristic
474C2×C4#52011yescharacteristic
475C2×C4#54611yescharacteristic
476C2×C4#54711yescharacteristic
477C2×C4#55111yescharacteristic
478C2×C4#55211yescharacteristic
479C2×C4C2×#13111yescharacteristic
480C2×C4C2×#13111yescharacteristic
481C2×C4C2×#13111yescharacteristic
482C2×C4#56811yescharacteristic
483C2×C4C2×#13711yescharacteristic
484C2×C4#57511yescharacteristic
485C2×C4#57611yescharacteristic
486C2×C4#57611yescharacteristic
487C8—24no—
488C8—22no—
489C8—22no—
490C8—42no—
491C8—41no—
492C8—41no—
493C8—41no—
494C8—41no—
495C8—41no—
496C8—41no—
497C8—42no—
498C8—81no—
499C8—81no—
500C8—81no—
501C22—41no—
502C22—42no—
503C22—41no—
504C22—42no—
505C22—41no—
506C22—41no—
507C22—41no—
508C22—41no—
509C22—41no—
510C22—42no—
511C22—81no—
512C22—81no—
513C22—81no—
514C22—81no—
515C22#235911yescharacteristic
516C22#292911yescharacteristic
517C22#293511yescharacteristic
518C22C4×#13711yescharacteristic
519C22#294811yescharacteristic
520C22C2×#44311yescharacteristic
521C22#295611yescharacteristic
522C4—21no—
523C4—21no—
524C4—22no—
525C4—42no—
526C4—41no—
527C4—41no—
528C4—81no—
529C4—81no—
530C4—81no—
531C4—81no—
532C4—81no—
533C4—81no—
534C4#296611yescharacteristic
535C4#296911yescharacteristic
536C4#299211yescharacteristic
537C4#299411yescharacteristic
538C4#330911yescharacteristic
539C4#331411yescharacteristic
540C4#335511yescharacteristic
541C4#335711yescharacteristic
542C2—81no—
543C2—82no—
544C2#1791311yescharacteristic
545C2#1792211yescharacteristic
546C2#1807711yescharacteristic
547C2#1808411yescharacteristic
548C2#2290111yescharacteristic
549C2#2293111yescharacteristic
550C2#2294811yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC23C2×C4×D4
prime radicalC2×#121C2×C4
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
3. #22626C2semidirect
4. #9594C2semidirect
5. #9560C2semidirect
6. #9545C2semidirect
7. #9536C2semidirect
8. #9533C2semidirect
10. #8845C2semidirect
11. #8831C2semidirect
12. #8288C2semidirect
13. C8×#102C2semidirect
14. #8146C2semidirect
15. #7896C2semidirect

Maximal subgroups

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

subgroupindex
#237542
#230832
#226262
#95942
#95602
#95452
#95362
#95332
#90782
#88452
#88312
#82882
C8×#1022
#81462
#78962