Finite Group Databasebeta

← #212388

Subgroups — #212388

Order 512.

694 conjugacy classes of subgroups.

The subgroup lattice has 527 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21713C211yesmaximal, characteristic, complemented
2C4×#474C211yesmaximal, characteristic
3D4×#97C211yesmaximal, characteristic
4#9869C211yesmaximal, characteristic, complemented
5#9605C211yesmaximal, characteristic, complemented
6#9540C211yesmaximal, characteristic, complemented
7#9540C211yesmaximal, characteristic, complemented
8#8845C211yesmaximal, characteristic
9#8845C211yesmaximal, characteristic
10#8563C211yesmaximal, characteristic
11#8288C211yesmaximal, characteristic, complemented
12#8287C211yesmaximal, characteristic, complemented
13#8263C211yesmaximal, characteristic, complemented
14#8204C211yesmaximal, characteristic
15#8200C211yesmaximal, characteristic
16C2×#474C411yescharacteristic
17C2×#446—21no—
18C2×C8×D4C2211yescharacteristic
19C2×C8×D4C2211yescharacteristic
20C22×#97—21no—
21C4×C4⋊Q8C2211yescharacteristic
22C42×D4C2211yescharacteristic
23#1988C411yescharacteristic
24#1985—21no—
25#1947C411yescharacteristic
26#1940—21no—
27#1939C2211yescharacteristic
28#1939C2211yescharacteristic
29#1918C411yescharacteristic
30#1918C411yescharacteristic
31#1862C411yescharacteristic
32#1862C411yescharacteristic
33#1842C2211yescharacteristic
34#1802C411yescharacteristic
35C4×#107C2211yescharacteristic
36C4×#107C2211yescharacteristic
37#1771—21no—
38#1770C2211yescharacteristic
39#1770C2211yescharacteristic
40#1770C2211yescharacteristic
41C8×C4⋊C4C2211yescharacteristic
42#1767C2211yescharacteristic
43C4×C4⋊C8C2211yescharacteristic
44C4×C4⋊C8C2211yescharacteristic
45C4×#102C2211yescharacteristic
46C4×#102C2211yescharacteristic
47#1752—21no—
48#1752—21no—
49#1749—21no—
50C4×#97C2211yescharacteristic
51C2×#334C2211yescharacteristic
52C2×#334C2211yescharacteristic
53C2×D4⋊C8C2211yescharacteristic
54C2×D4⋊C8C2211yescharacteristic
55C2×#322C2211yescharacteristic
56C2×#322C2211yescharacteristic
57#1279C2211yescharacteristic
58#1279C2211yescharacteristic
59#1279C2211yescharacteristic
60#1279C2211yescharacteristic
61#1279C2211yescharacteristic
62#1279C2211yescharacteristic
63#1277C2211yescharacteristic
64#1277C2211yescharacteristic
65#1273C2211yescharacteristic
66#1273C2211yescharacteristic
67C23×C8—21no—
68C23×C8—21no—
69C2×C4⋊Q8C2×C411yescharacteristic
70C2×C4×Q8—21no—
71C2×C4×D4C2×C411yescharacteristic
72C2×C4×D4C2×C411yescharacteristic
73C2×C4×D4C2311yescharacteristic
74C22×C42—21no—
75#474—24no—
76#446—44no—
77C8×D4—22no—
78C8×D4—22no—
79C8×D4—22no—
80C8×D4Q814yescomplemented
81C2×#107C2×C411yescharacteristic
82C2×#107C2×C411yescharacteristic
83C2×#107C2×C411yescharacteristic
84C2×#107C2×C411yescharacteristic
85C2×C4⋊C8—21no—
86C2×C4⋊C8—21no—
87C2×C4⋊C8—21no—
88C2×C4⋊C8—21no—
89C2×C4⋊C8—21no—
90C2×C4⋊C8C2×C411yescharacteristic
91C2×C4⋊C8C2×C411yescharacteristic
92C2×C4⋊C8C2311yescharacteristic
93C2×C4⋊C8C2311yescharacteristic
94C2×C4⋊C8C2311yescharacteristic
95C2×C4⋊C8C2311yescharacteristic
96C2×#102C2×C411yescharacteristic
97C2×#102C2×C411yescharacteristic
98C2×#102C2×C411yescharacteristic
99C2×#102C2×C411yescharacteristic
100C2×C22⋊C8—21no—
101C2×C22⋊C8—21no—
102C2×#97—21no—
103C2×#97—21no—
104C2×#97—21no—
105C2×#97—21no—
106C2×#97—44no—
107C2×#97D411yescharacteristic
108C2×#97D411yescharacteristic
109C2×#97D411yescharacteristic
110C2×#97D411yescharacteristic
111C2×C4×C8—21no—
112C2×C4×C8C2311yescharacteristic
113C2×C4×C8C2311yescharacteristic
114C2×C4×C8C2311yescharacteristic
115C2×C4×C8C2311yescharacteristic
116C2×C4×C8C2311yescharacteristic
117C2×C4×C8C2311yescharacteristic
118#391—21no—
119#389—21no—
120#389C2×C411yescharacteristic
121#389C2×C411yescharacteristic
122#384C2×C411yescharacteristic
123C4×C4⋊C4—21no—
124C4×C4⋊C4C2×C411yescharacteristic
125C4×C4⋊C4C2×C411yescharacteristic
126C4×C4⋊C4C2311yescharacteristic
127C4×C4⋊C4C2311yescharacteristic
128C4×C4⋊C4C2311yescharacteristic
129C4×C22⋊C4—21no—
130C4×C22⋊C4—21no—
131C4×C22⋊C4—21no—
132C43C2311yescharacteristic
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#336C2×C411yescharacteristic
150#336C2×C411yescharacteristic
151#334—22no—
152#334—22no—
153D4⋊C8—22no—
154D4⋊C8—22no—
155#322—22no—
156#322—22no—
157C22×Q8—21no—
158C22×D4C22×C411yescharacteristic
159C23×C4—21no—
160C23×C4—21no—
161C23×C4—21no—
162C22×C8—24no—
163C22×C8—21no—
164C22×C8—21no—
165C22×C8—21no—
166C22×C8—24no—
167C22×C8—21no—
168C22×C8—21no—
169C22×C8—21no—
170C22×C8—42no—
171C22×C8—42no—
172C22×C8—42no—
173C22×C8—42no—
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—41no—
184C4×Q8—42no—
185C4×Q8—42no—
186C4×D4—22no—
187C4×D4—22no—
188C4×D4—21no—
189C4×D4—21no—
190C4×D4—41no—
191C4×D4—41no—
192C4×D4C2×Q812yes—
193C4×D4#4112yes—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—21no—
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×C4⋊C4C22×C411yescharacteristic
206C2×C22⋊C4—21no—
207C2×C22⋊C4—21no—
208C2×C22⋊C4—21no—
209C2×C42—21no—
210C2×C42—21no—
211C2×C42—21no—
212C2×C42—21no—
213C2×C42—21no—
214C2×C42—42no—
215C2×C42—42no—
216C2×C42C22×C411yescharacteristic
217C2×C42C22×C411yescharacteristic
218C2×C42C2×D411yescharacteristic
219C2×C42C2×D411yescharacteristic
220C2×C42#4111yescharacteristic
221C2×C42#4111yescharacteristic
222C2×C42C2411yescharacteristic
223#107—22no—
224#107—22no—
225#107—41no—
226#107—41no—
227C4⋊C8—22no—
228C4⋊C8—21no—
229C4⋊C8—21no—
230C4⋊C8—21no—
231C4⋊C8—21no—
232C4⋊C8—21no—
233C4⋊C8—21no—
234C4⋊C8—41no—
235C4⋊C8—42no—
236C4⋊C8—42no—
237C4⋊C8—42no—
238C4⋊C8—42no—
239C4⋊C8—42no—
240C4⋊C8C2×Q812yes—
241C4⋊C8#4112yes—
242#102—22no—
243#102—22no—
244#102—22no—
245#102M1614yes—
246C22⋊C8—24no—
247C22⋊C8—24no—
248#97—22no—
249#97—22no—
250#97—22no—
251#97—41no—
252#97—42no—
253#97—42no—
254#97—42no—
255#97—42no—
256C4×C8—21no—
257C4×C8—21no—
258C4×C8—21no—
259C4×C8—21no—
260C4×C8—21no—
261C4×C8—21no—
262C4×C8—21no—
263C4×C8—21no—
264C4×C8—21no—
265C4×C8—21no—
266C4×C8—42no—
267C4×C8C2×Q812yes—
268C4×C8#4112yes—
269#95—21no—
270#95—21no—
271#95—21no—
272#95—21no—
273#95—21no—
274#95—21no—
275#95—21no—
276C24—21no—
277C2×Q8—42no—
278C2×Q8—42no—
279C2×Q8—81no—
280C2×D4—21no—
281C2×D4—21no—
282C2×D4—21no—
283C2×D4C4×Q812yes—
284C2×D4C2×M1612yes—
285C2×D4#13112yes—
286C22×C4—21no—
287C22×C4—21no—
288C22×C4—22no—
289C22×C4—22no—
290C22×C4—21no—
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—21no—
294C22×C4—21no—
295C22×C4—42no—
296C22×C4—42no—
297C22×C4—41no—
298C22×C4—41no—
299C22×C4—41no—
300C22×C4—41no—
301C22×C4—42no—
302C22×C4—42no—
303C22×C4—81no—
304C22×C4—81no—
305C22×C4C4×D411yescharacteristic
306C22×C4C4×D411yescharacteristic
307C22×C4C4×D411yescharacteristic
308C22×C4C4×D411yescharacteristic
309C22×C4C23×C411yescharacteristic
310C22×C4C22×D411yescharacteristic
311C22×C4C2×#4111yescharacteristic
312C2×C8—21no—
313C2×C8—21no—
314C2×C8—21no—
315C2×C8—21no—
316C2×C8—21no—
317C2×C8—21no—
318C2×C8—21no—
319C2×C8—21no—
320C2×C8—22no—
321C2×C8—21no—
322C2×C8—21no—
323C2×C8—21no—
324C2×C8—22no—
325C2×C8—21no—
326C2×C8—22no—
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—48no—
336C2×C8—48no—
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×C8C22⋊Q812yes—
353C2×C8C22⋊Q812yes—
354C2×C8C2×M1612yes—
355C2×C8#13712yes—
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—21no—
365C4⋊C4—21no—
366C4⋊C4—21no—
367C4⋊C4—21no—
368C4⋊C4—42no—
369C4⋊C4—42no—
370C4⋊C4—42no—
371C4⋊C4—42no—
372C4⋊C4—81no—
373C4⋊C4C4×Q812yes—
374C4⋊C4C2×M1612yes—
375C22⋊C4—22no—
376C22⋊C4—22no—
377C22⋊C4—42no—
378C22⋊C4—42no—
379C42—21no—
380C42—21no—
381C42—21no—
382C42—21no—
383C42—21no—
384C42—21no—
385C42—41no—
386C42—41no—
387C42—41no—
388C42—41no—
389C42—41no—
390C42—41no—
391C42—41no—
392C42—41no—
393C42—42no—
394C42—42no—
395C42AGL(1,ℤ/8)12yes—
396C42AGL(1,ℤ/8)12yes—
397C42C22×Q811yescharacteristic
398C42C2×#4111yescharacteristic
399C42C2×#4111yescharacteristic
400C42C2×#4111yescharacteristic
401C23—22no—
402C23—22no—
403C23—22no—
404C23—41no—
405C23—41no—
406C23—41no—
407C23—41no—
408C23C2×C4×D411yescharacteristic
409Q8—84no—
410D4—22no—
411D4—22no—
412D4—22no—
413D4#44614yescomplemented
414C2×C4—21no—
415C2×C4—21no—
416C2×C4—21no—
417C2×C4—21no—
418C2×C4—21no—
419C2×C4—21no—
420C2×C4—21no—
421C2×C4—21no—
422C2×C4—22no—
423C2×C4—21no—
424C2×C4—21no—
425C2×C4—21no—
426C2×C4—21no—
427C2×C4—22no—
428C2×C4—41no—
429C2×C4—41no—
430C2×C4—41no—
431C2×C4—41no—
432C2×C4—42no—
433C2×C4—42no—
434C2×C4—42no—
435C2×C4—42no—
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—41no—
448C2×C4—41no—
449C2×C4—41no—
450C2×C4—41no—
451C2×C4—84no—
452C2×C4—84no—
453C2×C4#44212yes—
454C2×C4#44212yes—
455C2×C4C2×C4×Q811yescharacteristic
456C2×C4C4×#4111yescharacteristic
457C2×C4C2×C22⋊Q811yescharacteristic
458C2×C4#52511yescharacteristic
459C2×C4#52911yescharacteristic
460C2×C4#53311yescharacteristic
461C2×C4C22×M1611yescharacteristic
462C2×C4C2×#13111yescharacteristic
463C2×C4#56811yescharacteristic
464C2×C4#56811yescharacteristic
465C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
466C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
467C2×C4C2×#13711yescharacteristic
468C2×C4#57511yescharacteristic
469C8—22no—
470C8—22no—
471C8—22no—
472C8—22no—
473C8—42no—
474C8—42no—
475C8—42no—
476C8—41no—
477C8—41no—
478C8—41no—
479C8—41no—
480C8—81no—
481C8—81no—
482C8—81no—
483C22—24no—
484C22—42no—
485C22—42no—
486C22—42no—
487C22—42no—
488C22—42no—
489C22—42no—
490C22C4×C22⋊Q811yescharacteristic
491C22#293311yescharacteristic
492C22D4×M1611yescharacteristic
493C22C2×#44211yescharacteristic
494C22C4×#13711yescharacteristic
495C22#295611yescharacteristic
496C22#295711yescharacteristic
497C4—22no—
498C4—22no—
499C4—21no—
500C4—21no—
501C4—41no—
502C4—42no—
503C4—42no—
504C4—41no—
505C4—81no—
506C4—81no—
507C4—81no—
508C4—81no—
509C4—81no—
510C4C2×#44611yescharacteristic
511C4#296111yescharacteristic
512C4#296811yescharacteristic
513C4#297011yescharacteristic
514C4#307711yescharacteristic
515C4#307911yescharacteristic
516C4#311011yescharacteristic
517C4#311511yescharacteristic
518C2—44no—
519C2#1800111yescharacteristic
520C2#1800911yescharacteristic
521C2#1804411yescharacteristic
522C2#1804711yescharacteristic
523C2#2159911yescharacteristic
524C2#2161311yescharacteristic
525C2#2162911yescharacteristic
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×#97 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #21713C2semidirect
4. #9869C2semidirect
5. #9605C2semidirect
6. #9540C2semidirect
7. #9540C2semidirect
11. #8288C2semidirect
12. #8287C2semidirect
13. #8263C2semidirect
80. C8×D4Q8semidirect
413. D4#446semidirect

Maximal subgroups

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

subgroupindex
#217132
C4×#4742
D4×#972
#98692
#96052
#95402
#88452
#85632
#82882
#82872
#82632
#82042
#82002