Finite Group Databasebeta

← #361352

Subgroups — #361352

Order 512.

1280 conjugacy classes of subgroups.

The subgroup lattice has 511 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1865C212yesmaximal
2C2×C4×#103C211yesmaximal, characteristic
3#13918C211yesmaximal, characteristic
4Q8×#95C211yesmaximal, characteristic
5#11293C212yesmaximal
6#11257C212yesmaximal
7#10861C211yesmaximal, characteristic
8#10806C211yesmaximal, characteristic
9#10797C211yesmaximal, characteristic
10#10789C211yesmaximal, characteristic
11#10789C212yesmaximal
12C22×C4×Q8C2212yes—
13C22×C4×Q8C2211yescharacteristic
14C22×#106C2212yes—
15C22×#103C2212yes—
16C22×#103C2211yescharacteristic
17C22×#103C2211yescharacteristic
18C22×C4×C8C2211yescharacteristic
19C2×#389C2212yes—
20C2×#382—21no—
21C2×C4×C4⋊C4C2211yescharacteristic
22#1865—28no—
23C4×#103C418yescomplemented
24C2×#340C2212yes—
25C2×#340C2212yes—
26C2×#340C2212yes—
27C2×#340C2212yes—
28C2×#340C2211yescharacteristic
29C2×#340C2211yescharacteristic
30C2×#336C2211yescharacteristic
31C2×#336C2212yes—
32C2×#336C2212yes—
33#1442—22no—
34C4⋊#95—21no—
35C4⋊#95—22no—
36C4⋊#95C2212yes—
37C4⋊#95C2211yescharacteristic
38C4⋊#95C2212yes—
39C4⋊#95C2211yescharacteristic
40#1434—21no—
41C4×#95—21no—
42C4×#95C2211yescharacteristic
43C23×Q8C2311yescharacteristic
44C23×C8C2312yes—
45C23×C8C2311yescharacteristic
46C23×C8C2311yescharacteristic
47C2×C4×Q8—24no—
48C2×C4×Q8—24no—
49C2×C4×Q8—22no—
50C2×C4×Q8—24no—
51C2×C4×Q8—24no—
52C2×C4×Q8—24no—
53C2×C4×Q8—42no—
54C2×C4×Q8—42no—
55C2×C4×Q8—42no—
56C2×C4×Q8C2×C414yes—
57C22×C4⋊C4—22no—
58C22×C4⋊C4—21no—
59C22×C4⋊C4—22no—
60C22×C4⋊C4C2312yes—
61C22×C4⋊C4C2311yescharacteristic
62C22×C4⋊C4C2311yescharacteristic
63C22×C4⋊C4C2312yes—
64C22×C4⋊C4C2311yescharacteristic
65C22×C42—22no—
66C22×C42—21no—
67C22×C42—21no—
68C22×C42C2312yes—
69C22×C42C2311yescharacteristic
70C2×#106—24no—
71C2×#106—24no—
72C2×#106—42no—
73C2×#103—24no—
74C2×#103—24no—
75C2×#103—24no—
76C2×#103—24no—
77C2×#103C2×C414yes—
78C2×#103C2×C414yes—
79C2×#103D414yes—
80C2×#103Q814yes—
81C2×C4×C8—42no—
82C2×C4×C8C2×C414yes—
83#389—24no—
84#389—42no—
85#382—44no—
86C4×C4⋊C4—22no—
87C4×C4⋊C4C2×C414yes—
88C2×#95—21no—
89C2×#95—22no—
90C2×#95—21no—
91C2×#95—22no—
92C2×#95—21no—
93C2×#95—22no—
94C2×#95—22no—
95C2×#95—22no—
96C2×#95—21no—
97C2×#95—21no—
98C2×#95D411yescharacteristic
99C2×#95D412yes—
100C2×#95D411yescharacteristic
101#340—24no—
102#340—24no—
103#340—42no—
104#340—42no—
105#340C2×C414yes—
106#340C2×C414yes—
107#336—24no—
108#336—22no—
109#336—42no—
110C22×Q8—21no—
111C22×Q8—22no—
112C22×Q8—21no—
113C22×Q8—22no—
114C22×Q8—21no—
115C22×Q8C22×C412yes—
116C22×Q8C2×D412yes—
117C22×Q8C2×Q812yes—
118C22×Q8#4114yes—
119C22×Q8#4114yes—
120C23×C4—21no—
121C23×C4—21no—
122C23×C4—21no—
123C23×C4—22no—
124C23×C4—21no—
125C23×C4—22no—
126C23×C4C2×D412yes—
127C23×C4C2×D411yescharacteristic
128C23×C4C2×D412yes—
129C23×C4C2×D411yescharacteristic
130C23×C4C2411yescharacteristic
131C22×C8—22no—
132C22×C8—22no—
133C22×C8—22no—
134C22×C8—22no—
135C22×C8—21no—
136C22×C8—21no—
137C22×C8—22no—
138C22×C8—22no—
139C22×C8—22no—
140C22×C8—22no—
141C22×C8—22no—
142C22×C8—22no—
143C22×C8—22no—
144C22×C8C22×C412yes—
145C22×C8C22×C412yes—
146C22×C8C2×D412yes—
147C22×C8C2×Q812yes—
148C4×Q8—28no—
149C4×Q8—44no—
150C4×Q8—48no—
151C4×Q8—48no—
152C4×Q8—48no—
153C4×Q8—48no—
154C4×Q8—48no—
155C2×C4⋊C4—21no—
156C2×C4⋊C4—21no—
157C2×C4⋊C4—21no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—28no—
167C2×C4⋊C4—24no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—44no—
178C2×C4⋊C4—44no—
179C2×C4⋊C4—44no—
180C2×C4⋊C4—44no—
181C2×C4⋊C4—44no—
182C2×C4⋊C4—82no—
183C2×C4⋊C4C22×C412yes—
184C2×C4⋊C4C22×C412yes—
185C2×C4⋊C4C22×C412yes—
186C2×C4⋊C4C2×D412yes—
187C2×C4⋊C4C2×Q812yes—
188C2×C4⋊C4#4114yes—
189C2×C42—22no—
190C2×C42—22no—
191C2×C42—28no—
192C2×C42—24no—
193C2×C42—24no—
194C2×C42—22no—
195C2×C42—22no—
196C2×C42—22no—
197C2×C42—22no—
198C2×C42—22no—
199C2×C42—22no—
200C2×C42—44no—
201C2×C42—44no—
202C2×C42—44no—
203C2×C42—42no—
204C2×C42—42no—
205C2×C42C22×C412yes—
206C2×C42#4112yes—
207#106—48no—
208#103—28no—
209#103—48no—
210#103C4⋊C4116yes—
211C4×C8—44no—
212#95—24no—
213#95—22no—
214#95—42no—
215#95—44no—
216#95—44no—
217#95—44no—
218#95—44no—
219#95—42no—
220#95—42no—
221#95—81no—
222#95—81no—
223#95—82no—
224#95SD1614yes—
225C24C22×D411yescharacteristic
226C2×Q8—22no—
227C2×Q8—22no—
228C2×Q8—22no—
229C2×Q8—24no—
230C2×Q8—24no—
231C2×Q8—24no—
232C2×Q8—24no—
233C2×Q8—22no—
234C2×Q8—24no—
235C2×Q8—24no—
236C2×Q8—24no—
237C2×Q8—22no—
238C2×Q8—24no—
239C2×Q8C2×C4⋊C414yes—
240C2×Q8#11714yes—
241C2×Q8#11714yes—
242C2×Q8#12418yes—
243C2×Q8#12518yes—
244C22×C4—22no—
245C22×C4—24no—
246C22×C4—22no—
247C22×C4—22no—
248C22×C4—24no—
249C22×C4—22no—
250C22×C4—22no—
251C22×C4—22no—
252C22×C4—22no—
253C22×C4—22no—
254C22×C4—22no—
255C22×C4—22no—
256C22×C4—22no—
257C22×C4—22no—
258C22×C4—22no—
259C22×C4—22no—
260C22×C4—21no—
261C22×C4—21no—
262C22×C4—22no—
263C22×C4—22no—
264C22×C4—22no—
265C22×C4—22no—
266C22×C4—22no—
267C22×C4—24no—
268C22×C4—22no—
269C22×C4—22no—
270C22×C4—22no—
271C22×C4—22no—
272C22×C4—22no—
273C22×C4—22no—
274C22×C4—41no—
275C22×C4—41no—
276C22×C4—41no—
277C22×C4—41no—
278C22×C4—42no—
279C22×C4—42no—
280C22×C4—42no—
281C22×C4—42no—
282C22×C4—41no—
283C22×C4—42no—
284C22×C4—41no—
285C22×C4—42no—
286C22×C4—42no—
287C22×C4—42no—
288C22×C4—42no—
289C22×C4—41no—
290C22×C4—41no—
291C22×C4—42no—
292C22×C4—42no—
293C22×C4—42no—
294C22×C4—42no—
295C22×C4—42no—
296C22×C4—42no—
297C22×C4—42no—
298C22×C4—42no—
299C22×C4C4×D412yes—
300C22×C4C4×D412yes—
301C22×C4C2×SD1614yes—
302C22×C4C2×SD1612yes—
303C22×C4#13512yes—
304C22×C4C23×C411yescharacteristic
305C22×C4C22×D411yescharacteristic
306C22×C4C22×Q811yescharacteristic
307C22×C4C2×#4112yes—
308C22×C4C2×#4112yes—
309C22×C4C2×#4112yes—
310C22×C4C2×#4111yescharacteristic
311C22×C4H(5,𝔽2)12yes—
312C22×C4H(5,𝔽2)11yescharacteristic
313C22×C42−1+411yescharacteristic
314C2×C8—22no—
315C2×C8—22no—
316C2×C8—24no—
317C2×C8—24no—
318C2×C8—22no—
319C2×C8—22no—
320C2×C8—22no—
321C2×C8—42no—
322C2×C8—42no—
323C2×C8—42no—
324C2×C8—42no—
325C2×C8—42no—
326C2×C8—42no—
327C2×C8—42no—
328C2×C8—42no—
329C2×C8—42no—
330C2×C8C2×C4⋊C414yes—
331C4⋊C4—22no—
332C4⋊C4—22no—
333C4⋊C4—22no—
334C4⋊C4—22no—
335C4⋊C4—22no—
336C4⋊C4—22no—
337C4⋊C4—22no—
338C4⋊C4—24no—
339C4⋊C4—24no—
340C4⋊C4—48no—
341C4⋊C4—416no—
342C4⋊C4—42no—
343C4⋊C4—42no—
344C4⋊C4—42no—
345C4⋊C4—42no—
346C4⋊C4—42no—
347C4⋊C4—42no—
348C4⋊C4—42no—
349C4⋊C4—42no—
350C4⋊C4—88no—
351C4⋊C4C2×C4⋊C414yes—
352C42—22no—
353C42—22no—
354C42—24no—
355C42—48no—
356C42—416no—
357C42—42no—
358C42—42no—
359C42—42no—
360C42—42no—
361C42—48no—
362C23C2×C4×D411yescharacteristic
363C23#52512yes—
364C23D4211yescharacteristic
365C23#54612yes—
366C23D4×Q811yescharacteristic
367C23C22×SD1611yescharacteristic
368C23C2×#13511yescharacteristic
369C23#57512yes—
370C23#57711yescharacteristic
371C23#57711yescharacteristic
372C23#57812yes—
373Q8—24no—
374Q8—24no—
375Q8—24no—
376Q8—28no—
377Q8—28no—
378Q8#38218yescomplemented
379C2×C4—22no—
380C2×C4—24no—
381C2×C4—24no—
382C2×C4—22no—
383C2×C4—28no—
384C2×C4—28no—
385C2×C4—24no—
386C2×C4—24no—
387C2×C4—24no—
388C2×C4—24no—
389C2×C4—22no—
390C2×C4—24no—
391C2×C4—22no—
392C2×C4—22no—
393C2×C4—24no—
394C2×C4—24no—
395C2×C4—42no—
396C2×C4—42no—
397C2×C4—42no—
398C2×C4—42no—
399C2×C4—42no—
400C2×C4—42no—
401C2×C4—42no—
402C2×C4—42no—
403C2×C4—42no—
404C2×C4—42no—
405C2×C4—42no—
406C2×C4—42no—
407C2×C4—42no—
408C2×C4—44no—
409C2×C4—42no—
410C2×C4—42no—
411C2×C4—44no—
412C2×C4—42no—
413C2×C4—42no—
414C2×C4—42no—
415C2×C4—42no—
416C2×C4—44no—
417C2×C4—42no—
418C2×C4—42no—
419C2×C4—42no—
420C2×C4—42no—
421C2×C4—42no—
422C2×C4—42no—
423C2×C4—82no—
424C2×C4—82no—
425C2×C4—82no—
426C2×C4—82no—
427C2×C4—82no—
428C2×C4—82no—
429C2×C4—82no—
430C2×C4—82no—
431C2×C4—82no—
432C2×C4—82no—
433C2×C4—82no—
434C2×C4—82no—
435C2×C4—82no—
436C2×C4—82no—
437C2×C4—82no—
438C2×C4C4×SD1614yes—
439C2×C4C22×C4⋊C411yescharacteristic
440C2×C4C2×#11711yescharacteristic
441C2×C4C2×#11711yescharacteristic
442C2×C4C4×#4111yescharacteristic
443C2×C4#51811yescharacteristic
444C2×C4#52011yescharacteristic
445C2×C4C2×#12412yes—
446C2×C4C2×#12512yes—
447C2×C4#52912yes—
448C2×C4#53812yes—
449C2×C4#53912yes—
450C2×C4#54612yes—
451C2×C4#54711yescharacteristic
452C2×C4#55112yes—
453C2×C4#55112yes—
454C2×C4#55212yes—
455C2×C4#55412yes—
456C2×C4#55411yescharacteristic
457C8—24no—
458C8—42no—
459C8—82no—
460C22#229011yescharacteristic
461C22#234211yescharacteristic
462C22D4×C4⋊C411yescharacteristic
463C22#247212yes—
464C22#247312yes—
465C22C2×C4×SD1611yescharacteristic
466C22#294311yescharacteristic
467C22#295011yescharacteristic
468C22#295211yescharacteristic
469C22#310212yes—
470C22#310312yes—
471C22#312012yes—
472C22#312112yes—
473C22#327712yes—
474C22D4×SD1611yescharacteristic
475C22#328411yescharacteristic
476C22#329812yes—
477C22#330012yes—
478C22#331712yes—
479C22#332112yes—
480C22Q8×SD1611yescharacteristic
481C22#338111yescharacteristic
482C22#339012yes—
483C4—24no—
484C4—28no—
485C4—42no—
486C4—44no—
487C4—44no—
488C4—82no—
489C4—82no—
490C4—84no—
491C4—82no—
492C4—162no—
493C4C2×#38211yescharacteristic
494C4#231811yescharacteristic
495C4#234311yescharacteristic
496C4#234911yescharacteristic
497C4#248812yes—
498C4#250512yes—
499C2#1389611yescharacteristic
500C2#2111511yescharacteristic
501C2#2113211yescharacteristic
502C2#2256511yescharacteristic
503C2#2257111yescharacteristic
504C2#2267511yescharacteristic
505C2C4⋊C4×SD1611yescharacteristic
506C2#2722612yes—
507C2#2723412yes—
508C2#2729812yes—
509C2#2730612yes—
subgroupG/H
center, socle, prime radicalC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C4×#95 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
23. C4×#103C4semidirect
378. Q8#382semidirect

Maximal subgroups

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

subgroupindex
C2×#18652
C2×C4×#1032
#139182
Q8×#952
#112932
#112572
#108612
#108062
#107972
#107892