Finite Group Databasebeta

← #397973

Subgroups — #397973

Order 512.

1434 conjugacy classes of subgroups.

The subgroup lattice has 534 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#477C211yesmaximal, characteristic
2C2×#1865C211yesmaximal, characteristic
3C2×#1805C212yesmaximal
4#15230C211yesmaximal, characteristic
5Q8×#95C211yesmaximal, characteristic
6#11384C211yesmaximal, characteristic
7#11325C212yesmaximal
8#11322C211yesmaximal, characteristic
9#11232C212yesmaximal
10#11227C211yesmaximal, characteristic
11#10789C212yesmaximal
12C22×C4⋊Q8C2211yescharacteristic
13C22×C4×Q8C2211yescharacteristic
14C22×C4×Q8C2212yes—
15C2×#477—28no—
16C2×#477—24no—
17C22×#107C2212yes—
18C22×#106C2211yescharacteristic
19C22×C4⋊C8C2211yescharacteristic
20C22×C4⋊C8C2212yes—
21C22×#103C2211yescharacteristic
22C22×#103C2212yes—
23C22×#103C2211yescharacteristic
24C2×#389C2211yescharacteristic
25C2×#382C2212yes—
26#1865—24no—
27#1805—28no—
28C2×#340C2212yes—
29C2×#340C2212yes—
30C2×#340C2211yescharacteristic
31C2×#340C2212yes—
32C2×#340C2212yes—
33C2×#336C2212yes—
34C2×#336C2211yescharacteristic
35#1446—22no—
36#1441—21no—
37#1441—21no—
38C4⋊#95—21no—
39C4⋊#95—22no—
40C4⋊#95C2212yes—
41C4⋊#95C2212yes—
42C4⋊#95C2211yescharacteristic
43C4×#95—21no—
44C4×#95C2211yescharacteristic
45C23×Q8—21no—
46C23×Q8C2311yescharacteristic
47C23×C8C2311yescharacteristic
48C23×C8C2312yes—
49C23×C8C2311yescharacteristic
50C2×C4⋊Q8—24no—
51C2×C4⋊Q8—22no—
52C2×C4⋊Q8—22no—
53C2×C4⋊Q8—42no—
54C2×C4×Q8—22no—
55C2×C4×Q8—24no—
56C2×C4×Q8—24no—
57C2×C4×Q8—24no—
58C2×C4×Q8—24no—
59C2×C4×Q8—42no—
60C2×C4×Q8—42no—
61C2×C4×Q8—42no—
62C2×C4×Q8D414yes—
63C2×C4×Q8Q818yescomplemented
64C22×C4⋊C4—21no—
65C22×C4⋊C4—22no—
66C22×C4⋊C4—21no—
67C22×C4⋊C4—21no—
68C22×C4⋊C4C2311yescharacteristic
69C22×C4⋊C4C2312yes—
70C22×C4⋊C4C2311yescharacteristic
71C22×C4⋊C4C2311yescharacteristic
72C22×C4⋊C4C2312yes—
73C22×C42—21no—
74C22×C42—22no—
75C22×C42C2311yescharacteristic
76C22×C42C2312yes—
77#477—232no—
78C2×#107—24no—
79C2×#107—24no—
80C2×#107—24no—
81C2×#106—22no—
82C2×#106—42no—
83C2×C4⋊C8—24no—
84C2×C4⋊C8—22no—
85C2×C4⋊C8—24no—
86C2×C4⋊C8—24no—
87C2×C4⋊C8—42no—
88C2×#103—24no—
89C2×#103—24no—
90C2×#103—22no—
91C2×#103—22no—
92C2×#103—24no—
93C2×#103—22no—
94C2×#103D418yes—
95C2×#103Q814yescomplemented
96#389—22no—
97#389—22no—
98#382—24no—
99#382—42no—
100C2×#95—22no—
101C2×#95—21no—
102C2×#95—22no—
103C2×#95—22no—
104C2×#95—21no—
105C2×#95—21no—
106C2×#95—22no—
107C2×#95—22no—
108C2×#95—22no—
109C2×#95D411yescharacteristic
110C2×#95D411yescharacteristic
111C2×#95D412yes—
112#340—24no—
113#340—24no—
114#340—22no—
115#340—42no—
116#340—42no—
117#336—22no—
118#336—24no—
119C22×Q8—21no—
120C22×Q8—22no—
121C22×Q8—22no—
122C22×Q8—21no—
123C22×Q8—21no—
124C22×Q8—44no—
125C22×Q8—44no—
126C22×Q8—42no—
127C22×Q8—81no—
128C22×Q8—81no—
129C22×Q8C2×D414yes—
130C22×Q8C2×D412yes—
131C22×Q8C2×Q814yes—
132C22×Q8C2×Q812yes—
133C22×Q8#4112yes—
134C23×C4—21no—
135C23×C4—22no—
136C23×C4—21no—
137C23×C4—21no—
138C23×C4—22no—
139C23×C4—21no—
140C23×C4C2×D411yescharacteristic
141C23×C4C2×D412yes—
142C23×C4C2×D412yes—
143C23×C4C2×D411yescharacteristic
144C23×C4C2411yescharacteristic
145C22×C8—21no—
146C22×C8—22no—
147C22×C8—22no—
148C22×C8—22no—
149C22×C8—22no—
150C22×C8—21no—
151C22×C8—22no—
152C22×C8—22no—
153C22×C8—22no—
154C22×C8—22no—
155C22×C8—21no—
156C22×C8—22no—
157C22×C8—22no—
158C22×C8—21no—
159C22×C8—21no—
160C22×C8C2×D414yes—
161C22×C8C2×Q812yes—
162C4⋊Q8—28no—
163C4⋊Q8—48no—
164C4⋊Q8—44no—
165C4×Q8—28no—
166C4×Q8—216no—
167C4×Q8—44no—
168C4×Q8—48no—
169C4×Q8—48no—
170C4×Q8—48no—
171C4×Q8—48no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—24no—
187C2×C4⋊C4—28no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—44no—
197C2×C4⋊C4—44no—
198C2×C4⋊C4—44no—
199C2×C4⋊C4—44no—
200C2×C4⋊C4—42no—
201C2×C4⋊C4—42no—
202C2×C4⋊C4—42no—
203C2×C4⋊C4—42no—
204C2×C4⋊C4C2×D412yes—
205C2×C4⋊C4C2×D414yes—
206C2×C4⋊C4C2×Q814yes—
207C2×C4⋊C4C2×Q812yes—
208C2×C42—22no—
209C2×C42—22no—
210C2×C42—22no—
211C2×C42—22no—
212C2×C42—22no—
213C2×C42—22no—
214C2×C42—24no—
215C2×C42—28no—
216C2×C42—44no—
217C2×C42—44no—
218C2×C42—44no—
219C2×C42C2×D412yes—
220C2×C42C2×Q814yes—
221C2×C42#4112yes—
222C2×C42#4114yes—
223#107—216no—
224#106—44no—
225C4⋊C8—28no—
226C4⋊C8—48no—
227#103—216no—
228#103—28no—
229#103—44no—
230#95—22no—
231#95—22no—
232#95—42no—
233#95—44no—
234#95—44no—
235#95—44no—
236#95—42no—
237#95—42no—
238#95—82no—
239#95—82no—
240#95—81no—
241#95—82no—
242C24C22×D411yescharacteristic
243C2×Q8—22no—
244C2×Q8—24no—
245C2×Q8—22no—
246C2×Q8—22no—
247C2×Q8—24no—
248C2×Q8—24no—
249C2×Q8—24no—
250C2×Q8—24no—
251C2×Q8—24no—
252C2×Q8—24no—
253C2×Q8—22no—
254C2×Q8—22no—
255C2×Q8—24no—
256C2×Q8—48no—
257C2×Q8—84no—
258C2×Q8—84no—
259C2×Q8—88no—
260C2×Q8—88no—
261C2×Q8#12014yes—
262C2×Q8C22⋊Q814yes—
263C2×Q8C22⋊Q818yes—
264C2×Q8C4⋊Q814yes—
265C2×Q8C4⋊Q818yes—
266C22×C4—22no—
267C22×C4—24no—
268C22×C4—22no—
269C22×C4—24no—
270C22×C4—22no—
271C22×C4—22no—
272C22×C4—22no—
273C22×C4—22no—
274C22×C4—22no—
275C22×C4—22no—
276C22×C4—22no—
277C22×C4—22no—
278C22×C4—22no—
279C22×C4—22no—
280C22×C4—22no—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—22no—
284C22×C4—21no—
285C22×C4—22no—
286C22×C4—21no—
287C22×C4—22no—
288C22×C4—22no—
289C22×C4—22no—
290C22×C4—24no—
291C22×C4—24no—
292C22×C4—22no—
293C22×C4—41no—
294C22×C4—42no—
295C22×C4—42no—
296C22×C4—42no—
297C22×C4—42no—
298C22×C4—42no—
299C22×C4—41no—
300C22×C4—42no—
301C22×C4—41no—
302C22×C4—41no—
303C22×C4—42no—
304C22×C4—42no—
305C22×C4—42no—
306C22×C4—42no—
307C22×C4—42no—
308C22×C4—41no—
309C22×C4—41no—
310C22×C4—42no—
311C22×C4—41no—
312C22×C4—42no—
313C22×C4—41no—
314C22×C4—42no—
315C22×C4—41no—
316C22×C4—42no—
317C22×C4—41no—
318C22×C4—41no—
319C22×C4—42no—
320C22×C4#12112yes—
321C22×C4#12112yes—
322C22×C4C22⋊Q814yes—
323C22×C4C22⋊Q814yes—
324C22×C4#13512yes—
325C22×C4#13712yes—
326C22×C4C22×D411yescharacteristic
327C22×C4C22×D412yes—
328C22×C4C22×Q812yes—
329C22×C4C22×Q811yescharacteristic
330C22×C4C2×#4111yescharacteristic
331C22×C4C2×#4112yes—
332C22×C4C2×#4111yescharacteristic
333C22×C4H(5,𝔽2)12yes—
334C22×C4H(5,𝔽2)11yescharacteristic
335C22×C42−1+411yescharacteristic
336C2×C8—24no—
337C2×C8—22no—
338C2×C8—24no—
339C2×C8—24no—
340C2×C8—24no—
341C2×C8—41no—
342C2×C8—42no—
343C2×C8—41no—
344C2×C8—42no—
345C2×C8—42no—
346C2×C8—42no—
347C2×C8—42no—
348C2×C8—41no—
349C2×C8—41no—
350C2×C8—41no—
351C2×C8—42no—
352C2×C8—42no—
353C4⋊C4—24no—
354C4⋊C4—22no—
355C4⋊C4—22no—
356C4⋊C4—24no—
357C4⋊C4—22no—
358C4⋊C4—24no—
359C4⋊C4—24no—
360C4⋊C4—22no—
361C4⋊C4—48no—
362C4⋊C4—416no—
363C4⋊C4—48no—
364C4⋊C4—42no—
365C4⋊C4—41no—
366C4⋊C4—42no—
367C4⋊C4—42no—
368C4⋊C4—42no—
369C4⋊C4—42no—
370C4⋊C4—41no—
371C4⋊C4—41no—
372C4⋊C4—41no—
373C4⋊C4—42no—
374C4⋊C4—84no—
375C42—22no—
376C42—24no—
377C42—22no—
378C42—24no—
379C42—24no—
380C42—24no—
381C42—24no—
382C42—48no—
383C42—416no—
384C23C2×#12111yescharacteristic
385C23C2×C22⋊Q812yes—
386C23D4212yes—
387C23#54611yescharacteristic
388C23D4×Q811yescharacteristic
389C23C2×#13511yescharacteristic
390C23C2×#13711yescharacteristic
391C23#57512yes—
392C23#57611yescharacteristic
393C23#57711yescharacteristic
394C23#57812yes—
395Q8—24no—
396Q8—24no—
397Q8—24no—
398Q8—28no—
399Q8—28no—
400Q8—816no—
401Q8#39518yes—
402C2×C4—24no—
403C2×C4—22no—
404C2×C4—24no—
405C2×C4—24no—
406C2×C4—28no—
407C2×C4—28no—
408C2×C4—24no—
409C2×C4—24no—
410C2×C4—22no—
411C2×C4—24no—
412C2×C4—24no—
413C2×C4—24no—
414C2×C4—24no—
415C2×C4—22no—
416C2×C4—24no—
417C2×C4—24no—
418C2×C4—22no—
419C2×C4—22no—
420C2×C4—42no—
421C2×C4—42no—
422C2×C4—42no—
423C2×C4—42no—
424C2×C4—42no—
425C2×C4—42no—
426C2×C4—42no—
427C2×C4—42no—
428C2×C4—42no—
429C2×C4—42no—
430C2×C4—44no—
431C2×C4—44no—
432C2×C4—42no—
433C2×C4—42no—
434C2×C4—42no—
435C2×C4—44no—
436C2×C4—42no—
437C2×C4—44no—
438C2×C4—44no—
439C2×C4—44no—
440C2×C4—42no—
441C2×C4—42no—
442C2×C4—81no—
443C2×C4—82no—
444C2×C4—82no—
445C2×C4—82no—
446C2×C4—82no—
447C2×C4—82no—
448C2×C4—82no—
449C2×C4—81no—
450C2×C4—82no—
451C2×C4—81no—
452C2×C4—82no—
453C2×C4—82no—
454C2×C4—81no—
455C2×C4—82no—
456C2×C4—81no—
457C2×C4—82no—
458C2×C4—82no—
459C2×C4—81no—
460C2×C4—81no—
461C2×C4#46414yes—
462C2×C4C2×#12011yescharacteristic
463C2×C4C2×C22⋊Q811yescharacteristic
464C2×C4C2×C22⋊Q812yes—
465C2×C4C2×C4⋊Q811yescharacteristic
466C2×C4C2×C4⋊Q812yes—
467C2×C4#53211yescharacteristic
468C2×C4#53312yes—
469C2×C4#53411yescharacteristic
470C2×C4#53711yescharacteristic
471C2×C4#54312yes—
472C2×C4#54412yes—
473C2×C4#54612yes—
474C2×C4#54612yes—
475C2×C4#54712yes—
476C2×C4#55112yes—
477C2×C4#55211yescharacteristic
478C2×C4#55412yes—
479C2×C4#55411yescharacteristic
480C8—42no—
481C8—44no—
482C8—81no—
483C22#241011yescharacteristic
484C22#241711yescharacteristic
485C22#243712yes—
486C22#244012yes—
487C22#285711yescharacteristic
488C22C2×#46411yescharacteristic
489C22#304011yescharacteristic
490C22#304411yescharacteristic
491C22#304511yescharacteristic
492C22#307612yes—
493C22#307812yes—
494C22#308212yes—
495C22#308312yes—
496C22#326812yes—
497C22#327012yes—
498C22#327512yes—
499C22#328512yes—
500C22#330511yescharacteristic
501C22#331312yes—
502C22#332911yescharacteristic
503C22#338111yescharacteristic
504C22#338511yescharacteristic
505C22#339012yes—
506C4—24no—
507C4—28no—
508C4—42no—
509C4—44no—
510C4—44no—
511C4—82no—
512C4—84no—
513C4—84no—
514C4—82no—
515C4—161no—
516C4C2×#39511yescharacteristic
517C4#243011yescharacteristic
518C4#243812yes—
519C4#254912yes—
520C4#256411yescharacteristic
521C4#265911yescharacteristic
522C2#1475811yescharacteristic
523C2#2436111yescharacteristic
524C2#2437111yescharacteristic
525C2#2481411yescharacteristic
526C2#2483211yescharacteristic
527C2#2595912yes—
528C2#2597912yes—
529C2#2605012yes—
530C2#2605512yes—
531C2#3320211yescharacteristic
532C2#3320611yescharacteristic
subgroupG/H
center, socle, prime radicalC24C22×D4
commutator, Frattini, cosocleC23×C4C24
LCSG ▷ C23×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
63. C2×C4×Q8Q8semidirect
95. C2×#103Q8semidirect

Maximal subgroups

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

subgroupindex
C22×#4772
C2×#18652
C2×#18052
#152302
Q8×#952
#113842
#113252
#113222
#112322
#112272
#107892