Finite Group Databasebeta

← #399609

Subgroups — #399609

Order 512.

1416 conjugacy classes of subgroups.

The subgroup lattice has 531 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×Q8×C4⋊C4C211yesmaximal, characteristic
2C2×#2084C211yesmaximal, characteristic
3C2×#1865C212yesmaximal
4C2×#1863C211yesmaximal, characteristic
5#15194C211yesmaximal, characteristic
6#11373C211yesmaximal, characteristic
7#11346C211yesmaximal, characteristic
8#11327C212yesmaximal
9#11219C212yesmaximal
10#10789C212yesmaximal
11#10789C211yesmaximal, characteristic
12C22×C4×Q8C2211yescharacteristic
13C22×C4×Q8C2212yes—
14C22×#107C2211yescharacteristic
15C22×#106C2212yes—
16C22×#103C2211yescharacteristic
17C22×#103C2211yescharacteristic
18C22×#103C2212yes—
19Q8×C4⋊C4—24no—
20Q8×C4⋊C4—24no—
21C2×#389—22no—
22C2×#389C2212yes—
23C2×#389C2211yescharacteristic
24C2×#389C2212yes—
25C2×#384—21no—
26C2×#384C2211yescharacteristic
27C2×C4×C4⋊C4—21no—
28C2×C4×C4⋊C4C2211yescharacteristic
29#2084—24no—
30#1865—28no—
31#1863—24no—
32C2×#340C2212yes—
33C2×#340C2212yes—
34C2×#340C2211yescharacteristic
35C2×#340C2212yes—
36C2×#340C2212yes—
37C2×#336C2211yescharacteristic
38C2×#336C2211yescharacteristic
39C2×#336C2212yes—
40C2×#336C2212yes—
41#1445—21no—
42#1444—21no—
43#1441—22no—
44C4⋊#95C2211yescharacteristic
45C4⋊#95C2212yes—
46C23×Q8C2311yescharacteristic
47C23×C8C2311yescharacteristic
48C23×C8C2311yescharacteristic
49C23×C8C2312yes—
50C2×C4×Q8—24no—
51C2×C4×Q8—24no—
52C2×C4×Q8—24no—
53C2×C4×Q8—24no—
54C2×C4×Q8—22no—
55C2×C4×Q8—24no—
56C2×C4×Q8—24no—
57C2×C4×Q8—42no—
58C2×C4×Q8—42no—
59C2×C4×Q8D414yes—
60C22×C4⋊C4—21no—
61C22×C4⋊C4—22no—
62C22×C4⋊C4—21no—
63C22×C4⋊C4—22no—
64C22×C4⋊C4—21no—
65C22×C4⋊C4—22no—
66C22×C4⋊C4—21no—
67C22×C4⋊C4D411yescharacteristic
68C22×C4⋊C4D411yescharacteristic
69C22×C4⋊C4C2311yescharacteristic
70C22×C4⋊C4C2311yescharacteristic
71C22×C4⋊C4C2312yes—
72C22×C4⋊C4C2312yes—
73C22×C4⋊C4C2311yescharacteristic
74C22×C42—21no—
75C22×C42—22no—
76C22×C42C2312yes—
77C22×C42C2311yescharacteristic
78C2×#107—22no—
79C2×#107—24no—
80C2×#106—24no—
81C2×#106—24no—
82C2×#106—42no—
83C2×#103—22no—
84C2×#103—24no—
85C2×#103—22no—
86C2×#103—24no—
87C2×#103—22no—
88C2×#103—24no—
89C2×#103D414yes—
90C2×#103Q818yes—
91#389—216no—
92#389—24no—
93#389—24no—
94#389—24no—
95#389—22no—
96#389—22no—
97#389—42no—
98#384—28no—
99#384—22no—
100#384D414yes—
101C4×C4⋊C4—28no—
102C4×C4⋊C4—22no—
103C4×C4⋊C4—22no—
104C2×#95—22no—
105C2×#95—21no—
106C2×#95—22no—
107C2×#95—21no—
108C2×#95—21no—
109C2×#95—22no—
110C2×#95—22no—
111C2×#95D412yes—
112#340—24no—
113#340—24no—
114#340—22no—
115#340—24no—
116#340—42no—
117#336—24no—
118#336—22no—
119#336—22no—
120#336—42no—
121C22×Q8—21no—
122C22×Q8—22no—
123C22×Q8—21no—
124C22×Q8—21no—
125C22×Q8—22no—
126C22×Q8C2×D412yes—
127C22×Q8C2×D412yes—
128C22×Q8C2×Q814yes—
129C22×Q8#4112yes—
130C22×Q8#4114yes—
131C23×C4—21no—
132C23×C4—21no—
133C23×C4—22no—
134C23×C4—21no—
135C23×C4—22no—
136C23×C4—21no—
137C23×C4C2×D412yes—
138C23×C4C2×D411yescharacteristic
139C23×C4C2×D412yes—
140C23×C4C2×D411yescharacteristic
141C23×C4C2411yescharacteristic
142C22×C8—22no—
143C22×C8—22no—
144C22×C8—21no—
145C22×C8—22no—
146C22×C8—21no—
147C22×C8—22no—
148C22×C8—22no—
149C22×C8—22no—
150C22×C8—22no—
151C22×C8—22no—
152C22×C8—21no—
153C22×C8—22no—
154C22×C8—21no—
155C22×C8—21no—
156C22×C8—22no—
157C22×C8C2×D412yes—
158C22×C8C2×Q814yes—
159C4×Q8—216no—
160C4×Q8—28no—
161C4×Q8—28no—
162C4×Q8—48no—
163C4×Q8—48no—
164C4×Q8—48no—
165C4×Q8—48no—
166C2×C4⋊C4—28no—
167C2×C4⋊C4—24no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—24no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—28no—
185C2×C4⋊C4—24no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—28no—
197C2×C4⋊C4—24no—
198C2×C4⋊C4—44no—
199C2×C4⋊C4—44no—
200C2×C4⋊C4—42no—
201C2×C4⋊C4—42no—
202C2×C4⋊C4—44no—
203C2×C4⋊C4—42no—
204C2×C4⋊C4—44no—
205C2×C4⋊C4—81no—
206C2×C4⋊C4—81no—
207C2×C4⋊C4—82no—
208C2×C4⋊C4—82no—
209C2×C4⋊C4—82no—
210C2×C4⋊C4—82no—
211C2×C4⋊C4C2×D414yes—
212C2×C4⋊C4C2×D412yes—
213C2×C4⋊C4C2×D412yes—
214C2×C4⋊C4C2×Q814yes—
215C2×C4⋊C4#4114yes—
216C2×C42—28no—
217C2×C42—24no—
218C2×C42—22no—
219C2×C42—22no—
220C2×C42—28no—
221C2×C42—22no—
222C2×C42—22no—
223C2×C42—22no—
224C2×C42—22no—
225C2×C42—22no—
226C2×C42—22no—
227C2×C42—44no—
228C2×C42—44no—
229C2×C42C2×D412yes—
230C2×C42C2×D412yes—
231#107—44no—
232#106—48no—
233#103—28no—
234#103—216no—
235#103—44no—
236#95—28no—
237#95—24no—
238#95—44no—
239#95—42no—
240#95—42no—
241#95—81no—
242#95—82no—
243#95—82no—
244C24C22×D411yescharacteristic
245C2×Q8—22no—
246C2×Q8—24no—
247C2×Q8—22no—
248C2×Q8—22no—
249C2×Q8—24no—
250C2×Q8—24no—
251C2×Q8—22no—
252C2×Q8—22no—
253C2×Q8—24no—
254C2×Q8—24no—
255C2×Q8—24no—
256C2×Q8—24no—
257C2×Q8—24no—
258C2×Q8#12114yes—
259C2×Q8C22⋊Q818yes—
260C2×Q8#12314yes—
261C2×Q8#12518yes—
262C2×Q8C4⋊Q814yes—
263C22×C4—22no—
264C22×C4—22no—
265C22×C4—24no—
266C22×C4—22no—
267C22×C4—22no—
268C22×C4—24no—
269C22×C4—22no—
270C22×C4—22no—
271C22×C4—22no—
272C22×C4—22no—
273C22×C4—24no—
274C22×C4—22no—
275C22×C4—22no—
276C22×C4—22no—
277C22×C4—21no—
278C22×C4—22no—
279C22×C4—22no—
280C22×C4—22no—
281C22×C4—21no—
282C22×C4—22no—
283C22×C4—24no—
284C22×C4—22no—
285C22×C4—22no—
286C22×C4—22no—
287C22×C4—24no—
288C22×C4—22no—
289C22×C4—22no—
290C22×C4—22no—
291C22×C4—22no—
292C22×C4—41no—
293C22×C4—41no—
294C22×C4—42no—
295C22×C4—41no—
296C22×C4—42no—
297C22×C4—42no—
298C22×C4—42no—
299C22×C4—42no—
300C22×C4—41no—
301C22×C4—42no—
302C22×C4—42no—
303C22×C4—42no—
304C22×C4—42no—
305C22×C4—42no—
306C22×C4—42no—
307C22×C4—42no—
308C22×C4—42no—
309C22×C4—42no—
310C22×C4—42no—
311C22×C4—41no—
312C22×C4—42no—
313C22×C4—41no—
314C22×C4—42no—
315C22×C4—41no—
316C22×C4#12012yes—
317C22×C4#12012yes—
318C22×C4#13512yes—
319C22×C4#13714yes—
320C22×C4#13712yes—
321C22×C4C22×D411yescharacteristic
322C22×C4C22×D411yescharacteristic
323C22×C4C22×D411yescharacteristic
324C22×C4C22×Q812yes—
325C22×C4C2×#4111yescharacteristic
326C22×C4C2×#4112yes—
327C22×C4C2×#4112yes—
328C22×C4H(5,𝔽2)11yescharacteristic
329C22×C4H(5,𝔽2)11yescharacteristic
330C22×C42−1+412yes—
331C2×C8—22no—
332C2×C8—24no—
333C2×C8—24no—
334C2×C8—24no—
335C2×C8—24no—
336C2×C8—41no—
337C2×C8—42no—
338C2×C8—41no—
339C2×C8—42no—
340C2×C8—42no—
341C2×C8—42no—
342C2×C8—42no—
343C2×C8—41no—
344C2×C8—41no—
345C2×C8—41no—
346C2×C8—42no—
347C2×C8—42no—
348C4⋊C4—28no—
349C4⋊C4—216no—
350C4⋊C4—22no—
351C4⋊C4—24no—
352C4⋊C4—22no—
353C4⋊C4—24no—
354C4⋊C4—22no—
355C4⋊C4—24no—
356C4⋊C4—24no—
357C4⋊C4—24no—
358C4⋊C4—416no—
359C4⋊C4—42no—
360C4⋊C4—41no—
361C4⋊C4—42no—
362C4⋊C4—42no—
363C4⋊C4—41no—
364C4⋊C4—42no—
365C4⋊C4—44no—
366C4⋊C4—48no—
367C4⋊C4—41no—
368C4⋊C4—41no—
369C4⋊C4—42no—
370C4⋊C4—84no—
371C4⋊C4—88no—
372C4⋊C4—84no—
373C4⋊C4—88no—
374C4⋊C4#12314yes—
375C42—216no—
376C42—22no—
377C42—22no—
378C42—24no—
379C42—416no—
380C42—42no—
381C42—42no—
382C42—42no—
383C42—42no—
384C23C2×#12011yescharacteristic
385C23#52512yes—
386C23D4211yescharacteristic
387C23#54611yescharacteristic
388C23D4×Q812yes—
389C23C2×#13511yescharacteristic
390C23C2×#13711yescharacteristic
391C23C2×#13712yes—
392C23#57611yescharacteristic
393C23#57712yes—
394C23#57711yescharacteristic
395Q8—24no—
396Q8—28no—
397Q8—28no—
398Q8—24no—
399Q8—24no—
400Q8#39818yes—
401C2×C4—22no—
402C2×C4—24no—
403C2×C4—24no—
404C2×C4—24no—
405C2×C4—24no—
406C2×C4—24no—
407C2×C4—28no—
408C2×C4—28no—
409C2×C4—22no—
410C2×C4—22no—
411C2×C4—24no—
412C2×C4—24no—
413C2×C4—22no—
414C2×C4—24no—
415C2×C4—22no—
416C2×C4—24no—
417C2×C4—42no—
418C2×C4—44no—
419C2×C4—42no—
420C2×C4—44no—
421C2×C4—42no—
422C2×C4—42no—
423C2×C4—42no—
424C2×C4—42no—
425C2×C4—42no—
426C2×C4—42no—
427C2×C4—42no—
428C2×C4—44no—
429C2×C4—44no—
430C2×C4—44no—
431C2×C4—42no—
432C2×C4—42no—
433C2×C4—44no—
434C2×C4—42no—
435C2×C4—44no—
436C2×C4—44no—
437C2×C4—42no—
438C2×C4—44no—
439C2×C4—42no—
440C2×C4—42no—
441C2×C4—42no—
442C2×C4—82no—
443C2×C4—82no—
444C2×C4—82no—
445C2×C4—81no—
446C2×C4—81no—
447C2×C4—82no—
448C2×C4—82no—
449C2×C4—82no—
450C2×C4—82no—
451C2×C4—82no—
452C2×C4—82no—
453C2×C4—81no—
454C2×C4—82no—
455C2×C4—82no—
456C2×C4—81no—
457C2×C4—81no—
458C2×C4#45214yes—
459C2×C4C2×#12111yescharacteristic
460C2×C4C2×C22⋊Q812yes—
461C2×C4C2×#12311yescharacteristic
462C2×C4C2×#12311yescharacteristic
463C2×C4#52512yes—
464C2×C4C2×#12512yes—
465C2×C4C2×C4⋊Q811yescharacteristic
466C2×C4#53511yescharacteristic
467C2×C4#53611yescharacteristic
468C2×C4#53912yes—
469C2×C4#53912yes—
470C2×C4D4211yescharacteristic
471C2×C4#54611yescharacteristic
472C2×C4#54712yes—
473C2×C4D4×Q812yes—
474C2×C4#55212yes—
475C2×C4#55412yes—
476C2×C4#55412yes—
477C8—44no—
478C8—42no—
479C8—81no—
480C22#240811yescharacteristic
481C22#243411yescharacteristic
482C22#244912yes—
483C22#247312yes—
484C22#284911yescharacteristic
485C22C2×#45211yescharacteristic
486C22#300411yescharacteristic
487C22#301011yescharacteristic
488C22#301211yescharacteristic
489C22#310812yes—
490C22#311312yes—
491C22#311912yes—
492C22#312512yes—
493C22#326612yes—
494C22#327811yescharacteristic
495C22#328711yescharacteristic
496C22#330411yescharacteristic
497C22#332012yes—
498C22#333311yescharacteristic
499C22#338112yes—
500C22#338512yes—
501C22#338712yes—
502C22#338712yes—
503C4—28no—
504C4—24no—
505C4—44no—
506C4—42no—
507C4—44no—
508C4—84no—
509C4—82no—
510C4—84no—
511C4—82no—
512C4—161no—
513C4C2×#39811yescharacteristic
514C4#244912yes—
515C4#247312yes—
516C4#248611yescharacteristic
517C4#250211yescharacteristic
518C4#262311yescharacteristic
519C2#1478611yescharacteristic
520C2#2424211yescharacteristic
521C2#2425111yescharacteristic
522C2#2583611yescharacteristic
523C2#2584711yescharacteristic
524C2#2637912yes—
525C2#2638912yes—
526C2#2729712yes—
527C2#2731312yes—
528C2#3300811yescharacteristic
529C2#3301311yescharacteristic
subgroupG/H
center, socle, prime radicalC24C22×D4
commutator, Frattini, cosocleC23×C4C24
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Maximal subgroups

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

subgroupindex
C2×Q8×C4⋊C42
C2×#20842
C2×#18652
C2×#18632
#151942
#113732
#113462
#113272
#112192
#107892