Finite Group Databasebeta

← #464324

Subgroups — #464324

Order 512.

1336 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×#2642C211yesmaximal, characteristic
2C2×#2056C212yesmaximal
3C2×#2026C211yesmaximal, characteristic
4C2×#2016C211yesmaximal, characteristic
5#14261C211yesmaximal, characteristic
6#11385C211yesmaximal, characteristic
7#11077C212yesmaximal
8#11067C212yesmaximal
9#11024C212yesmaximal
10#11010C211yesmaximal, characteristic
11#10833C211yesmaximal, characteristic
12C23×Q16C2212yes—
13C22×C4⋊Q8C2211yescharacteristic
14C22×#125C2211yescharacteristic
15C22×C4⋊C8C2212yes—
16C22×#103C2212yes—
17C22×#103C2212yes—
18C22×#103C2212yes—
19C22×#103C2212yes—
20#2642—28no—
21C2×#400—22no—
22C2×#400—22no—
23C2×#391C2212yes—
24C2×#391C2211yescharacteristic
25C2×#391C2212yes—
26C2×#391C2211yescharacteristic
27C2×#391C2211yescharacteristic
28C2×#389C2211yescharacteristic
29C2×#376C2211yescharacteristic
30#2056—28no—
31#2026—24no—
32#2016—24no—
33C2×#340C2212yes—
34C2×#340C2212yes—
35C2×#340C2212yes—
36C2×#336C2211yescharacteristic
37C2×#336C2211yescharacteristic
38C2×#336C2212yes—
39#1443—22no—
40#1442—22no—
41C4⋊#95C2211yescharacteristic
42C4⋊#95C2211yescharacteristic
43C23×Q8C2312yes—
44C23×Q8C2311yescharacteristic
45C22×Q16—24no—
46C22×Q16—24no—
47C22×Q16—24no—
48C22×Q16—42no—
49C22×Q16—42no—
50C22×Q16—42no—
51C22×Q16—42no—
52C23×C8C2312yes—
53C23×C8C2312yes—
54C2×C4⋊Q8—24no—
55C2×C4⋊Q8—22no—
56C2×C4⋊Q8—22no—
57C2×C4⋊Q8—42no—
58C2×#125—24no—
59C2×#125—22no—
60C2×#125—22no—
61C2×#125—42no—
62C22×C4⋊C4—22no—
63C22×C4⋊C4—22no—
64C22×C4⋊C4—21no—
65C22×C4⋊C4C2311yescharacteristic
66C22×C4⋊C4C2311yescharacteristic
67C22×C4⋊C4C2311yescharacteristic
68C22×C4⋊C4C2312yes—
69C22×C42C2311yescharacteristic
70C22×C42C2311yescharacteristic
71C22×C42C2311yescharacteristic
72C2×C4⋊C8—24no—
73C2×C4⋊C8—24no—
74C2×C4⋊C8—42no—
75C2×#103—24no—
76C2×#103—24no—
77C2×#103—24no—
78C2×#103—24no—
79C2×#103—24no—
80C2×#103—42no—
81C2×#103—42no—
82C2×#103—42no—
83C2×#103—42no—
84C2×#103—42no—
85C2×#103—42no—
86C2×#103—42no—
87#400—216no—
88#400—216no—
89#391—24no—
90#391—24no—
91#391—24no—
92#391—22no—
93#391—24no—
94#391—42no—
95#391D418yes—
96#391D414yes—
97#389—42no—
98#376—24no—
99C2×#95—22no—
100C2×#95—22no—
101C2×#95—24no—
102C2×#95—22no—
103C2×#95—21no—
104C2×#95—22no—
105C2×#95—21no—
106C2×#95—21no—
107C2×#95—22no—
108C2×#95—22no—
109C2×#95D412yes—
110C2×#95D412yes—
111#340—24no—
112#340—24no—
113#340—42no—
114#336—22no—
115#336—22no—
116#336—42no—
117C22×Q8—22no—
118C22×Q8—21no—
119C22×Q8—22no—
120C22×Q8—22no—
121C22×Q8—22no—
122C22×Q8—22no—
123C22×Q8—22no—
124C22×Q8—22no—
125C22×Q8—22no—
126C22×Q8—22no—
127C22×Q8—21no—
128C22×Q8—21no—
129C22×Q8—22no—
130C22×Q8—22no—
131C22×Q8—22no—
132C22×Q8—22no—
133C22×Q8—22no—
134C22×Q8—22no—
135C22×Q8—21no—
136C22×Q8—22no—
137C22×Q8—21no—
138C22×Q8—22no—
139C22×Q8C2×D414yes—
140C22×Q8C2×D412yes—
141C23×C4—22no—
142C23×C4—22no—
143C23×C4—21no—
144C23×C4—21no—
145C23×C4—21no—
146C23×C4—21no—
147C23×C4C2×D411yescharacteristic
148C23×C4C2×D411yescharacteristic
149C23×C4C2×D412yes—
150C23×C4C2×D412yes—
151C23×C4C2411yescharacteristic
152C2×Q16—48no—
153C2×Q16—48no—
154C2×Q16—48no—
155C2×Q16—48no—
156C2×Q16—48no—
157C2×Q16—48no—
158C2×Q16—48no—
159C22×C8—22no—
160C22×C8—22no—
161C22×C8—22no—
162C22×C8—22no—
163C22×C8—22no—
164C22×C8—22no—
165C22×C8—22no—
166C22×C8—22no—
167C22×C8—22no—
168C22×C8—22no—
169C22×C8—22no—
170C22×C8—22no—
171C22×C8—22no—
172C22×C8#4114yes—
173C4⋊Q8—48no—
174C4⋊Q8—48no—
175#125—48no—
176#125—48no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—28no—
184C2×C4⋊C4—28no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—44no—
196C2×C4⋊C4—44no—
197C2×C4⋊C4—44no—
198C2×C4⋊C4—44no—
199C2×C4⋊C4—42no—
200C2×C4⋊C4—82no—
201C2×C4⋊C4C2×D414yes—
202C2×C4⋊C4C2×D412yes—
203C2×C42—22no—
204C2×C42—22no—
205C2×C42—22no—
206C2×C42—21no—
207C2×C42—21no—
208C2×C42—22no—
209C2×C42—21no—
210C2×C42—22no—
211C2×C42—21no—
212C2×C42C2×D414yes—
213C2×C42C2×D412yes—
214C2×C42#4112yes—
215C4⋊C8—48no—
216#103—48no—
217#103—48no—
218#103—48no—
219#103—48no—
220#95—28no—
221#95—28no—
222#95—216no—
223#95—24no—
224#95—24no—
225#95—28no—
226#95—42no—
227#95—42no—
228#95—42no—
229#95—44no—
230#95—82no—
231#95—82no—
232C24C22×D411yescharacteristic
233C2×Q8—24no—
234C2×Q8—22no—
235C2×Q8—22no—
236C2×Q8—24no—
237C2×Q8—24no—
238C2×Q8—24no—
239C2×Q8—24no—
240C2×Q8—24no—
241C2×Q8—24no—
242C2×Q8—24no—
243C2×Q8—42no—
244C2×Q8—42no—
245C2×Q8—41no—
246C2×Q8—41no—
247C2×Q8—42no—
248C2×Q8—42no—
249C2×Q8—42no—
250C2×Q8—42no—
251C2×Q8—42no—
252C2×Q8—42no—
253C2×Q8—42no—
254C2×Q8—42no—
255C2×Q8—42no—
256C2×Q8—42no—
257C2×Q8—42no—
258C2×Q8—42no—
259C2×Q8—42no—
260C2×Q8—42no—
261C2×Q8—42no—
262C2×Q8—42no—
263C2×Q8—42no—
264C2×Q8—42no—
265C2×Q8—42no—
266C2×Q8—42no—
267C2×Q8—42no—
268C2×Q8—42no—
269C2×Q8—42no—
270C2×Q8—42no—
271C2×Q8—42no—
272C2×Q8—42no—
273C2×Q8—42no—
274C2×Q8—42no—
275C2×Q8—42no—
276C2×Q8—42no—
277C22×C4—22no—
278C22×C4—22no—
279C22×C4—22no—
280C22×C4—22no—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—22no—
284C22×C4—22no—
285C22×C4—22no—
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—22no—
289C22×C4—21no—
290C22×C4—21no—
291C22×C4—24no—
292C22×C4—24no—
293C22×C4—22no—
294C22×C4—22no—
295C22×C4—21no—
296C22×C4—21no—
297C22×C4—21no—
298C22×C4—21no—
299C22×C4—22no—
300C22×C4—22no—
301C22×C4—22no—
302C22×C4—22no—
303C22×C4—41no—
304C22×C4—42no—
305C22×C4—42no—
306C22×C4—41no—
307C22×C4—42no—
308C22×C4—41no—
309C22×C4—41no—
310C22×C4—41no—
311C22×C4—42no—
312C22×C4—42no—
313C22×C4—42no—
314C22×C4—41no—
315C22×C4—42no—
316C22×C4—41no—
317C22×C4—41no—
318C22×C4—42no—
319C22×C4—42no—
320C22×C4—42no—
321C22×C4—42no—
322C22×C4—42no—
323C22×C4—42no—
324C22×C4—42no—
325C22×C4—41no—
326C22×C4—41no—
327C22×C4—41no—
328C22×C4—41no—
329C22×C4—41no—
330C22×C4—41no—
331C22×C4—42no—
332C22×C4—42no—
333C22×C4—42no—
334C22×C4—42no—
335C22×C4#12014yes—
336C22×C4#12114yes—
337C22×C4C22×D411yescharacteristic
338C22×C4C22×D412yes—
339C22×C4C2×#4112yes—
340C22×C4C2×#4111yescharacteristic
341C22×C4H(5,𝔽2)11yescharacteristic
342C22×C4H(5,𝔽2)12yes—
343C22×C4H(5,𝔽2)11yescharacteristic
344C22×C4H(5,𝔽2)12yes—
345C22×C42−1+411yescharacteristic
346C22×C42−1+411yescharacteristic
347Q16—432no—
348C2×C8—24no—
349C2×C8—24no—
350C2×C8—24no—
351C2×C8—42no—
352C2×C8—42no—
353C2×C8—42no—
354C2×C8—42no—
355C2×C8—42no—
356C2×C8—42no—
357C2×C8—42no—
358C2×C8—42no—
359C2×C8—42no—
360C2×C8—42no—
361C2×C8—42no—
362C4⋊C4—24no—
363C4⋊C4—24no—
364C4⋊C4—24no—
365C4⋊C4—416no—
366C4⋊C4—416no—
367C4⋊C4—42no—
368C4⋊C4—42no—
369C4⋊C4—42no—
370C4⋊C4—42no—
371C4⋊C4—42no—
372C4⋊C4—42no—
373C4⋊C4—42no—
374C4⋊C4—84no—
375C42—24no—
376C42—24no—
377C42—24no—
378C42—24no—
379C42—42no—
380C42—42no—
381C23C2×#12011yescharacteristic
382C23C2×#12111yescharacteristic
383C23#53211yescharacteristic
384C23#53511yescharacteristic
385C23#53511yescharacteristic
386C23#53711yescharacteristic
387C23#53711yescharacteristic
388C23#57712yes—
389C23#57812yes—
390C23#57812yes—
391C23#57812yes—
392Q8—44no—
393Q8—44no—
394Q8—44no—
395Q8—44no—
396Q8—44no—
397Q8—44no—
398Q8—82no—
399Q8—82no—
400Q8—82no—
401Q8—82no—
402Q8—82no—
403Q8—82no—
404C2×C4—24no—
405C2×C4—24no—
406C2×C4—24no—
407C2×C4—24no—
408C2×C4—24no—
409C2×C4—24no—
410C2×C4—42no—
411C2×C4—42no—
412C2×C4—42no—
413C2×C4—42no—
414C2×C4—42no—
415C2×C4—42no—
416C2×C4—42no—
417C2×C4—42no—
418C2×C4—41no—
419C2×C4—41no—
420C2×C4—44no—
421C2×C4—42no—
422C2×C4—44no—
423C2×C4—44no—
424C2×C4—42no—
425C2×C4—44no—
426C2×C4—41no—
427C2×C4—41no—
428C2×C4—41no—
429C2×C4—41no—
430C2×C4—42no—
431C2×C4—42no—
432C2×C4—42no—
433C2×C4—44no—
434C2×C4—42no—
435C2×C4—44no—
436C2×C4—44no—
437C2×C4—42no—
438C2×C4—42no—
439C2×C4—81no—
440C2×C4—82no—
441C2×C4—81no—
442C2×C4—82no—
443C2×C4—82no—
444C2×C4—81no—
445C2×C4—82no—
446C2×C4—81no—
447C2×C4—81no—
448C2×C4—82no—
449C2×C4—81no—
450C2×C4—82no—
451C2×C4—82no—
452C2×C4—82no—
453C2×C4—82no—
454C2×C4—81no—
455C2×C4—81no—
456C2×C4—81no—
457C2×C4—81no—
458C2×C4—81no—
459C2×C4—81no—
460C2×C4—82no—
461C2×C4—82no—
462C2×C4—82no—
463C2×C4—82no—
464C2×C4#53511yescharacteristic
465C2×C4#53512yes—
466C2×C4#53611yescharacteristic
467C2×C4#53712yes—
468C2×C4#53812yes—
469C2×C4#54212yes—
470C2×C4#54211yescharacteristic
471C2×C4#54211yescharacteristic
472C2×C4D4212yes—
473C2×C4#54612yes—
474C2×C4#54612yes—
475C2×C4#54612yes—
476C2×C4#54712yes—
477C2×C4#55912yes—
478C2×C4H(3,𝔽4)12yes—
479C2×C4#56212yes—
480C8—44no—
481C8—82no—
482C22#240811yescharacteristic
483C22#243011yescharacteristic
484C22#254411yescharacteristic
485C22#261511yescharacteristic
486C22#262011yescharacteristic
487C22#266211yescharacteristic
488C22#281911yescharacteristic
489C22#301112yes—
490C22#301312yes—
491C22#304412yes—
492C22#304612yes—
493C22#317712yes—
494C22#317812yes—
495C22#322211yescharacteristic
496C22#322311yescharacteristic
497C22#322312yes—
498C22#322311yescharacteristic
499C22#322311yescharacteristic
500C22#322512yes—
501C22#325812yes—
502C22#325812yes—
503C22#326112yes—
504C22#326112yes—
505C4—44no—
506C4—44no—
507C4—84no—
508C4—82no—
509C4—84no—
510C4—82no—
511C4—81no—
512C4—81no—
513C4—161no—
514C4—161no—
515C4#267012yes—
516C4#267212yes—
517C4#271012yes—
518C4#271212yes—
519C2#1650411yescharacteristic
520C2#2422112yes—
521C2#2551212yes—
522C2#2862511yescharacteristic
523C2#2862911yescharacteristic
524C2#2967411yescharacteristic
525C2#2967611yescharacteristic
526C2#2983712yes—
527C2#3035112yes—
528C2#3281311yescharacteristic
529C2#3281611yescharacteristic
subgroupG/H
center, socle, prime radicalC24C22×D4
commutator, Frattini, cosocleC23×C4C24
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×#376 ◁ 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×#26422
C2×#20562
C2×#20262
C2×#20162
#142612
#113852
#110772
#110672
#110242
#110102
#108332