Finite Group Databasebeta

← #468322

Subgroups — #468322

Order 512.

1074 conjugacy classes of subgroups.

The subgroup lattice has 522 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2507C211yesmaximal, characteristic
2C2×#2083C211yesmaximal, characteristic
3C2×#1947C211yesmaximal, characteristic
4#15196C211yesmaximal, characteristic
5#11328C212yesmaximal
6#11328C212yesmaximal
7#11324C212yesmaximal
8#11293C212yesmaximal
9#11228C211yesmaximal, characteristic
10#11202C211yesmaximal, characteristic
11#10807C211yesmaximal, characteristic
12C22×#107C2212yes—
13C22×#107C2212yes—
14C22×C4⋊C8C2211yescharacteristic
15#2507—24no—
16#2507—24no—
17C2×#398—21no—
18C2×#398—21no—
19C2×#389—21no—
20C2×#389C2211yescharacteristic
21C2×#389C2211yescharacteristic
22C2×#389C2211yescharacteristic
23C2×#389C2211yescharacteristic
24C2×#389C2212yes—
25C2×#387—21no—
26C2×#384C2211yescharacteristic
27C2×#384C2211yescharacteristic
28C2×C4×C4⋊C4C2211yescharacteristic
29C2×C4×C4⋊C4C2211yescharacteristic
30#2083—24no—
31#1947—24no—
32C2×#340C2212yes—
33C2×#340C2212yes—
34C2×#340C2212yes—
35C2×#340C2212yes—
36C2×#340C2212yes—
37C2×#340C2212yes—
38C2×#336C2212yes—
39C2×#336C2211yescharacteristic
40C2×#336C2212yes—
41#1444—21no—
42#1444—21no—
43#1441—21no—
44#1441—21no—
45C4⋊#95C2212yes—
46C4⋊#95C2211yescharacteristic
47C23×C8C2312yes—
48C23×C8C2312yes—
49C22×C4⋊C4—21no—
50C22×C4⋊C4—21no—
51C22×C4⋊C4—21no—
52C22×C4⋊C4—21no—
53C22×C4⋊C4—21no—
54C22×C4⋊C4—21no—
55C22×C4⋊C4—21no—
56C22×C4⋊C4—21no—
57C22×C4⋊C4—22no—
58C22×C4⋊C4D411yescharacteristic
59C22×C4⋊C4D411yescharacteristic
60C22×C4⋊C4C2312yes—
61C22×C4⋊C4C2311yescharacteristic
62C22×C4⋊C4C2311yescharacteristic
63C22×C4⋊C4C2311yescharacteristic
64C22×C4⋊C4C2312yes—
65C22×C4⋊C4C2311yescharacteristic
66C22×C42—21no—
67C22×C42C2311yescharacteristic
68C22×C42C2311yescharacteristic
69C22×C42C2311yescharacteristic
70C2×#107—24no—
71C2×#107—24no—
72C2×#107—42no—
73C2×#107—42no—
74C2×#107—42no—
75C2×#107—42no—
76C2×C4⋊C8—24no—
77C2×C4⋊C8—22no—
78#398—28no—
79#398—28no—
80#389—28no—
81#389—22no—
82#389—22no—
83#389—22no—
84#389—22no—
85#389—22no—
86#389—22no—
87#389—22no—
88#389—42no—
89#389—42no—
90#389D414yes—
91#387—28no—
92#384—22no—
93#384—22no—
94#384—22no—
95#384—22no—
96C4×C4⋊C4—22no—
97C4×C4⋊C4—22no—
98C4×C4⋊C4—22no—
99C4×C4⋊C4D414yes—
100C2×#95—21no—
101C2×#95—21no—
102C2×#95—21no—
103C2×#95—21no—
104C2×#95—21no—
105C2×#95—21no—
106C2×#95—21no—
107C2×#95—22no—
108C2×#95—22no—
109C2×#95—22no—
110C2×#95D411yescharacteristic
111C2×#95D411yescharacteristic
112#340—24no—
113#340—24no—
114#340—24no—
115#340—42no—
116#340—42no—
117#340Q818yes—
118#336—24no—
119#336—22no—
120#336—42no—
121C23×C4—21no—
122C23×C4—21no—
123C23×C4—21no—
124C23×C4—21no—
125C23×C4—22no—
126C23×C4—22no—
127C23×C4C2×D411yescharacteristic
128C23×C4C2×D411yescharacteristic
129C23×C4C2×D411yescharacteristic
130C23×C4C2×D411yescharacteristic
131C23×C4C2×D411yescharacteristic
132C23×C4C2×D411yescharacteristic
133C23×C4C2411yescharacteristic
134C22×C8—22no—
135C22×C8—22no—
136C22×C8—22no—
137C22×C8—22no—
138C22×C8—22no—
139C22×C8—22no—
140C22×C8—22no—
141C22×C8—22no—
142C22×C8—22no—
143C22×C8—22no—
144C22×C8—22no—
145C22×C8—22no—
146C22×C8C2×Q814yes—
147C22×C8#4114yes—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—21no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—21no—
155C2×C4⋊C4—21no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—21no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4—21no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4—21no—
163C2×C4⋊C4—21no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—24no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—24no—
180C2×C4⋊C4—24no—
181C2×C4⋊C4—24no—
182C2×C4⋊C4—24no—
183C2×C4⋊C4—24no—
184C2×C4⋊C4—41no—
185C2×C4⋊C4—42no—
186C2×C4⋊C4—42no—
187C2×C4⋊C4—44no—
188C2×C4⋊C4—42no—
189C2×C4⋊C4—42no—
190C2×C4⋊C4—82no—
191C2×C4⋊C4—82no—
192C2×C4⋊C4—82no—
193C2×C4⋊C4—82no—
194C2×C4⋊C4—82no—
195C2×C4⋊C4—82no—
196C2×C4⋊C4—82no—
197C2×C4⋊C4—82no—
198C2×C4⋊C4—82no—
199C2×C4⋊C4Q1614yes—
200C2×C4⋊C4C2×D412yes—
201C2×C4⋊C4C2×D412yes—
202C2×C4⋊C4C2×D412yes—
203C2×C4⋊C4C2×D412yes—
204C2×C4⋊C4C2×Q814yes—
205C2×C4⋊C4C2×Q814yes—
206C2×C42—22no—
207C2×C42—22no—
208C2×C42—21no—
209C2×C42—21no—
210C2×C42—21no—
211C2×C42—21no—
212C2×C42—22no—
213C2×C42—21no—
214C2×C42—21no—
215C2×C42—21no—
216C2×C42—22no—
217C2×C42—21no—
218C2×C42—24no—
219C2×C42—82no—
220C2×C42C2×D412yes—
221C2×C42C2×D412yes—
222#107—48no—
223#107—48no—
224C4⋊C8—28no—
225#95—24no—
226#95—24no—
227#95—24no—
228#95—24no—
229#95—24no—
230#95—22no—
231#95—22no—
232#95—42no—
233#95—44no—
234#95—42no—
235#95—82no—
236#95—82no—
237C24C22×D411yescharacteristic
238C22×C4—22no—
239C22×C4—22no—
240C22×C4—21no—
241C22×C4—21no—
242C22×C4—22no—
243C22×C4—21no—
244C22×C4—21no—
245C22×C4—21no—
246C22×C4—22no—
247C22×C4—22no—
248C22×C4—22no—
249C22×C4—24no—
250C22×C4—21no—
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—21no—
256C22×C4—21no—
257C22×C4—22no—
258C22×C4—22no—
259C22×C4—22no—
260C22×C4—22no—
261C22×C4—21no—
262C22×C4—22no—
263C22×C4—22no—
264C22×C4—21no—
265C22×C4—21no—
266C22×C4—24no—
267C22×C4—22no—
268C22×C4—22no—
269C22×C4—41no—
270C22×C4—41no—
271C22×C4—42no—
272C22×C4—42no—
273C22×C4—41no—
274C22×C4—41no—
275C22×C4—41no—
276C22×C4—41no—
277C22×C4—42no—
278C22×C4—42no—
279C22×C4—42no—
280C22×C4—41no—
281C22×C4—42no—
282C22×C4—41no—
283C22×C4—42no—
284C22×C4—42no—
285C22×C4—42no—
286C22×C4—42no—
287C22×C4—42no—
288C22×C4—42no—
289C22×C4—42no—
290C22×C4—42no—
291C22×C4—42no—
292C22×C4—42no—
293C22×C4—42no—
294C22×C4—42no—
295C22×C4—42no—
296C22×C4—42no—
297C22×C4#12012yes—
298C22×C4#12012yes—
299C22×C4C2×Q1612yes—
300C22×C4C2×Q1612yes—
301C22×C4C2×Q1612yes—
302C22×C4#13712yes—
303C22×C4#13712yes—
304C22×C4C22×D411yescharacteristic
305C22×C4C22×D411yescharacteristic
306C22×C4C22×Q812yes—
307C22×C4C2×#4112yes—
308C22×C4H(5,𝔽2)11yescharacteristic
309C22×C4H(5,𝔽2)11yescharacteristic
310C22×C4H(5,𝔽2)11yescharacteristic
311C22×C42−1+412yes—
312C22×C42−1+412yes—
313C22×C42−1+411yescharacteristic
314C2×C8—24no—
315C2×C8—24no—
316C2×C8—24no—
317C2×C8—24no—
318C2×C8—24no—
319C2×C8—24no—
320C2×C8—42no—
321C2×C8—42no—
322C2×C8—42no—
323C2×C8—42no—
324C2×C8—42no—
325C2×C8—42no—
326C2×C8—42no—
327C2×C8—42no—
328C4⋊C4—22no—
329C4⋊C4—24no—
330C4⋊C4—28no—
331C4⋊C4—22no—
332C4⋊C4—22no—
333C4⋊C4—22no—
334C4⋊C4—22no—
335C4⋊C4—24no—
336C4⋊C4—24no—
337C4⋊C4—24no—
338C4⋊C4—44no—
339C4⋊C4—48no—
340C4⋊C4—41no—
341C4⋊C4—41no—
342C4⋊C4—41no—
343C4⋊C4—41no—
344C4⋊C4—41no—
345C4⋊C4—41no—
346C4⋊C4—42no—
347C4⋊C4—42no—
348C4⋊C4—42no—
349C4⋊C4—42no—
350C4⋊C4—42no—
351C4⋊C4—42no—
352C4⋊C4—84no—
353C4⋊C4—84no—
354C4⋊C4—84no—
355C4⋊C4—84no—
356C4⋊C4—84no—
357C4⋊C4—84no—
358C4⋊C4—84no—
359C4⋊C4—88no—
360C4⋊C4C4⋊Q814yes—
361C42—22no—
362C42—22no—
363C42—24no—
364C42—41no—
365C42—41no—
366C42—41no—
367C42—41no—
368C42—41no—
369C42—41no—
370C42—42no—
371C42—84no—
372C23C2×#12011yescharacteristic
373C23#53511yescharacteristic
374C23#53711yescharacteristic
375C23#54712yes—
376C23D4×Q812yes—
377C23C22×Q1611yescharacteristic
378C23C2×#13711yescharacteristic
379C23C2×#13711yescharacteristic
380C23#57511yescharacteristic
381C23#57612yes—
382C23#57612yes—
383C2×C4—24no—
384C2×C4—24no—
385C2×C4—22no—
386C2×C4—22no—
387C2×C4—24no—
388C2×C4—24no—
389C2×C4—22no—
390C2×C4—24no—
391C2×C4—22no—
392C2×C4—22no—
393C2×C4—22no—
394C2×C4—22no—
395C2×C4—22no—
396C2×C4—22no—
397C2×C4—22no—
398C2×C4—42no—
399C2×C4—42no—
400C2×C4—42no—
401C2×C4—42no—
402C2×C4—41no—
403C2×C4—44no—
404C2×C4—44no—
405C2×C4—41no—
406C2×C4—41no—
407C2×C4—41no—
408C2×C4—41no—
409C2×C4—41no—
410C2×C4—42no—
411C2×C4—42no—
412C2×C4—41no—
413C2×C4—41no—
414C2×C4—41no—
415C2×C4—42no—
416C2×C4—42no—
417C2×C4—44no—
418C2×C4—44no—
419C2×C4—42no—
420C2×C4—42no—
421C2×C4—44no—
422C2×C4—42no—
423C2×C4—42no—
424C2×C4—42no—
425C2×C4—41no—
426C2×C4—41no—
427C2×C4—42no—
428C2×C4—42no—
429C2×C4—42no—
430C2×C4—42no—
431C2×C4—82no—
432C2×C4—82no—
433C2×C4—82no—
434C2×C4—82no—
435C2×C4—82no—
436C2×C4—82no—
437C2×C4—82no—
438C2×C4—82no—
439C2×C4—82no—
440C2×C4—82no—
441C2×C4—82no—
442C2×C4—82no—
443C2×C4—82no—
444C2×C4—82no—
445C2×C4—82no—
446C2×C4—82no—
447C2×C4—82no—
448C2×C4#45114yes—
449C2×C4#52511yescharacteristic
450C2×C4C2×C4⋊Q811yescharacteristic
451C2×C4#53511yescharacteristic
452C2×C4#53511yescharacteristic
453C2×C4#53611yescharacteristic
454C2×C4#53611yescharacteristic
455C2×C4#53711yescharacteristic
456C2×C4#53711yescharacteristic
457C2×C4#53912yes—
458C2×C4#53912yes—
459C2×C4#54412yes—
460C2×C4#54412yes—
461C2×C4#54712yes—
462C2×C4D4×Q812yes—
463C2×C4#55412yes—
464C2×C4#55712yes—
465C2×C4#56312yes—
466C2×C4#56412yes—
467C8—44no—
468C8—44no—
469C22#240611yescharacteristic
470C22#243411yescharacteristic
471C22#261511yescharacteristic
472C22#272012yes—
473C22#272612yes—
474C22C2×#45111yescharacteristic
475C22#300711yescharacteristic
476C22#300811yescharacteristic
477C22#300811yescharacteristic
478C22#320911yescharacteristic
479C22#321611yescharacteristic
480C22#322012yes—
481C22#324911yescharacteristic
482C22#325511yescharacteristic
483C22#325912yes—
484C22#333812yes—
485C22#334612yes—
486C22#334612yes—
487C22#335512yes—
488C22Q8×Q1612yes—
489C22#338512yes—
490C22#338512yes—
491C22#338912yes—
492C4—24no—
493C4—42no—
494C4—42no—
495C4—42no—
496C4—42no—
497C4—44no—
498C4—82no—
499C4—84no—
500C4—81no—
501C4—82no—
502C4—82no—
503C4—162no—
504C4#258411yescharacteristic
505C4#263311yescharacteristic
506C4#266011yescharacteristic
507C4#272012yes—
508C4#272612yes—
509C4#282211yescharacteristic
510C2#1660711yescharacteristic
511C2#2409811yescharacteristic
512C2#2410211yescharacteristic
513C2#2584111yescharacteristic
514C2#2585011yescharacteristic
515C2#2966411yescharacteristic
516C2#2967011yescharacteristic
517C2#3234812yes—
518C2#3236612yes—
519C2#3250612yes—
520C2#3251412yes—
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×#25072
C2×#20832
C2×#19472
#151962
#113282
#113242
#112932
#112282
#112022
#108072