Finite Group Databasebeta

← #403716

Subgroups — #403716

Order 512.

1244 conjugacy classes of subgroups.

The subgroup lattice has 525 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2331C211yesmaximal, characteristic
2C2×#2058C211yesmaximal, characteristic
3C2×#2057C211yesmaximal, characteristic
4#14282C211yesmaximal, characteristic
5#11384C212yesmaximal
6#11325C212yesmaximal
7#11208C211yesmaximal, characteristic
8#11208C211yesmaximal, characteristic
9#10908C212yesmaximal
10#10907C212yesmaximal
11#10796C211yesmaximal, characteristic
12C22×C4⋊Q8C2211yescharacteristic
13C22×C4⋊Q8C2211yescharacteristic
14C22×#107C2212yes—
15C22×#106C2212yes—
16#2331—24no—
17#2331—24no—
18C2×#391—21no—
19C2×#391—21no—
20C2×#389C2211yescharacteristic
21C2×#389C2211yescharacteristic
22C2×#387—21no—
23C2×#387—21no—
24C2×#382C2211yescharacteristic
25C2×#382C2211yescharacteristic
26C2×C4×C4⋊C4C2211yescharacteristic
27C2×C4×C4⋊C4C2211yescharacteristic
28C2×#376C2211yescharacteristic
29#2058—24no—
30#2057—24no—
31C2×#340C2212yes—
32C2×#340C2212yes—
33C2×#340C2212yes—
34C2×#340C2212yes—
35C2×#340C2212yes—
36C2×#340C2212yes—
37C2×#336C2212yes—
38C2×#336C2211yescharacteristic
39C2×#336C2211yescharacteristic
40C2×#336C2212yes—
41#1446—21no—
42#1446—21no—
43#1445—21no—
44#1445—21no—
45C4⋊#95C2212yes—
46C4⋊#95C2212yes—
47C23×Q8—21no—
48C23×Q8—21no—
49C23×C8C2312yes—
50C23×C8C2312yes—
51C2×C4⋊Q8—22no—
52C2×C4⋊Q8—22no—
53C2×C4⋊Q8—22no—
54C2×C4⋊Q8—22no—
55C2×C4⋊Q8—24no—
56C2×C4⋊Q8—22no—
57C2×C4⋊Q8—22no—
58C2×C4⋊Q8—22no—
59C2×C4⋊Q8—42no—
60C2×C4⋊Q8Q814yescomplemented
61C22×C4⋊C4—21no—
62C22×C4⋊C4—21no—
63C22×C4⋊C4—21no—
64C22×C4⋊C4—21no—
65C22×C4⋊C4C2312yes—
66C22×C4⋊C4C2311yescharacteristic
67C22×C4⋊C4C2311yescharacteristic
68C22×C4⋊C4C2312yes—
69C22×C4⋊C4C2311yescharacteristic
70C22×C4⋊C4C2311yescharacteristic
71C22×C42—21no—
72C22×C42—21no—
73C22×C42C2311yescharacteristic
74C22×C42C2311yescharacteristic
75C22×C42C2311yescharacteristic
76C2×#107—24no—
77C2×#107—24no—
78C2×#107—42no—
79C2×#106—24no—
80C2×#106—24no—
81C2×#106—42no—
82#391—28no—
83#391—28no—
84#389—22no—
85#389—22no—
86#389D414yes—
87#389D414yes—
88#387—28no—
89#387—28no—
90#382—22no—
91#382—22no—
92#382—42no—
93C4×C4⋊C4—22no—
94C4×C4⋊C4—22no—
95C4×C4⋊C4—22no—
96C4×C4⋊C4—22no—
97#376—22no—
98#376—22no—
99C2×#95—21no—
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×#95—22no—
111C2×#95D411yescharacteristic
112C2×#95D411yescharacteristic
113C2×#95D411yescharacteristic
114C2×#95D411yescharacteristic
115#340—24no—
116#340—24no—
117#340—24no—
118#340—24no—
119#340—42no—
120#340—42no—
121#336—22no—
122#336—22no—
123#336—24no—
124#336Q818yescomplemented
125C22×Q8—24no—
126C22×Q8—24no—
127C22×Q8—24no—
128C22×Q8—24no—
129C22×Q8—42no—
130C22×Q8—42no—
131C22×Q8—82no—
132C22×Q8—82no—
133C22×Q8—82no—
134C22×Q8—82no—
135C23×C4—21no—
136C23×C4—21no—
137C23×C4—22no—
138C23×C4—21no—
139C23×C4—21no—
140C23×C4—22no—
141C23×C4C2×D411yescharacteristic
142C23×C4C2×D411yescharacteristic
143C23×C4C2×D411yescharacteristic
144C23×C4C2×D411yescharacteristic
145C23×C4C2×D411yescharacteristic
146C23×C4C2×D411yescharacteristic
147C23×C4C2411yescharacteristic
148C22×C8—22no—
149C22×C8—22no—
150C22×C8—22no—
151C22×C8—22no—
152C22×C8—22no—
153C22×C8—22no—
154C22×C8—22no—
155C22×C8—22no—
156C22×C8—22no—
157C22×C8—22no—
158C22×C8—22no—
159C22×C8—22no—
160C22×C8C2×Q814yes—
161C22×C8C2×Q814yes—
162C4⋊Q8—28no—
163C4⋊Q8—28no—
164C4⋊Q8—28no—
165C4⋊Q8—28no—
166C4⋊Q8—48no—
167C4⋊Q8—48no—
168C2×C4⋊C4—24no—
169C2×C4⋊C4—24no—
170C2×C4⋊C4—24no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—21no—
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—22no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—24no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—24no—
196C2×C4⋊C4—24no—
197C2×C4⋊C4—42no—
198C2×C4⋊C4—42no—
199C2×C4⋊C4—82no—
200C2×C4⋊C4—82no—
201C2×C4⋊C4C2×D412yes—
202C2×C4⋊C4C2×D412yes—
203C2×C4⋊C4C2×D412yes—
204C2×C4⋊C4C2×D412yes—
205C2×C4⋊C4C2×Q812yes—
206C2×C4⋊C4C2×Q812yes—
207C2×C4⋊C4#4114yes—
208C2×C4⋊C4#4112yes—
209C2×C4⋊C4#4112yes—
210C2×C42—21no—
211C2×C42—21no—
212C2×C42—21no—
213C2×C42—21no—
214C2×C42—21no—
215C2×C42—21no—
216C2×C42—22no—
217C2×C42—22no—
218C2×C42—22no—
219C2×C42—24no—
220C2×C42—24no—
221C2×C42—82no—
222C2×C42—82no—
223C2×C42C2×D412yes—
224C2×C42C2×D412yes—
225C2×C42C2×Q812yes—
226C2×C42C2×Q814yes—
227C2×C42#4112yes—
228#107—48no—
229#106—48no—
230#95—24no—
231#95—24no—
232#95—24no—
233#95—24no—
234#95—24no—
235#95—24no—
236#95—22no—
237#95—22no—
238#95—22no—
239#95—22no—
240#95—42no—
241#95—42no—
242#95—82no—
243#95—82no—
244#95—82no—
245#95—82no—
246C24C22×D411yescharacteristic
247C2×Q8—216no—
248C2×Q8—48no—
249C2×Q8—88no—
250C2×Q8—84no—
251C2×Q8—84no—
252C2×Q8—88no—
253C2×Q8—88no—
254C2×Q8—84no—
255C2×Q8—84no—
256C2×Q8—88no—
257C22×C4—21no—
258C22×C4—21no—
259C22×C4—21no—
260C22×C4—21no—
261C22×C4—22no—
262C22×C4—22no—
263C22×C4—22no—
264C22×C4—22no—
265C22×C4—21no—
266C22×C4—21no—
267C22×C4—21no—
268C22×C4—21no—
269C22×C4—22no—
270C22×C4—22no—
271C22×C4—21no—
272C22×C4—21no—
273C22×C4—21no—
274C22×C4—21no—
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—22no—
285C22×C4—22no—
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—22no—
289C22×C4—24no—
290C22×C4—42no—
291C22×C4—42no—
292C22×C4—42no—
293C22×C4—42no—
294C22×C4—42no—
295C22×C4—41no—
296C22×C4—41no—
297C22×C4—42no—
298C22×C4—41no—
299C22×C4—41no—
300C22×C4—42no—
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—42no—
312C22×C4—42no—
313C22×C4—42no—
314C22×C4—42no—
315C22×C4#12012yes—
316C22×C4#12012yes—
317C22×C4C22⋊Q812yes—
318C22×C4C22⋊Q812yes—
319C22×C4C4⋊Q812yes—
320C22×C4C4⋊Q812yes—
321C22×C4C22×D411yescharacteristic
322C22×C4C22×D411yescharacteristic
323C22×C4C22×Q811yescharacteristic
324C22×C4C22×Q812yes—
325C22×C4C2×#4111yescharacteristic
326C22×C4C2×#4111yescharacteristic
327C22×C4C2×#4111yescharacteristic
328C22×C4C2×#4112yes—
329C22×C4H(5,𝔽2)12yes—
330C22×C42−1+412yes—
331C2×C8—24no—
332C2×C8—24no—
333C2×C8—24no—
334C2×C8—24no—
335C2×C8—24no—
336C2×C8—24no—
337C2×C8—42no—
338C2×C8—42no—
339C2×C8—42no—
340C2×C8—42no—
341C2×C8—42no—
342C2×C8—42no—
343C2×C8—42no—
344C2×C8—42no—
345C4⋊C4—216no—
346C4⋊C4—22no—
347C4⋊C4—22no—
348C4⋊C4—22no—
349C4⋊C4—22no—
350C4⋊C4—22no—
351C4⋊C4—22no—
352C4⋊C4—24no—
353C4⋊C4—24no—
354C4⋊C4—24no—
355C4⋊C4—24no—
356C4⋊C4—48no—
357C4⋊C4—42no—
358C4⋊C4—42no—
359C4⋊C4—42no—
360C4⋊C4—42no—
361C4⋊C4—42no—
362C4⋊C4—42no—
363C4⋊C4—42no—
364C4⋊C4—42no—
365C4⋊C4—84no—
366C4⋊C4—84no—
367C4⋊C4C22⋊Q814yes—
368C4⋊C4C22⋊Q814yes—
369C42—22no—
370C42—22no—
371C42—22no—
372C42—22no—
373C42—24no—
374C42—24no—
375C42—24no—
376C42—24no—
377C42—84no—
378C42—84no—
379C23C2×#12011yescharacteristic
380C23C2×C22⋊Q811yescharacteristic
381C23#52511yescharacteristic
382C23#52511yescharacteristic
383C23C2×C4⋊Q811yescharacteristic
384C23#53511yescharacteristic
385C23#53611yescharacteristic
386C23#57612yes—
387C23#57712yes—
388C23#57712yes—
389C23#57812yes—
390Q8—816no—
391Q8—816no—
392C2×C4—24no—
393C2×C4—24no—
394C2×C4—22no—
395C2×C4—22no—
396C2×C4—24no—
397C2×C4—24no—
398C2×C4—24no—
399C2×C4—24no—
400C2×C4—22no—
401C2×C4—22no—
402C2×C4—22no—
403C2×C4—22no—
404C2×C4—22no—
405C2×C4—22no—
406C2×C4—22no—
407C2×C4—22no—
408C2×C4—44no—
409C2×C4—44no—
410C2×C4—42no—
411C2×C4—42no—
412C2×C4—44no—
413C2×C4—44no—
414C2×C4—42no—
415C2×C4—42no—
416C2×C4—42no—
417C2×C4—42no—
418C2×C4—42no—
419C2×C4—42no—
420C2×C4—41no—
421C2×C4—42no—
422C2×C4—42no—
423C2×C4—41no—
424C2×C4—41no—
425C2×C4—42no—
426C2×C4—42no—
427C2×C4—41no—
428C2×C4—41no—
429C2×C4—41no—
430C2×C4—41no—
431C2×C4—41no—
432C2×C4—44no—
433C2×C4—44no—
434C2×C4—44no—
435C2×C4—42no—
436C2×C4—42no—
437C2×C4—42no—
438C2×C4—42no—
439C2×C4—42no—
440C2×C4—42no—
441C2×C4—82no—
442C2×C4—82no—
443C2×C4—82no—
444C2×C4—82no—
445C2×C4—82no—
446C2×C4—82no—
447C2×C4—82no—
448C2×C4—82no—
449C2×C4—82no—
450C2×C4—82no—
451C2×C4—82no—
452C2×C4—82no—
453C2×C4—82no—
454C2×C4—82no—
455C2×C4—82no—
456C2×C4C2×C22⋊Q811yescharacteristic
457C2×C4C2×C22⋊Q811yescharacteristic
458C2×C4#52511yescharacteristic
459C2×C4#52511yescharacteristic
460C2×C4#53312yes—
461C2×C4#53312yes—
462C2×C4#54212yes—
463C2×C4#54212yes—
464C2×C4#54612yes—
465C2×C4#54612yes—
466C2×C4D4×Q812yes—
467C2×C4D4×Q812yes—
468C2×C4#55412yes—
469C2×C4#55612yes—
470C2×C4#55712yes—
471C2×C4Q8212yes—
472C8—44no—
473C8—44no—
474C22#239911yescharacteristic
475C22#240411yescharacteristic
476C22#243411yescharacteristic
477C22#256711yescharacteristic
478C22#258111yescharacteristic
479C22#263211yescharacteristic
480C22#284811yescharacteristic
481C22#300811yescharacteristic
482C22#300911yescharacteristic
483C22#301011yescharacteristic
484C22#301111yescharacteristic
485C22#308112yes—
486C22#308212yes—
487C22#311812yes—
488C22#311812yes—
489C22#312112yes—
490C22#312112yes—
491C22#316312yes—
492C22#316412yes—
493C22#322412yes—
494C22#322512yes—
495C22#324312yes—
496C22#324412yes—
497C4—42no—
498C4—42no—
499C4—42no—
500C4—44no—
501C4—44no—
502C4—42no—
503C4—42no—
504C4—42no—
505C4—82no—
506C4—84no—
507C4—82no—
508C4—162no—
509C4#245112yes—
510C4#245112yes—
511C4#246512yes—
512C4#246512yes—
513C2#1488111yescharacteristic
514C2#2384212yes—
515C2#2394511yescharacteristic
516C2#2394611yescharacteristic
517C2#2582011yescharacteristic
518C2#2582111yescharacteristic
519C2#2921712yes—
520C2#2928112yes—
521C2#3006812yes—
522C2#3296211yescharacteristic
523C2#3296411yescharacteristic
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

Complements

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

NH ≅ G/Ntype
60. C2×C4⋊Q8Q8semidirect
124. #336Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#23312
C2×#20582
C2×#20572
#142822
#113842
#113252
#112082
#109082
#109072
#107962