Finite Group Databasebeta

← #417208

Subgroups — #417208

Order 512.

1258 conjugacy classes of subgroups.

The subgroup lattice has 526 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1959C211yesmaximal, characteristic
2C2×#1958C211yesmaximal, characteristic
3C2×#1836C211yesmaximal, characteristic, complemented
4C2×#1834C211yesmaximal, characteristic, complemented
5#14354C211yesmaximal, characteristic, complemented
6#14273C211yesmaximal, characteristic
7#11052C212yesmaximal, complemented
8#11051C212yesmaximal
9#11050C212yesmaximal, complemented
10#11049C212yesmaximal
11#10694C211yesmaximal, characteristic, complemented
12C22×#124C2211yescharacteristic
13C22×#103C2211yescharacteristic
14C22×#103C2212yes—
15C22×#103C2211yescharacteristic
16C22×#102C2211yescharacteristic
17C22×#102C2212yes—
18C22×#102C2211yescharacteristic
19C22×#97C2211yescharacteristic
20C22×C4×C8C2211yescharacteristic
21C2×#391C2212yes—
22C2×#390C2212yes—
23C2×#384C2211yescharacteristic
24C2×#384C2211yescharacteristic
25C2×#384C2212yes—
26C2×#382C2211yescharacteristic
27C2×#382C2212yes—
28C2×#376C2211yescharacteristic
29#1959—24no—
30#1958—24no—
31#1836—24no—
32#1834—24no—
33C2×#340C2212yes—
34C2×#340C2212yes—
35C2×#340C2212yes—
36C2×#340C2212yes—
37C2×#336C2212yes—
38C2×#336C2212yes—
39#1444—22no—
40#1444—22no—
41#1439—22no—
42#1439—22no—
43C23×Q8C2311yescharacteristic
44C23×D4C2311yescharacteristic
45C23×C8C2311yescharacteristic
46C23×C8C2311yescharacteristic
47C23×C8C2312yes—
48C2×#124—22no—
49C2×#124—22no—
50C2×#124—22no—
51C2×#124—22no—
52C2×#124—22no—
53C2×#124Q814yes—
54C22×C4⋊C4—21no—
55C22×C4⋊C4—21no—
56C22×C4⋊C4—24no—
57C22×C4⋊C4—21no—
58C22×C4⋊C4—21no—
59C22×C4⋊C4C2311yescharacteristic
60C22×C4⋊C4C2311yescharacteristic
61C22×C4⋊C4C2312yes—
62C22×C4⋊C4C2312yes—
63C22×C22⋊C4—21no—
64C22×C22⋊C4—21no—
65C22×C22⋊C4—24no—
66C22×C42C2312yes—
67C22×C42C2311yescharacteristic
68C2×#103—22no—
69C2×#103—22no—
70C2×#103—24no—
71C2×#103—42no—
72C2×#103—42no—
73C2×#103—42no—
74C2×#103—42no—
75C2×#102—22no—
76C2×#102—22no—
77C2×#102—24no—
78C2×#102—42no—
79C2×#102—42no—
80C2×#102—42no—
81C2×#102—42no—
82C2×#97—22no—
83C2×#97—22no—
84C2×#97D414yes—
85C2×C4×C8—22no—
86C2×C4×C8—22no—
87C2×C4×C8D414yes—
88#391—24no—
89#391—24no—
90#390—24no—
91#390—24no—
92#384—22no—
93#384—22no—
94#384—22no—
95#384—22no—
96#384—42no—
97#384—42no—
98#382—22no—
99#382—22no—
100#382—42no—
101#382—42no—
102#376—22no—
103#376—22no—
104C2×#95—24no—
105C2×#95—24no—
106C2×#95—21no—
107C2×#95—21no—
108C2×#95D412yes—
109C2×#95D412yes—
110#340—24no—
111#340—24no—
112#340—24no—
113#340—24no—
114#336—24no—
115#336—24no—
116C25—21no—
117C22×Q8—21no—
118C22×Q8—21no—
119C22×Q8—21no—
120C22×Q8—21no—
121C22×Q8—21no—
122C22×Q8—22no—
123C22×Q8—22no—
124C22×Q8—22no—
125C22×Q8—22no—
126C22×Q8C2×Q812yes—
127C22×D4—21no—
128C22×D4—21no—
129C22×D4—21no—
130C22×D4—21no—
131C22×D4—22no—
132C22×D4—22no—
133C22×D4—21no—
134C22×D4—22no—
135C22×D4—22no—
136C22×D4C2×Q812yes—
137C23×C4—21no—
138C23×C4—22no—
139C23×C4—21no—
140C23×C4—22no—
141C23×C4—21no—
142C23×C4C2×D414yes—
143C23×C4C2×D411yescharacteristic
144C23×C4C2×D411yescharacteristic
145C23×C4C2411yescharacteristic
146C22×C8—22no—
147C22×C8—22no—
148C22×C8—22no—
149C22×C8—22no—
150C22×C8—21no—
151C22×C8—21no—
152C22×C8—21no—
153C22×C8—21no—
154C22×C8—22no—
155C22×C8—22no—
156C22×C8—22no—
157C22×C8—22no—
158C22×C8—22no—
159C22×C8—22no—
160C22×C8C2×D412yes—
161C22×C8C2×D412yes—
162C22×C8C2×D414yes—
163#124—28no—
164#124—28no—
165#124—44no—
166#124—44no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—22no—
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—22no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—48no—
188C2×C4⋊C4—42no—
189C2×C4⋊C4—42no—
190C2×C4⋊C4—42no—
191C2×C4⋊C4—42no—
192C2×C4⋊C4—82no—
193C2×C4⋊C4—82no—
194C2×C4⋊C4—82no—
195C2×C4⋊C4—82no—
196C2×C4⋊C4—84no—
197C2×C4⋊C4—84no—
198C2×C22⋊C4—24no—
199C2×C22⋊C4—24no—
200C2×C22⋊C4—42no—
201C2×C22⋊C4—42no—
202C2×C22⋊C4—42no—
203C2×C22⋊C4—42no—
204C2×C22⋊C4—48no—
205C2×C22⋊C4—84no—
206C2×C22⋊C4—84no—
207C2×C42—22no—
208C2×C42—22no—
209C2×C42—22no—
210C2×C42—22no—
211C2×C42C2×D412yes—
212C2×C42C2×D412yes—
213C2×C42C2×Q812yes—
214C2×C42#4114yes—
215C2×C42#4114yes—
216C2×C42#4112yes—
217C2×C42#4112yes—
218C2×C42#4112yes—
219#103—44no—
220#103—44no—
221#103—48no—
222#102—44no—
223#102—44no—
224#102—48no—
225#97—28no—
226C4×C8—28no—
227#95—24no—
228#95—24no—
229#95—48no—
230#95—48no—
231#95—42no—
232#95—42no—
233C24—22no—
234C24—41no—
235C24—42no—
236C24—41no—
237C24—42no—
238C24—42no—
239C24—42no—
240C24—42no—
241C24—42no—
242C24C22×D411yescharacteristic
243C2×Q8—22no—
244C2×Q8—22no—
245C2×Q8—24no—
246C2×Q8—24no—
247C2×Q8—41no—
248C2×Q8—41no—
249C2×Q8—42no—
250C2×Q8—42no—
251C2×Q8—41no—
252C2×Q8—41no—
253C2×Q8—42no—
254C2×Q8—41no—
255C2×Q8—41no—
256C2×Q8—42no—
257C2×Q8—42no—
258C2×Q8—42no—
259C2×Q8—42no—
260C2×Q8—42no—
261C2×D4—22no—
262C2×D4—22no—
263C2×D4—24no—
264C2×D4—24no—
265C2×D4—41no—
266C2×D4—41no—
267C2×D4—41no—
268C2×D4—41no—
269C2×D4—42no—
270C2×D4—41no—
271C2×D4—41no—
272C2×D4—42no—
273C2×D4—42no—
274C2×D4—42no—
275C2×D4—42no—
276C2×D4—42no—
277C2×D4—42no—
278C2×D4—42no—
279C22×C4—22no—
280C22×C4—22no—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—24no—
284C22×C4—24no—
285C22×C4—24no—
286C22×C4—24no—
287C22×C4—24no—
288C22×C4—24no—
289C22×C4—22no—
290C22×C4—41no—
291C22×C4—42no—
292C22×C4—41no—
293C22×C4—42no—
294C22×C4—42no—
295C22×C4—42no—
296C22×C4—41no—
297C22×C4—41no—
298C22×C4—42no—
299C22×C4—42no—
300C22×C4—42no—
301C22×C4—42no—
302C22×C4—42no—
303C22×C4—42no—
304C22×C4—44no—
305C22×C4—44no—
306C22×C4—42no—
307C22×C4—44no—
308C22×C4—44no—
309C22×C4—42no—
310C22×C4—42no—
311C22×C4—42no—
312C22×C4C22⋊Q812yes—
313C22×C4C22⋊Q812yes—
314C22×C4#12312yes—
315C22×C4#12312yes—
316C22×C4#12418yes—
317C22×C4#12712yes—
318C22×C4#12712yes—
319C22×C4C22×D411yescharacteristic
320C22×C4C22×D411yescharacteristic
321C22×C4C22×Q811yescharacteristic
322C22×C4C2×#4111yescharacteristic
323C22×C4C2×#4111yescharacteristic
324C22×C4C2×#4111yescharacteristic
325C22×C4C2×#4112yes—
326C22×C4C2×#4112yes—
327C22×C4H(5,𝔽2)12yes—
328C22×C4H(5,𝔽2)12yes—
329C2×C8—22no—
330C2×C8—22no—
331C2×C8—24no—
332C2×C8—22no—
333C2×C8—22no—
334C2×C8—24no—
335C2×C8—22no—
336C2×C8—22no—
337C2×C8—24no—
338C2×C8—42no—
339C2×C8—42no—
340C2×C8—42no—
341C2×C8—42no—
342C2×C8—42no—
343C2×C8—42no—
344C2×C8—42no—
345C2×C8—42no—
346C4⋊C4—42no—
347C4⋊C4—42no—
348C4⋊C4—42no—
349C4⋊C4—42no—
350C4⋊C4—41no—
351C4⋊C4—41no—
352C4⋊C4—42no—
353C4⋊C4—41no—
354C4⋊C4—41no—
355C4⋊C4—42no—
356C4⋊C4—42no—
357C4⋊C4—42no—
358C4⋊C4—42no—
359C4⋊C4—42no—
360C4⋊C4—84no—
361C4⋊C4—84no—
362C4⋊C4—84no—
363C4⋊C4—84no—
364C4⋊C4—816no—
365C22⋊C4—48no—
366C22⋊C4—48no—
367C22⋊C4—816no—
368C42—24no—
369C42—24no—
370C42—24no—
371C42—24no—
372C42C22⋊Q814yes—
373C42C22⋊Q814yes—
374C42#12314yes—
375C42#12314yes—
376C23—44no—
377C23—44no—
378C23—44no—
379C23—42no—
380C23—82no—
381C23—82no—
382C23—81no—
383C23—84no—
384C23—84no—
385C23—81no—
386C23—81no—
387C23—81no—
388C23—81no—
389C23—81no—
390C23—84no—
391C23—84no—
392C23—82no—
393C23C2×C22⋊Q811yescharacteristic
394C23C2×#12311yescharacteristic
395C23C2×#12412yes—
396C23C2×#12711yescharacteristic
397C23#53512yes—
398C23#57612yes—
399C23#57712yes—
400C23#57712yes—
401C23#57812yes—
402Q8—42no—
403Q8—42no—
404Q8—44no—
405Q8—81no—
406Q8—81no—
407Q8—82no—
408D4—42no—
409D4—42no—
410D4—44no—
411D4—81no—
412D4—81no—
413D4—82no—
414C2×C4—24no—
415C2×C4—24no—
416C2×C4—24no—
417C2×C4—24no—
418C2×C4—22no—
419C2×C4—22no—
420C2×C4—22no—
421C2×C4—22no—
422C2×C4—28no—
423C2×C4—28no—
424C2×C4—28no—
425C2×C4—44no—
426C2×C4—44no—
427C2×C4—44no—
428C2×C4—44no—
429C2×C4—44no—
430C2×C4—42no—
431C2×C4—81no—
432C2×C4—81no—
433C2×C4—82no—
434C2×C4—82no—
435C2×C4—84no—
436C2×C4—84no—
437C2×C4—82no—
438C2×C4—82no—
439C2×C4—81no—
440C2×C4—81no—
441C2×C4—82no—
442C2×C4—82no—
443C2×C4—81no—
444C2×C4—81no—
445C2×C4—81no—
446C2×C4—82no—
447C2×C4—81no—
448C2×C4—84no—
449C2×C4—84no—
450C2×C4—84no—
451C2×C4—84no—
452C2×C4#39914yes—
453C2×C4#39914yes—
454C2×C4C2×C22⋊Q811yescharacteristic
455C2×C4C2×C22⋊Q811yescharacteristic
456C2×C4C2×#12311yescharacteristic
457C2×C4C2×#12311yescharacteristic
458C2×C4#53211yescharacteristic
459C2×C4#53211yescharacteristic
460C2×C4#53212yes—
461C2×C4#53511yescharacteristic
462C2×C4#53511yescharacteristic
463C2×C4#53512yes—
464C2×C4#55312yes—
465C2×C4#55312yes—
466C2×C4#55312yes—
467C2×C4#55312yes—
468C2×C4#55412yes—
469C2×C4#55412yes—
470C2×C4#55512yes—
471C2×C4#55512yes—
472C8—42no—
473C8—42no—
474C8—44no—
475C22—82no—
476C22—82no—
477C22—82no—
478C22—88no—
479C22—161no—
480C22—161no—
481C22—164no—
482C22—161no—
483C22—161no—
484C22C2×#39911yescharacteristic
485C22#250412yes—
486C22#256312yes—
487C22#265711yescharacteristic
488C22#285811yescharacteristic
489C22#308312yes—
490C22#308412yes—
491C22#309812yes—
492C22#309912yes—
493C22#313614yes—
494C22#313714yes—
495C22#315411yescharacteristic
496C22#315411yescharacteristic
497C22#315511yescharacteristic
498C22#315611yescharacteristic
499C22#322414yes—
500C22#322514yes—
501C4—24no—
502C4—24no—
503C4—48no—
504C4—82no—
505C4—162no—
506C4—161no—
507C4—161no—
508C4—162no—
509C4#250011yescharacteristic
510C4#250011yescharacteristic
511C4#250012yes—
512C4#256611yescharacteristic
513C4#256611yescharacteristic
514C4#256612yes—
515C2—162no—
516C2#1524411yescharacteristic
517C2#2373711yescharacteristic
518C2#2373811yescharacteristic
519C2#2798414yes—
520C2#2915114yes—
521C2#3027411yescharacteristic
522C2#3027511yescharacteristic
523C2#3322811yescharacteristic
524C2#3322911yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC23×D4C23
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
3. C2×#1836C2semidirect
4. C2×#1834C2semidirect
5. #14354C2semidirect
7. #11052C2semidirect
9. #11050C2semidirect
11. #10694C2semidirect

Maximal subgroups

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

subgroupindex
C2×#19592
C2×#19582
C2×#18362
C2×#18342
#143542
#142732
#110522
#110512
#110502
#110492
#106942