Finite Group Databasebeta

← #382642

Subgroups — #382642

Order 512.

1392 conjugacy classes of subgroups.

The subgroup lattice has 575 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2319C211yesmaximal, characteristic
2C2×#1843C211yesmaximal, characteristic, complemented
3C2×#1804C211yesmaximal, characteristic
4C2×#1802C211yesmaximal, characteristic
5#14348C211yesmaximal, characteristic, complemented
6#11228C212yesmaximal, complemented
7#11227C212yesmaximal, complemented
8#11076C212yesmaximal
9#11076C212yesmaximal
10#10850C211yesmaximal, characteristic, complemented
11#10849C211yesmaximal, characteristic, complemented
12C22×C4×D4C2211yescharacteristic
13C22×C4⋊C8C2211yescharacteristic
14C22×C4⋊C8C2211yescharacteristic
15C22×C4⋊C8C2212yes—
16C22×C4⋊C8C2212yes—
17C22×#102C2212yes—
18C22×#102C2211yescharacteristic
19C22×#102C2211yescharacteristic
20#2319—24no—
21#2319C418yescomplemented
22C2×#390C2212yes—
23C2×#389C2212yes—
24C2×#388—22no—
25C2×#384C2211yescharacteristic
26C2×#382C2211yescharacteristic
27C2×#381—21no—
28C2×#380—21no—
29C2×C4×C4⋊C4C2211yescharacteristic
30C2×#376C2211yescharacteristic
31#1843—24no—
32#1804—24no—
33#1802—24no—
34C2×#340C2212yes—
35C2×#340C2211yescharacteristic
36C2×#340C2211yescharacteristic
37C2×#340C2212yes—
38C2×#340C2212yes—
39C2×#340C2212yes—
40C2×#340C2212yes—
41#1442—21no—
42#1442—21no—
43#1442—22no—
44C4⋊#95C2212yes—
45C4⋊#95C2212yes—
46C23×D4C2311yescharacteristic
47C24×C4—21no—
48C23×C8C2312yes—
49C23×C8C2311yescharacteristic
50C23×C8C2311yescharacteristic
51C2×C4×D4—22no—
52C2×C4×D4—22no—
53C2×C4×D4—22no—
54C2×C4×D4C2×C414yes—
55C2×C4×D4D414yes—
56C2×C4×D4Q814yes—
57C22×C4⋊C4—22no—
58C22×C4⋊C4—21no—
59C22×C4⋊C4—21no—
60C22×C4⋊C4C2311yescharacteristic
61C22×C4⋊C4C2311yescharacteristic
62C22×C4⋊C4C2312yes—
63C22×C4⋊C4C2312yes—
64C22×C4⋊C4C2311yescharacteristic
65C22×C22⋊C4—21no—
66C22×C22⋊C4—22no—
67C22×C22⋊C4—22no—
68C22×C42—21no—
69C22×C42C2312yes—
70C22×C42C2311yescharacteristic
71C2×C4⋊C8—22no—
72C2×C4⋊C8—22no—
73C2×C4⋊C8—24no—
74C2×C4⋊C8—24no—
75C2×C4⋊C8—22no—
76C2×C4⋊C8—22no—
77C2×C4⋊C8—22no—
78C2×C4⋊C8—22no—
79C2×C4⋊C8—42no—
80C2×C4⋊C8—42no—
81C2×C4⋊C8—42no—
82C2×C4⋊C8—42no—
83C2×#102—24no—
84C2×#102—22no—
85C2×#102—22no—
86C2×#102—42no—
87C2×#102—42no—
88C2×#102—42no—
89C2×#102—42no—
90#390—24no—
91#390C2×C418yes—
92#389—24no—
93#389C2×C418yes—
94#388—216no—
95#384—22no—
96#384—22no—
97#382—22no—
98#382—22no—
99#381—28no—
100#380—28no—
101C4×C4⋊C4—22no—
102C4×C4⋊C4C2×C414yes—
103#376C2×C414yes—
104#376D414yes—
105C2×#95—21no—
106C2×#95—22no—
107C2×#95—21no—
108C2×#95—21no—
109C2×#95—22no—
110C2×#95—22no—
111C2×#95—22no—
112C2×#95—22no—
113C2×#95D411yescharacteristic
114C2×#95D411yescharacteristic
115C2×#95D412yes—
116#340—24no—
117#340—22no—
118#340—22no—
119#340—42no—
120#340—42no—
121#340—42no—
122#340—42no—
123C25—21no—
124C22×D4—21no—
125C22×D4—21no—
126C22×D4—21no—
127C22×D4—22no—
128C22×D4—22no—
129C22×D4—22no—
130C22×D4—22no—
131C22×D4C22×C412yes—
132C22×D4C2×D412yes—
133C22×D4C2×Q812yes—
134C23×C4—21no—
135C23×C4—24no—
136C23×C4—24no—
137C23×C4—24no—
138C23×C4—22no—
139C23×C4—22no—
140C23×C4—21no—
141C23×C4—21no—
142C23×C4—82no—
143C23×C4—82no—
144C23×C4C2×D412yes—
145C23×C4C2×D412yes—
146C23×C4C2×D411yescharacteristic
147C23×C4C2×D411yescharacteristic
148C23×C4C2411yescharacteristic
149C22×C8—22no—
150C22×C8—21no—
151C22×C8—21no—
152C22×C8—22no—
153C22×C8—21no—
154C22×C8—21no—
155C22×C8—22no—
156C22×C8—21no—
157C22×C8—21no—
158C22×C8—22no—
159C22×C8—22no—
160C22×C8—22no—
161C22×C8—22no—
162C22×C8—22no—
163C22×C8—22no—
164C22×C8—22no—
165C22×C8—22no—
166C4×D4—28no—
167C4×D4—28no—
168C4×D4—28no—
169C4×D4C4⋊C4116yes—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—28no—
193C2×C4⋊C4—42no—
194C2×C4⋊C4—42no—
195C2×C4⋊C4—82no—
196C2×C4⋊C4—82no—
197C2×C4⋊C4—81no—
198C2×C4⋊C4—82no—
199C2×C4⋊C4—81no—
200C2×C4⋊C4C22×C414yes—
201C2×C4⋊C4C22×C412yes—
202C2×C4⋊C4C2×D412yes—
203C2×C4⋊C4C2×Q812yes—
204C2×C22⋊C4—24no—
205C2×C22⋊C4—24no—
206C2×C22⋊C4—24no—
207C2×C22⋊C4—28no—
208C2×C22⋊C4—28no—
209C2×C22⋊C4—82no—
210C2×C22⋊C4—82no—
211C2×C22⋊C4—82no—
212C2×C22⋊C4—82no—
213C2×C42—22no—
214C2×C42—22no—
215C2×C42—22no—
216C2×C42—22no—
217C2×C42—24no—
218C2×C42—81no—
219C2×C42—81no—
220C2×C42C22×C414yes—
221C2×C42C22×C412yes—
222C2×C42C2×D414yes—
223C2×C42C2×D412yes—
224C2×C42C2×D412yes—
225C2×C42C2×Q812yes—
226C2×C42#4112yes—
227C2×C42#4112yes—
228C4⋊C8—48no—
229C4⋊C8—48no—
230C4⋊C8—44no—
231C4⋊C8—44no—
232#102—48no—
233#102—44no—
234#102—44no—
235#95—24no—
236#95—28no—
237#95—42no—
238#95—42no—
239#95—82no—
240#95—82no—
241#95—82no—
242#95—82no—
243#95C22⋊C414yes—
244#95C22⋊C414yes—
245#95C22⋊C418yes—
246C24—22no—
247C24—22no—
248C24—22no—
249C24—42no—
250C24—42no—
251C24—41no—
252C24—42no—
253C24—41no—
254C24—42no—
255C24—41no—
256C24—41no—
257C24C22×D411yescharacteristic
258C2×D4—22no—
259C2×D4—24no—
260C2×D4—24no—
261C2×D4—22no—
262C2×D4—22no—
263C2×D4—22no—
264C2×D4—22no—
265C2×D4—22no—
266C2×D4—22no—
267C2×D4—24no—
268C2×D4—24no—
269C2×D4—42no—
270C2×D4—42no—
271C2×D4—42no—
272C2×D4—42no—
273C2×D4—42no—
274C2×D4—42no—
275C2×D4C2×C4⋊C414yes—
276C22×C4—21no—
277C22×C4—22no—
278C22×C4—21no—
279C22×C4—21no—
280C22×C4—22no—
281C22×C4—21no—
282C22×C4—22no—
283C22×C4—22no—
284C22×C4—22no—
285C22×C4—22no—
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—22no—
289C22×C4—22no—
290C22×C4—22no—
291C22×C4—22no—
292C22×C4—22no—
293C22×C4—22no—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—216no—
298C22×C4—24no—
299C22×C4—42no—
300C22×C4—42no—
301C22×C4—42no—
302C22×C4—42no—
303C22×C4—42no—
304C22×C4—42no—
305C22×C4—42no—
306C22×C4—42no—
307C22×C4—41no—
308C22×C4—41no—
309C22×C4—42no—
310C22×C4—42no—
311C22×C4—42no—
312C22×C4—42no—
313C22×C4—42no—
314C22×C4—41no—
315C22×C4—42no—
316C22×C4—42no—
317C22×C4—42no—
318C22×C4—42no—
319C22×C4—41no—
320C22×C4—42no—
321C22×C4—41no—
322C22×C4—42no—
323C22×C4—41no—
324C22×C4—88no—
325C22×C4—88no—
326C22×C4—84no—
327C22×C4—84no—
328C22×C4C2×C22⋊C414yes—
329C22×C4C2×C22⋊C414yes—
330C22×C4C2×C22⋊C412yes—
331C22×C4C2×C22⋊C412yes—
332C22×C4#12112yes—
333C22×C4#12112yes—
334C22×C4C22⋊Q812yes—
335C22×C4C22⋊Q812yes—
336C22×C4C23×C411yescharacteristic
337C22×C4C22×D411yescharacteristic
338C22×C4C22×D411yescharacteristic
339C22×C4C22×Q811yescharacteristic
340C22×C4C2×#4111yescharacteristic
341C22×C4C2×#4111yescharacteristic
342C22×C4H(5,𝔽2)12yes—
343C22×C4H(5,𝔽2)12yes—
344C22×C42−1+412yes—
345C22×C42−1+412yes—
346C2×C8—42no—
347C2×C8—41no—
348C2×C8—41no—
349C2×C8—42no—
350C2×C8—41no—
351C2×C8—41no—
352C2×C8—42no—
353C2×C8—42no—
354C2×C8—42no—
355C2×C8—42no—
356C2×C8—41no—
357C2×C8—41no—
358C2×C8—42no—
359C2×C8—42no—
360C2×C8—42no—
361C2×C8—42no—
362C2×C8—42no—
363C4⋊C4—22no—
364C4⋊C4—22no—
365C4⋊C4—22no—
366C4⋊C4—24no—
367C4⋊C4—24no—
368C4⋊C4—42no—
369C4⋊C4—42no—
370C4⋊C4—41no—
371C4⋊C4—41no—
372C4⋊C4—42no—
373C4⋊C4—41no—
374C4⋊C4—41no—
375C4⋊C4—42no—
376C4⋊C4—42no—
377C4⋊C4—42no—
378C4⋊C4—42no—
379C4⋊C4—42no—
380C4⋊C4—88no—
381C4⋊C4—84no—
382C4⋊C4—84no—
383C4⋊C4C2×C4⋊C414yes—
384C22⋊C4—216no—
385C22⋊C4—88no—
386C22⋊C4—88no—
387C42—24no—
388C42—24no—
389C42—24no—
390C42—24no—
391C42—84no—
392C42C2×C4⋊C414yes—
393C42#11714yes—
394C42#12114yes—
395C42C22⋊Q814yes—
396C23—24no—
397C23—44no—
398C23—44no—
399C23—42no—
400C23—42no—
401C23—42no—
402C23—42no—
403C23—44no—
404C23—42no—
405C23—44no—
406C23—42no—
407C23—42no—
408C23—42no—
409C23—42no—
410C23—42no—
411C23—82no—
412C23—82no—
413C23—82no—
414C23—82no—
415C23—82no—
416C23—82no—
417C23—82no—
418C23—82no—
419C23C22×C22⋊C411yescharacteristic
420C23C2×#12111yescharacteristic
421C23C2×C22⋊Q811yescharacteristic
422C23#53712yes—
423C23#53712yes—
424C23#57612yes—
425C23#57611yescharacteristic
426C23#57712yes—
427C23#57711yescharacteristic
428C23#57711yescharacteristic
429C23#57811yescharacteristic
430D4—24no—
431D4—24no—
432D4—44no—
433D4—42no—
434D4—42no—
435D4—44no—
436C2×C4—24no—
437C2×C4—22no—
438C2×C4—22no—
439C2×C4—22no—
440C2×C4—24no—
441C2×C4—22no—
442C2×C4—22no—
443C2×C4—22no—
444C2×C4—22no—
445C2×C4—22no—
446C2×C4—24no—
447C2×C4—24no—
448C2×C4—24no—
449C2×C4—24no—
450C2×C4—24no—
451C2×C4—24no—
452C2×C4—24no—
453C2×C4—44no—
454C2×C4—44no—
455C2×C4—42no—
456C2×C4—42no—
457C2×C4—42no—
458C2×C4—42no—
459C2×C4—42no—
460C2×C4—42no—
461C2×C4—42no—
462C2×C4—42no—
463C2×C4—42no—
464C2×C4—42no—
465C2×C4—44no—
466C2×C4—44no—
467C2×C4—44no—
468C2×C4—816no—
469C2×C4—82no—
470C2×C4—82no—
471C2×C4—82no—
472C2×C4—82no—
473C2×C4—82no—
474C2×C4—81no—
475C2×C4—81no—
476C2×C4—82no—
477C2×C4—82no—
478C2×C4—82no—
479C2×C4—82no—
480C2×C4—82no—
481C2×C4—82no—
482C2×C4—82no—
483C2×C4—82no—
484C2×C4—82no—
485C2×C4—81no—
486C2×C4—81no—
487C2×C4—82no—
488C2×C4—81no—
489C2×C4—81no—
490C2×C4#38014yes—
491C2×C4#38014yes—
492C2×C4C22×C4⋊C411yescharacteristic
493C2×C4C2×#11711yescharacteristic
494C2×C4#52012yes—
495C2×C4#52012yes—
496C2×C4C2×#12111yescharacteristic
497C2×C4C2×C22⋊Q811yescharacteristic
498C2×C4#53511yescharacteristic
499C2×C4#53511yescharacteristic
500C2×C4#53512yes—
501C2×C4#53611yescharacteristic
502C2×C4#53611yescharacteristic
503C2×C4#53612yes—
504C2×C4#54012yes—
505C2×C4#54112yes—
506C2×C4#54212yes—
507C2×C4#54212yes—
508C2×C4#54412yes—
509C2×C4#54412yes—
510C8—82no—
511C8—81no—
512C8—81no—
513C22—44no—
514C22—44no—
515C22—44no—
516C22—84no—
517C22—84no—
518C22—84no—
519C22—84no—
520C22—82no—
521C22—82no—
522C22—82no—
523C22—82no—
524C22C2×#38011yescharacteristic
525C22#231112yes—
526C22#262011yescharacteristic
527C22#262011yescharacteristic
528C22#262912yes—
529C22#289712yes—
530C22#289711yescharacteristic
531C22#289811yescharacteristic
532C22#304111yescharacteristic
533C22#304211yescharacteristic
534C22#304511yescharacteristic
535C22#304611yescharacteristic
536C22#308112yes—
537C22#308311yescharacteristic
538C22#308411yescharacteristic
539C22#325712yes—
540C22#325712yes—
541C22#325712yes—
542C22#325812yes—
543C22#325912yes—
544C22#326012yes—
545C22#326212yes—
546C22#326312yes—
547C4—24no—
548C4—24no—
549C4—44no—
550C4—44no—
551C4—44no—
552C4—84no—
553C4—162no—
554C4—161no—
555C4—161no—
556C4#231412yes—
557C4#231711yescharacteristic
558C4#231711yescharacteristic
559C4#261911yescharacteristic
560C4#262111yescharacteristic
561C4#262812yes—
562C2—84no—
563C2#1434011yescharacteristic
564C2#2065011yescharacteristic
565C2#2065211yescharacteristic
566C2#2189612yes—
567C2#2189712yes—
568C2#2983411yescharacteristic
569C2#2983611yescharacteristic
570C2#2983811yescharacteristic
571C2#2983911yescharacteristic
572C2#3000212yes—
573C2#3000312yes—
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC23×D4C23
LCSG ▷ C22×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
2. C2×#1843C2semidirect
5. #14348C2semidirect
6. #11228C2semidirect
7. #11227C2semidirect
10. #10850C2semidirect
11. #10849C2semidirect
21. #2319C4semidirect

Maximal subgroups

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

subgroupindex
C2×#23192
C2×#18432
C2×#18042
C2×#18022
#143482
#112282
#112272
#110762
#108502
#108492