Finite Group Databasebeta

← #356195

Subgroups — #356195

Order 512.

1866 conjugacy classes of subgroups.

The subgroup lattice has 638 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2349C211yesmaximal, characteristic
2C2×#1804C211yesmaximal, characteristic
3C2×#1770C211yesmaximal, characteristic, complemented
4C2×#1762C212yesmaximal
5C2×C4×#102C211yesmaximal, characteristic
6#13804C211yesmaximal, characteristic, complemented
7#10884C212yesmaximal, complemented
8#10881C212yesmaximal, complemented
9#10850C211yesmaximal, characteristic, complemented
10#10842C211yesmaximal, characteristic, complemented
11#10788C212yesmaximal
12C22×C4×D4C2212yes—
13C22×C4×D4C2211yescharacteristic
14C22×C4⋊C8C2211yescharacteristic
15C22×#102C2211yescharacteristic
16C22×#102C2212yes—
17C22×#102C2211yescharacteristic
18C22×#97C2212yes—
19C22×C4×C8C2211yescharacteristic
20#2349—24no—
21#2349C418yescomplemented
22C2×#389C2212yes—
23C2×#385—22no—
24C2×#382C2211yescharacteristic
25C2×#382C2211yescharacteristic
26C2×#380—21no—
27C2×C4×C4⋊C4C2212yes—
28C2×C4×C4⋊C4C2211yescharacteristic
29C2×C4×C4⋊C4C2211yescharacteristic
30C22×#95—21no—
31#1804C418yescomplemented
32#1770—24no—
33#1762—28no—
34C4×#102C418yescomplemented
35C2×#340C2212yes—
36C2×#340C2212yes—
37C2×#340C2212yes—
38C2×#340C2212yes—
39C2×#340C2211yescharacteristic
40C2×#340C2212yes—
41C2×#340C2211yescharacteristic
42C2×#336C2212yes—
43#1436—22no—
44C4⋊#95C2212yes—
45#1434—21no—
46C4×#95—21no—
47C23×D4C2311yescharacteristic
48C24×C4—21no—
49C24×C4—22no—
50C23×C8C2311yescharacteristic
51C23×C8C2312yes—
52C23×C8C2311yescharacteristic
53C2×C4×D4—24no—
54C2×C4×D4—24no—
55C2×C4×D4—22no—
56C2×C4×D4—24no—
57C2×C4×D4—22no—
58C2×C4×D4—22no—
59C2×C4×D4—42no—
60C2×C4×D4—42no—
61C2×C4×D4C2×C418yes—
62C2×C4×D4C2×C414yes—
63C2×C4×D4C2×C414yes—
64C2×C4×D4C2×C414yes—
65C22×C4⋊C4—22no—
66C22×C4⋊C4D412yes—
67C22×C4⋊C4C2311yescharacteristic
68C22×C4⋊C4C2312yes—
69C22×C4⋊C4C2312yes—
70C22×C4⋊C4C2311yescharacteristic
71C22×C4⋊C4C2311yescharacteristic
72C22×C22⋊C4—21no—
73C22×C22⋊C4—22no—
74C22×C42—21no—
75C22×C42—22no—
76C22×C42C2312yes—
77C22×C42C2311yescharacteristic
78C2×C4⋊C8—22no—
79C2×C4⋊C8—22no—
80C2×C4⋊C8C2×C414yescomplemented
81C2×#102—24no—
82C2×#102—22no—
83C2×#102—22no—
84C2×#102—24no—
85C2×#102C2×C418yes—
86C2×#102C2×C414yes—
87C2×#102C2×C414yes—
88C2×#102D414yes—
89C2×#102Q814yes—
90C2×#97—24no—
91C2×#97—24no—
92C2×#97—42no—
93C2×C4×C8—22no—
94C2×C4×C8—22no—
95C2×C4×C8C2×C414yescomplemented
96#389—24no—
97#389C2×C418yes—
98#385—216no—
99#382—22no—
100#382—22no—
101#382C2×C414yes—
102#382C2×C414yescomplemented
103#380—28no—
104C4×C4⋊C4—22no—
105C4×C4⋊C4—24no—
106C4×C4⋊C4—42no—
107C4×C4⋊C4C2×C414yes—
108C4×C4⋊C4C2×C414yescomplemented
109C4×C4⋊C4D414yes—
110C2×#95—21no—
111C2×#95—21no—
112C2×#95—21no—
113C2×#95—21no—
114C2×#95—22no—
115C2×#95—21no—
116C2×#95—28no—
117C2×#95—22no—
118C2×#95—22no—
119C2×#95—22no—
120C2×#95D411yescharacteristic
121C2×#95D411yescharacteristic
122#340—24no—
123#340—24no—
124#340—42no—
125#340—42no—
126#340C2×C418yescomplemented
127#340C2×C414yescomplemented
128#340C2×C414yescomplemented
129#336—24no—
130C25—21no—
131C22×D4—21no—
132C22×D4—21no—
133C22×D4—21no—
134C22×D4—21no—
135C22×D4—21no—
136C22×D4—21no—
137C22×D4—21no—
138C22×D4C22×C412yes—
139C22×D4C22×C412yes—
140C22×D4C22×C412yes—
141C22×D4C2×D412yes—
142C22×D4C2×Q812yes—
143C22×D4#4112yes—
144C22×D4#4112yes—
145C23×C4—21no—
146C23×C4—21no—
147C23×C4—24no—
148C23×C4—24no—
149C23×C4—24no—
150C23×C4—22no—
151C23×C4—22no—
152C23×C4—21no—
153C23×C4—28no—
154C23×C4—44no—
155C23×C4—44no—
156C23×C4—44no—
157C23×C4—44no—
158C23×C4—82no—
159C23×C4—82no—
160C23×C4—82no—
161C23×C4—82no—
162C23×C4C2×D412yes—
163C23×C4C2×D412yes—
164C23×C4C2×D411yescharacteristic
165C23×C4C2×D411yescharacteristic
166C23×C4C2411yescharacteristic
167C22×C8—21no—
168C22×C8—21no—
169C22×C8—21no—
170C22×C8—21no—
171C22×C8—21no—
172C22×C8—22no—
173C22×C8—22no—
174C22×C8—21no—
175C22×C8—21no—
176C22×C8—21no—
177C22×C8—22no—
178C22×C8—22no—
179C22×C8—21no—
180C22×C8—22no—
181C22×C8—22no—
182C22×C8—21no—
183C22×C8C22×C412yes—
184C22×C8C22×C414yes—
185C22×C8C22×C412yes—
186C22×C8C2×D412yes—
187C22×C8C2×Q812yes—
188C4×D4—216no—
189C4×D4—28no—
190C4×D4—28no—
191C4×D4—28no—
192C4×D4—48no—
193C4×D4—48no—
194C4×D4—48no—
195C4×D4C42116yescomplemented
196C2×C4⋊C4—21no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—28no—
218C2×C4⋊C4—42no—
219C2×C4⋊C4—42no—
220C2×C4⋊C4—82no—
221C2×C4⋊C4—82no—
222C2×C4⋊C4C22⋊C418yes—
223C2×C4⋊C4C22×C414yes—
224C2×C4⋊C4C22×C412yes—
225C2×C4⋊C4C22×C414yes—
226C2×C4⋊C4C22×C412yes—
227C2×C4⋊C4C22×C412yes—
228C2×C4⋊C4C22×C412yes—
229C2×C4⋊C4C22×C412yes—
230C2×C4⋊C4C2×D412yes—
231C2×C4⋊C4C2×Q812yes—
232C2×C22⋊C4—24no—
233C2×C22⋊C4—24no—
234C2×C22⋊C4—24no—
235C2×C22⋊C4—28no—
236C2×C22⋊C4—44no—
237C2×C22⋊C4—44no—
238C2×C42—22no—
239C2×C42—22no—
240C2×C42—22no—
241C2×C42—22no—
242C2×C42—24no—
243C2×C42—44no—
244C2×C42—44no—
245C2×C42—82no—
246C2×C42—82no—
247C2×C42C22×C414yes—
248C2×C42C22×C412yes—
249C2×C42C22×C412yes—
250C2×C42C22×C412yes—
251C2×C42C2×D414yes—
252C2×C42C2×D412yes—
253C2×C42#4112yes—
254C2×C42#4112yes—
255C4⋊C8—28no—
256#102—28no—
257#102—216no—
258#102C4⋊C4116yescomplemented
259#97—48no—
260C4×C8—28no—
261#95—24no—
262#95—28no—
263#95—24no—
264#95—24no—
265#95—24no—
266#95—24no—
267#95—44no—
268#95—44no—
269#95—82no—
270#95C22⋊C414yes—
271#95C22⋊C414yes—
272C24—22no—
273C24—22no—
274C24—22no—
275C24—22no—
276C24—22no—
277C24—22no—
278C24—22no—
279C24—42no—
280C24—42no—
281C24—42no—
282C24—42no—
283C24C22×D411yescharacteristic
284C2×D4—22no—
285C2×D4—22no—
286C2×D4—22no—
287C2×D4—22no—
288C2×D4—22no—
289C2×D4—22no—
290C2×D4—22no—
291C2×D4—22no—
292C2×D4—22no—
293C2×D4—22no—
294C2×D4—22no—
295C2×D4—22no—
296C2×D4—22no—
297C2×D4—22no—
298C2×D4—22no—
299C2×D4—22no—
300C2×D4—22no—
301C2×D4—22no—
302C2×D4—22no—
303C2×D4—22no—
304C2×D4—22no—
305C2×D4C2×C4214yes—
306C2×D4C2×C4⋊C414yes—
307C2×D4#11714yes—
308C2×D4C4×D414yes—
309C2×D4C4×D414yes—
310C2×D4C4×Q814yes—
311C2×D4C4×Q814yes—
312C22×C4—22no—
313C22×C4—22no—
314C22×C4—22no—
315C22×C4—22no—
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—22no—
325C22×C4—22no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—216no—
335C22×C4—24no—
336C22×C4—24no—
337C22×C4—22no—
338C22×C4—416no—
339C22×C4—416no—
340C22×C4—42no—
341C22×C4—42no—
342C22×C4—42no—
343C22×C4—42no—
344C22×C4—42no—
345C22×C4—42no—
346C22×C4—42no—
347C22×C4—41no—
348C22×C4—41no—
349C22×C4—42no—
350C22×C4—42no—
351C22×C4—42no—
352C22×C4—42no—
353C22×C4—42no—
354C22×C4—42no—
355C22×C4—42no—
356C22×C4—42no—
357C22×C4—42no—
358C22×C4—42no—
359C22×C4—42no—
360C22×C4—88no—
361C22×C4—88no—
362C22×C4—84no—
363C22×C4—84no—
364C22×C4—88no—
365C22×C4—88no—
366C22×C4—88no—
367C22×C4—88no—
368C22×C4—88no—
369C22×C4C2×C22⋊C414yes—
370C22×C4C2×C22⋊C414yes—
371C22×C4C2×C22⋊C412yes—
372C22×C4C2×C22⋊C412yes—
373C22×C4C4×D412yes—
374C22×C4C4×D412yes—
375C22×C4C4×D412yes—
376C22×C4C4×D412yes—
377C22×C4C23×C411yescharacteristic
378C22×C4C23×C411yescharacteristic
379C22×C4C23×C411yescharacteristic
380C22×C4C22×D411yescharacteristic
381C22×C4C22×D411yescharacteristic
382C22×C4C22×Q811yescharacteristic
383C22×C4C2×#4111yescharacteristic
384C22×C4C2×#4111yescharacteristic
385C22×C4C2×#4111yescharacteristic
386C22×C4C2×#4111yescharacteristic
387C22×C4H(5,𝔽2)11yescharacteristic
388C22×C4H(5,𝔽2)11yescharacteristic
389C22×C42−1+411yescharacteristic
390C22×C42−1+411yescharacteristic
391C2×C8—22no—
392C2×C8—22no—
393C2×C8—22no—
394C2×C8—22no—
395C2×C8—22no—
396C2×C8—22no—
397C2×C8—22no—
398C2×C8—22no—
399C2×C8—24no—
400C2×C8—24no—
401C2×C8—22no—
402C2×C8—24no—
403C2×C8—42no—
404C2×C8—41no—
405C2×C8—42no—
406C2×C8—41no—
407C2×C8—42no—
408C2×C8—41no—
409C2×C8—41no—
410C2×C8—42no—
411C2×C8C2×C4⋊C414yes—
412C4⋊C4—22no—
413C4⋊C4—22no—
414C4⋊C4—22no—
415C4⋊C4—22no—
416C4⋊C4—22no—
417C4⋊C4—22no—
418C4⋊C4—22no—
419C4⋊C4—24no—
420C4⋊C4—24no—
421C4⋊C4—24no—
422C4⋊C4—22no—
423C4⋊C4—24no—
424C4⋊C4—48no—
425C4⋊C4—42no—
426C4⋊C4—42no—
427C4⋊C4—42no—
428C4⋊C4—41no—
429C4⋊C4—41no—
430C4⋊C4—42no—
431C4⋊C4—88no—
432C4⋊C4C2×C4214yes—
433C4⋊C4C2×C4⋊C414yes—
434C22⋊C4—216no—
435C22⋊C4—416no—
436C42—24no—
437C42—24no—
438C42—24no—
439C42—24no—
440C42—84no—
441C42—88no—
442C42C2×C4214yes—
443C42#11714yes—
444C42C4×D414yes—
445C42C4×D414yes—
446C23—24no—
447C23—24no—
448C23—24no—
449C23—24no—
450C23—24no—
451C23—24no—
452C23—24no—
453C23—44no—
454C23—44no—
455C23—44no—
456C23—44no—
457C23—44no—
458C23—44no—
459C23—42no—
460C23—42no—
461C23—44no—
462C23—44no—
463C23—42no—
464C23—42no—
465C23—44no—
466C23—44no—
467C23—44no—
468C23—44no—
469C23C22×C22⋊C411yescharacteristic
470C23C2×C4×D411yescharacteristic
471C23C2×C4×D411yescharacteristic
472C23D4211yescharacteristic
473C23#54611yescharacteristic
474C23#54811yescharacteristic
475C23D4×Q811yescharacteristic
476C23#57512yes—
477C23#57512yes—
478C23#57612yes—
479C23#57712yes—
480D4—24no—
481D4—24no—
482D4—24no—
483D4—24no—
484D4—24no—
485D4—24no—
486D4—24no—
487D4C4×C4⋊C418yes—
488C2×C4—24no—
489C2×C4—24no—
490C2×C4—22no—
491C2×C4—22no—
492C2×C4—24no—
493C2×C4—24no—
494C2×C4—22no—
495C2×C4—22no—
496C2×C4—24no—
497C2×C4—24no—
498C2×C4—24no—
499C2×C4—24no—
500C2×C4—24no—
501C2×C4—24no—
502C2×C4—24no—
503C2×C4—24no—
504C2×C4—44no—
505C2×C4—44no—
506C2×C4—42no—
507C2×C4—44no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—44no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—44no—
521C2×C4—42no—
522C2×C4—44no—
523C2×C4—44no—
524C2×C4—44no—
525C2×C4—44no—
526C2×C4—42no—
527C2×C4—44no—
528C2×C4—42no—
529C2×C4—816no—
530C2×C4—832no—
531C2×C4—82no—
532C2×C4—82no—
533C2×C4—82no—
534C2×C4—82no—
535C2×C4—82no—
536C2×C4—82no—
537C2×C4—82no—
538C2×C4—82no—
539C2×C4—82no—
540C2×C4—82no—
541C2×C4C4×C22⋊C414yes—
542C2×C4C4×C22⋊C414yes—
543C2×C4C22×C4211yescharacteristic
544C2×C4C22×C4⋊C411yescharacteristic
545C2×C4C2×#11711yescharacteristic
546C2×C4C2×#11711yescharacteristic
547C2×C4C2×C4×D411yescharacteristic
548C2×C4C2×C4×D411yescharacteristic
549C2×C4C2×C4×D411yescharacteristic
550C2×C4C2×C4×D411yescharacteristic
551C2×C4C2×C4×Q811yescharacteristic
552C2×C4C2×C4×Q811yescharacteristic
553C2×C4#51811yescharacteristic
554C2×C4#51911yescharacteristic
555C2×C4#52011yescharacteristic
556C2×C4#52011yescharacteristic
557C2×C4#52011yescharacteristic
558C2×C4#52011yescharacteristic
559C2×C4D4211yescharacteristic
560C2×C4#54611yescharacteristic
561C2×C4#54711yescharacteristic
562C2×C4#54711yescharacteristic
563C2×C4#54811yescharacteristic
564C2×C4D4×Q811yescharacteristic
565C2×C4#55211yescharacteristic
566C2×C4#55211yescharacteristic
567C2×C4#55311yescharacteristic
568C2×C4#55311yescharacteristic
569C2×C4#55411yescharacteristic
570C2×C4#55411yescharacteristic
571C8—24no—
572C8—42no—
573C8—44no—
574C22—28no—
575C22—44no—
576C22—44no—
577C22—48no—
578C22—44no—
579C22—48no—
580C22—44no—
581C22—48no—
582C22—44no—
583C22—44no—
584C22—48no—
585C22C2×C4×C22⋊C411yescharacteristic
586C22D4×C22⋊C411yescharacteristic
587C22Q8×C22⋊C411yescharacteristic
588C22D4×C4⋊C411yescharacteristic
589C22#235211yescharacteristic
590C22#236011yescharacteristic
591C22#236111yescharacteristic
592C22#289612yes—
593C22#289712yes—
594C22#294311yescharacteristic
595C22#294311yescharacteristic
596C22#294312yes—
597C22#294612yes—
598C22#294811yescharacteristic
599C22#294911yescharacteristic
600C22#327312yes—
601C22#327512yes—
602C22#332212yes—
603C22#332312yes—
604C22#337012yes—
605C22#337612yes—
606C22#338812yes—
607C22#338912yes—
608C4—24no—
609C4—24no—
610C4—44no—
611C4—44no—
612C4—44no—
613C4—44no—
614C4—84no—
615C4—84no—
616C4—82no—
617C4C2×C4×C4⋊C411yescharacteristic
618C4#227411yescharacteristic
619C4#234411yescharacteristic
620C4#234511yescharacteristic
621C4D4×C4⋊C411yescharacteristic
622C4#235211yescharacteristic
623C4#236011yescharacteristic
624C4#236111yescharacteristic
625C2—48no—
626C2C22⋊C4×C4⋊C411yescharacteristic
627C2#2041111yescharacteristic
628C2#2041611yescharacteristic
629C2#2242712yes—
630C2#2252612yes—
631C2#2267211yescharacteristic
632C2#2267311yescharacteristic
633C2#2276011yescharacteristic
634C2#2276411yescharacteristic
635C2#2297212yes—
636C2#2300512yes—
subgroupG/H
center, socleC24C22×D4
commutatorC2×C4C22×C42
Frattini, cosocleC23×C4C24
prime radicalC23×D4C23
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. C2×#1770C2semidirect
6. #13804C2semidirect
7. #10884C2semidirect
8. #10881C2semidirect
9. #10850C2semidirect
10. #10842C2semidirect
21. #2349C4semidirect
31. #1804C4semidirect
34. C4×#102C4semidirect
80. C2×C4⋊C8C2×C4semidirect
95. C2×C4×C8C2×C4semidirect
102. #382C2×C4semidirect
108. C4×C4⋊C4C2×C4semidirect
126. #340C2×C4semidirect
127. #340C2×C4semidirect
128. #340C2×C4semidirect
195. C4×D4C42semidirect
258. #102C4⋊C4semidirect

Maximal subgroups

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

subgroupindex
C2×#23492
C2×#18042
C2×#17702
C2×#17622
C2×C4×#1022
#138042
#108842
#108812
#108502
#108422
#107882