Finite Group Databasebeta

← #362109

Subgroups — #362109

Order 512.

1562 conjugacy classes of subgroups.

The subgroup lattice has 597 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2351C211yesmaximal, characteristic
2C2×#1940C211yesmaximal, characteristic
3C2×#1803C212yesmaximal
4C2×#1803C211yesmaximal, characteristic
5C2×#1763C211yesmaximal, characteristic
6#13910C211yesmaximal, characteristic
7#11227C212yesmaximal
8#11207C212yesmaximal
9#10850C211yesmaximal, characteristic
10#10843C211yesmaximal, characteristic
11#10789C212yesmaximal
12C22×C4×Q8C2212yes—
13C22×C4×Q8C2211yescharacteristic
14C22×C4⋊C8C2211yescharacteristic
15C22×C4⋊C8C2212yes—
16C22×#103C2212yes—
17C22×#103C2211yescharacteristic
18C22×#103C2211yescharacteristic
19C22×#97C2211yescharacteristic
20#2351—24no—
21#2351C418yescomplemented
22C2×#389—22no—
23C2×#389—22no—
24C2×#389C2212yes—
25C2×#384C2211yescharacteristic
26C2×#384C2212yes—
27C2×#382—21no—
28C2×#382C2211yescharacteristic
29C2×C4×C4⋊C4—21no—
30C2×C4×C4⋊C4C2211yescharacteristic
31C2×C4×C4⋊C4C2211yescharacteristic
32#1940—24no—
33#1803—28no—
34#1803—24no—
35#1763—24no—
36C2×#340C2212yes—
37C2×#340C2212yes—
38C2×#340C2212yes—
39C2×#340C2211yescharacteristic
40C2×#340C2211yescharacteristic
41C2×#340C2212yes—
42C2×#340C2212yes—
43C2×#336C2212yes—
44#1444—21no—
45#1444—21no—
46C4⋊#95C2212yes—
47C23×Q8C2311yescharacteristic
48C23×C8C2312yes—
49C23×C8C2311yescharacteristic
50C23×C8C2311yescharacteristic
51C2×C4×Q8—22no—
52C2×C4×Q8—24no—
53C2×C4×Q8—22no—
54C2×C4×Q8—22no—
55C2×C4×Q8—24no—
56C2×C4×Q8—24no—
57C2×C4×Q8—42no—
58C2×C4×Q8—42no—
59C2×C4×Q8C2×C418yes—
60C2×C4×Q8C2×C414yes—
61C2×C4×Q8D414yes—
62C2×C4×Q8Q814yes—
63C22×C4⋊C4—21no—
64C22×C4⋊C4—22no—
65C22×C4⋊C4—22no—
66C22×C4⋊C4—21no—
67C22×C4⋊C4—22no—
68C22×C4⋊C4—22no—
69C22×C4⋊C4—21no—
70C22×C4⋊C4D412yes—
71C22×C4⋊C4C2312yes—
72C22×C4⋊C4C2312yes—
73C22×C4⋊C4C2311yescharacteristic
74C22×C4⋊C4C2311yescharacteristic
75C22×C4⋊C4C2311yescharacteristic
76C22×C42—21no—
77C22×C42—22no—
78C22×C42—21no—
79C22×C42C2312yes—
80C22×C42C2311yescharacteristic
81C2×C4⋊C8—24no—
82C2×C4⋊C8—22no—
83C2×C4⋊C8—24no—
84C2×C4⋊C8—22no—
85C2×C4⋊C8—22no—
86C2×C4⋊C8—42no—
87C2×#103—22no—
88C2×#103—22no—
89C2×#103—22no—
90C2×#103—22no—
91C2×#103—24no—
92C2×#103—24no—
93C2×#103—24no—
94C2×#103D414yes—
95C2×#103Q814yes—
96C2×#97—22no—
97C2×#97—22no—
98C2×#97—22no—
99#389—216no—
100#389—24no—
101#389—48no—
102#389C2×C418yes—
103#384—24no—
104#384—22no—
105#384—22no—
106#384—42no—
107#382—22no—
108#382—28no—
109#382C2×C414yes—
110C4×C4⋊C4—28no—
111C4×C4⋊C4—22no—
112C4×C4⋊C4—22no—
113C4×C4⋊C4C2×C414yes—
114C4×C4⋊C4D414yes—
115C2×#95—21no—
116C2×#95—22no—
117C2×#95—21no—
118C2×#95—22no—
119C2×#95—22no—
120C2×#95—21no—
121C2×#95D411yescharacteristic
122C2×#95D411yescharacteristic
123#340—24no—
124#340—24no—
125#340—24no—
126#340—22no—
127#340—22no—
128#340—42no—
129#340—42no—
130#336—24no—
131C22×Q8—21no—
132C22×Q8—21no—
133C22×Q8—21no—
134C22×Q8—21no—
135C22×Q8—21no—
136C22×Q8—21no—
137C22×Q8—21no—
138C22×Q8C22×C412yes—
139C22×Q8C2×D412yes—
140C22×Q8C2×D412yes—
141C22×Q8C2×Q812yes—
142C22×Q8C2×Q812yes—
143C22×Q8#4112yes—
144C22×Q8#4112yes—
145C23×C4—21no—
146C23×C4—21no—
147C23×C4—21no—
148C23×C4—22no—
149C23×C4—21no—
150C23×C4—22no—
151C23×C4C2×D412yes—
152C23×C4C2×D411yescharacteristic
153C23×C4C2×D411yescharacteristic
154C23×C4C2×D412yes—
155C23×C4C2411yescharacteristic
156C22×C8—21no—
157C22×C8—21no—
158C22×C8—21no—
159C22×C8—21no—
160C22×C8—21no—
161C22×C8—21no—
162C22×C8—21no—
163C22×C8—21no—
164C22×C8—21no—
165C22×C8—21no—
166C22×C8—21no—
167C22×C8—21no—
168C22×C8—22no—
169C22×C8—22no—
170C22×C8—22no—
171C22×C8—22no—
172C22×C8—22no—
173C22×C8—22no—
174C22×C8—22no—
175C22×C8C2×D412yes—
176C22×C8C2×Q812yes—
177C4×Q8—28no—
178C4×Q8—28no—
179C4×Q8—28no—
180C4×Q8—216no—
181C4×Q8—48no—
182C4×Q8—48no—
183C4×Q8—48no—
184C4×Q8C4⋊C4116yescomplemented
185C2×C4⋊C4—24no—
186C2×C4⋊C4—24no—
187C2×C4⋊C4—24no—
188C2×C4⋊C4—24no—
189C2×C4⋊C4—24no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—28no—
206C2×C4⋊C4—28no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—44no—
217C2×C4⋊C4—44no—
218C2×C4⋊C4—44no—
219C2×C4⋊C4—44no—
220C2×C4⋊C4—44no—
221C2×C4⋊C4—44no—
222C2×C4⋊C4—44no—
223C2×C4⋊C4—44no—
224C2×C4⋊C4—42no—
225C2×C4⋊C4—42no—
226C2×C4⋊C4—44no—
227C2×C4⋊C4—82no—
228C2×C4⋊C4—82no—
229C2×C4⋊C4—82no—
230C2×C4⋊C4C22⋊C418yes—
231C2×C4⋊C4C22×C414yes—
232C2×C4⋊C4C22×C412yes—
233C2×C4⋊C4C2×D412yes—
234C2×C4⋊C4C2×D412yes—
235C2×C4⋊C4C2×Q812yes—
236C2×C4⋊C4C2×Q812yes—
237C2×C42—24no—
238C2×C42—24no—
239C2×C42—24no—
240C2×C42—22no—
241C2×C42—22no—
242C2×C42—22no—
243C2×C42—22no—
244C2×C42—28no—
245C2×C42—44no—
246C2×C42—44no—
247C2×C42—44no—
248C2×C42—42no—
249C2×C42C22×C414yes—
250C2×C42C22×C412yes—
251C2×C42C2×D414yes—
252C2×C42C2×D412yes—
253C2×C42C2×D412yes—
254C2×C42C2×Q812yes—
255C2×C42#4112yes—
256C2×C42#4112yes—
257C4⋊C8—48no—
258C4⋊C8—44no—
259#103—28no—
260#103—28no—
261#103—48no—
262#97—44no—
263#95—24no—
264#95—28no—
265#95—24no—
266#95—44no—
267#95—42no—
268#95—44no—
269#95C22⋊C414yes—
270#95C22⋊C414yes—
271C24C22×D411yescharacteristic
272C2×Q8—22no—
273C2×Q8—22no—
274C2×Q8—22no—
275C2×Q8—22no—
276C2×Q8—22no—
277C2×Q8—22no—
278C2×Q8—22no—
279C2×Q8—22no—
280C2×Q8—22no—
281C2×Q8—22no—
282C2×Q8—22no—
283C2×Q8—22no—
284C2×Q8—22no—
285C2×Q8—22no—
286C2×Q8—22no—
287C2×Q8—22no—
288C2×Q8—22no—
289C2×Q8—22no—
290C2×Q8—22no—
291C2×Q8—22no—
292C2×Q8—22no—
293C2×Q8C2×C4⋊C414yes—
294C2×Q8C4×D414yes—
295C2×Q8C4×Q814yes—
296C2×Q8#12114yes—
297C2×Q8C22⋊Q814yes—
298C2×Q8#12514yes—
299C2×Q8C4⋊Q814yes—
300C22×C4—22no—
301C22×C4—22no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—22no—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—22no—
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—24no—
325C22×C4—22no—
326C22×C4—22no—
327C22×C4—42no—
328C22×C4—42no—
329C22×C4—41no—
330C22×C4—41no—
331C22×C4—42no—
332C22×C4—42no—
333C22×C4—41no—
334C22×C4—41no—
335C22×C4—42no—
336C22×C4—42no—
337C22×C4—42no—
338C22×C4—42no—
339C22×C4—42no—
340C22×C4—42no—
341C22×C4—42no—
342C22×C4—42no—
343C22×C4—42no—
344C22×C4—42no—
345C22×C4—42no—
346C22×C4—42no—
347C22×C4—42no—
348C22×C4—42no—
349C22×C4—42no—
350C22×C4C2×C22⋊C414yes—
351C22×C4C2×C22⋊C412yes—
352C22×C4C2×C22⋊C412yes—
353C22×C4C2×C22⋊C414yes—
354C22×C4#12112yes—
355C22×C4#12112yes—
356C22×C4C22⋊Q812yes—
357C22×C4C22⋊Q812yes—
358C22×C4AGL(1,ℤ/8)14yes—
359C22×C4#13714yes—
360C22×C4C23×C411yescharacteristic
361C22×C4C22×D411yescharacteristic
362C22×C4C22×D411yescharacteristic
363C22×C4C22×D411yescharacteristic
364C22×C4C22×Q811yescharacteristic
365C22×C4C22×Q811yescharacteristic
366C22×C4C2×#4111yescharacteristic
367C22×C4C2×#4111yescharacteristic
368C22×C4C2×#4111yescharacteristic
369C22×C4C2×#4111yescharacteristic
370C22×C4H(5,𝔽2)11yescharacteristic
371C22×C4H(5,𝔽2)11yescharacteristic
372C22×C42−1+411yescharacteristic
373C22×C42−1+411yescharacteristic
374C2×C8—22no—
375C2×C8—22no—
376C2×C8—22no—
377C2×C8—22no—
378C2×C8—22no—
379C2×C8—22no—
380C2×C8—41no—
381C2×C8—41no—
382C2×C8—41no—
383C2×C8—41no—
384C2×C8—41no—
385C2×C8—41no—
386C2×C8—41no—
387C2×C8—41no—
388C2×C8—42no—
389C2×C8—42no—
390C2×C8—42no—
391C2×C8—42no—
392C2×C8—42no—
393C2×C8—42no—
394C2×C8—42no—
395C4⋊C4—22no—
396C4⋊C4—22no—
397C4⋊C4—22no—
398C4⋊C4—216no—
399C4⋊C4—216no—
400C4⋊C4—24no—
401C4⋊C4—24no—
402C4⋊C4—22no—
403C4⋊C4—22no—
404C4⋊C4—22no—
405C4⋊C4—22no—
406C4⋊C4—416no—
407C4⋊C4—41no—
408C4⋊C4—41no—
409C4⋊C4—42no—
410C4⋊C4—42no—
411C4⋊C4—41no—
412C4⋊C4—41no—
413C4⋊C4—416no—
414C4⋊C4—42no—
415C4⋊C4—42no—
416C4⋊C4—42no—
417C4⋊C4—42no—
418C4⋊C4—88no—
419C4⋊C4—84no—
420C4⋊C4—84no—
421C4⋊C4—88no—
422C4⋊C4C2×C4⋊C414yes—
423C42—24no—
424C42—24no—
425C42—24no—
426C42—24no—
427C42—216no—
428C42—416no—
429C42—84no—
430C42C2×C4⋊C414yes—
431C42#11714yes—
432C42#12114yes—
433C42C22⋊Q814yes—
434C23C22×C22⋊C411yescharacteristic
435C23C2×#12111yescharacteristic
436C23C2×C22⋊Q811yescharacteristic
437C23D4211yescharacteristic
438C23#54611yescharacteristic
439C23#54811yescharacteristic
440C23D4×Q811yescharacteristic
441C23C2×AGL(1,ℤ/8)12yes—
442C23C2×#13712yes—
443C23#57712yes—
444C23#57812yes—
445Q8—24no—
446Q8—24no—
447Q8—24no—
448Q8—24no—
449Q8—24no—
450Q8—24no—
451Q8—24no—
452Q8#38918yescomplemented
453C2×C4—24no—
454C2×C4—24no—
455C2×C4—24no—
456C2×C4—24no—
457C2×C4—24no—
458C2×C4—24no—
459C2×C4—24no—
460C2×C4—24no—
461C2×C4—24no—
462C2×C4—24no—
463C2×C4—24no—
464C2×C4—24no—
465C2×C4—24no—
466C2×C4—24no—
467C2×C4—22no—
468C2×C4—22no—
469C2×C4—22no—
470C2×C4—22no—
471C2×C4—24no—
472C2×C4—24no—
473C2×C4—24no—
474C2×C4—24no—
475C2×C4—24no—
476C2×C4—24no—
477C2×C4—24no—
478C2×C4—44no—
479C2×C4—44no—
480C2×C4—42no—
481C2×C4—42no—
482C2×C4—42no—
483C2×C4—42no—
484C2×C4—42no—
485C2×C4—42no—
486C2×C4—44no—
487C2×C4—44no—
488C2×C4—44no—
489C2×C4—44no—
490C2×C4—42no—
491C2×C4—42no—
492C2×C4—44no—
493C2×C4—82no—
494C2×C4—82no—
495C2×C4—82no—
496C2×C4—81no—
497C2×C4—81no—
498C2×C4—81no—
499C2×C4—81no—
500C2×C4—82no—
501C2×C4—82no—
502C2×C4—82no—
503C2×C4—82no—
504C2×C4—82no—
505C2×C4—82no—
506C2×C4—82no—
507C2×C4—82no—
508C2×C4—82no—
509C2×C4—82no—
510C2×C4#38014yes—
511C2×C4#38014yes—
512C2×C4#41718yes—
513C2×C4C22×C4⋊C411yescharacteristic
514C2×C4C2×#11711yescharacteristic
515C2×C4C2×C4×D411yescharacteristic
516C2×C4C2×C4×Q811yescharacteristic
517C2×C4#51811yescharacteristic
518C2×C4#51911yescharacteristic
519C2×C4C2×#12111yescharacteristic
520C2×C4C2×#12111yescharacteristic
521C2×C4C2×C22⋊Q811yescharacteristic
522C2×C4C2×C22⋊Q811yescharacteristic
523C2×C4C2×#12511yescharacteristic
524C2×C4C2×C4⋊Q811yescharacteristic
525C2×C4#53711yescharacteristic
526C2×C4#53711yescharacteristic
527C2×C4#54411yescharacteristic
528C2×C4#54411yescharacteristic
529C2×C4D4211yescharacteristic
530C2×C4#54611yescharacteristic
531C2×C4#54711yescharacteristic
532C2×C4#54711yescharacteristic
533C2×C4#54811yescharacteristic
534C2×C4D4×Q811yescharacteristic
535C2×C4#55211yescharacteristic
536C2×C4#55211yescharacteristic
537C2×C4#55311yescharacteristic
538C2×C4#55311yescharacteristic
539C2×C4#55411yescharacteristic
540C2×C4#55411yescharacteristic
541C8—42no—
542C8—42no—
543C8—82no—
544C22C2×#38011yescharacteristic
545C22D4×C22⋊C411yescharacteristic
546C22Q8×C22⋊C411yescharacteristic
547C22#241611yescharacteristic
548C22#241711yescharacteristic
549C22#244011yescharacteristic
550C22#244211yescharacteristic
551C22C2×#41712yes—
552C22#289812yes—
553C22#303811yescharacteristic
554C22#303911yescharacteristic
555C22#304311yescharacteristic
556C22#304411yescharacteristic
557C22#307912yes—
558C22#308212yes—
559C22#326512yes—
560C22#327012yes—
561C22#330812yes—
562C22#331612yes—
563C22#336812yes—
564C22#337412yes—
565C22#338612yes—
566C22#338712yes—
567C4—28no—
568C4—28no—
569C4—24no—
570C4—24no—
571C4—44no—
572C4—44no—
573C4—84no—
574C4—82no—
575C4—82no—
576C4—162no—
577C4C2×#38911yescharacteristic
578C4#231611yescharacteristic
579C4#234411yescharacteristic
580C4#234511yescharacteristic
581C4#241611yescharacteristic
582C4#241711yescharacteristic
583C4#244011yescharacteristic
584C4#244211yescharacteristic
585C2#1390711yescharacteristic
586C2#2064811yescharacteristic
587C2#2065111yescharacteristic
588C2#2242112yes—
589C2#2252412yes—
590C2#2474211yescharacteristic
591C2#2474611yescharacteristic
592C2#2482711yescharacteristic
593C2#2483111yescharacteristic
594C2#2605612yes—
595C2#2612212yes—
subgroupG/H
center, socle, prime radicalC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
LCSG ▷ C22×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
21. #2351C4semidirect
184. C4×Q8C4⋊C4semidirect
452. Q8#389semidirect

Maximal subgroups

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

subgroupindex
C2×#23512
C2×#19402
C2×#18032
C2×#17632
#139102
#112272
#112072
#108502
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