Finite Group Databasebeta

← #195345

Subgroups — #195345

Order 512.

1506 conjugacy classes of subgroups.

The subgroup lattice has 562 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3226C211yesmaximal, characteristic
2#34206C211yesmaximal, characteristic
3#21833C211yesmaximal, characteristic, complemented
4C2×#1502C211yesmaximal, characteristic
5#14155C211yesmaximal, characteristic, complemented
6#9043C218yesmaximal, complemented
7#8014C211yesmaximal, characteristic, complemented
8#7908C211yesmaximal, characteristic, complemented
9C2×#532C2211yescharacteristic
10C22×C4⋊Q8C2211yescharacteristic
11C2×#520—21no—
12#3226—24no—
13#3226C418yescomplemented
14C2×#500—21no—
15C2×#486—21no—
16C2×#480C2214yes—
17C2×#474—22no—
18C2×#431C2211yescharacteristic
19C2×#431C2211yescharacteristic
20#2367—22no—
21#2366—21no—
22#2365—21no—
23#2354—21no—
24#2352—22no—
25D4×C4⋊C4—21no—
26#2325C2214yes—
27#2311—21no—
28C4×#121—22no—
29C2×#391—21no—
30C2×#391—21no—
31#2285C2211yescharacteristic
32C2×C4×C4⋊C4C2211yescharacteristic
33C2×C4×C4⋊C4C2211yescharacteristic
34#1990—21no—
35#1990—21no—
36#1988—22no—
37#1823C2214yes—
38#1502—24no—
39C2×#327C2214yes—
40#1445C2214yes—
41C4⋊#95—22no—
42#1294C2214yes—
43#1280C2214yes—
44C22×#41—21no—
45C23×Q8—21no—
46#532—22no—
47#532—22no—
48#532C2×C414yes—
49#532D414yes—
50C2×C4⋊Q8—21no—
51C2×C4⋊Q8—22no—
52C2×C4⋊Q8—21no—
53C2×C4⋊Q8—21no—
54C2×C4⋊Q8—21no—
55C2×C4⋊Q8—22no—
56C2×C4⋊Q8—22no—
57C2×C4⋊Q8C2×C414yescomplemented
58C2×C4⋊Q8D411yescharacteristic
59C2×#127D411yescharacteristic
60C2×#124—21no—
61C2×#121—21no—
62C2×#121C2312yes—
63#520—44no—
64#520—44no—
65C2×C4×Q8—22no—
66C2×C4×D4—22no—
67C2×C4×D4—24no—
68C2×C4×D4—22no—
69C2×C4×D4—21no—
70C2×C4×D4D412yes—
71C2×#117—22no—
72C2×#117—21no—
73C22×C4⋊C4—22no—
74C22×C4⋊C4—21no—
75C22×C4⋊C4—21no—
76C22×C4⋊C4D411yescharacteristic
77C22×C4⋊C4D411yescharacteristic
78C22×C4⋊C4C2312yes—
79C22×C42—21no—
80C22×C42D412yes—
81C22×C42C2311yescharacteristic
82#500—28no—
83#486—28no—
84#480—28no—
85#480C2×C4116yescomplemented
86#474—216no—
87#431—22no—
88#431—22no—
89#431—22no—
90#431C2×C414yescomplemented
91C2×#107—24no—
92C2×C4⋊C8—21no—
93C2×C4⋊C8—21no—
94C2×#102—24no—
95C2×C22⋊C8C2312yes—
96C2×C22⋊C8C2312yes—
97C2×C4×C8—21no—
98C2×C4×C8—21no—
99#391—22no—
100#391—44no—
101#391—44no—
102#390—22no—
103#390—22no—
104#389—22no—
105#388—24no—
106#388—24no—
107#387—24no—
108#386—24no—
109#386—24no—
110#385—24no—
111#384D411yescharacteristic
112#384D411yescharacteristic
113#382—21no—
114#380—22no—
115#380—22no—
116#380—21no—
117#380C2312yes—
118C4×C4⋊C4—22no—
119C4×C4⋊C4—21no—
120C4×C4⋊C4—21no—
121C4×C4⋊C4—21no—
122C4×C4⋊C4—21no—
123C4×C4⋊C4—21no—
124C4×C4⋊C4—22no—
125C4×C4⋊C4—22no—
126C4×C4⋊C4D412yes—
127C4×C22⋊C4—24no—
128C2×#95—24no—
129C2×#95—24no—
130C2×#95C2312yes—
131C2×#95C2312yes—
132#340—24no—
133#340—24no—
134#327—28no—
135C2×#41—24no—
136C2×#41—24no—
137C2×#41—42no—
138C2×#41—42no—
139C2×#41—42no—
140C2×#41—42no—
141C22×Q8—24no—
142C22×Q8—21no—
143C22×Q8—41no—
144C22×Q8—41no—
145C22×Q8—42no—
146C22×Q8—42no—
147C22×Q8—42no—
148C22×Q8—42no—
149C22×Q8—42no—
150C22×D4—22no—
151C22×D4—21no—
152C22×D4—21no—
153C22×D4C2×D412yes—
154C23×C4—22no—
155C23×C4—22no—
156C23×C4—22no—
157C23×C4—24no—
158C23×C4C2×D411yescharacteristic
159C23×C4C2×D412yes—
160C23×C4C2×D412yes—
161C23×C4C2×D411yescharacteristic
162C23×C4C2411yescharacteristic
163C22×C8—22no—
164C22×C8—22no—
165C4⋊Q8—24no—
166C4⋊Q8—24no—
167C4⋊Q8—24no—
168C4⋊Q8—24no—
169C4⋊Q8—22no—
170C4⋊Q8—42no—
171C4⋊Q8—42no—
172C4⋊Q8—42no—
173C4⋊Q8—42no—
174C4⋊Q8C22⋊C414yes—
175#127—22no—
176#127C22⋊C414yes—
177#124—24no—
178#124—24no—
179#121—24no—
180#121—24no—
181#121—22no—
182#121—22no—
183#121C22×C414yes—
184#121C2×D414yes—
185C4×Q8—48no—
186C4×D4—24no—
187C4×D4—24no—
188C4×D4—24no—
189C4×D4—48no—
190C4×D4—48no—
191C4×D4—48no—
192C4×D4—84no—
193C4×D4C22⋊C418yes—
194#117—48no—
195#117—42no—
196#117—42no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—24no—
206C2×C4⋊C4—24no—
207C2×C4⋊C4—24no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—48no—
218C2×C4⋊C4—42no—
219C2×C4⋊C4—41no—
220C2×C4⋊C4—41no—
221C2×C4⋊C4—42no—
222C2×C4⋊C4—42no—
223C2×C4⋊C4D814yes—
224C2×C4⋊C4C22×C414yes—
225C2×C4⋊C4C2×D412yes—
226C2×C4⋊C4C2×D412yes—
227C2×C4⋊C4C2×D412yes—
228C2×C22⋊C4—22no—
229C2×C22⋊C4—22no—
230C2×C22⋊C4—24no—
231C2×C22⋊C4—22no—
232C2×C22⋊C4—24no—
233C2×C22⋊C4—24no—
234C2×C42—24no—
235C2×C42—22no—
236C2×C42—22no—
237C2×C42—22no—
238C2×C42—44no—
239C2×C42—42no—
240C2×C42—42no—
241C2×C42—42no—
242C2×C42C22×C412yes—
243C2×C42C2×D412yes—
244C2×C42C2×D411yescharacteristic
245C2×C42C2×D411yescharacteristic
246C2×C42C2×D411yescharacteristic
247C2×C42C2×D411yescharacteristic
248C2×C42#4112yes—
249C2×C42#4112yes—
250#107—216no—
251C4⋊C8—24no—
252C4⋊C8—42no—
253#102—216no—
254C22⋊C8—22no—
255C22⋊C8—22no—
256C22⋊C8—22no—
257C22⋊C8C22×C414yes—
258C4×C8—24no—
259C4×C8—42no—
260#95—24no—
261#95—24no—
262#95—22no—
263#95—22no—
264#95—22no—
265#95—22no—
266#95—22no—
267#95—22no—
268#95—48no—
269#95—48no—
270C24—22no—
271C24C22×D411yescharacteristic
272#41—48no—
273#41—48no—
274#41—48no—
275#41—48no—
276C2×Q8—24no—
277C2×Q8—44no—
278C2×Q8—44no—
279C2×Q8—48no—
280C2×Q8—48no—
281C2×Q8—42no—
282C2×Q8—42no—
283C2×Q8—82no—
284C2×Q8—82no—
285C2×Q8—82no—
286C2×Q8—82no—
287C2×Q8—84no—
288C2×Q8—84no—
289C2×D4—24no—
290C2×D4—24no—
291C2×D4—22no—
292C2×D4—24no—
293C2×D4—24no—
294C2×D4—42no—
295C2×D4—42no—
296C2×D4—42no—
297C2×D4—42no—
298C2×D4—42no—
299C2×D4—42no—
300C2×D4—42no—
301C2×D4—42no—
302C2×D4—42no—
303C2×D4—42no—
304C2×D4C2×C22⋊C414yes—
305C22×C4—22no—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—28no—
311C22×C4—24no—
312C22×C4—24no—
313C22×C4—22no—
314C22×C4—22no—
315C22×C4—22no—
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—41no—
319C22×C4—41no—
320C22×C4—48no—
321C22×C4—44no—
322C22×C4—44no—
323C22×C4—42no—
324C22×C4—42no—
325C22×C4—42no—
326C22×C4—42no—
327C22×C4—42no—
328C22×C4—42no—
329C22×C4—42no—
330C22×C4—42no—
331C22×C4—42no—
332C22×C4—42no—
333C22×C4—42no—
334C22×C4—42no—
335C22×C4—42no—
336C22×C4—42no—
337C22×C4—42no—
338C22×C4—42no—
339C22×C4—84no—
340C22×C4—84no—
341C22×C4C4×D412yes—
342C22×C4C4×D412yes—
343C22×C4#12011yescharacteristic
344C22×C4#12011yescharacteristic
345C22×C4#12011yescharacteristic
346C22×C4#12011yescharacteristic
347C22×C4#12112yes—
348C22×C4#12112yes—
349C22×C4C2×D814yes—
350C22×C4C2×D812yes—
351C22×C4#13512yes—
352C22×C4AGL(1,ℤ/8)12yes—
353C22×C4AGL(1,ℤ/8)12yes—
354C22×C4C23×C411yescharacteristic
355C22×C4C22×D411yescharacteristic
356C22×C4C22×D411yescharacteristic
357C22×C4C22×D411yescharacteristic
358C22×C4C2×#4111yescharacteristic
359C22×C4C2×#4111yescharacteristic
360C2×C8—24no—
361C2×C8—42no—
362C2×C8—42no—
363C2×C8—42no—
364C2×C8—42no—
365C2×C8—42no—
366C4⋊C4—22no—
367C4⋊C4—24no—
368C4⋊C4—24no—
369C4⋊C4—24no—
370C4⋊C4—24no—
371C4⋊C4—24no—
372C4⋊C4—42no—
373C4⋊C4—42no—
374C4⋊C4—48no—
375C4⋊C4—48no—
376C4⋊C4—42no—
377C4⋊C4—42no—
378C4⋊C4—44no—
379C4⋊C4—44no—
380C4⋊C4—42no—
381C4⋊C4—42no—
382C4⋊C4—42no—
383C4⋊C4—42no—
384C4⋊C4—42no—
385C4⋊C4—42no—
386C4⋊C4C2×C22⋊C414yes—
387C22⋊C4—24no—
388C22⋊C4—24no—
389C22⋊C4—28no—
390C22⋊C4—48no—
391C22⋊C4—42no—
392C22⋊C4—42no—
393C22⋊C4—84no—
394C22⋊C4—84no—
395C42—48no—
396C42—44no—
397C42—44no—
398C42—82no—
399C42—84no—
400C42C2×C22⋊C412yes—
401C42C2×C22⋊C412yes—
402C42C4×D412yes—
403C42C4×D412yes—
404C42#12112yes—
405C42#12112yes—
406C42#12312yes—
407C42#12312yes—
408C23—21no—
409C23—21no—
410C23—21no—
411C23—21no—
412C23—24no—
413C23—42no—
414C23—42no—
415C23—42no—
416C23—42no—
417C23—42no—
418C23—42no—
419C23C2×C4×D411yescharacteristic
420C23C2×#12011yescharacteristic
421C23C2×#12111yescharacteristic
422C23C22×D811yescharacteristic
423C23C2×#13511yescharacteristic
424C23C2×AGL(1,ℤ/8)11yescharacteristic
425C23C2×AGL(1,ℤ/8)11yescharacteristic
426Q8—48no—
427Q8—88no—
428Q8—88no—
429D4—44no—
430D4—44no—
431D4—44no—
432D4—44no—
433D4—44no—
434D4—48no—
435D4—48no—
436D4—82no—
437C2×C4—21no—
438C2×C4—21no—
439C2×C4—21no—
440C2×C4—21no—
441C2×C4—21no—
442C2×C4—21no—
443C2×C4—24no—
444C2×C4—24no—
445C2×C4—24no—
446C2×C4—24no—
447C2×C4—41no—
448C2×C4—41no—
449C2×C4—41no—
450C2×C4—41no—
451C2×C4—42no—
452C2×C4—42no—
453C2×C4—42no—
454C2×C4—42no—
455C2×C4—416no—
456C2×C4—44no—
457C2×C4—44no—
458C2×C4—44no—
459C2×C4—44no—
460C2×C4—44no—
461C2×C4—44no—
462C2×C4—44no—
463C2×C4—42no—
464C2×C4—42no—
465C2×C4—42no—
466C2×C4—42no—
467C2×C4—42no—
468C2×C4—42no—
469C2×C4—42no—
470C2×C4—42no—
471C2×C4—42no—
472C2×C4—44no—
473C2×C4—82no—
474C2×C4—82no—
475C2×C4—82no—
476C2×C4—82no—
477C2×C4—82no—
478C2×C4—82no—
479C2×C4—82no—
480C2×C4—82no—
481C2×C4—82no—
482C2×C4—82no—
483C2×C4—168no—
484C2×C4#38612yes—
485C2×C4#38612yes—
486C2×C4#38612yes—
487C2×C4#38612yes—
488C2×C4C4×D814yes—
489C2×C4#44214yes—
490C2×C4#44712yes—
491C2×C4#44712yes—
492C2×C4#44912yes—
493C2×C4#44912yes—
494C2×C4#45512yes—
495C2×C4#45612yes—
496C2×C4#45914yes—
497C2×C4#46314yes—
498C2×C4C22×C22⋊C411yescharacteristic
499C2×C4C2×C4×D411yescharacteristic
500C2×C4C2×#12111yescharacteristic
501C2×C4C2×#12311yescharacteristic
502C8—44no—
503C8—82no—
504C22—21no—
505C22—21no—
506C22—21no—
507C22—21no—
508C22—21no—
509C22—21no—
510C22—21no—
511C22—21no—
512C22—21no—
513C22—21no—
514C22—21no—
515C22—21no—
516C22—44no—
517C22—44no—
518C22—44no—
519C22—82no—
520C22—82no—
521C22—82no—
522C22—82no—
523C22C2×#38611yescharacteristic
524C22C2×C4×D811yescharacteristic
525C22C2×#44211yescharacteristic
526C22C2×#44711yescharacteristic
527C22C2×#44911yescharacteristic
528C22C2×#45511yescharacteristic
529C22C2×#45611yescharacteristic
530C22#302811yescharacteristic
531C22#302811yescharacteristic
532C22C2×#45911yescharacteristic
533C22C2×#46311yescharacteristic
534C4—22no—
535C4—21no—
536C4—21no—
537C4—21no—
538C4—21no—
539C4—44no—
540C4—44no—
541C4—84no—
542C4—84no—
543C4—82no—
544C4—82no—
545C4—82no—
546C4#187914yes—
547C4#187914yes—
548C4#188214yes—
549C2—21no—
550C2—21no—
551C2—21no—
552C2—21no—
553C2—84no—
554C2#1839411yescharacteristic
555C2#1839811yescharacteristic
556C2#1840811yescharacteristic
557C2#1841211yescharacteristic
558C2C2×#187911yescharacteristic
559C2C2×#188211yescharacteristic
560C2#2096211yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC2×#532C22
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #21833C2semidirect
5. #14155C2semidirect
6. #9043C2semidirect
7. #8014C2semidirect
8. #7908C2semidirect
13. #3226C4semidirect
57. C2×C4⋊Q8C2×C4semidirect
85. #480C2×C4semidirect
90. #431C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#32262
#342062
#218332
C2×#15022
#141552
#90432
#80142
#79082