Finite Group Databasebeta

← #194032

Subgroups — #194032

Order 512.

1306 conjugacy classes of subgroups.

The subgroup lattice has 553 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3079C211yesmaximal, characteristic
2#34188C211yesmaximal, characteristic
3#21804C211yesmaximal, characteristic, complemented
4C2×#1990C211yesmaximal, characteristic, complemented
5C2×#1476C211yesmaximal, characteristic
6#8984C214yesmaximal, complemented
7#8982C214yesmaximal, complemented
8#8043C211yesmaximal, characteristic, complemented
9#7897C211yesmaximal, characteristic, complemented
10C22×C4⋊Q8C2211yescharacteristic
11C2×C4×#41C2211yescharacteristic
12#3079—24no—
13#3079C418yescomplemented
14C2×#477—21no—
15C2×#475C2212yes—
16C2×#476—21no—
17C2×#474C2212yes—
18C2×#431C2211yescharacteristic
19C2×#428—21no—
20C2×#423C2211yescharacteristic
21C2×#417—21no—
22C22×C4×C8C2211yescharacteristic
23#2355—21no—
24#2355C2212yes—
25#2354—21no—
26#2354C2212yes—
27#2352—22no—
28#2352—22no—
29#2314—22no—
30#2311—21no—
31#2285—21no—
32#2285—21no—
33#2285C2211yescharacteristic
34C2×#384C2211yescharacteristic
35#2274—21no—
36#1990—22no—
37#1990C2214yes—
38#1981C2214yes—
39#1956—21no—
40#1956—21no—
41#1787—22no—
42C2×#340—22no—
43#1476—24no—
44C2×Q8⋊C8C2212yes—
45C2×D4⋊C8C2212yes—
46#1296C2214yes—
47#1279C2214yes—
48C22×#41D411yescharacteristic
49C23×Q8—21no—
50C22×M16—21no—
51C23×C8—21no—
52C2×C4⋊Q8—21no—
53C2×C4⋊Q8—22no—
54C2×C4⋊Q8—21no—
55C2×C4⋊Q8—21no—
56C2×C4⋊Q8—22no—
57C2×C4⋊Q8—22no—
58C2×C4⋊Q8C2×C414yes—
59C2×C4⋊Q8C2312yes—
60C4×#41—22no—
61C4×#41—22no—
62C4×#41C2×C414yes—
63C4×#41D414yes—
64C2×C4×Q8D411yescharacteristic
65C2×C4×Q8C2311yescharacteristic
66C2×C4×D4—21no—
67C2×C4×D4—21no—
68C2×C4×D4D411yescharacteristic
69C2×C4×D4C2311yescharacteristic
70C2×#117—21no—
71C2×#117—22no—
72C2×#117—22no—
73C2×#117—21no—
74C2×#117D411yescharacteristic
75C22×C4⋊C4—21no—
76C22×C4⋊C4—21no—
77C22×C4⋊C4—21no—
78C22×C4⋊C4—21no—
79C22×C4⋊C4D412yes—
80C22×C42—21no—
81C22×C42C2311yescharacteristic
82#477—28no—
83#476—28no—
84#475—24no—
85#475C2×C418yescomplemented
86#474—24no—
87#474C2×C418yescomplemented
88#431—22no—
89#431—22no—
90#428—28no—
91C2×#107—22no—
92C2×#106—22no—
93#423—22no—
94#423C2×C414yes—
95C2×C4⋊C8—21no—
96C2×C4⋊C8C2312yes—
97C2×C4⋊C8C2312yes—
98#417—28no—
99C2×#103—22no—
100C2×#103—24no—
101C2×#103—24no—
102C2×#102—22no—
103C2×C22⋊C8—21no—
104C2×C4×C8—22no—
105C2×C4×C8—22no—
106C2×C4×C8—21no—
107C2×C4×C8—21no—
108C2×C4×C8C2312yes—
109#391—22no—
110#390—22no—
111#388—24no—
112#388—24no—
113#387—24no—
114#387—24no—
115#386—24no—
116#384—21no—
117#384—21no—
118#384D412yes—
119#384D411yescharacteristic
120#384D411yescharacteristic
121#384C2312yes—
122#382—22no—
123#382—22no—
124#382—21no—
125#380—22no—
126#380—22no—
127#380—22no—
128#380—21no—
129#380—22no—
130#380D412yes—
131C4×C4⋊C4—22no—
132C4×C4⋊C4—22no—
133C4×C4⋊C4—22no—
134C4×C4⋊C4—22no—
135C4×C4⋊C4—22no—
136C4×C4⋊C4—22no—
137C4×C4⋊C4—21no—
138C4×C4⋊C4C2312yes—
139C4×C22⋊C4—24no—
140#340—28no—
141Q8⋊C8—24no—
142D4⋊C8—24no—
143C2×#41—22no—
144C2×#41—22no—
145C2×#41—22no—
146C2×#41—22no—
147C2×#41—22no—
148C2×#41C22⋊C414yes—
149C22×Q8—24no—
150C22×Q8—22no—
151C22×Q8—41no—
152C22×Q8—42no—
153C22×Q8—82no—
154C22×Q8—82no—
155C22×Q8C2×D411yescharacteristic
156C22×D4—21no—
157C22×D4C2×D411yescharacteristic
158C23×C4—21no—
159C23×C4—21no—
160C23×C4—21no—
161C23×C4—22no—
162C23×C4—41no—
163C23×C4C2×D411yescharacteristic
164C23×C4C2×D411yescharacteristic
165C23×C4C2×D411yescharacteristic
166C2×M16—24no—
167C2×M16—42no—
168C2×M16—42no—
169C22×C8—24no—
170C22×C8—22no—
171C22×C8—22no—
172C22×C8—22no—
173C22×C8—42no—
174C22×C8—42no—
175C4⋊Q8—24no—
176C4⋊Q8—24no—
177C4⋊Q8—24no—
178C4⋊Q8—22no—
179C4⋊Q8—22no—
180C4⋊Q8—22no—
181C4⋊Q8—42no—
182C4⋊Q8—42no—
183C4⋊Q8—42no—
184C4⋊Q8C22×C414yes—
185C4×Q8—22no—
186C4×Q8—21no—
187C4×Q8—21no—
188C4×Q8C22⋊C414yes—
189C4×Q8C22×C412yes—
190C4×Q8C2×D412yes—
191C4×D4—24no—
192C4×D4—24no—
193C4×D4—24no—
194C4×D4—24no—
195C4×D4—22no—
196C4×D4—21no—
197C4×D4—21no—
198C4×D4C22⋊C414yes—
199C4×D4C22×C412yes—
200C4×D4C2×D412yes—
201#117—22no—
202#117—24no—
203#117—24no—
204#117—42no—
205#117—42no—
206#117—84no—
207#117—84no—
208#117C22⋊C414yes—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—24no—
212C2×C4⋊C4—24no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4—21no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—41no—
225C2×C4⋊C4—41no—
226C2×C4⋊C4—42no—
227C2×C4⋊C4—42no—
228C2×C4⋊C4—42no—
229C2×C4⋊C4—42no—
230C2×C4⋊C4—44no—
231C2×C4⋊C4—82no—
232C2×C4⋊C4C2×D412yes—
233C2×C4⋊C4C2×D412yes—
234C2×C4⋊C4C2×D411yescharacteristic
235C2×C4⋊C4C2×D411yescharacteristic
236C2×C4⋊C4C2×D414yes—
237C2×C22⋊C4—21no—
238C2×C22⋊C4—21no—
239C2×C22⋊C4—21no—
240C2×C22⋊C4—22no—
241C2×C22⋊C4—24no—
242C2×C22⋊C4—24no—
243C2×C42—21no—
244C2×C42—21no—
245C2×C42—21no—
246C2×C42—24no—
247C2×C42—24no—
248C2×C42—22no—
249C2×C42—24no—
250C2×C42—24no—
251C2×C42C22×C412yes—
252C2×C42C2×D411yescharacteristic
253C2×C42C2×D412yes—
254C2×C42C2×D411yescharacteristic
255C2×C42C2×D411yescharacteristic
256C2×C42#4112yes—
257C2×C42#4112yes—
258C2×C42C2411yescharacteristic
259#107—28no—
260#106—28no—
261C4⋊C8—24no—
262C4⋊C8—22no—
263C4⋊C8—22no—
264C4⋊C8—22no—
265C4⋊C8C22×C414yes—
266#103—28no—
267#103—84no—
268#103—84no—
269#102—28no—
270C22⋊C8—42no—
271C4×C8—22no—
272C4×C8—22no—
273C4×C8—42no—
274C4×C8—42no—
275#95—24no—
276#95—24no—
277#95—24no—
278#95—24no—
279#95—22no—
280#95—22no—
281C24—21no—
282C24#12011yescharacteristic
283#41—28no—
284#41—28no—
285#41—44no—
286#41—44no—
287C2×Q8—21no—
288C2×Q8—24no—
289C2×Q8—21no—
290C2×Q8—21no—
291C2×Q8—21no—
292C2×Q8—21no—
293C2×Q8—42no—
294C2×Q8—44no—
295C2×Q8—44no—
296C2×Q8—44no—
297C2×Q8—88no—
298C2×Q8—82no—
299C2×Q8—82no—
300C2×Q8—88no—
301C2×Q8C2×C22⋊C412yes—
302C2×D4—24no—
303C2×D4—21no—
304C2×D4—21no—
305C2×D4—21no—
306C2×D4—21no—
307C2×D4—21no—
308C2×D4—42no—
309C2×D4—42no—
310C2×D4C2×C22⋊C412yes—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—22no—
314C22×C4—22no—
315C22×C4—21no—
316C22×C4—21no—
317C22×C4—21no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—24no—
321C22×C4—24no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—24no—
325C22×C4—42no—
326C22×C4—42no—
327C22×C4—42no—
328C22×C4—42no—
329C22×C4—42no—
330C22×C4—42no—
331C22×C4—42no—
332C22×C4—44no—
333C22×C4—44no—
334C22×C4—82no—
335C22×C4—82no—
336C22×C4—82no—
337C22×C4—82no—
338C22×C4C2×C22⋊C412yes—
339C22×C4C4×D412yes—
340C22×C4C4×D412yes—
341C22×C4#12011yescharacteristic
342C22×C4#12011yescharacteristic
343C22×C4#12011yescharacteristic
344C22×C4#12112yes—
345C22×C4#12112yes—
346C22×C4#12312yes—
347C22×C4#13512yes—
348C22×C4#13512yes—
349C22×C4#13712yes—
350C22×C4#13712yes—
351C22×C4C22×D411yescharacteristic
352C22×C4C22×D411yescharacteristic
353C22×C4C22×D411yescharacteristic
354M16—48no—
355C2×C8—24no—
356C2×C8—24no—
357C2×C8—42no—
358C2×C8—48no—
359C2×C8—42no—
360C2×C8—42no—
361C2×C8—42no—
362C2×C8—42no—
363C2×C8—42no—
364C2×C8—42no—
365C4⋊C4—22no—
366C4⋊C4—22no—
367C4⋊C4—21no—
368C4⋊C4—21no—
369C4⋊C4—24no—
370C4⋊C4—24no—
371C4⋊C4—24no—
372C4⋊C4—24no—
373C4⋊C4—24no—
374C4⋊C4—44no—
375C4⋊C4—42no—
376C4⋊C4—42no—
377C4⋊C4—42no—
378C4⋊C4—42no—
379C4⋊C4—44no—
380C4⋊C4—44no—
381C4⋊C4—44no—
382C4⋊C4—44no—
383C4⋊C4—42no—
384C4⋊C4—42no—
385C4⋊C4—44no—
386C4⋊C4—42no—
387C4⋊C4—42no—
388C4⋊C4—82no—
389C4⋊C4C2×C22⋊C412yes—
390C4⋊C4C2×C22⋊C412yes—
391C22⋊C4—24no—
392C22⋊C4—22no—
393C22⋊C4—22no—
394C22⋊C4—24no—
395C22⋊C4—44no—
396C22⋊C4—84no—
397C22⋊C4—84no—
398C42—22no—
399C42—22no—
400C42—24no—
401C42—24no—
402C42—42no—
403C42—42no—
404C42—84no—
405C42—84no—
406C42C2×C22⋊C412yes—
407C42C4×D412yes—
408C42C4×D412yes—
409C42#12112yes—
410C42#12112yes—
411C42#12312yes—
412C42C23×C411yescharacteristic
413C42C22×D411yescharacteristic
414C42C2×#4111yescharacteristic
415C42C2×#4111yescharacteristic
416C23—21no—
417C23—21no—
418C23—21no—
419C23—21no—
420C23—22no—
421C23—41no—
422C23—41no—
423C23—41no—
424C23—41no—
425C23—41no—
426C23—41no—
427C23#38612yes—
428C23#45212yes—
429C23#45212yes—
430C23C2×#12011yescharacteristic
431Q8—22no—
432Q8—22no—
433Q8—41no—
434Q8—41no—
435Q8—48no—
436Q8—88no—
437Q8—84no—
438D4—22no—
439D4—22no—
440D4—48no—
441D4—41no—
442D4—41no—
443C2×C4—21no—
444C2×C4—21no—
445C2×C4—21no—
446C2×C4—21no—
447C2×C4—24no—
448C2×C4—24no—
449C2×C4—24no—
450C2×C4—24no—
451C2×C4—22no—
452C2×C4—22no—
453C2×C4—22no—
454C2×C4—41no—
455C2×C4—41no—
456C2×C4—41no—
457C2×C4—41no—
458C2×C4—41no—
459C2×C4—41no—
460C2×C4—48no—
461C2×C4—48no—
462C2×C4—44no—
463C2×C4—44no—
464C2×C4—42no—
465C2×C4—42no—
466C2×C4—44no—
467C2×C4—42no—
468C2×C4—44no—
469C2×C4—42no—
470C2×C4—42no—
471C2×C4—44no—
472C2×C4—42no—
473C2×C4—44no—
474C2×C4—88no—
475C2×C4—82no—
476C2×C4—82no—
477C2×C4—164no—
478C2×C4#38612yes—
479C2×C4#38612yes—
480C2×C4#38612yes—
481C2×C4#41614yes—
482C2×C4#41914yes—
483C2×C4#45212yes—
484C2×C4#45212yes—
485C2×C4#45512yes—
486C2×C4#45612yes—
487C2×C4#48514yes—
488C2×C4#48514yes—
489C2×C4C22×C22⋊C411yescharacteristic
490C2×C4C2×C4×D411yescharacteristic
491C2×C4C2×C4×D411yescharacteristic
492C2×C4C2×#12111yescharacteristic
493C2×C4C2×#12111yescharacteristic
494C2×C4C2×#12311yescharacteristic
495C2×C4C2×#13511yescharacteristic
496C2×C4C2×#13511yescharacteristic
497C2×C4C2×#13711yescharacteristic
498C2×C4C2×#13711yescharacteristic
499C2×C4#57612yes—
500C2×C4#57712yes—
501C8—44no—
502C8—44no—
503C8—82no—
504C22—22no—
505C22—22no—
506C22—22no—
507C22—22no—
508C22—22no—
509C22—22no—
510C22—42no—
511C22—42no—
512C22—42no—
513C22—81no—
514C22—81no—
515C22—81no—
516C22—81no—
517C22#187314yes—
518C22C2×#38611yescharacteristic
519C22C2×#45211yescharacteristic
520C22C2×#45211yescharacteristic
521C22#300811yescharacteristic
522C22#301011yescharacteristic
523C22C2×#45511yescharacteristic
524C22C2×#45611yescharacteristic
525C4—22no—
526C4—21no—
527C4—21no—
528C4—42no—
529C4—42no—
530C4—42no—
531C4—84no—
532C4—84no—
533C4—82no—
534C4—82no—
535C4#187314yes—
536C4#188214yes—
537C4C2×#41611yescharacteristic
538C4C2×#41911yescharacteristic
539C4#294612yes—
540C4#304512yes—
541C4C2×#48511yescharacteristic
542C4C2×#48511yescharacteristic
543C2—24no—
544C2—82no—
545C2#1863511yescharacteristic
546C2#1863711yescharacteristic
547C2#1865511yescharacteristic
548C2#1865711yescharacteristic
549C2C2×#187311yescharacteristic
550C2#2093211yescharacteristic
551C2C2×#188211yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
prime radicalC22×#41D4
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #21804C2semidirect
4. C2×#1990C2semidirect
6. #8984C2semidirect
7. #8982C2semidirect
8. #8043C2semidirect
9. #7897C2semidirect
13. #3079C4semidirect
85. #475C2×C4semidirect
87. #474C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#30792
#341882
#218042
C2×#19902
C2×#14762
#89842
#89822
#80432
#78972