Finite Group Databasebeta

← #197238

Subgroups — #197238

Order 512.

1269 conjugacy classes of subgroups.

The subgroup lattice has 570 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3077C211yesmaximal, characteristic
2#34186C211yesmaximal, characteristic
3#17575C211yesmaximal, characteristic, complemented
4#17575C211yesmaximal, characteristic, complemented
5C2×#1478C211yesmaximal, characteristic
6#9177C218yesmaximal, complemented
7#8095C211yesmaximal, characteristic, complemented
8#7957C211yesmaximal, characteristic, complemented
9C2×#533C2211yescharacteristic
10C22×C4×D4C2211yescharacteristic
11#3077—22no—
12#3077—22no—
13#3077C414yescomplemented
14#3077C2214yes—
15C2×#478—22no—
16C2×#476C412yescomplemented
17C2×#474C412yescomplemented
18C2×#431C2211yescharacteristic
19C2×#431C2211yescharacteristic
20C2×#428—21no—
21C2×#423C2211yescharacteristic
22C2×#418—21no—
23#2357—22no—
24#2357C2214yes—
25#2354—22no—
26#2354—22no—
27#2312—22no—
28#2309—21no—
29#2285C2211yescharacteristic
30#2285C2211yescharacteristic
31#2280—22no—
32#2274—21no—
33#1520—22no—
34#1520—22no—
35#1518—22no—
36#1518C2214yescomplemented
37#1503—22no—
38#1503C2214yescomplemented
39#1478—22no—
40#1478C2214yes—
41C2×D4⋊C8—22no—
42#1303C2214yescomplemented
43#1286C2214yescomplemented
44C23×D4D411yescharacteristic
45C24×C4—21no—
46C22×M16—21no—
47#533—21no—
48#533—21no—
49#533—21no—
50#533—21no—
51#533—21no—
52#533—21no—
53#533C2×C412yes—
54#533C2312yes—
55C2×C4⋊Q8C2×C411yescharacteristic
56C2×C4⋊Q8C2×C411yescharacteristic
57C2×#125—21no—
58C2×C22⋊Q8—21no—
59C2×C22⋊Q8—21no—
60C2×C4×Q8—21no—
61C2×C4×Q8—21no—
62C2×C4×D4—21no—
63C2×C4×D4—21no—
64C2×C4×D4—21no—
65C2×C4×D4—21no—
66C2×C4×D4—21no—
67C2×C4×D4—21no—
68C2×C4×D4C2×C412yes—
69C2×C4×D4C2×C412yes—
70C2×C4×D4D412yes—
71C2×C4×D4D412yes—
72C2×C4×D4D412yes—
73C2×C4×D4C2312yes—
74C2×#117—21no—
75C2×#117—21no—
76C2×#117—24no—
77C2×#117—21no—
78C22×C4⋊C4D411yescharacteristic
79C22×C22⋊C4—21no—
80C22×C42C2311yescharacteristic
81#478—28no—
82#478—28no—
83#476—24no—
84#476C2×C418yes—
85#474—24no—
86#474C2×C418yes—
87#431—21no—
88#431—21no—
89#431—21no—
90#431—21no—
91#431—21no—
92#431—21no—
93#431C2312yes—
94#431C2312yes—
95#428—24no—
96#428—24no—
97C2×#107—22no—
98C2×#106—22no—
99#423—21no—
100#423—21no—
101#423C2×C412yes—
102#423C2312yes—
103C2×C4⋊C8—21no—
104C2×C4⋊C8—21no—
105C2×C4⋊C8—21no—
106C2×C4⋊C8C2×C412yes—
107#418—24no—
108#418—24no—
109C2×#102—22no—
110C2×#102—22no—
111C2×C22⋊C8—21no—
112C2×C22⋊C8—21no—
113C2×C4×C8—21no—
114C2×C4×C8—21no—
115#390—24no—
116#390—24no—
117#388—28no—
118#388—28no—
119#384—21no—
120#384—21no—
121#382D412yes—
122#382C2312yes—
123#381—24no—
124#380—24no—
125#380—21no—
126#380—21no—
127#380D412yes—
128#380D412yes—
129#379—24no—
130C4×C4⋊C4—24no—
131C4×C4⋊C4—24no—
132C4×C4⋊C4—21no—
133C4×C4⋊C4—21no—
134C4×C4⋊C4D412yes—
135C4×C4⋊C4C2312yes—
136C4×C22⋊C4—24no—
137C4×C22⋊C4—24no—
138#328—28no—
139#328—28no—
140D4⋊C8—28no—
141C25—21no—
142C22×Q8—21no—
143C22×Q8—21no—
144C22×D4—21no—
145C22×D4—21no—
146C22×D4—21no—
147C22×D4—22no—
148C22×D4—22no—
149C22×D4—22no—
150C22×D4—22no—
151C22×D4C22⋊C412yes—
152C22×D4C22⋊C412yes—
153C22×D4C2×D412yes—
154C23×C4—21no—
155C23×C4—22no—
156C23×C4—22no—
157C23×C4—22no—
158C23×C4—22no—
159C23×C4—22no—
160C23×C4—22no—
161C23×C4—84no—
162C23×C4C2×D411yescharacteristic
163C23×C4C2×D411yescharacteristic
164C23×C4C2×D411yescharacteristic
165C2×M16—22no—
166C2×M16—22no—
167C2×M16—42no—
168C2×M16—42no—
169C22×C8—22no—
170C22×C8—41no—
171C22×C8—41no—
172C4⋊Q8—21no—
173C4⋊Q8—21no—
174C4⋊Q8—21no—
175C4⋊Q8—21no—
176C4⋊Q8—21no—
177C4⋊Q8—21no—
178C4⋊Q8C22×C412yes—
179C4⋊Q8C22×C412yes—
180#125—22no—
181#125—22no—
182#125—41no—
183#125—41no—
184C22⋊Q8—22no—
185C22⋊Q8—22no—
186C22⋊Q8—22no—
187C22⋊Q8—22no—
188C22⋊Q8—41no—
189C22⋊Q8—41no—
190C22⋊Q8—41no—
191C22⋊Q8—41no—
192C4×Q8—22no—
193C4×Q8—22no—
194C4×Q8—22no—
195C4×Q8—22no—
196C4×Q8—41no—
197C4×Q8—41no—
198C4×Q8—41no—
199C4×Q8—41no—
200C4×D4—22no—
201C4×D4—22no—
202C4×D4—22no—
203C4×D4—22no—
204C4×D4—22no—
205C4×D4—22no—
206C4×D4—22no—
207C4×D4—22no—
208C4×D4—22no—
209C4×D4—22no—
210C4×D4—22no—
211C4×D4—22no—
212C4×D4C22⋊C414yes—
213C4×D4C22⋊C414yes—
214C4×D4C22×C414yes—
215C4×D4C2×D414yes—
216#117—22no—
217#117—22no—
218#117—41no—
219#117—41no—
220#117—42no—
221#117—42no—
222#117—42no—
223#117—42no—
224#117—88no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4—21no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4—21no—
229C2×C4⋊C4—21no—
230C2×C4⋊C4—21no—
231C2×C4⋊C4—21no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—22no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—41no—
239C2×C4⋊C4—41no—
240C2×C4⋊C4C22⋊C412yes—
241C2×C4⋊C4C22⋊C412yes—
242C2×C4⋊C4C2×D412yes—
243C2×C22⋊C4—22no—
244C2×C22⋊C4—22no—
245C2×C22⋊C4—22no—
246C2×C22⋊C4—22no—
247C2×C22⋊C4—22no—
248C2×C22⋊C4—22no—
249C2×C22⋊C4—22no—
250C2×C22⋊C4—22no—
251C2×C22⋊C4—28no—
252C2×C22⋊C4—28no—
253C2×C22⋊C4—41no—
254C2×C42—22no—
255C2×C42—22no—
256C2×C42—28no—
257C2×C42—41no—
258C2×C42C22×C411yescharacteristic
259C2×C42C22×C411yescharacteristic
260C2×C42C2×D411yescharacteristic
261C2×C42C2×D411yescharacteristic
262C2×C42C2×D411yescharacteristic
263C2×C42C2×D411yescharacteristic
264C2×C42C2×D411yescharacteristic
265C2×C42#4111yescharacteristic
266C2×C42#4111yescharacteristic
267C2×C42#4111yescharacteristic
268C2×C42#4111yescharacteristic
269C2×C42C2411yescharacteristic
270#107—28no—
271#106—28no—
272C4⋊C8—22no—
273C4⋊C8—22no—
274C4⋊C8—22no—
275C4⋊C8—22no—
276C4⋊C8—22no—
277C4⋊C8—41no—
278C4⋊C8—41no—
279C4⋊C8C22×C414yes—
280#102—28no—
281#102—28no—
282C22⋊C8—22no—
283C22⋊C8—22no—
284C22⋊C8—41no—
285C22⋊C8—41no—
286C4×C8—22no—
287C4×C8—22no—
288C4×C8—41no—
289C4×C8—41no—
290#95—28no—
291#95—22no—
292#95—22no—
293#95C2×D414yes—
294#95C2×D414yes—
295C24—22no—
296C24—22no—
297C24—22no—
298C24—42no—
299C24—42no—
300C24—82no—
301C24—82no—
302C24#12011yescharacteristic
303C2×Q8—22no—
304C2×Q8—22no—
305C2×Q8—41no—
306C2×Q8—41no—
307C2×Q8—82no—
308C2×Q8—82no—
309C2×D4—22no—
310C2×D4—22no—
311C2×D4—22no—
312C2×D4—22no—
313C2×D4—22no—
314C2×D4—22no—
315C2×D4—22no—
316C2×D4—24no—
317C2×D4—24no—
318C2×D4—24no—
319C2×D4—24no—
320C2×D4—42no—
321C2×D4—42no—
322C2×D4—42no—
323C2×D4—42no—
324C2×D4—42no—
325C2×D4—42no—
326C2×D4C2×C22⋊C414yes—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—24no—
331C22×C4—24no—
332C22×C4—24no—
333C22×C4—24no—
334C22×C4—24no—
335C22×C4—24no—
336C22×C4—42no—
337C22×C4—48no—
338C22×C4—41no—
339C22×C4—41no—
340C22×C4—84no—
341C22×C4—84no—
342C22×C4—81no—
343C22×C4—84no—
344C22×C4—84no—
345C22×C4—84no—
346C22×C4—82no—
347C22×C4C2×C22⋊C411yescharacteristic
348C22×C4C2×C22⋊C411yescharacteristic
349C22×C4C4×D411yescharacteristic
350C22×C4C4×D411yescharacteristic
351C22×C4C4×D411yescharacteristic
352C22×C4C4×D411yescharacteristic
353C22×C4#12011yescharacteristic
354C22×C4#12011yescharacteristic
355C22×C4#12011yescharacteristic
356C22×C4#12111yescharacteristic
357C22×C4#12111yescharacteristic
358C22×C4#12111yescharacteristic
359C22×C4#12111yescharacteristic
360C22×C4#12311yescharacteristic
361C22×C4#12311yescharacteristic
362C22×C4#13512yes—
363C22×C4#13512yes—
364C22×C4#13712yes—
365C22×C4#13712yes—
366C22×C4C22×D411yescharacteristic
367C22×C4C22×D411yescharacteristic
368C22×C4C22×D411yescharacteristic
369M16—42no—
370M16—42no—
371M16—44no—
372C2×C8—24no—
373C2×C8—44no—
374C2×C8—42no—
375C2×C8—42no—
376C2×C8—82no—
377C2×C8—82no—
378C4⋊C4—22no—
379C4⋊C4—22no—
380C4⋊C4—22no—
381C4⋊C4—22no—
382C4⋊C4—22no—
383C4⋊C4—22no—
384C4⋊C4—22no—
385C4⋊C4—44no—
386C4⋊C4—44no—
387C4⋊C4—42no—
388C4⋊C4—42no—
389C4⋊C4—42no—
390C4⋊C4—42no—
391C4⋊C4—41no—
392C4⋊C4—41no—
393C4⋊C4—41no—
394C4⋊C4—41no—
395C4⋊C4—42no—
396C4⋊C4—42no—
397C4⋊C4—44no—
398C4⋊C4—82no—
399C4⋊C4—82no—
400C4⋊C4C2×C22⋊C414yes—
401C22⋊C4—24no—
402C22⋊C4—24no—
403C22⋊C4—24no—
404C22⋊C4—24no—
405C22⋊C4—42no—
406C22⋊C4—42no—
407C22⋊C4—44no—
408C22⋊C4—44no—
409C22⋊C4—88no—
410C22⋊C4—88no—
411C42—42no—
412C42—42no—
413C42—42no—
414C42—42no—
415C42—44no—
416C42—88no—
417C42C2×C22⋊C411yescharacteristic
418C42C2×C22⋊C411yescharacteristic
419C42C4×D411yescharacteristic
420C42C4×D411yescharacteristic
421C42C4×D411yescharacteristic
422C42C4×D411yescharacteristic
423C42#12111yescharacteristic
424C42#12111yescharacteristic
425C42#12111yescharacteristic
426C42#12111yescharacteristic
427C42#12311yescharacteristic
428C42#12311yescharacteristic
429C42C23×C411yescharacteristic
430C42C22×D411yescharacteristic
431C42C2×#4111yescharacteristic
432C42C2×#4111yescharacteristic
433C23—24no—
434C23—41no—
435C23—41no—
436C23—44no—
437C23—44no—
438C23—44no—
439C23—44no—
440C23—42no—
441C23—42no—
442C23—82no—
443C23—82no—
444C23—82no—
445C23—82no—
446C23—88no—
447C23—88no—
448C23#38611yescharacteristic
449C23#38611yescharacteristic
450C23#45212yes—
451C23#45212yes—
452C23C2×#12011yescharacteristic
453Q8—82no—
454Q8—82no—
455Q8—82no—
456Q8—82no—
457D4—24no—
458D4—24no—
459D4—42no—
460D4—42no—
461D4—44no—
462D4—44no—
463C2×C4—22no—
464C2×C4—22no—
465C2×C4—22no—
466C2×C4—24no—
467C2×C4—41no—
468C2×C4—41no—
469C2×C4—41no—
470C2×C4—41no—
471C2×C4—41no—
472C2×C4—41no—
473C2×C4—41no—
474C2×C4—41no—
475C2×C4—41no—
476C2×C4—48no—
477C2×C4—44no—
478C2×C4—48no—
479C2×C4—44no—
480C2×C4—44no—
481C2×C4—44no—
482C2×C4—44no—
483C2×C4—44no—
484C2×C4—44no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—84no—
488C2×C4—84no—
489C2×C4—82no—
490C2×C4—82no—
491C2×C4—82no—
492C2×C4—82no—
493C2×C4#38611yescharacteristic
494C2×C4#38611yescharacteristic
495C2×C4#38611yescharacteristic
496C2×C4#38611yescharacteristic
497C2×C4#38611yescharacteristic
498C2×C4#38611yescharacteristic
499C2×C4#41612yes—
500C2×C4#41612yes—
501C2×C4#41912yes—
502C2×C4#41912yes—
503C2×C4#45212yes—
504C2×C4#45212yes—
505C2×C4#48512yes—
506C2×C4#48512yes—
507C2×C4#48512yes—
508C2×C4#48512yes—
509C2×C4C22×C22⋊C411yescharacteristic
510C2×C4C2×C4×D411yescharacteristic
511C2×C4C2×C4×D411yescharacteristic
512C2×C4C2×#12111yescharacteristic
513C2×C4C2×#12111yescharacteristic
514C2×C4C2×#12311yescharacteristic
515C2×C4C2×#13511yescharacteristic
516C2×C4C2×#13511yescharacteristic
517C2×C4C2×#13711yescharacteristic
518C2×C4C2×#13711yescharacteristic
519C8—44no—
520C8—82no—
521C8—82no—
522C22—21no—
523C22—21no—
524C22—21no—
525C22—21no—
526C22—21no—
527C22—21no—
528C22—42no—
529C22—42no—
530C22—44no—
531C22—44no—
532C22—44no—
533C22—84no—
534C22—84no—
535C22—88no—
536C22—88no—
537C22#187312yes—
538C22#187312yes—
539C22C2×#38611yescharacteristic
540C22C2×#45211yescharacteristic
541C22C2×#45211yescharacteristic
542C4—21no—
543C4—21no—
544C4—21no—
545C4—21no—
546C4—21no—
547C4—21no—
548C4—21no—
549C4—44no—
550C4—84no—
551C4—84no—
552C4—82no—
553C4—82no—
554C4#187312yes—
555C4#187312yes—
556C4#190412yes—
557C4C2×#41611yescharacteristic
558C4C2×#41911yescharacteristic
559C4C2×#48511yescharacteristic
560C4C2×#48511yescharacteristic
561C2—21no—
562C2—21no—
563C2—22no—
564C2—22no—
565C2—84no—
566C2C2×#187311yescharacteristic
567C2C2×#190411yescharacteristic
568C2#2100011yescharacteristic
subgroupG/H
center, socleC22C2×#386
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
prime radicalC23×D4D4
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #17575C2semidirect
4. #17575C2semidirect
6. #9177C2semidirect
7. #8095C2semidirect
8. #7957C2semidirect
13. #3077C4semidirect
16. C2×#476C4semidirect
17. C2×#474C4semidirect
36. #1518C22semidirect
38. #1503C22semidirect
42. #1303C22semidirect
43. #1286C22semidirect

Maximal subgroups

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

subgroupindex
C2×#30772
#341862
#175752
C2×#14782
#91772
#80952
#79572