Finite Group Databasebeta

← #198398

Subgroups — #198398

Order 512.

1278 conjugacy classes of subgroups.

The subgroup lattice has 569 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3227C211yesmaximal, characteristic
2#34216C211yesmaximal, characteristic, complemented
3#21834C211yesmaximal, characteristic, complemented
4#21212C211yesmaximal, characteristic
5#17574C211yesmaximal, characteristic
6#9261C218yesmaximal, complemented
7#7981C211yesmaximal, characteristic, complemented
8#7924C211yesmaximal, characteristic, complemented
9C2×#532C2211yescharacteristic
10C22×C4⋊Q8C2211yescharacteristic
11#3227—22no—
12#3227—22no—
13#3227—22no—
14#3227—22no—
15C2×#498C412yescomplemented
16C2×#486C412yescomplemented
17C2×#481C2214yes—
18C2×#476C414yescomplemented
19C2×#431C2211yescharacteristic
20C2×#431C2211yescharacteristic
21C2×#410—21no—
22C2×#410—21no—
23#2369—22no—
24#2365—22no—
25#2345C2214yes—
26Q8×C22⋊C4—22no—
27#2331—22no—
28#2316—21no—
29C4×C4⋊Q8—22no—
30#2285C2211yescharacteristic
31#2285C2211yescharacteristic
32#2274—21no—
33C4×#117C2211yescharacteristic
34#1990—22no—
35#1987—22no—
36#1964—22no—
37#1964C2214yes—
38#1824C2214yes—
39#1519C2214yes—
40#1517—22no—
41#1502—22no—
42#1289C2214yes—
43#1282C2214yes—
44C22×#41—21no—
45C23×Q8—21no—
46#532—21no—
47#532—21no—
48#532—21no—
49#532—21no—
50#532—21no—
51#532—21no—
52#532—21no—
53#532—21no—
54C2×C4⋊Q8—21no—
55C2×C4⋊Q8—21no—
56C2×C4⋊Q8—21no—
57C2×C4⋊Q8—21no—
58C2×C4⋊Q8—21no—
59C2×C4⋊Q8—21no—
60C2×C4⋊Q8C2×C411yescharacteristic
61C2×C4⋊Q8C2×C412yes—
62C2×C4⋊Q8C2×C412yes—
63C2×C4⋊Q8D412yes—
64C2×C4⋊Q8D412yes—
65C2×C4⋊Q8D412yes—
66C2×C4⋊Q8D412yes—
67C2×#127C2×C411yescharacteristic
68C2×#124—21no—
69C2×#121—21no—
70C2×#121C2312yes—
71C2×C4×Q8—24no—
72C2×C4×D4—21no—
73C2×C4×D4C2×C412yes—
74C2×#117—21no—
75C2×#117—21no—
76C2×#117D411yescharacteristic
77C2×#117D411yescharacteristic
78C22×C4⋊C4—21no—
79C22×C4⋊C4C2312yes—
80C22×C42C2311yescharacteristic
81#498—24no—
82#498—24no—
83#486—24no—
84#486—24no—
85#481—28no—
86#481—28no—
87#476—28no—
88#476—28no—
89#431—21no—
90#431—21no—
91#431—21no—
92#431—21no—
93#431—21no—
94#431—21no—
95#431—21no—
96#431—21no—
97C2×#106C2×C418yes—
98C2×C4⋊C8—21no—
99C2×C4⋊C8C2×C412yes—
100C2×#102C2×C418yes—
101#410—44no—
102#410—44no—
103C2×#99—24no—
104C2×#99—24no—
105C2×C22⋊C8C2312yes—
106C2×C22⋊C8C2312yes—
107C2×C4×C8—21no—
108C2×C4×C8C2×C412yes—
109#391—24no—
110#391—28no—
111#389—24no—
112#389—24no—
113#389—24no—
114#387—28no—
115#384—21no—
116#384—21no—
117#382—21no—
118#382D412yes—
119#380—21no—
120#380—24no—
121#380—21no—
122#380C2312yes—
123#380C2312yes—
124C4×C4⋊C4—24no—
125C4×C4⋊C4—24no—
126C4×C4⋊C4—24no—
127C4×C4⋊C4—21no—
128C4×C4⋊C4—21no—
129C4×C4⋊C4—21no—
130C4×C4⋊C4—21no—
131C4×C4⋊C4—21no—
132C4×C4⋊C4D412yes—
133C4×C22⋊C4—24no—
134C4×C22⋊C4D412yes—
135C4×C22⋊C4C2312yes—
136#376—21no—
137C43—21no—
138#340—28no—
139#327—28no—
140C2×#41—41no—
141C2×#41—41no—
142C2×#41—41no—
143C2×#41—41no—
144C2×#41—42no—
145C2×#41—42no—
146C2×#41—42no—
147C2×#41—42no—
148C22×Q8—22no—
149C22×Q8—22no—
150C22×Q8—21no—
151C22×Q8—22no—
152C22×Q8—22no—
153C22×Q8—22no—
154C22×Q8—22no—
155C22×Q8—84no—
156C22×D4—22no—
157C22×D4—21no—
158C22×D4—21no—
159C22×D4C22×C412yes—
160C23×C4—22no—
161C23×C4—22no—
162C23×C4C2×D411yescharacteristic
163C23×C4C2×D411yescharacteristic
164C23×C4C2411yescharacteristic
165C22×C8—22no—
166C22×C8C22×C414yes—
167C4⋊Q8—21no—
168C4⋊Q8—21no—
169C4⋊Q8—22no—
170C4⋊Q8—22no—
171C4⋊Q8—22no—
172C4⋊Q8—22no—
173C4⋊Q8—21no—
174C4⋊Q8—21no—
175C4⋊Q8—22no—
176C4⋊Q8—22no—
177C4⋊Q8—22no—
178C4⋊Q8—22no—
179C4⋊Q8—22no—
180C4⋊Q8—22no—
181C4⋊Q8—22no—
182C4⋊Q8—22no—
183C4⋊Q8C22⋊C414yes—
184C4⋊Q8C22⋊C414yes—
185C4⋊Q8C22⋊C414yes—
186C4⋊Q8C22⋊C414yes—
187#127—21no—
188#127—21no—
189#127—21no—
190#127—21no—
191#124—41no—
192#124—41no—
193#124—41no—
194#124—41no—
195#121—22no—
196#121—22no—
197#121—22no—
198#121—22no—
199#121—41no—
200#121—41no—
201#121—41no—
202#121—41no—
203C4×Q8—88no—
204C4×D4—22no—
205C4×D4—22no—
206C4×D4—22no—
207C4×D4—22no—
208C4×D4—41no—
209C4×D4—41no—
210C4×D4—41no—
211C4×D4—41no—
212#117—22no—
213#117—22no—
214#117—22no—
215#117—42no—
216#117—42no—
217#117—42no—
218#117—42no—
219#117SD1614yes—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—21no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—21no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—21no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—22no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—28no—
237C2×C4⋊C4—41no—
238C2×C4⋊C4—41no—
239C2×C4⋊C4C22×C412yes—
240C2×C4⋊C4C22×C414yes—
241C2×C4⋊C4C2×D414yes—
242C2×C4⋊C4C2×D414yes—
243C2×C22⋊C4—22no—
244C2×C22⋊C4—22no—
245C2×C22⋊C4—22no—
246C2×C22⋊C4—22no—
247C2×C22⋊C4C2×D412yes—
248C2×C22⋊C4C2×D412yes—
249C2×C42—21no—
250C2×C42—22no—
251C2×C42—21no—
252C2×C42—22no—
253C2×C42—22no—
254C2×C42—28no—
255C2×C42—41no—
256C2×C42C22×C411yescharacteristic
257C2×C42C22×C411yescharacteristic
258C2×C42C2×D411yescharacteristic
259C2×C42C2×D411yescharacteristic
260C2×C42C2×D411yescharacteristic
261C2×C42C2×D411yescharacteristic
262C2×C42C2×D411yescharacteristic
263C2×C42C2×D411yescharacteristic
264C2×C42#4111yescharacteristic
265C2×C42#4111yescharacteristic
266C2×C42#4111yescharacteristic
267C2×C42#4111yescharacteristic
268#106—216no—
269C4⋊C8—22no—
270C4⋊C8—22no—
271C4⋊C8—41no—
272C4⋊C8—41no—
273#102—216no—
274#99—48no—
275#99—48no—
276C22⋊C8—22no—
277C22⋊C8—22no—
278C22⋊C8—22no—
279C22⋊C8—22no—
280C4×C8—22no—
281C4×C8—22no—
282C4×C8—41no—
283C4×C8—41no—
284#95—28no—
285#95—28no—
286#95—22no—
287#95—22no—
288#95—22no—
289#95—22no—
290#95—41no—
291#95C2×D414yes—
292C24—22no—
293C24C22×D411yescharacteristic
294#41—82no—
295#41—82no—
296#41—82no—
297#41—82no—
298#41—82no—
299#41—82no—
300#41—82no—
301#41—82no—
302C2×Q8—24no—
303C2×Q8—24no—
304C2×Q8—24no—
305C2×Q8—24no—
306C2×Q8—48no—
307C2×Q8—41no—
308C2×Q8—41no—
309C2×Q8—42no—
310C2×Q8—42no—
311C2×Q8—84no—
312C2×Q8—84no—
313C2×Q8—84no—
314C2×Q8—84no—
315C2×Q8—84no—
316C2×D4—22no—
317C2×D4—22no—
318C2×D4—44no—
319C2×D4—42no—
320C2×D4—42no—
321C2×D4—42no—
322C2×D4—42no—
323C2×D4—42no—
324C2×D4—42no—
325C2×D4—41no—
326C2×D4—41no—
327C2×D4—41no—
328C2×D4—41no—
329C2×D4—42no—
330C2×D4—42no—
331C2×D4—82no—
332C22×C4—22no—
333C22×C4—24no—
334C22×C4—24no—
335C22×C4—24no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—24no—
339C22×C4—42no—
340C22×C4—42no—
341C22×C4—42no—
342C22×C4—42no—
343C22×C4—44no—
344C22×C4—44no—
345C22×C4—42no—
346C22×C4—84no—
347C22×C4C4×D411yescharacteristic
348C22×C4C4×D411yescharacteristic
349C22×C4C4×D411yescharacteristic
350C22×C4C4×D411yescharacteristic
351C22×C4#12011yescharacteristic
352C22×C4#12011yescharacteristic
353C22×C4#12011yescharacteristic
354C22×C4#12011yescharacteristic
355C22×C4#12111yescharacteristic
356C22×C4#12111yescharacteristic
357C22×C4#12111yescharacteristic
358C22×C4#12111yescharacteristic
359C22×C4C2×SD1612yes—
360C22×C4#13512yes—
361C22×C4AGL(1,ℤ/8)12yes—
362C22×C4#13712yes—
363C22×C4C23×C411yescharacteristic
364C22×C4C22×D411yescharacteristic
365C22×C4C22×D411yescharacteristic
366C22×C4C22×D411yescharacteristic
367C22×C4C2×#4111yescharacteristic
368C22×C4C2×#4111yescharacteristic
369C2×C8—24no—
370C2×C8—44no—
371C2×C8—44no—
372C2×C8—42no—
373C4⋊C4—24no—
374C4⋊C4—24no—
375C4⋊C4—24no—
376C4⋊C4—24no—
377C4⋊C4—24no—
378C4⋊C4—24no—
379C4⋊C4—24no—
380C4⋊C4—24no—
381C4⋊C4—22no—
382C4⋊C4—22no—
383C4⋊C4—44no—
384C4⋊C4—44no—
385C4⋊C4—44no—
386C4⋊C4—41no—
387C4⋊C4—41no—
388C4⋊C4—42no—
389C4⋊C4—44no—
390C4⋊C4—82no—
391C4⋊C4—88no—
392C4⋊C4—82no—
393C4⋊C4C2×C22⋊C418yes—
394C22⋊C4—22no—
395C22⋊C4—22no—
396C22⋊C4—22no—
397C22⋊C4—42no—
398C22⋊C4—42no—
399C22⋊C4—42no—
400C22⋊C4—42no—
401C22⋊C4—42no—
402C22⋊C4—42no—
403C22⋊C4—42no—
404C22⋊C4—42no—
405C22⋊C4C2×SD1614yes—
406C42—24no—
407C42—44no—
408C42—44no—
409C42—42no—
410C42—44no—
411C42—88no—
412C42—82no—
413C42C2×C22⋊C411yescharacteristic
414C42C2×C22⋊C411yescharacteristic
415C42C2×C22⋊C411yescharacteristic
416C42C2×C22⋊C411yescharacteristic
417C42C4×D411yescharacteristic
418C42C4×D411yescharacteristic
419C42C4×D411yescharacteristic
420C42C4×D411yescharacteristic
421C42#12111yescharacteristic
422C42#12111yescharacteristic
423C42#12111yescharacteristic
424C42#12111yescharacteristic
425C42#12311yescharacteristic
426C42#12311yescharacteristic
427C42#12311yescharacteristic
428C42#12311yescharacteristic
429C23—41no—
430C23—41no—
431C23—42no—
432C23—42no—
433C23—42no—
434C23—82no—
435C23—82no—
436C23C2×C4×D411yescharacteristic
437C23C2×#12011yescharacteristic
438C23C2×#12111yescharacteristic
439C23C22×SD1611yescharacteristic
440C23C2×#13511yescharacteristic
441C23C2×AGL(1,ℤ/8)11yescharacteristic
442C23C2×#13711yescharacteristic
443Q8—48no—
444Q8—48no—
445Q8—84no—
446Q8—84no—
447Q8—82no—
448Q8—82no—
449D4—44no—
450D4—82no—
451D4—82no—
452D4—82no—
453D4—82no—
454D4—82no—
455D4—82no—
456D4—82no—
457D4—82no—
458D4—82no—
459C2×C4—22no—
460C2×C4—22no—
461C2×C4—22no—
462C2×C4—22no—
463C2×C4—22no—
464C2×C4—22no—
465C2×C4—24no—
466C2×C4—28no—
467C2×C4—41no—
468C2×C4—41no—
469C2×C4—41no—
470C2×C4—41no—
471C2×C4—41no—
472C2×C4—41no—
473C2×C4—42no—
474C2×C4—42no—
475C2×C4—48no—
476C2×C4—42no—
477C2×C4—42no—
478C2×C4—44no—
479C2×C4—44no—
480C2×C4—42no—
481C2×C4—42no—
482C2×C4—42no—
483C2×C4—44no—
484C2×C4—42no—
485C2×C4—84no—
486C2×C4—84no—
487C2×C4—84no—
488C2×C4—84no—
489C2×C4—82no—
490C2×C4—82no—
491C2×C4—82no—
492C2×C4#38611yescharacteristic
493C2×C4#38611yescharacteristic
494C2×C4#38611yescharacteristic
495C2×C4#38611yescharacteristic
496C2×C4#38611yescharacteristic
497C2×C4#38611yescharacteristic
498C2×C4#38611yescharacteristic
499C2×C4#38611yescharacteristic
500C2×C4C4×SD1612yes—
501C2×C4C4×SD1612yes—
502C2×C4#44012yes—
503C2×C4#44012yes—
504C2×C4#44812yes—
505C2×C4#44812yes—
506C2×C4#44912yes—
507C2×C4#44912yes—
508C2×C4#46012yes—
509C2×C4#46012yes—
510C2×C4#46412yes—
511C2×C4#46412yes—
512C2×C4C22×C22⋊C411yescharacteristic
513C2×C4C2×C4×D411yescharacteristic
514C2×C4C2×#12111yescharacteristic
515C2×C4C2×#12311yescharacteristic
516C8—44no—
517C8—82no—
518C22—21no—
519C22—21no—
520C22—21no—
521C22—21no—
522C22—21no—
523C22—21no—
524C22—41no—
525C22—41no—
526C22—41no—
527C22—41no—
528C22—82no—
529C22—82no—
530C22—82no—
531C22—162no—
532C22—162no—
533C22C2×#38611yescharacteristic
534C22C2×C4×SD1611yescharacteristic
535C22C2×#44011yescharacteristic
536C22C2×#44811yescharacteristic
537C22C2×#44911yescharacteristic
538C22C2×#46011yescharacteristic
539C22C2×#46411yescharacteristic
540C4—21no—
541C4—21no—
542C4—21no—
543C4—21no—
544C4—21no—
545C4—21no—
546C4—44no—
547C4—48no—
548C4—82no—
549C4—84no—
550C4—82no—
551C4—162no—
552C4#188012yes—
553C4#188012yes—
554C4#188012yes—
555C4#188012yes—
556C4#190412yes—
557C4#288412yes—
558C2—21no—
559C2—21no—
560C2—21no—
561C2—21no—
562C2—21no—
563C2—21no—
564C2—162no—
565C2C2×#188011yescharacteristic
566C2#2098611yescharacteristic
567C2C2×#190411yescharacteristic
subgroupG/H
center, socleC22C2×#386
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC2×#532C22
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C22×C42 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #34216C2semidirect
3. #21834C2semidirect
6. #9261C2semidirect
7. #7981C2semidirect
8. #7924C2semidirect
15. C2×#498C4semidirect
16. C2×#486C4semidirect
18. C2×#476C4semidirect

Maximal subgroups

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

subgroupindex
C2×#32272
#342162
#218342
#212122
#175742
#92612
#79812
#79242