Finite Group Databasebeta

← #198451

Subgroups — #198451

Order 512.

1171 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×#3218C211yesmaximal, characteristic
2#34371C211yesmaximal, characteristic, complemented
3#21875C211yesmaximal, characteristic
4#21804C211yesmaximal, characteristic, complemented
5#17567C211yesmaximal, characteristic
6#9262C214yesmaximal
7#9258C214yesmaximal, complemented
8#7984C211yesmaximal, characteristic, complemented
9#7926C211yesmaximal, characteristic, complemented
10C22×C4⋊Q8C2211yescharacteristic
11C2×#529C2211yescharacteristic
12#3218—22no—
13#3218—22no—
14#3218—22no—
15#3218—22no—
16C2×#497C412yescomplemented
17C2×#491C412yescomplemented
18C2×#462C412yescomplemented
19C2×#461C412yescomplemented
20C2×#451C2212yes—
21C2×#448C2212yes—
22C2×#432C2211yescharacteristic
23C2×#431C2211yescharacteristic
24#2381—22no—
25#2365—22no—
26#2345—22no—
27Q8×C22⋊C4C2214yes—
28#2331—22no—
29#2327—22no—
30#2313—22no—
31#2285C2211yescharacteristic
32#2283C2211yescharacteristic
33#2283C2211yescharacteristic
34#1996C2212yes—
35#1981—22no—
36#1964C2212yes—
37#1956—22no—
38#1899—22no—
39#1799—22no—
40#1787C2214yes—
41#1521—22no—
42#1520C2212yes—
43#1519C2212yes—
44#1500—22no—
45#1289C2214yes—
46#1282C2214yes—
47C23×Q8C2312yes—
48C22×Q16C2×C414yes—
49C22×SD16C2×C414yes—
50C2×C4⋊Q8—21no—
51C2×C4⋊Q8—21no—
52C2×C4⋊Q8—21no—
53C2×C4⋊Q8—21no—
54C2×C4⋊Q8—21no—
55C2×C4⋊Q8—21no—
56C2×C4⋊Q8C2×C412yes—
57C2×C4⋊Q8C2×C412yes—
58C2×C4⋊Q8D412yes—
59C2×C4⋊Q8D412yes—
60C2×C4⋊Q8D412yes—
61C2×C4⋊Q8D412yes—
62#529—21no—
63#529—21no—
64#529—21no—
65#529—21no—
66#529—21no—
67#529—21no—
68#529—21no—
69#529—21no—
70C2×#126—21no—
71C2×#125C2×C411yescharacteristic
72C2×#124C2×C411yescharacteristic
73C2×#123—21no—
74C2×C22⋊Q8C2311yescharacteristic
75C2×#121C2311yescharacteristic
76C2×C4×Q8—24no—
77C2×C4×Q8C2×C411yescharacteristic
78C2×C4×D4—21no—
79C2×C4×D4C2×C411yescharacteristic
80C2×#117—21no—
81C2×#117—21no—
82C22×C4⋊C4—21no—
83C22×C4⋊C4—21no—
84C22×C42C2311yescharacteristic
85#497—24no—
86#497—24no—
87#491—24no—
88#491—24no—
89#462—24no—
90#462—24no—
91#461—24no—
92#461—24no—
93#451—24no—
94#451—24no—
95#448—24no—
96#448—24no—
97#432—21no—
98#432—21no—
99#432—21no—
100#432—21no—
101#431—21no—
102#431—21no—
103#431—21no—
104#431—21no—
105C2×C4⋊C8—21no—
106C2×C4⋊C8C2×C412yes—
107C2×#103—28no—
108C2×#103C2×C414yes—
109C2×#103C2×C414yes—
110C2×#99—24no—
111C2×C22⋊C8C2312yes—
112C2×C22⋊C8C2312yes—
113C2×#97C2×C412yes—
114C2×C4×C8—21no—
115#391—24no—
116#391—28no—
117#389—24no—
118#387—28no—
119#387—28no—
120#387—28no—
121#384—21no—
122#383—21no—
123#383D412yes—
124#382—21no—
125#382—21no—
126#382—21no—
127#381—21no—
128#381D412yes—
129#380—24no—
130#380—21no—
131#380D412yes—
132#380C2312yes—
133#380C2312yes—
134C4×C4⋊C4—24no—
135C4×C4⋊C4—21no—
136C4×C4⋊C4—21no—
137C4×C4⋊C4—21no—
138C4×C4⋊C4—21no—
139C4×C22⋊C4—21no—
140C4×C22⋊C4—24no—
141C4×C22⋊C4C2312yes—
142#376—21no—
143#376D412yes—
144#342—24no—
145#328—24no—
146#327—24no—
147C22×Q8—22no—
148C22×Q8—22no—
149C22×Q8—22no—
150C22×Q8—84no—
151C22×Q8C22×C411yescharacteristic
152C22×Q8C22×C414yes—
153C22×Q8C2×D414yes—
154C22×Q8C2×D414yes—
155C22×D4—21no—
156C22×D4C22×C411yescharacteristic
157C23×C4—22no—
158C23×C4—21no—
159C23×C4C2×D411yescharacteristic
160C23×C4C2×D411yescharacteristic
161C23×C4C2411yescharacteristic
162C2×Q16—28no—
163C2×Q16—84no—
164C2×SD16—28no—
165C2×SD16—84no—
166C22×C8—22no—
167C22×C8C22×C414yes—
168C4⋊Q8—22no—
169C4⋊Q8—22no—
170C4⋊Q8—22no—
171C4⋊Q8—22no—
172C4⋊Q8—22no—
173C4⋊Q8—22no—
174C4⋊Q8—22no—
175C4⋊Q8—22no—
176C4⋊Q8—22no—
177C4⋊Q8—22no—
178C4⋊Q8—22no—
179C4⋊Q8—22no—
180C4⋊Q8C22⋊C414yes—
181C4⋊Q8C22⋊C414yes—
182C4⋊Q8C22⋊C414yes—
183C4⋊Q8C22⋊C414yes—
184#126—41no—
185#126—41no—
186#126—41no—
187#126—41no—
188#125—21no—
189#125—21no—
190#125—21no—
191#125—21no—
192#124—21no—
193#124—21no—
194#124—21no—
195#124—21no—
196#123—41no—
197#123—41no—
198#123—41no—
199#123—41no—
200C22⋊Q8—21no—
201C22⋊Q8—21no—
202C22⋊Q8—21no—
203C22⋊Q8—21no—
204#121—21no—
205#121—21no—
206#121—21no—
207#121—21no—
208C4×Q8—21no—
209C4×Q8—21no—
210C4×Q8—21no—
211C4×Q8—21no—
212C4×Q8—88no—
213C4×D4—21no—
214C4×D4—21no—
215C4×D4—21no—
216C4×D4—21no—
217C4×D4—41no—
218C4×D4—41no—
219C4×D4—41no—
220C4×D4—41no—
221#117—41no—
222#117—41no—
223#117—41no—
224#117—41no—
225#117—82no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—22no—
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—21no—
239C2×C4⋊C4—21no—
240C2×C4⋊C4—21no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—21no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—28no—
246C2×C4⋊C4—41no—
247C2×C4⋊C4—41no—
248C2×C4⋊C4C22×C411yescharacteristic
249C2×C4⋊C4C22×C411yescharacteristic
250C2×C22⋊C4—21no—
251C2×C22⋊C4—22no—
252C2×C22⋊C4—21no—
253C2×C22⋊C4—21no—
254C2×C22⋊C4—21no—
255C2×C22⋊C4—21no—
256C2×C22⋊C4—22no—
257C2×C22⋊C4C2×D414yes—
258C2×C42—21no—
259C2×C42—22no—
260C2×C42—22no—
261C2×C42—28no—
262C2×C42—41no—
263C2×C42C22×C411yescharacteristic
264C2×C42C22×C411yescharacteristic
265C2×C42C2×D411yescharacteristic
266C2×C42C2×D411yescharacteristic
267C2×C42C2×D411yescharacteristic
268C2×C42C2×D411yescharacteristic
269C2×C42C2×D411yescharacteristic
270C2×C42C2×D411yescharacteristic
271C2×C42#4111yescharacteristic
272C2×C42#4111yescharacteristic
273C2×C42#4111yescharacteristic
274C2×C42#4111yescharacteristic
275C4⋊C8—22no—
276C4⋊C8—22no—
277C4⋊C8—41no—
278C4⋊C8—41no—
279#103—28no—
280#103—28no—
281#103—88no—
282#99—84no—
283C22⋊C8—22no—
284C22⋊C8—22no—
285C22⋊C8—22no—
286C22⋊C8—22no—
287#97—22no—
288#97—22no—
289C4×C8—41no—
290C4×C8—41no—
291#95—28no—
292#95—28no—
293#95—22no—
294#95—22no—
295#95—22no—
296#95—22no—
297#95—22no—
298#95—22no—
299#95—22no—
300#95—22no—
301#95—41no—
302#95C2×D414yes—
303C24—21no—
304C24C22×D411yescharacteristic
305C2×Q8—24no—
306C2×Q8—24no—
307C2×Q8—24no—
308C2×Q8—21no—
309C2×Q8—21no—
310C2×Q8—48no—
311C2×Q8—48no—
312C2×Q8—44no—
313C2×Q8—44no—
314C2×Q8—42no—
315C2×Q8—84no—
316C2×Q8—84no—
317C2×Q8—84no—
318C2×Q8C2×C22⋊C418yes—
319C2×D4—21no—
320C2×D4—21no—
321C2×D4—42no—
322C2×D4—41no—
323C2×D4—41no—
324C2×D4—42no—
325C2×D4—42no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—24no—
329C22×C4—24no—
330C22×C4—24no—
331C22×C4—21no—
332C22×C4—24no—
333C22×C4—24no—
334C22×C4—21no—
335C22×C4—21no—
336C22×C4—42no—
337C22×C4—42no—
338C22×C4—41no—
339C22×C4—41no—
340C22×C4—42no—
341C22×C4—82no—
342C22×C4—84no—
343C22×C4C4×D411yescharacteristic
344C22×C4C4×D411yescharacteristic
345C22×C4C4×D411yescharacteristic
346C22×C4C4×D411yescharacteristic
347C22×C4#12011yescharacteristic
348C22×C4#12011yescharacteristic
349C22×C4#12011yescharacteristic
350C22×C4#12011yescharacteristic
351C22×C4#12111yescharacteristic
352C22×C4#12111yescharacteristic
353C22×C4#12111yescharacteristic
354C22×C4#12111yescharacteristic
355C22×C4C23×C411yescharacteristic
356C22×C4C22×D411yescharacteristic
357C22×C4C22×D411yescharacteristic
358C22×C4C22×D411yescharacteristic
359C22×C4C2×#4111yescharacteristic
360C22×C4C2×#4111yescharacteristic
361Q16—88no—
362SD16—88no—
363C2×C8—24no—
364C2×C8—44no—
365C2×C8—44no—
366C2×C8—42no—
367C4⋊C4—24no—
368C4⋊C4—24no—
369C4⋊C4—24no—
370C4⋊C4—24no—
371C4⋊C4—24no—
372C4⋊C4—24no—
373C4⋊C4—24no—
374C4⋊C4—24no—
375C4⋊C4—21no—
376C4⋊C4—21no—
377C4⋊C4—21no—
378C4⋊C4—21no—
379C4⋊C4—44no—
380C4⋊C4—44no—
381C4⋊C4—44no—
382C4⋊C4—41no—
383C4⋊C4—41no—
384C4⋊C4—41no—
385C4⋊C4—41no—
386C4⋊C4—41no—
387C4⋊C4—41no—
388C4⋊C4—41no—
389C4⋊C4—41no—
390C4⋊C4—82no—
391C4⋊C4—82no—
392C4⋊C4—88no—
393C22⋊C4—24no—
394C22⋊C4—24no—
395C22⋊C4—41no—
396C22⋊C4—41no—
397C22⋊C4—42no—
398C22⋊C4—42no—
399C22⋊C4—41no—
400C22⋊C4—41no—
401C22⋊C4—41no—
402C22⋊C4—41no—
403C22⋊C4—41no—
404C22⋊C4—41no—
405C22⋊C4—41no—
406C22⋊C4—41no—
407C22⋊C4—44no—
408C42—44no—
409C42—44no—
410C42—41no—
411C42—41no—
412C42—82no—
413C42—88no—
414C42C2×C22⋊C411yescharacteristic
415C42C2×C22⋊C411yescharacteristic
416C42C2×C22⋊C411yescharacteristic
417C42C2×C22⋊C411yescharacteristic
418C42C4×D411yescharacteristic
419C42C4×D411yescharacteristic
420C42C4×D411yescharacteristic
421C42C4×D411yescharacteristic
422C42#12111yescharacteristic
423C42#12111yescharacteristic
424C42#12111yescharacteristic
425C42#12111yescharacteristic
426C42#12311yescharacteristic
427C42#12311yescharacteristic
428C42#12311yescharacteristic
429C42#12311yescharacteristic
430C23—41no—
431C23—41no—
432C23—41no—
433C23—42no—
434C23—81no—
435C23—81no—
436C23C2×C4×D411yescharacteristic
437C23C2×#12011yescharacteristic
438C23C2×#12111yescharacteristic
439C23#57712yes—
440C23#57812yes—
441Q8—28no—
442Q8—28no—
443Q8—48no—
444Q8—42no—
445Q8—84no—
446Q8—84no—
447Q8—81no—
448D4—42no—
449D4—81no—
450D4—82no—
451D4—81no—
452D4—81no—
453C2×C4—22no—
454C2×C4—22no—
455C2×C4—22no—
456C2×C4—22no—
457C2×C4—21no—
458C2×C4—21no—
459C2×C4—21no—
460C2×C4—21no—
461C2×C4—22no—
462C2×C4—22no—
463C2×C4—22no—
464C2×C4—22no—
465C2×C4—22no—
466C2×C4—22no—
467C2×C4—28no—
468C2×C4—41no—
469C2×C4—41no—
470C2×C4—41no—
471C2×C4—41no—
472C2×C4—41no—
473C2×C4—41no—
474C2×C4—48no—
475C2×C4—44no—
476C2×C4—44no—
477C2×C4—44no—
478C2×C4—44no—
479C2×C4—42no—
480C2×C4—44no—
481C2×C4—44no—
482C2×C4—41no—
483C2×C4—41no—
484C2×C4—41no—
485C2×C4—82no—
486C2×C4—84no—
487C2×C4—84no—
488C2×C4—82no—
489C2×C4—81no—
490C2×C4—81no—
491C2×C4—82no—
492C2×C4—82no—
493C2×C4—82no—
494C2×C4#38611yescharacteristic
495C2×C4#38611yescharacteristic
496C2×C4#38611yescharacteristic
497C2×C4#38611yescharacteristic
498C2×C4#38611yescharacteristic
499C2×C4#38611yescharacteristic
500C2×C4#38611yescharacteristic
501C2×C4#38611yescharacteristic
502C2×C4C22×C22⋊C411yescharacteristic
503C2×C4C2×C4×D411yescharacteristic
504C2×C4C2×#12111yescharacteristic
505C2×C4C2×#12311yescharacteristic
506C8—44no—
507C8—82no—
508C22—21no—
509C22—21no—
510C22—21no—
511C22—21no—
512C22—21no—
513C22—21no—
514C22—42no—
515C22—42no—
516C22—81no—
517C22—81no—
518C22—81no—
519C22—162no—
520C22C2×#38611yescharacteristic
521C22#294712yes—
522C22#301312yes—
523C22#304311yescharacteristic
524C22#304411yescharacteristic
525C4—21no—
526C4—21no—
527C4—21no—
528C4—21no—
529C4—21no—
530C4—21no—
531C4—21no—
532C4—21no—
533C4—21no—
534C4—21no—
535C4—48no—
536C4—84no—
537C4—84no—
538C4—81no—
539C4—81no—
540C4—81no—
541C4—162no—
542C4#190512yes—
543C4#288312yes—
544C2—22no—
545C2—21no—
546C2—22no—
547C2—21no—
548C2—161no—
549C2#2093011yescharacteristic
550C2#2098511yescharacteristic
551C2C2×#190511yescharacteristic
subgroupG/H
center, socleC22C2×#386
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC2×#121C23
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. #34371C2semidirect
4. #21804C2semidirect
7. #9258C2semidirect
8. #7984C2semidirect
9. #7926C2semidirect
16. C2×#497C4semidirect
17. C2×#491C4semidirect
18. C2×#462C4semidirect
19. C2×#461C4semidirect

Maximal subgroups

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

subgroupindex
C2×#32182
#343712
#218752
#218042
#175672
#92622
#92582
#79842
#79262