Finite Group Databasebeta

← #198450

Subgroups — #198450

Order 512.

1123 conjugacy classes of subgroups.

The subgroup lattice has 546 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3239C211yesmaximal, characteristic
2#34371C211yesmaximal, characteristic, complemented
3#21875C211yesmaximal, characteristic
4#21834C211yesmaximal, characteristic, complemented
5#17567C211yesmaximal, characteristic
6#9261C214yesmaximal
7#9257C214yesmaximal, complemented
8#7984C211yesmaximal, characteristic, complemented
9#7926C211yesmaximal, characteristic, complemented
10C22×C4⋊Q8C2211yescharacteristic
11C2×#529C2211yescharacteristic
12#3239—22no—
13#3239—22no—
14#3239—22no—
15#3239—22no—
16C2×#501C412yescomplemented
17C2×#489C412yescomplemented
18C2×#484C2212yes—
19C2×#481C2212yes—
20C2×#475C412yescomplemented
21C2×#474C412yescomplemented
22C2×#432C2211yescharacteristic
23C2×#431C2211yescharacteristic
24#2381—22no—
25#2365—22no—
26#2345C2214yes—
27Q8×C22⋊C4—22no—
28#2331—22no—
29#2327—22no—
30#2313—22no—
31#2285C2211yescharacteristic
32#2283C2211yescharacteristic
33#2283C2211yescharacteristic
34#1996C2212yes—
35#1990—22no—
36#1987—22no—
37#1964C2212yes—
38#1899—22no—
39#1824C2214yes—
40#1799—22no—
41#1521—22no—
42#1520C2212yes—
43#1519C2212yes—
44#1500—22no—
45#1289C2214yes—
46#1282C2214yes—
47C23×Q8—21no—
48C2×C4⋊Q8—21no—
49C2×C4⋊Q8—21no—
50C2×C4⋊Q8—21no—
51C2×C4⋊Q8—21no—
52C2×C4⋊Q8—21no—
53C2×C4⋊Q8—21no—
54C2×C4⋊Q8C2×C412yes—
55C2×C4⋊Q8C2×C412yes—
56C2×C4⋊Q8D412yes—
57C2×C4⋊Q8D412yes—
58C2×C4⋊Q8D412yes—
59C2×C4⋊Q8D412yes—
60#529—21no—
61#529—21no—
62#529—21no—
63#529—21no—
64#529—21no—
65#529—21no—
66#529—21no—
67#529—21no—
68C2×#126—21no—
69C2×#125C2×C411yescharacteristic
70C2×#124C2×C411yescharacteristic
71C2×#123—21no—
72C2×C22⋊Q8C2311yescharacteristic
73C2×#121C2311yescharacteristic
74C2×C4×Q8—24no—
75C2×C4×Q8C2×C411yescharacteristic
76C2×C4×D4—21no—
77C2×C4×D4C2×C411yescharacteristic
78C2×#117—21no—
79C2×#117—21no—
80C22×C4⋊C4—21no—
81C22×C4⋊C4C2312yes—
82C22×C42C2311yescharacteristic
83#501—24no—
84#501—24no—
85#489—24no—
86#489—24no—
87#484—24no—
88#484—24no—
89#481—24no—
90#481—24no—
91#475—24no—
92#475—24no—
93#474—24no—
94#474—24no—
95#432—21no—
96#432—21no—
97#432—21no—
98#432—21no—
99#431—21no—
100#431—21no—
101#431—21no—
102#431—21no—
103C2×#107C2×C414yes—
104C2×#106C2×C414yes—
105C2×C4⋊C8—21no—
106C2×C4⋊C8C2×C412yes—
107C2×#103C2×C414yes—
108C2×#102C2×C414yes—
109C2×#99—24no—
110C2×C22⋊C8C2312yes—
111C2×C22⋊C8C2312yes—
112C2×#97C2×C412yes—
113C2×C4×C8—21no—
114#391—24no—
115#391—28no—
116#389—24no—
117#387—28no—
118#387—28no—
119#387—28no—
120#384—21no—
121#383—21no—
122#383D412yes—
123#382—21no—
124#382—21no—
125#382—21no—
126#381—21no—
127#381D412yes—
128#380—24no—
129#380—21no—
130#380D412yes—
131#380C2312yes—
132#380C2312yes—
133C4×C4⋊C4—24no—
134C4×C4⋊C4—21no—
135C4×C4⋊C4—21no—
136C4×C4⋊C4—21no—
137C4×C4⋊C4—21no—
138C4×C22⋊C4—21no—
139C4×C22⋊C4—24no—
140C4×C22⋊C4C2312yes—
141#376—21no—
142#376D412yes—
143#342—24no—
144#340—28no—
145#328—24no—
146#327—24no—
147C22×Q8—22no—
148C22×Q8—22no—
149C22×Q8—22no—
150C22×Q8—22no—
151C22×Q8—22no—
152C22×Q8—22no—
153C22×Q8—84no—
154C22×Q8C22×C411yescharacteristic
155C22×D4—21no—
156C22×D4C22×C411yescharacteristic
157C23×C4—22no—
158C23×C4—21no—
159C23×C4C2×D411yescharacteristic
160C23×C4C2×D411yescharacteristic
161C23×C4C2411yescharacteristic
162C22×C8—22no—
163C22×C8C22×C414yes—
164C4⋊Q8—22no—
165C4⋊Q8—22no—
166C4⋊Q8—22no—
167C4⋊Q8—22no—
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⋊Q8C22⋊C414yes—
177C4⋊Q8C22⋊C414yes—
178C4⋊Q8C22⋊C414yes—
179C4⋊Q8C22⋊C414yes—
180#126—41no—
181#126—41no—
182#126—41no—
183#126—41no—
184#125—21no—
185#125—21no—
186#125—21no—
187#125—21no—
188#124—21no—
189#124—21no—
190#124—21no—
191#124—21no—
192#123—41no—
193#123—41no—
194#123—41no—
195#123—41no—
196C22⋊Q8—21no—
197C22⋊Q8—21no—
198C22⋊Q8—21no—
199C22⋊Q8—21no—
200#121—21no—
201#121—21no—
202#121—21no—
203#121—21no—
204C4×Q8—21no—
205C4×Q8—21no—
206C4×Q8—21no—
207C4×Q8—21no—
208C4×Q8—88no—
209C4×D4—21no—
210C4×D4—21no—
211C4×D4—21no—
212C4×D4—21no—
213C4×D4—41no—
214C4×D4—41no—
215C4×D4—41no—
216C4×D4—41no—
217#117—41no—
218#117—41no—
219#117—41no—
220#117—41no—
221#117—82no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—21no—
232C2×C4⋊C4—21no—
233C2×C4⋊C4—21no—
234C2×C4⋊C4—22no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—21no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—28no—
239C2×C4⋊C4—41no—
240C2×C4⋊C4—41no—
241C2×C4⋊C4C22×C411yescharacteristic
242C2×C4⋊C4C22×C411yescharacteristic
243C2×C4⋊C4C22×C414yes—
244C2×C4⋊C4C2×D414yes—
245C2×C4⋊C4C2×D414yes—
246C2×C22⋊C4—21no—
247C2×C22⋊C4—22no—
248C2×C22⋊C4—21no—
249C2×C22⋊C4—21no—
250C2×C22⋊C4—21no—
251C2×C22⋊C4—21no—
252C2×C22⋊C4—22no—
253C2×C22⋊C4C2×D414yes—
254C2×C42—21no—
255C2×C42—22no—
256C2×C42—22no—
257C2×C42—28no—
258C2×C42—41no—
259C2×C42C22×C411yescharacteristic
260C2×C42C22×C411yescharacteristic
261C2×C42C2×D411yescharacteristic
262C2×C42C2×D411yescharacteristic
263C2×C42C2×D411yescharacteristic
264C2×C42C2×D411yescharacteristic
265C2×C42C2×D411yescharacteristic
266C2×C42C2×D411yescharacteristic
267C2×C42#4111yescharacteristic
268C2×C42#4111yescharacteristic
269C2×C42#4111yescharacteristic
270C2×C42#4111yescharacteristic
271#107—28no—
272#106—28no—
273C4⋊C8—22no—
274C4⋊C8—22no—
275C4⋊C8—41no—
276C4⋊C8—41no—
277#103—28no—
278#102—28no—
279#99—84no—
280C22⋊C8—22no—
281C22⋊C8—22no—
282C22⋊C8—22no—
283C22⋊C8—22no—
284#97—22no—
285#97—22no—
286C4×C8—41no—
287C4×C8—41no—
288#95—28no—
289#95—28no—
290#95—22no—
291#95—22no—
292#95—22no—
293#95—22no—
294#95—22no—
295#95—22no—
296#95—22no—
297#95—22no—
298#95—41no—
299#95C2×D414yes—
300C24—21no—
301C24C22×D411yescharacteristic
302C2×Q8—24no—
303C2×Q8—24no—
304C2×Q8—24no—
305C2×Q8—24no—
306C2×Q8—21no—
307C2×Q8—21no—
308C2×Q8—48no—
309C2×Q8—42no—
310C2×Q8—84no—
311C2×Q8—84no—
312C2×Q8—84no—
313C2×Q8—84no—
314C2×Q8—84no—
315C2×D4—21no—
316C2×D4—21no—
317C2×D4—42no—
318C2×D4—41no—
319C2×D4—41no—
320C2×D4—42no—
321C2×D4—42no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—24no—
325C22×C4—24no—
326C22×C4—24no—
327C22×C4—21no—
328C22×C4—24no—
329C22×C4—24no—
330C22×C4—21no—
331C22×C4—21no—
332C22×C4—42no—
333C22×C4—42no—
334C22×C4—41no—
335C22×C4—41no—
336C22×C4—42no—
337C22×C4—82no—
338C22×C4—84no—
339C22×C4C4×D411yescharacteristic
340C22×C4C4×D411yescharacteristic
341C22×C4C4×D411yescharacteristic
342C22×C4C4×D411yescharacteristic
343C22×C4#12011yescharacteristic
344C22×C4#12011yescharacteristic
345C22×C4#12011yescharacteristic
346C22×C4#12011yescharacteristic
347C22×C4#12111yescharacteristic
348C22×C4#12111yescharacteristic
349C22×C4#12111yescharacteristic
350C22×C4#12111yescharacteristic
351C22×C4C23×C411yescharacteristic
352C22×C4C22×D411yescharacteristic
353C22×C4C22×D411yescharacteristic
354C22×C4C22×D411yescharacteristic
355C22×C4C2×#4111yescharacteristic
356C22×C4C2×#4111yescharacteristic
357C2×C8—24no—
358C2×C8—44no—
359C2×C8—44no—
360C2×C8—42no—
361C4⋊C4—24no—
362C4⋊C4—24no—
363C4⋊C4—24no—
364C4⋊C4—24no—
365C4⋊C4—24no—
366C4⋊C4—24no—
367C4⋊C4—24no—
368C4⋊C4—21no—
369C4⋊C4—21no—
370C4⋊C4—21no—
371C4⋊C4—21no—
372C4⋊C4—44no—
373C4⋊C4—44no—
374C4⋊C4—44no—
375C4⋊C4—41no—
376C4⋊C4—41no—
377C4⋊C4—41no—
378C4⋊C4—41no—
379C4⋊C4—41no—
380C4⋊C4—41no—
381C4⋊C4—41no—
382C4⋊C4—41no—
383C4⋊C4—82no—
384C4⋊C4—82no—
385C4⋊C4—88no—
386C4⋊C4C2×C22⋊C418yes—
387C22⋊C4—24no—
388C22⋊C4—24no—
389C22⋊C4—41no—
390C22⋊C4—41no—
391C22⋊C4—42no—
392C22⋊C4—42no—
393C22⋊C4—41no—
394C22⋊C4—41no—
395C22⋊C4—41no—
396C22⋊C4—41no—
397C22⋊C4—41no—
398C22⋊C4—41no—
399C22⋊C4—41no—
400C22⋊C4—41no—
401C22⋊C4—44no—
402C42—44no—
403C42—44no—
404C42—41no—
405C42—41no—
406C42—82no—
407C42—88no—
408C42C2×C22⋊C411yescharacteristic
409C42C2×C22⋊C411yescharacteristic
410C42C2×C22⋊C411yescharacteristic
411C42C2×C22⋊C411yescharacteristic
412C42C4×D411yescharacteristic
413C42C4×D411yescharacteristic
414C42C4×D411yescharacteristic
415C42C4×D411yescharacteristic
416C42#12111yescharacteristic
417C42#12111yescharacteristic
418C42#12111yescharacteristic
419C42#12111yescharacteristic
420C42#12311yescharacteristic
421C42#12311yescharacteristic
422C42#12311yescharacteristic
423C42#12311yescharacteristic
424C23—41no—
425C23—41no—
426C23—41no—
427C23—42no—
428C23—81no—
429C23—81no—
430C23C2×C4×D411yescharacteristic
431C23C2×#12011yescharacteristic
432C23C2×#12111yescharacteristic
433C23#57712yes—
434C23#57812yes—
435Q8—48no—
436Q8—48no—
437Q8—42no—
438Q8—84no—
439Q8—84no—
440Q8—81no—
441D4—42no—
442D4—81no—
443D4—82no—
444D4—81no—
445D4—81no—
446C2×C4—22no—
447C2×C4—22no—
448C2×C4—22no—
449C2×C4—22no—
450C2×C4—21no—
451C2×C4—21no—
452C2×C4—21no—
453C2×C4—21no—
454C2×C4—22no—
455C2×C4—22no—
456C2×C4—22no—
457C2×C4—22no—
458C2×C4—22no—
459C2×C4—22no—
460C2×C4—28no—
461C2×C4—41no—
462C2×C4—41no—
463C2×C4—41no—
464C2×C4—41no—
465C2×C4—41no—
466C2×C4—41no—
467C2×C4—48no—
468C2×C4—44no—
469C2×C4—44no—
470C2×C4—44no—
471C2×C4—44no—
472C2×C4—42no—
473C2×C4—44no—
474C2×C4—44no—
475C2×C4—41no—
476C2×C4—41no—
477C2×C4—41no—
478C2×C4—82no—
479C2×C4—84no—
480C2×C4—84no—
481C2×C4—82no—
482C2×C4—81no—
483C2×C4—81no—
484C2×C4—82no—
485C2×C4—82no—
486C2×C4—82no—
487C2×C4#38611yescharacteristic
488C2×C4#38611yescharacteristic
489C2×C4#38611yescharacteristic
490C2×C4#38611yescharacteristic
491C2×C4#38611yescharacteristic
492C2×C4#38611yescharacteristic
493C2×C4#38611yescharacteristic
494C2×C4#38611yescharacteristic
495C2×C4C22×C22⋊C411yescharacteristic
496C2×C4C2×C4×D411yescharacteristic
497C2×C4C2×#12111yescharacteristic
498C2×C4C2×#12311yescharacteristic
499C8—44no—
500C8—82no—
501C22—21no—
502C22—21no—
503C22—21no—
504C22—21no—
505C22—21no—
506C22—21no—
507C22—42no—
508C22—42no—
509C22—81no—
510C22—81no—
511C22—81no—
512C22—162no—
513C22C2×#38611yescharacteristic
514C22#294712yes—
515C22#301312yes—
516C22#304311yescharacteristic
517C22#304411yescharacteristic
518C4—21no—
519C4—21no—
520C4—21no—
521C4—21no—
522C4—21no—
523C4—21no—
524C4—21no—
525C4—21no—
526C4—21no—
527C4—21no—
528C4—48no—
529C4—84no—
530C4—84no—
531C4—81no—
532C4—81no—
533C4—81no—
534C4—162no—
535C4#190412yes—
536C4#288412yes—
537C2—22no—
538C2—21no—
539C2—22no—
540C2—21no—
541C2—161no—
542C2#2093011yescharacteristic
543C2#2098611yescharacteristic
544C2C2×#190411yescharacteristic
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. #21834C2semidirect
7. #9257C2semidirect
8. #7984C2semidirect
9. #7926C2semidirect
16. C2×#501C4semidirect
17. C2×#489C4semidirect
20. C2×#475C4semidirect
21. C2×#474C4semidirect

Maximal subgroups

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

subgroupindex
C2×#32392
#343712
#218752
#218342
#175672
#92612
#92572
#79842
#79262