Finite Group Databasebeta

← #189410

Subgroups — #189410

Order 512.

803 conjugacy classes of subgroups.

The subgroup lattice has 532 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60288C211yesmaximal, characteristic
2#21370C211yesmaximal, characteristic, complemented
3#17816C211yesmaximal, characteristic, complemented
4#17669C211yesmaximal, characteristic
5#17642C211yesmaximal, characteristic
6#8693C212yesmaximal, complemented
7#8692C212yesmaximal, complemented
8#8691C212yesmaximal, complemented
9#8690C212yesmaximal, complemented
10#7997C211yesmaximal, characteristic, complemented
11C2×#1291C211yesmaximal, characteristic, complemented
12C2×#529C2211yescharacteristic
13#2994C412yes—
14#2994C412yes—
15#2992—22no—
16#2984C412yes—
17#2974C412yes—
18#2973—21no—
19#2972—21no—
20#2968—21no—
21#2968—21no—
22#2966C2212yes—
23#2966C2212yes—
24#2924—22no—
25#2920C2211yescharacteristic
26#2920C2211yescharacteristic
27C2×#432C2211yescharacteristic
28C2×#432C2211yescharacteristic
29C2×#431C2211yescharacteristic
30C2×C4×M16C2211yescharacteristic
31#1950—22no—
32#1855—21no—
33#1855—21no—
34C4×#108C2212yescomplemented
35C4×#108C2212yescomplemented
36#1572—22no—
37#1571—21no—
38#1570—21no—
39#1568C2212yescomplemented
40#1568C2212yescomplemented
41#1534—21no—
42#1533—21no—
43#1532—22no—
44#1529C2212yes—
45#1529C2212yes—
46#1529C2212yes—
47#1529C2212yes—
48#1528—21no—
49#1527—21no—
50#1524—22no—
51#1291C2212yescomplemented
52#1291C2212yescomplemented
53#1291C2211yescharacteristic, complemented
54#1291C2211yescharacteristic, complemented
55#1291C2211yescharacteristic, complemented
56#1291C2211yescharacteristic, complemented
57C22×M16—21no—
58#529—21no—
59#529—21no—
60#529—21no—
61#529—21no—
62#529C2×C412yes—
63#529D412yes—
64#529Q812yes—
65#529C2312yes—
66C2×#126—21no—
67C2×#125D411yescharacteristic
68C2×#124D411yescharacteristic
69C2×#123—21no—
70C2×C22⋊Q8Q811yescharacteristic
71C2×#121Q811yescharacteristic
72C2×C4×Q8C2×C411yescharacteristic
73C2×C4×D4—21no—
74C2×C4×D4C2×C411yescharacteristic
75C2×#117—21no—
76C22×C42C2311yescharacteristic
77#446—22no—
78#446—22no—
79#446C2×C412yes—
80#446C2×C412yes—
81C8×Q8C2×C412yes—
82C8×Q8C2×C412yes—
83#436C2×C412yes—
84#436C2×C412yes—
85#435—22no—
86#435—22no—
87#435—22no—
88#435—22no—
89#435—22no—
90#435—22no—
91C8×D4—22no—
92C8×D4—22no—
93C8×D4—22no—
94C8×D4—22no—
95C8×D4C2×C412yes—
96C8×D4C2×C412yes—
97#433—22no—
98#433—22no—
99#433—22no—
100#433—22no—
101#432—21no—
102#432—21no—
103#432—22no—
104#432—21no—
105#432—21no—
106#432—21no—
107#432—21no—
108#432—21no—
109#432—21no—
110#432C2×C412yes—
111#432C2312yes—
112#432C2311yescharacteristic
113#432C2311yescharacteristic
114#432C2311yescharacteristic
115#432C2311yescharacteristic
116#431—22no—
117#431—21no—
118#431—21no—
119#431—21no—
120#431—21no—
121#431C2×C412yes—
122#431C2×C411yescharacteristic
123#431C2×C411yescharacteristic
124#431D411yescharacteristic
125#431D411yescharacteristic
126#431D412yes—
127C2×#108—22no—
128C2×#108—22no—
129#424—22no—
130#424—22no—
131#423—21no—
132#423—21no—
133C2×C4⋊C8—21no—
134C2×C4⋊C8C2×C412yes—
135C2×C4⋊C8D412yes—
136#407—22no—
137#407—21no—
138#407—21no—
139C2×C22⋊C8—21no—
140C2×C22⋊C8—22no—
141C2×C22⋊C8—21no—
142C2×C22⋊C8D411yescharacteristic
143C2×C22⋊C8D411yescharacteristic
144#405—22no—
145#405—22no—
146C4×M16—21no—
147C4×M16—21no—
148C4×M16—21no—
149C4×M16—21no—
150C4×M16C2312yes—
151C4×M16C2311yescharacteristic
152C4×M16C2311yescharacteristic
153C2×#97—21no—
154C2×#97—21no—
155C2×#97—21no—
156C2×C4×C8—21no—
157C2×C4×C8C2312yes—
158#341—22no—
159#341—22no—
160#335—22no—
161#335—22no—
162#334—22no—
163#334—22no—
164#330—22no—
165#330—22no—
166#330—22no—
167#330—22no—
168#329—22no—
169#329—22no—
170#329—22no—
171#329—22no—
172C22×Q8C4⋊C411yescharacteristic
173C22×D4—21no—
174C22×D4C4⋊C411yescharacteristic
175C23×C4—21no—
176C23×C4C2×D411yescharacteristic
177C23×C4C2×Q811yescharacteristic
178C23×C4#4111yescharacteristic
179C2×M16—22no—
180C2×M16—22no—
181C2×M16—22no—
182C2×M16—21no—
183C2×M16—21no—
184C2×M16—44no—
185C2×M16—41no—
186C2×M16—41no—
187C22×C8—22no—
188C22×C8—22no—
189C22×C8—22no—
190C22×C8—41no—
191C22×C8—41no—
192C22×C8C2×D412yes—
193C22×C8C2×D412yes—
194#126—22no—
195#126—22no—
196#126—22no—
197#126—22no—
198#125—21no—
199#125—21no—
200#125C4⋊C412yes—
201#125C2×D412yes—
202#124—21no—
203#124—21no—
204#124C4⋊C412yes—
205#124C2×D412yes—
206#123—22no—
207#123—22no—
208#123—22no—
209#123—22no—
210C22⋊Q8—21no—
211C22⋊Q8—21no—
212C22⋊Q8C4⋊C412yes—
213C22⋊Q8C2×Q812yes—
214#121—21no—
215#121—21no—
216#121C4⋊C412yes—
217#121C2×Q812yes—
218C4×Q8—21no—
219C4×Q8—21no—
220C4×Q8C4⋊C412yes—
221C4×Q8C22×C412yes—
222C4×D4—22no—
223C4×D4—22no—
224C4×D4—22no—
225C4×D4—22no—
226C4×D4—21no—
227C4×D4—21no—
228C4×D4C4⋊C412yes—
229C4×D4C22×C412yes—
230#117—22no—
231#117—22no—
232#117—22no—
233#117—22no—
234C2×C4⋊C4—21no—
235C2×C4⋊C4—21no—
236C2×C4⋊C4—21no—
237C2×C4⋊C4—21no—
238C2×C4⋊C4C4⋊C411yescharacteristic
239C2×C4⋊C4C4⋊C411yescharacteristic
240C2×C22⋊C4—21no—
241C2×C22⋊C4—21no—
242C2×C22⋊C4—21no—
243C2×C22⋊C4—21no—
244C2×C22⋊C4—21no—
245C2×C42—21no—
246C2×C42C22×C411yescharacteristic
247C2×C42C22×C411yescharacteristic
248C2×C42C2×D411yescharacteristic
249C2×C42C2×D411yescharacteristic
250C2×C42C2×D411yescharacteristic
251C2×C42C2×D411yescharacteristic
252C2×C42C2×D411yescharacteristic
253C2×C42C2×Q811yescharacteristic
254C2×C42#4111yescharacteristic
255C2×C42#4111yescharacteristic
256C2×C42#4111yescharacteristic
257C2×C42C2411yescharacteristic
258#108—44no—
259#108—44no—
260C4⋊C8—22no—
261C4⋊C8—22no—
262C4⋊C8—22no—
263C4⋊C8—22no—
264C4⋊C8—22no—
265C4⋊C8—22no—
266C4⋊C8—21no—
267C4⋊C8—21no—
268C4⋊C8—41no—
269C4⋊C8—41no—
270C4⋊C8—41no—
271C4⋊C8—41no—
272C4⋊C8C22⋊C412yes—
273C4⋊C8C22⋊C412yes—
274C4⋊C8C22⋊C412yes—
275C4⋊C8C22⋊C412yes—
276C4⋊C8C22×C412yes—
277C4⋊C8C22×C412yes—
278C22⋊C8—22no—
279C22⋊C8—21no—
280C22⋊C8—21no—
281C22⋊C8—22no—
282C22⋊C8—22no—
283C22⋊C8—22no—
284C22⋊C8—22no—
285C22⋊C8—22no—
286C22⋊C8—21no—
287C22⋊C8—21no—
288C22⋊C8—41no—
289C22⋊C8—41no—
290C22⋊C8—41no—
291C22⋊C8—41no—
292C22⋊C8C22⋊C412yes—
293C22⋊C8C22⋊C412yes—
294C22⋊C8C22⋊C411yescharacteristic
295C22⋊C8C22⋊C411yescharacteristic
296C22⋊C8C22⋊C411yescharacteristic
297C22⋊C8C22⋊C411yescharacteristic
298#97—22no—
299#97—22no—
300#97—22no—
301#97—21no—
302#97—21no—
303#97—22no—
304#97—22no—
305#97—21no—
306#97—21no—
307#97—21no—
308#97—21no—
309#97—41no—
310#97—41no—
311C4×C8—22no—
312C4×C8—22no—
313C4×C8—22no—
314C4×C8—21no—
315C4×C8—21no—
316C4×C8—41no—
317C4×C8—41no—
318C4×C8C22×C412yes—
319C4×C8C22×C412yes—
320C4×C8C2×D412yes—
321C4×C8C2×D412yes—
322C24—21no—
323C24C22⋊Q811yescharacteristic
324C2×Q8—21no—
325C2×Q8—81no—
326C2×Q8C2×C4⋊C412yes—
327C2×D4—22no—
328C2×D4—22no—
329C2×D4—21no—
330C2×D4—44no—
331C2×D4—81no—
332C2×D4C2×C4⋊C412yes—
333C22×C4—21no—
334C22×C4—21no—
335C22×C4—21no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—42no—
339C22×C4—42no—
340C22×C4—82no—
341C22×C4—81no—
342C22×C4C2×C22⋊C411yescharacteristic
343C22×C4C2×C22⋊C411yescharacteristic
344C22×C4C2×C4⋊C411yescharacteristic
345C22×C4C2×C4⋊C411yescharacteristic
346C22×C4#11711yescharacteristic
347C22×C4#11711yescharacteristic
348C22×C4#12111yescharacteristic
349C22×C4#12111yescharacteristic
350C22×C4#12111yescharacteristic
351C22×C4#12111yescharacteristic
352C22×C4C22⋊Q811yescharacteristic
353C22×C4C22⋊Q811yescharacteristic
354C22×C4C22⋊Q811yescharacteristic
355C22×C4C22⋊Q811yescharacteristic
356C22×C4C22⋊Q811yescharacteristic
357C22×C4C22×D411yescharacteristic
358C22×C4C22×Q811yescharacteristic
359C22×C4C2×#4111yescharacteristic
360M16—24no—
361M16—42no—
362M16—44no—
363M16—42no—
364M16—42no—
365M16—82no—
366M16—82no—
367M16—82no—
368C2×C8—22no—
369C2×C8—22no—
370C2×C8—42no—
371C2×C8—42no—
372C2×C8—44no—
373C2×C8—42no—
374C2×C8—41no—
375C2×C8—41no—
376C2×C8—41no—
377C2×C8—41no—
378C2×C8—42no—
379C2×C8—82no—
380C2×C8—82no—
381C2×C8—82no—
382C2×C8—82no—
383C2×C8C2×C22⋊C412yes—
384C2×C8C2×C22⋊C412yes—
385C2×C8C2×C22⋊C412yes—
386C2×C8C2×C22⋊C412yes—
387C4⋊C4—22no—
388C4⋊C4—22no—
389C4⋊C4—22no—
390C4⋊C4—22no—
391C4⋊C4—22no—
392C4⋊C4—22no—
393C4⋊C4—22no—
394C4⋊C4—22no—
395C4⋊C4—21no—
396C4⋊C4—21no—
397C4⋊C4C2×C4⋊C412yes—
398C4⋊C4C2×C4⋊C412yes—
399C22⋊C4—22no—
400C22⋊C4—22no—
401C22⋊C4—22no—
402C22⋊C4—22no—
403C22⋊C4—22no—
404C22⋊C4—22no—
405C22⋊C4—22no—
406C22⋊C4—22no—
407C22⋊C4—22no—
408C22⋊C4—22no—
409C42—22no—
410C42—22no—
411C42C2×C22⋊C411yescharacteristic
412C42C2×C22⋊C411yescharacteristic
413C42C2×C4⋊C411yescharacteristic
414C42C2×C4⋊C411yescharacteristic
415C42#11711yescharacteristic
416C42#11711yescharacteristic
417C42#12111yescharacteristic
418C42#12111yescharacteristic
419C42#12111yescharacteristic
420C42#12111yescharacteristic
421C42C22⋊Q811yescharacteristic
422C42C22⋊Q811yescharacteristic
423C42C23×C411yescharacteristic
424C42C22×D411yescharacteristic
425C42C22×D411yescharacteristic
426C42C2×#4111yescharacteristic
427C23—22no—
428C23—22no—
429C23—41no—
430C23—41no—
431C23—42no—
432C23—81no—
433C23—81no—
434C23#38011yescharacteristic
435C23#38011yescharacteristic
436C23C2×C22⋊Q811yescharacteristic
437Q8—42no—
438Q8—81no—
439D4—42no—
440D4—44no—
441D4—44no—
442D4—81no—
443C2×C4—22no—
444C2×C4—22no—
445C2×C4—22no—
446C2×C4—22no—
447C2×C4—22no—
448C2×C4—22no—
449C2×C4—22no—
450C2×C4—22no—
451C2×C4—41no—
452C2×C4—41no—
453C2×C4—41no—
454C2×C4—41no—
455C2×C4—41no—
456C2×C4—41no—
457C2×C4—41no—
458C2×C4—41no—
459C2×C4—41no—
460C2×C4—41no—
461C2×C4—41no—
462C2×C4—41no—
463C2×C4—41no—
464C2×C4—42no—
465C2×C4—42no—
466C2×C4—42no—
467C2×C4—82no—
468C2×C4—81no—
469C2×C4—81no—
470C2×C4#38011yescharacteristic
471C2×C4#38011yescharacteristic
472C2×C4#38011yescharacteristic
473C2×C4#38011yescharacteristic
474C2×C4#38011yescharacteristic
475C2×C4#38011yescharacteristic
476C2×C4C22×C22⋊C411yescharacteristic
477C2×C4C22×C4⋊C411yescharacteristic
478C2×C4C2×#11711yescharacteristic
479C2×C4C2×#12111yescharacteristic
480C2×C4C2×#12111yescharacteristic
481C2×C4C2×C22⋊Q811yescharacteristic
482C8—22no—
483C8—22no—
484C8—22no—
485C8—22no—
486C8—44no—
487C8—42no—
488C8—42no—
489C8—82no—
490C8—81no—
491C8—81no—
492C8—81no—
493C8—81no—
494C22—21no—
495C22—21no—
496C22—21no—
497C22—21no—
498C22—21no—
499C22—21no—
500C22—21no—
501C22—21no—
502C22—42no—
503C22—42no—
504C22—42no—
505C22—82no—
506C22—162no—
507C22#181412yes—
508C22C2×#38011yescharacteristic
509C4—21no—
510C4—21no—
511C4—21no—
512C4—21no—
513C4—21no—
514C4—21no—
515C4—21no—
516C4—21no—
517C4—21no—
518C4—21no—
519C4—21no—
520C4—42no—
521C4—42no—
522C4—42no—
523C4#290112yes—
524C2—22no—
525C2—41no—
526C2—41no—
527C2—82no—
528C2#2065511yescharacteristic
529C2#2067311yescharacteristic
530C2C2×#181411yescharacteristic
subgroupG/H
center, socleC22C2×#380
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
prime radicalC2×#121Q8
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
2. #21370C2semidirect
3. #17816C2semidirect
6. #8693C2semidirect
7. #8692C2semidirect
8. #8691C2semidirect
9. #8690C2semidirect
10. #7997C2semidirect
11. C2×#1291C2semidirect
34. C4×#108C22semidirect
35. C4×#108C22semidirect
39. #1568C22semidirect
40. #1568C22semidirect
51. #1291C22semidirect
52. #1291C22semidirect
53. #1291C22semidirect
54. #1291C22semidirect
55. #1291C22semidirect
56. #1291C22semidirect

Maximal subgroups

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

subgroupindex
#602882
#213702
#178162
#176692
#176422
#86932
#86922
#86912
#86902
#79972
C2×#12912