Finite Group Databasebeta

← #192475

Subgroups — #192475

Order 512.

869 conjugacy classes of subgroups.

The subgroup lattice has 560 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60279C211yesmaximal, characteristic
2#21993C211yesmaximal, characteristic, complemented
3#21975C211yesmaximal, characteristic, complemented
4#21856C211yesmaximal, characteristic
5#17573C211yesmaximal, characteristic
6#8875C214yesmaximal, complemented
7#8874C214yesmaximal, complemented
8#8052C211yesmaximal, characteristic, complemented
9#7924C211yesmaximal, characteristic, complemented
10#3444C2211yescharacteristic
11#2993—21no—
12#2993—21no—
13#2992—21no—
14#2992—21no—
15#2985—21no—
16#2981—21no—
17#2976C2212yes—
18#2976C2212yes—
19#2965—22no—
20#2965—21no—
21C8×#41—21no—
22#2922—22no—
23#2920C2211yescharacteristic
24C2×#431C2211yescharacteristic
25C2×#431C2211yescharacteristic
26C2×#423C2211yescharacteristic
27#2283C2211yescharacteristic
28C4×#117C2211yescharacteristic
29#1994—21no—
30#1962—21no—
31#1950—22no—
32#1939—21no—
33#1939—21no—
34#1871C2214yes—
35#1855C2214yes—
36#1851C2212yes—
37#1851C2212yes—
38#1848—21no—
39#1843—21no—
40#1837—21no—
41#1832—21no—
42#1806—22no—
43#1521C414yescomplemented
44#1518C411yescharacteristic, complemented
45#1518C411yescharacteristic, complemented
46#1517C411yescharacteristic, complemented
47#1517C411yescharacteristic, complemented
48#1501C2214yes—
49#1297C2214yes—
50#1282C2212yes—
51#1282C2212yes—
52C22×M16C2312yes—
53#533C2×C411yescharacteristic
54#533D411yescharacteristic
55#533Q811yescharacteristic
56#532C2×C411yescharacteristic
57#532D411yescharacteristic
58#532Q811yescharacteristic
59#529—21no—
60#529—21no—
61#529—21no—
62#529—21no—
63#529C2×C412yes—
64#529Q812yes—
65#525—21no—
66#525—21no—
67#525C2312yes—
68C4×#41—21no—
69C4×#41—21no—
70C4×#41D412yes—
71C2×#117—21no—
72C22×C42C2311yescharacteristic
73#446—22no—
74#446—22no—
75#446—21no—
76#446—21no—
77C8×Q8—21no—
78C8×Q8—21no—
79#436—21no—
80#436—21no—
81#436—22no—
82#436—22no—
83#435—22no—
84#435—22no—
85#435—22no—
86#435—22no—
87#435—22no—
88#435—22no—
89C8×D4—21no—
90C8×D4—21no—
91C8×D4—22no—
92C8×D4—22no—
93#433—22no—
94#433—22no—
95#433—22no—
96#433—22no—
97#432—21no—
98#432—21no—
99#431—22no—
100#431—22no—
101#431—21no—
102#431—21no—
103#431—21no—
104#431—21no—
105#431—21no—
106#431—21no—
107#431—21no—
108#431—21no—
109#431C2×C411yescharacteristic
110#431C2×C411yescharacteristic
111#431C2×C412yes—
112C2×#108—24no—
113#424—22no—
114#424—22no—
115#423—21no—
116#423—21no—
117#423—21no—
118#423—21no—
119#423D411yescharacteristic
120#423D411yescharacteristic
121C2×C4⋊C8—21no—
122C2×C4⋊C8—21no—
123C2×C4⋊C8—21no—
124C2×C4⋊C8—22no—
125C2×C4⋊C8—22no—
126C2×C4⋊C8D411yescharacteristic
127C2×C4⋊C8D411yescharacteristic
128C2×#104—24no—
129#408—22no—
130#408—22no—
131#407C2312yes—
132C2×C22⋊C8C2311yescharacteristic
133C2×C22⋊C8C2311yescharacteristic
134C2×C22⋊C8C2312yes—
135C4×M16—22no—
136C4×M16D412yes—
137C2×C4×C8—22no—
138C2×C4×C8—21no—
139C2×C4×C8D411yescharacteristic
140C2×C4×C8D411yescharacteristic
141#383C2×C412yes—
142#382C2×C411yescharacteristic
143#382C2×C411yescharacteristic
144#381—21no—
145#380—21no—
146C4×C4⋊C4—21no—
147C4×C4⋊C4—21no—
148C4×C4⋊C4—21no—
149C4×C22⋊C4—21no—
150C4×C22⋊C4C2312yes—
151C4×C22⋊C4C2312yes—
152#376—21no—
153#376—21no—
154C43—21no—
155#341—24no—
156#339—24no—
157#336—22no—
158#336—22no—
159#328C2×C414yes—
160#328C2×C414yes—
161#327C2×C414yes—
162#327C2×C414yes—
163C2×#41—21no—
164C2×#41—21no—
165C2×#41C2×D412yes—
166C23×C4#4111yescharacteristic
167C23×C4#4111yescharacteristic
168C23×C4C2411yescharacteristic
169C2×M16—22no—
170C2×M16—22no—
171C2×M16—22no—
172C2×M16—22no—
173C2×M16—82no—
174C2×M16C2×D412yes—
175C2×M16C2×D412yes—
176C22×C8—22no—
177C22×C8—22no—
178C22×C8—22no—
179C22×C8—22no—
180C22×C8C2×D411yescharacteristic
181C22×C8C2×D411yescharacteristic
182C22×C8C2×D411yescharacteristic
183C22×C8C2×D411yescharacteristic
184C4⋊Q8C4⋊C411yescharacteristic
185C4⋊Q8C4⋊C411yescharacteristic
186C4⋊Q8C4⋊C411yescharacteristic
187#127C4⋊C411yescharacteristic
188#126—22no—
189#126—22no—
190#125—21no—
191#125—21no—
192#125C4⋊C412yes—
193#124—21no—
194#124—21no—
195#124C4⋊C412yes—
196#123—22no—
197#123—22no—
198#123—22no—
199C22⋊Q8—21no—
200C22⋊Q8—21no—
201C22⋊Q8—21no—
202C22⋊Q8—21no—
203C22⋊Q8C22×C412yes—
204C22⋊Q8C2×Q812yes—
205#121—21no—
206#121—21no—
207#121—21no—
208#121—21no—
209#121C22×C412yes—
210#121C2×Q812yes—
211#120—22no—
212C4×Q8—21no—
213C4×Q8—21no—
214C4×Q8—21no—
215C4×Q8—21no—
216C4×Q8C4⋊C412yes—
217C4×Q8C4⋊C412yes—
218C4×D4—22no—
219C4×D4—22no—
220C4×D4—22no—
221C4×D4—22no—
222C4×D4—22no—
223C4×D4—22no—
224C4×D4—21no—
225C4×D4—21no—
226C4×D4—21no—
227C4×D4—21no—
228C4×D4C4⋊C412yes—
229C4×D4C4⋊C412yes—
230#117—21no—
231#117—21no—
232#117—22no—
233#117—22no—
234#117—22no—
235#117—24no—
236#117—42no—
237#117C2×D412yes—
238C2×C4⋊C4—21no—
239C2×C4⋊C4—21no—
240C2×C4⋊C4—41no—
241C2×C22⋊C4—22no—
242C2×C22⋊C4—22no—
243C2×C42—22no—
244C2×C42—22no—
245C2×C42—22no—
246C2×C42—41no—
247C2×C42—41no—
248C2×C42C22×C411yescharacteristic
249C2×C42C22×C411yescharacteristic
250C2×C42C2×D411yescharacteristic
251C2×C42C2×D411yescharacteristic
252C2×C42C2×D411yescharacteristic
253C2×C42C2×D411yescharacteristic
254C2×C42C2×D411yescharacteristic
255C2×C42C2×D411yescharacteristic
256C2×C42C2×Q811yescharacteristic
257C2×C42C2×Q811yescharacteristic
258C2×C42#4111yescharacteristic
259C2×C42#4111yescharacteristic
260#108—84no—
261C4⋊C8—22no—
262C4⋊C8—22no—
263C4⋊C8—22no—
264C4⋊C8—22no—
265C4⋊C8—22no—
266C4⋊C8—21no—
267C4⋊C8—21no—
268C4⋊C8—21no—
269C4⋊C8—21no—
270C4⋊C8—21no—
271C4⋊C8—21no—
272C4⋊C8—21no—
273C4⋊C8—21no—
274C4⋊C8—21no—
275C4⋊C8—21no—
276C4⋊C8—41no—
277C4⋊C8—41no—
278C4⋊C8—41no—
279C4⋊C8—41no—
280C4⋊C8—82no—
281C4⋊C8—82no—
282#104—84no—
283C22⋊C8—22no—
284C22⋊C8—22no—
285C22⋊C8—22no—
286C22⋊C8—22no—
287C22⋊C8—21no—
288C22⋊C8—21no—
289C22⋊C8—21no—
290C22⋊C8—21no—
291C22⋊C8—22no—
292C22⋊C8—22no—
293C22⋊C8—22no—
294C22⋊C8C22×C412yes—
295C22⋊C8C22×C412yes—
296#97—22no—
297#97—22no—
298#97—22no—
299C4×C8—22no—
300C4×C8—22no—
301C4×C8—21no—
302C4×C8—21no—
303C4×C8—22no—
304C4×C8—21no—
305C4×C8—22no—
306C4×C8—21no—
307C4×C8—21no—
308C4×C8—21no—
309#95—22no—
310#95—21no—
311#95—21no—
312#95—22no—
313#95—22no—
314#95—41no—
315#95C22×C412yes—
316#95C22×C412yes—
317C24C2×#4111yescharacteristic
318#41—42no—
319#41—42no—
320#41—42no—
321#41—42no—
322#41—82no—
323C2×Q8—21no—
324C2×Q8—21no—
325C2×Q8C2×C4⋊C412yes—
326C2×D4—22no—
327C2×D4—22no—
328C2×D4—22no—
329C2×D4—21no—
330C2×D4—21no—
331C2×D4C2×C4⋊C412yes—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—41no—
340C22×C4—41no—
341C22×C4—42no—
342C22×C4—81no—
343C22×C4C4×D411yescharacteristic
344C22×C4C4×D411yescharacteristic
345C22×C4C4×D411yescharacteristic
346C22×C4C4×D411yescharacteristic
347C22×C4C22⋊Q811yescharacteristic
348C22×C4C22⋊Q811yescharacteristic
349C22×C4C22⋊Q811yescharacteristic
350C22×C4C22⋊Q811yescharacteristic
351C22×C4#12311yescharacteristic
352C22×C4#12311yescharacteristic
353C22×C4#12311yescharacteristic
354C22×C4#12311yescharacteristic
355C22×C4C23×C411yescharacteristic
356C22×C4C22×D411yescharacteristic
357C22×C4C22×D411yescharacteristic
358C22×C4C22×D411yescharacteristic
359C22×C4C22×Q811yescharacteristic
360C22×C4C2×#4111yescharacteristic
361M16—42no—
362M16—42no—
363M16—44no—
364M16—44no—
365M16—82no—
366M16—82no—
367C2×C8—22no—
368C2×C8—22no—
369C2×C8—21no—
370C2×C8—21no—
371C2×C8—22no—
372C2×C8—22no—
373C2×C8—21no—
374C2×C8—21no—
375C2×C8—42no—
376C2×C8—42no—
377C2×C8—41no—
378C2×C8—41no—
379C2×C8—41no—
380C2×C8—41no—
381C2×C8—42no—
382C2×C8—42no—
383C2×C8—42no—
384C2×C8—82no—
385C2×C8—82no—
386C4⋊C4—22no—
387C4⋊C4—22no—
388C4⋊C4—21no—
389C4⋊C4—21no—
390C4⋊C4—21no—
391C4⋊C4—21no—
392C4⋊C4—22no—
393C4⋊C4—22no—
394C4⋊C4—22no—
395C4⋊C4—22no—
396C4⋊C4—44no—
397C4⋊C4—42no—
398C4⋊C4—42no—
399C4⋊C4C2×C4⋊C412yes—
400C4⋊C4C2×C4⋊C412yes—
401C22⋊C4—22no—
402C22⋊C4—22no—
403C22⋊C4—22no—
404C22⋊C4—22no—
405C22⋊C4—22no—
406C22⋊C4—22no—
407C22⋊C4—22no—
408C22⋊C4—22no—
409C22⋊C4—22no—
410C22⋊C4—24no—
411C22⋊C4—42no—
412C22⋊C4—82no—
413C42—22no—
414C42—22no—
415C42—22no—
416C42—42no—
417C42—42no—
418C42—44no—
419C42—42no—
420C42—42no—
421C42—82no—
422C42C2×C4⋊C411yescharacteristic
423C42C2×C4⋊C411yescharacteristic
424C42C2×C4⋊C411yescharacteristic
425C42C2×C4⋊C411yescharacteristic
426C42C4×D411yescharacteristic
427C42C4×D411yescharacteristic
428C42C4×D411yescharacteristic
429C42C4×D411yescharacteristic
430C42#12011yescharacteristic
431C42#12011yescharacteristic
432C42#12011yescharacteristic
433C42#12011yescharacteristic
434C42C22⋊Q811yescharacteristic
435C42C22⋊Q811yescharacteristic
436C42C22⋊Q811yescharacteristic
437C42C22⋊Q811yescharacteristic
438C23—21no—
439C23—21no—
440C23—22no—
441C23—41no—
442C23—41no—
443C23C2×C4×D411yescharacteristic
444C23C2×C22⋊Q811yescharacteristic
445C23C2×#12311yescharacteristic
446Q8—42no—
447Q8—42no—
448Q8—81no—
449D4—42no—
450D4—42no—
451D4—42no—
452D4—42no—
453D4—81no—
454D4—82no—
455D4—82no—
456C2×C4—21no—
457C2×C4—21no—
458C2×C4—22no—
459C2×C4—22no—
460C2×C4—22no—
461C2×C4—22no—
462C2×C4—21no—
463C2×C4—21no—
464C2×C4—21no—
465C2×C4—21no—
466C2×C4—21no—
467C2×C4—21no—
468C2×C4—22no—
469C2×C4—22no—
470C2×C4—22no—
471C2×C4—41no—
472C2×C4—41no—
473C2×C4—41no—
474C2×C4—41no—
475C2×C4—41no—
476C2×C4—41no—
477C2×C4—41no—
478C2×C4—41no—
479C2×C4—41no—
480C2×C4—44no—
481C2×C4—44no—
482C2×C4—42no—
483C2×C4—42no—
484C2×C4—44no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—82no—
492C2×C4—82no—
493C2×C4#38511yescharacteristic
494C2×C4#38511yescharacteristic
495C2×C4#38511yescharacteristic
496C2×C4#38511yescharacteristic
497C2×C4#38511yescharacteristic
498C2×C4#38511yescharacteristic
499C2×C4#38511yescharacteristic
500C2×C4#38511yescharacteristic
501C2×C4#44312yes—
502C2×C4#44412yes—
503C2×C4C22×C4⋊C411yescharacteristic
504C2×C4C2×C4×D411yescharacteristic
505C2×C4C2×#12011yescharacteristic
506C2×C4C2×C22⋊Q811yescharacteristic
507C8—42no—
508C8—42no—
509C8—41no—
510C8—41no—
511C8—42no—
512C8—41no—
513C8—41no—
514C8—42no—
515C8—82no—
516C22—21no—
517C22—21no—
518C22—21no—
519C22—21no—
520C22—21no—
521C22—21no—
522C22—21no—
523C22—21no—
524C22—42no—
525C22—42no—
526C22—42no—
527C22—81no—
528C22—81no—
529C22C2×#38511yescharacteristic
530C22C2×#44311yescharacteristic
531C22C2×#44411yescharacteristic
532C4—21no—
533C4—21no—
534C4—21no—
535C4—21no—
536C4—21no—
537C4—21no—
538C4—21no—
539C4—21no—
540C4—21no—
541C4—42no—
542C4—42no—
543C4—42no—
544C4—82no—
545C4—82no—
546C4—84no—
547C4#186612yes—
548C4#186612yes—
549C4#302711yescharacteristic
550C4#302811yescharacteristic
551C2—21no—
552C2—21no—
553C2—41no—
554C2—41no—
555C2—82no—
556C2#2086711yescharacteristic
557C2C2×#186611yescharacteristic
558C2#2089011yescharacteristic
subgroupG/H
centerC2×C4C2×#120
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
socleC22C2×#385
prime radical#3444C22
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C22×C42 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #21993C2semidirect
3. #21975C2semidirect
6. #8875C2semidirect
7. #8874C2semidirect
8. #8052C2semidirect
9. #7924C2semidirect
43. #1521C4semidirect
44. #1518C4semidirect
45. #1518C4semidirect
46. #1517C4semidirect
47. #1517C4semidirect

Maximal subgroups

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

subgroupindex
#602792
#219932
#219752
#218562
#175732
#88752
#88742
#80522
#79242